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[
[
"States with identical steady dissipation rate: Role of kinetic constants\n in enzyme catalysis"
],
[
"Abstract A non-equilibrium steady state is characterized by a non-zero steady dissipation rate.",
"Chemical reaction systems under suitable conditions may generate such states.",
"We propose here a method that is able to distinguish states with identical values of the steady dissipation rate.",
"This necessitates a study of the variation of the entropy production rate with the experimentally observable reaction rate in regions close to the steady states.",
"As an exactly-solvable test case, we choose the problem of enzyme catalysis.",
"Link of the total entropy production with the enzyme efficiency is also established, offering a desirable connection with the inherent irreversibility of the process.",
"The chief outcomes are finally noted in a more general reaction network with numerical demonstrations."
],
[
"Introduction",
"A major shift in the field of thermodynamics in the last century was from idealized equilibrium processes to natural irreversible processes [1-4].Chemical reactions continue to play a pivotal role in this development and provide significant motivation in studying the non-equilibrium thermodynamic properties of systems in vitro as well as in vivo [5-10].",
"Since a closed system always tends to thermodynamic equilibrium (TE), a natural generalization in the theory of irreversible thermodynamics has been achieved via the concept of a steady state [1], [11].",
"In this regard, the quantity of primary importance is the entropy production rate (EPR) [12], [13], [14].",
"The EPR vanishes for a closed system in the long-time limit that reaches a true TE.",
"On the other hand, EPR is positive definite for a steady state that can emerge in an open system.",
"The easiest way to model such a system in the context of chemical reactions is to assume that concentrations of some of the reacting species are held fixed [15], [16].",
"Under this condition, aptly known as the chemiostatic condition [17], EPR tends to a non-zero constant, reflecting a steady dissipation rate (SDR) to sustain the system away from equilibrium [18].",
"The corresponding steady state is denoted as the non-equilibrium steady state (NESS) [19], [20], [21], [22].",
"This concept has been extensively used in analyzing single-molecule kinetic experiments [16], [17], [23].",
"The NESS also includes the TE as a special case when detailed balance (DB) is obeyed [24], thus providing a very general framework.",
"Recently, an important progress was made in the theory and characterization of NESS, considering a master equation formalism [25], [26], [27].",
"These studies have established that the classification of NESS requires not only the steady distribution (as in TE) but also the stationary fluxes or probability currents.",
"This approach enables one to identify all possible combinations of transition rates that ultimately lead the system to the same NESS.",
"However, these NESSs in general have different values of the EPR, and hence the SDR.",
"This proposition prompts one to check (i) how states with the same EPR at NESS can be generated and (ii) whether there exist ways to distinguish these states.",
"Here, we shall address both the issues by considering an enzyme-catalyzed reaction under chemiostatic condition.",
"Expressing the EPR as a function of experimentally measurable reaction rate, we emphasize also that, the quantity that identifies the various NESSs having the same EPR is linked with the enzyme efficiency, a useful measure that is expressible in terms of enzyme kinetic constants."
],
[
"The system",
"The basic scheme of enzyme catalysis within the Michaelis-Menten (MM) framework with reversible product formation step is shown in Fig.REF .",
"Under chemiostatic condition, $[{\\rm S}]$ and $[{\\rm P}]$ are kept constant by continuous injection and withdrawal, respectively.",
"This is the simplest model to mimic an open reaction system.",
"Unlike the usual case of full enzyme recovery with total conversion of substrate into product in a closed system, here both the concentrations of free enzyme E and the enzyme-substrate complex ES reach a steady value.",
"Also, instead of the rate of product formation, the progress of reaction is characterized by the rate of evolution of $[{\\rm E}]$ (or $[{\\rm ES}]$ ).",
"Figure: Schematic diagram of MM kinetics ofenzyme catalysis with reversible product formation stepunder chemiostatic condition."
],
[
"Kinetics",
"We define the pseudo-first-order rate constants as $k_1=k^{\\prime }_1[{\\rm S}]$ and $k_{-2}=k^{\\prime }_{-2}[{\\rm P}].$ Concentration of E is denoted by $c_1(t)$ and that of ES is given by $c_2(t).$ We have then $c_1(t)+c_2(t)=z.$ Here $z$ is a constant that stands for the total enzyme concentration.",
"Then the rate of the reaction, $v(t)$ , is written as $v(t)=\\dot{c_1}=-Kc_1(t)+(k_{-1}+k_2)z,$ where $K=(k_1+k_{-1}+k_2+k_{-2}).$ With the initial condition, $c_1(0)=z$ , the time-dependent solution is given as $c_1(t)=\\frac{z}{K}\\left((k_{-1}+k_2)+(k_1+k_{-2})e^{-Kt}\\right).$ The steady state enzyme concentration corresponds to the long-time limit of Eq.",
"(REF ): $c_1^s=\\left((k_{-1}+k_2)z\\right)/K.$ At any steady state, we thus note $v(t)=\\dot{c_1}\\,=0\\,=\\,\\dot{c_2}.$"
],
[
"Non-equilibrium thermodynamics",
"The fluxes of the reaction system are defined pairwise as [2], [13], [14] $J_1(t)=k_1c_1(t)-k_{-1}c_2(t),$ $J_2(t)=k_2c_2(t)-k_{-2}c_1(t).$ From Eq.",
"(REF ), Eq.",
"(REF ), Eq.",
"(REF ) and Eq.",
"(REF ), one gets $\\dot{c_1}=J_2(t)-J_1(t).$ At the steady state, Eq.",
"(REF ) leads to $J_1^s=J_2^s=J^s.$ An NESS is characterized by a non-zero flux, $J^s\\ne 0$ .",
"At TE, the fluxes vanish for both the reactions.",
"One may note, then the system satisfies DB.",
"The conjugate forces of the fluxes given in Eqs (REF )-(REF ) are defined as [2] $X_1(t)=\\mu _E+\\mu _S-\\mu _{ES}=T{\\rm ln}\\frac{k_1c_1(t)}{k_{-1}c_2(t)},$ $X_2(t)=\\mu _{ES}-\\mu _E-\\mu _P=T{\\rm ln}\\frac{k_2c_2(t)}{k_{-2}c_1(t)}.$ Corresponding to the scheme depicted in Fig.REF , the EPR is then given by [1], [2] $\\sigma (t)=\\frac{1}{T}\\sum _{i=1}^{2} J_i(t)X_i(t).$ We set here (and henceforth) the Boltzmann constant $k_B=1$ .",
"In the present case, the steady value of EPR becomes $\\sigma ^s=\\frac{1}{T}J^s(\\mu _S-\\mu _P).$ Therefore, unless the substrate and the product take part in equilibrium, the reaction system reaches an NESS with a SDR equal to $\\sigma ^s$ ."
],
[
"EPR close to NESS",
"The problem is now transparent.",
"If the rate constants become different, the steady concentrations will also differ.",
"But, one can adjust them in such a way that $\\sigma ^s$ remains the same.",
"In these situations, one needs an additional parameter to distinguish these states.",
"To proceed, we define a small deviation in $c_1(t)$ around NESS as $\\delta =c_1(t)-c_1^s.$ It then follows from Eq.",
"(REF ) that $c_2(t)=c_2^s-\\delta .$ From Eq.",
"(REF ) and Eq.",
"(REF ), the reaction rate becomes $v(t)=-K\\delta .$ Now, putting Eqs (REF )-(REF ) and Eqs (REF )-(REF ) in Eq.",
"(REF ) and taking only the first terms of the logarithmic parts, we obtain the EPR close to NESS as $\\sigma (t)={\\rm A}_0+{\\rm A}_1 v(t)+{\\rm A}_2 v^2(t).$ Here ${\\rm A}_0=J^s{\\rm ln}\\frac{k_1k_2}{k_{-1}k_{-2}},$ ${\\rm A}_1=-\\frac{1}{K}\\left((k_1+k_{-1}){\\rm ln}\\frac{k_1c_1^s}{k_{-1}c_2^s}+(k_2+k_{-2}){\\rm ln}\\frac{k_{-2}c_1^s}{k_2c_2^s}\\right),$ ${\\rm A}_2=\\frac{1}{K}\\left(\\frac{1}{c_1^s}+\\frac{1}{c_2^s}\\right).$ As $v(t)$ vanishes at any steady state, the SDR at NESS is given by $\\sigma ^s={\\rm A}_0>0.$ However, at TE, $\\sigma ^s={\\rm A}_0=0;$ one may check that here DB holds: $\\frac{k_1k_2}{k_{-1}k_{-2}}=1.$ Inspection of Eq.",
"(REF ) reveals that, near NESS, $\\sigma (t)$ varies linearly with $v(t)$ with a slope ${\\rm A}_1$ .",
"Thus, while ${\\rm A}_0$ distinguishes an NESS from a true TE, ${\\rm A}_1$ plays the same role in identifying systems with the same SDR but having different time profiles."
],
[
"Results and discussion",
"In this section, we consider various situations where the reaction system reaches NESS with the same SDR.",
"Focusing on Eq.",
"(REF ), the different cases that keep ${\\rm A}_0$ invariant are discussed next."
],
[
"Variants with same SDR",
"Case A: Any parent choice of rate constants.",
"Case B: Only $k_1$ and $k_2$ are exchanged.",
"Case C: Only $k_{-1}$ and $k_{-2}$ are exchanged.",
"Case D: Both $k_1,k_2$ and $k_{-1},k_{-2}$ are exchanged.",
"Case E: Both $k_1,k_{-1}$ and $k_2,k_{-2}$ are exchanged.",
"Case F: Both $k_1,k_{-2}$ and $k_2,k_{-1}$ are exchanged.",
"Case G: $k_1$ changed to $\\alpha k_1$ , $k_{-1}$ changed to $\\alpha k_{-1}$ , $k_2$ changed to $\\beta k_2$ and $k_{-2}$ changed to $\\beta k_{-2}$ , such that $\\beta =\\frac{1}{\\alpha }=\\frac{k_1+k_{-1}}{k_2+k_{-2}}.$ It can be easily verified that cases D and E possess not only identical ${\\rm A}_0$ but also the same ${\\rm A}_1$ and ${\\rm A}_2$ .",
"This is true for cases A and F as well.",
"So, we do not consider cases E and F any further.",
"A simple explanation of the equivalence is given in Fig.REF schematically, based on reflection symmetry.",
"Figure: Schematic diagram showing the equivalence ofthe pairs A and F, and D and E, based on reflection symmetry."
],
[
"Temporal profiles",
"To explore the characteristics of various cases given above, we take the rate constants from the single molecule experimental study of English et al.",
"[23] on the Escherichia coli $\\beta $ -galactosidase enzyme.",
"They are as follows: $k^{\\prime }_1=5.0$ E07 $\\,{\\rm M^{-1}s^{-1}},\\,k_{-1}=1.83$ E04 ${\\rm s^{-1}},\\,k_2=7.3$ E02 ${\\rm s^{-1}}.$ We clarify that, in their study [23], $k_2$ had actually been shown to be a fluctuating quantity with a distribution.",
"However, only an average value of $k_2$ will suffice our purpose.",
"The constant substrate concentration is set at $[{\\rm S}]=1.0$ E02 ${\\rm \\mu M}$ and thus, $k_1=k^{\\prime }_1[{\\rm S}]=5.0$ E03 ${\\rm s^{-1}}$ .",
"We choose $k_{-2}=1.0$ E-05 ${\\rm s^{-1}}$ to make the reaction scheme almost identical to the conventional MM kinetics.",
"Here $\\lbrace k_i\\rbrace $ $(i=\\pm 1,\\,\\pm 2)$ with magnitudes given above represents the parent choice of rate constants, i.e., case A.",
"The value of the constant $\\beta =1/\\alpha =3.2$ E01, in case G. Figure: Evolution of EPR σ(t)\\sigma (t) with time forvarious cases determined using the exact (Eq.",
"()) as well asthe approximate (Eq.",
"()) expressions.In panel (b), the EPR of case Gis plotted as (σ(t)-700.0\\sigma (t)-700.0) for clarity.The time-evolution of EPR $\\sigma (t)$ , determined using both the exact (Eq.",
"(REF )) and the approximate (Eq.",
"(REF )) expressions, are shown in Fig.REF , for the various cases.",
"The concentrations $c_1,\\,c_2$ are made dimensionless by scaling with respect to the total enzyme concentration $z$ .",
"This ensures that $\\sigma (t)$ has the unit of ${\\rm s^{-1}}$ .",
"From the figure, it is evident that Eq.",
"(REF ) nicely approximates the behavior near NESS.",
"Specifically, the curves of exact and approximate cases merge quite well for any $t\\ge \\,1.5$ E-04 s. Figure: Variation of reaction rate v(t)v(t) as a functionof time for different cases indicated in the plot.The evolution of reaction rate $v(t)$ is shown in Fig.REF for all the distinct cases.",
"The curves are displayed over a time-span where Eq.",
"(REF ) is valid, as mentioned above.",
"This gives us a quantitative understanding of the magnitude of $v(t)$ up to which the close to NESS approximation, and hence Eq.",
"(REF ), is valid.",
"We note the variation of $\\sigma (t)$ as a function of $v(t)$ in all the relevant cases in Fig.REF .",
"Both the exact (Fig.REF (a)) as well as the approximate results (Fig.REF (b)) are shown.",
"Two features are interesting.",
"First, in all the situations, the system reaches an NESS with identical $\\sigma ^s={\\rm A}_0=2.553$ E03 ${\\rm s^{-1}}$ .",
"Secondly, the quantity that distinguishes one case from the other is the slope ${\\rm A}_1$ of $\\sigma (t)$ vs. $v(t)$ curve near the NESS.",
"This slope can be positive as well as negative.",
"Figure: Variation of EPR σ(t)\\sigma (t) as a function ofreaction rate v(t)v(t) for different cases indicated in the plotusing (a) exact (Eq.",
"()) and (b) approximate(Eq.",
"()) expressions."
],
[
"Total entropy production and enzyme efficiency",
"One may like to next investigate the role of the rate constants in governing the overall dissipation in various cases.",
"Specifically, we like to enquire if the efficiency of the enzyme has anything to do with the total dissipation.",
"In this context, it may be recalled that, the conventional MM kinetics requires the rate constant $k_{-2}$ to be negligible compared with the others.",
"So, the enzyme kinetic constants, like the MM constant ${\\rm K_M}=\\frac{k_{-1}+k_2}{k^{\\prime }_1}$ and catalytic efficiency $\\eta =k_2/{\\rm K_M}$ , are meaningful in the limit $k_{-2}\\rightarrow 0$ .",
"Our choice of parent rate constants ensures that in case A, the system follows MM kinetics.",
"Case B, which leaves $k_{-2}$ unchanged and case G, which changes $k_{-2}$ to $\\beta k_{-2}$ (with $\\beta =3.2$ E01), can also be included within the MM scheme.",
"But, cases C to F, which exchange $k_{-2}$ with any one of the other bigger rate constants, can not follow the usual MM kinetics.",
"Therefore, we focus on cases A,B and G in finding any possible connection between the kinetic constants of the enzyme and the total dissipation.",
"While the SDR $\\sigma ^s$ is the same for all of them, the time-integrated EPR, giving the total entropy production, is different.",
"We define it as $S_I=\\int _{0}^{\\tau }\\sigma (t)dt.$ The upper limit $\\tau $ is fixed at such a time when all the systems reach NESS.",
"In the present set of cases, we find that setting $\\tau =1.0$ E-03 s is satisfactory.",
"The values of ${\\rm K_M},\\,\\eta $ and $S_I$ (determined by integrating $\\sigma (t)$ from Eq.",
"(REF )) are listed in Table REF , along with the slope ${\\rm A}_1$ [see Eq.",
"(REF )].",
"It is clear from the data that, in going from case A to case G, ${\\rm K_M}$ gradually increases, whereas $\\eta $ falls.",
"Both these features indicate that the enzyme becomes less efficient.",
"More interesting is to note that the corresponding $S_I$ values also exhibit a decreasing trend from case A to case G. Thus, we can say that, with identical SDR, the more efficient enzyme (bigger $\\eta $ and smaller ${\\rm K_M}$ ) involves higher total dissipation.",
"This can be rationalized by the fact that, higher efficiency corresponds to a faster conversion of substrate into product.",
"This implies an increased irreversibility in the process.",
"Consequently, a higher entropy production is noted.",
"Table: Values of the quantities A 1 {\\rm A}_1, K M {\\rm K_M},η\\eta and S I S_I for cases A,B and G.Before ending this section, we mention briefly the fate of the different situations when DB, Eq.",
"(REF ), gets satisfied.",
"In this scenario, whatever be the values of the rate constants, the final EPR is trivially zero as the reaction system reaches TE [see Eq.",
"(REF )].",
"For the same reason, ${\\rm A}_1$ also becomes zero [see Eq.",
"(REF )].",
"However, it follows from Eq.",
"(REF ) that, EPR varies quadratically with $v(t)$ near TE.",
"Then, in principle, ${\\rm A}_2$ can distinguish systems reaching TE.",
"It is easy to see from Eq.",
"(REF ) that, cases A, B and G possess different values for ${\\rm A}_2$ and hence they can be identified by following the behavior of EPR with the reaction rate."
],
[
"Extension to general reaction systems",
"The MM kinetics, shown in Fig.REF , with a single intermediate in the form of the ES complex, is exactly solvable.",
"We now generalize this scheme to an enzyme catalysis reaction having N number of species.",
"These include the free enzyme E and (N-1) intermediates, under similar chemiostatic condition as discussed in Section II.",
"The reaction scheme is depicted in Fig.REF .",
"Essentially, the species ${\\rm ES_j},\\,(j=1,\\cdots ,N-1)$ refer to the various conformers of the enzyme-substrate complex.",
"The corresponding rate equations are given as $\\dot{c_i}=-(k_i+k_{-(i-1)})c_i(t)+k_{i-1}c_{i-1}(t)+k_{-i}c_{i+1}(t),$ with $c_i(t) \\,(i=1,\\cdots ,N)$ being the concentration of species ${\\rm ES}_{(i-1)}$ at time $t$ .",
"The following periodic boundary conditions hold: $i-1=N,\\, {\\rm for}\\, i=1,$ $i+1=1,\\, {\\rm for}\\, i=N.$ We have set $k_1=k^{\\prime }_1[{\\rm S}]$ and $k_{-N}=k^{\\prime }_{-N}[{\\rm P}].$ The flux $J_i$ due to the i-th reaction is defined as $J_i(t)=k_i c_i(t)-k_{-i}c_{i+1}(t).$ The expression of EPR then becomes $\\sigma _N(t)=\\sum _{i=1}^{N}J_i(t){\\rm ln}\\frac{k_ic_i(t)}{k_{-i}c_{i+1}(t)}.$ Figure: Schematic diagram of enzyme kinetics with (N-1) numberof intermediates under chemiostatic condition."
],
[
"EPR as a functional of reaction rate near NESS",
"It is generally not possible to solve the set of coupled equations analytically for a system of arbitrary size.",
"However, again focusing on a situation close to the NESS, one can get some insights.",
"For that purpose, we define small deviations in species concentrations from their respective NESS values as $\\delta _i(t)=c_i(t)-c_i^s.$ For a short time interval $\\tau $ , using finite difference approximation, one gets $\\dot{\\delta _i}=\\dot{c_i} \\approx \\delta _i/\\tau .$ Putting Eqs (REF )-(REF ) in Eq.",
"(REF ), we get $\\left(1-(k_i+k_{-(i-1)})\\tau \\right)\\delta _i(t)+k_{i-1}\\tau \\delta _{i-1}(t)+k_{-i}\\tau \\delta _{i+1}(t)=0.$ As the reactions are coupled, so the $\\delta _i$ s are related to each other and can be expressed in terms of any one of them, say $\\delta _1.$ Then, one can write $\\delta _i=f_i\\delta _1, \\quad {\\rm with}\\,\\, f_1=1.$ Next we will discuss the scheme to determine the $f_i$ s. The set of coupled equations (REF ), with the help of Eq.",
"(REF ), can be cast in the matrix form ${\\bf Mf=0}.$ Here ${\\bf f}$ is a $N\\times 1$ matrix with ${\\bf f^T}=\\left(f_1, f_2,\\cdots ,f_N\\right)$ and ${\\bf M}$ is a $N\\times N$ matrix with the property $M_{ij}\\ne 0,\\, {\\rm for}\\, j=i,i-1,i+1$ $M_{ij}=0, \\, {\\rm otherwise}.$ The non-zero matrix elements are $M_{ii}=\\left(1-(k_i+k_{-(i-1)})\\tau \\right),$ $M_{i,i-1}=k_{i-1}\\tau ,$ $M_{i,i+1}=k_{-i}\\tau .$ From Eq.",
"(REF ) and Eq.",
"(REF ), we obtain a recursion relation $f_j=-\\frac{(M_{j-1,j-2}f_{j-2}+M_{j-1,j-1}f_{j-1})}{M_{j-1,j}},\\,{j=2,3,\\cdots ,N},$ with the boundary conditions: $f_0=f_N,\\,\\,M_{j,0}=M_{j,N}.$ The first of the relations becomes $f_2=-\\frac{M_{1N}f_N+M_{11}}{M_{12}}.$ Then, it is easy to follow from Eq.",
"(REF ) that, all the other $f_j$ s can be expressed in terms of $f_N.$ From the condition $\\sum _{i=1}^{N}c_i={\\rm constant},$ we get $\\sum _{i=1}^{N}\\delta _i=0,$ and using Eq.",
"(REF ), we have $\\sum _{i=2}^{N}f_i=-1.$ From Eqs (REF )-(REF ) and Eq.",
"(REF ), one can determine the $f_i$ s in Eq.",
"(REF ).",
"We are now ready to explore the EPR near the NESS.",
"From Eq.",
"(REF ), we have $J_i^s=J^s,\\quad (i=1,\\cdots ,N)$ at NESS.",
"As we have chosen to express all the deviations in concentration from the NESS in terms of $\\delta _1$ , so we take the reaction rate as $v(t)=\\dot{a_1}$ .",
"Then, from Eq.",
"(REF ) with $i=1$ and using Eq.",
"(REF ) along with the periodic boundary conditions, we get near NESS $v(t)=R\\delta _1,$ where $R=-(k_1+k_{-N})+k_{N}f_N+k_{-1}f_2.$ Now putting Eq.",
"(REF ), Eq.",
"(REF ), Eq.",
"(REF ) and Eq.",
"(REF ) in Eq.",
"(REF ) and also using the smallness of $\\delta _i$ s, the EPR near NESS becomes $\\sigma _N(t)=\\sum _{i=1}^{N}\\left(J^s+k_i\\delta _i-k_{-i}\\delta _{i+1}\\right)\\left({\\rm ln}\\frac{k_ic_i^s}{k_{-i}c_{i+1}^s}+\\frac{\\delta _i}{c_i^s}-\\frac{\\delta _{i+1}}{c_{i+1}^s}\\right)$ $\\sigma _N(t)={\\rm A}_N^{(0)}+{\\rm A}_N^{(1)} v(t)+{\\rm A}_N^{(2)} v^2(t),$ with ${\\rm A}_N^{(0)}=J^s{\\rm ln}\\frac{\\prod _{i=1}^Nk_i}{\\prod _{i=1}^Nk_{-i}},$ ${\\rm A}_N^{(1)}=\\frac{1}{R}\\sum _{i=1}^N \\left(k_if_i-k_{-i}f_{i+1}\\right){\\rm ln}\\frac{k_ic_i^s}{k_{-i}c_{i+1}^s},$ ${\\rm A}_N^{(2)}=\\frac{1}{R^2}\\sum _{i=1}^N \\left(k_if_i-k_{-i}f_{i+1}\\right)\\left(\\frac{f_i}{c_i^s}-\\frac{f_{i+1}}{c_{i+1}^s}\\right).$ Eq.",
"(REF ) is the generalized version of Eq.",
"(REF ), confirming that expression of the EPR as a functional of reaction rate possesses a universal character."
],
[
"Cases with invariant SDR",
"The next task is, whether states having the same SDR, i.e., identical ${\\rm A}_N^{(0)}$ , can be generated for the N-cycle.",
"An obvious clue comes from the invariance of a cycle under rotation.",
"Thus, if the steady concentrations $c_i^s$ are represented as N points uniformly placed on a circle, then rotations by an angle $\\theta $ , defined as $\\theta =\\frac{2\\pi j}{N},\\quad j=1,\\cdots ,(N-1),$ will just redistribute the $c_i^s$ values.",
"This keeps the steady flux $J^s$ in Eq.",
"(REF ) unchanged.",
"Therefore, for a N-cycle, there are at least (N-1) ways to interchange the rate constants $k_{\\pm i}$ that will lead the reaction system to states with the same SDR.",
"We illustrate this result here by taking the simplest non-trivial case of a triangular network as an example.",
"One can see from Eq.",
"(REF ) that, for a triangular network with $N=3$ , at least two kinds of changes of the rate constants keep the SDR unchanged.",
"They are given below: Case 1.",
"Any parent choice of rate constants.",
"Case 2.",
"Change $k_{\\pm i}\\rightarrow k_{\\pm (i+1)},\\, (i=1,\\cdots ,N)$ with the boundary condition $k_{\\pm (N+1)}=k_{\\pm 1}$ .",
"Case 3.",
"Change $k_{\\pm 1}\\rightarrow k_{\\pm 3}$ , $k_{\\pm 2}\\rightarrow k_{\\pm 1}$ and $k_{\\pm 3}\\rightarrow k_{\\pm 2}$ .",
"One can generate additional ways to keep ${\\rm A}_N^{(0)}$ fixed with some added constraints on the rate constants.",
"Two pairs of situations [cases 4 and 5, and 6 and 7] are the following: Case 4.",
"Any parent choice of rate constants with $k_1=k_{-1}$ .",
"Case 5.",
"Change $k_1\\rightarrow k_2$ , $k_2\\rightarrow k_3$ , $k_3\\rightarrow k_{-1}$ , $k_{-1}\\rightarrow k_{-2}$ , $k_{-2}\\rightarrow k_{-3}$ and $k_{-3}\\rightarrow k_{1}$ .",
"Case 6.",
"Any parent choice of rate constants with $k_3=k_{-3}$ .",
"Case 7.",
"Change $k_1\\rightarrow k_{-3}$ , $k_{-3}\\rightarrow k_{-2}$ , $k_{-2}\\rightarrow k_{-1}$ , $k_{-1}\\rightarrow k_{3}$ , $k_{3}\\rightarrow k_{2}$ and $k_{2}\\rightarrow k_{1}$ .",
"All the above variants have been numerically studied and shown in Fig.REF where the EPR, determined exactly by Eq.",
"(REF ), is plotted as a function of reaction rate $v(t)=\\dot{a_1}$ for each of the cases.",
"It is evident from the figure that the SDR are identical for the respective bunch of cases.",
"But they can be distinguished by following the $\\sigma _3(t)$ vs. $v(t)$ curve in the small-$v(t)$ regime.",
"Figure: Variation of EPR σ 3 (t)\\sigma _3(t) (Eq.",
"())as a function of reaction rate v(t)v(t) forcases (a) 1,2 and 3, (b) 4 and 5, (c) 6 and 7."
],
[
"Conclusion",
"In summary, the present endeavor has been to characterize steady states with the same non-zero SDR.",
"We have found that the variation of EPR with the reaction rate near completion of the reaction is a nice indicator to distinguish such states.",
"Particularly important is the role of the slope of $\\sigma (t)$ vs. $v(t)$ curve near $v(t)=0$ .",
"This has been substantiated by studying enzyme-catalysed reactions as an exactly-solvable test case.",
"We have also noticed, the leading term that accounts for the variation depends on the rate constants, more specifically on the enzyme efficiency.",
"It is gratifying to observe that the more efficient enzyme incurs higher total dissipation.",
"The physical appeal is immediate.",
"A more efficient enzyme approaches the steady state more quickly.",
"This implies the process becomes more irreversible.",
"Hence, $S_I$ becomes higher.",
"One more notable point is the following.",
"The SDR is equal to the steady heat dissipation rate.",
"Our study reveals that enzymes with very different efficiencies can show the same heat dissipation rate at steady state.",
"An extension to cases of higher complexities involving various conformers of the enzyme-substrate complex has also been envisaged.",
"Further studies along this line on enzymes with multiple sites may be worthwhile."
],
[
"Acknowledgment",
"K. Banerjee acknowledges the University Grants Commission (UGC), India for Dr. D. S. Kothari Fellowship.",
"K. Bhattacharyya thanks CRNN, CU, for partial financial support."
]
] | 1403.0336 |
[
[
"Topological quantum phases of ${^4}$He confined to nanoporous materials"
],
[
"Abstract The ground state of $^4$He confined in a system with the topology of a cylinder can display properties of a solid, superfluid and liquid crystal.",
"This phase, which we call compactified supersolid (CSS), originates from wrapping the basal planes of the bulk hcp solid into concentric cylindrical shells, with several central shells exhibiting superfluidity along the axial direction.",
"Its main feature is the presence of a topological defect which can be viewed as a disclination with Frank index $n=1$ observed in liquid crystals, and which, in addition, has a superfluid core.",
"The CSS as well as its transition to an insulating compactified solid with a very wide hysteresis loop are found by ab initio Monte Carlo simulations.",
"A simple analytical model captures qualitatively correctly the main property of the CSS -- a gradual decrease of the superfluid response with increasing pressure."
],
[
"The surface and the low density phases",
"At low $\\mu $ , at most, the first two outer shells are formed.",
"The columnar view of such a typical atomic configuration (at $\\mu =-9.375$ K) is shown in Fig.",
"REF .",
"In this phase the superfluid response comes from the second shell (farthest from the wall).",
"Accordingly, the first shell is ordered and the second one is disordered.",
"Figure: (Color online) The columnar view along the pore axis of a typical atomic positions in the μ=-9.375\\mu =-9.375K sample.",
"The red circle marks the position of the hard wall.Figure: (Color online) The density modulations n(r)n(r) and the c-map in the sample, μ=-3.0\\mu =-3.0K, where the pore bulk is occupied by a low density superfluid.",
"The first two strong peaks in n(r)n(r) correspond to the two ordered surface shells, and the weaker peaks are induced by the roton feature of the spectrum.",
"Inset: the columnar view of a typical configuration along the pore axis, μ=-3.0\\mu =-3.0K.",
"The radial density modulations in the bulk, r<18 r< 18Å, cannot be distinguished visually.The bulk phase exists at $\\mu \\ge -7$ K. It can be viewed as two outer shells coexisting with the low density superfluid filling the pore bulk.",
"The bulk density $n(r)$ and the superfluid density (shown by the c-map) are both modulated in the radial direction at the wavelength corresponding to the roton.",
"These modulations observed in a sample $\\mu =-3.0$ K are shown in Fig.",
"REF .",
"These are the precursors of the shells which eventually form the CS and CSS."
],
[
"The CS vs non-CS geometries",
"A possible fitting of the $hcp$ structure into a cylinder with least of the bulk strain is shown in the top panel of Fig.",
"REF .",
"In this structure, the strong attractive wall potential creates several (here we show two) outmost hexagonal shells wrapped around the wall.",
"Since being closely packed in 2D, such shells minimize the surface energy.",
"The $C_6$ axis in these shells is oriented radially with respect to the pore axis.",
"In the inner part of the pore, however, the $C_6$ axis is aligned with the cylinder axis similarly to the director in the non-singular nematic disclination solution (see in Ref. [14]).",
"Simulations of pores with radii $< 15$ Å[18], [19] as well as our present work with the pore of almost twice that radius show that this configuration is not realized at least in pores with radii less than $\\sim $ 30Å.",
"The preferred configuration is the compactified $hcp$ solid shown in the bottom panel in Fig.",
"REF .",
"This configuration hosts the Frank disclination of index $n=1$ , with its core coinciding with the cylinder axis at $r=0$ .",
"There is a string of atoms arranged along a very narrow line at $r=0$ .",
"As we will estimate below, the compactified configuration, CS, has lower energy than the standard $hcp$ in pores with radii, at least, up to $R_0 \\sim 300$ Å.",
"Figure: (Color online) The columnar views along the pore axis of two possible configurations.",
"Top panel: the non-compactified hcphcp solid.",
"The C 6 C_6 axis is in the radial direction in the outer shells (purple dots) and it becomes along the cylinder axis (perpendicular to the page plane) in the inner part of the pore (blue dots).",
"Bottom panel: The compactified hcphcp solid.",
"The C 6 C_6 axis is along the radial direction (in the page plane) in the whole sample.",
"It winds in a manner similar to the director in the Frank nematic disclination with the index n=1n=1 (see in Ref.",
").Figure: (Color online) The unrolled second outer shell (counted from the wall).",
"The green stars show the ideal CS positions and the red dots are the atomic positions from a typical configurations from the simulations.",
"The vertical axis is the cylindrical z-coordinate, and the horizontal axis ll is the coordinate along the shell circumference.",
"The five-fold angular modulation along the z-axis with the amplitude ≈0.5\\approx 0.5Å is clearly seen.Figure: (Color online) The same pattern as in Fig.",
"is shown on smaller scale so that the triangular layer structure is obvious.Let's consider in detail the ideal compactified $hcp$ geometry.",
"In order to produce minimal residual strain the A-B hexagonal (basal) planes of the standard $hcp$ structure should be rolled into concentric cylinders along the direction of the elementary cell vector belonging to the basal plane so that the orthogonal direction is aligned with the cylinder axis.",
"Eight such shells are seen in the bottom panel in Fig.",
"REF (plus the central core).",
"The actual structure of the CS found in the simulations is very close to the one formed by this procedure, as is demonstrated in Figs.",
"REF and REF .",
"The number of unit cells in the $N$ th shell with radius $R_N$ is given by $M(N)= 2\\pi R_N/a(N)$ , where the length of the unit cell $a(N)$ may vary from shell to shell.",
"The next shell has radius $R_{N+1}=R_N + a_z(N)$ , where $a_z(N)$ is the radial distance between the $N$ th and $(N+1)$ th shells.",
"In a perfect hcp crystal $a_z = \\sqrt{2/3} a$ , where $a\\approx 3.6-3.7$ Å is the unit cell length in the basal plane.",
"In the compactified hcp solid this relation needs to be relaxed to $a_z(N)=\\gamma (N) \\sqrt{2/3} a(N)$ with $\\gamma (N) \\approx 1$ in order to minimize the strain.",
"Thus, the radius of the $N$ -th shell becomes $R_N= \\sum _{N^{\\prime }=1}^N \\gamma (N^{\\prime }) \\sqrt{2/3} a(N^{\\prime })$ .",
"Expressing $a(N)$ in terms of the integer number $M(N)$ , we obtain the equation for the shell radii $R_N = 2\\pi \\sqrt{\\frac{2}{3}} \\sum _{N^{\\prime }=1}^N \\gamma (N) \\frac{R_{N^{\\prime }}}{M(N^{\\prime })}.$ This equation has a solution $M(N)=5N,\\, \\gamma (N)= 5/( 2\\pi \\sqrt{\\frac{2}{3}})\\approx 0.975$ .",
"Thus, $a_z(N)$ is compressed (radially) by about $2.5\\%$ when compared with the standard hcp crystal.",
"In addition to the radial compression, there is shear strain of one shell with respect to its neighbor.",
"The \"quantization\" rule $M(N)=5N$ , Eq.",
"(REF ) implies that the circumference of each shell is broken into 5 equal angular segments, each subtended by an angle $2\\pi /5$ .",
"Within each segment, the smallest distance between two atoms from neigboring shells reaches the minimum $\\sqrt{3}a/2\\approx 0.87a$ along one radial line.",
"[There are five of such lines forming the $C_5$ symmetric pattern].",
"Thus, this strain can be estimated as $\\approx 1-0.87=0.13$ at its maximum, and about $0.13/2\\sim 0.065$ on average for the whole sample.",
"In simulations we have observed that such strain has been relaxed to about $0.04$ by the static angular modulation of atomic displacement about $0.5$ Å along the pore axis with the angular period $2\\pi /5$ .",
"Fig.",
"REF shows this pattern (see also Fig.",
"REF ).",
"Thus, the CS structure can be characterized by the $0.025$ compression strain and by about $0.04$ of the shear strain.",
"We estimate the resulting energy change as being due to the elastic energy $\\delta E_{el} \\sim (0.025^2+ 0.04^4) E_D$ , where $E_D$ is determined by elastic constants defining the Debye energy $\\sim 30$ K of solid $^4$ He.Thus, the compactification costs about extra $0.07$ K per particle.",
"In other words, the non-compactified $hcp$ $^4$ He of the same average density represented in the upper panel in Fig.",
"REF has less energy $\\sim 0.07KR_0^2$ if one ignores the boundary.",
"The boundary between the ideal $hcp$ and the outer shells are characterized by maximal possible misfit: the $C_6$ axis must rotate by 90 degrees in order to be aligned with the cylinder axis.",
"We estimate the energy of such misfit as being larger than $\\sim 0.1E_D\\sim 3$ K. Thus, the total excess energy of the CS can be written as $ \\sim - 3K\\cdot 2\\pi (R_0/a) + 0.07K\\cdot \\pi (R_0/a)^2$ .",
"It becomes larger than that of the non-compactified structure at radii larger than $R_0\\sim 90 a$ .",
"For typical values of $a$ this estimate gives about $R_0\\approx 300$ Å."
]
] | 1403.0106 |
[
[
"On the (in)variance of the dust-to-metals ratio in galaxies"
],
[
"Abstract Recent works have demonstrated a surprisingly small variation of the dust-to-metals ratio in different environments and a correlation between dust extinction and the density of stars.",
"Naively, one would interpret these findings as strong evidence of cosmic dust being produced mainly by stars.",
"But other observational evidence suggest there is a significant variation of the dust-to-metals ratio with metallicity.",
"As we demonstrate in this paper, a simple star-dust scenario is problematic also in the sense that it requires that destruction of dust in the interstellar medium (e.g., due to passage of supernova shocks) must be highly inefficient.",
"We suggest a model where stellar dust production is indeed efficient, but where interstellar dust growth is equally important and acts as a replenishment mechanism which can counteract the effects of dust destruction.",
"This model appears to resolve the seemingly contradictive observations, given that the ratio of the effective (stellar) dust and metal yields is not universal and thus may change from one environment to another, depending on metallicity."
],
[
"Introduction",
"The variation of the overall dust-to-metals ratios between galaxies of vastly different morphology, ages and metallicities appears surprisingly small in many cases, with a mean value close to the Galactic ratio ($\\sim 0.5$ ).",
"The relatively tight correlation between the dust-to-gas ratio and the metallicity (yielding an almost invariant dust-to-metals ratio) in the Local Group galaxies has been known for quite a while [49], [22], [53].",
"Indirect evidence for a `universal' mean value is also provided by the almost linear relation between $B$ -band optical depth and stellar surface density in spiral galaxies [16].",
"But recent results based on gamma-ray burst (GRB) afterglows, quasar foreground damped Ly$\\alpha $ -absorbers [54] and distant lens galaxies [6], [5] now seem to extend this correlation beyond the local Universe and down to metallicties just a few percent of the solar value.",
"[54] argue this can only be explained by either rapid dust enrichment by supernovae or very rapid interstellar grain growth by accretion of metals.",
"However, there can be significant variations within a galaxy [32], [33], although the existence of dust-to-metals gradients is somewhat difficult to establish observationally with reliable independent methods.",
"If, on the other hand, the dust-to-metals ratio does not vary much at all, in any environment, one may assume dust grains as well as atomic metals are mainly produced by stars.",
"Recent findings of large amounts of cold dust in supernova (SN) remnants [29], [15] seem to support this hypothesis, although the exact numbers can be disputed [44], [34], [35].",
"In other words: the overall picture is not consistent.",
"A new study by [7] seems to confirm the rising trend with metallicity of the dust-to-metals ratio in quasar DLAs found in previous studies [50], [51].",
"Furthermore, [11] derived a dust mass in the local starburst dwarf I Zw 18, as well as a high-redshift object of similar character, which clearly indicate a dust-to-metals ratio below the Galactic value.",
"These results, together with the likely existence of dust-to-metals gradients along galaxy discs [33], suggest the variance (or invariance) of the dust-to-metals ratio may depend on the environment.",
"In such case, there may exist an equilibrium mechanism that keeps the dust-to-metals ratio close to constant if certain conditions are fulfilled, while a metallicity dependence may occur as a result of deviations from those conditions in other environments.",
"Recently, [25] have tried to alleviate the tension between the results from the GRB afterglows of [54] and other data (for local dwarf galaxies) by fine-tuning the parameters of their standard galactic dust evolution model including grain growth [18], [24].",
"What they suggest is a quite reasonable compromise, but a truly convincing explanation of the different trends (constant and rising dust-to-metals ratio) would require some modification of the dust-formation scenario.",
"In particular, a model in which inherent properties of a galaxy more or less uniquely determines its dust-to-metals ratio would be desirable.",
"Even if the models by [25] are marginally consistent with the data they compare with, there is obviously still some tension between models and observations.",
"The new results by [7] only act as to emphasise this.",
"In the standard picture of production and destruction of cosmic dust one is faced with the following two problems: (1) in metal-poor environments dust is only supplied by stars as the interstellar density of metals is too low for efficient grain growth, but still being destroyed by SN shockwaves (albeit with a relatively low efficiency); (2) to compensate the destruction of dust grains, which eventually becomes efficient, with grain growth requires that one pushes the boundaries of the model, i.e., to obtain a sufficiently short grain-growth timescale, one is forced to accept a very large span of gas densities (several orders of magnitude) and a very low star-formation efficiency.",
"These problems are discussed in more detail by [25].",
"In this paper we investigate a scenario for the evolution of the galactic dust component where destruction of grains due to sputtering in SN shockwaves is roughly balanced by grain growth by accretion of molecular gas.",
"This idea has also been put forth in other studies to improve models of the build-up of dust in the local as well as distant (early) Universe [21], [31], [47], but here we take it one step further and consider a model where there can be an exact balance.",
"Given a constant ratio of the effective dust yield and the total metal yield for a generation of stars, such a scenario will lead to an invariant dust-to-metals ratio.",
"We continue by discussing the possibility that young undeveloped (low metallicity) systems may have a different yield ratio due to different dust yields for individual stars (e.g., the expected metallicity dependence)."
],
[
"Observational clues and constraints on the dust-to-metals ratio",
"Recently, [16] derived a correlation between the optical depth in the $B$ -band $\\tau _B$ and the stellar-mass surface density $\\Sigma _\\star $ in nearby spiral galaxies selected from the Galaxy and Mass Assembly (GAMA) survey, which were detected in the FIR/sub-mm in the Herschel-ATLAS field.",
"They find a nearly linear relation, $\\log (\\tau _B) = (1.12\\pm 0.11) \\times \\log \\left({\\Sigma _\\star \\over M_\\odot \\,{\\rm kpc}^{-2}} \\right) - 8.6\\pm 0.8,$ where the errors reflect the $1\\sigma $ scatter in the data.",
"The regression is marginally consistent with an exactly linear correlation between $\\tau _B$ and $\\Sigma _\\star $ .",
"If the dust density $\\Sigma _{\\rm d}$ is proportional to the stellar-mass density $\\Sigma _{\\rm stars}$ , there should exist a linear correlation between the optical depth $\\tau _{\\lambda }$ and the corrected, de-projected surface density of a galaxy, much like the relation above, because of the connection with the dust abundance, i.e., $\\tau _\\lambda \\sim \\Sigma _{\\rm d}$ .",
"[16], argue that their relation is evidence of efficient interstellar grain growth.",
"This conclusion depends on whether stars, primarily massive stars, can make a significant contribution to the dust production and on the efficiency of dust destruction in the interstellar medium (ISM).",
"In principle, the $\\tau _B - \\Sigma _\\star $ connection only says that dust and stars 'go hand-in-hand' in local spiral galaxies, which seems to suggest significant stellar dust production and that destruction of dust must be balanced by grain growth in the ISM.",
"However, as we will go on to show later, this is not necessarily the case.",
"A constant dust-to-metals ratio does not only seem to apply in the local Universe, however.",
"[54] combine extinction ($A_V$ values) and abundance data from GRB afterglows with similar data from QSO foreground absorbers and multiply-imaged galaxy-lensed QSOs, to determine the dust-to-metals ratios for a wide range of galaxy types and redshifts of $z = 0.1- 6.3$ , and almost three orders in metal abundance.",
"The mean dust-to-metals ratio for their sample is very close to the Galactic value and the $1\\sigma $ deviation is no more than 0.3 dex, suggesting the dust-to-metals ratio may be fairly invariant throughout the observable Universe.",
"Chandra X-ray observations of distant lens galaxies lend further support to this picture [5], [6].",
"Taken at face value, these results would imply a very rapid dust-formation scenario that is roughly the same in any environment.",
"The number of data points at low metallicity is relatively small in the work by [54].",
"It is therefore not certain that the dust-to-metals ratio is nearly invariant also at low metallicities.",
"A very recent study by [7] has shown, using a different method, that there is likely a turn-down in the dust-to-metals ratio at low metallicity.",
"This is also consistent with the constraint on the dust-to-metals ratio derived for the local starburst galaxy I Zw 18 [17], [11].",
"[7] measure the degrees of depletion of gas-phase abundances in the ISM for various elements, particularly focusing on Fe and Zn, and infer the dust abundance from these depletions.",
"The dust-depletion patterns are observed in UV/optical GRB afterglows and QSO spectra, associated with the ISM of the GRB host-galaxies and QSO-DLAs, and are derived assuming that the depletion is entirely due to dust condensation, regardless of its origin.",
"In particular, the method used by [7] relies on the assumption that the observed [Zn/Fe] traces the overall dust content in the ISM, and thus that (1) the intrinsic relative abundance of Zn and Fe is solar and (2) a non-negligible amount of iron is present in the bulk of the dust.",
"This is not obviously the case due to uncertainties in the origins of Zn and Fe, but investigating the reliability of these assumptions - and thus the exact slope of the trend of the dust-to-metals ratio with metallicity - goes beyond the scope of this paper.",
"What is particularly interesting about the new results by [7] is that the dust-to-metals ratio increases with increasing metallicity and, even more important, with increasing metal density.",
"The latter is a clear indication of grain growth being an important part of the build up of the dust mass.",
"Further evidence from DLAs of a down-turn in the dust-to-metals ratio at low metallicity is seen in the works by, e.g., [50], [51].",
"A similar, although somewhat steeper, down-turn of the dust-to-gas ratio was also recently found by [42] for low-metallicity galaxies in the local Universe."
],
[
"Grain growth",
"In a gaseous medium of a given temperature and density, the rate of accretion of a gas-phase species $i$ onto a spherical dust grain is given by the surface area of the grain ($4\\pi a^2$ where $a$ is the grain radius) and the sticking coefficient (probability) $f_{\\rm s}$ for that species [9].",
"The mass density of a species $i$ locked up in dust $\\rho _{{\\rm d}, i}$ then grows at a rate ${1\\over \\rho _{{\\rm d},i}} {d\\rho _{{\\rm d},i}\\over dt} = 3f_{\\rm s} {\\langle v\\rangle \\over a_{\\rm eff}} {\\rho _i-\\rho _{{\\rm d},i}\\over \\rho _{\\rm gr}},$ where $\\rho _i$ denotes mass density per unit volume of the growth species $i$ , $\\langle v_{\\rm g}\\rangle $ is the mean thermal speed of the gas particles, $a_{\\rm eft}$ is the effective (average) grain size and $\\rho _{\\rm gr}$ is the material bulk density of the dust.",
"Thus, the overall timescale of grain growth $\\tau _{\\rm grow}$ is, to first approximation, inversely proportional to the difference between total metallicity $Z$ and the dust-to-gas ratio $Z_{\\rm d}$ and can therefore be approximated using [32] $\\tau _{\\rm grow} \\propto {1\\over Z\\,\\rho _{\\rm H_2}}\\left(1-{Z_{\\rm d}\\over Z}\\right)^{-1},$ where $\\rho _{\\rm H_2} $ is the density of molecular hydrogen.",
"Here, the grain size, sticking probability, thermal speed of the gas particles and their molecular composition have been regarded as more or less invariant quantities.",
"For simplicity we will assume that the star formation rate is proportional to the molecular gas abundance.",
"Thus, $d\\rho _{\\rm s}/dt \\propto \\rho _{\\rm H_2}$ .",
"We can then regard the timescale $\\tau _{\\rm grow}$ as essentially just a simple function of the metallicity, the gas abundance and the growth rate of the stellar component.",
"Following [32] we adopt $\\tau _{\\rm grow} ^{-1}= {\\epsilon Z \\over \\rho _{\\rm g}} \\left(1-{Z_{\\rm d}\\over Z}\\right) {d\\rho _{\\rm s}\\over dt},$ where the constant $\\epsilon $ can be treated as a unit less free (but constrained) parameter of the model, representing the overall efficiency of grain growth."
],
[
"Destruction by sputtering",
"The dominant mechanism for dust destruction is by sputtering in the high-velocity interstellar shocks driven by SNe, which can be directly related to the energy of the SNe [39].",
"Following [36], [10] the dust destruction time-scale is $\\tau _{\\rm d} = {\\rho _{\\rm g}\\over \\langle m_{\\rm ISM}\\rangle \\,R_{\\rm SN}},$ where $\\rho _{\\rm g}$ is the gas mass density, $\\langle m_{\\rm ISM}\\rangle $ is the effective gas mass cleared of dust by each SN event, and $R_{\\rm SN}$ is the SN rate per unit volume.",
"The latter may be approximated as $R_{\\rm SN}(t) \\approx \\dot{\\rho }_{\\rm sfr}(r,t)\\int _{8M_\\odot }^{100M_\\odot } \\phi (m)\\,dm,$ where $\\phi (m)$ is the stellar initial mass function (IMF) and $\\dot{\\rho }_{\\rm sfr}$ is the star-formation rate per unit volume.",
"For a non-evolving IMF the integral in equation (REF ) is a constant with respect to time, and is not expected to vary much spatially within a galaxy either.",
"Hence, the time scale $\\tau _{\\rm d}$ may be expressed as $\\tau _{\\rm d}^{-1} \\approx {\\delta \\over \\rho _{\\rm g}}{d\\rho _{\\rm s}\\over dt},$ where $\\delta $ will be referred to as the dust destruction parameter.",
"This parameter is dimensionless, and as such it can be seen as a measure of the overall efficiency of dust destruction.",
"Small grains are more susceptible to destruction by sputtering in SN shock waves than large grains [43].",
"This is due to the larger grains' tendency to decouple from the gas and thus being less exposed to ions.",
"This fact suggests the above model is, partially, an inadequate description of the effects of destruction due to SN shocks.",
"Grain-grain interaction may lead to shattering and thus creation of smaller grains [19], [2], which are then more likely to be sputtered away.",
"Hence, the timescale of dust destruction may not only be inversely proportional to the SN rate, but also the abundance of dust, since the rate of interactions (or collisions) is proportional to the number density $n_{\\rm d}$ .With the adaptations usually employed in chemical collision theory [3], the collision frequency is $R_{\\rm coll} \\equiv \\sigma _{\\rm coll} v_{\\rm rel}\\,n_{\\rm d}$ , where $\\sigma _{\\rm coll} = 2\\pi \\langle a^2\\rangle $ is the effective cross-section for grain-grain collisions, $n_{\\rm d}$ is the number density of dust grains and $v_{\\rm rel}$ is the typical relative velocity of two colliding grains.",
"Using $R_{\\rm coll}$ we may define the collision density as ${1\\over 2}R_{\\rm coll}\\,n_{\\rm d}$ .",
"The factor $1/2$ has been introduced to avoid double-counting the collisions.",
"Obviously, the collision density is proportional to $Z_{\\rm d}^2$ since $n_{\\rm d}\\propto Z_{\\rm d}$ .",
"The efficiency of dust destruction is roughly proportional to the shattering rate, since smaller fragments are more easily destroyed, and the shattering rate is to first order proportional to the collision rate, which sketchily motivates the modified model of dust destruction suggested above.",
"A reasonable modification to the dust-destruction timescale would then be to introduce a dependence on the dust-to-gas ratio $Z_{\\rm d}$ , i.e., $\\tau _{\\rm d}^{-1} \\approx {\\delta \\over \\rho _{\\rm g}}{Z_{\\rm d}\\over Z_{\\rm d,\\,G}}{d\\rho _{\\rm s}\\over dt},$ where $Z_{\\rm d,\\,G}$ is the present-day Galactic dust-to-gas ratio.",
"The dust-destruction efficiency $\\delta $ can be calibrated to the expected efficiency (timescale) for the Galaxy, which we take to be roughly 0.7 Gyr [23].",
"The effective Galactic gas-consumption rate is about $2\\,M_\\odot $ pc$^{-2}$ Gyr$^{-1}$ , and the gas density is $\\sim 8\\,M_\\odot $ pc$^{-2}$ [30], which implies $\\delta \\approx 5$ .",
"[31] estimated $\\delta \\approx 10$ based on a [27] IMF and that stars of initial masses above $10\\,M_\\odot $ become SNe.",
"We can thus assume $\\delta \\sim 5 - 10$ is a reasonable estimate of the expected range for $\\delta $ ."
],
[
"Simple models of dust evolution",
"In [32], [33] we showed that dust growth would be the most important mechanism for changing the dust-to-metals ratio $\\zeta $ in a galaxy throughout its course of evolution and/or create a dust-to-metals gradient along a galaxy disc.",
"Since, in the present work, we want to also consider the situations where $\\zeta $ is not changing much, we will focus on the two viable scenarios for dust production: (1) pure stellar dust production and inefficient dust destruction and (2) a scenario where dust destruction is balanced by dust growth in the ISM.",
"To simplify our model we make the same assumptions as in [32] and [33], i.e., a galaxy evolves effectively as a `closed box' and the stellar dust/metals production can be described under the instantaneous recycling approximation.",
"We also assume the effects of the inevitably chaining grain-size distribution are negligible on average, so that grain growth and destruction are functions of macroscopic properties only as described in the next subsection.",
"Furthermore, we make the assumption that the fraction of condensible metals (metals that may end up in dust grains) $Z_{\\rm c}$ is essentially the same as the total metallicity, i.e., $Z_{\\rm c} \\approx Z$ .",
"This assumption is quite reasonable as the observed depletion is surprisinglingy close to 100% for many of the most abundant metals except oxygen and the noble gases [41].",
"The equation for the evolution of the dust-to-metals ratio $\\zeta = Z_{\\rm d}/Z$ is then [32], $Z{d\\zeta \\over dZ} = {y_{\\rm d}\\over y_Z} + {\\zeta Z\\over y_Z}[G(Z)-D(Z)]- \\zeta ,$ where $G$ is the rate of increase of the dust mass due to grain growth relative to the rate of gas consumption due to star formation, $D$ is the corresponding function for dust destruction and $y_{\\rm d}$ , $y_Z$ is the effective stellar dust and metal yields, respectively.",
"The dust yield $y_{\\rm d}$ may have a significant dependence on the metallicity of the stellar population, which we will return to later.",
"In terms of the timescales for grain growth and destruction above, $G$ and $D$ can be defined as $G(Z) = \\epsilon Z\\,\\left[1-{Z_{\\rm d}(Z)\\over Z}\\right], \\quad D = {\\delta } \\quad {\\rm or}\\quad D(Z) = \\delta ^{\\prime } Z_{\\rm d}(Z),$ where $\\delta /\\delta ^{\\prime } = Z_{\\rm d,\\,G}$ ."
],
[
"Pure stellar dust production",
"We first consider the case where we have only stellar dust production and no destruction of dust in the ISM ($\\epsilon = \\delta = 0$ ).",
"For a `closed box', the dust-to-gas ratio $Z_{\\rm d}$ is simply given by $Z_{\\rm d} = y_{\\rm d}\\ln \\left(1+{\\Sigma _\\star \\over \\Sigma _{\\rm gas}} \\right),$ Note that by replacing $y_{\\rm d}$ with $y_Z$ , we would obtain the corresponding relations for metallicity.",
"Series expansion around $\\Sigma _\\star /\\Sigma _{\\rm gas} = 0$ yields $Z_{\\rm d} = y_{\\rm d}{\\Sigma _\\star \\over \\Sigma _{\\rm gas}} + {y_{\\rm d}\\over 2}\\left({\\Sigma _\\star \\over \\Sigma _{\\rm gas}} \\right)^2 - \\dots ,$ from which we may conclude that $\\Sigma _{\\rm d}\\approx y_{\\rm d}\\,\\Sigma _\\star $ for an unevolved galaxy where $\\Sigma _\\star /\\Sigma _{\\rm gas}\\ll 1$ .",
"Thus, the dust masses in young starbursts, like I Zw18, should give us a measure of the stellar dust yield $y_{\\rm d}$ (at least for low metallicities).",
"This may also give a hint about the origin of the $\\Sigma _{\\rm d} \\sim \\Sigma _\\star $ connection seen in the results by [16], i.e., that we should consider a model where a balance between growth and destruction leads to a similar $\\Sigma _{\\rm d}\\sim \\Sigma _\\star $ relation for more evolved systems.",
"If we include interstellar dust destruction with a timescale given by eq.",
"(REF ) ($D = \\delta $ , $G = 0$ ) the closed-box solution to (REF ) can be written in the form $\\zeta = {Z_{\\rm d}\\over Z} = {y_{\\rm d}\\over y_Z}{1\\over \\delta }\\left[1-\\left(1+{\\Sigma _\\star \\over \\Sigma _{\\rm gas}} \\right)^{-\\delta }\\right]\\ln \\left(1+{\\Sigma _\\star \\over \\Sigma _{\\rm gas}}\\right)^{-1}.$ Analysis of this solution shows that $\\Sigma _\\star /\\Sigma _{\\rm gas}\\gg 1$ requires $\\zeta \\ll 1$ [31].",
"Using the timescale given by eq.",
"(REF ), which is based on the suggested grain-grain interactions ($D = \\delta ^{\\prime } Z_{\\rm d} $ , $G = 0$ ), gives a solution of the form $\\zeta = {Z_{\\rm d}\\over Z} = {1 \\over y_Z}\\sqrt{y_{\\rm d}\\over \\delta ^{\\prime }}\\tanh \\left[\\sqrt{y_{\\rm d} \\delta ^{\\prime }} \\ln \\left(1+{\\Sigma _\\star \\over \\Sigma _{\\rm gas}}\\right)\\right],$ which suggests the same asymptotic behaviour, i.e., $\\Sigma _\\star /\\Sigma _{\\rm gas}\\gg 1$ requires $\\zeta \\ll 1$ .",
"This tells us that only stellar dust production cannot work if there is interstellar dust destruction on any level after the dust has become part of the diffuse ISM.",
"The dust-to-metals ratio $\\zeta $ will decrease monotonously unless the effective stellar dust yield $y_{\\rm d}$ increases in such a way that it compensates for the dust destruction.",
"Otherwise, if we are to maintain a roughly constant $\\zeta $ , there cannot be any significant destruction of dust in the ISM."
],
[
"Growth/destruction equilibrium model",
"With $G$ as in Eq.",
"(REF ) and $D=\\delta $ (the `canonical' model of dust destruction) we have an equation for $\\zeta $ which reads $Z{d\\zeta \\over dZ} = {y_{\\rm d}\\over y_Z} + {\\zeta Z\\over y_Z}\\left[\\epsilon \\left(1-\\zeta \\right)\\,Z -\\delta \\right] - \\zeta .$ The equilibrium case $d\\zeta /dZ = 0$ would correspond to $\\epsilon (1-\\zeta )\\,Z-\\delta = 0$ and $\\zeta = y_{\\rm d}/y_Z$ , which is equivalent to the criterion ${\\delta \\over \\epsilon }= Z\\,\\left(1-{y_{\\rm d}\\over y_Z}\\right).$ This is a problem, however, since $y_{\\rm d}$ , $y_Z$ , as well as $\\delta $ , $\\epsilon $ are constants by definition, while $Z$ cannot be constant, except under very special conditions.",
"It is therefore virtually impossible to keep $\\zeta $ more or less constant over a wide range of metallicities.",
"If we instead consider our second equation of dust evolution, $Z{d\\zeta \\over dZ} = {y_{\\rm d}\\over y_Z} + {\\zeta Z^2\\over y_Z}\\left[\\epsilon \\left(1-\\zeta \\right) -\\delta ^{\\prime } \\zeta \\right] - \\zeta ,$ for the case where the dust-destruction timescale depends on the dust-to-gas ratio $Z_{\\rm d}$ , i.e., $D(Z) = \\delta ^{\\prime } Z_{\\rm d}(Z)$ , where $\\delta ^{\\prime } = \\delta /Z_{\\rm d,\\,G}$ , we obtain a more realistic equilibrium condition.",
"More precisely, we have that $\\epsilon (1-\\zeta )-\\delta ^{\\prime } \\zeta = 0$ , which leads to ${\\delta ^{\\prime }\\over \\epsilon }= {y_Z \\over y_{\\rm d}}-1.$ This criterion is more useful than Eq.",
"(REF ), since it does not involve any variable.",
"If we adopt the Galactic dust-to-metals ratio, $\\zeta _{\\rm G}\\approx 0.5$ , we have $y_{\\rm d}/y_Z\\approx 0.5$ and thus $\\epsilon \\approx \\delta ^{\\prime }$ .",
"With $\\delta \\sim 5-10$ and $\\delta /\\delta ^{\\prime } \\approx 100$ (Galactic gas-to-dust ratio), we then have $\\epsilon \\sim 500-1000$ , which suggests a relatively high efficiency of grain growth is required to only maintain balance between growth and destruction.",
"A parameter range $\\epsilon \\sim 500-1000$ is consistent with the results by [33].",
"The special case $\\epsilon = \\delta ^{\\prime }$ is worth some further consideration.",
"Provided there is no dust if $Z=0$ , it follows directly from Eq.",
"(REF ) that $\\zeta (0) = y_{\\rm d}/y_Z$ regardless of whether $\\epsilon = \\delta ^{\\prime }$ or not.",
"In the opposite limit (large $Z$ ) the dust-to-metals ratio $\\zeta $ will approach its asymptotic value and thus be constant.",
"Hence, Eq.",
"(REF ) reduces to $0 = {\\epsilon \\zeta \\over y_Z} (1-2\\zeta ),$ which corresponds to $\\zeta \\rightarrow 1/2$ (the asymptotic value).",
"Thus, if $\\epsilon $ and $\\delta ^{\\prime }$ are similar, regardless of the actual value, we would have $\\zeta \\sim 0.5$ .",
"This result is particularly interesting since the dust-to-metals ratio in essentially all Local Group galaxies are close to $\\zeta \\approx 0.5$ [20], [8].",
"With the model suggested above, this ratio would be a universal ratio which all galaxies will evolve toward, while the dust-to-metals ratio at early times may be quite different.",
"A similar idea is discussed in [21].",
"The general solution to Eq.",
"(REF ) for the initial condition $Z_{\\rm d}(0) = Z(0) = 0$ and $\\epsilon > 0$ is $\\zeta = {y_{\\rm d}\\over y_Z} {M\\left[1+{1\\over 2}{y_{\\rm d}\\over y_Z}\\left(1+ {\\delta ^{\\prime }\\over \\epsilon } \\right), {3\\over 2}; {1\\over 2}{\\epsilon Z^2\\over y_Z}\\right]\\over M\\left[{1\\over 2}{y_{\\rm d}\\over y_Z}\\left(1+ {\\delta ^{\\prime }\\over \\epsilon } \\right), {1\\over 2}; {1\\over 2}{\\epsilon Z^2 \\over y_Z}\\right]},$ where $M(a,b;z)$ is the Kummer-Tricomi function of the first kind, which is identical to the confluent hypergeometric function $_1F_1(a,b;z)$ .",
"[32].",
"The growth/destruction equilibrium case, ${\\delta ^{\\prime } /\\epsilon }= {y_Z / y_{\\rm d}}-1$ , corresponds to $a=b$ , where we note that $M(a,a;z) = e^z$ .",
"Consequently, $\\zeta = y_{\\rm d}/y_Z$ , as discussed above.",
"In reality, one would expect deviations from an exactly constant dust-to-metals ratio to occur as a consequence of local variations of the yield ratio $y_{\\rm d}/y_Z$ together with $\\delta ^{\\prime }$ and $\\epsilon $ .",
"The latter two parameters are clearly different for different dust compositions and may also have implicit dependences on the gas density and, perhaps most importantly, on the grain-size distribution, which can only be `universal on average'.",
"For the case $\\epsilon = \\delta ^{\\prime } = 0$ (neither dust growth, nor destruction) we have the trivial solution $\\zeta = y_{\\rm d}/y_Z$ , which is of course identical to the equilibrium case above."
],
[
"Metallicity-dependent stellar dust production",
"The effective stellar dust yield $y_{\\rm d}$ has so far been treated as a constant.",
"To first order, this is an acceptable approximation, but as we are here interested in dust production at very low metallicity it is necessary to consider a scenario in which $y_{\\rm d}$ is a function of the metallicity $Z$ .",
"There are two reasons for this.",
"First, some key-elements for dust production (such as silicon) may be less abundant in low-metallicity stars.",
"This is obviously the case for the massive, short-lived, AGB stars which are producing mainly silicates, but has no (or very little) silicon production of their own.",
"Second, dust condensation is strongly dependent on the absolute abundance/density of the relevant metals.",
"That is, there may exist a critical metallicity below which dust condensation become inefficient due to low partial pressures for many metals, leading to less nucleation and slow accretion.",
"It is already well established that such a critical metallicity exists for grain growth in the ISM [1] This can be the case also in massive stars which, despite that they produce significant amounts of metals, may have too low partial pressures of certain key-elements to have efficient nucleation.",
"Figure: Effective stellar dust yield as a function of metallicity.",
"The solid red line shows the smooth `jump' from low to high degree of dust condensation according to Eq.",
".A very simple scenario would be the one where $y_{\\rm d}$ is simply proportional to the metallicity $Z$ .",
"Assuming interstellar dust processing has no effect on the dust mass fraction of the ISM ($G=D=0$ or $G=D\\ne 0$ ) and $y_{\\rm d}(Z) = y_{\\rm d,0} + k\\,Z$ , where $y_{\\rm d,0}$ , $k$ are constants, we have $Z{d\\zeta \\over dZ} = {y_{\\rm d,0} + k\\,Z\\over y_Z} - \\zeta ,$ which has the simple solution [with initial condition $\\zeta (0)=0$ ] $\\zeta (Z) = {1\\over 2 }{y_{\\rm d,0} + y_{\\rm d}(Z)\\over y_Z}.$ This model produces a rising trend as seen in several observations, but is otherwise not very realistic.",
"First, there is no `roof' in the solution above.",
"$\\zeta $ can continue to grow even beyond the absolute upper limit $\\zeta = 1$ .",
"Second, it is expected that there is critical/threshold metallicity for efficient dust formation rather than a linear rise as above.",
"Thus, a more realistic scenario is that in which stellar dust production becomes efficient at a certain metallicity, i.e., there is a smooth `jump' in $y_{\\rm d}$ at some metallicity $Z_{\\rm e}$ .",
"The transition from inefficient to efficient dust condensation is likely smooth, so it would be reasonable to adopt something of the form (see Fig.",
"REF ) $y_{\\rm d}(Z) = y_{\\rm d, 0} + \\Delta y_{\\rm d}\\exp \\left(-{Z_{\\rm e}\\over Z}\\right),$ where $y_{\\rm d, 0}$ is the minimum dust yield for inefficient dust condensation and $y_{\\rm d, max} = y_{\\rm d, 0} + \\Delta y_{\\rm d}$ is the maximum dust yield obtained at high efficiency.",
"Thus, we obtain the solution [with initial condition $\\zeta (Z_0)=\\zeta _0$ ] $\\zeta (Z) = {y_{\\rm d}(Z) \\over y_Z} + {\\Delta y_{\\rm d}\\over y_Z}{Z_{\\rm e}\\over Z}\\left[ {\\rm E}_1\\left({Z_{\\rm e}\\over Z_0} \\right) - {\\rm E}_1\\left({Z_{\\rm e}\\over Z} \\right) - {Z_0\\over Z_{\\rm e}}\\exp \\left(-{Z_{\\rm e}\\over Z_0}\\right)\\right]$ where we have defined the so-called exponential integral ${\\rm E}_n$ as ${\\rm E}_n(x) \\equiv \\int _1^\\infty \\frac{e^{-xt}}{t^n}\\, dt.$ If the initial metallicity $Z_0$ is very small, or, more precisely, if $Z_0\\ll Z_{\\rm e}$ , we can simplify the above expression into $\\zeta (Z) = {y_{\\rm d}(Z) \\over y_Z} - {\\Delta y_{\\rm d}\\over y_Z} {Z_{\\rm e}\\over Z} {\\rm E}_1\\left({Z_{\\rm e}\\over Z} \\right).$"
],
[
"Results and discussion",
"Below we consider the dust-to-metals trends derived from optical depth, extinction magnitude and depletion levels of certain metals and compare them with the simplistic models described in the previous section.",
"Furthermore, we present simple Monte Carlo simulations to demonstrate how such simple scenario would appear when allowing the model parameters to vary within a certain parameter space."
],
[
"B-band optical depth and dust abundance: do stars dominate cosmic dust production?",
"Due to the approximate proportionality between $A_V$ and the dust-to-gas ratio one would expect the $B$ -band optical depth $\\tau _B$ to be a simple function of dust density.",
"More precisely, $\\tau _B \\sim \\Sigma _{\\rm d}$ .",
"Given the result by [16], the dust mass density $\\Sigma _{\\rm d}$ is then simply proportional to the stellar mass density $\\Sigma _{\\star }$ .",
"Theoretically, this proportionality is expected for all unevovled (gas-rich) galaxies (see Section REF ).",
"But for it to hold also for more evolved galaxies, a balance between growth and destruction of dust in the ISM is necessary.",
"The trend obtained by [16] is fundamentally an empirical result, and consistent with a simple model where dust is produced by stars.",
"Nevertheless, the nearly linear $\\tau _B - \\Sigma _{\\star }$ is not answering the question whether dust is formed mainly in stars or grown in the ISM (growth and destruction can conspire to produce a $\\Sigma _{\\rm d}\\sim \\Sigma _\\star $ relation), but seems to favour models with significant stellar dust production.",
"The linear relation discussed above may suggest the dust-to-metals ratio is not showing large variations since the metal content of a galaxy is typically correlated with the stellar mass [26], [40], but there is still plenty of room for scatter and the relation is derived for local spiral galaxies which may be in similar evolutionary states where the dust-to-metals ratio has reached an `equilibrium plateau'.",
"A more diverse sample of objects would therefore provide a more useful statistical constraint."
],
[
"Invariant dust-to-metals ratio?",
"We have transformed the dust-to-metals ratios in [54] from observational units to unit less ratiosDefining the dust-to-metals ratio in observational units as $k/Z\\equiv \\log (N_{\\textsc {Hi}}) + [{\\rm X/H}] + \\log (A_V)$ , where $N_{\\textsc {Hi}}$ is the column density of neutral hydrogen and [X/H] is the abundance of X relative to the corresponding solar value, we adopted the Galactic value $(k/Z)_G = 21.3$ [54].",
"The unit less dust-to-metals ratio is obtained as $\\zeta = \\zeta _G 10^{[k/Z-(k/Z)_G]}$ , where $\\zeta _G \\approx 0.5$ .",
"Here, we adopt $\\zeta _G = 0.47$ to maintain consistency between the data sets.",
"But the exact value is not very important as long as the adopted value is the same for all data sets considered.",
"(as in the models discussed in previous sections) for those objects where all relevant quantities have been measured with sufficient accuracy (see Fig.",
"REF ).",
"The relatively small variation of the dust-to-metals ratio ($\\zeta = 0.47\\pm 0.13$ ) seen over such a wide range of redshifts and metallicities (and likely also galaxy types) in the work by [54] is a relatively strong constraint on the dust formation scenario, provided it can be trusted despite the rather small number of reliable measurements.",
"A dust-to-metals ratio $\\zeta = 0.47$ would correspond to $y_{\\rm d}/y_Z = 0.47$ in an `equilibrium model' where $\\delta ^{\\prime } = \\epsilon \\,(y_Z/y_{\\rm d}-1)$ (see Section REF ).",
"As shown by the different models (analytic solutions to Eq.",
"REF ) over-plotted in Fig.",
"REF , variation of the yield ratio $y_{\\rm d}/y_Z$ leads to a wide range of dust-to-metals ratios at low metallicity, but converges to the asymptotic value, which is $\\zeta = 0.5$ for the special case $\\delta ^{\\prime } = \\epsilon $ .",
"The $1\\sigma $ scatter in the [54] data suggest $\\zeta $ can vary at most about 30%, but it should be noted the observed values cover a range $\\zeta = 0.18 - 0.67$ , which indicates significant variations of $\\zeta $ cannot be ruled out due to small-number statistics.",
"The growth/destruction-equilibrium model suggested in Section REF is attractive as it would explain the existence of a characteristic, essentially universal, dust-to-metals ratio $\\zeta $ , as suggested by [54].",
"Deviations from this `universal' value could then be attributed to variations of the yield ratio $y_{\\rm d}/y_Z$ .",
"As we have mentioned in Section REF , there could exist a critical metallicity (or, more precisely, number density of certain key elements) in stars below which dust condensation is inefficient.",
"However, it could also be that $y_{\\rm d}/y_Z$ is a `universal constant' and that the limited variance in $\\zeta $ could be explained by the fact that the growth and destruction parameters, $\\epsilon $ and $\\delta ^{\\prime }$ , respectively, can vary between different environments.",
"Realistically, none of these parameters ($y_{\\rm d}$ , $y_Z$ , $\\epsilon $ , $\\delta ^{\\prime }$ ) should be viewed as `universal constants', of course.",
"We will return to this aspect of the variance in $\\zeta $ in Section REF .",
"Figure: Dust-to-metals ratio as a function of metallicity for a subset of the GRB and QSO-DLA sample and three QSO-DLAs used by .",
"The mean ratio (dashed line) is essentially identical to the Galactic dust-to-metals ratio.",
"The over-plottedfull-drawn lines show models with ϵ=δ ' =750\\epsilon = \\delta ^{\\prime } = 750 and various (constant) y d /y Z y_{\\rm d}/y_Z ratios ranging from 20-8020-80%.Figure: Dust-to-metals ratio as a function of metallicity for a subset (objects with silicon-based metallicities were excluded) the GRB and QSO-DLAs considered by .",
"The overalltrend is consistent with the dust-to-metals ratio derived for I Zw 18 by .",
"The over-plotted solid black lines show models with ϵ=δ ' \\epsilon = \\delta ^{\\prime } and various y d /y Z y_{\\rm d}/y_Z ratios ranging from10-5010-50%.",
"The blue line (grey in printed version) shows the best numerical solution with metallicity-dependent stellar dust yield, including grain processing in the ISM (see Sect.",
"), and thedotted line shows the corresponding (analytical) solution for the case of only metallicity-dependent stellar dust production (Eq.",
").",
"Note how the transition from stellar dust production to interstellar dust growth appears tohappen at roughly 1/10 of solar metallicity."
],
[
"Increasing dust-to-metals ratio?",
"At first glance, an invariant dust-to-metals ratio in one context [16], [54] seem to be inconsistent with a clearly rising trend with metallicity in another [7].",
"But as we have already discussed, the growth/destruction-equilibrium model with $\\epsilon = \\delta ^{\\prime }$ has an asymptotic dust-to-metals ratio $\\zeta _{\\rm A}$ which is eventually reached regardless of what $\\zeta $ is at early times.",
"But if $\\zeta $ shows a clear trend with metallicity, as in the results by [7], there cannot just be random variations of the yield ratio $y_{\\rm d}/y_Z$ .",
"[7], as well as, e.g., [50], find a $\\zeta $ increasing with metallicity, which is what one would expect in a scenario where the bulk of cosmic dust is grown in the ISM rather than produced directly by stars.",
"However, according to our simplistic model with a constant $y_{\\rm d}$ , the expected trend without growth/destruction-equilibrium is a steep rise in $\\zeta $ at some critical metallicity [32], which is not in agreement with the observed trend (see Fig.",
"REF , models with $y_{\\rm d}/y_Z < 0.5$ ).",
"The observed slower rise of the dust-to-metals ratio can thus be a result of a changing yield ratio.",
"If $y_{\\rm d}$ increases at a certain metallicity, as described in Section REF , the observed trend could easily be explained.",
"The analytic solutions for different $y_{\\rm d}/y_Z $ (and $\\epsilon = \\delta ^{\\prime }$ ) over-plotted in Fig.",
"REF show that if the yield ratio changes from a few percent at very low metallicity to $\\sim 0.5$ at moderately low metallicity ($Z\\sim 0.1\\,Z_\\odot $ ), the correct rising trend would be obtained.",
"Ultimately, this shows that we need to modify our model - a constant yield ratio $y_{\\rm d}/y_Z$ fails to reproduce the trend.",
"The blue line in Fig.",
"REF is a numerical solution (forth-order Runge-Kutta) using Eq.",
"(REF ) with the parameter values plotted in Fig.",
"REF to describe $y_{\\rm d}(Z)$ , which demonstrates exactly this point.",
"At the same time, there is always a $y_{\\rm d}/y_Z $ that will lead to a constant dust-to-metals ratio $\\zeta $ for any given $\\epsilon /\\delta ^{\\prime }$ .",
"We suggest this could be a good compromise in order to obtain a model that can explain why $\\zeta $ in some cases show very little variation and in other cases a trend with metallicity.",
"The case where interstellar dust processing has no effect on the dust mass fraction of the ISM ($G=D=0$ or $G=D\\ne 0$ ) is indicated by the dotted black line in Fig.",
"REF (corresponding to Eq.",
"REF ).",
"The effect of interstellar grain growth is the difference between the solid blue and dotted black lines, where the critical metallicity (the point where the lines diverge) occurs at $Z/Z_\\odot \\approx 0.1$ .",
"The most likely cause for a changing effective dust yield $y_{\\rm d}$ is the existence of a critical metallicity below which dust formation is significantly less efficient compared to the efficiency at higher metallicities.",
"As we have already mentioned, a lower number density of key-elements for dust condensation may be important in stars that do not produce much of these key-elements themselves.",
"But for most massive stars that undergo a core-collapse supernova explosion the amount of metals produced is significant even at $Z=0$ [38].",
"However, gas opacities and cooling rates may be lower at very low metallicities, which in turn may affect the heating and cooling of existing dust grains.",
"If the average grain temperature is high enough for sublimation to occur, the net efficiency of condensation may be low.",
"Thus, it is not clear that very metal-poor stars can be efficient dust producers even if raw material for dust formation is present.",
"Asymptotic giant branch (AGB) stars are probably not very important dust producers at low metallicity according to recent work in which a steep dependence on metallicty is found [48].",
"In addition, at really low metallicity of the interstellar gas, i.e., at very early times, low- and intermediate-mass stars have not had enough time to evolve into AGB stars either.",
"For example, metal-poor halo stars in the Galaxy appear to have been formed from gas that is mainly enriched by massive stars (supernovae with progenitor masses typically in the range $10-20\\,M_\\odot $ ), although variations in the abundance patterns sometimes occur [14].",
"Moreover, the destruction of dust in SNe is likely more efficient the more massive the progenitor star is (and the degree of dust condensation is likely lower), which means that a bias towards more massive stars at low metallicity may also lead to less stellar dust per unit stellar mass.",
"Numerical models of SN dust production do indeed confirm that the most massive stars have less surviving dust in their ejecta [4].",
"To summarise the above: oxygen-rich AGB stars (the more massive and short-lived ones) cannot produce very much dust at low metallicity since they do not produce the refractory elements needed for dust production, and the effective dust yield of massive stars is probably strongly metallicity dependent too.",
"Thus, a $y_{\\rm d}/y_Z $ increasing with metallicity seems reasonable.",
"The reason why the GRB and QSO-DLAs studied by [54], as well as local galaxies, show so little variation in their dust-to-metal ratios (despite a wide range of metallicities) is still not obvious.",
"But provided the effective dust yield $y_{\\rm d}$ depends on the metallicity, this invariant ratio as well as the rising trend found in quasar DLAs by measuring depletions [50], [51], [7] could be `two sides of the same coin'.",
"Statistical variations in the overall efficiencies of grain growth and destruction in the ISM, combined with some uncertainty in which metallicity $Z_{\\rm e}$ stellar dust production starts to become efficient, will allow for enough scatter in the dust-to-metals ratio as a function of metallicity to have one fundamental model which is consistent with both the flat and the rising trend.",
"This will be explored in the next section.",
"As an alternative hypothesis, one may consider the possibility that the $A_V$ -based dust abundance estimates in [54] are biased towards environments which have, relatively speaking, significant foreground contamination from intervening systems and therefore appear to have higher dust-to-metals ratios at low metallicity.",
"This possibility should of course be investigated, but goes beyond the scope of this paper.",
"Figure: Left panel: Monte Carlo simulation with stellar dust production and no interstellar growth and/or destruction.",
"Model parameters (random variables) according to Table .Right panel: same as the left panel but with interstellar growth and destruction included as well.",
"The solid black line shows the same numerical solution as in Fig.",
".",
"The over plotted observational data is taken from .Table: Random variables/parameters used for the Monte Carlo models."
],
[
"Monte Carlo simulation of the dust-to-metals ratio as a function of metallicity",
"We expect variations in not only the effective dust yield $y_{\\rm d}$ , but also in the timescales of grain growth and destruction ($\\epsilon $ and $\\delta $ , in practice).",
"To quantify the effects of such variations, to some extent, we have performed a couple of Monte Carlo simulations where we vary the parameters $\\delta ^{\\prime }$ and $\\epsilon $ within reasonable ranges as well as setting them to zero (see Table REF ).",
"The yield ratio $y_{\\rm d}/y_Z$ is not completely arbitrary either.",
"On the one hand, the fraction of metals being injected into the ISM in the form of dust grains cannot be 100%, since the degree of dust condensation must be limited by the physical conditions and the abundances of certain key elements (e.g., carbon or silicon) in the dust chemistry.",
"On the other hand, this fraction cannot be too small either, since it is an observational fact that low- and intermediate-mass stars as well as massive stars in the local Universe produce significant amounts of dust.",
"The fraction of dust that actually survive and eventually enrich the ISM is not known, but with the observed trend shown in Fig.",
"REF as reference we have calibrated the range of the effective yield ratio $y_{\\rm d}/y_Z$ to approximately $0.02 - 0.44$ .",
"Thus, two of the parameters of Eq.",
"(REF ) are fixed: $y_{\\rm d,0} = 2.0\\cdot 10^{-4}$ and $\\Delta y_{\\rm d} = 0.042$ , while $Z_{\\rm e}$ remains as a random variable of the Monte Carlo simulation together with $\\delta ^{\\prime }$ and $\\epsilon $ .",
"In Fig.",
"REF we have plotted the resultant probability density functions (PDF) of our simulation results.",
"To begin with, we performed a Monte Carlo simulation of the case of stellar dust production only, with the yield ratio $y_{\\rm d}/y_Z$ ($y_{\\rm d}$ not metallicity dependent) and the metallicity $Z$ as the only random variables.",
"For this simulation we assumed that $y_{\\rm d}/y_Z$ follows a normal distribution with standard deviation $0.1$ , centred at $y_{\\rm d}/y_Z = 0.5$ (model A in Table REF ).",
"The resultant PDF is consistent with data from [54], as can be seen in the left panel of Fig.",
"REF .",
"After establishing this `bench mark', we then considered the case of a metallicity dependent dust yield according to Eq.",
"(REF ) with the parameter values given above and $Z_{\\rm e} = 0.75-1.5\\cdot 10^{-4}$ .",
"The $\\epsilon $ and $\\delta ^{\\prime }$ ranges are difficult to define, but as we argued in Section REF , $\\delta \\sim 5 - 10$ ($\\delta ^{\\prime }\\sim 500-1000$ ) is a reasonable estimate of the expected range for $\\delta $ .",
"Under the assumption $\\epsilon \\approx \\delta ^{\\prime }$ , we may then assume $\\epsilon \\sim 500 - 1000$ (see model B in Table REF ).",
"All random variables were in this case assumed to follow uniform distributions.",
"As we showed in Section REF , the dust-to-metals ratio converges to $\\zeta = 0.5$ if $\\epsilon = \\delta ^{\\prime }$ , regardless of the value of $y_{\\rm d}/y_Z$ or whether $y_{\\rm d}$ is metallicity dependent or not.",
"Clearly, this is the reason why the scatter in $\\zeta $ becomes smaller at high metallicity when interstellar grain growth and destruction is included, compared to the case where $\\epsilon = \\delta ^{\\prime } = 0$ in which the scatter is the same regardless of metallicity (cf.",
"left and right panels in Fig.",
"REF ).",
"This inherent property of the model suggests one could, in principle, use the amount of scatter at approximately solar metallicity to constrain the width of the range of likely $\\epsilon $ and $\\delta ^{\\prime }$ values.",
"The observational data suggest a relatively small scatter (see Figs.",
"REF and REF ), albeit with large error bars on some data points.",
"The parameter ranges that we have used in our simple Monte Carlo simulation appears to give a result that is consistent with the spread and uncertainty of the data at solar-like metallicities.",
"Of course, one cannot draw very firm conclusions from a simplistic simulation like the present, but it seems that models which include interstellar grain growth and destruction is favoured by the fact that there appears to be significantly more scatter among the data points at low ($\\sim 1/10$ of solar) metallicity than near solar metallicity.",
"We therefore think our growth/destruction equilibrium model is plausible and may provide guidance towards a more consistent picture of the of the origin and evolution of cosmic dust."
],
[
"Conclusions",
"Several observational studies suggest a surprisingly small variation of the dust-to-metals ratio in vastly different environments.",
"It is worth stressing that the `trivial solution' to the problem, i.e., adopting a (constant) yield ratio of $y_{\\rm d}/y_Z \\sim 0.5$ , works for any model where there is a replenishment mechanism to counteract possible dust destruction [25].",
"But other observational evidence also suggest there is a significant variation of the dust-to-metals ratio between different environments, and an invariant dust-to-metals ratio is problematic also in the sense that it requires fine-tuning and is pushing the limits of the `standard models' of dust evolution in galaxies to explain all data [25].",
"We find that a reasonable way to resolve this apparent contradiction, and avoiding fine-tuning and extreme model parameters, is to assume that stellar dust production can be efficient, but that interstellar dust growth is equally important and act as a replenishment mechanism which can almost exactly counteract the dust destruction in the ISM.",
"In this scenario, the ratio of the effective (stellar) dust and metal yields is not likely a universal constant and may change due to some metallicity-dependence of the stellar dust yield.",
"We propose the existence of a critical stellar metallicity above which nucleation and condensation of dust in stars can be efficient.",
"We conclude that destruction and growth of grains in the ISM likely strives towards an equilibrium state, which mimics the general behaviour of the case of pure stellar dust production (and no destruction of grains).",
"This explains the relatively small variation of the dust-to-metals ratio seen in several observational studies of local galaxies, but allows also for a significantly lower ratio at low metallicity if the effective stellar dust yield can vary with metallicity.",
"The suggested scenario has important implications for the rapid build-up of large dust masses at high redshifts.",
"Instead of requiring an extreme efficiency of dust formation in massive stars (SNe) as suggested by, e.g., [10], the large dust masses seen in the quasar-host galaxy SDSS J1148+5251 (and other objects at high redshifts), follows naturally from the rapid production of metals that is expected in a massive starburst.",
"Just as [47] we are led to conclude that, though massive stars must produce significant amounts of dust, dust masses of the order $10^8-10^9\\,M_\\odot $ (as in SDSS J1148+5251) are not likely a result of stellar dust sources only (as a consequence of interstellar dust destruction) and the resultant dust component must therefore be dominated by grain growth in molecular clouds."
],
[
"Acknowledgments",
"The authors thank the anonymous reviewer for his/her constructive criticism which helped improve this paper.",
"Nordita is funded by the Nordic Council of Ministers, the Swedish Research Council, and the two host universities, the Royal Institute of Technology (KTH) and Stockholm University.",
"The Dark Cosmology Centre is funded by the Danish National Research Foundation.",
"ADC acknowledges support by the Weizmann Institute of Science Dean of Physics Fellowship and the Koshland Center for Basic Research."
]
] | 1403.0502 |
[
[
"Chaotic enhancement of dark matter density in binary systems"
],
[
"Abstract We study the capture of galactic dark matter particles (DMP) in two-body and few-body systems with a symplectic map description.",
"This approach allows modeling the scattering of $10^{16}$ DMPs after following the time evolution of the captured particle on about $10^9$ orbital periods of the binary system.",
"We obtain the DMP density distribution inside such systems and determine the enhancement factor of their density in a center vicinity compared to its galactic value as a function of the mass ratio of the bodies and the ratio of the body velocity to the velocity of the galactic DMP wind.",
"We find that the enhancement factor can be on the order of tens of thousands."
],
[
"Introduction",
"In 1890, Henri Poincaré proved that the dynamics of the three-body gravitational problem is generally non-integrable [18].",
"Even 125 years later, many aspects of this problem remain unsolved.",
"Thus the capture cross-section $\\sigma $ of a particle that scatters on the binary system of Sun and Jupiter has only recently been determined, and it has been shown that $\\sigma $ is much larger than the area of the Jupiter orbit [10], [11].",
"The capture mechanism is described by a symplectic dynamical map that generates a chaotic dynamics of a particle.",
"The scattering, capture, and dynamics of a particle in a binary system recently regained interest with the search for dark matter particles (DMP) in the solar system and the Universe [1], [7], [14].",
"Thus it is important to analyze the capture and ejection mechanisms of a DMP by a binary system.",
"Such a system can be viewed as a binary system with a massive star and a light body orbiting it.",
"This can be the Sun and Jupiter, a star and a giant planet, or a super massive black hole (SMBH) and a light star or black hole (BH).",
"In this work we analyze the scattering process of DMP galactic flow, with a constant space density, in a binary system.",
"One of the main questions here is whether the density of captured DMPs in a binary system can be enhanced compared to the DMP density of the scattering flow.",
"The results obtained by [11] show that a volume density of captured DMPs at a distance of the Jupiter radius $r<r_p=r_J $ is enhanced by a factor $\\zeta \\approx 4000$ compared to the density of Galactic DMPs which are captured after one one orbital period around the Sun and which have an energy corresponding to velocities $v < v_{cap} \\sim v_p \\sqrt{m_p/M} \\sim 1$ km.s$^{-1} \\ll u$ .",
"Here, $m_p, M$ are the masses of the light and massive bodies, respectively, $u \\approx 220$ km.s$^{-1}$ is the average velocity of a Galactic DMP wind for which, following [1], we assume a Maxwell velocity distribution: $f(v) dv = \\sqrt{54/\\pi } v^2/u^3 \\exp (-3v^2/2u^2) dv$ .",
"Our results presented below show that for an SMBH binary system with $v_{cap} > u$ there is a large enhancement factor $\\zeta _g \\sim 10^4$ of the captured DMP volume density, taken at a distance of about a binary system size, compared to its galactic value for all scattering energies (and not only for the DMP volume density at low velocities $v < v_{cap} \\ll u,$ as discussed by [11]).",
"We note that the Galactic DMP density is estimated at $\\rho _g \\sim 4 \\times 10^{-25}$ g.cm$^{-3}$ , while the typical intergalactic DMP density is estimated to be $\\rho _{g0} \\sim 2.5 \\times 10^{-30}$ g.cm$^{-3}$ [7], [14].",
"At first glance, this high enhancement factor $\\zeta _g \\sim 10^4$ seems to be rather unexpected because it apparently contradicts Liouville's theorem, according to which the phase space density is conserved during a Hamiltonian evolution.",
"Because of this, it is often assumed [8], [12] that the volume (or space) DMP density cannot be enhanced for DMPs captured by a binary system, and thus $\\zeta _g \\sim 1$ .",
"Below we show that this restriction is not valid for the following reasons: first, we have an open system where DMPs can escape to infinity, being ejected from the binary system by a time-dependent force induced by binary rotation.",
"This means that the dynamics is not completely Hamiltonian.",
"Second, DMPs are captured (or they linger, or are trapped) and are accumulated from continuum at negative coupled energies near the binary during a certain capture lifetime (although not forever).",
"Thus, the longer the capture lifetime, the higher the accumulated density.",
"Third, we obtain the enhancement for the volume density and not for the density in the phase space, for which the enhancement is indeed restricted by Liouville's theorem.",
"We discuss the details of this enhancement effect in the next sections.",
"The scattering and capture process of a DMP in a binary system can be an important element of galaxy formation.",
"This process can also be useful to analyze cosmic dust and DMP interaction with a supermassive black hole binary.",
"This is expected to play a prominent role in galaxy formation, see [9].",
"Thus we hope that analyzing this process will be useful for understanding the properties of velocity curves in galaxies, which was started by [24] and [20].",
"We note that the velocity curves of captured DMPs in our binary system have certain similarities with those found in real galaxies."
],
[
"Symplectic map description",
"Following the approach developed by [17], [5], [13], and [11], we used a symplectic dark map description of the DMP dynamics in one orbital period of a DMP in a binary system $w_{n+1}=w_n+F(x_n) \\; , \\;\\; x_{n+1}=x_n+w^{-3/2}_{n+1} \\; ,$ where $x_n= t_n/T_p \\; (mod \\; 1)$ is given by time $t_n$ taken at the moment of DMP $n-th$ passage through perihelion, $T_p$ is the planet period, and $w=-2E/m_d v_p^2$ .",
"Here $E, m_d, \\text{and } v_p$ are the energy, mass of the DMP, and the velocity of the planet or star.",
"The amplitude $J$ of the kick $F$ -function is proportional to the mass ratio $J \\sim m_p/M$ .",
"The shape of $F(x)$ depends on the DMP perihelion distance $q$ , the inclination angle $\\theta $ between the planetary plane $(x,y)$ and DMP plane, and the perihelion orientation angle $\\varphi $ , as discussed by [11].",
"In the following we use for convenience units with $m_d=v_p=r_p=1$ (here $m_d$ is the DMP mass, which does not affect the DMP dynamics in gravitational systems).",
"Figure: Poincaré sections for the dark map () (top)and the Kepler map () (bottom)for parameters of the Halley comet case in Eq.",
"()and J=0.007J=0.007 in Eq.",
"() (see text).For $q>r_p$ the amplitude $J$ drops exponentially with $q$ and $F(x)= J \\sin (2 \\pi x),$ as shown by [17].",
"This functional form of $F(x)$ is significantly simpler than the real one at $q < r_p$ , while it still produces chaotic dynamics at $0< w \\ll 1$ and integrable motion with invariant curves above a chaos border $w > w_{ch}$ .",
"In this regime the map takes the form $w_{n+1}=w_n+J\\sin (2\\pi x_n) \\; , \\;\\; x_{n+1}=x_n+w^{-3/2}_{n+1} \\; .$ The same map describes a microwave ionization of excited hydrogen atoms that is called the Kepler map, see [2], [21].",
"There, the Coulomb attraction plays the role of gravity, while a circular planet rotation is effectively created by the microwave polarization.",
"The microwave ionization experiments performed by [6] were made for three-dimensional atoms, but the ionization process is still well described by the Kepler map [3], [21].",
"These results provide additional arguments in favor of a simplified Kepler map description of DMP dynamics in binary systems.",
"The dynamics of the Kepler map can be locally described by the Chirikov standard map [4].",
"We note that the approach based on the Kepler map has recently been used to determine chaotic zones in gravitating binaries, see [22].",
"The similarity of dynamics of dark (REF ) and Kepler (REF ) maps is also well visible from comparing their Poincaré sections, shown in Fig.",
"REF , for the typical dark map parameters corresponding to the Halley comet [11] and the corresponding parameter $J$ of the Kepler map.",
"To take into account that $J$ decreases with $q,$ we use the relation $J=J_0=const$ for $q<q_b$ and $J=J_0 \\exp (-\\alpha (q-q_b)))$ for $q \\ge q_b$ (below $J$ is used instead of $J_0$ ).",
"We use $ q_b=1.5$ and $\\alpha =2.5$ , corresponding to typical dark map parameters [11], but we checked that the obtained enhancement is not affected by a moderate variation of $q_b \\text{ or } \\alpha $ .",
"The simplicity of map (REF ) allows increasing the number $N_p$ of injected DMPs by a factor one hundred compared to map (REF ).",
"The correspondence between (REF ) and (REF ) is established by the relation $J=5 m_p/M,$ which works approximately for the typical parameters of Halley comet case.",
"Of course, as discussed by [11], the dark map and moreover the Kepler map give an approximate description of DMP dynamics in binary systems.",
"However, this approach is much more efficient than the exact solution of Newton equations used by [15], [16], and [23] and allows obtaining results with very many DMPs injected during the lifetime of the solar system (SS) $t_S=4.5 \\times 10^9$ years.",
"The validity of such a map description is justified by the results obtained by [17], [5], [13], [11], [19], and [3]."
],
[
"Capture cross-section",
"The capture cross-section $\\sigma $ is computed as previously described by [11] with $\\sigma (w)/\\sigma _p = (\\pi ^2 r_p |w|)^{-1}\\int _0^{2\\pi } d \\theta \\int _0^\\pi d\\varphi \\int _0^{\\infty } dq h(q,\\theta ,\\varphi )$ , where $h$ is a fraction of DMPs captured after one map iteration from $w<0$ to $w>0$ , given by an interval length inside the $F(x)$ envelope at $|w|=const$ , $\\sigma _p=\\pi r_p^2$ .",
"The equation for $\\sigma (w)$ is based on the expression for the scattering impact parameter $ r_d^2=2qr_p/|w|$ .",
"For the Kepler map the $h-$ function only depends on $q,$ and the numerical computation is straightforward.",
"The differential energy distribution of captured DMPs is $dN/dw =\\sigma (w) n_g f(w)/2$ with $n_g=\\rho _g/m_d$ .",
"The results for $\\sigma (\\omega )$ and $dN/dw/N_p$ , obtained for maps (REF ) and (REF ), are shown in Fig.",
"REF .",
"Here $N_p=\\int _0^1 dw n_g \\sigma _p v_p^2 f(w)/2$ is the number of DMPs crossing the planet orbit area per unit of time.",
"The results of Fig.",
"REF show that both maps give similar results, which provides additional support for the Kepler map description.",
"The theoretical dependence $\\sigma \\propto 1/|w|$ , predicted by [10], is clearly confirmed.",
"The only difference between maps (REF ) and (REF ) is that the kick amplitude $J \\approx 5m_p/M$ for (REF ) is restricted, and thus after one kick we may have only $|w| \\le J$ , while for (REF ) some orbits can be captured with $|w| > J =5m_p/M$ as a result of close encounters.",
"However, the probability of such events is low."
],
[
"Chaotic dynamics",
"The injection, capture, evolution, and escape of DMPs is computed as described by [11]: we numerically modeled a constant flow of scattered DMPs with an energy distribution $d N_s = \\sigma (w) v_p^2f(w) dw/2$ per time unit (we used $q \\le q_{max}=4r_p$ ).",
"For Jupiter we have $u \\approx 17 \\gg 1$ and $d N_s \\propto dq d w$ .",
"However, for an SMBH we can have $u^2 < J$ so that one kick captures almost all the DMPs from the galactic distribution $f(w)$ .",
"In this case, we used the whole distribution $f(w)$ ($w=v^2$ ).",
"Map (REF ) is simpler than (REF ) since the kick function only depends on $q,$ which allows performing simulations with more DMPs.",
"Figure: (a) Number N cap N_{cap}of captured DMPs as a function of time tt in yearsfor the energy rangew>0w >0 (black curve), w>4·10 -5 w >4 \\cdot 10^{-5} corresponding tohalf the distance between Sun andthe Alpha Centauri system (red curve), w>1/20w >1/20corresponding to r<100 AU r<100 {\\rm AU}(blue curve); N J =4×10 11 N_J=4 \\times 10^{11} DMPs are injectedduring SS lifetime t S t_S; data are obtained fromthe map () at J=0.005J=0.005, u=17u=17corresponding to the Sun-Jupiter case.",
"(b) The top part shows the density distributionρ(w)∝dN/dw\\rho (w) \\propto dN/dw in energyat time t S t_S(normalized as ∫ 0 ∞ ρdw=1\\int _0^\\infty \\rho dw=1),the bottom part shows the Poincaré sectionof the map (); the inset showsthe density distribution of the captured DMPs in ww(black curve), the red line shows the slope -3/2.The scattering and evolution processes were followed during the whole lifetime $t_S$ of the SS.",
"The total number of DMPs, injected during time $t_{S}$ for $|w| \\le J$ and all $q$ is $N_J$ .",
"For the Kepler map the highest value is $N_J = 4 \\times 10^{11}$ , which is 100 times higher than for the dark map.",
"Figure: (a) Stationary radial density ρ(r)∝dN/dr\\rho (r) \\propto dN/drfrom the Kepler map at J=0.005J=0.005with u=17u=17 at time t S t_S (red curve) andu=0.035u=0.035 at time t u ≈4×10 8 T p t_u \\approx 4 \\times 10^8 T_p(black curve);data from the dark mapat m p /M=10 -3 m_p/M=10^{-3} are shown bythe blue curve at u=17u=17 and time t S t_S for the Sun-Jupiter case,and by the green curve at u=0.035u=0.035 and t S t_S for the SMBH;the normalization is fixed as ∫ 0 6r p ρdr=1\\int _0^{6r_p} \\rho dr =1, r p =1r_p=1.",
"(b) Volume density ρ v =ρ/r 2 \\rho _v=\\rho /r^2 fromthe data of panel (a),the dashed line shows the slope -2-2.The time dependence $N_{cap}(t)$ for the Kepler map, shown in Fig.",
"REF , is very similar to that found for the dark map by [11].",
"For a finite SS region $w> 1/20$ the growth of $N_{cap}(t)$ saturates after a time scale of $t_d \\approx 10^7$ years.",
"This scale approximately corresponds to a diffusive escape time $t_d \\sim 12\\text{ } {\\rm years} /D \\sim 10^6 {\\rm \\text{}years,}$ where the diffusion rate is taken in a random phase approximation to be $D \\approx J^2/2$ [2].",
"The diffusive spreading extends from $w \\sim 0$ up to chaos border $w_{ch} \\approx 0.3$ .",
"This value agrees well with the theoretical value $w_{ch}=(3\\pi J)^{2/5} = 0.29$ obtained from the Chirikov criterion [4] [17], [2], [10].",
"The validity of the Chirikov criterion in this system was also demonstrated in [22].",
"As for the dark map, we obtain a density distribution of $\\rho (w) \\propto 1/w^{3/2}$ , corresponding to the ergodic estimate according to which $\\rho (w)$ is proportional to time period at a given $w$ .",
"The results of Figs.",
"REF ,REF , and REF confirm the close similarity of dynamics described by maps (REF ) and (REF )."
],
[
"Radial variation of the dark matter density",
"To compute the DMP density, we considered captured orbits $N_{AC}$ with $w>4 \\times 10^{-5}$ .",
"The radial density $\\rho (r)$ was computed by the method described by [11]: $N_{AC}$ were determined at instant time $t_S$ ; for them the dynamics in real space was recomputed during a time period $\\Delta t \\sim 100 $ years of planet.",
"The value of $\\rho (r)$ was computed by averaging over $k=10^3$ points randomly distributed over $\\Delta t$ for all $ N_{AC}$ orbits.",
"We also checked that a semi-analytical averaging, using an exact density distribution over Kepler ellipses for each of $N_{AC}$ orbits, gives the same result: assuming ergodicity $\\rho _{w,q}(r)dr=w^{3/2}dt/2\\pi $ and using Kepler's equation, the radial density of the DMPs on a given orbit is $\\rho _{w,q}(r)=\\left(rw^2/2\\pi \\right)\\left(\\left(1-qw\\right)^2-\\left(1-rw\\right)^{2}\\right)^{-1/2}$ , then adding the radial density of each $N_{AC}$ orbit, we retrieve the DMP radial density $\\rho (r)$ shown in Fig.",
"REF .",
"From the obtained space distribution we determine a fraction $\\eta _{r_i}$ of $N_{AC}$ DMP orbits located inside a range $0 \\le r \\le r_i$ by computing $\\eta _{r_i} = \\Delta N_i/(k N_{AC}),$ where $\\Delta N_i$ is the number of points inside the above range (we used $r_i/r_p=0.2, 1, \\text{and} \\text{ }6$ ).",
"In Fig.",
"REF we show the dependence of radial $\\rho (r)$ and volume $\\rho _v=\\rho /r^2$ densities on distance $r$ .",
"For the Kepler map data, the density $\\rho (r)$ has a characteristic maximum at $r_{max}$ that is determined by the chaos border position $r_{max} \\approx 2/w_{ch}$ (this dependence, as well as the relation $w_{ch} = (3\\pi J)^{2/5}$ , is numerically confirmed for the studied range $10^{-3} < J < 10^{-2}$ for the Kepler map with a given fixed $J$ ).",
"The density profile $\\rho (r)$ is not sensitive to the value of $u$ and remains practically unchanged for $u=17, \\; 0.035$ .",
"For the dark map a variation of the kick function with $q$ and angles leads to a variation of $w_{ch}$ that leads to a slow growth of $\\rho $ at large $r$ .",
"A power-law fit of $\\rho _v \\propto 1/r^\\beta $ in a range $2 < r < 100$ gives $\\beta \\approx 2.25 \\pm 0.003$ for the Kepler map data and $\\beta = 1.52 \\pm 0.002$ for the dark map.",
"We attribute the difference in $\\beta $ values to a larger fraction of integrable islands for the dark map, as is visible in Fig.",
"REF for typical parameters.",
"We note that an effective range of radial variation is bounded by the kick amplitude with $r < r_{cap} \\approx 1/J,$ and in the range $r_p<r<r_{cap}$ the data are compatible with $\\rho \\sim const$ (dashed line in Fig.",
"REF b).",
"We note that the value of $u$ does not significantly affect the density variation with $r$ , as is clearly seen in Fig.",
"REF .",
"The spacial density distribution of computed from the dark map at $u=0.035$ shown in Fig.",
"REF is also very similar to those at $u=17$ [11].",
"This independence of $u$ arises because $\\rho (r)$ is determined by the dynamics at $w>0,$ which is practically insensitive to the DMP energies at $-J<w<0$ that are captured by one kick.",
"Figure: Density of captured DMPs at present time t S /T p ≈4×10 8 t_S/T_p \\approx 4 \\times 10^8for the dark map at m p /M=10 -3 m_p/M=10^{-3} and u/v p =0.035u/v_p=0.035Top panels: DMP surface densityρ s ∝dN/dzdr ρ \\rho _s \\propto dN/dzdr_\\rho shown at the leftin the cross plane (0,y,z)(0,y,z) perpendicular to the planetary orbit(data are averaged over r ρ =x 2 +y 2 =constr_\\rho =\\sqrt{x^2+y^2}=const),at the right in the planet plane (x,y,0)(x,y,0);only the range |r|≤6|r| \\le 6 around the center is shown.Bottom panels: correspondingDMP volume density ρ v ∝dN/dxdydz\\rho _v \\propto dN/dxdydz at the leftin the plane (0,y,z)(0,y,z), at the right in the planet plane(x,y,0)(x,y,0);only the range |r|≤2|r| \\le 2 around the SMBH is shown.The color is proportional to the densitywith yellow/black for maximum/zero density."
],
[
"Enhancement of dark matter density",
"To determine the enhancement of the DMP density captured by a binary system we followed the method developed by [11].",
"We computed the total mass of DMP flow crossing the range $q \\le 4 r_p$ during time $t_S$ : $M_{tot}= \\int _0^{\\infty } dv v f(v) \\sigma \\rho _g t_S\\approx 35 \\rho _g t_S k r_p M/u, $ where we used the cross-section $\\sigma = \\pi r_d^2 = 8\\pi k M r_p/v^2$ for injected orbits with $q \\le 4 r_p$ , $w=v^2$ , $k$ is the gravitational constant.",
"For SS at $u/v_p \\approx 17$ we have $M_{tot} \\approx 0.5 \\cdot 10^{-6} M$ .",
"From the numerically known fractions $\\eta _{ri}$ of the previous section and the fraction of captured orbits $\\eta _{AC}=N_{AC}/N_{tot}$ we find the mass $M_{ri}=\\eta _{ri} \\eta _{AC} M_{tot}$ inside the volume $V_i= 4\\pi r_i^3/3$ of radius $r<r_i$ ($r_i=0.2 r_p; r_p; 6 r_p$ ).",
"Here $N_{tot}$ is the total number of injected orbits during the time $t_S$ , while the number of orbits injected in the range $|w|<J$ (only those can be captured) is $N_J = N_{tot} ( \\int _0^{J} dw f(w)/w)/( \\int _0^{\\infty } dw f(w)/w)$ .",
"For $J \\ll u^2$ we have $\\kappa = N_{tot}/N_J = 2u^2/(3J) \\approx 3.8 \\times 10^4$ for $u/v_p =17$ and $\\kappa =1$ for $u/v_p=0.035$ at $J=0.005$ .",
"Thus for $u/v_p=17$ the number of orbits, injected at $0 <|w|<J$ , $N_J = 4 \\times 10^{11}$ , corresponds to the total number of injected orbits $N_{tot} \\approx 1.5 \\times 10^{16}$ .",
"Finally, we obtain the global density enhancement factor $\\zeta _g(r_i)=\\rho _v(r_i)/\\rho _g \\approx 16 \\pi \\eta _{ri} \\eta _{AC} (r_p/r_i)^3 \\tau _S v_p/u$ , where $\\tau _S=t_S/T_p$ is the injection time expressed in the number of planet periods $T_p=2\\pi r_p/v_p$ .",
"For $u^2 \\gg J$ it is useful to determine the enhancement $\\zeta = \\rho _v(r_i)/\\rho _{gJ}$ of the scattered galactic density in the range $0<|w|<J,$ whose density is $\\rho _{gJ} \\approx 1.38 \\rho _g J^{3/2} (v_p/u)^3$ .",
"Thus $\\zeta = 0.72 \\zeta _g (u/v_p)^3/J^{3/2}$ .",
"The results of the DMP density enhancement factors $\\zeta $ and $\\zeta _g$ are shown in Fig.",
"REF .",
"At $(u/v_p)^2 \\gg J$ we have $\\zeta \\gg 1$ and $\\zeta _g \\ll 1$ .",
"At $u/v_p=17$ we find that $\\zeta \\propto 1/J$ (the fit gives exponent $a=1.04 \\pm 0.01$ ) and $\\zeta _g \\propto \\sqrt{J}$ (the fit exponent is $a=0.46 \\pm 0.1$ ) in agreement with the above relation between $\\zeta $ and $\\zeta _g$ .",
"In general, we have $\\zeta _g \\propto 1/u$ for $u/v_p \\ll \\sqrt{J}$ and $ \\zeta _g \\propto 1/u^3$ for $u/v_p \\gg \\sqrt{J}$ .",
"There is only weak variation of $\\zeta _g$ with $J$ for $u/v_p \\ll \\sqrt{J}$ .",
"The values of $\\zeta $ and $\\zeta _g$ have similar values for the dark and Kepler maps (a part of the fact that at $r_i=0.2 r_p$ and $r_i=r_p$ the dark map has approximately the same $\\zeta $ since there $\\rho _v(r) \\sim const $ for $r \\le r_p$ ).",
"Figure: Dependence of the DMP density enhancementfactor ζ=ρ v (r i )/ρ gJ \\zeta =\\rho _v(r_i)/\\rho _{gJ}on JJ at u/v p =17u/v_p=17 (Jupiter); here ρ gJ \\rho _{gJ}is the galactic DMP volume densityfor an energy range of 0<|w|<J0<|w|<J andr i =0.2r p ,r p ,6r p r_i =0.2 r_p, r_p, 6 r_p (blue, black, red);points and squares show results for map ()with the number of injected particlesN J =4×10 9 N_J= 4 \\times 10^{9} and 4×10 11 4 \\times 10^{11}, respectively; crosses show data for map ()with N J =4×10 9 N_J= 4 \\times 10^{9} and J=5m p /MJ=5m_p/M.",
"(b) Dependence of the galactic enhancement factorζ g =ρ v (r i )/ρ g \\zeta _g=\\rho _v(r_i )/\\rho _{g}on u/v p u/v_p at r ζ =r p r_\\zeta =r_p and J=0.005J=0.005 in () (points)and m p /M=0.001m_p/M=0.001 in () (crosses),here ρ g \\rho _{g} is the global galactic density;lines show dependencies ζ g ∝1/u\\zeta _g \\propto 1/u (red)and ζ g ∝1/u 3 \\zeta _g \\propto 1/u^3 (blue).",
"(c) Dependence of ζ g \\zeta _g on JJ at u/v p =17u/v_p=17;(d) the same at u/v p =0.035u/v_p=0.035,parameters of symbols are as in (a),(b).The green curve shows theory () in all panels.All these results can be summarized by the following formula for the chaotic enhancement factor of DMP density in a binary system: $\\zeta _g = A \\sqrt{J} (v_p/u)^3/[1+B J (v_p/u)^2] \\; ,J=5 m_p/M \\; .$ Here $\\zeta _g$ is given for DMP density at $r_i=r_p$ and $A \\approx 15.5$ , $B \\approx 0.7$ .",
"This formula describes the numerical data of Fig.",
"REF well.",
"For $(u/v_p)^2 \\gg J$ we have $\\zeta _g \\ll 1,$ but we still have an enhancement of $\\zeta = 0.72 \\zeta _g (u/v_p)^3/J^{3/2} \\approx 0.72A/J \\gg 1$ .",
"For $(u/v_p)^2 \\ll J$ we have the global enhancement $\\zeta _g \\approx 22(v_p/u)/\\sqrt{J} \\gg 1$ .",
"The color representation of dependence (REF ) is shown in Fig.",
"REF .",
"Figure: Logarithm of DMP density enhancement factor log 10 ζ g \\log _{10}\\zeta _gfrom (), shown by color and log\\log value-levels,as a function of u/v p u/v_p andJJ; two points are for J=0.005J=0.005, u/v p =17u/v_p=17 (SS)and u/v p =0.035u/v_p=0.035 (SMBH; such v p v_p is about 2%2 \\% of thelight velocity)).Equation (REF ) can be understood on the basis of simple estimates.",
"The total captured mass $M_{cap} \\approx M_{AC}$ is accumulated during the diffusive time $t_d$ and hence $M_{cap} \\sim v_p^2 J t_d M_{tot}/(\\pi u^2 t_S) \\sim \\rho _g \\tau _d J (v_p/u)^3$ , where $\\tau _d=t_d/T_p$ , and we omit numerical coefficients.",
"This mass is concentrated inside a radius $r_{cap} \\sim 1/J$ so that at $r \\sim 1/J$ the volume density is $\\rho _v(r=1/J) \\sim M_{cap}/r_{cap}^3 \\sim \\rho _g J^2 w_{ch}^2 (v_p/u)^3 \\sim \\rho _{gJ} J^{1/2} w_{ch}^2 \\sim \\rho _{gJ} J^{1.3} $ , where we use a relation $\\tau _d \\sim w_{ch}^2/J^2 \\sim 1/J^{6/5}$ .",
"(Our modeling of the injection process in the Kepler map with a constant injection flow in time, counted as the number of map iterations, shows that the number of absorbed particles scales as $N_K \\sim \\tau _d \\sim J^{-6/5}$ at small $J$ .)",
"It is important to stress that $\\rho _v(r=1/J) \\ll \\rho _{gJ}$ in contrast to the naive expectation that $\\rho _v(r=1/J) \\sim \\rho _{gJ}$ .",
"Using our empirical density decay $\\rho _v \\propto 1/r^\\beta $ with $\\beta \\approx 2.25$ for the Kepler map, we obtain $\\zeta \\propto 1/J^{0.95}$ , which is close to the dependence $\\zeta \\sim 1/J$ and $\\zeta _g \\sim J^{1/2}/(u/v_p)^3$ from (REF ) at $u^2 \\gg J$ .",
"For the dark map we have $\\beta \\approx 1.5$ but $w_{ch} \\sim const$ as a result of the sharp variation of $F(x)$ with $x$ , which again gives $\\zeta \\sim 1/J$ .",
"It is difficult to obtain the exact analytical derivation of the relation $\\zeta \\sim 1/J$ due to contributions of different $q$ values (which have different $\\tau _d$ ) and different kick shapes in (REF ) that affect $\\tau _d$ and the structure of chaotic component.",
"In the regime $(u/v_p)^2 \\ll J$ the entire energy range of the scattering flow is absorbed by one kick, and $M_{cap}$ is increased by a factor $(u/v_p)^2/J,$ leading to an increase of $\\zeta _g$ by the same factor, which yields $\\zeta _g \\propto v_p/(u \\sqrt{J}),$ in agreement with (REF ).",
"We note that for galaxies the value of exponent $\\beta $ is debated [14].",
"For the adiabatic growth model, we have $2.25 \\le \\beta \\le 2.5, $ which is close to the value obtained from our symplectic map simulations.",
"Figure: Density distribution of DMPs ρ(q)\\rho (q) over qqobtained from the Kepler map at J=0.005J=0.005 andtime t u ≈4×10 8 T p t_u \\approx 4 \\times 10^8 T_p:(a) u/v p =17u/v_p=17;(b) u/v p =0.04u/v_p=0.04;the density is normalized to unity(∫ 0 ∞ ρdq/r p =1\\int _0^{\\infty } \\rho dq/r_p=1).The nontrivial properties of the distribution of the captured DMPs in $q$ are shown in Fig.",
"REF in a stationary regime at times $t_S/T_p \\approx 4 \\times 10^8$ for the Kepler map.",
"While for $u/v_p \\sim 17 \\gg 1$ we have a smooth drop of DMP density $\\rho (q)$ at $q>1.5r_p$ , for $u/v_p =0.04 \\ll 1$ we have an increase of $\\rho (q)$ by a factor 3 for $q/r_p \\approx 2.5$ compared to $q/r_p \\approx 1$ .",
"We attribute this variation to different capture conditions at $u \\gg \\sqrt{J} v_p$ , where only DMPs at low velocities are captured by one kick, and $u \\ll \\sqrt{J} v_p$ , where practically all DMPs are captured by one kick.",
"As a result of the dependence of $J$ on $q,$ we also have various diffusive timescales $t_d \\propto 1/J^2$ that can affect the contribution of the DMPs at different $q$ values in the volume density distribution on $r$ .",
"Finally, we stress the importance of the obtained result of large enhancement factors $\\zeta $ and $\\zeta _g$ .",
"This result is drastically different from the frequent claims that there is no enhancement of the DMP density in the center vicinity of a binary system compared to its galactic value because of the Liouville theorem, which implies that the density of DM in the phase space is conserved during the evolution [8], [12].",
"However, this statement does not take into account the actual dynamics of captured DMPs.",
"Indeed, the galactic space density $\\rho _g$ is obtained from all energies of DMPs in the Maxwell distribution.",
"The analysis of symplectic DMP dynamics shows that DMPs at large $q \\gg 1$ are not captured, while DMPs with $q \\sim 1$ are captured, and by diffusion, they penetrate up to high values $w \\sim w_{ch}$ , thus accumulating DMPs with typical distance values $r \\sim 1/w_{ch}$ .",
"The symplectic map approach also determines an effective size of our binary system of $r_{cap} \\sim 1/J$ corresponding to an energy range $w \\sim J$ .",
"If we assume that the DMP density in this range is the same as its galactic value, then we should conclude that the enhancement factor should be $\\zeta _g \\sim (r_{cap}/r_p)^\\beta \\sim 1/{w_{cap}}^{\\beta } \\sim 1/J^{\\beta } \\sim 1.5 \\times 10^5$ for typical values $J=0.005$ and $\\beta =2.25$ (we consider here the case $u/v_p \\ll \\sqrt{J}$ ).",
"This estimate gives a value $\\zeta _g$ that is even higher than that given by relation (REF ).",
"In fact, relation (REF ) takes into account that only bounded values of $q$ are captured, it also estimates the chaos region, where DMPs are accumulated during the chaotic diffusion process, populating a part of the phase space volume from $w \\sim 0$ up to $w \\sim w_{ch} \\sim 1$ .",
"This gives a lower value of $\\zeta _g$ than the above simplified estimate.",
"We also note that at $u/v_p \\ll \\sqrt{J}$ the typical kinetic energy of an ejected DMP $J v_p^2$ is significantly higher than the typical DMP energy $u^2$ in the galactic wind.",
"For these reasons, there is no contradiction with the Liouville theorem, and a large enhancement of the captured DMP density is possible."
],
[
"Few-body model",
"Above we considered the DMP capture in a two-body gravitating system.",
"We expect that a central SMBH binary dominating the galaxy potential can be viewed as a simplified galaxy model.",
"Recent observations of [9] indicate that such systems may exist.",
"Within the Kepler map approach it is easy to analyze the whole SS (an SMBH binary) including all $\\text{eight}$ planets ($\\text{eight}$ stars) with given positions $r_i$ and velocities $v_i$ measured in units of orbit radius $r_p$ and velocity $v_p$ of Jupiter for SS at $u/v_p=17$ (and of, e.g., the fifth star for an SMBH binary at $u/v_p = 0.035$ ).",
"Thus in (REF ) we have now for the SS $\\text{eight}$ kick terms with $J_i \\sim (m_i/M) (v_i/v_p)^2$ .",
"For the SMBH binary model we consider $\\text{eight}$ stars modeled by map (REF ) with the values $J_1=2.5 \\times 10^{-4}$ , $J_2=5 \\times 10^{-4}$ , $J_3=7.5 \\times 10^{-4}$ , $J_4=10^{-3}$ , $J_5=2.5 \\times 10^{-3}$ , $J_6=6.25 \\times 10^{-4}$ , $J_7=5 \\times 10^{-4}$ , and $J_8=1.25\\times 10^{-4}$ with the same ratio $r_i/r_p$ as for the SS.",
"In both cases we injected $N_J = 2.8 \\times 10^{10}$ particles considering evolution during $\\tau _S$ orbital periods of Jupiter (fifth star).",
"The steady-state density distribution is shown in Fig.",
"REF .",
"For the SS, $\\rho (r)$ is very close to the case of only one Jupiter discussed above.",
"This result is natural since its mass is dominant in the SS.",
"For the SMBH binary model we also find a similar distribution (see Fig.",
"REF ) with a slightly slower decay of $\\rho _v(r)$ with $r$ ($\\beta =2.06 \\pm 0.002$ ) due to the contribution of more stars.",
"We obtain $\\zeta =3000$ (SS) and $\\zeta _g=3 \\times 10^4$ (SMBH).",
"These two examples show that the binary model captures the main physical effects of the DMP capture and evolution."
],
[
"Discussion",
"Our results show that DMP capture and dynamics inside two-body and few-body systems can be efficiently described by symplectic maps.",
"The numerical simulations and analytical analysis show that in the center of these systems the DMP volume density can be enhanced by a factor $\\zeta _g \\sim 10^4$ compared to its galactic value.",
"The values of $\\zeta _g$ are highest for a high velocity $v_p $ of a planet or star rotating around the system center.",
"We note that our approach based on a symplectic map description of the restricted three-body problem is rather generic.",
"Thus it can also be used to analyze comet dynamics, cosmic dust, and free-floating constituents of the Galaxy."
]
] | 1403.0254 |
[
[
"Approximate Integrated Likelihood via ABC methods"
],
[
"Abstract We propose a novel use of a recent new computational tool for Bayesian inference, namely the Approximate Bayesian Computation (ABC) methodology.",
"ABC is a way to handle models for which the likelihood function may be intractable or even unavailable and/or too costly to evaluate; in particular, we consider the problem of eliminating the nuisance parameters from a complex statistical model in order to produce a likelihood function depending on the quantity of interest only.",
"Given a proper prior for the entire vector parameter, we propose to approximate the integrated likelihood by the ratio of kernel estimators of the marginal posterior and prior for the quantity of interest.",
"We present several examples."
],
[
"Introduction",
"Given a statistical model with generic density $p(x\\vert \\text{$\\theta $})$ , with $\\text{$\\theta $}\\in \\Theta \\subset \\mathbb {R}^d$ , one is often interested in a low dimensional function $\\text{$\\psi $}$ of the parameter vector $\\text{$\\theta $}$ , say $\\text{$\\psi $}=\\text{$\\psi $}(\\text{$\\theta $})\\in \\mathbb {R}^k$ , with $k<d$ .",
"Modern parametric or semi-parametric statistical theories, at least the approaches based on likelihood and Bayesian theories, aim at constructing a likelihood function which depends on $\\text{$\\psi $}$ only.",
"There is a huge literature on the problem of eliminating nuisance parameters, and we do not even try to summarize it.",
"Interested readers may refer to [7] and [19] for a Bayesian perspective, and to the comprehensive books by [22] and [29] or to [18] for a more classical point of view.",
"In a Bayesian framework the problem of eliminating the nuisance parameters is, at least in principle, trivial.",
"Let $\\text{$\\lambda $}=\\text{$\\lambda $}(\\text{$\\theta $})$ the complementary parameter transformation, such that $\\text{$\\theta $}=(\\text{$\\psi $}, \\text{$\\lambda $})$ and let $\\pi (\\text{$\\theta $}) = \\pi (\\text{$\\psi $}, \\text{$\\lambda $}) = \\pi (\\text{$\\psi $}) \\pi (\\text{$\\lambda $}\\vert \\text{$\\psi $})$ the prior distribution.",
"Then, after assuming we observe a data set $\\mathbf {x}=(x_1, \\dots ,x_n)$ from our working model, and computed the likelihood function $L(\\text{$\\psi $}, \\text{$\\lambda $}; \\mathbf {x})\\propto p(\\mathbf {x}; \\text{$\\psi $}, \\text{$\\lambda $})$ , the marginal posterior distribution of $\\text{$\\psi $}$ is $\\begin{split}\\pi (\\text{$\\psi $}\\vert \\mathbf {x}) = \\frac{\\int _\\Lambda \\pi (\\text{$\\psi $}, \\text{$\\lambda $}) L(\\text{$\\psi $}, \\text{$\\lambda $}; \\mathbf {x})d\\text{$\\lambda $}}{\\int _\\Lambda \\int _\\Psi \\pi (\\text{$\\psi $}, \\text{$\\lambda $}) L(\\text{$\\psi $}, \\text{$\\lambda $}; \\mathbf {x}) d\\text{$\\lambda $}d\\text{$\\psi $}}\\\\\\propto \\pi (\\text{$\\psi $}) \\int _\\Lambda \\pi (\\text{$\\lambda $}\\vert \\text{$\\psi $}) L(\\text{$\\psi $}, \\text{$\\lambda $};\\mathbf {x}) d\\text{$\\lambda $}.\\end{split}$ The integral in the right-hand side of (REF ) is, by definition, the integrated likelihood for the parameter of interest $\\text{$\\psi $}$ , where “integration” is meant with respect to the conditional prior distribution $\\pi (\\text{$\\lambda $}\\vert \\text{$\\psi $})$ ; it will be denoted by $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ .",
"The use of integrated likelihoods has become popular also among non Bayesian statisticians; there are several examples in which its use is clearly superior, or at least equivalent, even from a repeated sampling perspective, in reporting the actual uncertainty associated to the estimates.",
"See for example, [26], [27] and [28].",
"However the explicit calculation of the above integral might not be so easy, especially when the dimension $d-k$ is large.",
"Notice that the dimension $d$ may also include a possible latent structure which, from a strictly probabilistic perspective, is not different from a parameter vector.",
"In this paper we are interested to explore the use of approximate Bayesian computation (ABC, henceforth) methods in producing an approximate integrated likelihood function, in situations where a closed form expression of $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ is not available, or it is too costly even to evaluate the “global\" likelihood function $L(\\text{$\\psi $}, \\text{$\\lambda $}; \\mathbf {x})$ , like, for example, in many genetic applications or in the hidden (semi)-Markov literature.",
"These are situations where MCMC methods may not be satisfactory and completely reliable.",
"Another class of problems where an integrated likelihood would be of primary interest is that of semi-parametric problems, where the parameter of interest is a scalar - or a vector - quantity and the nuisance parameter is represented by the nonparametric part of the model; in such cases the integration step over the $\\Lambda $ space would be infinite dimensional, and very often infeasible to be solved in a closed form; we will discuss this issue in § .",
"Approximate Bayesian computation has now become an essential tool for the analysis of complex stochastic models when the likelihood function is unavailable.",
"It can be considered as a (class of) popular algorithms that achieves posterior simulation by avoiding the computation of the likelihood function (see [6] for a recent survey).",
"A crucial condition for the use of ABC algorithms is that it must be relatively easy to generate new pseudo-observations from the working model, for a fixed value of the parameter vector.",
"In its simplest form, the ABC algorithm is as follows (Algorithm 1 in [20]) In this paper we will argue, through several examples of increasing complexity, how the approximate integrated likelihood produced by ABC algorithms performs when compared with the existing methods.",
"We will also explore its use in particular examples where other methods simply fail to produce a useful and easy-to-use likelihood function for the parameter of interest.",
"The paper is organized as follows.",
"In the next section we describe our proposal in detail.",
"Section discusses some theoretical issues related to the precision of the ABC approximation.",
"Section compares the ABC integrated likelihood with other available approaches in a series of examples.",
"Section concludes with a final discussion of pros and cons of the method."
],
[
"The proposed method",
"The main goal of the paper is to obtain an approximation of the integrated likelihood $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ , for $\\text{$\\psi $}=\\text{$\\psi $}(\\text{$\\theta $})\\in \\mathbb {R}^k$ .",
"From expression (REF ) it is easy to see that $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x}) \\propto \\frac{\\pi (\\text{$\\psi $}\\vert \\mathbf {x})}{\\pi (\\text{$\\psi $})},$ that is the integrated likelihood function may be interpreted as the amount of experimental evidence which transforms our prior knowledge into posterior knowledge about the parameter of interest: from this perspective, we can interpret (REF ) as the Bayesian definition of the integrated likelihood function.",
"Suppose that $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ is hard or impossible to obtain in a closed form.",
"For example the nuisance parameter $\\text{$\\lambda $}$ might be infinite dimensional (see Example 4.4) or it may represent the non observable latent structure associated to the statistical model as in Hidden Markov or semi-Markov set-ups.",
"In these situations one can exploit the alternative expression (REF ) of $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ .",
"Of course, if $\\tilde{L}(\\text{$\\psi $}; \\mathbf {x})$ is not available, neither $\\pi (\\text{$\\psi $}\\vert \\mathbf {x})$ will be.",
"However it is possible to obtain an approximate posterior distribution $\\tilde{\\pi }(\\text{$\\psi $}\\vert \\mathbf {x})$ , by using some standard ABC algorithm; in § we will discuss some issues related to the precision of this approximation; for now, we describe the practical implementation of the method.",
"As in any ABC approach for the estimation of the posterior distribution, one has to select a number of summary statistics $\\eta _1(\\mathbf {x}), \\dots , \\eta _h(\\mathbf {x})$ ; select a distance $\\rho (\\cdot , \\cdot )$ to measure the distance between “true” and proposed data, or their summary statistics; select a tolerance threshold $\\varepsilon $ choose a (MC)MC algorithm which proposes values for the parameter vector $\\text{$\\theta $}$ .",
"Once the posterior is approximated by a size $M$ ABC posterior sample $(\\text{$\\theta $}_1^\\ast , \\text{$\\theta $}_2^\\ast , \\dots \\text{$\\theta $}^\\ast _M)$ , one can produce a non parametric kernel based density approximation of the marginal posterior distribution of $\\text{$\\psi $}$ , say $\\tilde{\\pi }^{ABC}(\\text{$\\psi $}\\vert \\mathbf {x})$ .",
"A similar operation can be done with the marginal prior $\\pi (\\text{$\\psi $})$ , by performing another - cheap - simulation from ${\\pi }(\\text{$\\psi $})$ to get another density approximation, say $\\tilde{\\pi }(\\text{$\\psi $})$ .",
"Notice that one is bound to use proper priors for all the involved parameters.",
"Then one can define the ABC integrated likelihood $\\tilde{L}^{ABC}(\\text{$\\psi $}; \\mathbf {x}) \\propto \\frac{\\tilde{\\pi }^{ABC}(\\text{$\\psi $}\\vert \\mathbf {x})}{\\tilde{\\pi }(\\text{$\\psi $})}.$"
],
[
"The quality of approximation",
"The gist of this note is to propose an approximate method for producing a likelihood function for a quantity of interest when the usual road of integrating with respect to the nuisance parameters cannot be followed.",
"There are two sources of error in (REF ).",
"The first type of approximate error is introduced by the ABC approximation in the numerator so the level of accuracy of (REF ) is of the same order of any ABC-type approximation.",
"We believe that the main difficulty with ABC methods is the choice of summary statistics.",
"However, while generic ABC methods have the goal of producing a “global” approximation to the posterior distribution, our particular use of the ABC approximation may suggest some alternative strategies for the choice of summary statistics.",
"Classical statistical theory on the elimination of nuisance parameters can be in fact of some guidance in the selection of summary statistics which are partially or conditionally sufficient for the parameter of interest.",
"[5] represents an excellent reading on these topics.",
"In particular, his Definition 5 of “Specific Sufficiency” can be used in semi-parametric set-ups, like Example 4.4 below, where the selected summary statistics are oriented towards the preservation of information about the parameter of interest.",
"In our notation a statistic $T$ is specific sufficient for $\\text{$\\psi $}$ if, for each fixed value of the nuisance parameter $\\text{$\\lambda $}$ , $T$ is sufficient for the restricted statistical model in which $\\text{$\\lambda $}$ is held fixed and known.",
"Another source of error in ABC is given by the tolerance threshold $\\varepsilon $ .",
"As stressed in [20], the choice of the tolerance level is mostly a matter of computational power: smaller $\\varepsilon $ 's are associated with higher computational costs and more precision.",
"It is enough to reproduce the argument in § 1.2 of [30] to see that for $\\varepsilon \\rightarrow 0$ , the error in (REF ), which is due to the tolerance, vanishes.",
"Then, there is a balance between the fact that $\\varepsilon $ has to be small and the fact that the simulation has to be practicable.",
"It could be useful to choose $\\varepsilon $ in a recursive way, by realizing a first simulation with a high tolerance level and then by choosing it in the left tail of the thresholds related to the accepted values.",
"However, it is always recommended to compare different levels.",
"The second main source of error is due to the kernel approximation step.",
"A second order expansion for a Gaussian kernel estimator provides that $\\mathbb {E}\\left[\\tilde{\\pi }^{ABC}\\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)\\right]=\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)+\\frac{ 1 }{ 2 } \\frac{\\partial ^2}{\\partial \\psi ^2}\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right) h_x^2 k_2+\\mathcal {O}\\left(h_x^4\\right)$ where $h_x$ is the bandwidth and $k_2=1$ in the case of Gaussian kernel.",
"A similar approximation holds for the prior distribution.",
"Then, using general results on a first order approximation for the ratio of functions of random variables ([17], pag.",
"351), one has $\\mathbb {E} \\left[ \\frac{ \\tilde{\\pi }^{ABC}\\left( \\text{$\\psi $}\\vert \\mathbf {x}\\right)}{\\tilde{\\pi }\\left(\\text{$\\psi $}\\right)} \\right]=\\frac{\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)+\\frac{1}{2} \\frac{\\partial ^2}{\\partial \\psi ^2}\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)h_x^2+\\mathcal {O}\\left(h_x^4\\right)}{\\pi \\left(\\text{$\\psi $}\\right)+\\frac{1}{2} \\frac{\\partial ^2}{\\partial \\psi ^2}\\pi \\left(\\text{$\\psi $}\\right)h_\\pi ^2+\\mathcal {O}\\left(h_\\pi ^4\\right)}$ where $h_x$ is the bandwidth chosen for the approximation of the posterior distribution and $h_\\pi $ is the one chosen for the approximation of the prior.",
"The prior distribution is often known in closed form or may be easily approximated with an higher accuracy than the posterior distribution.",
"The previous formula ensures that our estimator will be consistent provided that a sample size dependent bandwidth $h_n$ , converging to 0, is adopted.",
"It is a matter of calculation to show that the variance of the estimator is $&& \\phantom{ } \\mathbb {V}\\left[ \\frac{ \\tilde{\\pi }^{ABC}\\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)}{\\tilde{\\pi }\\left(\\text{$\\psi $}\\right)} \\right] \\\\&=&\\left[ \\frac{\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)+C_\\mathbf {x}}{\\pi \\left(\\text{$\\psi $}\\right)+C}\\right]^2\\nonumber \\\\&\\times &\\left[\\frac{\\frac{\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)}{2nh\\sqrt{\\pi }}+\\mathcal {O}\\left(n^{-1}\\right)}{\\left[\\pi \\left(\\text{$\\psi $}\\vert \\mathbf {x}\\right)+C_\\mathbf {x}\\right]^2}+\\frac{\\frac{\\pi \\left(\\text{$\\psi $}\\right)}{2nh\\sqrt{\\pi }}+\\mathcal {O}\\left(n^{-1}\\right)}{\\left[\\pi \\left(\\text{$\\psi $}\\right)+C\\right]^2}\\right] \\nonumber $ where $C_\\mathbf {x}=\\frac{h_x^2}{2}\\frac{\\partial ^2}{\\partial \\psi ^2}\\pi \\left( \\text{$\\psi $}\\vert \\mathbf {x}\\right) +\\mathcal {O}\\left( h_x^4\\right)$ and $C=\\frac{h_\\pi ^2}{2}\\frac{\\partial ^2}{\\partial \\psi ^2}\\pi \\left( \\text{$\\psi $}\\right) +\\mathcal {O}\\left(h_\\pi ^4\\right)$ .",
"Again, using a bandwidth $h_n$ , such that $h_n\\rightarrow 0$ , as $n\\rightarrow \\infty $ , one can see that the first factor of the variance is asymptotically equal to the square of the true unknown value, while the second factor vanishes like $n^{-1}$ .",
"In conclusion, the ABC approximation of the integrated likelihood function mainly depends on the ABC approximation and the kernel density estimate of the posterior distribution, whereas the prior distribution may be considered known, in general.",
"[8] shows that the asymptotic variance of the kernel density estimator of the posterior distribution inversely depends on the number of simulations $n$ and on the kernel bandwidth, while the bias is proportional to the bandwidth.",
"The mean squared error is minimized by $h_n=\\mathcal {O}\\left(n^{-\\frac{1}{d+5}} \\right)$ where $d$ is the dimension of the summary statistics.",
"Then the minimal MSE is $MSE^*=\\mathcal {O}\\left(n^{-\\frac{4}{d+5}}\\right)$ which shows that the accuracy in the approximation decreases as the dimension of the summary statistics increases.",
"This result may be used to define the number of simulations (and the burn-in) needed to reach the desired level of accuracy."
],
[
"Examples",
"In this section we illustrate our proposal throughout several examples of increasing complexity.",
"The first one is a toy example and it is included only to show - in a very simple situation - which are the crucial steps of the algorithm.",
"Example 4.1.",
"[Poisson means].",
"Suppose we observe a sample of size $n$ from $X\\sim \\mbox{Poi}(\\theta _1)$ and, independently of it, another sample of size $n$ from $Y\\sim \\mbox{Poi}(\\theta _2)$ .",
"The parameter of interest is $\\psi =\\theta _1/\\theta _2$ .",
"This is considered a benchmark example in partial likelihood literature since the conditional likelihood (see [16]), the profile likelihood and the integrated likelihood obtained using the conditional reference prior [7] are all proportional to $\\tilde{L}(\\psi ; \\mathbf {x}, \\mathbf {y}) \\propto \\frac{\\psi ^{n\\bar{x}}}{(1+\\psi )^{n(\\bar{x}+ \\bar{y})}},$ with the obvious meaning of the symbols above.",
"Without loss of generality, set $\\lambda =\\theta _2$ as the nuisance parameter.",
"In this situation, the ABC approximation of the integrated likelihood is, in some sense, not comparable with the “correct” integrated likelihood because the latter is obtained through the use of an improper conditional reference prior on $\\lambda $ given $\\psi $ , and, as already stressed, it is not possible to use improper priors in the ABC approach.",
"A solution may be using a prior which mimics the reference prior: we have taken $\\theta _1, \\theta _2 \\stackrel{\\texttt {iid}}{\\sim }\\mbox{Ga}\\left(0.1,\\,0.1\\right)$ .",
"Notice that, in the economy of the method, only the prior on $\\lambda $ , not on $\\psi $ is important.",
"The ABC algorithm has been implemented to obtain approximations for the posterior distributions of $\\theta _1$ and $\\theta _2$ .",
"The distance $\\rho $ has been taken as the Euclidean distance, different tolerance levels have been compared - $\\varepsilon =\\left(0.001,\\,0.01,\\,0.1,\\,0.5\\right)$ - and the sample means of the two samples have been taken as summary (sufficient) statistics.",
"Samples of $1,000$ simulations have been obtained to approximate the posterior distributions.",
"The approximation to the posterior distribution of $\\psi $ is then simply obtained as the ratio between the accepted values for $\\theta _1$ and $\\theta _2$ via ABC.",
"Given a sample from the prior distribution of $\\psi $ , the approximation of its integrated likelihood is obtained through the ratio between the kernel density estimates of both the prior and the posterior distribution.",
"Figure REF shows the approximations with different choices of the tolerance level: the approximations are close together and they are all close to the integrated likelihood; the choice for the tolerance level does not seem to have a strong influence; it is mostly a matter of computational power: the acceptance rate is generally very low (often under 1%), nevertheless it grows with the tolerance level.",
"As the threshold goes to zero, the approximation is closer to the integrated likelihood, although the computational time increases.",
"Simulations have been repeated for different scenarios, by changing the sample size and the number of simulations, however the results do not seem to change in a significant way.",
"In particular, as expected, the algorithm does not depend on the (induced) prior on $\\psi $ .",
"Figure: {{figure:86a6a641-a976-4a85-89d6-def9ea9af840}}Example 4.2.",
"[Neyman and Scott's class of problems].",
"This is a famous class of problems, where the number of parameters increases with the sample size ([21]; [18]).",
"Here we consider a specific example, already discussed in [12] and [19], namely matched pairs of Bernoulli observations: every subject is assigned to treatment or control group and the randomization occurs separately within each pair, i.e.",
"each data point in one data set is related to one and only one data point in the other data set.",
"Let $Y_{ij}$ 's be Bernoulli random variables, where $i=1,\\dots ,k$ represents the stratum and $j=0,\\,1$ indicates the observation within the pair.",
"The probability of success $p_{ij}$ follows a logit model: $\\mathrm {logit}\\, p_{ij}=\\lambda _{i}+\\psi _{j}$ For identifiability reasons, $\\psi _{0}$ is set equal to 0, while $\\psi _{1}=\\psi $ is considered constant across the $k$ strata; $\\psi $ is the parameter of interest.",
"To formalize the problem, assume $\\left(R_{i0},\\,R_{i1}\\right)$ are $k$ independent matched pairs such that, for each $i$ : $R_{i0}\\sim \\mathrm {Be}\\left(\\frac{e^{\\lambda _{i}}}{1+e^{\\lambda _{i}}}\\right),\\quad R_{i1}\\sim \\mathrm {Be}\\left(\\frac{e^{\\lambda _{i}+\\psi }}{1+e^{\\lambda _{i}+\\psi }}\\right).$ The complete likelihood for $\\mathbf {\\lambda }=\\left(\\lambda _{1},\\dots ,\\lambda _{k}\\right)$ and $\\psi $ is $L\\left(\\psi ,\\,\\lambda \\right)=\\frac{e^{\\sum _{i=1}^{k}\\lambda _{i}S_{i}+\\psi T}}{\\prod _{i=1}^{k}\\left(1+e^{\\lambda _{i}}\\right)\\left(1+e^{\\lambda _{i}+\\psi }\\right)}$ where $S_{i}=R_{i0}+R_{i1}$ for $i=1,\\dots ,k$ and $T=\\sum _{i=1}^{k}R_{i1}$ is the number of successes among the cases.",
"It is easy to show that the conditional maximum likelihood estimate of $\\lambda _{i}$ is infinite when $S_{i}=0$ or $S_{i}=2$ .",
"The classical solution to this problem is to eliminate the pairs where $S_{i}=0$ or $S_{i}=2$ from the analysis.",
"Nevertheless this is certainly a loss of information, because the fact that a pair gives the same result under both treatments may suggest a “not-so-big” difference between groups.",
"It is easy to show that the conditional maximum likelihood estimator is $\\left[\\hat{\\lambda }_{i,\\psi }\\mid \\left(S_{i}=1\\right)\\right]=-{\\psi }/{2}$ ; also, let $b$ be the number of pairs with $S_{i}=1$ .",
"The profile likelihood of $\\psi $ is $\\tilde{L}\\left(\\psi \\mid S_{i}=1\\right)=\\frac{e^{\\psi T}}{\\left(1+e^{\\frac{\\psi }{2}}\\right)^{2b}}$ This likelihood function is not useful, since the maximum likelihood estimate for $\\psi $ is inconsistent (see [12], Example 12.13): as $b$ increases, $\\hat{\\psi }\\rightarrow 2\\psi $ .",
"The modified version of the profile likelihood, proposed by [4] uses a multiplying factor: $M\\left(\\psi \\right)=\\left|J_{\\lambda \\lambda }\\left(\\psi ,\\,\\hat{\\lambda }_{\\psi }\\right)\\right|^{-\\frac{1}{2}}\\left|\\frac{\\partial \\hat{\\lambda }}{\\partial \\hat{\\lambda }_{\\psi }}\\right|=\\frac{e^{\\frac{b\\psi }{4}}}{\\left(1+e^{\\frac{\\psi }{2}}\\right)^{b}}$ where $J_{\\lambda \\lambda }\\left(\\psi ,\\,\\hat{\\lambda }_{\\psi }\\right)$ is the lower right corner of the observed Fisher information matrix.",
"The conditional distribution of $T$ given $S_{1}=S_{2}=\\dots =S_{b}=1$ is Binomial and depends on $\\psi $ only.",
"That is $T\\mid [S_{1}=S_{2}=\\dots =S_{b}=1,\\,\\psi ] \\sim \\mathrm {Bin}\\left(b,\\,\\frac{e^{\\psi }}{1+e^{\\psi }}\\right)$ ; we can use it to get a conditional likelihood function: $L_{C}\\left(\\psi \\right)\\propto \\tbinom{b}{T}\\,\\frac{e^{\\psi T}}{\\left(1+e^{\\psi }\\right)^{b}}$ which leads to a consistent maximum conditional likelihood estimator.",
"A Bayesian approach has the advantage that it does not need to discard the pairs with $S_{i}=0$ or 2.",
"The likelihood contribution for the $i$ -th pair is simply $L\\left(\\psi ,\\,\\lambda _{i}\\right)=\\frac{e^{\\lambda _{i}S_{i}+\\psi R_{i1}}}{\\left(1+e^{\\lambda _{i}}\\right)\\left(1+e^{\\lambda _{i}+\\psi }\\right)}.$ With a change of parametrization $\\omega _{i}={e^{\\lambda _{i}}}/{(1+e^{\\lambda _{i}})}$ and using a (proper) Jeffreys' prior for $\\omega _{i}\\vert \\psi $ (namely a $Beta\\left(\\frac{1}{2},\\,\\frac{1}{2}\\right)$ ), the integrated likelihood is $L_{i}\\left(\\psi \\right)=e^{\\psi R_{i1}}\\int _{0}^{1}\\frac{\\omega _{i}^{S_{i}-\\frac{1}{2}}\\left(1-\\omega _{i}\\right)^{\\frac{3}{2}-S_{i}}}{1-\\omega _{i}\\left(1-e^{\\psi }\\right)}d\\omega _{i}$ where the integral is one of the possible representation of the Hypergeometric or Gauss series, as shown in [1] (formula 15.3.1, pag.",
"558).",
"Therefore, the integrated likelihood is proportional to $L_{i}\\left(\\psi \\right)\\propto \\,_{2}F_{1}\\left(1,\\,S_{i}+\\frac{1}{2},\\,3,\\,1-e^{\\psi }\\right)\\, e^{\\psi R_{i1}}$ Define $L_{jl}\\left(\\psi \\right)$ as the integrated likelihood function associated with the $i$ -th pair for which $\\left(R_{i0},\\, R_{i1}\\right)=\\left(j,\\, l\\right)$ and $n_{jl}$ the number of pairs for which $\\left(R_{i0},\\, R_{i1}\\right)=\\left(j,\\,l\\right)$ , then the integrated likelihood function for $\\psi $ is $L_{int}\\left(\\psi \\right)\\propto \\prod _{j,l=0,1}L_{jl}\\left(\\psi \\right)^{n_{jl}}.$ It is worthwhile to notice that this likelihood is not, in some sense, comparable with profile and conditional likelihoods, because it also considers the pairs discarded by non-Bayesian methods.",
"The ABC approach has been used with simulated data, with a sample size $n$ equal to 30.",
"Simulations were performed by setting $\\psi =1$ , a value which is quite frequent in applications, when similar treatments are compared.",
"The values of $\\lambda =\\left(\\lambda _{1},\\dots ,\\,\\lambda _{n}\\right)$ has been generated by setting $\\xi _i=\\lambda _i/(1+\\lambda _i)$ and drawing the $\\xi ^{\\prime }$ s from a $ \\mbox{U}(0,1)$ distribution.",
"Again, we have used the Euclidean distance between summary statistics and different tolerance levels $\\varepsilon =\\left(0.001,\\,0.01,\\,0.1,\\,0.5\\right)$ .",
"The summary statistics are the sample means for $\\mathbf {R}_{0}$ and $\\mathbf {R}_{1}$ .",
"We have also assumed a normal prior for $\\psi $ with zero mean and standard deviation equal to 10.",
"The proposed values for $\\lambda _{i}$ 's have been generated from a $Beta\\left(\\frac{1}{2},\\,\\frac{1}{2}\\right)$ distribution for the above defined transformations $\\xi _i$ 's.",
"With a sample from the posterior distribution of $\\psi $ for each tolerance level and a sample from its prior distribution, we have obtained an approximation of the likelihood of $\\psi $ via density kernel estimation.",
"The results are shown in Figure REF : the approximations are quite good for tolerance levels below 0.1; on the other hand, when the threshold grows to 0.5 the approximate likelihood function is very flat and multi-modal, i.e.",
"too many proposed values, even very different from the true value of $\\psi $ , are misleadingly accepted; for example, a value of $\\psi $ around 43 has been accepted in one of our simulations.",
"Figure: {{figure:8cc4dbbf-3e6f-467c-8eaf-62de21d71248}}Once again, a fair comparison between Bayesian and non-Bayesian approaches is not strictly possible, nevertheless the various proposals are shown in Figure REF : all the proposed solutions are concentrated relatively close to the true value, although the profile likelihood seems to be biased towards large values of $\\psi $ : this behaviour is also present, although to a minor extent in the modified profile and the integrated likelihood solutions.",
"The ABC approximation closely mimics the integrated likelihood, obtained via a saddle-point approximation of the Hypergeometric series [10].",
"Figure: {{figure:6b3cd455-a416-40c7-bb2a-74ace91f18bc}}Similar conclusions are valid for different choices of the prior distribution, different sample sizes, and different numbers of simulations.",
"Just like in Example 4.1, the acceptance rates are typically very low (always under $1\\%$ for tolerance levels under $0.01$ and about $5\\%$ for a tolerance level of $0.1$ ).",
"Acceptance rates dramatically increase to about $60\\%$ , for $\\varepsilon =0.5$ ; however in these cases, approximations get much worse.",
"Example 4.3 [Likelihood function for the quantiles of a $g$ -and-$k$ distribution].",
"Quantile distributions are, in general, defined by the inverse of their cumulative distribution function.",
"They are characterized by a great flexibility of shapes obtained by varying parameters values.",
"They may easily model kurtotic or skewed data with the great advantage that they typically have a small number of parameters, unlike mixture models which are usually adopted to describe this kind of data.",
"An advantage of quantile distributions is that it is extremely easy to simulate from them by means of a simple inversion.",
"However, there are no free lunches, and the above advantages are paid with the fact that their probability density functions (and therefore, the implied likelihood functions) are often not available in a closed form expression.",
"One of the most interesting examples of quantile class of distributions is the so-called $g$ -and-$k$ distribution, described in [14], whose quantile function $Q$ is given by $\\begin{split}Q\\left(u;\\, A,\\, B,\\, g,\\, k\\right)=A+B\\left[1+c\\frac{1-\\exp \\left\\lbrace -g\\,z\\left(u\\right)\\right\\rbrace }{1+\\exp \\left\\lbrace -g\\, z\\left(u\\right)\\right\\rbrace }\\right]\\cdot \\\\\\left\\lbrace 1+z\\left(u\\right)^{2}\\right\\rbrace ^{k}z\\left(u\\right)\\end{split}$ where $z\\left(u\\right)$ is the $u$ -th quantile of the standard normal distribution; parameters $A$ , $B$ , $g$ and $k$ represent location, scale, skewness and kurtosis respectively; $c$ is an additional parameter which measures the overall asymmetry and it is generally fixed at $0.8$ , following [23].",
"The class of Normal distributions is a proper subset of this class; it is obtained by setting $g=k=0$ .",
"Suppose we are interested in one or more quantiles using this model.",
"There are no easy solution to the problem of constructing a partial likelihood for these quantiles.",
"The fact that the likelihood function is not available makes any classical approach practically impossible to implement.",
"[24] propose a numerical maximum likelihood approach; however they also explain that very large sample sizes are necessary to obtain reliable estimates of the parameters.",
"On the other hand, even though this quantile distributions have no explicit likelihood, simulation from these models is easy, and an approximate Bayesian computation approach, also for producing an integrated likelihood of the parameters of interest, seems reasonable.",
"For this specific problem, two types of ABC algorithms have been compared: the former is the usual ABC algorithm based on simulations from the prior distributions (with $10^{3}$ iterations); the latter is an ABC-MCMC algorithm ($10^{6}$ iterations, with a burn-in of $10^{5}$ simulations).",
"Two versions of ABC-MCMC have been used, the former described in [20] (see Algorithm REF ) and the latter described in [2] (see Algorithm REF ).",
"The main difference between these two versions of ABC-MCMC algorithm is that, in the first case, there is no rejection step; at each iteration a value is accepted (either the new proposed value or the value accepted in the previous iteration); in the second case, instead, it is possible to discard the current value and to propose a new one, so the chain always “moves”.",
"Data have been simulated from a $g$ -and-$k$ distribution with parameters $A=3$ , $B=1$ , $g=2$ and $k=0.5$ .",
"As previously said, $c$ is considered known and set equal to $0.8$ .",
"The sample size, has been set equal to $n=1000$ .",
"The empirical cumulative distribution function and the histogram of the simulated data are shown in Figure REF .",
"Figure: {{figure:259e981a-6373-4102-a3f2-334efba85ec8}}The transition kernel of the ABC-MCMC algorithm needs to be chosen having in mind two conflicting objectives: on one hand, full exploration of the parameter space, and, on the other hand, a reasonably high acceptance rate, which increases for proposals mostly concentrated where the posterior mass is present.",
"As described in [2] uniform priors with bounds $\\left(0,\\,10\\right)$ have been chosen for each parameter and a random walk-normal kernel with variance $0.1$ has been used together with a large number of iterations ($10^{6}$ ) so that the parameter space is likely to be fully investigated.",
"The vector of summary statistics consists of the sample mean, the standard deviation, and the sample skewness and kurtosis indexes.",
"The Euclidean distance has been used to compare summary statistics.",
"The tolerance level $\\varepsilon $ has been chosen in a recursive way: first, a very large value has been selected, and a histogram of all the distances has been drawn.",
"A reasonable value has been taken from the $5\\%$ left tail of this histogram.",
"Then, the chosen threshold has been compared with smaller values.",
"In particular, a threshold equal to 3 corresponds to $3.9\\%$ left tail.",
"This has been compared with tolerance levels equal to 2 and $0.5$ .",
"The analysis of the approximate posterior distributions shows that three out of four parameters ($A$ , $B$ and $k$ ) are well identified, while the posterior distribution of $g$ is rather flat.",
"In general, as the tolerance level decreases, results improve and posterior distributions tend to be more concentrated.",
"Nevertheless, even using the lowest tolerance level the posterior distribution of $g$ does not seem to concentrate around any value.",
"This suggests that the algorithm needs an even smaller value of the threshold.",
"A simulation with tolerance level equal to $0.25$ has been then performed using Algorithm REF : the approximation of the posterior distribution of $g$ is still not centred around its true value, even if there is a mode around it; nevertheless the problem with this so low tolerance level and this type of algorithm is that the acceptance rate of new proposed values is very low and the chain does not move too much.",
"This tolerance level is also so low to make the application of the other algorithms prohibitive in terms of computational time.",
"Our main goal of the analysis was to find an approximation of the integrated likelihood function for a given quantile: in particular, we have considered the percentiles of order $0.05$ , $0.10$ , $0.25$ and $0.50$ .",
"Notice that, in the $g$ -and-$k$ distribution model, the median is always equal to $A$ .",
"The results are shown in Figure REF , and REF .",
"The performance is in general very good: the approximations are always concentrated around the true values.",
"The ABC algorithm with simulations from the prior distribution has some apparent problems of multi-modality, which are however absent using Algorithm REF .",
"However, in this case, the obtained approximations are not very smooth, and they show more irregularities as the tolerance level decreases: as we have already remarked, a too low threshold leads to very low acceptance rates and this means that the chains do not move too much.",
"In this example, Algorithm REF has the best overall performance: the approximations are smooth and all concentrated around the true quantile values.",
"As the tolerance level decreases, the likelihood approximations are more concentrated; obviously the computational time gets larger.",
"The acceptance rates of these algorithms are in general very low: the basic ABC algorithm has an acceptance rate of $0.138\\%$ when the threshold is equal to 3, and it goes down to $0.041\\%$ and $0.007\\%$ with tolerance levels of 2 and $0.5$ respectively; the ABC-MCMC Algorithm REF needs, respectively, 187, 1487, about $500K$ and more than 3 millions of simulations for the initialization step for the different tolerance levels 3, 2, $0.5$ and $0.25$ .The acceptance rates of the proposed values is also very low: 18.41%, 9.90%, 0.47% and 0.046% respectively; it is clear that the acceptance rates relative to the smaller thresholds cannot lead to smooth approximations; the ABC-MCMC Algorithm REF needs 1104, 4383 and about $400K$ simulations for the initialization step for tolerance levels 3, 2 and 0.5 respectively; in this case every accepted value is a “new” value, and this solves the problems in Algorithm REF .",
"In conclusion, ABC-MCMC seems to perform better, although the versions we have implemented present some cons: the algorithm in [20] is faster but it must be calibrated in terms of the tolerance level, which has to be low in order to achieve good approximations, and the MCMC acceptance rate, which has to be sufficiently high in order to allow the chains to move.",
"Figure: {{figure:0a9bede8-291d-4bd4-810e-4ad25e949aee}}Figure: {{figure:0cd9ce03-c06c-4919-85f5-2f055d390f70}}Figure: {{figure:c58d5448-df79-4ad3-9cc7-d41a1fcc287f}}Example 4.4 [Semiparametric regression].",
"Consider the following model $\\mathbf {Y}=\\mathbf {X}\\text{$\\beta $}+ \\gamma \\left( \\mathbf {z}\\right) + \\varepsilon ,$ where $\\mathbf {Y}=(Y_1, Y_2,...,Y_n)^\\prime $ is a vector of $n$ real-valued variables, and $\\mathbf {x}=(x_1,x_2, \\dots , ,x_n)^\\prime $ , $\\gamma (\\mathbf {z})=(\\gamma (z_1), \\gamma (z_2), \\dots , \\gamma (z_n))^\\prime $ and $\\mathbf {z}=( z_1,z_2,\\dots , z_n)^\\prime $ are observed constants respectively taking values in $\\mathbb {R}^p$ and $\\mathcal {Z}$ , $\\varepsilon $ is the usual random component that we assume having multivariate normal distribution with mean $\\bf 0$ and covariance matrix $\\bf \\Omega _\\phi $ which depends on some parameters $\\phi $ , $\\text{$\\beta $}$ is a vector of unknown parameters taking values in $\\mathbb {R}^p$ and $\\gamma : \\mathcal {Z} \\rightarrow \\mathbb {R}$ is an unknown function.",
"If the analysis is focused on $\\text{$\\beta $}$ or $\\bf \\Omega _\\phi $ , $\\gamma $ may be considered a nuisance parameter and a method to remove it from the analysis is needed.",
"In particular, if a weight function for $\\gamma $ based on a zero-mean Gaussian stochastic process with covariance function $K_\\lambda \\left( \\cdot ,\\cdot \\right)$ with parameter $\\lambda $ is used, the vector $\\left( \\gamma \\left( z_1 \\right),...,\\gamma \\left( z_n \\right) \\right)$ has a multivariate Normal distribution with mean $\\bf 0$ and covariance $\\bf \\Sigma _\\lambda $ and the integrated likelihood function of $\\text{$\\beta $}$ is $\\left| \\bf \\Omega _\\phi + \\bf \\Sigma _\\lambda \\right|^{ -\\frac{1}{2}} \\exp \\left\\lbrace -\\frac{1}{2}\\left( \\mathbf {Y}-\\mathbf {X}\\beta \\right)^\\prime \\left(\\bf \\Omega _\\phi +\\Sigma _\\lambda \\right)^{-1} \\left( \\mathbf {Y}-\\mathbf {X}\\beta \\right) \\right\\rbrace $ where $\\Sigma _\\lambda $ is the $n\\times n$ matrix with $K_\\lambda \\left( z_i, z_j\\right)$ in the $\\left(i,j \\right)$ element.",
"This form may be obtained because of the assumption on the Normal distribution of the errors and the use of a Gaussian process weight function for $\\gamma $ ; more general cases are not so straightforward to handle outside the Normal set-up.",
"In [15] the Authors show that, for a given choice of $K_\\lambda \\left(\\cdot ,\\cdot \\right)$ , when the dispersion parameter, say $\\eta =\\left(\\phi ,\\lambda \\right)$ is known, $\\beta $ can be estimated by the generalized least-squares estimator: $\\hat{\\beta }=X^T \\left(X^T V^{-1} X \\right)^{-1}X^T V^{-1} Y$ where $V=\\bf \\Omega _\\phi +\\bf \\Sigma _\\lambda $ ; if the dispersion parameter is unknown, $\\beta $ can be estimated as a function of an estimator of $\\eta $ , $\\hat{\\beta }\\left( \\hat{\\eta }\\right)$ .",
"The method has been used with data from a survey of the fauna on the sea bed lying between the Queensland coast and the Great Barrier Reef; the response variable analysed is a score, on a log weight scale, which combines information across the captured species; this score value is considered dependent on the latitude $\\mathbf {x}$ in a linear way and on the longitude $\\mathbf {z}$ in an unknown way; see [9] for more details.",
"The model is $Y_j=\\beta _0+x_j\\beta _1 +\\gamma \\left(z_j\\right)+\\varepsilon _j,\\quad j=1,...,n$ where $\\varepsilon _1,...,\\varepsilon _j$ are independent normal errors with mean 0 and constant variance $\\sigma ^2_\\varepsilon $ .",
"Using the integrated likelihood approach, a Gaussian covariance function $K\\left(z,\\tilde{z} \\right)=\\tau ^2 \\exp \\left(-\\frac{1}{2} \\frac{\\left| z-\\tilde{z}\\right|^2}{\\alpha }\\right)$ and a restricted maximum likelihood estimate (REML, [13]) for the nuisance parameters, the estimates of $\\beta _1$ is 1.020, with a standard error of 0.356 (see [15]).",
"We have used our ABC approximation in order to find an integrated likelihood for $\\text{$\\beta $}$ .",
"It is then necessary to define proper prior distributions for all the parameters of the model, i.e.",
"$\\beta $ , $\\sigma ^2_\\varepsilon $ and the parameter of the covariance function of the Gaussian process, $\\alpha $ and $\\tau ^2$ .",
"For $\\beta $ a g-prior has been chosen such that $\\beta \\sim \\mathrm {N}_2 \\left(\\textbf {0},\\,g \\sigma ^2_\\varepsilon \\left( \\mathbf {X^T X} \\right)^{-1} \\right)$ , where $g\\sim \\mbox{U}\\left(0, 2n \\right)$ and $\\sigma ^2_\\varepsilon \\sim IG \\left(a,b \\right)$ with a, and b suitably small (as an approximation of the Jeffreys' prior).",
"A Gaussian process with squared exponential covariance function has been used as prior process for the function $\\gamma \\left(\\cdot \\right)$ .",
"The hyper-parameters of the Gaussian process have the following prior distributions: $\\tau ^2\\sim IG\\left(a,b \\right)$ , with $a=b=0.01$ and $\\alpha \\sim IG \\left( 2,\\nu \\right)$ with $\\nu =\\rho _0/\\left(-2 \\log (0.05) \\right)$ and $\\rho _0=\\max _{i,j=1...n}|z_i-z_j|$ ; see [25] and [3] for more details.",
"The choice of the summary statistics is not straightforward, because it is necessary to find statistics that take into account both the parametric and the nonparametric parts of the model, nevertheless sufficiency is not guaranteed.",
"A function of z has been considered and the maximum likelihood estimates of the coefficients of the new model have been used as summary statistics.",
"In particular, two choices of function has been considered: $g\\left(z_j\\right)=z_j$ and $h\\left(z_j\\right)=z^2_j$ for $j=1,...,n$ .",
"An analysis of the maximum likelihood estimates has shown that the estimate of the constant $\\beta _0$ is particularly unstable, therefore only the estimates for the predictor variables' coefficients contribute to the approximation as summary statistics.",
"In the MCMC step, normal transitional kernels have been used for all the parameters of the model, centred at the values accepted on the previous step and with small variance.",
"The results are shown in Figure REF : the ABC approximation with $10^6$ simulations are concentrated around the estimates obtained by maximizing the integrated likelihood of the model.",
"In this case, the ABC approach may be seen as a way to properly account for the uncertainty on the nuisance parameters that is not considered when REML estimates are used.",
"Figure REF compares different choices of summary statistics and prior distributions for the variance $\\sigma ^2_\\varepsilon $ : on the left a $ \\mbox{U}\\left(0,10 \\right)$ is used and on the right a proper approximation of the Jeffreys' prior is used ($\\mbox{Ga}\\left(a,b \\right)$ with $a,b$ small).",
"All the approximations are smooth and concentrated around the maximum likelihood estimate.",
"Moreover, Figure REF shows that using the summary statistics based on a quadratic approximation of $\\gamma \\left(\\cdot \\right)$ leads to better results, because they are all smooth.",
"On the other hand the approximations obtained by considering a linear model with respect to $z$ present slight multimodality problems.",
"Figure: {{figure:2fa24e65-a882-40c6-9835-ee580944a58f}}The number of simulations for the initialization step depends on the choice of the tolerance level: the approximation of the likelihood of $\\beta _1$ by using a Uniform prior for $\\sigma ^2_\\varepsilon $ needs 368, 2053 and 10945 simulations to accept the first value for tolerance levels of 1, 0.5 and 0.25 respectively; the approximation with Gamma prior with small parameters for $\\sigma ^2_\\varepsilon $ needs 40, 34 and 81 simulations to accept the first value.",
"These results refer to the summary statistics obtained with the quadratic approximation of $\\gamma \\left(\\cdot \\right)$ , the other choice of summary statistics considered has shown similar values.",
"The acceptance rates of ABC-MCMC algorithm are in general low, in particular with the lowest tolerance levels; they are around $25\\%$ for the highest thresholds considered."
],
[
"Discussion",
"We have explored the use of ABC methodology, a relatively new computational tools for Bayesian inference in complex models, in a rather classical inferential problem, namely the elimination of nuisance parameters.",
"We stress the fact that there are many situations where it is practically impossible to obtain a likelihood function for the parameter of interest in a closed form: in those cases the proposed method can be a competitive alternative to numerical methods.",
"As a technical aside one should note that, in many situations the prior $\\pi (\\text{$\\psi $})$ might be available in a closed form, so the kernel approximation of the prior is not necessary, and the accuracy of our method is even better.",
"However, we have preferred to present the method in its generality.",
"Another issue related to this last point is the approximation of the marginal posterior density of some components of the parameter.",
"Also in this case, the problem is made simpler by the fact that no approximation is needed for the prior and standard asymptotic arguments for kernel estimators hold for the approximation obtained from the posterior sample.",
"The main drawback of the present approach is that it requires the use of proper prior densities.",
"This can be a problem, especially when the nuisance parameter is high-dimensional and the elicitation process would be difficult.",
"A practical solution in these case is to adopt proper priors which approximate the appropriate improper noninformative prior for that model."
]
] | 1403.0387 |
[
[
"Transport in multiband systems with hot spots on the Fermi surface:\n Forward-scattering corrections"
],
[
"Abstract Multiband models with hot spots are of current interest partly because of their relevance for the iron-based superconductors.",
"In these materials, the momentum-dependent scattering off spin fluctuations and the ellipticity of the electron Fermi pockets are responsible for anisotropy of the lifetimes of excitations around the Fermi surface.",
"The deep minima of the lifetimes---the so-called hot spots---have been assumed to contribute little to the transport as is indeed predicted by a simple relaxation-time approach.",
"Calculating forward-scattering corrections to this approximation, we find that the effective transport times are much more isotropic than the lifetimes and that, therefore, the hot spots contribute to the transport even in the case of strong spin-fluctuation scattering.",
"We discuss this effect on the basis of an analytical solution of the Boltzmann equation and calculate numerically the temperature and doping dependence of the resistivity and the Hall, Seebeck, and Nernst coefficients."
],
[
"Introduction",
"Many materials of high current interest for condensed matter physics are metals with strong spin fluctuations, for example doped cuprates and iron pnictides.",
"In both classes, spin fluctuations are thought to mediate the superconducting pairing at relatively high temperatures.",
"[1], [2] Spin fluctuations are also crucial in the normal state, where they provide an important scattering mechanism and thus strongly affect transport.",
"The transport properties of the pnictides are nevertheless quite distinct from the cuprates and show unusual temperature dependences.",
"[3], [4], [5], [6], [7], [8], [9], [10], [11] The main ingredients needed for the description of transport in these systems have been controversially discussed.",
"[12], [13], [14], [15] The scattering of electrons off spin fluctuations is governed by the spin susceptibility.",
"Close to an antiferromagnetic instability, the susceptibility is strongly peaked in momentum space in the vicinity of the possible ordering vectors $\\mathbf {Q}$ .",
"Transport in such systems can thus often be understood based on the concept of hot and cold regions of the Fermi surfaces.",
"[16], [17] The hot regions are the parts of the Fermi surfaces that are connected by the possible ordering vectors $\\mathbf {Q}$ .",
"The scattering is particularly strong in these regions.",
"Conversely, in the cold regions not connected by ordering vectors the scattering rate is lower.",
"If the difference in the scattering rate is large, i.e., close to the instability, transport is thus dominated by the cold regions with high conductivity, and the hot regions are then said to be “short-circuited.” The concept of hot and cold regions generally explains the experimental observations for cuprates and was implicitly assumed to hold also for the pnictides.",
"[12], [8], [10], [11] An analysis of the lifetimes of excited electrons close to the Fermi surfaces seems to support this picture,[13] with the imperfect nesting of electron and hole Fermi pockets naturally leading to the appearance of hot and cold regions with short and long lifetimes, respectively.",
"Within the relaxation-time approximation (RTA), in which the complex relaxation dynamics of each state is modeled by a simple exponential decay, the transport relaxation time is approximated by the lifetime.",
"Since the conductivity is directly proportional to the relaxation time, the states with short lifetimes then do not contribute significantly to the transport.",
"In this paper we show that in multiband systems this effect can be compensated if the forward-scattering corrections to the RTA are taken into account.",
"Forward-scattering corrections, which are equivalent to vertex corrections in the Kubo formalism, have been studied extensively for one-band models relevant for cuprates and heavy-fermion systems.",
"[18] The pnictides are, in contrast, multiband systems with electron and hole Fermi pockets.",
"The study of two-band models with circular Fermi pockets has shown that forward-scattering corrections to the RTA are huge close to the antiferromagnetic instability and that they give rise to transport anomalies such as a large enhancement of the Hall coefficient[14], [15] and negative magnetoresistance.",
"[15] The minority carriers, i.e., the carriers on the smaller Fermi pocket, were found to exhibit negative transport times, indicating a drift in the direction opposite of what one would expect based on their charge.",
"However, in the simplified models with circular Fermi pockets all states on a given Fermi pocket are equivalent because of rotational symmetry.",
"They are thus unable to address the concept of hot and cold regions, which only appear for noncircular Fermi pockets.",
"In this article we present a semiclassical Boltzmann theory of transport for a two-band model with elliptical electron pockets relevant for the iron pnictides.",
"We show that due to the forward scattering, the hot-spot picture fails for the pnictides even for very strong spin fluctuations and highly elliptical electron pockets.",
"In contrast to the lifetimes, which are highly anisotropic around the Fermi pockets with deep minima at the hot spots, the effective transport relaxation times are found to be much more isotropic and to show no special features at the hot spots.",
"Our approximate analytical solution of the Boltzmann equation provides insight into the mechanism behind this effect: The anisotropy of the spin-fluctuation scattering extends the effective relaxation time.",
"At the hot spots, the reduction of the relaxation time due to the stronger scattering is thus compensated by the extension due to the higher anisotropy.",
"To elucidate the consequences of this mechanism, we calculate numerically the temperature-dependent transport coefficients from the full Boltzmann equation and compare them to the analytical solution and the RTA, finding that the RTA makes qualitatively incorrect predictions.",
"For strongly momentum-dependent scattering, we find large transport anomalies as well as a strong doping dependence.",
"The remainder of this paper is organized as follows: In Secs.",
"and , we present the two-band model, give expressions for the scattering rates, and set up the Boltzmann equation for our model.",
"To gain insight into the physics, we present in Sec.",
"an analytical solution to leading order in the ellipticities of the electron pockets.",
"Higher-order corrections are discussed in the appendix.",
"In Sec.",
", we present full numerical solutions of the Boltzmann equations.",
"We also calculate the temperature dependence of the resistivity and the Hall, Seebeck, and Nernst coefficients.",
"Finally, we draw some conclusions in Sec.",
"."
],
[
"Model",
"We model the FeAs layers of the iron pnictides by an effective two-dimensional two-band model with the dispersions in the single-iron unit cell [1] given by [19] $\\varepsilon _{h\\mathbf {k}}&=&\\varepsilon _{h}-\\mu +2t_{h}\\,(\\cos k_{x}a+\\cos k_{y}a), \\\\\\varepsilon _{e\\mathbf {k}}&=&\\varepsilon _{e}-\\mu +t_{e,1}\\cos k_{x}a\\,\\cos k_{y}a\\nonumber \\\\&& {}- t_{e,2}\\,\\xi \\,(\\cos k_{x}a+\\cos k_{y}a),$ where $a$ is the iron-iron separation.",
"As illustrated in Fig.",
"REF , the band $h$ gives rise to a nearly circular hole Fermi pocket at the center of the Brillouin zone, while the band $e$ forms two electron pockets $e1$ and $e2$ , displaced by $\\mathbf {Q}_{e1}=(0,\\pi /a)$ and $\\mathbf {Q}_{e2}=(\\pi /a,0)$ , respectively.",
"The parameter $\\xi $ controls the ellipticity of the electron pockets.",
"The chemical potential $\\mu $ is determined by the filling $n$ , i.e., the number of electrons per unit cell, which can be tuned by doping in the pnictides.",
"The filling $n$ determines the sizes of the Fermi pockets.",
"For $n\\approx 2.08$ the areas of the three pockets are nearly equal, while for smaller (larger) $n$ the hole pocket (electron pockets) become larger.",
"Following Ref.",
"Brydon2011, we take $\\varepsilon _h=-3.5\\,t_h$ , $\\varepsilon _e=3\\,t_h$ , $t_{e,1}=4\\,t_h$ and $t_{e,2}=t_h$ .",
"It is widely accepted that repulsive interactions between the nested electron and hole pockets drive a magnetic instability towards a stripe spin-density wave in the pnictide parent compounds with magnetic ordering vector ${\\bf Q}_{e1}$ or ${\\bf Q}_{e2}$ .",
"[20] Above the magnetic transition temperature, we therefore expect that the spin susceptibility will display pronounced peaks at these vectors.",
"Because of the ellipticity of the electron pockets, however, the nesting is imperfect and distinct hot spots develop at the points on the electron and hole Fermi pockets separated by ${\\bf Q}_{e1}$ or ${\\bf Q}_{e2}$ , see Fig.",
"REF .",
"The positions of the hot spots change with the doping: [19], [12] for underdoping ($n<2.08$ ) the hot spots are located near the major axis of the electron pockets, while at overdoping ($n>2.08$ ) the hots spots shift to the minor axis.",
"On the hole pocket, the hot spots shift from the axes to the diagonal and back again as one dopes across the antiferromagnetic dome.",
"We assume that the transport behavior is dominated by the scattering off spin fluctuations, which we model by the phenomenological susceptibility proposed by Millis, Monien, and Pines,[21] with temperature-dependent parameters based on neutron-scattering experiments.",
"[22] Although this ignores the anisotropy of the magnetic excitations in the pnictides caused by the ellipticity of the electron Fermi pockets,[23] we shall see that the precise form of the susceptibility is less important for the transport than the anisotropy of the scattering rate.",
"Together with momentum-independent impurity scattering, the scattering rate from a single-electron state $|b, \\mathbf {k}\\rangle $ to a state $|b^{\\prime }, \\mathbf {k}^{\\prime }\\rangle $ , where $b=e$ , $h$ denotes the band, can be written as[24] $W_{b\\mathbf {k}}^{b^{\\prime }\\mathbf {k}^{\\prime }} &=& (1-\\delta _{bb^{\\prime }})\\,W_{\\text{sf}}\\, \\frac{p_T(\\varepsilon _{b\\mathbf {k}}-\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }})}{\\left(\\varepsilon _{b\\mathbf {k}}-\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }}\\right)^{2}+ \\omega _{\\mathbf {k},\\mathbf {k}^{\\prime }}^{2}} \\nonumber \\\\&& {}+ \\delta (\\varepsilon _{b\\mathbf {k}}-\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }})\\, W_{\\text{imp}},$ where $W_{\\text{sf}}$ and $W_{\\text{imp}}$ represent the overall strength of the scattering off spin fluctuations and impurities, respectively, $p_T(x)\\equiv x\\left(\\coth x/2k_BT-\\tanh x/2k_BT\\right)$ , and $\\omega _{\\mathbf {k},\\mathbf {k}^{\\prime }} \\equiv \\Gamma _{T}\\, \\Big ( 1+\\xi _{T}^{2} \\min _{\\mathbf {Q}}[(\\mathbf {k}-\\mathbf {k}^{\\prime }+\\mathbf {Q})^{2}]\\Big ),$ where the four possible values for $\\mathbf {Q}$ are $\\pm \\mathbf {Q}_{e1}$ and $\\pm \\mathbf {Q}_{e2}$ .",
"With the Curie-Weiß temperature $-\\theta _\\mathrm {CW}<0$ , the frequency scale and the correlation length are given by [14], [22] $\\Gamma _T=\\Gamma _0\\,(T+\\theta _\\mathrm {CW})/\\theta _\\mathrm {CW}$ and $\\xi _T=\\xi _0\\,\\sqrt{\\theta _\\mathrm {CW}/(T+\\theta _\\mathrm {CW})}\\:\\exp (-T/T_0)$ , respectively.",
"Following Ref.",
"Fanfarillo2012, we here introduce an additional exponential decay of $\\xi _T$ to account for the high-temperature behavior and choose $T_0=200\\,\\mathrm {K}$ .",
"Following Ref.",
"Inosov2010, we take $\\xi _0=10\\,a$ , $\\theta _\\mathrm {CW}=30\\,\\mathrm {K}$ and $\\Gamma _0=4.2\\,\\mathrm {meV}$ .",
"The resulting form of $\\omega _{\\mathbf {k},\\mathbf {k}^{\\prime }}$ and thus $W_{b\\mathbf {k}}^{b^{\\prime }\\mathbf {k}^{\\prime }}$ is only valid as long as the system does not order antiferromagnetically or becomes superconducting.",
"The transport is governed by states on the Fermi pockets, denoted by $|s,\\theta \\rangle $ , where $s=h$ , $e1$ , $e2$ is the pocket index and $\\theta $ is the polar angle along the pocket, see Fig.",
"REF .",
"From Eq.",
"(REF ) we see that in the low-temperature regime, $k_BT\\ll \\varepsilon _F$ , the scattering rate is sharply peaked at $\\varepsilon _{b\\mathbf {k}}=\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }}$ so that scattering is nearly elastic.",
"We exploit this fact by writing $W_{b\\mathbf {k}_{F}}^{b^{\\prime }\\mathbf {k}^{\\prime }}\\approx \\delta (\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }}-\\varepsilon _{F})\\,W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }},$ where $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }} \\equiv (1-\\delta _{bb^{\\prime }})\\, W_{\\text{sf}}\\int d\\varepsilon \\,\\frac{p_T(\\varepsilon )}{\\varepsilon ^{2}+\\omega _{\\mathbf {k},\\mathbf {k}^{\\prime }}^{2}}+ W_{\\text{imp}}$ is the effective elastic scattering rate between states on the Fermi pockets $s$ , $s^{\\prime }$ belonging to the bands $b$ , $b^{\\prime }$ .",
"Since the spin susceptibility and thus $W_{b\\mathbf {k}}^{b^{\\prime }\\mathbf {k}^{\\prime }}$ is strongly momentum dependent, the elastic scattering rate $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}$ strongly depends on the angles $\\theta $ and $\\theta ^{\\prime }$ , in particular on the change in angle, $\\theta ^{\\prime }-\\theta $ .",
"This is what we call anisotropic scattering in the following.",
"More specifically, the scattering anisotropy stems from the peaks in the spin susceptibility at the wave vectors $\\pm \\mathbf {Q}_{e1}$ and $\\pm \\mathbf {Q}_{e2}$ .",
"For an initial state $|h,\\theta \\rangle $ with wave vector $\\mathbf {k}$ , the scattering rate has maxima for the final states $|e1,\\bar{\\theta }_{e1}\\rangle $ and $|e2,\\bar{\\theta }_{e2}\\rangle $ , defined as the states on the Fermi pockets $e1$ , $e2$ with wave vectors closest to $\\mathbf {k}+\\mathbf {Q}_{e1}$ and $\\mathbf {k}+\\mathbf {Q}_{e2}$ , respectively, see Fig.",
"REF .",
"Similarly, for an initial state $|e1, \\theta \\rangle $ ($|e2, \\theta \\rangle $ ) with wave vector $\\mathbf {k}$ , the scattering rate has a maximum for the final state $|h,\\bar{\\theta }_{h}\\rangle $ with wave vector closest to $\\mathbf {k}-\\mathbf {Q}_{e1}$ ($\\mathbf {k}-\\mathbf {Q}_{e2}$ ), where $\\bar{\\theta }_{h}\\approx \\theta $ since the hole pocket is nearly circular.",
"The scattering rate summed over all final states determines the characteristic lifetime of the state $|s,\\theta \\rangle $ , $\\tau _{s\\theta }=\\bigg (\\frac{1}{2\\pi }\\sum _{s^{\\prime }}\\int d\\theta ^{\\prime }\\, N_{s^{\\prime }\\theta ^{\\prime }}\\, W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}\\bigg )^{\\!-1},$ where $N_{s\\theta }=|d\\mathbf {k}_{F,s\\theta }/d\\theta |/\\pi \\hbar |\\mathbf {v}_{F,s\\theta }|$ is the density of states, with the spin degeneracy included, of pocket $s$ at the polar angle $\\theta $ and $\\mathbf {k}_{F,s\\theta }$ and $\\mathbf {v}_{F,s\\theta }$ are the Fermi momentum and the Fermi velocity, respectively.",
"In contrast to the transport relaxation time, which will be discussed below, the lifetime only depends on the integrated scattering strength and is independent of the precise shape of $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}$ as a function of $\\theta ^{\\prime }$ ."
],
[
"Boltzmann formalism",
"Our starting point is the semiclassical Boltzmann transport equation for a multiband system, ${-f^{\\prime }_0(\\varepsilon _{b\\mathbf {k}})\\, \\mathbf {E}\\cdot \\mathbf {v}_{b\\mathbf {k}}-\\frac{e}{\\hbar }\\, \\mathbf {B}\\cdot (\\mathbf {v}_{b\\mathbf {k}}\\times \\nabla _{\\mathbf {k}})\\, g_{b\\mathbf {k}} } \\nonumber \\\\&& \\qquad = \\sum _{b^{\\prime }\\mathbf {k}^{\\prime }} W_{b\\mathbf {k}}^{b^{\\prime }\\mathbf {k}^{\\prime }}\\,(g_{b\\mathbf {k}}-g_{b^{\\prime }\\mathbf {k}^{\\prime }}), \\hspace{40.0pt}$ where $\\mathbf {E}=(E_x,E_y,0)$ and $\\mathbf {B}=(0,0,B)$ are weak uniform electric and magnetic fields, respectively, $\\mathbf {v}_{b\\mathbf {k}} \\equiv \\hbar ^{-1}\\,\\nabla _{\\mathbf {k}}\\varepsilon _{b\\mathbf {k}}$ is the velocity and $g_{b\\mathbf {k}}\\equiv f_{b\\mathbf {k}}-f_0(\\varepsilon _{b\\mathbf {k}})$ is the difference between the non-equilibrium distribution function $f_{b\\mathbf {k}}$ and the Fermi-Dirac distribution $f_0(\\varepsilon _{b\\mathbf {k}})$ .",
"This difference is of the general form[25], [26], [27] $g_{b\\mathbf {k}}=-f^{\\prime }_0(\\varepsilon _{b\\mathbf {k}})\\,\\mathbf {E}\\cdot (\\mathbf {\\Lambda }_{b\\mathbf {k}}+\\delta \\mathbf {\\Lambda }_{b\\mathbf {k}}),$ with the as yet unknown vector mean free path $\\mathbf {\\Lambda }_{b\\mathbf {k}}+\\delta \\mathbf {\\Lambda }_{b\\mathbf {k}}$ .",
"Here, $\\mathbf {\\Lambda }_{b\\mathbf {k}}$ ($\\delta \\mathbf {\\Lambda }_{b\\mathbf {k}}$ ) is of zero (first) order in the magnetic field $\\mathbf {B}$ .",
"For states on the Fermi pockets we write $\\mathbf {\\Lambda }_{s\\theta }$ , $\\delta \\mathbf {\\Lambda }_{s\\theta }$ with obvious definitions.",
"Inserting Eqs.",
"(REF ), (REF ), and (REF ) into the Boltzmann equation (REF ) and using $\\sum _{b^{\\prime }\\mathbf {k}^{\\prime }}=\\sum _{s^{\\prime }}\\int \\frac{d\\theta ^{\\prime }}{2\\pi }N_{s^{\\prime }\\theta ^{\\prime }}\\int d\\varepsilon _{b^{\\prime }\\mathbf {k}^{\\prime }}$ , one finds for states at the Fermi energy[27] $\\mathbf {\\Lambda }_{s\\theta } &=& \\tau _{s\\theta }\\, \\mathbf {v}_{s\\theta }+ \\tau _{s\\theta }\\sum _{s^{\\prime }}\\int \\frac{d\\theta ^{\\prime }}{2\\pi }\\,N_{s^{\\prime }\\theta ^{\\prime }}\\,W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}\\,\\mathbf {\\Lambda }_{s^{\\prime }\\theta ^{\\prime }}, \\\\\\delta \\mathbf {\\Lambda }_{s\\theta } &=& \\tau _{s\\theta }\\, \\eta _s\\,\\frac{eB}{\\pi \\hbar ^2}\\, \\frac{1}{N_{s\\theta }}\\,\\frac{\\partial \\mathbf {\\Lambda }_{s\\theta }}{\\partial \\theta } \\nonumber \\\\&& {} + \\tau _{s\\theta }\\sum _{s^{\\prime }}\\int \\frac{d\\theta ^{\\prime }}{2\\pi }\\,N_{s^{\\prime }\\theta ^{\\prime }}\\,W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}\\,\\delta \\mathbf {\\Lambda }_{s^{\\prime }\\theta ^{\\prime }} ,\\hspace{40.0pt}$ where $\\eta _h=1$ and $\\eta _{e1}=\\eta _{e2}=-1$ .",
"The RTA consists of neglecting the forward-scattering corrections in Eqs.",
"(REF ) and (), i.e., the second terms on the right-hand sides.",
"Thus in the RTA one obtains $\\mathbf {\\Lambda }_{s\\theta } &=& \\mathbf {\\Lambda }_{s\\theta }^{(0)} \\;\\equiv \\;\\tau _{s\\theta }\\, \\mathbf {v}_{s\\theta }, \\\\\\delta \\mathbf {\\Lambda }_{s\\theta } &=& \\delta \\mathbf {\\Lambda }_{s\\theta }^{(0)} \\;\\equiv \\;\\tau _{s\\theta }\\, \\eta _s\\,\\frac{eB}{\\pi \\hbar ^2}\\, \\frac{1}{N_{s\\theta }}\\,\\frac{\\partial \\mathbf {\\Lambda }_{s\\theta }^{(0)}}{\\partial \\theta }.$ Evidently, within the RTA the solution is determined by the bare lifetimes $\\tau _{s\\theta }$ given in Eq.",
"(REF ).",
"The RTA becomes exact if the scattering rate is isotropic around the Fermi pockets so that the forward-scattering corrections average out.",
"For a nonzero scattering anisotropy, however, the result may differ significantly from the RTA.",
"[15] The charge current $\\mathbf {J}=\\sigma \\mathbf {E}$ is controlled by the conductivity tensor $\\sigma $ , which is in turn determined by the vector mean free path,[27] $\\sigma ^{ij} = e^2 \\sum _{s} \\int \\frac{d\\theta }{2\\pi }\\,N_{s\\theta }\\, v_{s\\theta }^{i}\\,\\big (\\Lambda _{s\\theta }^{j}+\\delta \\Lambda _{s\\theta }^{j}\\big )\\equiv \\sum _{s} \\int \\frac{d\\theta }{2\\pi }\\,\\sigma ^{ij}_{s\\theta }.$ Writing $\\mathbf {E}=E\\,(\\cos \\phi ,\\sin \\phi ,0)$ , we find the current parallel to the electric field as $\\frac{\\mathbf {J}\\cdot \\mathbf {E}}{E} &=& \\sum _{s} \\int \\frac{d\\theta }{2\\pi }\\, \\big (\\sigma _{s\\theta }^{xx}\\cos ^2\\phi + \\sigma _{s\\theta }^{yy}\\sin ^2\\phi \\nonumber \\\\&& {} + \\sigma _{s\\theta }^{xy}\\cos \\phi \\sin \\phi + \\sigma _{s\\theta }^{yx}\\cos \\phi \\sin \\phi \\big )\\nonumber \\\\&\\equiv & \\sum _{s} \\int \\frac{d\\theta }{2\\pi }\\,J_{s\\theta },$ where $J_{s\\theta }$ is the contribution of the state $|s,\\theta \\rangle $ to the current."
],
[
"Analytical results",
"To gain insight into transport beyond the RTA, we now construct an approximate analytical solution of Eqs.",
"(REF ) and () that fully accounts for the anisotropic scattering.",
"We will first discuss a few reasonable assumptions that make an analytical solution feasible.",
"The full numerical solution is discussed in Sec. .",
"As illustrated in Fig.",
"REF , the scattering rate $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}$ understood as a function of $\\theta ^{\\prime }$ has a maximum at $\\theta ^{\\prime }=\\bar{\\theta }_{s^{\\prime }}$ , which of course depends on $\\theta $ .",
"The small difference between $\\theta $ and $\\bar{\\theta }_{s^{\\prime }}$ stems from the ellipticity of the electron pockets.",
"We now make two simplifying assumptions: (i) The peak of the scattering rate $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}$ as a function of $\\theta ^{\\prime }$ is assumed to be symmetric around $\\theta ^{\\prime }=\\bar{\\theta }_{s^{\\prime }}$ , and (ii) the peak width is small on the scale on which the Fermi velocity $|\\mathbf {v}_{s\\theta }|$ and the density of states $N_{s\\theta }$ vary.",
"Both assumptions become exact in the limit of very strongly peaked spin susceptibility, i.e., as the magnetic instability is approached.",
"In the opposite limit of isotropic scattering, the forward-scattering corrections cancel out so that we also obtain the exact results.",
"On the right-hand side of Eq.",
"(REF ), we split $\\mathbf {\\Lambda }_{s^{\\prime }\\theta ^{\\prime }}$ into contributions parallel and perpendicular to $\\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}}$ , $\\mathbf {\\Lambda }_{s^{\\prime }\\theta ^{\\prime }} = \\frac{|\\mathbf {\\Lambda }_{s^{\\prime }\\theta ^{\\prime }}|}{|\\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}}|}\\big [\\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}}\\cos (\\theta ^{\\prime }-\\bar{\\theta }_{s^{\\prime }})+ \\hat{\\mathbf {z}}\\times \\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}}\\sin (\\theta ^{\\prime }-\\bar{\\theta }_{s^{\\prime }})\\big ] .$ By virtue of the assumptions (i) and (ii), the sine term drops out and we obtain $\\mathbf {\\Lambda }_{s\\theta } = \\mathbf {\\Lambda }_{s\\theta }^{(0)}+ \\left(1-\\frac{1}{2}\\, \\delta _{s,h}\\right)\\sum _{s^{\\prime }} a_{s\\theta }^{s^{\\prime }}\\, \\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}},$ where $a_{s\\theta }^{s^{\\prime }} \\equiv (1+\\delta _{s,h})\\,\\tau _{s\\theta } \\int \\frac{d\\theta ^{\\prime }}{2\\pi }\\, N_{s^{\\prime }\\theta ^{\\prime }}\\,W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }} \\cos (\\theta ^{\\prime }-\\bar{\\theta }_{s^{\\prime }})$ parametrizes the scattering anisotropy and in the following will be referred to as the anisotropy parameter.",
"The Kronecker symbols $\\delta _{s,h}$ appearing in Eqs.",
"(REF ) and (REF ) ensure that $a_{s\\theta }^{s^{\\prime }}\\in [0,1]$ and that $a_{s\\theta }^{s^{\\prime }}\\rightarrow 1$ corresponds to the limit of strong scattering anisotropy, $W_{s\\theta }^{s^{\\prime }\\theta ^{\\prime }}\\propto \\delta (\\theta ^{\\prime }-\\bar{\\theta }_{s^{\\prime }})$ , while $a_{s\\theta }^{s^{\\prime }}\\rightarrow 0$ gives the case of isotropic scattering, where the RTA result is recovered.",
"Figure: (Color online) Sketch of multiple scattering.",
"During the process, anelectron initially in state |s,θ〉|s,\\theta \\rangle effectively scatters betweenFermi pockets towards the closest hot spot (red/gray dot).",
"The sequenceof states (black dots) is given by the maximum of the scattering rate.",
"Theirdecreasing contribution to the vector mean free path Λ\\mathbf {\\Lambda } of theoriginal state |s,θ〉|s,\\theta \\rangle is indicated by the decreasing size of thedots.Iterating Eq.",
"(REF ), we obtain $\\mathbf {\\Lambda }$ in terms of $\\mathbf {\\Lambda }^{(0)}$ as a power series in the anisotropy parameter.",
"We now discuss the states appearing in this series.",
"The zero-order contribution to $\\mathbf {\\Lambda }_{s\\theta }$ is of course $\\mathbf {\\Lambda }_{s\\theta }^{(0)}$ , the RTA result for the same state $|s,\\theta \\rangle $ .",
"The first-order term involves $\\mathbf {\\Lambda }_{s^{\\prime }\\bar{\\theta }_{s^{\\prime }}}^{(0)}$ for the state $|s^{\\prime },\\bar{\\theta }_{s^{\\prime }}\\rangle $ .",
"This is the final state on the Fermi pocket $s^{\\prime }\\ne s$ to which the initial state $|s,\\theta \\rangle $ has the largest scattering rate.",
"Due to the ellipticity of the electron pockets, the shift of the angle, $\\bar{\\theta }_{s^{\\prime }}-\\theta $ , is always directed towards the closest hot spot, i.e., the intersection of the Fermi pocket $s$ with pocket $s^{\\prime }$ shifted by the appropriate vector $\\mathbf {Q}$ .",
"The state appearing in the second-order term is the one reached from $|s^{\\prime },\\bar{\\theta }_{s^{\\prime }}\\rangle $ with the largest scattering rate, again shifted towards the closest hot spot.",
"The states appearing in all higher-order terms are obtained in the same way.",
"The whole process can be interpreted as an effective hopping of the electron along a sequence of states, as illustrated by Fig.",
"REF .",
"The contribution to $\\mathbf {\\Lambda }_{s\\theta }$ from $\\mathbf {\\Lambda }_{s_\\nu \\theta _\\nu }^{(0)}$ of the state $|s_\\nu ,\\theta _\\nu \\rangle $ reached after $\\nu $ hopping events involves the product of $\\nu $ anisotropy parameters at $\\theta $ , $\\theta _1$ , $\\ldots $ , $\\theta _{\\nu -1}$ .",
"Since the angular shift between successive hopping events is due to the ellipticity of the electron pockets, and vanishes for a purely circular pocket, it is small for small ellipticities.",
"Indeed, in the appendix we show that for a circular hole pocket and a single elliptical electron pocket the error in the vector mean free path is of fourth order in the eccentricity of the electron pocket.",
"If we henceforth neglect this shift, i.e., let $\\theta _\\nu \\approx \\theta _{\\nu -1}$ for all $\\nu $ , we incur an error that is small for the moderate ellipticities of the electron Fermi pockets of the pnictides.",
"In the following section we shall see that this convenient approximation generally compares well with the full numerical solution of Eqs.",
"(REF ) and ().",
"Accordingly setting $\\bar{\\theta }_{s^{\\prime }}=\\theta $ in Eq.",
"(REF ), the vector mean free paths for different $\\theta $ decouple and we obtain $\\mathbf {\\Lambda }_{h\\theta } &=& \\frac{\\mathbf {\\Lambda }_{h\\theta }^{(0)}+\\frac{1}{2}\\, \\big (a_{h\\theta }^{e1}\\mathbf {\\Lambda }_{e1\\theta }^{(0)} + a_{h\\theta }^{e2}\\mathbf {\\Lambda }_{e2\\theta }^{(0)}\\big )}{1-\\frac{1}{2}\\,\\big (a_{h\\theta }^{e1}a_{e1\\theta }^{h}+a_{h\\theta }^{e2}a_{e2\\theta }^{h}\\big )}, \\\\\\mathbf {\\Lambda }_{e1\\theta } &=& \\mathbf {\\Lambda }_{e1\\theta }^{(0)} + a_{e1\\theta }^{h}\\,\\mathbf {\\Lambda }_{h\\theta }, \\\\\\mathbf {\\Lambda }_{e2\\theta } &=& \\mathbf {\\Lambda }_{e2\\theta }^{(0)} + a_{e2\\theta }^{h}\\,\\mathbf {\\Lambda }_{h\\theta }.$ Results for the magnetic part $\\delta \\mathbf {\\Lambda }_{s\\theta }$ can be found analogously by replacing $\\mathbf {\\Lambda }$ by $\\delta \\mathbf {\\Lambda }$ and $\\mathbf {\\Lambda }^{(0)}$ by $\\tau _{s\\theta }\\, \\eta _s\\,\\frac{eB}{\\pi \\hbar ^2}\\, \\frac{1}{N_{s\\theta }}\\,\\frac{\\partial \\mathbf {\\Lambda }_{s\\theta }}{\\partial \\theta } ,$ cf.",
"Eq.",
"().",
"Since the anisotropy parameters $a_{s\\theta }^{s^{\\prime }}$ are the only parameters in the solution, apart from the RTA vector mean free paths, we will refer to these expressions as the anisotropy approximation (AA).",
"Clearly, for $a_{s\\theta }^{s^{\\prime }}\\ne 0$ the vector mean free paths involve the RTA solutions of all three Fermi pockets.",
"This coupling between the pockets becomes stronger for larger anisotropy parameters.",
"Additionally, the denominator in Eq.",
"(REF ), which appears in all results, provides a factor that is larger than unity.",
"In the anisotropic limit, $a_{s\\theta }^{s^{\\prime }}\\rightarrow 1$ , the vector mean free paths $\\mathbf {\\Lambda }_{s\\theta }$ of all three pockets at a certain angle $\\theta $ become equal and diverge.",
"Thus, for strong scattering anisotropy the vector mean free path of the minority carriers must be inverted relative to the RTA result $\\mathbf {\\Lambda }_{s\\theta }^{(0)} \\propto \\mathbf {v}_{s\\theta }$ .",
"Semiclassically, we can interpret our results as follows.",
"The solution to the Boltzmann equation describes a non-equilibrium stationary state in which the acceleration of the electrons due to external forces is balanced by scattering.",
"The vector mean free path of state $|s,\\theta \\rangle $ can be understood as the displacement that an electron suffers until its velocity $\\mathbf {v}_{s\\theta }$ is randomized by scattering.",
"The lifetime $\\tau _{s\\theta }$ is the mean time between two scattering events.",
"If the scattering is isotropic the velocity is randomized after a single scattering event and the vector mean free path thus reads $\\tau _{s\\theta }\\mathbf {v}_{s\\theta }\\equiv \\mathbf {\\Lambda }^{(0)}_{s\\theta }$ .",
"On the other hand, anisotropic scattering only partially randomizes the velocity so that the effective relaxation time exceeds the lifetime $\\tau _{s\\theta }$ , giving rise to multiple scattering during the relaxation, see Fig.",
"REF .",
"The enhancement by denominator in Eq.",
"(REF ) accounts for this fact.",
"In the extreme limit of $a_{s\\theta }^{s^{\\prime }}\\rightarrow 1$ , the factor diverges, indicating that the velocities cannot relax at all and the vector mean free paths become infinite.",
"This physical picture also applies to the case of two circular Fermi pockets considered in Refs.",
"Fanfarillo2012 and Breitkreiz2013.",
"Because of rotational symmetry, the vector mean free path is parallel to the velocity in that case, and the AA becomes exact.",
"This permits a simple description in terms of transport times.",
"However, we are here concerned with noncircular Fermi pockets, which means that the vector mean free path is generally not parallel to the velocity.",
"The common feature is that strong anisotropic scattering forces the vector mean free path of electron and hole pockets at $\\theta $ to point in the same direction, which is set by the majority carriers.",
"In the relevant parameter range for our model, we will find that the direction is set by the electrons since there are two electron pockets.",
"A change of the dominant carrier type can only be achieved by strong hole doping."
],
[
"Numerical results",
"To obtain quantitative results without further approximations beyond the choice of the model and the semiclassical transport theory, we calculate the scattering rate given in Eq.",
"(REF ) by numerical integration.",
"Furthermore, we discretize the polar angle $\\theta $ , choosing 160 sites on each Fermi pocket.",
"We have checked that taking more points does not significantly change the results.",
"The lifetimes, Eq.",
"(REF ), and the anisotropy parameters, Eq.",
"(REF ), are obtained by summation over the discrete sites.",
"Finally, Eqs.",
"(REF ) and () are solved numerically by matrix inversion.",
"The numerical results will be compared to the AA, which is given by inserting the lifetimes and the anisotropy parameters into Eqs.",
"(REF )–()."
],
[
"Scattering rate",
"Figure REF (a) shows the temperature dependence of the scattering rate for $\\xi =1$ in Eq.",
"(REF ) and $W_{\\text{imp}}/W_{\\text{sf}}=10^{-3}$ .",
"While at high temperatures the scattering rate is isotropic, at lower temperatures a peak due to spin fluctuations develops corresponding to scattering vectors close to $\\mathbf {Q}_{e1}$ or $\\mathbf {Q}_{e2}$ .",
"The peak becomes sharper as the temperature is lowered so that the scattering anisotropy increases.",
"At very low temperatures spin fluctuations freeze out and only the isotropic impurity scattering remains so that the anisotropy vanishes again.",
"In Fig.",
"REF (b) we plot the anisotropy parameter corresponding to the scattering rate shown in Fig.",
"REF (a), averaged over the Fermi pocket.",
"It clearly exhibits the increase for decreasing temperature and the final sharp downturn at very low temperatures.",
"Note that in real pnictides, this low-temperature behavior will in most cases be preempted by antiferromagnetic or superconducting order, which are not described by our model spin susceptibility."
],
[
" Hot-spot picture",
"In this subsection we explore how different parts of the Fermi pockets contribute to the transport.",
"In particular, we want to find out to what extent the concept of hot and cold regions is applicable.",
"Choosing $T=1\\,\\mathrm {K}$ and $W_{\\text{imp}}/W_{\\text{sf}}=0$ , we focus on the regime of strong spin fluctuations with strong scattering anisotropy, where the difference between the RTA and the full result is the most striking.",
"The current parallel to the electric field is given by Eq.",
"(REF ).",
"The state-resolved current contributions $J_{s\\theta }$ depend on the direction of the electric field due to the noncircular Fermi pockets but we are here not interested in this dependence and therefore average $J_{s\\theta }$ over all directions of the electric field in the xy-plane.",
"For $\\mathbf {B}=0$ this gives $J_{s\\theta } \\equiv e^2N_{s\\theta } \\,\\frac{v_{s\\theta }^{x}\\Lambda _{s\\theta }^{x}+v_{s\\theta }^{y}\\Lambda _{s\\theta }^{y}}{2}\\, E.$ Figure REF shows the contributions $J_{s\\theta }$ resulting from the RTA as well as from the full numerical calculation.",
"The two are completely different.",
"Most prominently, the hot-spot picture[12], [8], [10], [11] is no longer valid if forward-scattering corrections are taken into account.",
"As discussed above, the scattering off spin fluctuations is strongest in the hot regions since the spin susceptibility is peaked at $\\mathbf {Q}_{e1}$ and $\\mathbf {Q}_{e2}$ , see Fig.",
"REF .",
"Thus the lifetimes are shorter and the RTA vector mean free paths given in Eqs.",
"(REF ) and () are smaller.",
"This is indeed reflected by the suppressed current contributions in the hot regions shown in Figs.",
"REF (a) and REF (b).",
"However, no signatures of hot regions are seen in the full results in Figs.",
"REF (c) and REF (d).",
"This is due to the anisotropy of the scattering rate.",
"In the hot regions, the anisotropy $a_{s\\theta }^{s^{\\prime }}$ is enhanced relative to the cold regions and, according to Eqs.",
"(REF )–(), this leads to an enhancement of the vector mean free path, as was discussed in section .",
"Thus the reduction of the lifetimes is compensated by the enhanced scattering anisotropy and the contribution of the hot regions to the current is comparable to that of other parts of the Fermi pockets, i.e., the short-circuiting of the hot spots does not occur.",
"This insight is a central result of our work.",
"Figure REF also shows that the holes contribute negatively to the total current in the full calculation.",
"In the semiclassical picture, this means that the holes drift in the same direction as the electrons.",
"The insights gained in section illuminate this behavior: For the set of parameters chosen in Fig.",
"REF , the scattering anisotropy averaged over all Fermi states is close to unity, $\\left\\langle a\\right\\rangle _\\theta =0.96$ .",
"As discussed in section , such a huge anisotropy leads to an effective relaxation time that is much longer than the lifetime.",
"In effect, during the relaxation, an electron initially on the hole Fermi pocket scatters multiple times between states on the hole pocket and states on the electron Fermi pockets, which have nearly opposite velocity.",
"Since there are more states on the electron pockets than on the hole pocket, the electron spends the larger part of the time on the electron pockets.",
"The electron thus on average drifts in the opposite direction to what one would get if it stayed on the hole pocket.",
"The RTA is not sensitive to the inversion of the velocity upon interpocket scattering and thus cannot account for this effect."
],
[
"Transport coefficients",
"The transport coefficients can be obtained from the vector mean free paths.",
"The conductivity tensor is given in Eq.",
"(REF ), while the thermoelectric tensor reads[27] $\\alpha ^{ij} = -\\frac{\\pi ^2k_B^2T}{3e}\\,\\frac{\\partial \\sigma ^{ij}}{\\partial \\mu } .$ We will focus on the resistivity $\\rho =\\frac{1}{\\sigma ^{xx}} ,$ the Hall coefficient, $R_H=\\frac{\\sigma ^{xy}}{(\\sigma ^{xx})^2B} ,$ the Seebeck coefficient (thermopower), $S=-\\frac{\\alpha ^{xx}}{\\sigma ^{xx}} ,$ and the Nernst coefficient, ${\\cal N}=\\frac{\\sigma ^{xy}\\alpha ^{xx}-\\sigma ^{xx}\\alpha ^{xy}}{(\\sigma ^{xx})^2B} .$ We give the resistivity in units of $\\rho _0 \\equiv \\frac{\\hbar }{e^2}\\, \\frac{\\hbar W_{\\text{sf}}}{{\\cal V}_0}\\times 10^{-2}\\,(\\mathrm {eV})^2 ,$ where ${\\cal V}_0$ is the volume of the unit cell, and the Nernst coefficient in units of ${\\cal N}_0 \\equiv \\frac{{\\cal V}_0}{e\\rho _0}\\times 10^{-5}\\, \\mathrm {V/K} .$ For the scattering strength ratio we choose in the following $W_{\\text{imp}}/W_{\\text{sf}}=10^{-3}$ ."
],
[
"Comparison of approximations",
"Figure REF shows the temperature dependence of the transport coefficients, comparing the full numerical result with the RTA and the AA.",
"We see that the RTA results tend to coincide with the full calculation only at very high and very low temperatures, where the scattering is nearly isotropic, see Fig.",
"REF .",
"In the temperature range with strong anisotropy (20–150 K) the deviations from the RTA are huge.",
"On the other hand, the AA shows qualitative agreement with the full results over all temperatures and for both ellipticities.",
"The agreement is even quantitative for the resistivity.",
"It is the worst for the Nernst coefficient ${\\cal N}$ but even here the positive and negative extrema in ${\\cal N}$ are predicted by the AA close to the correct temperatures.",
"For $\\xi =1$ the AA is slightly better than for $\\xi =2$ since the former value leads to less eccentric electron pockets.",
"The close agreement between the AA and the full numerical results shows that the transport behavior does not sensitively depend on the precise details of the anisotropic scattering, and thus justifies our use of the approximate susceptibility in Eq.",
"(REF ).",
"Both the RTA and the full results show strong temperature dependence.",
"For the RTA, this can be traced back to the nontrivial geometry of the Fermi pockets leading to the hot-spot structure for high scattering anisotropies.",
"However, as discussed in subsection REF , forward-scattering corrections invalidate the hot-spot picture for strong anisotropies.",
"The temperature dependence of the RTA results thus stems from the wrong origin.",
"The true temperature dependence can be understood on the basis of the AA, which gives qualitatively correct results.",
"Here, it is due to the strong temperature dependence of the anisotropy parameters $a_{s\\theta }^{s^{\\prime }}$ shown in Fig.",
"REF (b), i.e., it relies on the corrections to the RTA in Eqs.",
"(REF ) and () as well as (REF )–().",
"The differences between the RTA and the full results for the resistivity and the Hall coefficient are consistent with the predictions of Ref.",
"Breitkreiz2013 for two circular Fermi pockets.",
"In the resistivity, we note that the expected enhancement and reduction for high and low scattering anisotropies, respectively, lead to a more pronounced change of slope compared to the RTA.",
"Although the difference between the RTA and the full resistivity is relatively small compared to the large corrections to the electron and hole contributions shown in Fig.",
"REF , these corrections have opposite signs and thus partially compensate each other, as already found for circular Fermi pockets in Refs.",
"Fanfarillo2012,Breitkreiz2013.",
"The predicted enhancement of the Hall coefficient is also present.",
"[14], [15] However, the extremum of the Hall coefficient in Fig.",
"REF is due to the maximum in the scattering anisotropy (cf.",
"Fig.",
"REF ) and is thus of different origin than in Ref.",
"Breitkreiz2013, where a maximum in the Hall coefficient was predicted for the case that the anisotropy crosses a characteristic anisotropy level at which the mobilities of holes and electrons are of equal magnitude but opposite sign.",
"We do not see any signatures of such a crossing in the present results.",
"For the thermoelectric effects, Fig.",
"REF shows that the RTA results are even qualitatively incorrect, with the Seebeck and Nernst coefficients showing the wrong sign in the temperature range with strong scattering anisotropy.",
"According to Eqs.",
"(REF ) and (REF ), the Seebeck coefficient $S$ is proportional to $\\partial \\ln \\sigma ^{xx}/\\partial \\mu = - \\partial \\ln \\rho /\\partial \\mu $ .",
"In the RTA, it stems from the shift of the hot spots with the chemical potential, i.e., with doping.",
"In the full results and the AA, it is instead due to the change in the anisotropy parameters $a_{s\\theta }^{s^{\\prime }}$ with the chemical potential.",
"Figure REF shows that for the chosen parameters, the two effects contribute to $S$ with opposite sign.",
"The full results for the Nernst coefficient ${\\cal N}$ change sign between the ellipticities $\\xi =1$ and $\\xi =2$ .",
"This effect is missed by the RTA.",
"We return to the Nernst coefficient below.",
"Qualitative differences between the RTA and the full solution of the Boltzmann equation have also been reported for single-band cuprate models with strongly anisotropic scattering.",
"[27], [18] The physics discussed here, including the inverted vector mean free path of minority carriers, rely on the presence of multiple bands and Fermi pockets, though."
],
[
"Doping dependence",
"We now turn to the doping dependence of the transport coefficients.",
"Figures REF (a)–REF (d) show the full solutions at different fillings, while Fig.",
"REF (e) shows the current contributions of states on the Fermi surfaces at the two temperatures $T=100\\,\\mathrm {K}$ and $T=400\\,\\mathrm {K}$ with strong and weak scattering anisotropy, respectively.",
"Note that the current contributions from the hole pocket are negative for $T=100\\,\\mathrm {K}$ and $n\\gtrsim 1.99$ , i.e., towards the electron-doped side.",
"On the hole-doped side, the scattering is more isotropic due to the large discrepancy in size between the electron and hole pockets.",
"At high temperatures, the transport coefficients all show a smooth doping dependence resulting from the change in the Fermi surfaces and velocities in the presence of mostly isotropic scattering.",
"In the intermediate temperature range, where anisotropic scattering is strong, this is overlaid by nontrivial doping dependence due to the forward-scattering corrections.",
"The resistivity around $T\\approx 100\\,\\mathrm {K}$ is largest for intermediate fillings, for which the Fermi pockets are well nested.",
"This is because the narrow peaks in the spin susceptibilities at $\\mathbf {Q}_{e1}$ and $\\mathbf {Q}_{e2}$ lead to efficient scattering only for nested Fermi pockets.",
"The inefficiency of anisotropic scattering for small and large $n$ causes a rapid decrease in the resistivity with doping, as shown in the inset in Fig.",
"REF (a).",
"Note that the relative change in $\\rho $ with doping is much larger here than at high temperatures.",
"Since the Seebeck coefficient $S$ is proportional to $\\partial \\ln \\sigma ^{xx}/\\partial \\mu = -\\partial \\ln \\rho /\\partial \\mu = -\\rho ^{-1}\\partial \\rho /\\partial \\mu $ , it is sensitive to this relative change in $\\rho $ with $\\mu $ or $n$ and is, therefore, strongly enhanced in the intermediate temperature range with strong scattering anisotropy, as Fig.",
"REF (c) clearly shows.",
"For the Hall coefficient $R_H$ , Fig.",
"REF (b), one would naively expect the largest and smallest values for the most strongly hole-doped and electron-doped cases, respectively, since electrons and holes contribute with opposite signs.",
"This is indeed the case at $T\\approx 400\\,\\mathrm {K}$ , where the scattering is nearly isotropic and no negative current contributions occur.",
"At $T\\approx 100\\,\\mathrm {K}$ , however, Fig.",
"REF (b) shows a strong negative enhancement of $R_H$ for intermediate filling.",
"According to Fig.",
"REF (e), the contribution of the holes to the total current is negative in this range.",
"In the semiclassical picture this means that the holes drift in the same direction as the electrons, reducing the charge current.",
"Irrespective of that, the magnetic field deflects the holes and the electrons in the same direction.",
"Hence, the inverted sign of the hole contribution reduces the charge current without changing the Hall voltage.",
"This gives rise to an enhancement of the Hall coefficient defined as the Hall voltage relative to the charge current.",
"The Nernst coefficient ${\\cal N}$ plotted in Fig.",
"REF (d) is highly sensitive to small doping changes and also, as is evident from Fig.",
"REF , to changes in the band parameters.",
"Equations (REF )–(REF ) and (REF ) show that ${\\cal N} = \\frac{3e}{\\pi ^2k_B^2T}\\, \\frac{\\partial }{\\partial \\mu }\\,\\frac{R_H}{\\rho }= \\frac{3e}{\\pi ^2k_B^2T}\\, \\frac{\\partial n}{\\partial \\mu }\\,\\frac{\\partial }{\\partial n}\\, \\frac{R_H}{\\rho } .$ The Nernst coefficient is thus sensitive to the nonmonotonic doping dependence of both $\\rho $ and $R_H$ .",
"For the cases we have considered, the contributions from $\\rho $ and $R_H$ usually counteract each other.",
"The complicated behavior of ${\\cal N}$ , for example the different sign of ${\\cal N}$ for $n=2.05$ compared to the other fillings, is thus due to the quantitative competition of the doping dependences of $\\rho $ and $R_H$ and not to any clear qualitative features in the Fermi surfaces or the scattering.",
"This suggests that the other coefficients might be more advantageous as probes of the electronic system.",
"However, the detailed comparison of experimental transport coefficients and calculations for realistic models remains work for the future."
],
[
"Conclusions",
"We have studied transport in a two-band model relevant for the iron pnictides, using the semiclassical Boltzmann equation.",
"Forward-scattering corrections due to anisotropic interband scattering off spin fluctuations have been included.",
"Spin fluctuations have been described by a phenomenological Millis-Monien-Pines susceptibility,[21] with temperature-dependent parameters chosen based on neutron-scattering results for the pnictides.",
"[22] Our analytical and numerical investigations show that the anisotropic scattering gives rise to unusual transport behavior.",
"Most surprisingly, the hot spots are not short-circuited by the cold regions of the Fermi pockets even for very strong scattering.",
"The enhanced scattering rate in the hot regions indeed leads to a short lifetime there, but this effect is balanced by the enhanced vector mean free path due to the anisotropic scattering.",
"This breakdown of the concept of hot and cold regions is not found in a simple RTA neglecting forward-scattering corrections.",
"The nearly isotropic contribution of states around the Fermi pocket to the transport, even for strongly elliptical electron pockets, justifies the discussion of transport in terms of isotropic mobilities for each pocket.",
"However, as discussed for the case of circular pockets,[15], [14] the mobility of the minority carriers can turn negative in the regime of highly anisotropic scattering.",
"In the present work, negative mobility corresponds to inverted vector mean free paths and the resulting negative current contributions.",
"The contribution of hot spots to the transport and the occurrence of negative currents are the main features that distinguish the transport properties of pnictides from previously considered one-band systems with similarly anisotropic scattering.",
"In this work, we have presented unusual temperature and doping dependences of various transport coefficients.",
"Beyond this, negative current contributions can also lead to a negative magnetoresistance.",
"[15] However, the present model with two electron pockets and one hole pocket does not show negative magnetoresistance in the considered parameter range.",
"Calculations of transport coefficients for more realistic pnictide models are desirable to allow quantitative predictions.",
"Financial support by the Deutsche Forschungsgemeinschaft through Research Training Group GRK 1621 is gratefully acknowledged.",
"The authors thank J. Schmiedt, H. Kontani, and O. Kashuba for useful discussions."
],
[
"Appendix: Discussion of the angular shift",
"As discussed in the main text, the vector mean free path of a state $|s,\\theta \\rangle $ can be written as a power series in the anisotropy parameter, where the term of order $n$ contains the RTA vector mean free path of a state reached by $n$ hopping events towards the closest hot spot.",
"We have argued that the angular shift towards the hot spot is a small effect for the vector mean free path for realistic ellipticities of the electron pockets and have therefore ignored it above.",
"We here explore this effect analytically within a simple model.",
"To get an estimate for the upper limit of the correction to the vector mean free path, it is sufficient to consider only a single electron pocket.",
"Our simple model consists of a circular hole Fermi pocket with the Fermi wave number $k$ and an elliptical electron Fermi pocket described by the semi-major and semi-minor axis $k_a=k(1-\\epsilon ^2)^{-1/4}$ and $k_b=k(1-\\epsilon ^2)^{1/4}$ , respectively, where $\\epsilon $ is the eccentricity of the ellipse.",
"To focus on the shift effect we assume constant anisotropy, $a_{s\\theta }^{s^{\\prime }}=a$ .",
"For two Fermi pockets and constant anisotropy, Eq.",
"(REF ) takes the form $\\mathbf {\\Lambda }_{s\\theta }=\\mathbf {\\Lambda }_{s\\theta }^{(0)}+a\\, \\mathbf {\\Lambda }_{\\bar{s}\\bar{\\theta }},$ where $\\bar{h}=e$ , $\\bar{e}=h$ , and the RTA solution $\\mathbf {\\Lambda }^{(0)}$ is given by Eq.",
"(REF ).",
"Using simple trigonometry, we find that for the given geometry, the difference between $\\bar{\\theta }$ and $\\theta $ to leading order in the eccentricity $\\epsilon $ reads $\\frac{\\epsilon ^4}{16}\\sin 4\\theta $ .",
"Iterating Eq.",
"(REF ), we obtain the solution for the electron pocket as $\\mathbf {\\Lambda }_{e\\theta } = \\sum _{n=0}^{\\infty } a^{2n}\\,\\big (\\mathbf {\\Lambda }^{(0)}_{e\\theta _n}+a \\mathbf {\\Lambda }^{(0)}_{h\\theta _n}\\big ),$ with $\\theta _n=\\theta _{n-1} + \\frac{\\epsilon ^4}{16}\\, \\sin 4\\theta _{n-1} \\quad \\text{and}\\quad \\theta _0=\\theta .$ The solution for the hole pocket follows immediately from Eqs.",
"(REF ) and (REF ).",
"Replacing the discrete index $n$ by a continuous variable, we obtain $\\mathbf {\\Lambda }_{e\\theta } = -\\frac{2\\ln a}{1-a^{2}}\\int _{0}^{\\infty } dn\\, a^{2n} \\,\\big (\\mathbf {\\Lambda }_{e\\theta _{n}}^{(0)}+a\\mathbf {\\Lambda }_{h\\theta _{n}}^{(0)}\\big )+\\mathbf {R},$ with a correction $\\mathbf {R}$ .",
"By splitting the integration range into intervals $[m,m+1]$ with integer $m$ , one can easily show that $\\left|\\mathbf {R}\\right| \\le \\sum _n a^{2n}\\left| \\big (\\mathbf {\\Lambda }^{(0)}_{e\\theta _n}+a\\mathbf {\\Lambda }^{(0)}_{h\\theta _n}\\big )-\\big (\\mathbf {\\Lambda }^{(0)}_{e\\theta _{n+1}}+a\\mathbf {\\Lambda }^{(0)}_{h\\theta _{n+1}}\\big ) \\right|,$ which is obviously of higher order in $\\epsilon ^2$ because of Eq.",
"(REF ).",
"Substituting $n=4\\ln (1+z)/\\epsilon ^4$ we obtain $\\mathbf {\\Lambda }_{e\\theta } = \\frac{1}{1-a^{2}}\\int _{0}^{\\infty } dz\\, \\gamma \\, \\bigg (\\frac{1}{1+z}\\bigg )^{\\!\\gamma +1}\\big (\\mathbf {\\Lambda }_{e\\theta (z)}^{(0)}+a\\mathbf {\\Lambda }_{h\\theta (z)}^{(0)}\\big ) ,$ with $\\gamma \\equiv 8\\,\\frac{\\ln (1/a)}{\\epsilon ^4}$ and $\\theta (z) \\equiv \\frac{1}{2}\\arctan \\left[(z+1)\\tan 2\\theta \\right].$ In the integral in Eq.",
"(REF ), the factor $\\gamma \\,[1/(1+z)]^{\\gamma +1}$ acts as a distribution function which is normalized to unity and becomes a $\\delta $ -function in the limit of zero ellipticity, i.e., for $\\gamma \\rightarrow \\infty $ .",
"Hence, the largest shifts are achieved for small values of $\\gamma $ , which, according to Eq.",
"(REF ), correspond to large scattering anisotropy and large ellipticity.",
"The shift also depends on the position on the Fermi pocket.",
"There is no shift at the hot spots, $\\theta =(2n-1)\\,\\pi /4$ , and at the cold spots, $\\theta =n\\,\\pi /2$ .",
"The largest shift can be expected to occur between the hot and cold spots, in the vicinity of $(2n-1)\\,\\pi /8$ .",
"We can make further analytical progress by expanding the vector $\\big (\\mathbf {\\Lambda }_{e\\theta (z)}^{(0)}+a\\mathbf {\\Lambda }_{h\\theta (z)}^{(0)}\\big )$ to linear order in $\\theta (z)$ .",
"This is best justified if the total angular shift is small, i.e., if we start with $\\theta $ close to a hot spot.",
"However, the total shift can never be larger than $\\pi /4$ so that the approximation always gives at least qualitatively correct results for not excessive eccentricities.",
"Equation (REF ) can then be written as $\\mathbf {\\Lambda }_{e\\theta } = \\frac{1}{1-a^{2}}\\,\\big (\\mathbf {\\Lambda }_{e,\\theta +\\Delta _{\\theta ,\\gamma }}^{(0)}+a\\mathbf {\\Lambda }_{h,\\theta +\\Delta _{\\theta ,\\gamma }}^{(0)}\\big ),$ with the effective angular shift $\\Delta _{\\theta ,\\gamma } &=& \\int _{0}^{\\infty } dz\\,\\gamma \\, \\Big (\\frac{1}{1+z}\\Big )^{\\!\\gamma +1}\\theta (z)-\\theta \\nonumber \\\\&\\cong & \\frac{\\sin 4\\theta }{32}\\, \\frac{\\epsilon ^4}{\\ln (1/a)}+ \\frac{\\sin 8\\theta }{512}\\,\\bigg [ \\frac{\\epsilon ^4}{\\ln (1/a)} \\bigg ]^2\\nonumber \\\\&& {} + \\mathcal {O}\\bigg (\\bigg [\\frac{\\epsilon ^4}{\\ln (1/a)}\\bigg ]^3\\bigg ) .$ By neglecting the shift, $\\Delta _{\\theta ,\\gamma }=0$ , we would obtain the analogue of Eqs.",
"(REF )–() for the case of constant anisotropy and a single electron pocket.",
"In Fig.",
"REF we plot the angular shift at $\\theta =\\pi /8$ for different anisotropies as a function of the eccentricity squared, $\\epsilon ^2$ .",
"Realistic scattering anisotropies hardly exceed the value $a=0.95$ , for which the shift is small up to $\\epsilon ^2\\approx 0.5$ .",
"Stronger ellipticities might, however, lead to significant corrections."
]
] | 1403.0144 |
[
[
"Analysis of stable screw dislocation configurations in an anti--plane\n lattice model"
],
[
"Abstract We consider a variational anti-plane lattice model and demonstrate that at zero temperature, there exist locally stable states containing screw dislocations, given conditions on the distance between the dislocations, and on the distance between dislocations and the boundary of the crystal.",
"In proving our results, we introduce approximate solutions which are taken from the theory of dislocations in linear elasticity, and use the inverse function theorem to show that local minimisers lie near them.",
"This gives credence to the commonly held intuition that linear elasticity is essentially correct up to a few spacings from the dislocation core."
],
[
"Introduction",
"Plasticity in crystalline materials is a highly complex phenomenon, a key aspect of which is the movement of dislocations.",
"Dislocations are line defects within the crystal structure which were first hypothesised to act as carriers of plastic flow in [21], [23], [26], and later experimentally observed in [4], [16].",
"As they move through a crystal, dislocations interact with themselves and other defects via the orientation-dependent stress fields they induce [17].",
"This leads to complex coupled behaviour, and efforts to create accurate mathematical models to describe their motion and interaction, and so better engineer such materials are ongoing (see for example [6], [2]).",
"Over the last decade, a body of mathematical analysis of dislocation models has begun to develop which aims to derive models of crystal plasticity in a consistent way from models of dislocation motion and energetics.",
"Broadly, this work starts from either atomistic models, as in [1], [8], [24], [3], or `semidiscrete' models, where dislocations are lines or points in an elastic continuum, as in [13], [20], [25], [11], [28], [12], [10].",
"In the present work we focus on the analysis of dislocations at the atomistic level and therefore briefly recount recent achievements in this area.",
"In [8] the focus is the derivation of homogenised dynamical equations for dislocations and dislocation densities from a generalisation of the Frenkel–Kontorova model for edge dislocations.",
"In [3] a clear mathematical framework for describing the Burgers vector of dislocations in lattices was developed, and the asymptotic form of a discrete energy is given in the regime where dislocations are far from each other relative to the lattice spacing.",
"In [24] a rigorous asymptotic description of the energy in a finite crystal undergoing anti–plane deformation with screw dislocations present is developed.",
"[1] follows in the same vein, broadening the class of models considered, and also treating the asymptotics of a minimising movement of the dislocation energy.",
"In a similar anti–plane setting, but in an infinite crystal, [18] demonstrated that there are globally stable states with unit Burgers vector.",
"In the present contribution we demonstrate the existence of locally stable states containing multiple dislocations with arbitrary combinations of Burgers vector.",
"Once more, our analysis concerns crystals under anti–plane deformation, but in addition to the full lattice, we now consider finite convex domains with boundaries.",
"Recent results contained in [1] also address the question of local stability of dislocation configurations in finite domains, but under different assumptions to those employed here, and using a different set of analytical techniques.",
"In particular, our analysis employs discrete regularity results which enable us to provide quantitative estimates on the equilibrium configurations, while previous results only provide estimates on the energies."
],
[
"Outline",
"The setting for our results is similar to that described in [18]: our starting point is the energy difference functional $E^\\Omega (y;\\tilde{y}):=\\sum _{b\\in \\mathcal {B}^\\Omega }\\big [\\psi (Dy_b)-\\psi (D\\tilde{y}_b)\\big ],$ where $\\Omega \\subset \\Lambda $ is a subset of a Bravais lattice, $\\mathcal {B}^\\Omega $ is a set of pairs of interacting (lines of) atoms, $Dy_b$ is a finite difference, and $\\psi $ is a 1-periodic potential.",
"We call a deformation $y$ a locally stable equilibrium if $u=0$ minimises $E^\\Omega (y+u;y)$ among all perturbations $u$ which have finite energy, and are sufficiently small in the energy norm.",
"The key assumption upon which we base our analysis is the existence of a local equilibrium in the homogeneous infinite lattice containing a dislocation which satisfies a condition which we term strong stability — this notion is made precise in §REF .",
"Under this key assumption, our main result is Theorem REF .",
"This states that, given a number of positive and negative screw dislocations, there exist locally stable equilibria containing these dislocations in a given domain as long as the core positions satisfy a minimum separation criterion from each other and from the boundary of the domain.",
"Furthermore, these configurations may only be globally stable if there is one dislocation in an infinite lattice.",
"The proof of Theorem REF is divided into two cases, that in which $\\Omega =\\Lambda $ , and that in which $\\Omega $ is a finite convex lattice polygon: these are proved in § and § respectively."
],
[
"The lattice",
"Underlying the results presented in this paper is the structure of the triangular lattice $\\Lambda :={\\textstyle \\frac{a_1+a_2}{3}}+[a_1,a_2]\\cdot \\mathbb {Z}^2,\\quad \\text{where }a_1=(1,0)^T\\text{ and }a_2=\\big ({\\textstyle \\frac{1}{2}},{\\textstyle \\frac{\\sqrt{3}}{2}}\\big )^T.$ In this section we detail the main geometric and topological definitions we use to conduct the analysis."
],
[
"The lattice complex",
"For the purposes of providing a clear definition of the notion of Burgers vector in the lattice, we describe the construction of a CW complexFor further details on the definition of a CW complex and other aspects of algebraic topology, see for example [15].",
"for a general lattice subset.",
"Recall from [3] that we may define a lattice complex in 2D by first defining a set of lattice points (or 0–cells), $\\Lambda $ , then defining the bonds (or 1–cells), $\\mathcal {B}$ , and finally the cells (or 2–cells) $\\mathcal {C}$ , and the corresponding boundary operators, $\\partial $ .",
"Throughout the paper, $\\Lambda $ , $\\mathcal {B}$ and $\\mathcal {C}$ will refer to the lattice complex generated by $\\Lambda $ as defined in (REF ) — see also [18] for further details of this construction.",
"Here, we also consider subcomplexes generated by subsets $\\Omega \\subset \\Lambda $ .",
"Given $\\Omega \\subseteq \\Lambda $ , we define the corresponding sets of bonds and cells to be $\\mathcal {B}^\\Omega :=\\big \\lbrace (\\xi ,\\zeta )\\in \\mathcal {B}\\,\\big |\\,\\xi ,\\zeta \\in \\Omega \\big \\rbrace \\quad \\text{and} \\quad \\mathcal {C}^\\Omega :=\\big \\lbrace (\\xi ,\\zeta ,\\eta )\\in \\mathcal {C}\\,\\big |\\,\\xi ,\\zeta ,\\eta \\in \\Omega \\big \\rbrace .$ It is straightforward to check that this satisfies the definitions of a CW subcomplex of the full lattice complex presented in [18], and so we may make use of the definitions of integration and $p$ -forms as given in [3] restricted to this subcomplex.",
"To keep notation concise, we will frequently write $f_b:=f(b)\\qquad \\text{when}\\quad f:\\mathcal {B}\\rightarrow \\mathbb {R}\\quad \\text{is a 1--form.",
"}$ We note that we have chosen to define $\\Lambda $ such that $0\\in \\mathbb {R}^2$ lies at the barycentre of a cell which we will denote $C_0$ , and more generally we will use the notation $x^C\\in \\mathbb {R}^2$ to refer to the barycentre of $C\\in \\mathcal {C}$ ."
],
[
"Lattice symmetries",
"The triangular lattice is a highly symmetric structure, and all of its rotational symmetries can be described in terms of multiples of positive rotations by $\\pi /3$ about various points in $\\mathbb {R}^2$ .",
"We therefore fix ${\\sf R}_6$ to be the corresponding linear transformation.",
"We define two special classes of affine transformations on $\\mathbb {R}^2$ which are automorphisms of $\\Lambda $ , $G^C(x)&:={\\sf R}_6^i(x-x^C)=0\\quad \\text{where }i\\in \\lbrace 0,1\\rbrace \\text{ is chosen so that }G^C(\\Lambda )=\\Lambda ,\\\\H^C(x)&:=\\big (G^C\\big )^{-1}(x)={\\sf R}_6^{-i}x+x^C,$ and note that by definition, $G^C(C)=C_0$ and $H^C(C_0)=C$ .",
"We also understand $G^C,H^C$ as automorphisms on $\\mathcal {B}$ and $\\mathcal {C}$ in the following way: if $b=(\\xi ,\\zeta )\\in \\mathcal {B}$ and $C^{\\prime }=(\\xi ,\\zeta ,\\eta )\\in \\mathcal {C}$ , then $G^C(b):=\\big (G^C(\\xi ),G^C(\\zeta )\\big ),\\quad \\text{and}\\quad G^C(C^{\\prime }):=\\big (G^C(\\xi ),G^C(\\zeta ),G^C(\\eta )\\big ).$ Later, it will be important to consider the transformation of 1–forms under such automorphisms, and so we write $\\big (f\\circ G^C\\big )_b:=f\\big (G^C(b)\\big )\\qquad \\text{when}\\quad f:\\mathcal {B}\\rightarrow \\mathbb {R}\\quad \\text{is a 1--form.",
"}$"
],
[
"Nearest neighbours",
"We define the set of nearest neighbour directions by $\\mathcal {R}:=\\lbrace a_i\\in \\mathbb {R}^2\\,|\\,i\\in \\mathbb {Z}\\rbrace ,\\quad \\text{where}\\quad a_i := {\\sf R}_6^{i-1}a_1.$ Given $\\Omega \\subseteq \\Lambda $ and $\\xi \\in \\Omega $ , we define the nearest neighbour directions of $\\xi $ in $\\Omega $ to be $\\mathcal {R}_\\xi ^\\Omega :=\\big \\lbrace a_i\\in \\mathcal {R}\\,\\big |\\,\\xi +a_i\\in \\Omega \\rbrace \\subseteq \\mathcal {R}.$"
],
[
"Distance",
"To describe the distance between elements in the complex, we use the usual notion of Euclidean distance of sets, $\\mathrm {dist}(A,B):=\\inf \\big \\lbrace |x-y|\\,\\big |\\,x\\in A,\\,y\\in B\\big \\rbrace .$"
],
[
"Convex crystal domains",
"In addition to studying dislocations in the infinite lattice $\\Lambda $ , we will also consider dislocations in a convex lattice polygon: We say that $\\Omega \\subset \\Lambda $ is a convex lattice polygon if $C_0 \\in \\mathcal {C}^\\Omega , \\quad {\\rm conv}(\\Omega )\\cap \\Lambda =\\Omega ,\\quad \\text{and} \\quad \\Omega \\text{ is finite.",
"}$ Here and throughout the paper, ${\\rm conv}(U)$ means the closed convex hull of $U\\subset \\mathbb {R}^2$ , and $\\Omega $ will denote either a convex lattice polygon or $\\Lambda $ unless stated otherwise.",
"For a convex lattice polygon, we define corresponding `continuum' domains $U^\\Omega :={\\rm conv}(\\Omega )\\qquad \\text{and}\\qquad W^\\Omega :=\\mathrm {clos}\\Big (\\bigcup \\big \\lbrace C\\in \\mathcal {C}^\\Omega \\,\\big |\\,C\\text{ positively--oriented}\\big \\rbrace \\Big ).$ We note that $\\Omega \\subset W^\\Omega \\subseteq U^\\Omega $ ; for an illustration of an example of these definitions, see Figure REF ."
],
[
"Boundary and boundary index",
"We note that the positively–oriented boundary $\\partial W^\\Omega $ may be decomposed as $\\partial W^\\Omega = \\big \\lbrace \\xi \\in \\Omega \\,|\\,\\mathcal {R}_\\xi ^\\Omega \\ne \\mathcal {R}\\big \\rbrace \\cup \\Big \\lbrace b\\in \\mathcal {B}^\\Omega \\,\\Big |\\,b\\in \\partial \\sum \\lbrace C\\in \\mathcal {C}^\\Omega \\,|\\,C\\text{ positively--oriented}\\rbrace \\Big \\rbrace ;$ in other words, into the lattice points which do not have a full set of nearest neighbours, and hence lie on the `edge' of the set $\\Omega $ , and into the bonds which follow the positively–oriented boundary of the entire subcomplex within the full lattice.",
"Since it will be necessary to sum over these sets later, we write $\\xi \\in \\partial W^\\Omega $ to mean $\\xi \\in \\partial W^\\Omega \\cap \\Omega $ , and $b\\in \\partial W^\\Omega $ to mean $b\\in \\partial C\\cap \\partial W^\\Omega $ for some positively–oriented $C\\in \\mathcal {C}^\\Omega $ .",
"It is clear that since $\\Omega $ is a finite set, $U^\\Omega $ is a convex polygonal domain in $\\mathbb {R}^2$ , and $\\partial U^\\Omega $ is made up of finitely–many straight segments.",
"We number the corners of such polygons according to the positive orientation of $\\partial U^\\Omega $ as $\\kappa _m$ , $m= 1, \\dots , M$ , and $\\kappa _0:=\\kappa _M$ ; evidently, $\\kappa _m\\in \\Omega $ for all $m$ .",
"We further set $\\Gamma _m:=(\\kappa _{m-1},\\kappa _m)\\subset \\mathbb {R}^2$ , the straight segments of the boundary.",
"For each $m$ , $\\kappa _m-\\kappa _{m-1}$ is a lattice direction.",
"Since any pair $a_i,a_{i+1}$ with $i\\in \\mathbb {Z}$ form a basis for the lattice directions, there exists $i$ such that $\\kappa _m-\\kappa _{m-1} = j^{\\prime } a_i + k^{\\prime } a_{i+1},\\quad \\text{where}\\quad j^{\\prime },k^{\\prime } \\in \\mathbb {N}, j^{\\prime } > 0.$ Define the lattice tangent vector to $\\Gamma _m$ to be $\\tau _m:=j a_i+ k a_{i+1},\\quad \\text{where }\\gcd (j,k)=1\\quad \\text{and}\\quad \\kappa _m-\\kappa _{m-1} = n \\tau _m\\quad \\text{for some }n \\in \\mathbb {N}.$ This definition entails that $\\tau _m$ is irreducible in the sense that no lattice direction with smaller norm is parallel to $\\tau _m$ , and hence if $\\zeta \\in \\Gamma _m\\cap \\partial W^\\Omega $ , $\\zeta =\\kappa _m+j\\tau _m$ for some $j=\\lbrace 0,\\ldots J_m\\rbrace $ .",
"In addition to the decomposition of $\\partial W^\\Omega $ into lattice points and bonds, we may also decompose into `periods' $P_\\zeta $ indexed by $\\zeta \\in \\partial W^\\Omega \\cap \\partial U^\\Omega $ , so $\\partial W^\\Omega = \\bigcup _{m=1}^M\\bigcup _{j=0}^{J_m}P_{\\kappa _m+j\\tau _m}\\qquad \\text{where}\\quad P_{\\zeta }:=\\big \\lbrace x\\in \\partial W^\\Omega \\,\\big |\\,(x-\\zeta )\\cdot \\tau _m\\in \\big [0,|\\tau _m|^2\\big ]\\big \\rbrace .$ An illustration of the definition of $P_\\zeta $ may be found on the right–hand side of Figure REF .",
"Denoting the 1–dimensional Hausdorff measure as $\\mathcal {H}^1$ , we define the index of $\\Gamma _m$ , and of $\\partial W^\\Omega $ respectively, to be ${\\rm index}(\\Gamma _m) := \\mathcal {H}^1(P_\\zeta )\\quad \\text{for any }\\zeta \\in \\Gamma _m\\cap \\partial W^\\Omega \\quad \\text{and}\\quad {\\rm index}(\\partial W^\\Omega ):= \\max _{m = 1, \\dots , M} {\\rm index}(\\Gamma _m).$ Figure: The figure on the left shows an example of a convex latticepolygon.",
"Here, Ω\\Omega is the set of dark grey points, W Ω W^\\Omega is the lightgrey region and the dark grey region corresponds to U Ω ∖W Ω U^\\Omega \\setminus W^\\Omega .The boundaries of W Ω W^\\Omega and U Ω U^\\Omega are denoted by dashed and plain linesrespectively.The figure on the right illustrates the definition of P ζ P_\\zeta , clearlyshowing the periodic structure of ∂W Ω \\partial W^\\Omega ."
],
[
"Deformations and Burgers vector",
"The positions of deformed atoms will be described by maps $y\\in {W}(\\Omega ):=\\lbrace y:\\Omega \\rightarrow \\mathbb {R}\\rbrace $ .",
"For $y\\in {W}(\\Omega )$ and $b=(\\xi ,\\eta )\\in \\mathcal {B}^\\Omega $ , we define the finite difference $Dy_b = y(\\eta )-y(\\xi ).$"
],
[
"Function spaces",
"In addition to the space ${W}(\\Omega )$ , we define ${W}_0(\\Omega )&:=\\big \\lbrace v\\in {W}(\\Omega )\\,\\big |\\,v(\\xi _0)=0\\text{ and }{\\rm supp}(Dv)\\text{ is bounded.",
"}\\big \\rbrace ,\\\\\\dot{{W}}^{1,2}(\\Omega )&:=\\big \\lbrace v\\in {W}(\\Omega )\\,\\big |\\,v(\\xi _0)=0\\text{ and }Dv\\in \\ell ^2\\big (\\mathcal {B}^\\Omega \\big )\\big \\rbrace ,$ where $\\xi _0=(0,\\frac{\\sqrt{3}}{3})^T\\in \\Omega $ .",
"It is shown in [22] that $\\dot{{W}}^{1,2}$ is a Hilbert space and ${W}_0\\subset \\dot{{W}}^{1,2}$ is dense."
],
[
"Burgers vector",
"We now slightly generalise some key definitions from [18].",
"Given $y:\\Omega \\rightarrow \\mathbb {R}$ , the set of associated bond length 1-forms is defined to be $[Dy]:=\\big \\lbrace \\alpha :\\mathcal {B}^\\Omega \\rightarrow [-{\\textstyle \\frac{1}{2}},{\\textstyle \\frac{1}{2}}]\\,\\big |\\,\\alpha _{-b}=-\\alpha _b,\\,Dy_b-\\alpha _b\\in \\mathbb {Z}\\text{ for all }b\\in \\mathcal {B}^\\Omega ,\\,\\alpha _b\\in (-{\\textstyle \\frac{1}{2}},{\\textstyle \\frac{1}{2}}]\\text{ if }b\\in \\partial W^\\Omega \\big \\rbrace .$ A dislocation core of a bond length 1-form $\\alpha $ is a positively–oriented 2-cell $C\\in \\mathcal {C}^\\Omega $ such that $\\int _{\\partial C}\\alpha \\ne 0$ .",
"As remarked in [18], the Burgers vector of a single cell may only be $-1$ , 0 or 1, so we define the set of dislocation cores to be $\\mathcal {C}^\\pm [\\alpha ]:=\\Big \\lbrace C\\in \\mathcal {C}^\\Omega \\,\\Big |\\,C\\text{ positively--oriented, }\\int _{\\partial C}\\alpha =\\pm 1\\Big \\rbrace .$ We can now define the net Burgers vector of a deformation $y$ with $|\\mathcal {C}^\\pm [\\alpha ]| < \\infty $ (i.e., a finite number of dislocation cores) to be $B[y] := \\sum _{C\\in \\mathcal {C}^\\pm [\\alpha ]} \\int _{\\partial C} \\alpha .$ If $\\Omega $ is a convex lattice polygon, then it is straightforward to see that $B[y] = \\int _{\\partial W^\\Omega }\\alpha $ , and the requirement that $\\alpha _b\\in (-{\\textstyle \\frac{1}{2}},{\\textstyle \\frac{1}{2}}]$ for $b\\in \\partial W^\\Omega $ ensures that the net Burgers vector is independent of $\\alpha \\in [Dy]$ , since any two bond length 1-forms agree for all $b\\in \\partial W^\\Omega $ ."
],
[
"Dislocation configurations",
"In order to prescribe the location of an array of dislocations, we define a dislocation configuration (or simply, a configuration) to be a set $\\mathcal {D}$ of ordered pairs $(C,s)\\in \\mathcal {C}^\\Omega \\times \\lbrace -1,1\\rbrace $ , satisfying the condition that $(C,s)\\in \\mathcal {D}\\quad \\text{implies}\\quad (C,-s)\\notin \\mathcal {D}.$ Such sets $\\mathcal {D}$ should be thought of as a set of dislocation core positions with accompanying Burgers vector $\\pm 1$ .",
"We define the minimum separation distance of a configuration to be $L_\\mathcal {D}:=\\inf \\big \\lbrace \\mathrm {dist}(C,C^{\\prime })\\,\\big |\\,(C,s),(C^{\\prime },t)\\in \\mathcal {D}, C\\ne C^{\\prime }\\rbrace ,$ and in the case where $\\Omega $ is a convex lattice polygon, we define the minimum separation between the dislocations and the boundary to be $S_\\mathcal {D}:=\\inf \\big \\lbrace \\mathrm {dist}(C,\\partial W^\\Omega )\\,\\big |\\,(C,s)\\in \\mathcal {D}\\big \\rbrace .$"
],
[
"Energy difference functional and equilibria",
"We assume that lattice sites interact via a 1–periodic nearest–neighbour pair potential $\\psi \\in C^4(\\mathbb {R})$ , which is even about 0.",
"We discuss possible extensions in §REF .",
"For a pair of displacements $y,\\tilde{y}\\in {W}(\\Omega )$ , we define $E^\\Omega (y;\\tilde{y}):=\\sum _{b\\in \\mathcal {B}^\\Omega }\\psi (Dy_b)-\\psi (D\\tilde{y}_b),$ where we will drop the use of the superscript in the case where $\\Omega =\\Lambda $ .",
"We note immediately that this functional is well-defined whenever $y-\\tilde{y}\\in {W}_0(\\Omega )$ .",
"It is also clear that Gateaux derivatives in the first argument (in ${W}_0(\\Omega )$ directions) exist up to fourth order, and do not depend on the second argument.",
"We denote these derivatives $\\delta ^j E^\\Omega (y)$ , so that for $v, w \\in {W}_0(\\Omega )$ , we have $\\langle \\delta E^\\Omega (y),v\\rangle :=\\sum _{b\\in \\mathcal {B}^\\Omega }\\psi ^{\\prime }(Dy_b)\\cdot Dv_b,\\quad \\text{and} \\quad \\langle \\delta ^2E^\\Omega (y)v,w\\rangle :=\\sum _{b\\in \\mathcal {B}^\\Omega } \\psi ^{\\prime \\prime }(Dy_b)\\cdot Dv_bDw_b.$ In §REF , we will demonstrate that under certain conditions on $\\tilde{y}$ , $E(y;\\tilde{y})$ it may be extended by continuity in its first argument to a functional which is also well–defined for $y-\\tilde{y}\\in \\dot{{W}}^{1,2}(\\Omega )$ .",
"The following definition makes precise the various notions of equilibrium we will consider below.",
"[Stable Equilibrium] (i) A displacement $y\\in {W}(\\Omega )$ is a locally stable equilibrium if there exists $\\epsilon > 0$ such that $E^\\Omega (y + u; y) \\ge 0$ for all $u\\in {W}_0(\\Omega )$ with $\\Vert D u \\Vert _2 \\le \\epsilon $ .",
"(ii) We call a locally stable equilibrium $y$ strongly stable if, in addition, there exists $\\lambda > 0$ such that $\\langle \\delta ^2E^\\Omega (y) v, v \\rangle \\ge \\lambda \\Vert Dv\\Vert _{\\ell ^2}^2\\qquad \\forall v \\in {W}_0(\\Omega ).$ (iii) A displacement $y\\in {W}(\\Omega )$ is a globally stable equilibrium if $E^\\Omega (y + u; y) \\ge 0$ for all $u \\in {W}_0(\\Omega )$ ."
],
[
"Strong stability assumption",
"Here, we discuss the key assumption employed in proving the main results of the paper.",
"As motivation, we review a result from [18].",
"Let $\\hat{y}: \\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace \\rightarrow \\mathbb {R}$ be the dislocation solution for anti–plane linearised elasticity [17], i.e.",
"$\\hat{y}(x):={\\textstyle \\frac{1}{2\\pi }}\\arg (x)={\\textstyle \\frac{1}{2\\pi }}\\arctan \\big ({\\textstyle \\frac{x_2}{x_1}}\\big ),$ where we identify $x \\in \\mathbb {R}^2$ with the point $x_1+i x_2 \\in \\mathbb {C}$ , and the branch cut required to make this function single–valued is taken along the positive $x_1$ –axis.",
"The accepted intuition is that $\\hat{y}$ provides a good description of dislocation configurations, except in a `core' region which stores a finite amount of energy.",
"Reasonable candidates for equilibrium configurations are therefore of the form $y = \\hat{y}+ u$ where $u \\in \\dot{{W}}^{1,2}(\\Omega )$ .",
"This intuition is made precise as follows [18]: [Global stability of single dislocation $\\Lambda $ ] In addition to the foregoing assumptions, suppose that $\\psi (r) \\ge {\\textstyle \\frac{1}{2}}\\psi ^{\\prime \\prime }(0) r^2$ for $r\\in [-{\\textstyle \\frac{1}{2}},{\\textstyle \\frac{1}{2}}]$ where $\\psi ^{\\prime \\prime }(0) > 0$ , then there exists $u\\in \\dot{{W}}^{1,2}(\\Lambda )$ such that $\\hat{y}+u$ is a globally stable equilibrium of $E$ .",
"In the present work, where we focus on multiple dislocation cores, we remove the additional technical assumptions on $\\psi $ but instead directly assume the existence of a single stable core; i.e.",
"(STAB) there exists $u \\in \\dot{{W}}^{1,2}(\\Lambda )$ such that $y=\\hat{y}+u$ is a strongly stable equilibrium.",
"Throughout the rest of the paper, $u$ is fixed to satisfy (STAB).",
"We denote $\\lambda _d := \\lambda $ to be the stability constant from (REF ) with $y = \\hat{y}+u$ , and fix a finite collection of cells, $A$ , such that $\\mathcal {C}^\\pm [\\alpha ] \\subset A$ for any $\\alpha \\in [D\\hat{y}+Du]$ .",
"To demonstrate that (STAB) holds for a non-trivial class of potentials $\\psi $ satisfying our assumptions, we refer to Lemma 3.3 in [18], which states that, if $\\psi =\\psi _{\\mathrm {lin}}$ , where $\\psi _{\\mathrm {lin}}(x):={\\textstyle \\frac{1}{2}}\\lambda \\,\\mathrm {dist}(x,\\mathbb {Z})^2,$ then $\\delta ^2E > 0$ at $y=\\hat{y}+ u$ , a globally stable equilibrium.",
"(Theorem REF does not in fact require global smoothness of $\\psi $ .)",
"Furthermore, it immediately follows that $\\mathrm {dist}(Dy_b,{\\textstyle \\frac{1}{2}}+\\mathbb {Z})\\ge \\epsilon _0$ for some $\\epsilon _0>0$ .",
"Using the inverse function theorem as stated below in Lemma REF , it is fairly straightforward to see that if $\\psi \\in \\mathrm {C}^4(\\mathbb {R})$ satisfies $\\big | \\psi ^{(j)}(r) - \\psi _{\\mathrm {lin}}^{(j)}(r) \\big | \\le \\epsilon |r|^{p-j}\\text{ for } r \\in [-1/2, 1/2] \\text{ and } j = 1, 2,$ where $p > 2$ and $\\epsilon $ is sufficiently small, then there exists $w \\in \\dot{{W}}^{1,2}(\\Lambda )$ with $\\Vert Dw \\Vert _{\\ell ^2} \\le C \\epsilon $ such that $\\hat{y}+ u + w$ is a strongly stable equilibrium for $E$ .",
"Potentials constructed in this way are by no means the only possibilities — (STAB) can in fact be checked for any given potential by way of a numerical calculation, using for example the methods analysed in [7].",
"We also remark here that in §5 of [1], a demonstration of the existence of local minimisers is given under different assumptions.",
"Instead of (STAB), structural assumptions are made on the potential, which the example we provide here also satisfies.",
"Under these assumptions, Lemma 5.1 in [1] demonstrates that there exist energy barriers which dislocations must overcome in order to move from cell to cell.",
"This leads to the proof of Theorem 5.5, which includes the statement that there exist local minimisers containing dislocations in finite lattices."
],
[
"Existence Results",
"We state the existence result for stable dislocation configurations in the infinite lattice and in convex lattice polygons together.",
"To this end, we denote $S_\\mathcal {D}:= +\\infty $ for the case $\\Omega = \\Lambda $ .",
"We remark here that the main achievement of this analysis is to show the constants $L_0$ and $S_0$ depend only on the number of dislocations, the potential $\\psi $ and ${\\rm index}(\\partial W^\\Omega )$ , and not on the specific domain $\\Omega $ or its diameter.",
"Suppose that (STAB) holds and either $\\Omega = \\Lambda $ or $\\Omega $ is a convex lattice polygon.",
"(1) For each $N \\in \\mathbb {N}$ , there exist constants $L_0=L_0(N)$ and $S_0=S_0\\big ({\\rm index}(\\partial W^\\Omega ),N\\big )$ such that for any core configuration $\\mathcal {D}$ satisfying $|\\mathcal {D}| = N$ , $L_\\mathcal {D}\\ge L_0$ and $S_\\mathcal {D}\\ge S_0$ , there exists a strongly stable equilibrium $z \\in {W}(\\Omega )$ , and for any $\\alpha \\in [Dz]$ , $\\mathcal {C}^\\pm [\\alpha ] \\subset \\bigcup _{(C,s)\\in \\mathcal {D}} H^C(A) \\qquad \\text{and}\\qquad \\int _{\\partial H^C(A)} \\alpha = s,\\,\\text{ for all }(C,s)\\in \\mathcal {D}.$ In particular, $B[z] = \\sum _{(C,s)\\in \\mathcal {D}} s$ , and the conditions $L_\\mathcal {D}\\ge L_0$ and $S_\\mathcal {D}\\ge S_0$ entail that core regions $x^C + A$ do not overlap each other, or with the boundary.",
"(2) The equilibrium $z$ can be written as $z = \\sum _{(C, s) \\in \\mathcal {D}} s (\\hat{y}+ u) \\circ G^C + w,$ where $w \\in \\dot{{W}}^{1,2}(\\Omega )$ and $\\Vert Dw \\Vert _{\\ell ^2} \\le c(L_\\mathcal {D}^{-1} + S_\\mathcal {D}^{-1/2})$ , where $c$ is a constant depending only on $N$ and ${\\rm index}(\\partial W^\\Omega )$ .",
"(3) Unless $N = 1$ and $\\Omega = \\Lambda $ , $z$ cannot be a globally stable equilibrium."
],
[
"Strategy of the proof",
"In both cases, the overall strategy of proof is similar: We construct an approximate equilibrium $z$ , using the linear elasticity solution for the dislocation configuration and a truncated version of the core corrector whose existence we assumed in (STAB).",
"For given $\\mathcal {D}$ , we obtain bounds which demonstrate that $\\delta E^\\Omega (z)$ decays to zero as $L_\\mathcal {D}, S_\\mathcal {D}\\rightarrow \\infty $ , where $z$ are the approximate equilibria corresponding to $\\mathcal {D}$ .",
"We show that $\\delta ^2E^\\Omega (z) \\ge \\lambda _d-\\epsilon $ as $L_\\mathcal {D}$ , $S_\\mathcal {D}\\rightarrow \\infty $ .",
"We apply the Inverse Function Theorem to demonstrate the existence of a corrector $w \\in \\dot{{W}}^{1,2}(\\Omega )$ such that $z+w$ is a strongly stable equilibrium.",
"Since $\\Vert Dw \\Vert _{\\ell ^2}$ can be made arbitrarily small by making more stringent requirements on $\\mathcal {D}$ , we can demonstrate that the condition on the core position holds, which completes the proof of parts (1) and (2) of the statement.",
"Part (3) of the statement is proved by construction of explicit counterexamples.",
"The reduced rate $S_\\mathcal {D}^{-1/2}$ (as opposed to $L_\\mathcal {D}^{-1}$ ) with respect to separation from the boundary is due to surface stresses which are not captured by the standard linear elasticity theory that we use to construct the predictor $z$ .",
"By formulating a half-plane problem where $\\partial W^\\Omega $ is not identical to $\\partial U^\\Omega $ , one may readily check that the bound is in general sharp.",
"If ${\\rm index}(\\partial W^\\Omega ) = 1$ , then it may be possible to prove that the rate should be bounded by $S_\\mathcal {D}^{-1}$ .",
"Otherwise, it is necessary to add an additional boundary correction to the predictor which captures these surface stresses.",
"All of these routes seem to require improved regularity estimates of the boundary corrector $\\bar{y}$ that we construct in (REF )."
],
[
"Possible extensions",
"We have avoided the most difficult aspect of the analysis of dislocations by imposing the strong stability assumption (STAB) for a single core.",
"Once this is established (or assumed), several extensions of our analysis become possible, which we disuss in the following paragraphs.",
"Symmetry of $\\psi $ : An immediate extension is to drop the requirement that $\\psi $ is even about 0, which would be the case if the body was undergoing macroscopic shear.",
"This extension would require us to separately assume the existence of strongly stable positive and negative dislocation cores in the full lattice, as they would no longer necessarily be symmetric.",
"Apart from the introduction of logarithmic factors into some of the bounds we obtain, it appears that the analysis would be analogous to that contained here.",
"General domains: It is straightforward to generalise the analysis carried out in § to `half–plane' lattices, since the linear elastic corrector $\\bar{y}$ may be explicitly constructed via a reflection principle.",
"This suggests that in fact the analysis could be extended to hold in any convex domain with a finite number of corners $\\kappa _m\\in \\Omega $ and tangent vectors $\\tau _m$ which are lattice directions — in effect, an `infinite' polygon.",
"The key technical ingredient required here would be to prove decay results for the corrector problem analysed in §REF in such domains, which we were not able to find in the literature.",
"It is unclear to us, though, to what extent extensions to non-convex domains are feasible.",
"Interaction potential: The assumption that interactions are governed by nearest–neighbour pair potentials only is easily lifted as well.",
"A generalisation to many–body interactions with a finite range beyond nearest neighbours is conceptually straightforward (though would add some technical, and in particular notational difficulties), as long as suitable symmetry assumptions are placed on the many-body site potential.",
"Note, in particular, that the crucial decay estimates from [7] that we employ are still valid in this case.",
"In-plane models: Generalisations to in–plane models seem to be relatively straightforward only in the infinite-lattice case.",
"In the finite domain case, one would need to account for surface relaxation effects, which we have entirely avoided here by choosing an anti–plane model (however, see Remark REF .",
"The phenomenon of surface relaxation in discrete problems seems a difficult one, and to the authors' knowledge, has yet to be addressed systematically in the Applied Analysis literature, but for some results in this direction, see [27].",
"A possible way forward would be to impose an additional stability condition on the boundary, similar to our condition (STAB), which could then be investigated separately."
],
[
"Extension of the energy difference functional",
"The following is a slight variation of [18].",
"Let $y\\in {W}(\\Omega )$ , and suppose that $\\delta E^\\Omega (y)$ is a bounded linear functional.",
"Then $u\\mapsto E^\\Omega (y+u;y)$ is continuous as a map from ${W}_0(\\Omega )$ to $\\mathbb {R}$ with respect to the norm $\\Vert D\\cdot \\Vert _2$ ; hence there exists a unique continuous extension of $u \\mapsto E^\\Omega (y+u;y)$ to a map defined on $\\dot{{W}}^{1,2}(\\Omega )$ .",
"The extended functional $u \\mapsto E^\\Omega (y+u; y)$ , $u \\in \\dot{{W}}^{1,2}(\\Omega )$ is three time continuously Frechet differentiable.",
"The proof of this statement is almost identical to the proof of [18] and hence we omit it.",
"We note that in a finite domain, the condition that $\\delta E^\\Omega (y)$ is a bounded linear functional is always satisfied, since $\\dot{{W}}^{1,2}(\\Omega )$ is a finite dimensional space."
],
[
"Stability of the homogeneous lattice",
"The following lemma demonstrates that $y=0$ is a globally stable as well as strongly stable equilibrium.",
"In particular, this shows that $\\hat{y}+u$ cannot be a unique stable equilibrium among all $y\\in {W}(\\Lambda )$ .",
"Suppose that ${\\bf (STAB)}$ holds, then the deformation $y \\equiv 0$ is a strongly stable equilibrium for any $\\Omega \\subset \\Lambda $ .",
"Precisely, $\\langle \\delta ^2E^\\Omega (0)v,v\\rangle = \\psi ^{\\prime \\prime }(0) \\sum _{b \\in \\mathcal {B}^\\Omega }Dv_b^2 \\qquad \\text{and } \\quad \\psi ^{\\prime \\prime }(0) \\ge \\lambda _d.$ Suppose that $v\\in {W}_0(\\Omega )$ and $C^i\\in \\mathcal {C}$ is a sequence such that $\\mathrm {dist}(C^i,0)\\rightarrow \\infty $ as $i\\rightarrow \\infty $ .",
"Define $v^i:=v\\circ G^{C^i}$ ; if $y=\\hat{y}+u$ , $\\lambda _d\\Vert Dv\\Vert _2^2=\\lambda _d\\Vert Dv^i\\Vert _2^2\\le \\langle \\delta ^2E^\\Lambda (y)v^i,v^i\\rangle =\\sum _{b\\in \\mathcal {B}} \\psi ^{\\prime \\prime }(Dy_b)\\big (Dv^i_b\\big )^2=\\sum _{b\\in \\mathcal {B}}\\psi ^{\\prime \\prime }\\big (D(y\\circ H^{C^i})_b\\big )Dv^2_b,$ and since $\\mathrm {dist}(Dy_b,\\mathbb {Z})\\rightarrow 0$ as $\\mathrm {dist}(b,0)\\rightarrow \\infty $ , it follows that $0<\\lambda _d\\le \\psi ^{\\prime \\prime }(0)$ .",
"Since $\\psi $ is even about 0 it must be that $\\psi ^{\\prime }(0)=0$ , and the statement follows trivially."
],
[
"The linear elasticity residual",
"We now prove a result estimating the residual of the pure linear elasticity predictor.",
"Let $\\mathcal {D}$ be a dislocation configuration in $\\Lambda $ and $z := \\sum _{(C,s) \\in \\mathcal {D}} \\hat{y}\\circ H^{C}$ .",
"For $L_\\mathcal {D}$ sufficiently large, there exists $g : \\mathcal {B}\\rightarrow \\mathbb {R}$ such that $\\langle \\delta E(z), v \\rangle = \\sum _{b \\in \\mathcal {B}} g_b Dv_b\\quad \\text{and} \\quad |g_b| \\le c \\sum _{(C,s) \\in \\mathcal {D}} \\mathrm {dist}(b, C)^{-3}.$ The canonical form for $\\delta E(z)$ is $\\langle \\delta E(z), v \\rangle = \\sum \\psi ^{\\prime }(\\alpha _b) Dv_b$ , where $\\alpha $ is a bond length one-form associated with $Dz$ .",
"For $L_\\mathcal {D}$ sufficiently large, arguing as in [18] we obtain that $\\alpha _b \\in (-1/2,1/2)$ , which entails that $\\alpha \\in [Dz]$ is unique, and may be written in the form $\\alpha _{(\\xi ,\\xi +a_i)} = \\int _0^1 \\nabla _{a_i} z(\\xi +t a_i)\\,{\\rm d}t$ , where here and below $\\nabla z$ will mean the extension of the gradient of $z$ to a function in $\\mathrm {C}^\\infty \\big (\\mathbb {R}^2\\setminus \\bigcup _{(C,s)\\in \\mathcal {D}}\\lbrace x^C\\rbrace ;\\mathbb {R}^2\\big )$ .",
"We note that $|\\psi ^{\\prime }(\\alpha _b)| \\lesssim \\sum \\mathrm {dist}(b,C)^{-1}$ only, so we must remove a “divergence-free component”.",
"To that end, let $\\omega _b := \\bigcup \\lbrace C^{\\prime } \\in \\mathcal {C}\\,|\\,\\pm b \\in \\partial C^{\\prime }, C^{\\prime }\\text{ positively--oriented}\\rbrace $ and let $V := |\\omega _b|$ for some arbitrary $b \\in \\mathcal {B}$ .",
"Further, let $\\bar{C}_\\epsilon := \\bigcup _{(C,s) \\in \\mathcal {D}} B_\\epsilon (x^C)$ .",
"Then, for $b = (\\xi , \\xi +a_i)$ , we define $h_b := \\frac{\\psi ^{\\prime \\prime }(0)}{V} \\lim _{\\epsilon \\rightarrow 0}\\int _{\\omega _b \\setminus \\bar{C}_\\epsilon } \\nabla z \\cdot a_i \\,{\\rm d}x\\qquad \\text{and} \\qquad g_b := \\psi ^{\\prime }(\\alpha _b) - h_b.$ It is fairly straightforward to show that the limit exists by applying the divergence theorem, which entails that $h_b$ and $g_b$ are well–defined and $\\sum _{b \\in \\mathcal {B}} h_b Dv_b =\\lim _{\\epsilon \\rightarrow 0} \\frac{\\psi ^{\\prime \\prime }(0)}{V} \\int _{\\mathbb {R}^2 \\setminus \\bar{C}_\\epsilon }\\nabla z \\cdot \\nabla Iv \\,{\\rm d}x= 0$ for all $v \\in {W}_0(\\Lambda )$ , where $Iv$ denotes the continuous and piecewise affine interpolant of $v$ .",
"Thus, we obtain that $\\langle \\delta E(z), v \\rangle = \\sum _{b \\in \\mathcal {B}} g_b Dv_b$ as desired.",
"It remains to prove the estimate on $g_b$ .",
"Taylor expanding, we obtain $\\psi ^{\\prime }(\\alpha _b)-h_b = \\psi ^{\\prime }(0)+\\psi ^{\\prime \\prime }(0)\\bigg (\\alpha _b-\\frac{1}{V}\\lim _{\\epsilon \\rightarrow 0}\\int _{\\omega _b \\setminus \\bar{C}_\\epsilon }\\nabla z \\cdot a_i \\,{\\rm d}x\\bigg )+{\\textstyle \\frac{1}{2}}\\psi ^{\\prime \\prime \\prime }(0)|\\alpha _b|^2+O\\big (|\\alpha _b|^3\\big ).$ The first and third terms vanish since $\\psi $ is even.",
"Note that $\\alpha _b=\\frac{1}{V}\\int _{\\omega _b} \\nabla z\\cdot a_i\\,{\\rm d}x$ , where $a_i$ is the direction of the bond $b$ , so Taylor expanding about the midpoint of $b$ and using the symmetry of $b$ and $\\omega _b$ to eliminate the term involving $\\nabla ^2z$ , we obtain $\\int _b\\nabla z\\cdot a_i\\,{\\rm d}x-\\frac{1}{V}\\lim _{\\epsilon \\rightarrow 0}\\int _{\\omega _b \\setminus \\bar{C}_\\epsilon }\\nabla z \\cdot a_i \\,{\\rm d}x= O\\big (|\\nabla ^3z|\\big ).$ Finally, as $|\\alpha _b|\\lesssim \\mathrm {dist}(b,C)^{-1}$ and $|\\nabla ^3 z|\\lesssim \\mathrm {dist}(b,C)^{-3}$ for all $(C,s)\\in \\mathcal {D}$ , the stated estimate follows."
],
[
"Regularity of the corrector",
"We now slightly refine the general regularity result of Theorem 3.1 in [7], exploiting the evenness of the potential $\\psi $ .",
"Let $u$ be the core corrector whose existence postulated in (STAB): then there exists a constant $C_{\\rm reg}$ such that $|Du_b| \\le C_{\\rm reg}\\,\\mathrm {dist}(b,C)^{-2}\\qquad \\text{ for all } b \\in \\mathcal {B}\\text{ and }(C,s)\\in \\mathcal {D}.$ Our setting satisfies all assumptions of the $d = 2, m = 1$ (anti–plane) case described in Section 2.1 of [7] with $\\mathcal {N}_\\xi = \\lbrace a_i \\,|\\,i = 1, \\dots , 6 \\rbrace $ for all $\\xi \\in \\Lambda $ , and the complete set of assumptions summarized in (pD) in Section 2.4.5 of [7].",
"Using Lemma REF , we may apply Lemma 3.4 [7] with $p=3$ , implying $|Du_b| \\lesssim \\mathrm {dist}(b,C)^{-2}$ ."
],
[
"Approximation by truncation",
"Following [7] we define a family of truncation operators $\\Pi ^C_R$ , which we will apply to $u\\in \\dot{{W}}^{1,2}(\\Omega )$ .",
"Let $\\eta \\in \\mathrm {C}^1(\\mathbb {R}^2)$ be a cut off function which satisfies $\\eta (x):=\\left\\lbrace \\begin{array}{rl} 1, & |x|\\le {\\textstyle \\frac{3}{4}},\\\\0, & |x|\\ge 1.",
"\\end{array} \\right.$ Let $Iu$ be the piecewise affine interpolant of $u$ over the triangulation given by $\\mathcal {T}_\\Lambda =\\mathcal {C}$ .",
"For $R > 2$ let $A_R:=B_R\\setminus B_{R/2+1}$ , an annulus over which $\\eta (x/R)$ is not constant.",
"Define $\\Pi ^C_R:\\dot{{W}}^{1,2}\\rightarrow {W}_0$ by $\\Pi ^C_Ru(\\xi ):=\\eta \\Big ({\\textstyle \\frac{\\xi -x^C}{R}}\\Big )\\big (u(\\xi )-a^C_R\\big ),\\qquad \\text{where}\\qquad a^C_R:=\\mathchoice{{\\displaystyle {\\textstyle -}{\\int }}\\hbox{$\\textstyle -$}}{\\hspace{0.0pt}}{-}{.",
"}5$ xC+AR Iu(x) dx.",
"In addition, we define $\\Pi _R := \\Pi _R^{C_0}$ .",
"We now state the following result concerning the approximation property of the family of truncation operators $\\Pi ^C_R$ , which follows from results in [7].",
"Let $v \\in \\dot{{W}}^{1,2}(\\Lambda )$ and $C \\in \\mathcal {C}$ , then $\\big \\Vert D \\Pi _R^C v - D v \\big \\Vert _{\\ell ^2(\\mathcal {B})}\\le \\gamma _1 \\Vert Dv\\Vert _{\\ell ^2(\\mathcal {B}\\setminus B_{R/2}(x^C))}.$ where $\\gamma _1$ is independent of $R, v$ and $C$ .",
"In particular, if $u \\in \\dot{{W}}^{1,2}(\\Lambda )$ is the core corrector from ${\\bf (STAB)}$ , then $\\big \\Vert D\\Pi ^C_R (u \\circ G^C) - D (u \\circ G^C) \\big \\Vert _{\\ell ^2(\\mathcal {B})}\\le \\gamma _2 R^{-1},$ where $\\gamma _2$ is independent of $R$ and $C$ .",
"Since $\\Vert \\cdot \\Vert _{\\ell ^2(\\mathcal {B})}$ is invariant under composition of functions with lattice automorphisms we can assume, without loss of generality, that $C = C_0$ .",
"The estimate (REF ) then is simply a restatement of [7].",
"The second estimate (REF ) then follows immediately from Lemma REF .",
"Next, we show that the assumption (STAB) implies that that $\\delta ^2E^\\Lambda (\\hat{y}+\\Pi _Ru)$ is positive for sufficiently large $R$ .",
"There exist constants $\\lambda _{d,R}$ such that $\\langle \\delta ^2E^\\Lambda \\big (\\hat{y}+\\Pi _Ru\\big )v,v\\rangle \\ge \\lambda _{d,R}\\Vert Dv\\Vert _2^2\\qquad \\text{for all}\\quad v\\in {W}_0,$ and $\\lambda _{d,R}\\rightarrow \\lambda _d> 0$ as $R\\rightarrow \\infty $ .",
"Noting that $\\Vert Dv\\Vert _\\infty \\le \\Vert Dv\\Vert _2$ for any $v\\in \\dot{{W}}^{1,2}(\\Lambda )$ , $\\langle \\delta ^2 E^\\Lambda (\\hat{y}+\\Pi _Ru)v,v\\rangle &= \\big \\langle [\\delta ^2 E^\\Lambda (\\hat{y}+\\Pi _Ru)-\\delta ^2 E^\\Lambda (\\hat{y}+u)]v,v\\big \\rangle +\\big \\langle \\delta ^2 E^\\Lambda (\\hat{y}+u)v,v\\big \\rangle ,\\\\&\\ge \\big (\\lambda _d-\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty \\big \\Vert D\\Pi _Ru-Du\\big \\Vert _\\infty \\big ) \\Vert Dv\\Vert _2^2,\\\\&\\ge \\big (\\lambda _d-\\epsilon _R\\big ) \\Vert Dv\\Vert _2^2,$ where $\\epsilon _R\\lesssim R^{-1}$ as $R\\rightarrow \\infty $ by Lemma REF ."
],
[
"Inverse Function Theorem",
"We review a quantitative version of the inverse function theorem, adapted from [19].",
"Let $X,Y$ be Hilbert spaces, $w\\in X$ , $F\\in C^2(B_R^X(w);Y)$ with Lipschitz continuous Hessian, $\\Vert \\delta ^2F(x)-\\delta ^2F(y)\\Vert _{L(X,Y)}\\le M\\Vert x-y\\Vert _X$ for any $x,y\\in B_R^X(w)$ .",
"Furthermore, suppose that there exist constants $\\mu ,r>0$ such that $\\langle \\delta ^2F(w)v,v\\rangle \\ge \\mu \\Vert v\\Vert _X^2,\\quad \\Vert \\delta F(w)\\Vert _Y\\le r,\\quad \\text{and}\\quad {\\textstyle \\frac{2Mr}{\\mu ^2}}<1,$ then there exists a locally unique $\\bar{w}\\in B_R^X(w)$ such that $\\delta F(\\bar{w})=0$ , $\\Vert w-\\bar{w}\\Vert _X\\le \\frac{2r}{\\mu }$ and $\\langle \\delta ^2F(\\bar{w})v,v\\rangle \\ge \\big (1-{\\textstyle \\frac{2Mr}{\\mu ^2}}\\big )\\mu \\Vert v\\Vert _X^2.$"
],
[
"Proof for the Infinite Lattice",
"Before considering the case of finite lattice domains, we first set out to prove Theorem REF in the case when $\\Omega =\\Lambda $ .",
"In this case we are able to give a substantially simplified argument that concerns only the interaction between dislocations rather than the additional difficulty of the interaction of dislocations with the boundary which is present in the finite domain case."
],
[
"Analysis of the predictor",
"Suppose that $\\mathcal {D}$ is a dislocation configuration in $\\Lambda $ : we define an approximate solution (predictor) with truncation radius $R$ to be $z := \\sum _{(C,s)\\in \\mathcal {D}} s\\,\\big (\\hat{y}+ \\Pi _R u\\big ) \\circ G^C.$ The following lemma provides an estimate on the residual of such approximate solutions in terms of $L_\\mathcal {D}$ .",
"Suppose $z$ is the approximate solution for a dislocation configuration $\\mathcal {D}$ in $\\Lambda $ as defined in (REF ) with truncation radius $R = L_\\mathcal {D}/5$ .",
"Then there exists $L_0 =L_0(N)$ and a constant $c = c(N)$ , such that, whenever $L_\\mathcal {D}> L_0$ , $\\big \\Vert \\delta E(z) \\big \\Vert _{\\dot{{W}}^{1,2}(\\Lambda )^*} \\le c L_\\mathcal {D}^{-1}.$ Enumerate the elements of $\\mathcal {D}$ as $(C^i,s^i)$ where $i=1,\\ldots ,N$ .",
"Setting $G^i:=G^{C^i}$ , let $y^i := (\\hat{y}+ \\Pi _R u) \\circ G^i$ and $\\hat{y}^i := \\hat{y}\\circ G^i$ .",
"Let $r := 2 (R+1) = 2 (L_\\mathcal {D}/5+1)$ and $v$ be any test function in $\\dot{{W}}^{1,2}(\\Lambda )$ .",
"Define $v^i:=\\Pi ^{C^i}_{r} \\!",
"v\\quad \\text{for }i=1,\\ldots ,N,\\qquad \\text{and} \\qquad v^0 := v -\\sum _{i = 1}^N v^i.$ Lemma REF implies that $\\Vert Dv^i\\Vert _{2} \\lesssim \\Vert Dv\\Vert _2$ for $i = 0, \\dots , N$ .",
"Assumption (STAB) implies $\\delta E(\\hat{y}+u) = 0$ , so we may decompose the residual into $\\langle \\delta E(z),v\\rangle &= \\sum _{i = 0}^N \\langle \\delta E(z),v^i\\rangle ,\\\\&= \\big \\langle \\delta E(z),v^0 \\big \\rangle +\\sum _{i\\ne 0}\\big \\langle \\delta E(z)-\\delta E(y^i),v^i \\big \\rangle \\\\&\\qquad \\qquad +\\sum _{i\\ne 0}\\big \\langle \\delta E(y^i)-\\delta E\\big (\\hat{y}^i+u\\circ G^i\\big ),v^i \\big \\rangle \\\\&=: {\\rm T}_1 + {\\rm T}_2 + {\\rm T}_3.$ The term ${\\rm T}_1$ : Employing Lemma REF , and using the fact that $z = \\sum _{i =1}^N \\hat{y}\\circ G^i$ in ${\\rm supp}(v^0)$ we obtain that $\\big |{\\rm T}_1\\big | &= \\big |\\langle \\delta E(z), v^0 \\rangle \\big | \\le \\sum _{b \\in \\mathcal {B}} |g_b| |Dv^0_b|\\lesssim \\sum _{b \\in \\mathcal {B}} \\sum _{i = 1}^N \\mathrm {dist}(b, C^i)^{-3}|Dv^0_b| \\\\&\\lesssim \\sum _{i = 1}^N \\bigg ( \\sum _{\\begin{array}{c}b \\in \\mathcal {B}\\\\\\mathrm {dist}(b, C^i) \\ge r/2-1\\end{array}}\\mathrm {dist}(b, C^i)^{-6} \\bigg )^{1/2}\\Vert Dv^0\\Vert _2 \\lesssim r^{-2} \\Vert D v\\Vert _2.$ The term ${\\rm T}_2$ : Here, we have $z- y^i = \\sum _{j \\ne i}\\hat{y}^j$ in the support of $v^i$ .",
"We expand $\\big \\langle \\delta E(z)-\\delta E\\big (y^i\\big ),v^i\\big \\rangle &=\\sum _{b \\in \\mathcal {B}} \\psi ^{\\prime \\prime }(s_b) \\sum _{j \\ne i} D \\hat{y}^j_b Dv^i_b \\\\&= \\psi ^{\\prime \\prime }(0) \\sum _{j \\ne i} \\sum _{b \\in \\mathcal {B}} D \\hat{y}^j_b Dv^i_b+ \\sum _{b \\in \\mathcal {B}} h_b Dv^i_b,$ where $|s_b| \\lesssim (1+\\mathrm {dist}(b, C^i))^{-1}$ and $|h_b| = \\Big |\\big (\\psi ^{\\prime \\prime }(s_b) - \\psi ^{\\prime \\prime }(0)\\big ) \\sum _{j \\ne i} D\\hat{y}^j_b\\Big | \\lesssim (1+\\mathrm {dist}(b, C^i))^{-2} L_\\mathcal {D}^{-1}.$ We have Taylor expanded and used the evenness of $\\psi $ to arrive at the estimate on the right.",
"The first group of terms in (REF ) can be estimated as in (REF ) to obtain $|\\sum _{b \\in \\mathcal {B}}D \\hat{y}^j_b Dv^i_b| \\lesssim L_\\mathcal {D}^{-2}\\Vert Dv\\Vert _2 $ for all $j \\ne i$ .",
"For the second group in (REF ), we have $\\Big |\\sum _{b \\in \\mathcal {B}} h_b Dv^i_b\\Big | \\lesssim L_\\mathcal {D}^{-1}\\bigg (\\sum _{\\begin{array}{c}b\\in \\mathcal {B}\\\\ \\mathrm {dist}(b, C^i) \\le r+1\\end{array}}(1+\\mathrm {dist}(b, C^i))^{-4} \\bigg )^{1/2} \\Vert D v^i \\Vert _2\\lesssim L_\\mathcal {D}^{-1}\\Vert Dv \\Vert _2.$ The term ${\\rm T}_3$ : The final group in (REF ) is straightforward to estimate using the truncation result of Lemma REF , giving $\\big |\\big \\langle \\delta E(y^i)-\\delta E(\\hat{y}^i+u\\circ G^i),v^i\\big \\rangle \\big | \\le \\Vert \\psi ^{\\prime \\prime }\\Vert _\\infty \\Vert D\\Pi _Ru - Du\\Vert _2 \\Vert Dv^i\\Vert _2 \\lesssim R^{-1}\\Vert Dv \\Vert _2.$ Conclusion: Inserting the estimates for ${\\rm T}_1, {\\rm T}_2, {\\rm T}_3$ into (REF ) we obtain $\\big |\\langle \\delta E(z), v\\rangle \\big | \\lesssim \\Big (r^{-2} + L_\\mathcal {D}^{-1} + R^{-1}\\Big )\\Vert Dv\\Vert _2 \\lesssim L_\\mathcal {D}^{-1} \\Vert Dv\\Vert _2.$"
],
[
"Stability of the predictor",
"We proceed to prove that $\\delta ^2E(y)$ is positive, where $y$ is the predictor constructed in (REF ).",
"This result employs ideas similar to those used in the proof of [7], modified here to an aperiodic setting and extended to cover the case of multiple defect cores.",
"Let $z$ be a predictor for a dislocation configuration $\\mathcal {D}$ in $\\Lambda $ , as defined in (REF ), where $|\\mathcal {D}|=N$ .",
"Then there exist postive constants $R_0 = R_0(N)$ and $L_0 = L_0(N)$ such that if $R\\ge R_0$ and $L_\\mathcal {D}\\ge L_0$ , there exists $\\lambda _{L,R}\\ge \\lambda _d / 2$ so that $\\langle \\delta ^2 E(z)v,v\\rangle \\ge \\lambda _{L,R}\\Vert Dv\\Vert _2^2 \\qquad \\text{for all } v \\in \\dot{{W}}^{1,2}(\\Lambda ).$ Lemma REF implies the existence of $R_0$ such that $\\lambda _{d,R}\\ge 3\\lambda _d/4>0$ for all $R\\ge R_0$ , thus we choose a truncation radius $R\\ge R_0$ which will remain fixed for the rest of the proof.",
"We now argue by contradiction.",
"Suppose that there exists no $L_0$ satisfying the statement; it follows that there exists $\\mathcal {D}^n$ , a sequence of dislocation configurations such that $N:=|\\mathcal {D}^n|$ , $\\big |\\big \\lbrace (C,+1)\\in \\mathcal {D}^n\\big \\rbrace \\big |$ and $\\big |\\big \\lbrace (C,-1)\\in \\mathcal {D}^n\\big \\rbrace \\big |$ are constant, $L^n:=L_{\\mathcal {D}^n}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ and $\\delta ^2E(z^n)<\\lambda _d/2$ for all $n$ , where $z^n$ is the approximate solution corresponding to the configuration $\\mathcal {D}^n$ in $\\Lambda $ with truncation radius $R$ , as defined in (REF ).",
"The first condition may be assumed without loss of generality by taking subsequences.",
"We enumerate the elements $(C^{n,i},s^{n,i})$ of $\\mathcal {D}^n$ , and write $G^{n,i}:=G^{C^{n,i}}$ and $H^{n,i}:=H^{C^{n,i}}$ .",
"By translation invariance and the fact that $\\psi $ is even, we may assume without further loss of generality that $(C^{n,1},s^{n,1})=(C_0,+1)$ .",
"For each $n$ , $\\lambda _n:=\\inf _{\\begin{array}{c}v\\in \\dot{{W}}^{1,2}(\\Lambda )\\\\\\Vert Dv\\Vert _2 =1\\end{array}}\\langle \\delta ^2E(z^n)v,v\\rangle <\\lambda _d/2$ exists since, for any $z\\in {W}(\\Lambda )$ , $\\delta ^2E(z)$ is a bounded bilinear form on $\\dot{{W}}^{1,2}(\\Lambda )$ .",
"Let $v^n\\in {W}_0(\\Lambda )$ be a sequence of test functions such that $\\Vert Dv^n\\Vert _2=1$ and $\\lambda _n\\le \\langle \\delta ^2E(z^n)v^n,v^n\\rangle \\le \\lambda _n + n^{-1}.$ Since $v^n$ is bounded in $\\dot{{W}}^{1,2}(\\Lambda )$ , it has a weakly convergent subsequence.",
"By the translation invariance of the norm and taking further subsequences without relabelling, we further assume that $\\bar{v}^{n,i}:=v^n\\circ H^{n,i}$ weakly converges for each $i$ .",
"We now employ the result of [7].",
"This states that there exists a sequence of radii, $r^n\\rightarrow \\infty $ , for which we may also assume $r^n\\le L^n/3$ , so that for each $i=1,\\ldots ,N$ , $w^{n,i}:=\\Pi ^{C^{n,i}}_{r^n}v^n\\quad \\text{satisfies}\\quad w^{n,i}\\circ H^{n,i}\\rightarrow \\bar{w}^i\\quad \\text{and}\\quad (v^n-w^{n,i})\\circ H^{n,i}\\rightharpoonup 0\\quad \\text{in }\\dot{{W}}^{1,2}(\\Lambda ).$ Writing $\\bar{w}^{n,i} := w^{n,i}\\circ H^{n,i}$ , and defining $w^{n,0}:=v^n-\\sum _{i=1}^Nw^{n,i}$ , it follows that $\\langle \\delta ^2E(z^n)v^n,v^n\\rangle = \\sum _{i,j=0}^N\\langle \\delta ^2E(z^n)w^{n,i},w^{n,j}\\rangle =\\sum _{i=0}^N\\langle \\delta ^2E(z^n)w^{n,i},w^{n,i}\\rangle +2\\sum _{i=1}^N\\langle \\delta ^2E(z^n)w^{n,0},w^{n,i}\\rangle ,$ where, by choosing $r^n\\le L^n/3$ , we have ensured that ${\\rm supp}\\lbrace w^{n,i}\\rbrace $ for $i=1,\\ldots ,N$ only overlaps with ${\\rm supp}\\lbrace w^{n,0}\\rbrace $ , and hence all other `cross–terms' vanish.",
"For $i=1,\\ldots ,N$ , $\\langle \\delta ^2E(z^n)w^{n,i},w^{n,i}\\rangle &=\\langle [\\delta ^2E(z^n\\circ H^{n,i})-\\delta ^2E(\\hat{y}+\\Pi _Ru)]\\bar{w}^{n,i},\\bar{w}^{n,i}\\rangle +\\langle \\delta ^2E(\\hat{y}+\\Pi _Ru)\\bar{w}^{n,i},\\bar{w}^{n,i}\\rangle ,\\\\&\\ge \\Big (\\lambda _{d,R}-{\\textstyle \\frac{N\\,\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty }{2L^n/3}}\\Big )\\Vert Dw^{n,i}\\Vert _2^2.$ For the $i=0$ term, we have $\\langle \\delta ^2E(z^n)w^{n,0},w^{n,0}\\rangle &=\\langle [\\delta ^2E(z^n)-\\delta ^2E(0)]w^{n,0},w^{n,0}\\rangle +\\langle \\delta ^2E(0)w^{n,0},w^{n,0}\\rangle ,\\\\&\\ge \\Big (\\psi ^{\\prime \\prime }(0)-{\\textstyle \\frac{N\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty }{r^n}}\\Big )\\Vert Dw^{n,0}\\Vert _2^2.$ For the cross–terms, Since we assumed that $r^n\\le L^n/3$ , we deduce that $\\langle \\delta ^2E(z^n)w^{n,0},w^{n,i}\\rangle =\\langle \\delta ^2E(z^n)(v^n-w^{n,i}),w^{n,i}\\rangle .$ Using the translation invariance of $E$ , and adding and subtracting terms, we therefore write $\\big \\langle \\delta ^2E(z^n)w^{n,0},w^{n,i}\\big \\rangle &=\\big \\langle \\big [\\delta ^2E(z^n\\circ H^{n,i})-\\delta ^2E(\\hat{y}+\\Pi _Ru)\\big ](\\bar{v}^{n,i}-\\bar{w}^{n,i}),\\bar{w}^{n,i}\\big \\rangle \\\\&\\qquad +\\big \\langle \\delta ^2E(\\hat{y}+\\Pi _Ru)(\\bar{v}^{n,i}-\\bar{w}^{n,i}),\\bar{w}^{n,i}-\\bar{w}^i\\big \\rangle \\\\&\\qquad \\qquad +\\big \\langle \\delta ^2E(\\hat{y}+\\Pi _Ru)(\\bar{v}^{n,i}-\\bar{w}^{n,i}),\\bar{w}^i\\big \\rangle ,\\\\&=:{\\rm T}_1+{\\rm T}_2+{\\rm T}_3.$ Estimating the first two terms on the right hand side, we obtain: ${\\rm T}_1\\le {\\textstyle \\frac{N\\,\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty }{2L^n/3}}\\Big (\\Vert Dv^{n,i}\\Vert _2+\\Vert Dw^{n,i}\\Vert _2\\Big )\\Vert Dw^{n,i}\\Vert _2 \\le {\\textstyle \\frac{N\\,\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty }{L^n/3}}\\qquad \\text{and}\\\\{\\rm T}_2\\le \\Vert \\psi ^{\\prime \\prime }\\Vert _\\infty \\Big (\\Vert Dv^{n,i}\\Vert _2+\\Vert Dw^{n,i}\\Vert _2\\Big )\\Vert D\\bar{w}^{n,i}-D\\bar{w}^i \\Vert _2,$ both of which converge to 0 as $n\\rightarrow \\infty $ .",
"Since $\\bar{v}^{n,i}-\\bar{w}^{n,i}\\rightharpoonup 0$ as $n\\rightarrow \\infty $ , it follows that ${\\rm T}_3\\rightarrow 0$ as well, and hence $\\langle \\delta ^2E(z^n)w^{n,0},w^{n,i}\\rangle \\rightarrow 0$ as $n\\rightarrow \\infty $ for each $i$ .",
"Putting (REF ) and the result of Lemma REF together with (REF ) and (REF ), $\\langle \\delta ^2E(z^n)v^n,v^n\\rangle \\ge (\\lambda _{d,R}-\\epsilon ^n)\\sum _i\\Vert Dw^{n,i}\\Vert _2^2 + \\epsilon ^n,$ where $\\epsilon ^n\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"All that remains is to verify that $\\liminf _{n\\rightarrow \\infty } \\Big (\\sum _i\\Vert Dw^{n,i}\\Vert _2^2-\\Vert Dv^n\\Vert _2^2\\Big )\\ge 0.$ By definition, $\\sum _i |Dw^{n,i}_b|^2 \\ne |Dv^n_b|^2$ only when $b\\in {\\rm supp}\\lbrace Dw^{n,i}\\rbrace \\cap {\\rm supp}\\lbrace Dw^{n,0}\\rbrace $ for some $i=1,\\ldots ,N$ .",
"In such cases, $|Dw^{n,0}_b|^2+|Dw^{n,i}_b|^2-|Dv^n_b|^2 = -2\\,Dw^{n,0}_bDw^{n,i}_b.$ Therefore, consider $\\delta ^{n,i}&:=\\sum _{b\\in \\mathcal {B}}Dw^{n,0}_bDw^{n,i}_b=\\sum _{b\\in \\mathcal {B}}\\big (Dw^{n,0}\\circ H^{n,i}\\big )_b\\big (D\\bar{w}^{n,i}_b-D\\bar{w}^i_b\\big )+\\sum _{b\\in \\mathcal {B}}\\big (Dw^{n,0}\\circ H^{n,i}\\big )_bD\\bar{w}^i_b.$ Since $w^{n,0}\\circ H^{n,i}\\rightharpoonup 0$ and $\\bar{w}^{n,i}\\rightarrow \\bar{w}^i$ , it follows that $\\delta ^{n,i}\\rightarrow 0$ , and thus (REF ) holds.",
"Further, $\\lambda _n + n^{-1} \\ge \\langle \\delta ^2E(z^n)v^n,v^n\\rangle \\ge (\\lambda _{d,R}-\\epsilon ^n)\\Big (1-\\sum _i\\delta ^{n,i}\\Big )+\\epsilon ^n,$ and so for $n$ sufficiently large, it is clear that $\\lambda _n\\ge 2\\lambda _{d,R}/3\\ge \\lambda _d/2>0$ , which contradicts the assumption that $\\lambda _n<\\lambda _d/2$ for all $n$ ."
],
[
"Proof of (2)",
"Lemma REF and Lemma REF now enable us to state that there exist $L_0$ and $R_0$ depending only on $N=|\\mathcal {D}|$ such that whenever $\\mathcal {D}$ satisfies $L_\\mathcal {D}\\ge L_0$ , $R\\ge R_0$ , and $z$ is an approximate solution corresponding to $\\mathcal {D}$ with truncation radius $R$ , $\\lambda _{L,R}\\ge \\mu :=\\frac{\\lambda _d}{2}>0,\\qquad \\text{and}\\qquad \\Vert \\delta E(z)\\Vert < r:=\\min \\bigg \\lbrace \\frac{c\\lambda _d}{4L_\\mathcal {D}},\\frac{\\lambda _d^2}{16\\,\\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty }\\bigg \\rbrace .$ We note that $\\Vert \\delta ^2E(z+u)-\\delta ^2E(z+v)\\Vert \\le \\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty \\Vert Du-Dv\\Vert _2,$ so setting $M := \\Vert \\psi ^{\\prime \\prime \\prime }\\Vert _\\infty $ , we may apply Lemma REF , since $\\frac{2Mr}{\\mu ^2}\\le {\\textstyle \\frac{1}{2}}<1$ .",
"It follows that there exists $w\\in \\dot{{W}}^{1,2}(\\Lambda )$ with $\\Vert Dw\\Vert _2\\le c^{\\prime }L_\\mathcal {D}^{-1}$ such that $\\delta E(z+w)=0,\\qquad \\langle \\delta ^2E(z+w)v,v\\rangle \\ge \\frac{\\lambda _d}{4}\\Vert Dv\\Vert _2^2,$ and so $z+w$ is a strongly stable equilibrium.",
"The constant $c^{\\prime }$ depends only on $N$ , the number of dislocation cores, establishing item (2) of Theorem REF ."
],
[
"Proof of (1)",
"We begin by increasing $R_0$ if necessary to ensure that $\\frac{N}{2\\pi R_0}\\le \\frac{1}{4}$ .",
"Suppose that $z$ is a predictor for a configuration $\\mathcal {D}$ in $\\Lambda $ satisfying $L_\\mathcal {D}\\ge L_0$ and $R\\ge R_0$ .",
"If $\\alpha \\in [Dz]$ , by increasing $R_0$ , we have ensured that $\\alpha _b\\in [-{\\textstyle \\frac{1}{4}},{\\textstyle \\frac{1}{4}}]\\qquad \\text{for any }b\\notin \\bigcup _{(C,s)\\in \\mathcal {D}}{\\rm supp}\\lbrace D\\Pi _R^Cu\\rbrace ,$ and furthermore $\\alpha _b = \\sum _{(C,s)\\in \\mathcal {D}} s(\\hat{\\alpha }\\circ G^C)_b \\qquad \\text{for any }b\\notin \\bigcup _{(C,s)\\in \\mathcal {D}}{\\rm supp}\\lbrace D\\Pi _R^Cu\\rbrace .$ Let $\\alpha ^{\\prime }\\in [Dz+Dw]$ , and so if $L_\\mathcal {D}> 4c^{\\prime }$ , where $c^{\\prime }$ is the constant arising in the proof of (2), $z+w$ is a strongly stable local equilibrium such that $\\Vert Dw\\Vert _\\infty \\le \\Vert Dw\\Vert _2<{\\textstyle \\frac{1}{4}}$ .",
"When $b\\notin {\\rm supp}\\lbrace D\\Pi _R^{C}u\\rbrace $ for any $(C,s)\\in \\mathcal {D}$ , this choice entails that $\\alpha ^{\\prime }_b = \\alpha _b+Dw_b$ .",
"Taking $A$ to be a collection of positively-oriented cells such that $B_R(0)\\subset \\mathrm {clos}(A)\\subset B_{L_\\mathcal {D}/2}(0)$ and setting $A^C:=H^C(A)$ , $\\int _{\\partial A^{C^{\\prime }}}\\alpha ^{\\prime }= \\int _{\\partial A^{C^{\\prime }}}\\sum _{(C,s)\\in \\mathcal {D}}s\\,(\\hat{\\alpha }\\circ G^C) +Dw= s^{\\prime }\\qquad \\text{for any }(C^{\\prime },s^{\\prime })\\in \\mathcal {D},\\qquad \\text{and}\\\\\\int _{\\partial C} \\alpha ^{\\prime } = 0\\qquad \\text{for any}\\quad C\\notin \\bigcup _{(C,s)\\in \\mathcal {D}} A^C,\\quad \\text{implying}\\quad \\mathcal {C}^\\pm [\\alpha ^{\\prime }]\\subset \\bigcup _{(C,s)\\in \\mathcal {D}} H^C(A).$"
],
[
"Proof of (3)",
"We divide the proof of statement (3) into two cases: $B[z+w]=0$ , and $|B[z+w]|>1$ .",
"Suppose $z+w$ is a strongly stable equilibrium such that $B[z+w]=0$ , arising from statement (2) of Theorem REF .",
"It follows that $|\\lbrace (C,1)\\in \\mathcal {D}\\rbrace |=|\\lbrace (C,-1)\\in \\mathcal {D}\\rbrace |$ , so enumerating pairs $(C^i_+,1),(C^i_-,-1)\\in \\mathcal {D}$ , we define $v^i(x):={\\textstyle \\frac{1}{2\\pi }}\\big [\\arg \\big (x-x^{C^i_+}\\big )-\\arg \\big (x-x^{C^i_-}\\big )\\big ],\\quad \\text{and}\\quad v(x):=\\sum _iv^i(x),$ where $v^i$ is a function with a branch cut of finite length.",
"As for approximate solutions $z$ , we may extend $\\nabla v^i$ to a function which is $\\mathrm {C}^\\infty (\\mathbb {R}^2\\setminus \\lbrace x^{C^i_+},x^{C^i_-}\\rbrace ;\\mathbb {R}^2)$ .",
"It may then be directly verified that $|\\nabla v^i(x)| \\lesssim |x|^{-2}$ for $|x|$ suitably large, and hence when $v^i$ is understood as a function in ${W}(\\Lambda )$ , it follows that $v^i\\in \\dot{{W}}^{1,2}(\\Omega )$ .",
"It may now be checked that $Dz_b-Dv_b\\in \\mathbb {Z}$ for all $b\\in \\mathcal {B}$ , and hence $E(z-v;z+w) = E(0;z+w) =-E(z+w;0) <0,$ as Lemma REF implies that $0=\\operatornamewithlimits{{\\rm argmin}}_{u\\in \\dot{{W}}^{1,2}(\\Omega )} E(u;0)$ , which contradicts the assumption that $z+w$ was a globally stable equilibrium.",
"If $|B[z+w]|>1$ , then without loss of generality, we suppose $B[z+w]>1$ .",
"We will only consider the case where $B[z+w]=2$ here, leaving the general case for the interested reader.",
"Suppose for contradiction that $z+w$ is a strongly stable equilibrium given by (2) in Theorem REF with $B[z+w]=2$ , and that $z+w$ is additionally globally stable.",
"If true, then any configuration of the form $y = \\hat{y}+\\hat{y}\\circ G^C $ must satisfy $E(y;z+w)\\ge 0$ , since by a similar argument to that used in the previous case, we may define $y$ such that $y-z\\in \\dot{{W}}^{1,2}(\\Lambda )$ .",
"Our strategy is to construct a sequence $y^n$ of the form (REF ) such that $E(y^{n+1};y^n)\\le -C<0$ , and hence prove a contradiction.",
"To that end, define a sequence of cells $C^n$ such that $x^{C^n}$ lies on the positive $x$ -axis for all $n$ , with $\\mathrm {dist}(C^n,C^{n+1})<\\mathrm {dist}(0,C^n),\\quad \\mathrm {dist}(0,C^0)\\ge K\\quad \\text{and}\\quad \\mathrm {dist}(C^{n-1},C^n)\\ge K,$ where $K$ is a parameter we will choose later.",
"Our choice of $y^n$ is then $y^n:=\\hat{y}+\\hat{y}\\circ G^{C^n}.$ If $K$ is sufficiently large, we note that $\\alpha ^n\\in [Dy^n]$ is unique.",
"Letting $v^n:=y^n-y^{n-1}$ , decompose $Dv^n=\\beta ^n+Z^n_b$ , where $\\beta ^n = \\alpha ^n_b-\\alpha ^{n-1}_b$ , and $Z^n_b=Dv^n_b-\\beta ^n_b$ has support only on bonds crossing the $x$ -axis between $x^{C^{n-1}}$ and $x^{C^n}$ .",
"We consider $E(y^{n-1};y^n)=-E(y^n;y^{n-1})$ .",
"It may be checked directly that $\\beta ^n\\in \\ell ^2(\\mathcal {B})$ for any $n$ , and is uniformly bounded in $\\ell ^p(\\mathcal {B})$ for any $p>2$ .",
"Using these facts, the decay of $\\hat{\\alpha }_b$ , and Taylor expanding, we obtain that for any $\\epsilon >0$ , there exists a constant $C$ depending only on $\\psi $ and $\\epsilon $ such that $E(y^{n-1};y^n) &= \\sum _{b\\in \\mathcal {B}}\\psi (\\alpha ^{n-1}_b)-\\psi (\\alpha ^n_b)\\\\&= \\sum _{b\\in \\mathcal {B}}\\psi (\\alpha ^n_b-\\beta ^n_b)-\\psi (\\alpha ^n_b)-\\psi ^{\\prime }(\\alpha ^n_b)(-\\beta ^n_b)+\\langle \\delta E(y^n),-Dv^n\\rangle -\\sum _{b\\in \\mathcal {B}}\\psi ^{\\prime }(\\alpha ^n_b)(-Z^n_b),\\\\&\\ge {\\textstyle \\frac{1}{2}}(\\psi ^{\\prime \\prime }(0)-\\epsilon )\\Vert \\beta ^n\\Vert _2^2-C-\\langle \\delta E(y^n),Dv^n\\rangle +\\sum _{b\\in \\mathcal {B}}\\psi ^{\\prime }(\\alpha ^n_b)Z^n_b.$ Employing the result of Lemma REF , we find that we may write $-\\langle \\delta E(y^n),Dv^n\\rangle +\\sum _{b\\in \\mathcal {B}}\\psi ^{\\prime }(\\alpha ^n_b)Z^n_b= \\sum _{b\\in \\mathcal {B}}-g^n_b \\beta ^n_b + h^n_b Z^n_b\\ge -\\Vert g^n\\Vert _2\\Vert \\beta ^n\\Vert _2+\\sum _{b\\in \\mathcal {B}} h^n_bZ^n_b.$ It may be verified that $\\Vert g^n\\Vert _2$ is uniformly bounded in $n$ , using the properties demonstrated in Lemma REF , and that $Z^n_b$ is negative on bonds of the form $(\\xi ,\\xi +a_2)$ or $(\\xi ,\\xi +a_3)$ crossing the $x$ -axis.",
"Since by assumption $\\mathrm {dist}(C^{n-1},C^n)<\\mathrm {dist}(0,C^{n-1})$ , $h^n_b$ is negative for all bonds in ${\\rm supp}\\lbrace Z^n\\rbrace $ — in particular, these assertions imply that $\\sum _{b\\in \\mathcal {B}}h^n_bZ^n_b\\ge 0,\\quad \\text{and so}\\quad E(y^{n-1};y^n)\\ge c_0 \\Vert \\beta ^n\\Vert _2^2-c_1$ for some constants $c_0,c_1>0$ which depend only on $\\psi $ .",
"Applying Jensen's inequality to $\\beta ^n$ on a series of closed curves around $C^n$ , we find that $\\Vert \\beta ^n\\Vert _2^2\\ge c\\,\\log (\\mathrm {dist}(C^{n-1},C^n))\\ge c^{\\prime }\\log (K),$ where $c,c^{\\prime }$ are constants depending only on the lattice, and hence as long as $K$ is suitably large, we have that $E(y^{n-1};y^n)\\ge C\\ge 0$ .",
"Thus $E(y^k;z+w) = \\sum _{n=1}^k E(y^n;y^{n-1})+ E(y^0;z+w) \\le -Ck + E(y^0;z+w),$ and letting $k\\rightarrow \\infty $ , we have a contradiction to the fact that $z+w$ is a globally stable equilibrium."
],
[
"Proof for Finite Lattice Polygons",
"As in §, we construct approximate solutions and prove estimates on the derivative and Hessian of the energy evaluated at these points so that we may apply Lemma REF .",
"The two main differences between this and the preceding analysis are (i) $z$ defined in REF does not satisfy the natural boundary conditions of Laplace's equation in a finite domain, and (ii) we must estimate residual force contributions at the boundary, which cannot be achieved by a simple truncation argument as used in §REF — at this stage the fact that $\\Omega $ has a boundary plays a crucial role.",
"To obtain a predictor satisfying the natural boundary conditions we introduce a boundary corrector, $\\bar{y}\\in C^1(U^\\Omega ) \\cap C^2({\\rm int}(U^\\Omega ))$ , corresponding to a configuration $\\mathcal {D}$ in $\\Omega $ which satisfies $-\\Delta \\bar{y}= 0\\quad \\text{in }U^\\Omega ,\\qquad \\nabla \\bar{y}\\cdot \\nu =-\\sum _{(C,s)\\in \\mathcal {D}}s\\nabla (\\hat{y}\\circ G^C)\\cdot \\nu \\quad \\text{on }\\partial U^\\Omega ,$ where $\\nu $ is the outward unit normal on $\\partial U^\\Omega $ .",
"§REF is devoted to a study of this problem and its solution.",
"We then define an approximate solution (predictor) corresponding to $\\mathcal {D}$ in $\\Omega $ with truncation radius $R$ as $z := \\sum _{(C,s)\\in \\mathcal {D}}s\\,\\big (\\hat{y}+ \\Pi _R u\\big ) \\circ G^C+\\bar{y}.$"
],
[
"The continuum boundary corrector",
"Here, as remarked above, we give proofs of several important facts about the boundary corrector.",
"Since we are considering a boundary value problem in a polygonal domain, we use the theory developed in [14] to obtain regularity of solutions to (REF ).",
"Noting that the boundary corrector problem is linear, it suffices to analyse the problem when only one positive dislocation is present at a point $x^{\\prime }\\in U^\\Omega $ .",
"We therefore consider the problem $-\\Delta \\bar{y}= 0\\quad \\text{in }U^\\Omega ,\\qquad \\nabla \\bar{y}\\cdot \\nu =g_m\\quad \\text{on }\\Gamma _m,$ where as in §REF , $\\Gamma _m$ are the straight segments of $\\partial U^\\Omega $ between corners $(\\kappa _{m-1},\\kappa _m)$ , $\\nu $ is the outward unit normal, and $g_m(s):=-\\nabla \\hat{y}(s-x^{\\prime })\\cdot \\nu \\qquad \\text{for }s\\in \\Gamma _m.$ As before, by $\\nabla \\hat{y}(x-x^{\\prime })$ we mean the extension of the gradient of $\\hat{y}(x-x^{\\prime })$ to a function in $\\mathrm {C}^\\infty \\big (\\mathbb {R}^2\\setminus \\lbrace x^{\\prime }\\rbrace \\big )$ .",
"Since $\\nu $ is constant along $\\Gamma _m$ , it follows that $g_m\\in \\mathrm {C}^\\infty (\\Gamma _m)$ , and so applying Corollary 4.4.3.8 in [14], it may be seen that this problem has a solution in $\\mathrm {H}^2(U^\\Omega )$ which is unique up to an additive constant, as long as $\\int _{\\partial U^\\Omega }g=0$ .",
"This condition may be verified by standard contour integration techniques, for example.",
"Furthermore, $\\bar{y}$ is harmonic in the interior of $U^\\Omega $ , and hence analytic on the same set.",
"We now obtain several bounds for solutions of the problem (REF ) in terms of $\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )$ , taking note of the domain dependence of any constants.",
"The key fact used to construct these estimates is that $\\hat{y}+\\bar{y}$ is a harmonic conjugate of the Green's function for the Laplacian with Dirichlet boundary conditions on $U^\\Omega $ .",
"Suppose $U^\\Omega $ is a convex lattice polygon, and $\\bar{y}$ solves (REF ).",
"Then there exist constants $c_1$ and $c_2$ which are independent of the domain such that $|\\nabla \\bar{y}(x)|\\le c_1\\,\\mathrm {dist}(x,x^{\\prime })^{-1}\\quad \\text{for any }x\\in U^\\Omega ,\\qquad \\Vert \\nabla \\bar{y}\\Vert _\\infty \\le c_1\\,\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )^{-1},\\\\\\text{and}\\quad \\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(U^\\Omega )}\\le c_3 \\frac{\\log (\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega ))}{\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )}.$ We begin by noting that $\\hat{y}(x-x^{\\prime })={\\textstyle \\frac{1}{2\\pi }}\\arg (x-x^{\\prime })$ is a harmonic conjugate of ${\\textstyle \\frac{1}{2\\pi }}\\log (|x-x^{\\prime }|)$ , and we will further demonstrate that $\\bar{y}$ is a harmonic conjugate of $\\Psi $ , the solution of the Dirichlet boundary value problem $-\\Delta \\Psi (x)=0\\quad \\text{in }U^\\Omega ,\\qquad \\Psi (s) = -{\\textstyle \\frac{1}{2\\pi }}\\log (|x-x^{\\prime }|)\\quad \\text{on }\\partial U^\\Omega .$ By virtue of Corollary 4.4.3.8 in [14], there exists a unique $\\Psi \\in \\mathrm {H}^2(U^\\Omega )$ solving this problem, and since $\\Psi $ is harmonic in $U^\\Omega $ , a simply connected region, a harmonic conjugate $\\Psi ^*$ exists.",
"By definition, $\\Psi ^*$ satisfies the Cauchy–Riemann equations $\\nabla \\Psi ^*(x) = {\\sf R}_4^T\\nabla \\Psi (x)\\quad \\text{for all }x\\in U^\\Omega ,$ where ${\\sf R}_4$ is the matrix corresponding a positive rotation through ${\\textstyle \\frac{\\pi }{2}}$ about the origin.",
"In particular, $\\frac{\\partial \\Psi ^*}{\\partial \\nu } = \\frac{\\partial \\Psi }{\\partial \\tau }=\\frac{(x-x^{\\prime })}{2\\pi |x-x^{\\prime }|^2}\\cdot {\\sf R}_4\\nu =-\\nabla \\hat{y}(x-x^{\\prime })\\cdot \\nu \\quad \\text{on }\\partial U^\\Omega ,\\quad \\text{and} \\quad -\\Delta \\Psi ^*=0\\text{ in }U^\\Omega ,$ where $\\tau $ is the unit tangent vector to $\\partial U^\\Omega $ with the positive orientation.",
"By uniqueness of solutions for (REF ), it follows that $\\Psi ^*=\\bar{y}$ up to an additive constant, and hence $\\bar{y}$ is a harmonic conjugate of $\\Psi $ .",
"Furthermore, by differentiating (REF ), $\\Vert \\nabla ^2\\Psi \\Vert _{\\mathrm {L}^2(U^\\Omega )}=\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(\\Omega )}.$ The identities (REF ) and (REF ) will allow us to use estimates on the derivatives of $\\Psi $ to directly deduce (REF ) and ().",
"To prove (REF ), we rely upon Proposition 1 in [9], which states that there exists a constant $c_1$ depending only on $\\mathrm {diam}(U^\\Omega )$ such that $|\\nabla \\Psi (x)|\\le c_1\\,\\mathrm {dist}(x^{\\prime },x)^{-1}.$ However, as $U^\\Omega \\subset \\mathbb {R}^2$ , it is straightforward to see by a change of variables and a scaling argument that the constant $c_1$ cannot depend on $\\mathrm {diam}(U^\\Omega )$ , and is therefore independent of the domain (as long as it remains convex).",
"Taking the Euclidean norm of both sides in (REF ) now implies the pointwise bound in (REF ), and the $\\mathrm {L}^\\infty $ bound follows immediately as the partial derivatives of $\\bar{y}$ satisfy the strong maximum principle.",
"To prove (), we use the classical a priori bounds for the Poisson problem.",
"In order to do so, we must introduce an auxiliary problem with homogeneous boundary conditions.",
"We therefore seek a solution to $-\\Delta (\\Psi -\\Phi )=\\Delta \\Phi \\quad \\text{in }U^\\Omega ,\\qquad \\Psi -\\Phi \\in \\mathrm {H}^2(U^\\Omega )\\cap \\mathrm {H}^1_0(U^\\Omega ),$ where the function $\\Phi :\\mathbb {R}^2\\rightarrow \\mathbb {R}$ is defined to be $\\Phi (x)&:=-{\\textstyle \\frac{1}{2\\pi }}\\phi \\big (|x-x^{\\prime }|/\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )\\big )\\log (|x-x^{\\prime }|),\\\\\\text{where}\\quad \\phi &\\in \\mathrm {C}^\\infty ([0,+\\infty ))\\quad \\text{and}\\quad \\phi (r)=\\left\\lbrace \\begin{array}{rl} 0 & r\\in [0,{\\textstyle \\frac{1}{4}}],\\\\1&r\\ge 1.",
"\\end{array} \\right.$ By construction $\\Phi \\in \\mathrm {C}^\\infty (U^\\Omega )$ , and $\\nabla ^2\\Phi \\in \\mathrm {L}^2(\\mathbb {R}^2)$ .",
"Thus $\\Delta \\Phi \\in \\mathrm {L}^2(U^\\Omega )$ , and the existence of a unique solution $\\Psi -\\Phi \\in \\mathrm {H}^2(U^\\Omega )\\cap \\mathrm {H}^1_0(U^\\Omega )$ follows from [14]).",
"Furthermore, inspecting the proof of [14], we see that $\\Vert \\nabla ^2(\\Psi -\\Phi )\\Vert _{\\mathrm {L}^2(U^\\Omega )} = \\Vert \\Delta \\Phi \\Vert _{\\mathrm {L}^2(U^\\Omega )}\\le \\Vert \\nabla ^2\\Phi \\Vert _{\\mathrm {L}^2(\\mathbb {R}^2)},$ and thus a straightforward integral estimate yields $\\Vert \\nabla ^2 \\Psi \\Vert _{\\mathrm {L}^2(U^\\Omega )}\\le \\Vert \\nabla ^2(\\Psi -\\Phi )\\Vert _{\\mathrm {L}^2(U^\\Omega )}+\\Vert \\nabla ^2\\Phi \\Vert _{\\mathrm {L}^2(\\mathbb {R}^2)} \\le c_2 \\frac{\\log \\big (\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )\\big )}{\\mathrm {dist}(x^{\\prime },\\partial U^\\Omega )},$ where $c_2$ is independent of the domain."
],
[
"Analysis of the predictor",
"Here, we prove that the predictor defined in (REF ) is indeed an approximate equilibrium.",
"Our first step is to formulate the analogue of Lemma REF in the polygonal case.",
"[Finite domain stress lemma] Let $\\Omega $ be a convex lattice polygon, $\\mathcal {D}$ a dislocation configuration in $\\Omega $ and $z:=\\sum _{(C,s)\\in \\mathcal {D}} \\hat{y}\\circ G^C+\\bar{y}$ , where $\\bar{y}$ solves (REF ).",
"Then there exist $L_0$ and $S_0$ which depend only on $N=|\\mathcal {D}|$ such that whenever $L_\\mathcal {D}\\ge L_0$ and $S_\\mathcal {D}\\ge S_0$ , there exist $g:\\mathcal {B}^\\Omega \\rightarrow \\mathbb {R}$ and $\\Sigma :\\lbrace b\\in \\partial W^\\Omega \\rbrace \\rightarrow \\mathbb {R}$ such that $\\langle \\delta E^\\Omega (z),v\\rangle = \\sum _{b\\in \\mathcal {B}^\\Omega } g_b Dv_b+\\sum _{b\\in \\partial W^\\Omega }\\Sigma _bDv_b,$ and furthermore $|g_b|&\\le c_1\\sum _{(C,s)\\in \\mathcal {D}}\\big (1+\\mathrm {dist}(b,C)\\big )^{-3}+c_1\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(\\omega _b)}&&\\text{for all}\\quad b\\notin \\partial W^\\Omega ,\\\\\\text{and}\\quad |g_b+\\Sigma _b|&\\le c_2\\sum _{(C,s)\\in \\mathcal {D}}\\big (1+\\mathrm {dist}(P_\\zeta ,C)\\big )^{-1}+c_1\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(\\omega _b)}&&\\text{for all}\\quad b\\in P_\\zeta \\subset \\partial W^\\Omega .$ The constant $c_1$ is independent of the domain, and $c_2$ depends linearly on ${\\rm index}(\\partial W^\\Omega )$ .",
"We begin by choosing $L_0$ and $S_0$ to ensure that $\\alpha \\in [Dz]$ is unique: since the constant in (REF ) is independent of the domain, and $D\\hat{y}$ has a fixed rate of decay, this choice depends only on $N$ as stated.",
"Furthermore, we have the representation $\\alpha _{(\\xi ,\\xi +a_i)} = \\int _0^1 \\nabla z(\\xi +t a_i)\\cdot a_i\\,{\\rm d}t$ , where $\\nabla z$ is to be understood as the extension of the gradient of $z$ to a function in $\\mathrm {C}^\\infty (U^\\Omega \\setminus \\bigcup _{(C,s)\\in \\mathcal {D}}\\lbrace x^C\\rbrace )$ .",
"Let $\\omega _b := \\bigcup \\lbrace C\\in \\mathcal {C}^\\Omega \\,|\\,\\pm b\\in \\partial C,C\\text{ positively oriented}\\rbrace $ , the union of any cells which $b$ lies in the boundary of.",
"For $b\\notin \\partial W^\\Omega $ , $\\omega _b$ is always a pair of cells, and we set $V:=|\\omega _b|$ for any $b\\notin \\partial W^\\Omega $ .",
"Let $\\bar{C}_\\epsilon := \\bigcup _{(C,s) \\in \\mathcal {D}} B_\\epsilon (x^C)$ .",
"If $b = (\\xi , \\xi +a_i)$ , define $h_b := \\frac{\\psi ^{\\prime \\prime }(0)}{V} \\lim _{\\epsilon \\rightarrow 0}\\int _{\\omega _b \\setminus \\bar{C}_\\epsilon } \\nabla z \\cdot a_i \\,{\\rm d}x\\qquad \\text{and} \\qquad g_b := \\psi ^{\\prime }(\\alpha _b) - h_b.$ As in the proof of Lemma REF , an application of the divergence theorem demonstrates that the former (and hence the latter) definition makes sense.",
"Let $v\\in {W}(\\Omega )$ , and denote its piecewise linear interpolant $Iv$ ; applying the divergence theorem once more, $\\sum _{b \\in \\mathcal {B}^\\Omega } h_b Dv_b =\\lim _{\\epsilon \\rightarrow 0} \\frac{\\psi ^{\\prime \\prime }(0)}{V} \\int _{W^\\Omega \\setminus \\bar{C}_\\epsilon } \\nabla z \\cdot \\nabla Iv \\,{\\rm d}x= \\frac{\\psi ^{\\prime \\prime }(0)}{V}\\int _{\\partial W^\\Omega }Iv\\,\\nabla z\\cdot \\nu \\,{\\rm d}s.$ Recalling the definition of $P_\\zeta $ from (REF ), we find that $\\sum _{b \\in \\mathcal {B}^\\Omega } h_b Dv_b =\\!\\!\\sum _{\\zeta \\in \\partial W^\\Omega \\cap \\partial U^\\Omega }\\frac{\\psi ^{\\prime \\prime }(0)}{V}\\int _{P_\\zeta } Iv\\,\\nabla z\\cdot \\nu \\,{\\rm d}s.$ By considering the integral over a single period, we may integrate by parts $\\int _{P_\\zeta } Iv\\,\\nabla z\\cdot \\nu \\,{\\rm d}s= Iv(\\zeta +\\tau )\\int _{P_\\zeta } \\nabla z\\cdot \\nu \\,{\\rm d}s-\\int _{P_\\zeta } Iv^{\\prime }\\bigg (\\int _{\\gamma _\\zeta ^s}\\nabla z\\cdot \\nu \\,{\\rm d}t\\bigg )\\,{\\rm d}s,$ where $\\gamma _\\zeta ^s$ is the arc–length parametrisation of the Lipschitz curve following $P_\\zeta $ between $\\zeta $ and $s$ , $\\tau $ is the relevant lattice tangent vector, and $Iv^{\\prime }$ is the derivative along the curve following $P_\\zeta $ .",
"Applying the divergence theorem to the region bounded by $P_\\zeta $ and $\\partial U^\\Omega $ (as seen on the right of Figure REF ) and using the boundary conditions $\\nabla z\\cdot \\nu =0$ on $\\partial U^\\Omega $ , it follows that $\\int _{P_\\zeta } \\nabla z\\cdot \\nu =0$ .",
"Splitting the domain of integration $P_\\zeta $ into individual bonds and noting that $\\nabla Iv$ is constant along each bond, $\\int _{P_\\zeta } Iv\\,\\nabla z\\cdot \\nu \\,{\\rm d}s&=-\\sum _{b\\in P_\\zeta } \\int _{b=(\\xi ,\\xi +a_i)} \\hspace{-22.76219pt}\\nabla Iv\\cdot a_i \\bigg (\\int _{\\gamma _\\zeta ^s}\\nabla z\\cdot \\nu \\,{\\rm d}t\\bigg ) \\,{\\rm d}s,\\\\&=\\sum _{b\\in P_\\zeta } \\Sigma _bDv_b,\\quad \\text{where}\\quad \\Sigma _b:=-\\int _b\\int _{\\gamma _\\zeta ^s}\\nabla z\\cdot \\nu \\,{\\rm d}t\\,{\\rm d}s.$ This concludes the proof of the first part of the statement.",
"To obtain (REF ), we Taylor expand the potential to obtain $g_b=\\psi ^{\\prime \\prime }(0)\\bigg (\\int _b\\nabla z\\cdot a_i\\,{\\rm d}x-\\lim _{\\epsilon \\rightarrow 0}\\frac{1}{V}\\int _{\\omega _b\\setminus \\bar{C}_\\epsilon }\\nabla z\\cdot a_i\\,{\\rm d}x\\bigg ) +O\\big (|Dz_b|^3\\big ).$ Since $\\nabla z = \\sum _{(C,s)\\in \\mathcal {D}}\\nabla \\hat{y}\\circ G^C+\\nabla \\bar{y}$ , the only change to the analysis carried out in the proof of Lemma REF is to estimate the terms involving $\\nabla \\bar{y}$ .",
"As $\\int _b \\nabla \\bar{y}\\cdot a_i\\,{\\rm d}x= {\\textstyle \\frac{1}{|\\omega _b|}}\\int _{\\omega _b} \\nabla I\\bar{y}\\cdot a_i\\,{\\rm d}x$ , applying Jensen's inequality and standard interpolation error estimates (see for example §4.4 of [5]) gives $\\int _b\\nabla \\bar{y}\\cdot a_i\\,{\\rm d}x-\\frac{1}{V}\\!\\int _{\\omega _b}\\nabla \\bar{y}\\cdot a_i\\,{\\rm d}x&= \\frac{1}{V}\\int _{\\omega _b} \\big (\\nabla I\\bar{y}-\\nabla \\bar{y}\\big )\\cdot a_i\\,{\\rm d}x\\le \\frac{1}{\\sqrt{V}}\\big \\Vert \\nabla I\\bar{y}-\\nabla \\bar{y}\\big \\Vert _{\\mathrm {L}^2(\\omega _b)}\\le c\\big \\Vert \\nabla ^2 \\bar{y}\\big \\Vert _{\\mathrm {L}^2(\\omega _b)}$ where $c>0$ is a fixed constant.",
"Applying Young's inequality and (REF ) to estimate $|Dz_b|^3$ now leads immediately to (REF ).",
"Estimate () follows in a similar way: Taylor expanding $g_b$ , but noting that $|\\omega _b|=V/2$ and $\\omega _b$ is no longer symmetric, the same argument used above gives $|g_b+\\Sigma _b| \\le \\int _{b}\\bigg |{\\textstyle \\frac{1}{2}}\\nabla z\\cdot a_i-\\int _{\\gamma _\\zeta ^s}\\nabla z\\cdot \\nu \\,{\\rm d}t\\bigg |\\,{\\rm d}s+c\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(\\omega _b)}+\\sum _{(C,s)\\in \\mathcal {D}}\\big \\Vert \\nabla ^2\\hat{y}\\circ G^C\\big \\Vert _{\\mathrm {L}^\\infty (\\omega _b)}+O(|Dz_b|^3).$ Applying (REF ) to the first and last terms and and using the decay of $\\nabla \\hat{y}$ now yields $|g_b+\\Sigma _b|\\le c\\big (1+\\mathcal {H}^1(P_\\zeta )\\big )\\sum _{(C,s)\\in \\mathcal {D}}\\big (1+\\mathrm {dist}(P_\\zeta ,C)\\big )^{-1}+c\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(\\omega _b)}.$ Upon recalling the definition of ${\\rm index}(\\partial W^\\Omega )$ from (REF ), the proof is complete.",
"We can now deduce a residual estimate for the predictor in the finite domain case.",
"Suppose $\\Omega $ is a convex lattice polygon, and $z$ is the approximate solution corresponding to a dislocation configuration $\\mathcal {D}$ in $\\Omega $ defined in (REF ) with truncation radius $R=\\min \\big \\lbrace L_\\mathcal {D}/5,S_\\mathcal {D}^{1/2}\\big \\rbrace $ .",
"Then there exist constants $L_0$ , $S_0$ and $c$ depending only on $N=|\\mathcal {D}|$ and ${\\rm index}(\\partial W^\\Omega )$ such that whenever $L_\\mathcal {D}\\ge L_0$ and $S_\\mathcal {D}\\ge S_0$ , $\\big \\Vert \\delta E^{\\Omega }(z)\\big \\Vert _{(\\dot{{W}}^{1,2}(\\Omega ))^*}\\le c\\Big (L_\\mathcal {D}^{-1}+S_\\mathcal {D}^{-1/2}\\Big ).$ We begin by enumerating the elements $(C^i,s^i)\\in \\mathcal {D}$ , and set $G^i:=G^{C^i}$ .",
"For $i=1,\\ldots ,N$ , we let $\\hat{y}^i = \\hat{y}\\circ G^i$ , let $y^i=(\\hat{y}+\\Pi _Ru)\\circ G^i$ , and let $\\bar{y}^i$ be the corrector solving (REF ) with $x^{\\prime }=x^{C^i}$ .",
"Define $r:=2(R+1)=2\\big (\\!\\min \\big \\lbrace L_\\mathcal {D}/5,S_\\mathcal {D}^{1/2}\\big \\rbrace +1\\big )$ .",
"Taking a test function $v\\in \\dot{{W}}^{1,2}(\\Omega )$ , let $v^i(\\xi ):=\\Pi _r^{C^i}v(\\xi )\\qquad \\text{and}\\qquad v^0(\\xi ):=v(\\xi )-\\sum _iv^i(\\xi ).$ Lemma REF implies there is a universal constant independent of $\\Omega $ such that $\\Vert Dv^i\\Vert _2\\le C\\Vert Dv\\Vert _2$ for any $i=0,\\ldots ,N$ .",
"Adding and subtracting terms, we write $\\langle \\delta E^{\\Omega }(z),v\\rangle &=\\langle \\delta E^{\\Omega }(z),v^0\\rangle +\\sum _i\\big \\langle \\big [\\delta E^\\Omega (z)-\\delta E^\\Lambda (y^i)\\big ],v^i\\big \\rangle \\\\&\\qquad \\qquad +\\sum _i\\big \\langle \\big [\\delta E^\\Lambda (y^i)-\\delta E^\\Lambda \\big (\\hat{y}^i+ u\\circ G^i\\big )\\big ],v^i\\big \\rangle ,\\\\&=:\\mathrm {T}_1+\\mathrm {T}_2+\\mathrm {T}_3.$ We estimate each of these terms in turn.",
"The term ${\\rm T}_1$ : Applying Lemma REF and the fact that $z=\\sum _{i=1}^N\\hat{y}^i+\\bar{y}^i$ in ${\\rm supp}(v^0)$ , we make a similar estimate to that in Lemma REF : $\\big |\\mathrm {T}_1\\big |&= \\bigg |\\sum _{b\\in \\mathcal {B}^\\Omega }g_b Dv^0_b+\\sum _{b\\in \\partial W^\\Omega }\\Sigma _bDv^0_b\\bigg |\\\\&\\le c_1\\Bigg (\\bigg (\\sum _{\\begin{array}{c}(C,s)\\in \\mathcal {D},\\,b\\in \\mathcal {B}^\\Omega \\\\\\mathrm {dist}(b,C)\\ge r/2-1\\end{array}} \\big (1+\\mathrm {dist}(b,C)\\big )^{-6}\\bigg )^{1/2}+\\Vert \\nabla ^2\\bar{y}\\Vert _{\\mathrm {L}^2(W^\\Omega )}\\Bigg ) \\Vert Dv^0\\Vert _2\\\\&\\qquad \\quad +c_2\\bigg (\\sum _{\\zeta \\in \\partial W^\\Omega \\cap \\partial U^\\Omega }\\big (1+\\mathrm {dist}(P_\\zeta ,C^i)\\big )^{-2}\\bigg )^{1/2}\\Vert Dv^0\\Vert _2,\\\\&\\le c\\Big (r^{-2}+S_\\mathcal {D}^{-1}\\log (S_\\mathcal {D})+{\\rm index}\\big (\\partial W^\\Omega \\big )^{1/2}S_\\mathcal {D}^{-1/2}\\Big )\\Vert Dv^0\\Vert _2.$ To arrive at the final line we have used (), and the constant $c$ here is independent of the domain and the index.",
"The term ${\\rm T}_2$ : For the second set of terms, we have $z-y^i=\\sum _{j\\ne i}\\hat{y}^j+\\sum _j\\bar{y}^j$ in the support of $v^i$ .",
"We expand as in Lemma REF to obtain $\\big \\langle \\delta E^\\Omega (z)-\\delta E\\big (y^i\\big ),v^i\\big \\rangle =\\sum _{b\\in \\mathcal {B}^\\Omega }\\psi ^{\\prime \\prime }(s_b)\\bigg (\\sum _{j\\ne i} D\\hat{y}^j_b+\\sum _{j=0}^N D\\bar{y}^j_b\\bigg )Dv^i_b,\\\\=\\psi ^{\\prime \\prime }(0)\\sum _{b\\in \\mathcal {B}^\\Omega }\\bigg (\\sum _{j\\ne i} D\\hat{y}^j_b+D\\bar{y}^j_b\\bigg )Dv^i_b +\\psi ^{\\prime \\prime }(0)\\sum _{b\\in \\mathcal {B}^\\Omega } D\\bar{y}^i_bDv^i_b+\\sum _{b\\in \\mathcal {B}^\\Omega }h_bDv^i_b,$ where $|s_b|\\lesssim \\sum _j(1+\\mathrm {dist}(b,C^j))^{-1}+S_\\mathcal {D}^{-1}$ and a Taylor expansion yields $|h_b| = \\Big |\\big (\\psi ^{\\prime \\prime }(s_b) - \\psi ^{\\prime \\prime }(0)\\big )\\Big (\\sum _{j \\ne i} D\\hat{y}^j_b+\\sum _{j=0}^N D\\bar{y}^j_b\\Big )\\Big | \\lesssim |s_b|^2\\,r^{-1}.$ Applying Lemma REF to the first term in (REF ), a similar argument to that used to arrive at (REF ) gives $\\sum _{b\\in \\mathcal {B}^\\Omega }\\bigg (\\sum _{j\\ne i} D\\hat{y}^j_b+D\\bar{y}^j_b\\bigg )Dv^i_b\\le c\\Big (r^{-2}+S_\\mathcal {D}^{-1}\\log (S_\\mathcal {D})\\Big )\\Vert Dv^i\\Vert _2.$ Applying the global form of (REF ) to the second term in (REF ), $\\sum _{b\\in \\mathcal {B}^\\Omega }D\\bar{y}^i_bDv^i_b\\le c_1\\,rS_\\mathcal {D}^{-1}\\Vert Dv^i\\Vert _2,$ and finally, $\\sum _{b\\in \\mathcal {B}^\\Omega }h_bDv^i_b\\le r^{-1}\\Bigg (\\bigg (\\sum _{\\begin{array}{c}b\\in \\mathcal {B}^\\Omega ,\\,(C,s)\\in \\mathcal {D}\\\\\\mathrm {dist}(b,C^i)\\le r+1\\end{array}}\\big (1+\\mathrm {dist}(b,C)\\big )^{-4}\\bigg )^{1/2}+rS_\\mathcal {D}^{-2}\\Bigg )\\Vert Dv^i\\Vert _2\\le c\\Big (r^{-1}+S_\\mathcal {D}^{-2}\\Big )\\Vert Dv^i\\Vert _2.$ Combining these estimates gives $\\langle \\delta E^\\Omega (z)-\\delta E(y^i),v^i\\rangle \\le c\\big (r^{-1}+S_\\mathcal {D}^{-1}\\log (S_\\mathcal {D})+r S_\\mathcal {D}^{-1}\\big )\\Vert Dv^i\\Vert _2.$ The term ${\\rm T}_3$ : The final group may be once more estimated using the truncation result of Lemma REF , giving $\\big |\\big \\langle \\delta E(y^i)-\\delta E(\\hat{y}^i+u\\circ G^i),v^i\\big \\rangle \\big |\\lesssim R^{-1}\\Vert Dv^i\\Vert _2.$ Conclusion: Inserting the estimates (REF ), (REF ) and (REF ) into (REF ), and using the fact that $\\Vert Dv^i\\Vert _2\\lesssim \\Vert Dv\\Vert _2$ , we obtain the bound $\\big |\\langle \\delta E^\\Omega (z),v\\rangle \\big |\\lesssim \\Big (L_\\mathcal {D}^{-1}+S_\\mathcal {D}^{-1/2}\\Big )\\Vert Dv\\Vert _2.$"
],
[
"Stability of the predictor",
"Next we prove the stability of the predictor configuration defined in (REF ).",
"Given $I_0$ and $N\\in \\mathbb {N}$ , there exist $R_0=R_0(N)$ , $L_0=L_0(N)$ and $S_0=S_0(N,I_0)$ such that whenever $z$ is the approximate solution corresponding to a dislocation configuration $\\mathcal {D}$ in a convex lattice polygon $\\Omega $ with truncation radius $R$ given in (REF ), and furthermore: ${\\rm index}(\\partial W^\\Omega )\\le I_0$ , $S_\\mathcal {D}\\ge S_0$ , $L_\\mathcal {D}\\ge L_0$ and $R\\ge R_0$ , then there exists $\\lambda \\ge \\lambda _d/2$ such that $\\langle \\delta ^2E^\\Omega (z)v,v\\rangle \\ge \\lambda \\Vert Dv\\Vert _2^2\\qquad \\text{for all}\\quad v\\in \\dot{{W}}^{1,2}(\\Omega ).$ Fixing $I_0$ and $N$ , we choose $R_0$ and $L_0$ such that the conclusion of Lemma REF holds for any dislocation configuration $\\mathcal {D}$ in $\\Lambda $ with $|\\mathcal {D}|=N$ .",
"Throughout the proof, we fix $R$ to be any number with $R\\ge R_0$ , and we will consider only configurations such that $L_\\mathcal {D}\\ge L_0$ .",
"Suppose for contradiction that there exists a sequence of domains $\\Omega ^n$ with accompanying dislocation configurations $\\mathcal {D}^n$ which together satisfy ${\\rm index}(\\partial W^{\\Omega ^n})=I_0$ , $N:=|\\mathcal {D}^n|$ , $|\\lbrace (C,+1)\\in \\mathcal {D}^n\\rbrace |$ and $|\\lbrace (C,-1)\\in \\mathcal {D}^n\\rbrace |$ are constant, $(C_0,+1)\\in \\mathcal {D}^n$ , $S^n:=S_{\\mathcal {D}^n}\\rightarrow \\infty $ as $n\\rightarrow \\infty $ and $\\delta ^2E^{\\Omega ^n}(z^n)<\\lambda _d/2$ for all $n$ , where $z^n:=\\sum _{(C,s)\\in \\mathcal {D}^n}s(\\hat{y}+\\Pi _Ru)\\circ G^C+\\bar{y}^n,$ and $\\bar{y}^n$ solves (REF ) with $\\Omega =\\Omega ^n$ .",
"We note that condition (3) may be assumed without loss of generality by applying lattice symmetries.",
"Condition (5) implies that there exists $v^n\\in \\dot{{W}}^{1,2}(\\Omega ^n)$ such that $\\Vert Dv^n\\Vert _2=1$ and $\\lambda ^n:=\\inf _{\\begin{array}{c}v\\in \\dot{{W}}^{1,2}(\\Omega ^n)\\\\\\Vert Dv\\Vert _2=1\\end{array}}\\langle \\delta ^2E^{\\Omega ^n}(z^n)v,v\\rangle =\\langle \\delta ^2E^{\\Omega ^n}(z^n)v^n,v^n\\rangle <\\lambda _d/2,$ since this is a minimisation problem for a continuous function over a compact set.",
"For each $n$ , enumerate $(C^{n,i},s^{n,i})\\in \\mathcal {D}^n$ , and let $G^{n,i}:= G^{C^{n,i}}$ and $H^{n,i}:= H^{C^{n,i}}$ .",
"Considering $Dv^n$ as an element of $\\ell ^2(\\mathcal {B})$ by extending $Dv^n_b:=\\left\\lbrace \\begin{array}{rl} Dv^n_b & b\\in \\mathcal {B}^\\Omega ,\\\\0 & b\\in \\mathcal {B}\\setminus \\mathcal {B}^\\Omega , \\end{array} \\right.$ there exists a subsequence such that $Dv^n\\circ H^{n,i}$ is weakly convergent for each $i$ .",
"For given $i$ and $j$ , $\\mathrm {dist}(C^{n,i},C^{n,j})$ either remains bounded or tends to infinity, and so define an equivalence relation $i\\sim j$ if and only if $\\mathrm {dist}(C^{n,i},C^{n,j})$ is uniformly bounded as $n\\rightarrow \\infty $ .",
"By possibly taking further subsequences, we may assume that if $i\\sim j$ then $Q^{ji}:=G^{n,j}\\circ H^{n,i}$ is constant along the sequence, and hence if $Dv^n\\circ H^{n,i}\\rightharpoonup D\\bar{v}^i$ for each $i$ , $D\\bar{v}^j\\circ Q^{ji} = D\\bar{v}^i\\qquad \\text{when}\\quad i\\sim j.$ For each equivalence class, $[i]$ , define $y^{n,[i ]}:= \\sum _{j\\in [i]}s^j(\\hat{y}+\\Pi _Ru)\\circ G^{n,j}.$ Using the result of [7], there exists a sequence $r^n\\rightarrow \\infty $ which we may also assume satisfies $r^n\\le \\min _{i\\nsim j}\\big \\lbrace \\mathrm {dist}(C^{n,i},C^{n,j})\\big \\rbrace /5\\quad \\text{and}\\quad r^n\\le S^n/5,$ so that, defining $w^{n,[i]} := \\Pi _{r^n}^{C^{n,i}}v^n$ , $w^{n,[i]}\\circ H^{n,i}\\rightarrow \\bar{w}^{[i]}\\text{ in }\\dot{{W}}^{1,2}(\\Lambda )\\qquad \\text{and}\\qquad (Dv^n-Dw^{n,[i]})\\circ H^{n,i}\\rightharpoonup 0\\text{ in }\\ell ^2(\\mathcal {B}),$ where $i$ is a fixed representative of $[i]$ .",
"Further defining $Dw^{n,0}:=Dv^n-\\sum _{[i ]} Dw^{n,[i]}$ , we have $\\langle \\delta ^2E^{\\Omega ^n}(z^n)v^n,v^n\\rangle =\\langle \\delta ^2E^{\\Omega ^n}(z^n)w^{n,0},w^{n,0}\\rangle +\\sum _{[i]}\\Big (2\\langle \\delta ^2E^{\\Omega ^n}(z^n)w^{n,[i]},w^{n,0}\\rangle +\\langle \\delta ^2E^{\\Omega ^n}(z^n)w^{n,[i]},w^{n,[i]}\\rangle \\Big ).$ The definition of $r^n$ and (REF ) imply that $\\Vert \\nabla \\bar{y}^n\\Vert _{\\mathrm {L}^\\infty (U^{\\Omega ^n})}\\le c_1/r^n$ , where $c_1$ is independent of $n$ , so in a similar fashion to the proof of Lemma REF , we obtain: $\\big \\langle \\delta ^2E^{\\Omega ^n}(z^n) w^{n,0},w^{n,0}\\big \\rangle &=\\big \\langle [\\delta ^2E^{\\Omega ^n}(z^n)-\\delta ^2E^{\\Omega ^n}(0)]w^{n,0},w^{n,0}\\big \\rangle +\\big \\langle \\delta ^2E^{\\Omega ^n}(0) w^{n,0},w^{n,0}\\big \\rangle \\\\&\\ge \\big (\\psi ^{\\prime \\prime }(0)-c/r^n\\big )\\big \\Vert Dw^{n,0}\\big \\Vert _2^2,\\\\\\big \\langle \\delta ^2E^{\\Omega ^n}(z^n) w^{n,[i]},w^{n,[i]}\\big \\rangle &=\\big \\langle [\\delta ^2E^{\\Omega ^n}(z^n)-\\delta ^2E^{\\Lambda }(y^{n,[i]})]w^{n,[i]},w^{n,[i]}\\big \\rangle +\\big \\langle \\delta ^2E^{\\Lambda }(y^{n,[i]}) w^{n,[i]},w^{n,[i]}\\big \\rangle ,\\\\&\\ge \\big (\\lambda _{L,R}-c/r^n\\big )\\big \\Vert Dw^{n,[i]}\\big \\Vert _2^2,\\qquad \\text{and}\\\\\\big \\langle \\delta ^2E^{\\Omega ^n}(z^n) w^{n,0},w^{n,[i]}\\big \\rangle &\\rightarrow 0\\qquad \\text{as}\\quad n\\rightarrow \\infty ,$ where $c$ represents a constant independent of $n$ .",
"Furthermore, the arguments of the proof of Lemma REF imply that $\\liminf _{n\\rightarrow \\infty }\\Big (\\sum _i \\Vert Dw^{n,[i]}\\Vert _2^2-\\Vert Dv^n\\Vert _2^2\\Big )\\ge 0,$ and so we deduce that $\\lambda _n = \\big \\langle \\delta ^2E^{\\Omega ^n}(z^n)v^n,v^n\\big \\rangle \\ge \\lambda _d/2>0$ for $n$ sufficiently large, providing the required contradiction."
],
[
"Conclusion of the proof of Theorem ",
"To conclude the proof of conclusions (1) and (2) of Theorem REF , we may apply small modifications of the arguments used in §REF and §REF , and hence we omit these.",
"To prove conclusion (3), recall the result of Lemma REF , which states that $y\\equiv 0$ is a globally stable equilibrium in any lattice domain.",
"When $\\Omega $ is a convex lattice polygon, ${W}(\\Omega )\\subset \\dot{{W}}^{1,2}(\\Omega )$ , so if $z+w$ is the local equilibrium for $E^\\Omega $ constructed in (1), then $-z-w\\in \\dot{{W}}^{1,2}(\\Omega )$ , and furthermore $E(z+w-z-w;z+w) = E(0;z+w) = - E(z+w;0) < 0,\\qquad \\text{as}\\qquad 0=\\operatornamewithlimits{{\\rm argmin}}_{u\\in \\dot{{W}}^{1,2}(\\Omega )} E(u;0).$"
]
] | 1403.0518 |
[
[
"Recent Star Formation in the Leading Arm of the Magellanic Stream"
],
[
"Abstract Strongly interacting galaxies undergo a short-lived but dramatic phase of evolution characterized by enhanced star formation, tidal tails, bridges and other morphological peculiarities.",
"The nearest example of a pair of interacting galaxies is the Magellanic Clouds, whose dynamical interaction produced the gaseous features known as the Magellanic Stream trailing the pair's orbit about the Galaxy, the Bridge between the Clouds, and the Leading Arm, a wide and irregular feature leading the orbit.",
"Young, newly formed stars in the Bridge are known to exist, giving witness to the recent interaction between the Clouds.",
"However, the interaction of the Clouds with the Milky Way is less well understood.",
"In particular, the Leading Arm must have a tidal origin, however no purely gravitational model is able to reproduce its morphology and kinematics.",
"A hydrodynamical interaction with the gaseous hot halo and disk of the Galaxy is plausible as suggested by some models and supporting neutral hydrogen observations.",
"Here we show for the first time that young, recently formed stars exist in the Leading Arm, indicating that the interaction between the Clouds and our Galaxy is strong enough to trigger star formation in certain regions of the Leading Arm --- regions in the outskirts of the Milky Way disk (R ~ 18 kpc), far away from the Clouds and the Bridge."
],
[
"Introduction",
"The Magellanic Clouds (MCs) offer a unique opportunity to study galaxy interactions in unprecedented detail due to their proximity to the Milky Way (MW).",
"Thus, detailed mapping of their gaseous content, the 3D kinematics of their stellar content, and the chemical-abundance makeup of these components are readily available for the Clouds.",
"The most obvious features of their interaction are the H I structures known as the $\\sim 200^\\circ $ -long Magellanic Stream (MS), the Bridge, and the Leading Arm (LA) (Nidever et al.",
"2010).",
"The recent work on the modeling of the Clouds' interaction by Diaz & Bekki (2012) makes a compelling case for tidal model, where the MS, Bridge and LA are made primarily of material pulled out from the SMC during two close encounters between the two Clouds.",
"The first encounter took place $\\sim 2$ Gyr ago, and the second $\\sim 200$ Myr ago.",
"This work used the most recent absolute proper-motion determinations for the Clouds: one HST-based (Kallivayalil et al.",
"2006), the other ground-based (Vieira et al.",
"2010).",
"Both determinations imply exactly two encounters between the Clouds to reproduce the MS, LA and Bridge.",
"As for their motion relative to the MW, HST measurements(Kallivayalil 2006, 2013) favor the scenario where the MCs are on the first passage about the MW, with the MS and LA determined solely by the tidal interaction between the Clouds.",
"The ground-based proper-motion measurement allows for two pericentric passages of the Clouds about the MW in the past 2.5 Gyr, and thus some tidal influence of the Galaxy in the formation of the MS, LA and Bridge is expected.",
"The main drawback of the tidal models is that, while they produce a leading arm, all fail to reproduce the observed multi-branches morphology of the LA, and its kinematics.",
"A model by Diaz & Bekki (2011) that also includes a hydrodynamical interaction of the LA with the diffuse, hot gaseous halo of the MW, better reproduces the kinematics along the LA, but not its morphology.",
"The LA has a complex structure, possibly made of as many as four substructures according to For et al.",
"(2013) and Venzmer et al.",
"(2012), situated above and below the Galactic plane, and encompassing $\\sim 60^\\circ $ in width.",
"It has been argued that there is a strong drag exerted by the MW gaseous disk on the H I substructures in the LA (McClure et al.",
"2008, Venzmer et al.",
"2012).",
"This is implied by the head-tail velocity structure of the H I clouds in the LA, as well as the velocity gradient seen in a given LA substructure/arm (Venzmer et al.",
"2012).",
"Thus it would be enlightening to search for newly formed stars in the LA, an expected result of the hydrodynamical interaction between the MS gas and the MW gaseous disk and halo.",
"We also note that the most recent ($\\sim 200 $ Myr ago) encounter between the two Clouds which created the Bridge, is abundantly accompanied by recent star formation, a fact well known since the work by Irwin et al.",
"(1990) and subsequent follow-up by, e.g., Demers et al.",
"(1998).",
"In a recent study Casetti-Dinescu et al.",
"(2012) listed 567 OB-type star candidates in a $\\sim $ 7900 square degree area encompassing the periphery of the Clouds, the Bridge, the LA, and part of the MS.",
"The photometric and proper-motion selection was aimed at finding hot (earlier than B5) and distant stars.",
"Also, the proper-motion selection was aimed at selecting stars with motions consistent with membership to the Magellanic system.",
"In the LA region, three stellar overdensities were found comprising a total of 45 candidates.",
"This is a lower limit of such candidates, since the study is area-wise incomplete (Casetti-Dinescu et al.",
"2012).",
"Here, we have spectroscopically observed 42 of the 45 candidates.",
"Their spatial distribution is shown in Figure 1.",
"Also shown is the H I distribution from the GASS survey (McClure-Griffith et al.",
"2009, Kalberla et al.",
"2010) for which we have restricted the velocity with respect to the Local Standard of Rest, to be $ 150 \\le V_{LSR} \\le 400$ km/s.",
"The three candidate overdensities we label: A at $(\\Lambda _M, B_M) \\sim (15^\\circ ,-22^\\circ )$ , B at $(\\Lambda _M,B_M) \\sim (42^\\circ ,-8^\\circ )$ , and C at $(\\Lambda _M,B_M) \\sim (52^\\circ ,28^\\circ )$ .",
"In what follows, we dscribe the spectroscopic observations and the results."
],
[
"Observations",
"Intermediate-resolution spectra were obtained with the IMACS spectrograph on the 6.5m Baade telescope at Las Campanas Observatory.",
"The setup gave a resolution of 1.3$~Å$ (R$\\approx $ 3500) in the range 3650 to 5230 $Å$ .",
"The 1200 l/mm grating at the f/4 camera was employed at first order, with a blaze angle of $17^\\circ $ and 0.75”-wide slit, for a resulting resolution of 1.3$~Å$ (R$\\approx $ 3500) in the range 3650 to 5230 $Å$ .",
"The average seeing during observations was 0.7”, and the resulting spectral The spectral signal-to-noise ratio was higher than 50 for all the targets.",
"Cross-correlation techniques (Tonry & Davis 1979) as implemented in the IRAF fxcor task, were used to measure heliocentric radial velocities (RVs).",
"In absence of a prior knowledge of the exact temperature and gravity of the targets, the synthetic spectrum (Munari et al.",
"2005) of a main-sequence B-type star was adopted as the template A mismatch between the parameters of the template and object spectra enhances the uncertainties but does not affect the results, especially for hot stars (Moni Bidin et al.",
"2011)..",
"The final uncertainty, taking into account the relevant sources of errors, are estimated to be between 3 and 14 km s$^{-1}$ , typically $\\approx $ 5 km s$^{-1}$ for most of the targets.",
"The spectra are also fitted with standard routines (Bergeron et al.",
"1992, Saffer et al.",
"1994, Napiwotzki et al.",
"1999) to derive the temperature, gravity, surface helium abundance, and, in some cases, rotational velocityRotational velocities were not fitted for, but rather used as an input parameter.",
"By using different inputs, the one that gave the lowest chi square, was adopted as the value of the $vsini$ .",
"These values should be regarded as indicative, with errors $\\sim 30$ km s$^{-1}$.",
"At these colors, our major source of contamination is foreground subdwarf O and B stars (sdBs) and white dwarfs.",
"Close binaries are extremely common among sdB's (Maxted et al.",
"2001), hence, RVs alone are not conclusive to assess the membership of our targets to the LA.",
"To distinguish main sequence stars from sdBs, besides the surface gravity, we can also use the surface helium abundance because the atmosphere of sdB's in the temperature range $T_{eff} > 11500$ K is depleted of helium by a factor between 10 and 100 due to gravitational settling (Baschek 1975, Moni Bidin et al.",
"2012).",
"Rotational velocity is also indicative, as fast rotators are common among early-type main sequence stars, but not among sdB's (Geier & Heber 2012)."
],
[
"Results",
"Of the 42 candidates observed, we find 19 young, massive stars, together with 22 foreground sdB and white dwarf stars, and one uncertain object.",
"The density of young stars in regions A and B is higher than in C, at a significance of $2.6\\sigma $ in A, for instance, after correcting for areal incompleteness.",
"We adopt RV = 150 km s$^{-1}$ as the lower limit for kinematical membership to the LA based on H I velocities in the LA (e.g., Venzmer et al.",
"2012).",
"Note that, in this region of the sky, heliocentric and LSR RVs differ by a very small amount (at most 14 km s$^{-1}$ in region C).",
"We find a total of six stars with RV $ > 150$ km s$^{-1}$ : four in region B, two in region A, and none in region C (Fig.",
"1).",
"These are listed in Table 1, along with one other star of interest.",
"Table: Spectral Parameters for Stars of InterestRemarkably, of the six stars with RV $ > 150$ km s$^{-1}$ , five are young, massive objects, as inferred from $T_{eff}, log~g$ and helium abundance.",
"Three of these stars are also fast rotators.",
"The RV average and dispersion of these five stars is $201\\pm 14$ km s$^{-1}$ , and 32 km s$^{-1}$ respectively.",
"Even more remarkably, we find one candidate star to be a very hot, main sequence star with spectral type O6V (Tab.",
"1), and thus a massive ($\\sim 40~M_{\\odot }$ ), short-lived (1-2 Myr) star at a heliocentric distance of $\\sim 40$ kpc.",
"In Figure 1, we highlight the 19 young stars, the six stars with RV $ > 150$ km s$^{-1}$ , and the one O6V type star.",
"We plot the RVs as a function of surface gravity, color-coded by effective temperature, in Figure 2, and highlight the fast rotators.",
"As a guide, we also indicate the mean RV of the thin+thick disk populations and its $\\pm 2\\sigma $ standard-deviation range as derived from the Besancon galactic model (Robin et al.",
"2003).",
"The group of stars with RV $ > 150$ km s$^{-1}$ and $log~g \\le 4$ stands out in this plot.",
"A possible non-LA origin for the young stars must be considered, i.e., that these are runaway stars.",
"Such stars are believed to originate in star-forming regions of the Galactic disk as massive binaries that are disrupted by either a supernova, or by a three- or four-body dynamical interaction, probably within a young star cluster (e.g., Bromely et al.",
"2009) In principle, all of our 19 young stars could be suspected of having such an origin.",
"However, the five young B-type stars with RV $ > 150$ km s$^{-1}$ do not fit such a scenario, since their RV dispersion of 33 km s$^{-1}$ is too low compared to that of runaway stars, $\\sim 130$ km s$^{-1}$ (Bromely et al.",
"2009).",
"To obtain such a low velocity dispersion, the ejection mechanism would have to be directionally coherent, which is highly unlikely.",
"For the remaining young stars, it is difficult to distinguish, on an individual basis, between a runaway star and an LA member.",
"This is because the RV is compatible with both the kinematics of a runaway star and the orbital motion of a binary star in the LA.",
"Massive young stars are known to form predominantly in binary and multiple systems (Sana et al.",
"2012), thus it is likely that our single-epoch RVs are affected.",
"Given these two stochastic effects on RVs — contamination by runaway stars and binary orbital motion — it is remarkable that we have found five young stars exhibiting a dispersion of only 30 km s$^{-1}$ , and a mean RV compatible with LA kinematics.",
"Considering also the areal incompleteness (Casetti-Dinescu et al.",
"2012), what we have discovered is probably the “tip of the iceberg”.",
"Absolute magnitudes and ages have been derived for all the young stars based on isochrones in the $log~g-T_{eff}$ plane (Bressan et al.",
"2012).",
"In Figure 3, we plot distance modulus versus age for these stars.",
"Stars with RV $ > 150$ km s$^{-1}$ are once again highlighted.",
"As a guide, the gray band indicates the kinematical distance (21 kpc) with uncertainty, derived for a high velocity cloud in the LA by McClure et al (2008) that crosses the Galactic disk.",
"Clearly, this kinematic distance is within the range of our stellar distances.",
"A number of more nearby stars are also present in this young sample, and they may be more readily explained as having originated in the disk.",
"Finally, we discuss the notable O6V star (Tab.",
"1).",
"Its origin in the MW disk is doubtful, since it is too young (1-2 Myr) to have traveled at a reasonable speed to a Galactocentric distance of $\\sim 39$ kpc.",
"To demonstrate this, we calculate its orbit using a three-component Galactic potential model and find its last disk crossing occured over 500 Myr ago.",
"Even allowing for uncertainties in the distance, the shortest time since last disk crossing in the orbit is 385 Myr.",
"Therefore, this star could not have been born in the Galactic disk.",
"Another possiblity is that it was born in the LMC and subsequently ejected, but this would require a velocity of the order of $10^4$ km s$^{-1}$ for such a young star to reach its current position.",
"This being an unrealistic value, the only viable possibility is that it was born in situ, far away from both the Galactic disk and the LMC.",
"These observations establish that conditions were met for recent star formation in the LA material located in the outskirts of the Galactic disk ($R\\sim 18$ kpc), most likely as a consequence of the interaction between the Galactic disk and portions of the LA.",
"We note that the most distant HI structure associated with the MW disk, is a spiral arm at $R = 18$ to 24 kpc (McClure-Griffiths et al.",
"2004), while stellar samples indicate shorter distances for the “edge” of the disk of $\\sim 14$ kpc (e.g., Minniti et al.",
"2011) Our findings cast new light on the interaction of the Clouds with the MW, perhaps making a first infall scenario less likely.",
"Whether this is the case remains to be established by more complex models, and in light of the lower velocity of the Clouds as indicated by the Vieira et al.",
"(2010), Costa et al.",
"(2009) and the Kallivayalil et al.",
"(2013) studies compared to the first HST study by Kallivayalil et al.",
"(2006).",
"Figure: The spatial distribution in Magellanic coordinates of our OB candidates (crosses)The background color map shows the H I density distribution for 150≤RV LSR ≤400 150 \\le RV_{LSR} \\le 400 km/s,with the main LA branches(Venzmer et al.",
"2012)indicated.The 42 OB candidates observed spectroscopically are shown with circles.Filled green symbols indicate the young stars, while symbols highlighted with red squaresindicate stars with radial velocity RV>150RV > 150 km s -1 ^{-1}.",
"The black star symbol representsthe most massive, young star in our sample (sp.",
"type O6V).",
"Our threeregions of interest (A, B and C) are also labeled.The dashed line represents the Galactic plane.Figure: Kinematic and spectral properties of our OB candidates.Heliocentric radial velocity as a function of surface gravity for 42 observed stars.Each symbols' color represents the effective temperature as indicated.The mean and ±2σ\\pm 2\\sigma standard deviation for the Galactic thin+thick disk(Robin et al.",
"2003) are indicated with a hatched area.The horizontal line at 150 km s -1 ^{-1} showsthe limit for LA RV-member candidates.",
"Fast rotators (vsini >100>100 km s -1 ^{-1}) arehighlighted with a black circle.",
"Note, the group of six stars with velocities in excess of150 km s -1 ^{-1} and log g smaller than ∼4.2\\sim 4.2 dex.",
"Five of these are classified asmassive, young stars, and only one as a sdB, primarilyon account of its low He abundance (see Table 1).",
"Note also the hot, relatively lowlog g star at RV ∼70\\sim 70 km s -1 ^{-1} (star symbol).",
"This is the earliest spectral type star foundwith this surface gravity, and thus classified as O6V, a massive, young star located at ∼40\\sim 40 kpcfrom the Sun.Figure: Distances and ages for the young stars.Distance moduli versus ages are shown for our nineteen massive, young stars.The stars with RV>150>150 km s -1 ^{-1} are shown with filled squares.",
"The star symbolindicates the O6V star.The gray band represents the kinematical distance of one high velocity cloud member of the LA(McClure-Griffiths et al.",
"2008);the width of the band corresponds to a 20%20\\% error in the distance.This investigation is based on data gathered with the 6.5-m Baade telescope, located at Las Campanas Observatory, Chile (program ID: CN2013A-152).",
"D.I.C.",
"acknoweledges partial support by the NSF through the grant 0908996.",
"R.A.M.",
"acknowledges partial support from Project IC120009 “Millennium Institute of Astrophysics (MAS)” of the Iniciativa Cientifica Milenio del Ministero de Economia, Fomento y Turismo de Chile, and from project PFB-06 CATA.",
"Baschek, B.",
"1975, in: Problems in stellar atmospheres and envelopes New York, Springer-Verlag New York Inc. 1975, 101-148 Bergeron, P., Saffer, R. A., & Liebert, J.",
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"2009, , 706, 925 Casetti-Dinescu, D. I., Vieira, K., Girard, T. M., & van Altena, W. F. 2012, , 753, 123 Costa, E., Méndez, R. A., Pedreros, M. H., Moyano, M., Gallart, C., Noel, N., Baume, G. & Carraro, G. 2009, , 137, 4339.",
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"2012, , 543, 149 Irwin, M. J., Demers, S., & Kunkel, W. E. 1990 , 99, 191 Kalberla, P. M., W., et al.",
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"2004, , 607, L127 McClure-Griffiths, N. M., et al., 2008, , 673, L143 McClure-Griffith, N. M., Staveley-Smith, L., Lockman, F. J., Calabretta, M. R., Ford, H. A., Kalberla, P. M. W., Murphy, T., Nakanishi, H., and Pisano, D. J., 2009, , 181, 398 Minniti, D., Saito, R. K., Alonso-Garcia, J., Lucas, P. W., & Hempel, M. 2011, , 733, L43 Moni Bidin, C., Villanova, S., Piotto, G., & Momany, Y.",
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]
] | 1403.0517 |
[
[
"On the C*-algebra Generated by Toeplitz Operators and Fourier\n Multipliers on the Hardy Space of a Locally Compact Group"
],
[
"Abstract Let $G$ be a locally compact abelian Hausdorff topological group which is non-compact and whose Pontryagin dual $\\Gamma$ is partially ordered.",
"Let $\\Gamma^{+}\\subset\\Gamma$ be the semigroup of positive elements in $\\Gamma$.",
"The Hardy space $H^{2}(G)$ is the closed subspace of $L^{2}(G)$ consisting of functions whose Fourier transforms are supported on $\\Gamma^{+}$.",
"In this paper we consider the C*-algebra $C^{*}(\\mathcal{T}(G)\\cup F(C(\\dot{\\Gamma^{+}})))$ generated by Toeplitz operators with continuous symbols on $G$ which vanish at infinity and Fourier multipliers with symbols which are continuous on one point compactification of $\\Gamma^{+}$ on the Hilbert-Hardy space $H^{2}(G)$.",
"We characterize the character space of this C*-algebra using a theorem of Power."
],
[
"introduction",
"For a locally compact abelian Hausdorff topological group $G$ whose Pontryagin dual $\\Gamma $ is partially ordered, one can define the positive elements of $\\Gamma $ as $\\Gamma ^{+}=\\lbrace \\gamma \\in \\Gamma :\\gamma \\ge e\\rbrace $ where $e$ is the identity of the group $G$ and the Hardy space $H^{2}(G)$ as $H^{2}(G)=\\lbrace f\\in L^{2}(G):\\hat{f}(\\gamma )=0\\quad \\forall \\gamma \\notin \\Gamma ^{+}\\rbrace $ where $\\hat{f}$ is the Fourier transform of $f$ .",
"It is not difficult to see that $H^{2}(G)$ is a closed subspace of $L^{2}(G)$ and since $L^{2}(G)$ is a Hilbert space there is a unique orthogonal projection $P:L^{2}(G)\\rightarrow H^{2}(G)$ onto $H^{2}(G)$ .",
"This definition of the Hardy space $H^{2}(G)$ is motivated by Riesz theorem in the classical cases when $G=\\mathbb {T}$ i.e when $G$ is the unit circle, which characterizes the Hardy class functions among $f\\in L^{2}(\\mathbb {T})$ as the space of functions whose negative Fourier coefficients vanish and by the Paley-Wiener theorem when $G=\\mathbb {R}$ , the real line since the group Fourier transform is the Fourier series when $G=\\mathbb {T}$ and coincides with the Euclidean Fourier transform when $G=\\mathbb {R}$ .",
"One can extend the theory of Toeplitz operators to this setting by defining a Toeplitz operator with symbol $\\phi \\in L^{\\infty }(G)$ as $T_{\\phi }=P M_{\\phi }$ where $M_{\\phi }$ is the multiplication by $\\phi $ and $P$ is the orthogonal projection of $L^{2}$ onto $H^{2}$ .",
"Such a definition was first considered by Coburn and Douglas in [2].",
"However the Toeplitz operators considered in [2] were more general since no partial order was assumed on the dual $\\Gamma $ whereas the Hardy space was defined as the space of functions whose Fourier transforms are supported on a fixed sub-semigroup $\\Gamma _{0}$ of $\\Gamma $ .",
"The definition of Hardy space of groups whose duals are partially ordered and their Toeplitz operators were introduced and studied by Murphy in [7] and [8].",
"However in these papers [7] and [8], Murphy studies the case where $G$ is compact.",
"In this paper we will study the case where $G$ is not compact.",
"One very important assumption that we will make is that $\\Gamma ^{+}$ separates the points of $G$ , i.e.",
"for any $t_{1},t_{2}\\in G$ satisfying $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"The Toeplitz C*-algebra of a locally compact group is defined as $\\mathcal {T}(G)=C^{*}(\\lbrace T_{\\phi }:\\phi \\in C_{0}(G)\\rbrace \\cup \\lbrace I\\rbrace )$ where $C_{0}(G)$ is the space of continuous functions vanishing at infinity and $I$ is the identity operator.",
"In the study of this Toeplitz C*-algebra, the most important notions are the commutator ideal $com(G)=I^{*}(\\lbrace T_{\\phi }T_{\\psi }-T_{\\psi }T_{\\phi }:\\phi ,\\psi \\in C_{0}(G)\\rbrace )$ , the semi-commutator ideal $scom(G)=I^{*}(\\lbrace T_{\\phi \\psi }-T_{\\psi }T_{\\phi }:\\phi ,\\psi \\in C_{0}(G)\\rbrace )$ and the symbol map $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/com(G)$ , $\\Sigma (\\phi )=[T_{\\phi }]$ where $\\dot{G}$ is the one point compactification of $G$ and $[T_{\\phi }]$ denotes the equivalence class of $T_{\\phi }$ modulo $com(G)$ .",
"It is not difficult to see that $com(G)\\subseteq scom(G)$ .",
"We start by proving the following important result whose proof is adapted from [6]: Lemma 1 Let $G$ be a locally compact abelian Hausdorff topological group whose Pontryagin dual $\\Gamma $ is partially ordered and let $\\Gamma ^{+}$ be the semigroup of positive elements of $\\Gamma $ .",
"Suppose that $\\Gamma ^{+}$ separates the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Let $com(G)$ and $scom(G)$ be the commutator and the semi-commutator ideal of the Toeplitz C*-algebra $\\mathcal {T}(G)$ respectively.",
"Then $com(G)=scom(G)$ It is shown in [2] and [7] that $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/com(G)$ is an isometry but is not a homomorphism since it may not preserve the multiplication.",
"However $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/scom(G)$ is a homomorphism and combining this fact with Lemma 1 above we deduce that the symbol map $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/com(G)$ is an isometric isomorphism which means that $M(\\mathcal {T}(G))=\\dot{G}$ where $M(A)$ is the character space of a C*-algebra $A$ .",
"We introduce another class of operators acting on $H^{2}(G)$ which are called “Fourier multipliers\".",
"These operators in the classical case $G=\\mathbb {R}$ were introduced in [4].",
"The space of Fourier multipliers is defined as $F(C(\\dot{\\Gamma ^{+}}))=\\lbrace D_{\\theta }=\\mathcal {F}^{-1}M_{\\theta }\\mathcal {F}\\mid _{H^{2}(G)}:\\theta \\in C(\\dot{\\Gamma ^{+}})\\rbrace $ where $\\mathcal {F}:L^{2}(G)\\rightarrow L^{2}(\\Gamma )$ is the Fourier transform.",
"By Plancherel theorem it is not difficult to see that the image $\\mathcal {F}(H^{2}(G))$ of $H^{2}(G)$ under the Fourier transform is equal to $L^{2}(\\Gamma ^{+})$ .",
"Again it is not difficult to see that $F(C(\\dot{\\Gamma ^{+}}))$ is isometrically isomorphic to $C(\\dot{\\Gamma ^{+}})$ .",
"This means that $M(F(C(\\dot{\\Gamma ^{+}})))=\\dot{\\Gamma ^{+}}$ Lastly we consider the C*-algebra generated by $\\mathcal {T}(G)$ and $F(C(\\dot{\\Gamma ^{+}}))$ which we denote by $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))$ i.e.",
"$\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))=C^{*}(\\mathcal {T}(G)\\cup F(C(\\dot{\\Gamma ^{+}})))$ Using a Theorem of Power [9],[10] which characterizes the character space of the C*-algebra generated by two C*-algebras as a certain subset of the cartesian product of character spaces of these two C*-algebras, we prove following theorem: Main Theorem Let $G$ be a non-compact,locally compact abelian Hausdorff topological group whose Pontryagin dual $\\Gamma $ is partially ordered and let $\\Gamma ^{+}$ be the semigroup of positive elements of $\\Gamma $ .",
"Suppose that $\\Gamma ^{+}$ separates the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Let $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))=C^{*}(\\mathcal {T}(G)\\cup F(C(\\dot{\\Gamma ^{+}})))$ be the C*-algebra generated by Toeplitz operators and Fourier multipliers on $H^{2}(G)$ .",
"Then for the character space $M(\\Psi )$ of $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))$ we have $M(\\Psi )\\cong (\\dot{G}\\times \\lbrace \\infty \\rbrace )\\cup (\\lbrace \\infty \\rbrace \\times \\dot{\\Gamma ^{+}})$"
],
[
"preliminaries",
"In this section we fix the notation that we will use throughout and recall some preliminary facts that will be used in the sequel.",
"Let $S$ be a compact Hausdorff topological space.",
"The space of all complex valued continuous functions on $S$ will be denoted by $C(S)$ .",
"For any $f\\in C(S)$ , $\\parallel f\\parallel _{\\infty }$ will denote the sup-norm of $f$ , i.e.",
"$\\parallel f\\parallel _{\\infty }=\\sup \\lbrace \\mid f(s)\\mid :s\\in S\\rbrace .$ If $S$ is a locally compact Hausdorff topological space, $C_{0}(S)$ will denote the space of continuous functions $f$ which vanish at infinity i.e.",
"for any $\\varepsilon >0$ there is a compact subset $K\\subset S$ such that $\\mid f(x)\\mid <\\varepsilon $ for all $x\\notin K$ .",
"For a Banach space $X$ , $K(X)$ will denote the space of all compact operators on $X$ and $\\mathcal {B}(X)$ will denote the space of all bounded linear operators on $X$ .",
"The real line will be denoted by $\\mathbb {R}$ , the complex plane will be denoted by $\\mathbb {C}$ and the unit circle group will be denoted by $\\mathbb {T}$ .",
"The one point compactification of a locally compact Hausdorff topological space $S$ will be denoted by $\\dot{S}$ .",
"For any subset $S\\subset $ $\\mathcal {B}(H)$ , where $H$ is a Hilbert space, the C*-algebra generated by $S$ will be denoted by $C^{*}(S)$ and for any subset $S\\subset A$ where $A$ is a C*-algebra, the closed two-sided ideal generated by $S$ will be denoted by $I^{*}(S)$ .",
"For any $\\phi \\in $ $L^{\\infty }(G)$ where $G$ is a Borel space(a topological space with a regular measure on it), $M_{\\phi }$ will be the multiplication operator on $L^{2}(G)$ defined as $M_{\\phi }(f)(t)=\\phi (t)f(t).$ For convenience, we remind the reader of the rudiments of theory of Banach algebras, some basic abstract harmonic analysis and Toeplitz operators.",
"Let $A$ be a Banach algebra.",
"Then its character space $M(A)$ is defined as $M(A)=\\lbrace x\\in A^{*}:x(ab)=x(a)x(b)\\quad \\forall a,b\\in A\\rbrace $ where $A^{*}$ is the dual space of $A$ .",
"If $A$ has identity then $M(A)$ is a compact Hausdorff topological space with the weak* topology.",
"When $A$ is commutative $M(A)$ is called the maximal ideal space of $A$ .",
"For a commutative Banach algebra $A$ the Gelfand transform $\\Gamma :A\\rightarrow C(M(A))$ is defined as $\\Gamma (a)(x)=x(a).$ If $A$ is a commutative C*-algebra with identity, then $\\Gamma $ is an isometric *-isomorphism between $A$ and $C(M(A))$ .",
"If $A$ is a C*-algebra and $I$ is a two-sided closed ideal of $A$ , then the quotient algebra $A/I$ is also a C*-algebra (see [5]).",
"For a Banach algebra $A$ , we denote by $com(A)$ the closed ideal in $A$ generated by the commutators $\\lbrace a_{1}a_{2}-a_{2}a_{1}:a_{1},a_{2}\\in A\\rbrace $ .",
"It is an algebraic fact that the quotient algebra $A/com(A)$ is a commutative Banach algebra.",
"The reader can find detailed information about Banach and C*-algebras in [11] and [5] related to what we have reviewed so far.",
"On a locally compact abelian Hausdorff topological group $G$ there is a unique(up to multiplication by a constant) translation invariant measure $\\lambda $ on $G$ i.e.",
"for any Borel subset $E\\subset G$ and for any $x\\in G$ , $\\lambda (xE)=\\lambda (E)$ where $xE=\\lbrace xy:y\\in E\\rbrace $ is the translate of $E$ by $x$ .",
"This measure is called the Haar measure of $G$ .",
"Let $L^{1}(G)$ be the space of integrable functions with respect to this measure.",
"Then $L^{1}(G)$ becomes a commutative Banach algebra with multiplication as the convolution defined as $(f\\ast g)(t)=\\int _{G}f(ts^{-1})g(s)d\\lambda (s)$ The Pontryagin dual $\\Gamma $ of $G$ is defined to be the set of all continuous homomorphisms from $G$ to the circle group $\\mathbb {T}$ : $\\Gamma =\\lbrace \\gamma :G\\rightarrow \\mathbb {T}:\\gamma (st)=\\gamma (s)\\gamma (t)\\quad \\textrm {and}\\quad \\gamma \\quad \\textrm {iscontinuous}\\rbrace $ It is a well known fact that $\\Gamma $ is in one to one correspondence with the maximal ideal space $M(L^{1}(G))$ of $L^{1}(G)$ via the Fourier transform: $<\\gamma ,f>=\\hat{f}(\\gamma )=\\int _{G}\\overline{\\gamma (t)}f(t)d\\lambda (t)$ When $\\Gamma $ is topologized by the weak* topology coming from $M(L^{1}(G))$ , $\\Gamma $ becomes a locally compact abelian Hausdorff topological group with point-wise multiplication as the group operation: $(\\gamma _{1}\\gamma _{2})(t)=\\gamma _{1}(t)\\gamma _{2}(t)$ Let $\\tilde{\\lambda }$ be a fixed Haar measure on $\\Gamma $ .",
"Plancherel theorem asserts that the Fourier transform $\\mathcal {F}$ is an isometric isomorphism of $L^{2}(G)$ onto $L^{2}(\\Gamma )$ : $\\mathcal {F}(f)(\\gamma )=\\hat{f}(\\gamma )=\\int _{G}\\overline{\\gamma (t)}f(t)d\\lambda (t)$ with inverse $\\mathcal {F}^{-1}$ defined as $\\mathcal {F}^{-1}(f)(t)=\\check{f}(t)=\\int _{\\Gamma }\\gamma (t)f(\\gamma )d\\tilde{\\lambda }(\\gamma )$ Here we note that $\\tilde{\\lambda }$ is normalized so that the above formula for the inverse Fourier transform holds.",
"For detailed information on abstract harmonic analysis consult [12].",
"A partially ordered group $\\Gamma $ is a group with partial order $\\ge $ on it satisfying $\\gamma _{1}\\ge \\gamma _{2}$ implies $\\gamma \\gamma _{1}\\ge \\gamma \\gamma _{2}$ $\\forall \\gamma \\in \\Gamma $ .",
"This definition of the ordered group was given in [8].",
"Let $\\Gamma ^{+}=\\lbrace \\gamma \\in \\Gamma :\\gamma \\ge e\\rbrace $ be the semi-group of positive elements of $\\Gamma $ where $e$ is the unit of the group $\\Gamma $ .",
"Let $G$ be a locally compact abelian Hausdorff topological group and let $\\Gamma $ be the Pontryagin dual of $G$ .",
"Then the Hardy space $H^{2}(G)$ is defined as $H^{2}(G)=\\lbrace f\\in L^{2}(G):\\hat{f}(\\gamma )=0\\quad \\forall \\gamma \\notin \\Gamma ^{+}\\rbrace $ The Hardy space $H^{2}(G)$ is a closed subspace of $L^{2}(G)$ and since $L^{2}(G)$ is a Hilbert space, there is a unique orthogonal projection $P:L^{2}(G)\\rightarrow H^{2}(G)$ .",
"For any $\\phi \\in L^{\\infty }(G)$ the Toeplitz operator $T_{\\phi }:H^{2}(G)\\rightarrow H^{2}(G)$ is defined as $T_{\\phi }=P M_{\\phi }$ Toeplitz operators satisfy the following algebraic properties: $T_{c\\phi +\\psi }=cT_{\\phi }+T_{\\psi }$ $\\forall c\\in \\mathbb {C}$ , $\\forall \\phi ,\\psi \\in C(\\dot{G})$ $T_{\\phi }^{*}=T_{\\bar{\\phi }}$ $\\forall \\phi \\in C(\\dot{G})$ The proofs of these properties are the same as in the classical case where $G=\\mathbb {T}$ (or $G=\\mathbb {R}$ ) and can be found in [3].",
"The Toeplitz C*-algebra $\\mathcal {T}(G)$ is defined to be the C*-algebra generated by continuous symbols on $G$ : $\\mathcal {T}(G)=C^{*}(\\lbrace T_{\\phi }:\\phi \\in C_{0}(G)\\rbrace \\cup \\lbrace I\\rbrace )$ where $I$ is the identity operator and $C_{0}(G)$ is the space of continuous functions which vanish at infinity: $C_{0}(G)=\\lbrace f:G\\rightarrow \\mathbb {C}:f\\quad \\textrm {is continuousand}\\quad \\forall \\epsilon >0\\quad \\exists K\\subset \\subset G\\mid f(t)\\mid <\\epsilon \\quad \\forall t\\notin K\\rbrace $ where $K\\subset \\subset G$ denotes a compact subset of $G$ .",
"Actually one has $\\mathcal {T}(G)=C^{*}(\\lbrace T_{\\phi }:\\phi \\in C(\\dot{G})\\rbrace )$ where $\\dot{G}$ is the one-point compactification of $G$ .",
"In the case where $G$ is compact one has $G=\\dot{G}$ and the most prototypical concrete example of this case is $G=\\mathbb {T}$ .",
"This case was analyzed by Coburn in [1].",
"The famous result of Coburn asserts that for any $T\\in \\mathcal {T}(\\mathbb {T})$ there are unique $K\\in K(H^{2}(\\mathbb {T}))$ and $\\phi \\in C(\\mathbb {T})$ such that $T=T_{\\phi }+K$ .",
"Hence the quotient algebra $\\mathcal {T}(\\mathbb {T})/K(H^{2}(\\mathbb {T}))$ modulo the compact operators is isometrically isomorphic to $C(\\mathbb {T})$ .",
"The two sided closed *-ideal $com(G)$ generated by the commutators is called the commutator ideal of $\\mathcal {T}(G)$ : $com(G)=I^{*}(\\lbrace T_{\\phi }T_{\\psi }-T_{\\psi }T_{\\phi }:\\phi ,\\psi \\in C(\\dot{G})\\rbrace )$ and the semi-commutator ideal $scom(G)$ is defined as $scom(G)=I^{*}(\\lbrace T_{\\phi \\psi }-T_{\\psi }T_{\\phi }:\\phi ,\\psi \\in C(\\dot{G})\\rbrace )$ The symbol map $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/com(G)$ is defined as $\\Sigma (\\phi )=[T_{\\phi }]$ where $[.",
"]$ denotes the equivalence class modulo $com(G)$ .",
"In [2] and [8] it is shown that $\\Sigma $ is an isometry.",
"The symbol map $\\Sigma $ also preserves the * operation however is not a homomorphism i.e does not preserve multiplication.",
"But if $com(G)=scom(G)$ then it is an isometric isomorphism.",
"We will show under certain conditions that $com(G)=scom(G)$ .",
"We introduce another class of operators which we call the “Fourier multipliers\".",
"This class of operators in the case $G=\\mathbb {R}$ was introduced in [4] and proved to be useful in calculating the essential spectra of a class of composition operators.",
"The Fourier multiplier $D_{\\theta }:H^{2}(G)\\rightarrow H^{2}(G)$ with symbol $\\theta \\in C(\\dot{\\Gamma ^{+}})$ is defined as $D_{\\theta }(f)(t)=(\\mathcal {F}^{-1}M_{\\theta }\\mathcal {F}(f))(t)$ The most prototypical example of a Fourier multiplier is a convolution operator with kernel $k\\in L^{1}(G)$ : $(T_{k}f)(t)=\\int _{G}k(ts^{-1})f(s)d\\lambda (s)$ It is not difficult to see that actually $T_{k}=D_{\\hat{k}}$ where $\\hat{k}$ denotes the Fourier transform of $k$ .",
"The set of all Fourier multipliers $F(C(\\dot{\\Gamma ^{+}}))$ defined as $F(C(\\dot{\\Gamma ^{+}}))=\\lbrace D_{\\theta }:\\theta \\in C(\\dot{\\Gamma ^{+}})\\rbrace $ is a commutative C*-algebra since the map $D:C(\\dot{\\Gamma ^{+}})\\rightarrow F(C(\\dot{\\Gamma ^{+}}))$ defined as $D(\\theta )=D_{\\theta }$ is an isometric *-isomorphism.",
"Lastly we consider the C*-algebra generated by Toeplitz operators and Fourier multipliers.",
"Let $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))$ be the C*-algebra $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))=C^{*}(\\mathcal {T}(G)\\cup F(C(\\dot{\\Gamma ^{+}})))$ generated by Toeplitz operators with continuous symbols and continuous Fourier multipliers.",
"The main result of this paper is a characterization of the character space $M(\\Psi )$ of $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))$ .",
"We know that $M(F(C(\\dot{\\Gamma ^{+}})))\\cong \\dot{\\Gamma ^{+}},$ under certain conditions we have $scom(G)=com(G)$ and this implies that $M(\\mathcal {T}(G))\\cong \\dot{G}.$ We will use the following theorem due to Power [9],[10] in identifying the character space of $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))$ : Power's Theorem Let $C_{1}$ , $C_{2}$ be C*-subalgebras of $B(H)$ with identity, where $H$ is a separable Hilbert space, such that $M(C_{i})\\ne $ $\\emptyset $ , where $M(C_{i})$ is the space of multiplicative linear functionals of $C_{i}$ , $i= 1,\\,2$ and let $C$ be the C*-algebra that they generate.",
"Then for the commutative C*-algebra $\\tilde{C}=$ $C/com(C)$ we have $M(\\tilde{C})=$ $P(C_{1},C_{2})\\subset $ $M(C_{1})\\times M(C_{2})$ , where $P(C_{1},C_{2})$ is defined to be the set of points $(x_{1},x_{2})\\in $ $M(C_{1})\\times M(C_{2})$ satisfying the condition: Given $0\\le a_{1} \\le 1$ , $0 \\le a_{2} \\le 1$ , $a_{1}\\in C_{1}$ , $a_{2}\\in C_{2}$ , $x_{i}(a_{i})=1\\quad \\textrm {with}\\quad i=1,2\\quad \\Rightarrow \\quad \\Vert a_{1}a_{2}\\Vert =1.$ The proof of this theorem can be found in [9].",
"Power's theorem will give the character space $M(\\Psi )$ of $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))$ as a certain subset of the cartesian product $\\dot{G}\\times \\dot{\\Gamma ^{+}}$ ."
],
[
"the character space of $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))$ ",
"In this section we will concentrate on the C*-algebra $\\Psi (C_{0}(G),C(\\dot{\\Gamma }))$ .",
"But before that we will identify the character space $M(\\mathcal {T}(G))$ of $\\mathcal {T}(G)$ under certain conditions.",
"The condition that we will pose on $G$ is that $\\Gamma ^{+}$ separate the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Under this condition we show that $scom(G)=com(G)$ and this implies that $M(\\mathcal {T}(G))\\cong \\dot{G}$ .",
"Hence we begin by proving the following lemma whose proof is adapted from the proof of Theorem 2.2 of [6]: Proposition 2 Let $G$ be a locally compact abelian Hausdorff topological group whose Pontryagin dual $\\Gamma $ is partially ordered and let $\\Gamma ^{+}$ be the semigroup of positive elements of $\\Gamma $ .",
"Suppose that $\\Gamma ^{+}$ separates the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Let $com(G)$ and $scom(G)$ be the commutator and the semi-commutator ideal of the Toeplitz C*-algebra $\\mathcal {T}(G)$ respectively.",
"Then $com(G)=scom(G)$ It is trivial that $com(G)\\subseteq scom(G)$ hence we need to show that $scom(G)\\subseteq com(G)$ : Let $B=\\lbrace \\phi \\in C_{0}(G):T_{\\phi }T_{\\psi }-T_{\\phi \\psi }\\in com(G)\\rbrace $ then $B$ is a self-adjoint subalgebra of $C_{0}(G)$ : Let $\\psi \\in B$ then since $T_{\\phi }T_{\\bar{\\psi }}-T_{\\phi \\bar{\\psi }}=(T_{\\psi }T_{\\bar{\\phi }}-T_{\\bar{\\phi }}T_{\\psi })^{*}+(T_{\\bar{\\phi }}T_{\\psi }-T_{\\bar{\\phi }\\psi })^{*}$ we have $(T_{\\psi }T_{\\bar{\\phi }}-T_{\\bar{\\phi }}T_{\\psi })^{*}\\in com(G)$ , $(T_{\\bar{\\phi }}T_{\\psi }-T_{\\bar{\\phi }\\psi })^{*}\\in com(G)$ and hence $T_{\\phi }T_{\\bar{\\psi }}-T_{\\phi \\bar{\\psi }}\\in com(G)$ $\\forall \\phi \\in C(\\dot{G})$ .",
"This implies that $\\bar{\\psi }\\in B$ .",
"It is clear that $\\psi _{1},\\psi _{2}\\in B$ implies that $\\psi _{1}+\\psi _{2}\\in B$ .",
"Let us check that $\\psi _{1}\\psi _{2}\\in B$ : we have $T_{\\phi }T_{\\psi _{1}\\psi _{2}}-T_{\\phi \\psi _{1}\\psi _{2}}=T_{\\phi }(T_{\\psi _{1}\\psi _{2}}-T_{\\psi _{1}}T_{\\psi _{2}})+(T_{\\phi }T_{\\psi _{1}}-T_{\\phi \\psi _{1}})T_{\\psi _{2}}+(T_{\\phi \\psi _{1}}T_{\\psi _{2}}-T_{\\phi \\psi _{1}\\psi _{2}}).$ Since $\\psi _{1}\\in B$ and $com(G)$ is an ideal we have $T_{\\phi }(T_{\\psi _{1}\\psi _{2}}-T_{\\psi _{1}}T_{\\psi _{2}})\\in com(G)$ , $(T_{\\phi }T_{\\psi _{1}}-T_{\\phi \\psi _{1}})T_{\\psi _{2}}\\in com(G)$ and $(T_{\\phi \\psi _{1}}T_{\\psi _{2}}-T_{\\phi \\psi _{1}\\psi _{2}})\\in com(G)$ which implies that $T_{\\phi }T_{\\psi _{1}\\psi _{2}}-T_{\\phi \\psi _{1}\\psi _{2}}\\in com(G)$ $\\forall \\phi \\in C_{0}(G)$ .",
"So we have $\\psi _{1}\\psi _{2}\\in B$ .",
"Now we need to show that $B$ separates the points of $G$ to conclude the proof since in that case $B$ is closed and by Stone-Weierstrass theorem we will have $B=C_{0}(G)$ : Now let $A(G)=\\lbrace \\psi \\in C_{0}(G):\\psi f\\in H^{2}(G)\\quad \\forall f\\in H^{2}(G)\\rbrace $ .",
"Clearly $A(G)\\subseteq B$ , hence if we show that $A(G)$ separates the points of $G$ we are done.",
"For any $k\\in L^{1}(\\Gamma ^{+})$ consider $\\check{k}(t)=\\int _{\\Gamma ^{+}}k(\\gamma )\\gamma (t)d\\tilde{\\lambda }(\\gamma )$ then since for any $f\\in H^{2}(G)$ we have $\\mathcal {F}(\\check{k}f)=k\\ast \\hat{f}$ the Fourier transform of $\\check{k}f$ will be supported in $\\Gamma ^{+}$ .",
"This implies that $\\check{k}\\in A(G)$ .",
"Since $\\Gamma ^{+}$ separates the points of $G$ , $\\lbrace \\check{k}:k\\in L^{1}(\\Gamma ^{+})\\rbrace $ also separates the points of $G$ and hence $A(G)$ separates the points of $G$ .",
"This implies that $B$ separates the points of $G$ .",
"This proves our lemma.",
"We have the following corollary of proposition 2: Corollary 3 Let $G$ be a locally compact abelian Hausdorff topological group whose Pontryagin dual $\\Gamma $ is partially ordered and let $\\Gamma ^{+}$ be the semigroup of positive elements of $\\Gamma $ .",
"Suppose that $\\Gamma ^{+}$ separates the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Let $\\mathcal {T}(G)$ be the Toeplitz C*-algebra with symbols in $C(\\dot{G})$ acting on $H^{2}(G)$ .",
"Then we have $M(\\mathcal {T}(G))\\cong \\dot{G}$ The symbol map $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/com(G)$ , $\\Sigma (\\phi )=[T_{\\phi }]$ is an isometry that preserves the *-operation and $\\Sigma :C(\\dot{G})\\rightarrow \\mathcal {T}(G)/scom(G)$ is multiplicative.",
"Since $com(G)=scom(G)$ , $\\Sigma $ is an isometric isomorphism.",
"Since characters kill the commutators we have $M(\\mathcal {T}(G))=M(\\mathcal {T}(G)/com(G))\\cong M(C(\\dot{G}))=\\dot{G}$ We will need the following small observation in proving our main theorem: Lemma 4 Let $G$ be a locally compact,non-compact,abelian Hausdorff topological group and let $K_{1},K_{2}\\subset G$ be two non-empty compact subsets of $G$ .",
"Then there is $t_{0}\\in G$ such that $K_{1}\\cap (t_{0}K_{2})=\\emptyset $ where $t_{0}K_{2}=\\lbrace t_{0}t:t\\in K_{2}\\rbrace $ .",
"Since $G$ is non-compact and locally compact there is a one-point compactification $\\dot{G}$ of $G$ .",
"Hence there is a point at infinity $\\infty \\in \\dot{G}$ such that $\\infty \\notin G$ .",
"Assume that the lemma does not hold i.e.",
"there are two non-empty compact subsets $K_{1},K_{2}\\subset G$ such that $K_{1}\\cap (tK_{2})\\ne \\emptyset $ for all $t\\in G$ .",
"Now take a net $\\lbrace t_{\\alpha }\\rbrace _{\\alpha \\in \\mathcal {I}}\\subset G$ such that $\\lim _{\\alpha \\in \\mathcal {I}}t_{\\alpha }=\\infty $ .",
"Since $\\forall \\alpha \\in \\mathcal {I}$ we have $K_{1}\\cap (t_{\\alpha }K_{2})\\ne \\emptyset $ , there are $x_{\\alpha }\\in K_{1}$ and $y_{\\alpha }\\in K_{2}$ such that $x_{\\alpha }=t_{\\alpha }y_{\\alpha }$ .",
"Since $K_{1}$ and $K_{2}$ are compact there are $x_{0}\\in K_{1}$ , $y_{0}\\in K_{2}$ and sub-nets $x_{\\alpha _{1}}\\in K_{1}$ , $y_{\\alpha _{2}}\\in K_{2}$ such that $\\lim x_{\\alpha _{1}}=x_{0}$ and $\\lim y_{\\alpha _{2}}=y_{0}$ .",
"One can further find a common sub-net index set $\\mathcal {I}_{0}\\subset \\mathcal {I}$ such that $\\lim _{\\beta \\in \\mathcal {I}_{0}}x_{\\beta }=x_{0}$ and $\\lim _{\\beta \\in \\mathcal {I}_{0}}y_{\\beta }=y_{0}$ .",
"Since $x_{\\beta }=t_{\\beta }y_{\\beta }$ , $\\lim _{\\beta \\in \\mathcal {I}_{0}}t_{\\beta }=\\infty $ and multiplication is continuous this implies that $x_{0}=\\lim _{\\beta \\in \\mathcal {I}_{0}}x_{\\beta }=\\lim _{\\beta \\in \\mathcal {I}_{0}}t_{\\beta }y_{\\beta }=\\infty $ but this contradicts to the fact that $x_{0}\\in K_{1}$ .",
"This contradiction proves the lemma.",
"Now we will show the following lemma which will shorten the proof of our main theorem.",
"The proof of the following lemma is adapted from [13]: Lemma 5 Let $G$ be a locally compact abelian Hausdorff topological group with Pontryagin dual $\\Gamma $ .",
"Let $\\phi \\in C_{0}(G)$ and $\\theta \\in C_{0}(\\Gamma )$ each have compact supports.",
"Then $D_{\\theta }M_{\\phi }$ is a compact operator on $L^{2}(G)$ where $D_{\\theta }=\\mathcal {F}^{-1}M_{\\theta }\\mathcal {F}$ .",
"Let $K_{1}\\subset G$ and $K_{2}\\subset \\Gamma $ be compact supports of $\\phi $ and $\\theta $ respectively.",
"Then for any $f\\in L^{2}(G)$ we have $&&(D_{\\theta }M_{\\phi }f)(t)=\\int _{K_{2}}\\gamma (t)\\theta (\\gamma )\\left(\\int _{K_{1}}\\overline{\\gamma (\\tau )}\\phi (\\tau )f(\\tau )d\\lambda (\\tau )\\right)d\\tilde{\\lambda }(\\gamma )\\\\&&=\\int _{K_{1}}\\left(\\phi (\\tau )\\int _{K_{2}}\\gamma (t\\tau ^{-1})\\theta (\\gamma )d\\tilde{\\lambda }(\\gamma )\\right)f(\\tau )d\\lambda (\\tau )=\\int _{K_{1}}k(t,\\tau )f(\\tau )d\\lambda (\\tau )$ where $k(t,\\tau )=\\phi (\\tau )\\int _{K_{2}}\\gamma (t\\tau ^{-1})\\theta (\\gamma )d\\tilde{\\lambda }(\\gamma )$ Now consider $& &\\int _{G}\\int _{G}\\mid k(t,\\tau )\\mid ^{2}d\\lambda (t)d\\lambda (\\tau )=\\int _{G}\\int _{G}\\mid \\phi (\\tau )\\int _{K_{2}}\\gamma (t\\tau ^{-1})\\theta (\\gamma )d\\tilde{\\lambda }(\\gamma )\\mid ^{2}d\\lambda (t)d\\lambda (\\tau )\\\\&&\\le \\parallel \\phi \\parallel _{\\infty }^{2}\\int _{K_{1}}\\int _{G}\\mid \\check{\\theta }(t\\tau ^{-1})\\mid ^{2}d\\lambda (t)d\\lambda (\\tau )=\\parallel \\phi \\parallel _{\\infty }^{2}\\int _{K_{1}}\\parallel \\check{\\theta }\\parallel _{2}d\\lambda (\\tau )\\\\&&=\\parallel \\phi \\parallel _{\\infty }^{2}\\int _{K_{1}}\\parallel \\theta \\parallel _{2}d\\lambda (\\tau )=\\parallel \\phi \\parallel _{\\infty }^{2}\\parallel \\theta \\parallel _{2}\\lambda (K_{1})<\\infty $ This implies that $D_{\\theta }M_{\\phi }$ is Hilbert-Schmidt and hence compact.",
"Now we are ready to prove our main theorem as follows: Main Theorem Let $G$ be a non-compact,locally compact abelian Hausdorff topological group whose Pontryagin dual $\\Gamma $ is partially ordered and let $\\Gamma ^{+}$ be the semigroup of positive elements of $\\Gamma $ .",
"Suppose that $\\Gamma ^{+}$ separates the points of $G$ i.e.",
"for any $t_{1},t_{2}\\in G$ with $t_{1}\\ne t_{2}$ there is $\\gamma \\in \\Gamma ^{+}$ such that $\\gamma (t_{1})\\ne \\gamma (t_{2})$ .",
"Let $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))=C^{*}(\\mathcal {T}(G)\\cup F(C(\\dot{\\Gamma ^{+}})))$ be the C*-algebra generated by Toeplitz operators and Fourier multipliers on $H^{2}(G)$ .",
"Then for the character space $M(\\Psi )$ of $\\Psi (C_{0}(G),C(\\dot{\\Gamma ^{+}}))$ we have $M(\\Psi )\\cong (\\dot{G}\\times \\lbrace \\infty \\rbrace )\\cup (\\lbrace \\infty \\rbrace \\times \\dot{\\Gamma ^{+}})$ We will use Power's Theorem.",
"In the setup of Power's theorem $C_{1}=\\mathcal {T}(G)$ and $C_{2}=F(C(\\dot{\\Gamma ^{+}}))$ .",
"By corollary 3 we have $M(C_{1})=\\dot{G}$ and we have $M(C_{2})=\\dot{\\Gamma ^{+}}$ .",
"So we need to determine $(t,\\gamma )\\in \\dot{G}\\times \\dot{\\Gamma ^{+}}$ satisfying for $0\\le \\phi ,\\theta \\le 1$ , $\\phi (t)=\\theta (\\gamma )=1$ implies $\\parallel T_{\\phi }D_{\\theta }\\parallel =1$ .",
"Let $(t,\\gamma )\\in G\\times \\Gamma ^{+}$ .",
"Let $\\phi \\in C(\\dot{G})$ , $\\theta \\in C(\\dot{\\Gamma ^{+}})$ such that $0\\le \\phi ,\\theta \\le 1$ and $\\phi (t)=\\theta (\\gamma )=1$ .",
"Let us also assume that $\\theta $ and $\\phi $ have compact supports.",
"Let $\\tilde{\\theta }\\in C(\\dot{\\Gamma })$ such that $0\\le \\tilde{\\theta }\\le 1$ , $\\tilde{\\theta }$ has compact support and $\\tilde{\\theta }\\mid _{C(\\dot{\\Gamma ^{+}})}=\\theta $ .",
"Since $\\parallel T_{\\phi }D_{\\theta }\\parallel \\le \\parallel M_{\\phi }D_{\\tilde{\\theta }}\\parallel _{L^{2}(G)}$ it suffices to show that $\\parallel M_{\\phi }D_{\\tilde{\\theta }}\\parallel _{L^{2}(G)}<1$ .",
"We will also assume that $\\phi (s)<1$ $\\forall s\\in G-\\lbrace t\\rbrace $ .",
"Since $(M_{\\phi }D_{\\tilde{\\theta }})^{*}=D_{\\tilde{\\theta }}M_{\\phi }$ and $D_{\\tilde{\\theta }}M_{\\phi }$ is compact by Lemma 5, $M_{\\phi }D_{\\tilde{\\theta }}$ is also compact.",
"Hence $M_{\\phi }D_{\\tilde{\\theta }}(M_{\\phi }D_{\\tilde{\\theta }})^{*}=M_{\\phi }D_{\\tilde{\\theta }}^{2}M_{\\phi }$ is a compact self-adjoint operator on $L^{2}(G)$ and this implies that $\\parallel M_{\\phi }D_{\\tilde{\\theta }}^{2}M_{\\phi }\\parallel =\\mu $ where $\\mu $ is the largest eigenvalue of $M_{\\phi }D_{\\tilde{\\theta }}^{2}M_{\\phi }$ .",
"Let $f\\in L^{2}(G)$ be the corresponding eigenvector such that $\\parallel f\\parallel _{2}=1$ , then we have $\\mu =\\parallel \\mu f\\parallel _{2}=\\parallel (M_{\\phi }D_{\\tilde{\\theta }}^{2}M_{\\phi }f)\\parallel _{2}<\\parallel D_{\\tilde{\\theta }}^{2}M_{\\phi }f\\parallel _{2}\\le 1$ since $\\phi (s)<1$ $\\forall s\\in G-\\lbrace t\\rbrace $ .",
"This implies that $\\parallel D_{\\tilde{\\theta }}M_{\\phi }\\parallel ^{2}=\\parallel M_{\\phi }D_{\\tilde{\\theta }}^{2}M_{\\phi }\\parallel <1$ .",
"This means that $(t,\\gamma )\\notin M(\\Psi )$ $\\forall (t,\\gamma )\\in G\\times \\Gamma ^{+}$ .",
"So if $(t,\\gamma )\\in M(\\Psi )$ then either $t=\\infty $ or $\\gamma =\\infty $ .",
"Now let $t\\in \\dot{G}$ and $\\gamma =\\infty $ .",
"Let $\\phi \\in C(\\dot{G})$ and $\\theta \\in C(\\dot{\\Gamma ^{+}})$ such that $0\\le \\phi ,\\theta \\le 1$ and $\\phi (t)=\\theta (\\infty )=1$ .",
"Observe that $P=D_{\\chi _{\\Gamma ^{+}}}$ where $\\chi _{\\Gamma ^{+}}$ is the characteristic function of $\\Gamma ^{+}$ .",
"So we have $D_{\\theta }T_{\\phi }=D_{\\theta }D_{\\chi _{\\Gamma ^{+}}}M_{\\phi }=D_{\\chi _{\\Gamma ^{+}}}D_{\\theta }M_{\\phi }=D_{\\theta }M_{\\phi }$ .",
"Since $\\mathcal {F}$ is unitary we have $\\parallel D_{\\theta }M_{\\phi }\\parallel _{H^{2}(G)}=\\parallel \\mathcal {F}D_{\\theta }M_{\\phi }\\mathcal {F}^{-1}\\parallel _{L^{2}(\\Gamma ^{+})}=\\parallel M_{\\theta }\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1}\\parallel _{L^{2}(\\Gamma ^{+})}$ Since $\\theta (\\infty )=1$ we have $\\forall \\epsilon >0$ , $\\exists \\gamma _{0}\\in \\Gamma ^{+}$ such that $1-\\epsilon \\le \\theta (\\gamma )\\le 1$ $\\forall \\gamma \\ge \\gamma _{0}$ .",
"Consider the operator $S_{\\gamma _{0}}:L^{2}(\\Gamma ^{+})\\rightarrow L^{2}(\\Gamma ^{+})$ defined as $(S_{\\gamma _{0}}f)(\\gamma )=f(\\gamma _{0}^{-1}\\gamma )$ , then $S_{\\gamma _{0}}$ is an isometry.",
"Observe that $& &(\\mathcal {F}^{-1}S_{\\gamma _{0}}f)(t)=(\\check{S_{\\gamma _{0}}f})(t)=\\int _{\\Gamma ^{+}}\\gamma (t)f(\\gamma _{0}^{-1}\\gamma )d\\tilde{\\lambda }(\\gamma )\\\\& &=\\int _{\\Gamma ^{+}}\\gamma _{0}(t)u(t)f(u)d\\tilde{\\lambda }(u)=\\gamma _{0}(t)\\check{f}(t)=(M_{\\gamma _{0}}\\check{f})(t)$ Hence we have $S_{\\gamma _{0}}=\\mathcal {F}M_{\\gamma _{0}}\\mathcal {F}^{-1}$ which implies that $S_{\\gamma _{0}}(\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1})=(\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1})S_{\\gamma _{0}}.$ Now let $f\\in L^{2}(\\Gamma ^{+})$ such that $\\parallel (\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1})f\\parallel _{2}>1-\\epsilon $ and $\\parallel f\\parallel _{2}=1$ then for $g=\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1}f$ we have $\\parallel M_{\\theta }S_{\\gamma _{0}}g\\parallel _{2}\\ge (1-\\epsilon )^{2}$ since $S_{\\gamma _{0}}g$ is supported on $\\lbrace \\gamma :\\gamma \\ge \\gamma _{0}\\rbrace $ , $\\theta (\\gamma )\\ge 1-\\epsilon $ $\\forall \\gamma \\ge \\gamma _{0}$ and $\\parallel S_{\\gamma _{0}}g\\parallel _{2}\\ge 1-\\epsilon $ .",
"Since $S_{\\gamma _{0}}g=(\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1})S_{\\gamma _{0}}f$ we have $\\parallel (M_{\\theta }\\mathcal {F}M_{\\gamma _{0}}\\mathcal {F}^{-1})(S_{\\gamma _{0}}f)\\parallel _{2}\\ge (1-\\epsilon )^{2}.$ Since $S_{\\gamma _{0}}$ is an isometry we have $\\parallel S_{\\gamma _{0}}f\\parallel _{2}=1$ and this implies that $\\parallel M_{\\theta }\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1}\\parallel \\ge (1-\\epsilon )^{2}$ $\\forall \\epsilon >0$ .",
"Therefore we have $\\parallel M_{\\theta }\\mathcal {F}M_{\\phi }\\mathcal {F}^{-1}\\parallel =\\parallel D_{\\theta }T_{\\phi }\\parallel =1$ .",
"Hence $(t,\\infty )\\in M(\\Psi )$ $\\forall t\\in \\dot{G}$ .",
"Now let $\\gamma \\in \\dot{\\Gamma ^{+}}$ and $t=\\infty $ .",
"Let $\\phi \\in C(\\dot{G})$ and $\\theta \\in C(\\dot{\\Gamma ^{+}})$ such that $0\\le \\phi ,\\theta \\le 1$ and $\\phi (\\infty )=\\theta (\\gamma )=1$ .",
"Since $\\phi (\\infty )=1$ ,for any $\\epsilon >0$ there is a compact subset $K_{1}\\subset G$ such that $1-\\epsilon \\le \\phi (t)\\le 1$ $\\forall t\\notin K_{1}$ .",
"Let $\\tilde{\\theta }=\\chi _{\\Gamma ^{+}}\\theta $ .",
"Then we have $D_{\\theta }T_{\\phi }=D_{\\theta }D_{\\chi _{\\Gamma ^{+}}}M_{\\phi }=D_{\\chi _{\\Gamma ^{+}}\\theta }M_{\\phi }=D_{\\tilde{\\theta }}M_{\\phi }.$ Let $\\epsilon >0$ be given.",
"Let $g\\in H^{2}(G)$ so that $\\parallel g\\parallel _{2}=1$ and $\\parallel D_{\\tilde{\\theta }}g\\parallel _{2}\\ge 1-\\epsilon $ .",
"Let $K_{2}\\subset G$ be a compact subset of $G$ so that $(\\int _{K_{2}}\\mid g(t)\\mid ^{2}d\\lambda (t))^{\\frac{1}{2}}\\ge 1-\\epsilon .$ By Lemma 4 we have $t_{0}\\in G$ such that $K_{1}\\cap (t_{0}K_{2})=\\emptyset $ .",
"Let $(S_{t_{0}}g)(t)=g(tt_{0}^{-1})$ then $(\\int _{t_{0}K_{2}}\\mid S_{t_{0}}g(t)\\mid ^{2}d\\lambda (t))^{\\frac{1}{2}}=(\\int _{K_{2}}\\mid g(t)\\mid ^{2}d\\lambda (t))^{\\frac{1}{2}}\\ge 1-\\epsilon .$ and this implies that $\\parallel S_{t_{0}}g-M_{\\phi }S_{t_{0}}g\\parallel _{2}\\le 2\\epsilon .$ We observe that $S_{t_{0}}=\\mathcal {F}^{-1}M_{\\hat{t_{0}}}\\mathcal {F}$ where $\\hat{t_{0}}:\\Gamma \\rightarrow \\mathbb {C}$ defined as $\\hat{t_{0}}(\\gamma )=\\gamma (t_{0})$ .",
"This implies that $S_{t_{0}}$ is unitary and we have $D_{\\tilde{\\theta }}S_{t_{0}}=S_{t_{0}}D_{\\tilde{\\theta }}$ .",
"Since $\\parallel D_{\\tilde{\\theta }}\\parallel =1$ we have $\\parallel D_{\\tilde{\\theta }}S_{t_{0}}g-D_{\\tilde{\\theta }}M_{\\phi }S_{t_{0}}g\\parallel _{2}\\le 2\\epsilon .$ Since $S_{t_{0}}$ is unitary for $f=S_{t_{0}}g$ we have $\\parallel f\\parallel _{2}=1$ and $D_{\\tilde{\\theta }}S_{t_{0}}=S_{t_{0}}D_{\\tilde{\\theta }}$ together with $\\parallel D_{\\tilde{\\theta }}g\\parallel _{2}\\ge 1-\\epsilon $ implies that $\\parallel D_{\\tilde{\\theta }}M_{\\phi }f\\parallel _{2}\\ge 1-3\\epsilon .$ Since $\\epsilon >0$ is arbitrary we have $\\parallel D_{\\tilde{\\theta }}M_{\\phi }\\parallel =\\parallel D_{\\theta }T_{\\phi }\\parallel =1$ .",
"Therefore we have $(\\infty ,\\gamma )\\in M(\\Psi )$ , $\\forall \\gamma \\in \\dot{\\Gamma ^{+}}$ .",
"Our theorem is thus proven."
],
[
"Acknowledgements",
"The author wishes to express his sincere thanks to Prof. Rıza Ertürk of Hacettepe University for useful discussions on Lemma 4. tocpart tocchapterBIBLIOGRAPHY"
]
] | 1403.0253 |
[
[
"Stored Electromagnetic Energy and Quality Factor of Radiating Structures"
],
[
"Abstract This paper deals with the old yet unsolved problem of defining and evaluating the stored electromagnetic energy - a quantity essential for calculating the quality factor, which reflects the intrinsic bandwidth of the considered electromagnetic system.",
"A novel paradigm is proposed to determine the stored energy in the time domain leading to the method, which exhibits positive semi-definiteness and coordinate independence, i.e.",
"two key properties actually not met by the contemporary approaches.",
"The proposed technique is compared with two up-to-date frequency domain methods that are extensively used in practice.",
"All three concepts are discussed and compared on the basis of examples of varying complexity, starting with lumped RLC circuits and finishing with realistic radiators."
],
[
"Introduction",
"In physics, an oscillating system is traditionally characterized [1] by its oscillation frequency and quality factor $Q$ , which gives a measure of the lifetime of free oscillations.",
"At its high values, the quality factor $Q$ is also inversely proportional to the intrinsic bandwidth in which the oscillating system can effectively be driven by external sources [2], [3].",
"The concept of quality factor $Q$ as a single frequency estimate of relative bandwidth is most developed in the area of electric circuits [4] and electromagnetic radiating systems [3].",
"Its evaluation commonly follows two paradigms.",
"As far as the first one is concerned, the quality factor is evaluated from the knowledge of the frequency derivative of input impedance [5], [6], [7].",
"As for the second paradigm, the quality factor is defined as $2\\pi $ times the ratio between the cycle mean stored energy and the cycle mean lost energy [5], [8].",
"Generally, these two concepts yield distinct results, but come to identical results in the case of vanishingly small losses, the reason being the Foster's reactance theorem [9], [10].",
"The evaluation of quality factor by means of frequency derivative of input impedance was made very popular by the work of Yaghjian and Best [11] and is widely used in engineering practice [12], [13] thanks to its property of being directly measurable.",
"Recently, this concept of quality factor has also been expressed as a bilinear form of source current densities [14], which is very useful in connection with modern numerical software tools [15].",
"Regardless of the mentioned advantages, the impedance concept of quality factor suffers from a serious drawback of being zero in specific circuits [16], [17] and/or radiators [18], [17] with evidently non-zero energy storage.",
"This unfortunately prevents its usage in modern optimization techniques [19].",
"The second paradigm, in which the quality factor is evaluated via the stored energy and lost energy, is not left without difficulties either.",
"In the case of non-dispersive components, the cycle mean lost energy does not pose a problem and may be evaluated as a sum of the cycle mean radiated energy and the cycle mean energy dissipated due to material losses [20].",
"Unfortunately, in the case of a non-stationary electromagnetic field associated with radiators, the definition of stored (non-propagating) electric and magnetic energies presents a problem that has not yet been satisfactorily solved [3].",
"The issue comes from the radiation energy, which does not decay fast enough in radial direction, and is in fact infinite in stationary state [21].",
"In order to overcome the infinite values of total energy, the evaluation of stored energy in radiating systems is commonly accompanied by the technique of extracting the divergent radiation component from the well-known total energy of the system [20].",
"This method is somewhat analogous to the classical field [22] re-normalization.",
"Most attempts in this direction have been performed in the domain of time-harmonic fields.",
"The pioneering work in this direction is the equivalent circuit method of Chu [21], in which the radiation and energy storage are represented by resistive and reactive components of a complex electric circuit describing each spherical mode.",
"This method was later generalized by several works of Thal [23], [24].",
"Although powerful, this method suffers from fundamental drawback of spherical harmonics expansion, which is unique solely in the exterior of the sphere bounding the sources.",
"Therefore, the circuit method cannot provide any information on the radiation content of the interior region, nor on the connection of energy storage with the actual shape of radiator.",
"The radiation extraction for spherical harmonics has also been performed directly at the field level.",
"The classical work in this direction comes from Collin and Rothschild [25].",
"Their proposal leads to good results for canonical systems [25], [26], [27], and has been analytically shown self-consistent outside the radian-sphere [28].",
"Similarly to the work of Chu, this procedure is limited by the use of spherical harmonics to the exterior of the circumscribing sphere.",
"The problem of radiation extraction around radiators of arbitrary shape has been for the first time attacked by Fante [29] and Rhodes [30], giving the interpretation to the Foster's theorem [10] in open problems.",
"The ingenious combination of the frequency derivative of input impedance and the frequency derivative of far-field radiation pattern led to the first general evaluation of stored energy.",
"Fante's and Rhodes' works have been later generalized by Yaghjian and Best [11], who also pointed out an unpleasant fact that this method is coordinate-dependent.",
"A scheme for minimisation of this dependence has been developed [11], but it was not until the work of Vandenbosch [31] who, generalizing the expressions of Geyi for small radiators [32] and rewriting the extraction technique into bilinear forms of currents, was able to reformulate the original extraction method into a coordinate-independent scheme.",
"A noteworthy discussion of various forms of this extraction technique can be found in the work of Gustafsson and Jonsson [33].",
"It was also Gustafsson et al.",
"who emphasized [19] that under certain conditions, this extraction technique fails, giving negative values for specific current distributions.",
"Hence the aforementioned approach remains incomplete too [34].",
"The problem of stored energy has seldom been addressed directly in the time domain.",
"Nevertheless, there are some interesting works dealing with time-dependent energies.",
"Shlivinski expanded the fields into spherical waves in time domain [35], [36], introducing time domain quality factor that qualifies the radiation efficiency of pulse-fed radiators.",
"Collarday [37] proposed a brute force method utilizing the finite differences technique.",
"In [38], Vandenbosch derived expressions for electric and magnetic energies in time domain that however suffer from an unknown parameter called storage time.",
"A notable work of Kaiser [39] then introduced the concept of rest electromagnetic energy, which resembles the properties of stored energy, but is not identical to it [40].",
"The knowledge of the stored electromagnetic energy and the capability of its evaluation are also tightly connected with the question of its minimization [21], [41], [42], [43], [44].",
"Such lower bound of the stored energy would imply the upper bound to the available bandwidth, a parameter of great importance for contemporary communication devices.",
"In this paper, a scheme for radiation energy extraction is proposed following a novel line of reasoning in the time domain.",
"The scheme aims to overcome the handicaps of the previously published works, and furthermore is able to work with general time-dependent source currents of arbitrary shape.",
"It is presented together with the two most common frequency domain methods, the first being based on the time-harmonic expressions of Vandenbosch [31] and the second using the input impedance approximation introduced by Yaghjian and Best [11].",
"All three concepts are closely investigated and compared on the basis of examples of varying complexity.",
"The working out of all three concepts starts solely from the currents flowing on a radiator, which are usually given as a result in modern electromagnetic simulators.",
"This raises challenging possibilities of modal analysis [45] and optimization [46].",
"The paper is organized as follows.",
"Two different concepts of quality factor $Q$ that are based on electromagnetic energies (in both, the frequency and time domain), are introduced in §.",
"Subsequently, the quality factor $Q$ derived from the input impedance is formulated in terms of currents on a radiator in §.",
"The following two sections present numerical examples: § treats non-radiating circuits and § deals with radiators.",
"The results are discussed in § and the paper is concluded in §."
],
[
"Energy concept of quality factor $Q$",
"In the context of energy, the quality factor is most commonly defined as $Q= 2 \\pi \\frac{\\langle \\mathcal {W}_\\mathrm {sto}\\left( t \\right)\\rangle }{W_\\mathrm {lost}} = 2 \\pi \\frac{W_\\mathrm {sto}}{W_\\mathrm {lost}},$ where a time-harmonic steady state with angular frequency $\\omega _0$ is assumed, with $\\mathcal {W}_\\mathrm {sto}\\left( t \\right)$ as the electromagnetic stored energy, $\\langle \\mathcal {W}_\\mathrm {sto}\\left( t \\right) \\rangle = W_\\mathrm {sto}$ as the cycle mean of $\\mathcal {W}_\\mathrm {sto}\\left( t \\right)$ and $W_\\mathrm {lost}$ as the lost electromagnetic energy during one cycle [20].",
"In conformity with the font convention introduced above, in the following text, the quantities defined in the time domain are stated in calligraphic font, while the frequency domain quantities are indicated in the roman font.",
"A typical $Q$ -measurement scenario is depicted in figure REF , which shows a radiator fed by a shielded power source.",
"Figure: A device with unknown QQ that is fed by a shielded power source with internal resistance R 0 R_0.The input impedance in the time-harmonic steady state at the frequency $\\omega _0$ seen by the source is $Z_\\mathrm {in}$ .",
"Assuming that the radiator is made of conductors with ideal non-dispersive conductivity $\\sigma $ and lossless non-dispersive dielectrics, we can state that the lost energy during one cycle, needed for (REF ), can be evaluated as $W_\\mathrm {lost}= \\int \\limits _\\alpha ^{\\alpha +T} i_0 \\left( t\\right) u_0 \\left( t \\right) t̥ = \\frac{\\pi }{\\omega _0} \\mathrm {Re}\\lbrace Z_\\mathrm {in}\\rbrace I_0^2 = W_\\mathrm {r}+ W_\\sigma ,$ where $I_0$ is the amplitude of $i_0 \\left( t \\right)$ (see figure REF ), $W_\\mathrm {r}$ represents the cycle mean radiation loss and $W_\\sigma $ stands for the energy lost in one cycle via conduction.",
"The part $W_\\sigma $ of (REF ) can be calculated as $W_\\sigma = \\frac{\\pi }{\\omega _0} \\int \\limits _{V} \\sigma \\left| {E} \\left({r}, \\omega _0 \\right) \\right|^2 V̥,$ with $V$ being the shape of radiator and ${E}$ being the time-harmonic electric field intensity under the convention ${\\mathcal {E}} \\left(t \\right) = \\mathrm {Re}\\lbrace {E} \\left( \\omega \\right) \\mathrm {e}^{J\\omega t}\\rbrace $, $J= \\sqrt{-1}$ .",
"At the same time, the near-field of the radiator [47] contains the stored energy $\\mathcal {W}_\\mathrm {sto}\\left( t \\right)$ , which is bound to the sources and does not escape from the radiator towards infinity.",
"The evaluation of the cycle mean energy $W_\\mathrm {sto}$ is the goal of the following §REF and §REF , in which the power balance [10] is going to be employed."
],
[
"Stored energy in time domain",
"This subsection presents a new paradigm of stored energy evaluation.",
"The first step consists in imagining the spherical volume $V_1$ (see figure REF ) centred around the system, whose radius is large enough to lie in the far-field region [47].",
"The total electromagnetic energy content of the sphere (it also contains heat $W_\\sigma $ ) is $\\mathcal {W} \\left( V_1, t \\right) = \\mathcal {W}_\\mathrm {sto} \\left( t \\right) + \\mathcal {W}_\\mathrm {r} \\left( V_1, t \\right),$ where $\\mathcal {W}_\\mathrm {r} \\left( V_1, t \\right)$ is the energy contained in the radiation fields that have already escaped from the sources.",
"Let us assume that the power source is switched on at $t = -\\infty $ , bringing the system into a steady state, and then switched off at $t = t_\\mathrm {off}$ .",
"For $t \\in [ t_\\mathrm {off},\\infty )$ the system is in a transient state, during which all the energy $\\mathcal {W} \\left( V_1, t_\\mathrm {off} \\right)$ will either be transformed into heat at the resistor $R_0$ and the radiator's conductors or radiated through the bounding envelope $S_1$ .",
"Explicitly, Poynting's theorem [10] states that the total electromagnetic energy at time $t_\\mathrm {off}$ can be calculated as $\\begin{split}\\mathcal {W} \\left( V_1, t_\\mathrm {off} \\right) = & \\,\\, R_0 \\int \\limits _{t_\\mathrm {off}}^{\\infty } i_{\\mathrm {R}_0}^2 (t) t̥ + \\int \\limits _{t_{\\mathrm {off}}}^{\\infty } \\int \\limits _{V} {\\mathcal {E}} \\left({r}, t \\right) \\cdot {\\mathcal {J}} \\left({r}, t \\right) V̥ t̥ \\\\& + \\int \\limits _{t_{\\mathrm {off}}}^{\\infty } \\oint \\limits _{S_1} \\Big ( {\\mathcal {E}}_\\mathrm {far} \\left({r}, t \\right) \\times {\\mathcal {H}}_\\mathrm {far} \\left({r}, t \\right) \\Big ) \\cdot t̥,\\end{split}$ in which $S_1$ lies in the far-field region.",
"As a special yet important example, let us assume a radiating device made exclusively of perfect electric conductors (PEC).",
"In that case, the far-field can be expressed as [20] Hfar ( r, t ) = -14 c0 V ' n0 J ( r ', t ' )R , Efar ( r, t ) = -4 V ' J ( r ', t ' ) - ( n0 J ( r ', t ' )) n0R in which $c_0$ is the speed of light, $R = \\left| {r} - {r} ^{\\prime } \\right|$ , ${n}_ 0 = \\left( {r} - {r} ^{\\prime } \\right) / R$, $t ^{\\prime } = t - R / c_0$ stands for the retarded time and the dot represents the derivative with respect to the time argument, i.e.",
"${\\mathcal {\\dot{J}}} \\left( {r} ^{\\prime }, t ^{\\prime } \\right) = \\frac{\\partial {\\mathcal {J}} \\left( {r} ^{\\prime }, \\tau \\right)}{\\partial \\tau } \\Bigg |_{\\tau = t ^{\\prime }}.$ Since we consider the far-field, we can further write [48] $R \\approx r$ for amplitudes, $R \\approx r - {r}_0 \\cdot {r} ^{\\prime }$ for time delays, with ${n}_0 \\approx {r}_0$ and $r = \\left|{r} \\right|$ .",
"Using (REF )–(REF ) and the above-mentioned approximations, the last term in (REF ) can be written as $\\begin{split}\\int \\limits _{t_{\\mathrm {off}}}^{\\infty } & \\oint \\limits _{S_1} \\Big ( {\\mathcal {E}}_\\mathrm {far} \\left( {r}, t \\right) \\times {\\mathcal {H}}_\\mathrm {far} \\left( {r}, t \\right) \\Big ) \\cdot {r}_0 t̥ = \\frac{1}{Z_0} \\int \\limits _{t_{\\mathrm {off}}}^{\\infty } \\oint \\limits _{S_1} \\left| {\\mathcal {E}}_\\mathrm {far} \\left( {r}, t \\right) \\right|^2 t̥ \\\\&= \\frac{\\mu ^2}{Z_0 \\left( 4 \\pi \\right)^2} \\int \\limits _{t_{\\mathrm {off}}}^{\\infty } \\int \\limits _0^\\pi \\int \\limits _0^{2 \\pi } \\left| \\int \\limits _{V ^{\\prime }} \\Big ({\\mathcal {\\dot{J}}} \\left( {r} ^{\\prime }, t ^{\\prime } \\right) - \\left( {r}_0 \\cdot {\\mathcal {\\dot{J}}} \\left( {r} ^{\\prime }, t ^{\\prime } \\right)\\right) {r}_0 \\Big ) \\right|^2 \\sin \\theta t̥,\\end{split}$ where $t ^{\\prime } = t - r / c_0 + {r}_0 \\cdot {r} ^{\\prime } / c_0$ , where $Z_0$ is the free space impedance and where the relation ${\\mathcal {H}}_\\mathrm {far} \\left( {r}, t \\right) = \\frac{{r}_0 \\times {\\mathcal {E}}_\\mathrm {far} \\left( {r}, t \\right)}{Z_0}$ has been used.",
"Utilizing (REF ) and (REF ), we are thus able to find the total electromagnetic energy inside $S_1$ , see figure REF for graphical representation.",
"Figure: Graphical representation of the total electromagnetic energy evaluation via () for a loss-less radiator excited by ideal voltage source.",
"Panel (a) shows a steady state just before t=t off t=t_{\\mathrm {off}}, when the steady state radiation (orange wavelets) as well as the steady state stored energy (blue cloud) were maintained by the source.",
"Panel (b) shows that after the source is switched off, the existing radiation travels to S 1 S_1 (and some of it also passes S 1 S_1) while a new radiation (blue wavelets) emerges at the expense of the stored energy.",
"Panel (c) depicts the time t≫t off t \\gg t_{\\mathrm {off}} when the stored energy is almost exhausted.",
"Capturing all wavelets for t>t off t > t_{\\mathrm {off}} by means of integral () gives the total energy within the capturing surface S 1 S_1.Note here that the total electromagnetic energy content of the sphere could also be expressed as $\\mathcal {W} \\left( V_1 , t_\\mathrm {off} \\right) = \\frac{1}{2} \\int \\limits _{V_1} \\left( \\mu \\left| {\\mathcal {H}} \\left( {r}, t_\\mathrm {off} \\right) \\right|^2 + \\epsilon \\left| {\\mathcal {E}} \\left( {r}, t_\\mathrm {off} \\right) \\right|^2\\right) V̥$ which can seem to be simpler than the aforementioned scheme.",
"The simplicity is, however, just formal.",
"The main disadvantage of (REF ) is that the integration volume includes also the near-field region, where the fields are rather complex (and commonly singular).",
"Furthermore, contrary to (REF ), the radius of the sphere plays an important role in (REF ) unlike in (REF ), where it appears only via a static time shift $r / c_0$ .",
"In fact, it will be shown later on that this dependence can be completely eliminated in the calculation of stored energy.",
"In order to obtain the stored energy $\\mathcal {W}_\\mathrm {sto}\\left( t_\\mathrm {off}\\right)$ inside $S_1$ we, however, need to know the radiation content of the sphere at $t = t_\\mathrm {off}$ .",
"A thought experiment aimed at attaining it is presented in figure REF .",
"It exploits the properties of (REF ).",
"Figure: Graphical representation of the radiated energy evaluation via () for a loss-less radiator excited by ideal voltage source.",
"Panel (a) shows a steady state just before t=t off t=t_{\\mathrm {off}}, when the steady state radiation (orange wavelets) as well as the steady state stored energy (blue cloud) were maintained by the source.",
"Panel (b) shows that at t≥t off t \\ge t_{\\mathrm {off}}, the radiating currents are modified so to inhibit any radiation, although they possibly create a new energy storage (green cloud).",
"The radiation emitted before t=t off t=t_{\\mathrm {off}} (orange wavelets) is unaffected by this modification.",
"Panel (c) depicts the time t≫t off t \\gg t_{\\mathrm {off}} when almost all radiation passed S 1 S_1.",
"The radiation content of the sphere S 1 S_1 is evaluated via ().",
"The green stored energy does not participate as it is not represented by radiation, and is consequently not captured by the integral ().Consulting the figure, let us imagine that during the calculation of $\\mathcal {W} \\left( V_1, t_\\mathrm {off} \\right)$ we were capturing the time course of the current ${\\mathcal {J}} \\left( {r} ^{\\prime }, t \\right)$ at every point.",
"In addition, let us assume that we define an artificial current ${\\mathcal {J}}_{\\mathrm {freeze}} \\left( {r} ^{\\prime }, t \\right)$ as ${\\mathcal {J}}_{\\mathrm {freeze}} \\left( {r} ^{\\prime }, t \\right) = \\left\\lbrace \\begin{array}{ll}{\\mathcal {J}} \\left( {r} ^{\\prime }, t \\right), & t < t_\\mathrm {off}\\\\& \\\\{\\mathcal {J}} \\left( {r} ^{\\prime }, t_\\mathrm {off} \\right), & t \\ge t_\\mathrm {off}\\\\\\end{array} \\right.$ and use it inside (REF ) instead of the true current ${\\mathcal {J}} \\left( {r} ^{\\prime }, t \\right)$ .",
"The expression (REF ) then claims that for $t < t_\\mathrm {off}$ the artificial current ${\\mathcal {J}}_{\\mathrm {freeze}} \\left( {r} ^{\\prime }, t \\right)$ is radiating in the same way as in the case of the original problem, but for $t > t_\\mathrm {off}$ , the radiation is instantly stopped.",
"Therefore, if we now evaluate (REF ) over the new artificial current, it will give exactly the radiation energy $\\mathcal {W}_\\mathrm {r} \\left( V_1, t_\\mathrm {off} \\right)$ , which has escaped from the sources before $t_\\mathrm {off}$ .",
"Subtracting it from $\\mathcal {W} \\left( V_1, t_\\mathrm {off} \\right)$ , we obtain the stored energy $\\mathcal {W}_\\mathrm {sto}\\left( t_\\mathrm {off}\\right)$ and averaging over one period, we obtain the cycle mean stored energy $W_\\mathrm {sto}= \\langle \\mathcal {W}_\\mathrm {sto}\\left( t_\\mathrm {off} \\right) \\rangle = \\frac{1}{T} \\int \\limits _{\\alpha }^{\\alpha +T} \\mathcal {W}_\\mathrm {sto}\\left(t_\\mathrm {off}\\right) .$ With respect to the freezing of the current, it is important to realize that this could mean an indefinite accumulation of charge at a given point.",
"However, it is necessary to consider this operation as to be performed on the artificial impressed sources, which can be chosen freely.",
"Figure: Sketch of the far-field cancellation.",
"The circumscribing sphere S 1 {S}_1 can be advantageously stretched right around the radiator, since outside this smallest sphere, the first and the second run are identically subtracted.When subtracting the radiated energy from the total energy, it is important to take into account that for $t < t_\\mathrm {off}$ , the currents were the same in both situations.",
"Thus defining $D = \\max \\lbrace \\left| {r}^{\\prime }\\right|\\rbrace $, we can state that for $t < t_\\mathrm {off} + \\left( r - D \\right) / c_0$ , the integrals (REF ) will exactly cancel during the subtraction, see figure REF .",
"The relation (REF ) can then be safely evaluated only for $t ^{\\prime } = t - D / c_0 + {r}_0 \\cdot {r} ^{\\prime } / c_0$ (the worst-case scenario depicted in figure REF b), which means that the currents need to be saved only for $t > t_\\mathrm {off} - 2 D / c_0$.",
"It is crucial to take into consideration that this is equivalent to say that, after all, the bounding sphere $S_1$ does not need to be situated in the far-field.",
"It is sufficient (and from the computational point of view also advantageous), if $S_1$ is the smallest circumscribing sphere centred in the coordinate system, for the rest of the far-field is cancelled anyhow, see figure REF a.",
"As a final note, we mention that even though the above-described method relies on the integration on a spherical surface, the resulting stored energy properly takes into account the actual geometry of the radiator, representing thus a considerable generalization of the time domain prescription for the stored energy proposed in [28] which is able to address only the regions outside the smallest circumscribing sphere.",
"Further properties of the method are going to be presented on numerical results in § and will be detailed in §."
],
[
"Stored Energy in Frequency Domain",
"This subsection rephrases the stored energy evaluation by Vandenbosch [31], which approaches the issue in the frequency domain, utilizing the complex Poynting's theorem that states [20] that $- \\frac{1}{2} \\left\\langle {E}, {J} \\right\\rangle = P_\\mathrm {m}- P_\\mathrm {e}+ 2J\\omega \\left(W_\\mathrm {m}- W_\\mathrm {e}\\right) = P_\\mathrm {in},$ in which $P_\\mathrm {in}$ is the cycle mean complex power, the terms $P_\\mathrm {m}$ and $P_\\mathrm {e}$ form the cycle mean radiated power $P_\\mathrm {m}- P_\\mathrm {e}$ and $2\\omega \\left(W_\\mathrm {m}-W_\\mathrm {e}\\right)$ is the cycle mean reactive net power, and $\\left\\langle {u}, {v} \\right\\rangle = \\int \\limits _{V} {u} \\left({r} \\right) \\cdot {v}^\\ast \\left({r} \\right) $ is the inner product [49].",
"In the classical treatment of (REF ), $W_\\mathrm {m}$ and $W_\\mathrm {e}$ are commonly taken [20] as $\\mu \\left| {H} \\right|^2 / 4$ and $\\epsilon \\left| {E} \\right|^2 / 4$ that are integrated over the entire space.",
"Both of them are infinite for the radiating system.",
"Nonetheless, when electromagnetic potentials are utilized [50], the complex power balance (REF ) can be rewritten as $P_\\mathrm {in}= P_\\mathrm {m}^{{A}}-P_\\mathrm {e}^\\varphi +2J\\omega \\left(W_\\mathrm {m}^{{A}}-W_\\mathrm {e}^\\varphi \\right) = \\frac{J\\omega }{2} \\left( \\left\\langle {A}, {J} \\right\\rangle - \\left\\langle \\varphi , \\rho \\right\\rangle \\right),$ where ${A}$ represents the vector potential, $\\varphi $ represents the scalar potential, and $\\rho $ stands for the charge density.",
"As an alternative to the classical treatment, it is then possible to write $W_\\mathrm {m}^{{A}}- J\\frac{P_\\mathrm {m}^{{A}}}{2\\omega } = \\displaystyle \\frac{1}{4} \\left\\langle {A} , {J} \\right\\rangle $ and $W_\\mathrm {e}^\\varphi - J\\frac{P_\\mathrm {e}^\\varphi }{2\\omega } = \\displaystyle \\frac{1}{4} \\left\\langle \\varphi , \\rho \\right\\rangle $ without altering (REF ).",
"However, it is important to stress that in such case, $W_\\mathrm {m}^{{A}}$ in (REF ) and $W_\\mathrm {e}^\\varphi $ in (REF ) generally represent neither stored nor total magnetic and electric energies [20].",
"Some attempts have been undertaken to use (REF ) and (REF ) as stored magnetic and electric energies even in non-stationary cases [51].",
"These attempts were however faced with extensive criticism [52], [53], mainly due to the variance of separated energies under gauge transformations.",
"Regardless of the aforementioned issues, (REF ) and (REF ) were modified [31] in an attempt to obtain the stored magnetic and electric energies.",
"This modification reads WmWmA+ Wrad2, WeWe+ Wrad2, where the particular term $W_\\mathrm {rad}= \\displaystyle \\mathrm {Im}\\left\\lbrace k \\left( k^2 \\left\\langle {L}_{\\mathrm {rad}} {J}, {J} \\right\\rangle - \\left\\langle {L}_{\\mathrm {rad}} \\nabla \\cdot {J} , \\nabla \\cdot {J} \\right\\rangle \\right) \\right\\rbrace $ is associated with the radiation field, and the operator ${L}_{\\mathrm {rad}} {U} = \\frac{1}{16 \\pi \\epsilon \\omega ^2} \\int \\limits _{V ^{\\prime }} {U} \\left( {r} ^{\\prime } \\right) \\mathrm {e}^{-Jk R} \\, \\mathrm {d} V ^{\\prime }$ is defined using $k = \\omega / c_0$ as the wavenumber.",
"The electric currents ${J}$ are assumed to flow in a vacuum.",
"For computational purposes, it is also beneficial to use the radiation integrals for vector and scalar potentials [47], and rewrite (REF ), (REF ) as [14] $W_\\mathrm {m}^{{A}}- J\\frac{P_\\mathrm {m}^{{A}}}{2\\omega } = \\displaystyle k^2 \\left\\langle {L} {J}, {J} \\right\\rangle $ and $W_\\mathrm {e}^\\varphi - J\\frac{P_\\mathrm {e}^\\varphi }{2\\omega } = \\displaystyle \\left\\langle {L} \\nabla \\cdot {J}, \\nabla \\cdot {J} \\right\\rangle ,$ with ${L} {U} = \\frac{1}{16 \\pi \\epsilon \\omega ^2} \\int \\limits _{V ^{\\prime }} {U} \\left( {r} ^{\\prime } \\right) \\frac{\\mathrm {e}^{-Jk R}}{R} .$ It is suggested in [31] that $\\widetilde{W}_\\mathrm {sto}= \\widetilde{W}_\\mathrm {m}+ \\widetilde{W}_\\mathrm {e}$ is the stored energy $W_\\mathrm {sto}$ .",
"Yet this statement cannot be considered absolutely correct, since as it was shown in [19], [54], $\\widetilde{W}_\\mathrm {sto}$ can be negative.",
"Consequently, it is necessary to conclude that $\\widetilde{W}_\\mathrm {sto}$ , defined by the frequency domain concept [31], can only approximately be equal to the stored energy $W_\\mathrm {sto}$ , resulting in $\\widetilde{W}_\\mathrm {sto}\\approx W_\\mathrm {sto},$ and then by analogy with (REF ) $\\widetilde{Q} = 2 \\pi \\frac{\\widetilde{W}_\\mathrm {sto}}{W_\\mathrm {lost}} = 2 \\pi \\frac{\\widetilde{W}_\\mathrm {m}+ \\widetilde{W}_\\mathrm {e}}{W_\\mathrm {lost}} \\approx Q$ is defined."
],
[
"Fractional bandwidth concept of quality factor $Q$",
"It is well-known that for $Q \\gg 1$, the quality factor $Q$ is approximately inversely proportional to the fractional bandwidth (FBW) $Q_{Z}\\approx \\frac{\\chi }{\\mathrm {FBW}},$ where $\\chi $ is a given constant and $\\mathrm {FBW} = (\\omega ^+ - \\omega ^-) / \\omega _0$ , [11].",
"The quality factor $Q$ , which is known to fulfil (REF ), was found by Yaghjian and Best [11] utilizing an analogy with RLC circuits and using the transition from conductive to voltage standing wave ratio bandwidth.",
"Its explicit definition reads $Q_{Z}= \\frac{\\omega }{2\\, \\mathrm {Re}\\lbrace P_{\\mathrm {in}}\\rbrace } \\left|\\frac{\\partial P_\\mathrm {in}}{\\partial \\omega }\\right| = \\left| Q_R + JQ_X \\right|,$ where the total input current at the radiator's port is assumed to be normalized to $I_0 = 1$ A.",
"The differentiation of the complex power in the form of (REF ) can be used to find the source definition of (REF ), and leads to [14] QR = PmA+ Pe+ Prad+ P Wlost, QX = 2Wsto+ WWlost, in which $\\frac{P_\\mathrm {rad}}{2\\omega } = \\displaystyle \\mathrm {Re}\\left\\lbrace k \\left( k^2 \\left\\langle {L}_{\\mathrm {rad}} {J} , {J} \\right\\rangle - \\left\\langle {L}_{\\mathrm {rad}} \\nabla \\cdot {J}, \\nabla \\cdot {J} \\right\\rangle \\right) \\right\\rbrace ,$ and $W_{\\partial \\omega }- J\\frac{P_{\\partial \\omega }}{2\\omega } = k^2 \\left(\\left\\langle {L} {J}, D {J} \\right\\rangle + \\left\\langle L {J}^\\ast , D {J}^\\ast \\right\\rangle \\right)- \\left( \\left\\langle {L} \\nabla \\cdot {J} , D \\nabla \\cdot {J} \\right\\rangle + \\left\\langle L \\nabla \\cdot {J}^\\ast , D \\nabla \\cdot {J}^\\ast \\right\\rangle \\right).$ The operator $D$ is defined as $D {U} = \\omega \\frac{\\partial {U} }{\\partial \\omega }.$ As particular cases of (), we obtain the Rhodes' definition [5] of the quality factor $Q$ as $|Q_X|$ and the definition (REF ) as $Q_X$ , omitting the $W_{\\partial \\omega }$ term from ().",
"For the purposes of this paper, we can observe in (REF ), (REF ), () and () that the stored energy in the case of the FBW concept is equivalent to $W_\\mathrm {sto}^\\mathrm {FBW}\\equiv \\frac{1}{2} \\left| \\frac{\\partial P_\\mathrm {in}}{\\partial \\omega } \\right| =\\left|\\widetilde{W}_\\mathrm {sto}+ W_{\\partial \\omega }-J\\frac{P_\\mathrm {m}^{{A}}+ P_\\mathrm {e}^\\varphi + P_\\mathrm {rad}+ P_{\\partial \\omega }}{2\\omega }\\right|,$ but we remark here that (REF ) was not intended to be the stored energy [11]."
],
[
"Non-radiating circuits",
"The previous §§ and have defined three generally different concepts of stored energy, namely $W_\\mathrm {sto}$ , $\\widetilde{W}_\\mathrm {sto}$ and $W_\\mathrm {sto}^\\mathrm {FBW}$ .",
"Given that $W_\\mathrm {lost}$ is uniquely defined, we can benefit from the use of the corresponding dimensionless quality factors $Q$ , $\\widetilde{Q}$ and $Q_{Z}$ for comparing them.",
"This is performed in § for non-radiating circuits and in § for radiating systems.",
"Particularly, in §, we assume passive lossy but non-dispersive and non-radiating one-ports."
],
[
"Time domain stored energy for lumped elements",
"Following the general procedure indicated in §REF , let us assume a general RLC circuit that was for $t \\in \\left(-\\infty ,t_\\mathrm {off}\\right)$ fed by a time-harmonic source (current or voltage) $s\\left( t \\right) = \\sin \\left(\\omega _0 t \\right)$ which was afterwards switched off for $t \\in \\left[ t_\\mathrm {off},\\infty \\right)$ .",
"Since the circuit is non-radiating, the total energy $\\mathcal {W} \\left( V_1, t_\\mathrm {off}\\right)$ is directly equal to $\\mathcal {W}_\\mathrm {sto}\\left( t_\\mathrm {off} \\right)$ .",
"Furthermore, a careful selection of the voltage (or current) source for a given circuit helps us to eliminate the internal resistance of the source.",
"So we get $W_\\mathrm {sto}= \\sum \\limits _k \\frac{R_k}{T} \\int \\limits _{\\alpha }^{\\alpha +T} \\int \\limits _{t_\\mathrm {off}}^{\\infty } i_{\\mathrm {R},k}^2 \\left( t \\right) t̥ ,$ where $i_{\\mathrm {R},k} \\left( t \\right)$ is the transient current in the $k$ -th resistor.",
"The currents $i_{\\mathrm {R},k}$ are advantageously evaluated in the frequency domain.",
"The Fourier transform of the source reads [55] $S\\left(\\omega \\right) = \\frac{J\\pi }{2} \\left( \\delta \\left(\\omega +\\omega _0\\right) - \\delta \\left(\\omega -\\omega _0\\right) \\right) + \\frac{\\mathrm {e}^{-J\\omega t_\\mathrm {off}}}{2} \\left( \\frac{\\mathrm {e}^{J\\omega _0 t_\\mathrm {off}}}{\\omega - \\omega _0} - \\frac{\\mathrm {e}^{-J\\omega _0 t_\\mathrm {off}}}{\\omega + \\omega _ 0}\\right).$ We can then write $I_{\\mathrm {R},k} \\left(\\omega \\right) = T_{\\mathrm {R}_k}\\left(\\omega \\right) S\\left(\\omega \\right)$, where $T_{\\mathrm {R}_k} \\left(\\omega \\right)$ represents the transfer function.",
"Consequently $\\begin{split}i_{\\mathrm {R},k}\\left( t \\right) &= \\frac{1}{2 \\pi } \\int \\limits _{-\\infty }^{\\infty } T_{\\mathrm {R}_k}\\left(\\omega \\right) S\\left(\\omega \\right) \\mathrm {e}^{J\\omega t} \\\\& = \\frac{1}{2} \\mathrm {Im}\\left\\lbrace T_{\\mathrm {R}_k}\\left(\\omega _0\\right) \\mathrm {e}^{J\\omega _0 t} \\right\\rbrace + \\frac{\\omega }{4 \\pi } \\int \\limits _{-\\infty }^{\\infty } T_{\\mathrm {R}_k}\\left(\\omega \\right) \\left( \\frac{\\mathrm {e}^{J\\omega _0 t_\\mathrm {off}}}{\\omega - \\omega _0} - \\frac{\\mathrm {e}^{-J\\omega _0 t_\\mathrm {off}}}{\\omega + \\omega _0} \\right) \\mathrm {e}^{J\\omega \\left(t- t_\\mathrm {off}\\right)} .\\end{split}$ As the studied circuit is lossy, $T_{\\mathrm {R}_k} \\left(\\omega \\right)$ has no poles on the real $\\omega $ -axis and the second integral can be evaluated by the standard contour integration in the complex plane of $\\omega $ along the semi-circular contour in the upper $\\omega $ half-plane, while omitting the points $\\omega = \\pm \\omega _0$ .",
"The result of the contour integration for $t > t_\\mathrm {off}$ can be written as $i_{\\mathrm {R},k}\\left( t \\right) = \\frac{J}{2} \\sum \\limits _m \\mathop {\\mathrm {res}}\\limits _{\\omega \\rightarrow \\omega _{m,k}} \\Bigg \\lbrace T_{\\mathrm {R}_k} \\left(\\omega \\right) \\Bigg (\\frac{\\mathrm {e}^{J\\omega _0 t_\\mathrm {off}}}{\\omega - \\omega _0} - \\frac{\\mathrm {e}^{- J\\omega _0 t_\\mathrm {off}}}{\\omega + \\omega _0} \\Bigg ) \\mathrm {e}^{J\\omega \\left(t- t_\\mathrm {off}\\right)} \\Bigg \\rbrace ,$ where $\\omega _{m,k}$ are the poles of $T_{\\mathrm {R}_k}\\left(\\omega \\right)$ with $\\mathrm {Im}\\left\\lbrace \\omega _{m,k}\\right\\rbrace > 0$ .",
"The substitution of (REF ) into (REF ) gives the mean stored energy.",
"It is also important to realize that in this case, it is easy to analytically carry out both integrations involved in (REF ).",
"The result is obviously identical to the cycle mean of the classical definition of stored energy.",
"$\\mathcal {W}_\\mathrm {sto}\\left( t_\\mathrm {off} \\right) = \\frac{1}{2} \\left(\\sum \\limits _m L_m i_{\\mathrm {L},m}^2 \\left(t_\\mathrm {off} \\right) + \\sum \\limits _n C_n u_{\\mathrm {C},n}^2 \\left( t_\\mathrm {off} \\right) \\right),$ which is the lumped circuit form of (REF ), with $i_{\\mathrm {L},m} \\left( t \\right)$ being the current in the $m$ -th inductor $L_m$ and $u_{\\mathrm {C},n} \\left( t \\right)$ being the voltage on the $n$ -th capacitor $C_n$ ."
],
[
"Frequency domain stored energy for lumped elements",
"Without the radiation ($P_\\mathrm {rad}= 0, \\omega W_\\mathrm {rad}= 0$), the cycle mean of (REF ), which is also equal to the cycle mean (REF ), is identical to the frequency domain expression $\\widetilde{W}_\\mathrm {sto}= W_\\mathrm {m}^{{A}}+ W_\\mathrm {e}^\\varphi = \\frac{1}{4} \\left(\\sum \\limits _m L_m |I_{\\mathrm {L},m}|^2 + \\sum \\limits _n C_n |U_{\\mathrm {C},n}|^2 \\right) = \\frac{1}{4} \\int \\limits _{V} \\left( \\mu \\left|{H}\\right|^2 + \\epsilon \\left|{E} \\right|^2\\right)V̥,$ where $W_\\mathrm {m}^{{A}}$ and $W_\\mathrm {e}^\\varphi $ are defined by (REF ) and (REF ) respectively.",
"We thus conclude that $W_\\mathrm {sto}= \\widetilde{W}_\\mathrm {sto}$ and $Q= \\widetilde{Q}$ for non-radiating circuits."
],
[
"Frequency domain stored energy for lumped elements derived from FBW concept",
"In order to evaluate (REF ), the same procedure as in the derivation of Foster's reactance theorem [10] can be employed (keeping in mind the unitary input current, no radiation and assuming non-zero conductivity).",
"It results in $\\begin{split}W_\\mathrm {sto}^\\mathrm {FBW} &= \\left| \\frac{1}{4} \\int \\limits _{V} \\left( \\mu \\left|{H}\\right|^2 + \\epsilon \\left|{E} \\right|^2\\right)V̥ - \\frac{J\\sigma }{2}\\int \\limits _{V} {E}^\\ast \\cdot \\frac{\\partial {E}}{\\partial \\omega }V̥ \\right| \\\\&= \\Bigg | \\frac{1}{4} \\left( \\sum \\limits _m L_m |I_{\\mathrm {L},m}|^2 + \\sum \\limits _n C_n |U_{\\mathrm {C},n}|^2 \\right) - \\frac{J}{2} \\sum \\limits _k R_k I_{\\mathrm {R},k}^\\ast \\frac{\\partial I_{\\mathrm {R},k}}{\\partial \\omega }\\Bigg |,\\end{split}$ where $I_{\\mathrm {R},k}$ is the amplitude of the current through the $k$ -th resistor.",
"The formula indicated above clearly reveals the fundamental difference between $W_\\mathrm {sto}$ and $W_\\mathrm {sto}^\\mathrm {FBW}$ , which consists in the last term of RHS in (REF ).",
"It means that, in general, $W_\\mathrm {sto}^\\mathrm {FBW}$ does not represent the time-averaged stored energy."
],
[
"Results",
"In the previous §§REF –REF we have shown that for non-radiating circuits there is no difference between the quality factor defined in the time domain ($Q$ ) and the one defined in the frequency domain ($\\widetilde{Q}$ ).",
"Nevertheless, there is a substantial difference between $Q$ and $Q_{Z}$ , which is going to be presented in §REF using two representative examples depicted in figure REF .",
"Figure: Studied RLC circuits: (a) C 1 C_1 in series with parallel L 1 L_1 and R 1 R_1, and (b) C 2 C_2 in parallel with serial R 2 R_2 and L 2 L_2We do not explicitly consider simple series and parallel RLC circuits in this paper, since the three definitions of the stored energy and quality factor $Q$ deliver exactly the same results, i.e.",
"$Q = \\omega _0 L / R = \\omega _0 R C$.",
"This is attributable to the frequency independence of the current flowing through the resistor (the series resonance circuit), or of the voltage on the resistor (the parallel resonance circuit).",
"In those cases, the last term of (REF ) vanishes identically.",
"This fact is the very reason why the FBW approach works perfectly for radiators that can be approximated around resonance by a parallel or series RLC circuit.",
"However, it also means that for radiators that need to be approximated by other circuits, the approach may not deliver the correct energy.",
"This is probably the reason why this method seems to fail in the case of wideband radiators and radiators with slightly separated resonances.",
"In the case of circuits depicted in figure REF , the input impedances are $Z_\\mathrm {in}^{(a)} = \\displaystyle \\frac{1}{J\\omega C_1} + \\frac{1}{\\displaystyle \\frac{1}{R_1} + \\frac{1}{J\\omega L_1}}, \\quad Z_\\mathrm {in}^{(b)} = \\displaystyle \\frac{1}{\\displaystyle \\frac{1}{R_2 + J\\omega L_2} + J\\omega C_2},$ and the corresponding resonance frequencies read $\\omega _0^{(a)} = \\frac{R_1}{L_1}\\frac{1}{\\displaystyle \\sqrt{\\frac{C_1 R_1^2}{L_1}-1}},\\quad \\omega _0^{(b)} = \\frac{R_2}{L_2}\\displaystyle \\sqrt{\\frac{L_2}{C_2 R_2^2}-1},$ respectively.",
"Utilizing the method from §REF , it can be demonstrated that the energy quality factors are $Q^{(a)} = \\widetilde{Q}^{(a)} = \\frac{R_1}{\\omega _0^{(a)} L_1}, \\quad Q^{(b)} = \\widetilde{Q}^{(b)} = \\frac{\\omega _0^{(b)} L_2}{R_2},$ while the FBW quality factors equal $Q_{Z}^{(a)} = \\kappa ^{(a)} Q^{(a)}, \\quad Q_{Z}^{(b)} = \\kappa ^{(b)} Q^{(b)},$ where $\\kappa ^{(a)} = \\frac{1}{\\omega _0^{(a)} \\sqrt{L_1 C_1}}, \\quad \\kappa ^{(b)} = \\omega _0^{(b)} \\sqrt{L_2 C_2}.$ For the sake of completeness, it is useful to indicate that the quality factors proposed by Rhodes [5] are found to be $\\left|Q_X^{(a)} \\right| = \\left(\\kappa ^{(a)} \\right)^2 \\mathcal {Q}^{(a)}, \\quad \\left| Q_X^{(b)} \\right| = \\left(\\kappa ^{(b)} \\right)^2 \\mathcal {Q}^{(b)}.$ The comparison of the above-mentioned quality factors is depicted in figure REF using the parametrization by $R_i / L_i$ and $R_i C_i$ , where $i \\in \\left\\lbrace 1, 2\\right\\rbrace $ .",
"The circuit (a) in figure REF is resonant for $R_1 C_1 > L_1 / R_1$, whilst the circuit (b) in the same figure is resonant for $R_1 C_1 < L_1 / R_1$.",
"It can be observed that the difference between the depicted quality factors decreases as the quality factor rises and finally vanishes for $Q \\rightarrow \\infty $ .",
"On the other hand, there are significant differences for $Q < 2$ .",
"Figure: The quality factors for the two particular lumped RLC circuits of figure .",
"The curves correspond to R i /L i =1s -1 R_i / L_i = 1 \\,\\mathrm {s}^{-1}, with i∈{1,2}i \\in \\lbrace 1,2\\rbrace .Therefore, we can conclude that for general RLC circuits made of lumped (non-radiating) elements $W_\\mathrm {sto}\\equiv \\widetilde{W}_\\mathrm {sto}\\ne W_\\mathrm {sto}^\\mathrm {FBW}\\Longrightarrow Q\\equiv \\widetilde{Q} \\ne Q_{Z}.$"
],
[
"Radiating structures",
"The evaluation of the quality factor $Q$ for radiating structures is far more involved than for non-radiating circuits.",
"This is due to the fact that the radiating energy should be subtracted correctly.",
"Hence, the method proposed in §REF was implemented according to the flowchart depicted in figure REF .",
"The evaluation is done in Matlab [56].",
"The current density ${\\mathcal {J}} \\left({r} ^{\\prime } ,t\\right)$ and the current $i_{\\mathrm {R}_0} \\left(t\\right)$ flowing through the internal resistance of the source are the only input quantities used, see figure REF .",
"In particular cases treated in this section, we utilize the ideal voltage source that invokes $i_{\\mathrm {R}_0} \\left(t\\right) \\equiv 0$ , and thus the first integral in RHS of (REF ) vanishes.",
"Figure: Flowchart of the method proposed in §.",
"The time domain currents are processed according to the right-hand side of the flowchart.In order to verify the proposed approach, three types of radiators are going to be calculated, namely the centre-fed dipole, off-centre-fed dipole and Yagi-Uda antenna.",
"All these radiators are made of an infinitesimally thin-strip perfect electric conductor and operate in vacuum background.",
"Consequently, the second integral in RHS of (REF ) also vanishes.",
"The quality factor $Q$ calculated with the help of the novel method is going to be compared with the results of two remaining classical approaches detailed in §REF and §, which produced the quality factors $\\widetilde{Q}$ (REF ) and $Q_{Z}$ (REF ) respectively.",
"All essential steps of the method are going to be explained using the example of a centre-fed dipole in §REF .",
"Subsequently, in §REF and §REF , the method is going to be directly applied to more complicated radiators.",
"The most important properties of the novel method are going to be examined in the subsequent discussion §."
],
[
"Centre-fed thin-strip dipole",
"The first structure to be calculated is a canonical radiator: a dipole of the length $L$ and width $w = L/200$.",
"The dipole is fed by a voltage source [47] located in its centre.",
"The calculation starts in FEKO commercial software [57] in which the dipole is simulated.",
"The dipole is fed by a unitary voltage and the currents ${J} \\left({r} ^{\\prime }, \\omega \\right)$ are evaluated within the frequency span from $ka = 0$ to $ka \\approx 325$ for 8192 samples.",
"The resulting currents are imported into Matlab.",
"We define the normalized time $t_\\mathrm {n} = t \\omega _0 / \\left( 2 \\pi \\right)$ (see x-axis of figures REF and REF ), where $\\omega _0$ is the angular frequency that the quality factor $Q$ is going to be calculated at.",
"Then iFFT over $S \\left(\\omega \\right) {J} \\left({r} ^{\\prime }, \\omega \\right) $ , see (REF ), is applied, and the time domain currents ${\\mathcal {J}} \\left({r}^{\\prime },t\\right)$ with $\\Delta t_n = 0.02$ for $t_n \\in \\left(0 , 163\\right) $ are obtained.",
"The implementation details of iFFT, which must also contain singularity extraction of the source spectrum $S \\left(\\omega \\right)$ , are not discussed here, as they are not of importance to the method of quality factor calculation itself.",
"The next step consists in the evaluation of (REF ) for both, the original currents ${J} \\left({r} ^{\\prime }, t\\right)$ and frozen currents ${J}_\\mathrm {freeze} \\left({r} ^{\\prime }, t\\right)$, see figure REF .",
"Figure: Current flowing through the voltage gap of dipole (exact proportions of the antenna are indicated in the inset).",
"The blue line shows the steady state and the transient state of the original currents 𝒥r ' ,t{\\mathcal {J}} \\left({r}^{\\prime },t \\right), whilst the red line corresponds to the modified currents 𝒥 freeze r ' ,t{\\mathcal {J}}_\\mathrm {freeze}\\left( {r}^{\\prime },t\\right).",
"The depicted curves correspond to the source with the voltage ut=U 0 sinω 0 tHt off -tu\\left( t \\right) = U_0 \\sin \\left(\\omega _0 t \\right) H\\left(t_\\mathrm {off} - t\\right), where the U 0 U_0 was chosen so that the mean radiated power equals 0.50.5 W.Figure: Radiated power passing through the surface S 1 {S}_1 for a centre-fed dipole.",
"The meaning of the blue and red lines as well as the normalization of input voltage is the same as in figure .At this point, it is highly instructive to explicitly show the time course of the current at the centre of the dipole (see figure REF ), as well as the time course of the power passing through the surface $S_1$ in both aforementioned scenarios (original and frozen currents), see figure REF .",
"The source was switched off at $t_\\mathrm {n} = 0$.",
"During the following transient (blue lines in figure REF and figure REF ), all energy content of the sphere is lost by the radiation.",
"Within the second scenario, with all currents constant for $t \\ge t_\\mathrm {off}$ , the radiation of the dipole is instantaneously stopped at $t_\\mathrm {off} = 0$ .",
"The power radiated for $t_\\mathrm {off}>0$ (red line in figure REF ) then represents the radiation that existed at $t = t_\\mathrm {off}$ within the sphere, but needed some time to leave the volume.",
"Subtracting the blue and red curves in figure REF and integrating in time for $t \\ge t_\\mathrm {off}$ then gives the stored energy at $t = t_\\mathrm {off}$.",
"In order to construct the course of $\\mathcal {Q}\\left(t_\\mathrm {off}\\right)$ , the stored energy is evaluated for six different switch-off times $t_\\mathrm {off}$ .",
"The resulting $\\mathcal {Q}\\left(t_\\mathrm {off}\\right)$ is then fitted by $\\mathcal {Q}\\left(t_\\mathrm {off}, \\omega _0\\right) = A + B \\sin \\left(2\\omega _0 t_\\mathrm {off}+\\beta \\right).$ The fitting was exact (within the used precision) in all fitted points, which allowed us to consider (REF ) as an exact expression for all $t_\\mathrm {off}$ .",
"The constant $A$ then equals $Q \\left(\\omega _0\\right)$ .",
"We are typically interested in the course of $Q$ with respect to the frequency.",
"Repeating the above-explained procedure for varying $\\omega _0$ , we obtain the red curve in figure REF .",
"In the same figure, the comparison with $\\widetilde{Q}$ from §REF (blue curve) and $Q_{Z}$ from § (green curve) is depicted.",
"To calculate $\\widetilde{Q}$ by means of (REF ), (REF ), (REF ), (REF ) in the frequency domain, we used the currents ${J} \\left({r} ^{\\prime }, \\omega \\right)$ from FEKO and renormalized them with respect to the input current $I_0 = 1$ A.",
"Similarly, the calculation of the FBW quality factor $Q_{Z}$ (REF ) is performed for the same source currents with identical normalization, and is based on expression (REF ) and all subsequent relations integrated in Matlab.",
"Figure: Frequency dependence of the quality factors for a centre-fed dipole."
],
[
"Off-centre-fed thin-strip dipole",
"The second example is represented by an off-centre-fed dipole, which is known to exhibit the zero value of $Q_{Z}$ [18], for $ka \\approx 6.2$ provided that the delta gap is placed at $0.23 L$ from the bottom of the dipole.",
"The dipole has the same parameters as in the previous example, except the position of feeding (see the inset in figure REF ).",
"Figure: Comparison of all quality factors in the case of off-centre-fed dipole (see the inset for exact proportions of the antenna).It is apparent from figure REF that the quality factor $Q$ based on the new stored energy evaluation does not suffer from drop-off around $ka \\approx 6.2$ , and in fact yields similar values as $\\widetilde{Q}$ , including the same trend."
],
[
"Yagi-Uda antenna",
"Yagi-Uda antenna was selected as a representative of quite complex structure that the method can ultimately be tested on.",
"The antenna has the same dimensions as in [11] and is depicted in the inset in figure REF .",
"Since this antenna has non-unique phase centre, it can serve as an ideal candidate for verification of the coordinate independence of the novel method.",
"Figure: Radiated power passing through the surface S 1 {S}_1 in the case of Yagi-Uda antenna.",
"The meaning of the blue and red lines as well as the normalization of input voltage is the same as in figure .",
"The antenna proportions are depicted in the inset of figure .The results were calculated in the same way as in the previous examples, and are indicated in figures REF and REF .",
"The comparison between the results in figure REF and those related to the dipole in figure REF clearly reveals that the transient state is remarkably longer in the case of Yagi-Uda antenna, which means that the longer integration time is required.",
"Furthermore, it can be seen (red curve for $t > t_\\mathrm {off}$ ) that the bounding sphere contains a considerable amount of radiation that should be subtracted.",
"The accuracy of this subtraction is embodied in figure REF , which shows the quality factors $Q$ , $\\widetilde{Q}$ , and $Q_{Z}$ .",
"Notice the similarity between $Q$ and $\\widetilde{Q}$ .",
"Figure: Comparison of all Q factors of Yagi-Uda antenna (antenna proportions are stated in the inset)."
],
[
"Discussion",
"Based on the previous sections, important properties of the novel time domain technique can be isolated and discussed.",
"This discussion also poses new and so far unanswered questions that can be addressed in future.",
"The coordinate independence / dependence constitutes an important issue of many similar techniques evaluating the stored electromagnetic energy.",
"Contrary to the radiation energy subtraction of Fante [29], Rhodes [30], Yaghjian and Best [11], or Gustafsson and Jonsson [33], the new time-domain method can be proved to be coordinate-independent.",
"It means that the same results are obtained irrespective of the position and rotation of the coordinate system.",
"Due to the explicit reference to coordinates, this statement in question may not be completely obvious from (REF ).",
"However it should be noted that any potential spatial shift or rotation of coordinate system emerges only as a static time shift of the received signal at the capturing sphere.",
"Such static shift is irrelevant to the energy evaluation due to the integration over semi-infinite time interval.",
"The positive semi-definiteness represents another essential characteristic.",
"It should be immanent in all theories concerning the stored energy.",
"Although (REF ) contains the absolute value, it is difficult to mathematically prove the positive semi-definiteness of the stored energy evaluation as a whole, because it is not automatically granted that the integration during the second run integrates smaller amount of energy than the integration during the first run.",
"Despite that, we can anticipate the expected behaviour from the physical interpretation of the method, which stipulates that the energy integrated in the second run must have been part of the first run as well.",
"At worst, the subtraction of both runs can give null result.",
"This observation is in perfect agreement with the numerical results.",
"Nevertheless, the exact and rigorous proof admittedly remains an unresolved issue that is to be addressed in the future.",
"Unlike the methods of Fante [29], Rhodes [5] or Collin and Rothschild [25], the obvious benefit of the novel method consists in its ability to account for a shape of the radiator, not being restricted to the exterior of circumscribing sphere.",
"Finally, it is crucial to realize that the novel method is not restricted to the time-harmonic domain, but can evaluate the stored energy in any general time-domain state of the system.",
"This raises new possibilities for analyzing radiators in the time domain, namely the ultra-wideband radiators and other systems working in the pulse regime."
],
[
"Conclusion",
"Three different concepts aiming to evaluate the stored electromagnetic energy and the resulting quality factor $Q$ of radiating system were investigated.",
"The novel time domain scheme constitutes the first one, while the second one utilizes time-harmonic quantities and classical radiation energy extraction.",
"The third one is based on the frequency variation of radiator?s input impedance.",
"All methods were subject to in-depth theoretical comparison and their differences were presented on general non-radiating RLC networks as well as common radiators.",
"It was explicitly shown that the most practical scheme based on the frequency derivative of the input impedance generally fails to give the correct quality factor, but may serve as a very good estimate of it for structures that are well approximated by series or parallel resonant circuits.",
"In contrast, the frequency domain concept with far-field energy extraction was found to work correctly in the case of general RLC circuits and simple radiators.",
"Unlike the newly proposed time domain scheme, it could however yield negative values of stored energy, which is actually known to happen for specific current distributions.",
"In this respect, the novel time domain method proposed in this paper could be denoted as reference, since it exhibits the coordinate independence, positive semi-definiteness, and most importantly, takes into account the actual radiator shape.",
"Another virtue of the novel scheme is constituted by the possibility to use it out of the time-harmonic domain, e.g.",
"in the realm of radiators excited by general pulse.",
"The follow-up work should focus on the radiation characteristics of separated parts of radiators or radiating arrays, the investigation of different time domain feeding pulses and their influence on performance of ultra-wideband radiators and, last but not least, on the theoretical formulation of the stored energy density generated by the new time domain method.",
"This manuscript does not contain primary data and as a result has no supporting material associated with the results presented.",
"All authors contributed to the formulation, did numerical simulations and drafted the manuscript.",
"All authors gave final approval for publication.",
"The authors are grateful to Ricardo Marques (Department of Electronics and Electromagnetism, University of Seville) and Raul Berral (Department of Applied Physics, University of Seville) for many valuable discussions that stimulated some of the core ideas of this contribution.",
"The authors are also grateful to Jan Eichler (Department of Electromagnetic Field, Czech Technical University in Prague) for his help with the simulations.",
"The authors would like to acknowledge the support of COST IC1102 (VISTA) action and of project 15-10280Y funded by the Czech Science Foundation.",
"We declare to have no competing interests."
]
] | 1403.0572 |
[
[
"Search for $W^\\prime$ signal via $tW^\\prime$ associated production at\n LHC"
],
[
"Abstract A variety of new physics models predict the existence of extra charged gauge bosons ($W^\\prime$).",
"It is verified that $W^\\prime$ and top quark associated production is a promising process to search for $W^\\prime$ signal at the LHC.",
"We study the collider signatures of multi-jets$+$lepton$+\\met$ by reconstructing the $tW'$ intermediate state through the decay modes of $W^\\prime \\to tb$, $W^\\prime \\to q \\bar q'$ and $W^\\prime \\to l \\nu$ respectively.",
"An angular distribution related to charged lepton and top quark moving direction is provided to distinguish alternative left-handed or right-handed chiral couplings of $W^\\prime$ to quarks in the hadronic decay modes.",
"The superiority on the search of $W^\\prime$ is demonstrated in the leptonic decay channel for the left-handed type interaction, which is forbidden for the right-handed type interaction in the most theories.",
"To be more realistic, the relevant standard model backgrounds are simulated.",
"We adopt various kinematic cuts to suppress the backgrounds and collect the integral luminosity needed for the corresponding process detected at the LHC with $3\\sigma$ sensitivity.",
"We also provide a forward-backward asymmetry which is related to the chirality of $W^\\prime$.",
"The successive studies can shed light on the potential searching for $W^\\prime$ signal as well as distinguishing typical new physics models."
],
[
"Introduction",
"Though the standard model (SM) has gained great success and SM-like Higgs boson is discovered at the LHC, it is still in progress to investigate the new phenomena induced by many new physics models, such as extra dimensional models [1], [2], [3], [4], grand unified theories [5], [6], [7] and left-right symmetric models [8], [9], [10], [11], [12], etc.",
"Among this kind of models, a new extra charged gauge boson $W^\\prime $ is proposed.",
"To discriminate different new physics models beyond the SM, it is crucial to search for the $W^\\prime $ production signal and study its properties at the LHC.",
"Recently, the latest experimental results have explored the potential to observe the heavy gauge bosons at the LHC.",
"Searching for a $W^\\prime $ boson with a signature of lepton and missing transverse energy has been performed by ATLAS and CMS collaborations.",
"The results show that no significant excess over the SM expectation has been observed through $W^\\prime \\rightarrow e\\nu $ or $W^\\prime \\rightarrow \\tau \\nu $ decay [13], [14].",
"On the other hand, the signature of $W^\\prime \\rightarrow tb$ has also been investigated at the center of mass energy of 7 and 8 TeV, and the observed limits are displayed on the cross section as well as $W^\\prime $ mass [15], [16], [17].",
"In general, $W^\\prime \\rightarrow l \\nu $ is the most prospective channel for its research at the LHC due to the unambiguous backgrounds, while $W^\\prime \\rightarrow tb$ decay channel also becomes important, especially within the models in which the couplings of $W^\\prime $ to leptons are extremely suppressed.",
"The collider signature of a top-philic $W^\\prime $ , which couples only to the third generation quarks of the SM, produced in association with a top quark is investigated at the LHC [18].",
"In addition, the investigation on the properties of $W^\\prime $ also contributes to searching for its signal at the LHC, such as the chirality of $W^\\prime $ coupling to SM particles.",
"In reference [19], $pp\\rightarrow W^\\prime \\rightarrow tb$ process is analyzed and the authors propose that the angular distribution for the charged lepton decayed from the top quarks can be a feature of the chirality of $W^\\prime $ with $W^\\prime \\rightarrow tb$ decay mode.",
"The extensively studies on the chiral property are carried out in the gauge interaction of $W^\\prime $ decaying into $W$ and Higgs boson, which can be used to distinguish $W^\\prime $ from the charged Higgs boson [20], [21].",
"In this paper we study the $W^\\prime $ production in association with top quark at the LHC and the future hadron colliders.",
"The dominant three decay modes of $W^\\prime \\rightarrow tb$ , $W^\\prime \\rightarrow q \\bar{q}^{\\prime }$ ($q$ stands for the light quark) and $W^\\prime \\rightarrow l \\nu $ are investigated with the collider signature of $5{\\rm jets}+l^{\\pm }+\\displaystyle {\\lnot }E_T$ , $3{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T$ and $3{\\rm jets}+l^{-}+\\displaystyle {\\lnot }E_T$ .",
"Searching for a massive $W^\\prime $ signature with the electroweak coupling to fermions through the $W^\\prime \\rightarrow tb$ decay channel is hard due to the large backgrounds.",
"The $W^\\prime \\rightarrow q \\bar{q}^{\\prime }$ channel shows evidence in the search of $W^\\prime $ at $\\sqrt{s}=33$ TeV.",
"Because the highly boosted charged lepton decayed from $W^{\\prime }$ differs from the backgrounds, the investigation of $W^\\prime \\rightarrow l \\nu $ channel contributes to the search of $W^{\\prime }$ signal.",
"Moreover, we provide the charged lepton angular distribution to distinguish the chirality of $W^\\prime $ .",
"This paper is organized as follows.",
"The couplings of $W^\\prime $ to SM particles are discussed in Sec.",
"II together with the mass constraints.",
"In Sec.",
"III, the numerical results with $W^\\prime \\rightarrow tb$ , $W^\\prime \\rightarrow q \\bar{q}^{\\prime }$ and $W^\\prime \\rightarrow l \\nu $ decay modes are listed respectively and the corresponding SM backgrounds are simulated.",
"Finally, we give a brief summary in Sec.",
"IV."
],
[
"Theoretical Framework and Mass Constraints",
"The heavy charged gauge boson appears in various models with different couplings.",
"We extract the $W^\\prime $ couplings to quark and lepton ($\\psi $ ) from the following general formula, ${\\cal L} = \\frac{ g_{L} }{\\sqrt{2}}V_{L}^{\\prime ij}\\bar{\\psi _u^i}\\gamma _{\\mu }P_{L}\\psi _d^j{{W^{\\prime +}_{L}}^{\\mu }}+\\frac{ g_{R}}{\\sqrt{2}}V_{R}^{\\prime ij}\\bar{\\psi _u^i}\\gamma _{\\mu }P_{R}\\psi _d^j{{W^{\\prime +}_{R}}^{\\mu }}+ {\\rm h.c.} \\ ,$ where $g_{L} (g_{R})$ is the coupling constant, $V_{L}^{\\prime }$ ($V_{R}^{\\prime }$ ) is a unitary matrix representing the fermion flavor mixing, and $P_{L,R}=(1\\mp \\gamma _5)/2$ is the left-, right-handed chiral projection operator.",
"To give a simplified result, we set $g_L=g_2,~g_R=0$ with the pure left-handed gauge boson $W^\\prime _L$ , $g_L=0,~g_R=g_2$ with the pure right-handed gauge boson $W^\\prime _R$ , where $g_2$ is the electro-weak coupling constant in the SM.",
"The mass parameter is one of the most crucial factors in the search of $W^\\prime $ as well as the interaction couplings.",
"From the proton anti-proton collisions, CDF and D$\\emptyset $ collaborations obtain the lower bound for a SM-like $W^\\prime $ at 0.8 and 1 TeV [22], [23], respectively.",
"The investigations on the K and B meson demonstrate that the $W^\\prime $ lower mass limit is around 2.5 TeV [24].",
"The recent searches of $W^\\prime $ have been performed by the ATLAS and CMS detectors at the LHC.",
"Based on the results of $W^\\prime \\rightarrow l \\nu $ decay channel, a $W^\\prime $ with sequential SM couplings is excluded at the 95$\\%$ credibility level for masses up to 2.55 and 2.90 TeV [13], [14].",
"As discussed in the references [9], [10], [11], [12], one can imagine neutrino being Dirac particle.",
"In this case, $W^{\\prime }_R$ would have the usual leptonic decay channel, which implies a lower limit on its mass about 3 TeV.",
"Independent of the nature of neutrino mass, the left-right symmetric theory implies a theoretical lower limit on the $W^\\prime _R$ mass of about 2.5 TeV [24], [25], [26].",
"Additionally, the right-handed $W^\\prime $ with the mass below 1.13 and 1.85 TeV is excluded through the reconstructed $tb$ resonances [15], [16], [17].",
"The search for the dijet mass spectrum, including quark-quark, quark-gluon, and gluon-gluon pairs, also presents mass bound $M_{W^{\\prime }}>1.51$ TeV [27].",
"In particular, the CMS detector illustrates the mass of right-handed $W^\\prime $ should be larger than 0.84 TeV through the $tW^{\\prime }$ associated production [28].",
"Figure: The branching fractions of W ' →tbW^\\prime \\rightarrow tb,W ' →qq ¯ ' W^\\prime \\rightarrow q\\bar{q}^{\\prime },and W ' →lvW^\\prime \\rightarrow lv.The branching fractions of $W^\\prime $ decay into fermions are shown in Fig.",
"REF .",
"Whatever the left-handed or right-handed $W^\\prime $ decay into light quarks leads to more than a half fractions in the massive region.",
"A disparate decay mode between the left-handed and right-handed $W^\\prime $ is that the $W^\\prime _R \\rightarrow l \\nu $ is forbidden due to only left-handed neutrino existed in the SM.",
"We completely study the signatures of $W^\\prime $ production in association with top quarks via the three decay modes, $W^\\prime \\rightarrow tb$ , $W^\\prime \\rightarrow q\\bar{q}^{\\prime }$ and $W^\\prime \\rightarrow l\\nu $ in the following section."
],
[
"Numerical Results and discussion",
"The tree level predominant partonic sub-process for $tW^\\prime $ production is the following process $g(p_1)+b(p_2)\\rightarrow t(p_3)+W^{\\prime -}(p_4),$ as depicted in Fig.",
"REF , where $p_i$ denotes the 4-momentum of the corresponding particle.",
"The $g \\bar{b} \\rightarrow \\bar{t} W^{\\prime +}$ process is not referred in this paper for the similar character under the $CP-$ invariance.",
"The LHC leads to a large probability to study the $W^\\prime $ for the larger gluon parton distribution than Tevatron.",
"Figure: Feynman diagrams for gb→tW '- gb\\rightarrow tW^{\\prime -} process at the tree level.The total cross section for the process $gb\\rightarrow tW^{\\prime {-}}$ can be expressed as $\\sigma =\\int f_{g}(x_{1})f_{b}(x_{2})\\hat{\\sigma }_{gb\\rightarrow tW^{\\prime ^{-}}}(\\hat{s})dx_{1}dx_{2},$ where $f_{g}(x_{1})(f_{b}(x_{2}))$ is the parton distribution function (PDF) of gluon (quark), $\\hat{s}$ is the partonic center of mass energy squared, and $\\hat{\\sigma }$ is the partonic level cross section for $gb\\rightarrow tW^{\\prime {-}}$ process.",
"To obtain the numerical results we set $V_{tb}=1$ , $M_{W}=80.399$ GeV, and $m_{t}=173.1$ GeV.",
"For PDF, we use CTEQ6L1[29].",
"The total cross section for $pp\\rightarrow tW^{\\prime {-}}$ process is displayed in Fig.",
"REF , as a function of $M$ representing the mass of $W^{\\prime -}_L$ and $W^{\\prime -}_R$ at the LHC with 14 TeV and 33 TeV.",
"In addition, we also give the results at the future hadron collider with 50 TeV and 80 TeV as a reference.",
"There is no discrepancy in the cross section of left-handed $W^\\prime $ production from right-handed.",
"The $tW^{\\prime -}$ cross section can be up to 15 (300) $fb$ for $M=1$ TeV with $\\sqrt{s}=14$ (33) TeV at the LHC.",
"While the observation of the $W^\\prime $ signal at 14 TeV LHC for the mass heavier than 3.3 TeV is difficult in the $tW^{\\prime -}$ associated production on the condition of the couplings chosen in this paper.",
"The $W^\\prime $ and top quark can not be observed directly at the LHC, thus we analyze the multi-jets plus lepton and missing transverse energy signal with different decay modes as the follows.",
"Figure: The total cross section as a function of W ' W^\\prime mass MM for pp→tW '- pp\\rightarrow tW^{\\prime -}process at the LHC for 14 TeV and 33 TeV, and future hadron collider for 50 TeV and 80 TeV."
],
[
"$W^{\\prime }\\rightarrow tb$ channel for {{formula:d1e8b321-afd0-4488-9513-c9800d05290e}} production ",
"First we explore the $W^{\\prime }\\rightarrow tb$ channel for the $pp\\rightarrow tW^{\\prime -}$ process at the LHC with $\\sqrt{s}=14$ and 33 TeV as the following process $pp\\rightarrow tW^{\\prime -}\\,\\rightarrow \\, t\\bar{t}b\\,\\rightarrow \\, bl^{+}\\,+\\, b\\bar{b}jj\\,+\\,\\displaystyle {\\lnot }E_T,\\\\$ $pp\\rightarrow tW^{\\prime -}\\,\\rightarrow \\, t\\bar{t}b\\,\\rightarrow \\, bjj\\,+\\, b\\bar{b}l^{-}\\,+\\,\\displaystyle {\\lnot }E_T,$ where the charged lepton is an excellent trigger for the event search.",
"Corresponding to process (REF ), the associated top quark semi-leptonically decays as $t\\rightarrow bl^{+}\\nu _{l}$ and the anti-top quark hadronically decays as $\\bar{t}\\rightarrow \\bar{b}jj$ .",
"While in process (REF ), the decay modes of top and anti-top quark are exchanged.",
"To be more realistic, the simulation at the detector is performed by smearing the leptons and jets energies according to the assumption of the Gaussian resolution parameterization $\\frac{\\delta (E)}{E}=\\frac{a}{\\sqrt{E}}\\oplus b,$ where $\\delta (E)/E$ is the energy resolution, $a$ is a sampling term, $b$ is a constant term, and $\\oplus $ denotes a sum in quadrature.",
"We take $a=5\\%$ , $b=0.55\\%$ for leptons and $a=100\\%$ , $b=5\\%$ for jets respectively[30].",
"The transverse momentum distributions of jets and charged lepton are shown in Fig.",
"REF (a) and (b) as well as the missing transverse energy for the precess $pp\\rightarrow tW^{\\prime -}\\rightarrow t\\bar{t}b\\rightarrow bl^{+}+b\\bar{b}jj+\\displaystyle {\\lnot }E_T$ .",
"In order to identify the isolated jet (lepton), the angular distribution between particle i and particle j is defined by $\\Delta R_{ij}=\\sqrt{\\Delta \\phi _{ij}^{2}+\\Delta \\eta _{ij}^{2}},$ where $\\Delta \\phi _{i j}$ denotes the difference between the particles' azimuthal angle, and $\\Delta \\eta _{ij}$ the difference between the particles' rapidity.",
"In Fig.",
"REF (c), we display the differential distributions $\\sigma ^{-1}d\\sigma /d\\Delta R$ for $\\Delta R=min(\\Delta R_{ij})$ .",
"Figure: (a) The normalized differential distributions with the transverse momentum of the jets (j 1 ,j 2 ,j 3 ,j 4 ,j 5 j_1,j_2,j_3,j_4,j_5)with p Tj 1 >p Tj 2 >p Tj 3 >p Tj 4 >p Tj 5 p_{Tj_1}>p_{Tj_2}>p_{Tj_3}>p_{Tj_4}>p_{Tj_5} for M=3M=3 TeVin the process of pp→tW L '- →bl + +bb ¯jj+¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow bl^{+}+b\\bar{b}jj+\\displaystyle {\\lnot }E_T at s=33\\sqrt{s}=33 TeV.",
"(b) The same as (a) but for the charged lepton (solid line) and transverse missing energy ¬E T \\displaystyle {\\lnot }E_T (dashed line).",
"(c) The minimal angular separation distributions between jets (solid line) and that between jets and the charged lepton (dashed line).The analysis of the whole process including the reconstruction of the intermediate resonances is propitious to select the $tW^{\\prime }$ production process from the substantial backgrounds.",
"Theoretically, the top (anti-top) quark with semi-leptonical decay can be reconstructed by one of the five jets, the charged lepton and the neutrino, while three of remaining jets can be used to reconstruct the anti-top (top) quark with hadronical decay.",
"Although the momentum of neutrino can not be directly recorded at the detector, one can resolve it by the kinematical constraint.",
"The neutrinos' transverse momentum is determined by the sum of the observable particles' transverse momentum according to the momentum conservation, and the longitudinal part can be solved through the on shell condition for the W-boson, $&&{\\bf p}_{\\nu T}=-({\\bf p}_{lT}+\\sum _{j=1}^{5}{\\bf p}_{jT}),\\nonumber \\\\&&m_{W}^{2}=(p_{\\nu }+p_{l})^{2}.$ Once the neutrino's momentum reconstructed, we can reconstruct the top or anti-top quark invariant mass $M_{jl\\nu }^{2}=(p_{l}+p_{\\nu }+p_{j})^{2},$ where j refers to any one of the five jets that makes the $M_{jl\\nu }$ be closest to the top quark mass.",
"Based on the above discussion, we employ the basic cuts to outstand the $tW^{\\prime }$ process as Cut $A_{1}$ : $&{\\rm For~~ 14~~and~~33~~ TeV}\\left\\lbrace \\begin{array}{ll}& p_{lT}>50~{\\rm GeV},~~~~p_{jT}>50~{\\rm GeV},~~~~\\displaystyle {\\lnot }E_T>50~{\\rm GeV},\\nonumber \\\\& |\\eta _{l}|<2.5,~~~~|\\eta _{j}|<2.5,~~~~\\Delta R_{jj(lj)}>0.4,\\nonumber \\\\& |M_{j_{a}l\\nu }-m_{t}|\\le 30~{\\rm GeV},~~~~|M_{j_{b}j_{c}j_{d}}-m_{t}|\\le 30~{\\rm GeV},\\nonumber \\\\&|M_{j_{b}j_{c}}-m_{W}|\\le 10~{\\rm GeV}.\\end{array} \\right.&$ Then the $W^\\prime $ mass peak can be reconstructed through the momentum of all the particles except the ones used to reconstruct the associated top or anti-top quark.",
"The distributions of $1/\\sigma (d\\sigma /dM_{tb}+d\\sigma /dM_{\\bar{t}b})$ are shown in Fig.",
"REF .",
"The resonance peak is significant in the invariant mass distribution of $tb$ , which will be a direct signal in the search of $W^\\prime $ .",
"In order to further purify the signal, we require the following cut Cut $A_{2}$ :$~~|M_{jj_{b}j_{c}j_{d}}-M|\\le 10\\%\\, M$ or $|M_{jj_{a}l\\nu }-M|\\le 10\\%\\, M$ , together with the remaining jet that not be used to reconstruct top or anti-top quark tagged as a b-jet.",
"The $b$ -tagging efficiency is assumed to be 60% while the miss-tagging efficiency of a light jet as a $b$ jet is taken as transverse momentum dependent [30]: $\\epsilon _l =\\left\\lbrace \\begin{array}{ll}\\displaystyle {\\frac{1}{150}}, &P_T< 100\\, {\\rm GeV},\\\\\\displaystyle {\\frac{1}{450}}\\,\\left(\\frac{P_T}{25\\,{\\rm GeV}}-1\\right), &100 \\,{\\rm GeV}\\le P_T<250\\, {\\rm GeV},\\\\\\displaystyle {\\frac{1}{50}}, & P_T\\ge 250\\, {\\rm GeV}.\\end{array} \\right.$ Figure: The distributions of 1/σ(dσ/dM tb +dσ/dM t ¯b )1/\\sigma (d\\sigma /dM_{tb}+d\\sigma /dM_{\\bar{t}b}) with respect tothe invariant mass reconstructed by top (anti-top) quark and the remaining jetfor M=1,2,3M=1,~2,~3 TeV for the process ofpp→tW L '- →bl + +bb ¯jj+¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow bl^{+}+b\\bar{b}jj+\\displaystyle {\\lnot }E_T after cut A 1 A_{1} at the LHC for 33 TeV.The cross section of process (REF ) at the LHC with $\\sqrt{s}=14 $ and 33 TeV after Cut $A_{1}$ and $A_{2}$ are displayed as a function of $M$ in Fig.",
"REF (a), which deriving from $W^\\prime _R$ is larger than $W^\\prime _L$ due to the discrepancy of the branching fraction $W^\\prime \\rightarrow tb$ .",
"If the mass of $W^\\prime $ is larger than 1 TeV, it will be negative to observe $W^\\prime $ from the process at the 14 TeV LHC, while the cross section can be enhanced more than one order of magnitude at 33 TeV.",
"Considering the effect from the background of $t\\bar{t} j$ , we give the integral luminosity needed at the LHC with $S/\\sqrt{B}=3\\sigma $ for various $M$ in Fig .REF (b).",
"The result shows that the $W^\\prime $ with mass below 1 TeV can be observed from the process (REF ) at the LHC with $\\sqrt{s}=33 $ TeV.",
"For a $W^\\prime $ with mass up to 3 TeV, only with the sizable couplings it can be detected in the $tW^{\\prime }$ associated production via the $W^\\prime \\rightarrow tb$ decay at the future LHC.",
"It makes no difference to the results for the process (REF ) as displayed in Fig.",
"REF .",
"Figure: (a) The total cross section as a function ofthe charged gauge bosons mass MM at s=14\\sqrt{s}=14 and 33 TeVfor the process of pp→tW '- →5 jets +l + +¬E T pp\\rightarrow tW^{\\prime -}\\rightarrow 5{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T after all the cuts.",
"(b) The integral luminosity needed at the LHC for s=14\\sqrt{s}=14 and 33 TeVat S/B=3σS/\\sqrt{B}=3\\sigma sensitivity.",
"The solid and dashed lines stand for W L '- W_L^{\\prime -} and W R '- W_R^{\\prime -} respectively, and the two straight dotted lines present the luminosity of 300 and 1000fb -1 1000~fb^{-1}.Figure: (a) The total cross section as a function ofthe charged gauge bosons mass MM at s=14\\sqrt{s}=14 and 33 TeVfor the process of pp→tW '- →5 jets +l - +¬E T pp\\rightarrow tW^{\\prime -}\\rightarrow 5{\\rm jets}+l^{-}+\\displaystyle {\\lnot }E_T after all the cuts.",
"(b) The integral luminosity needed at the LHC for s=14\\sqrt{s}=14 and 33 TeVat S/B=3σS/\\sqrt{B}=3\\sigma sensitivity.",
"The solid and dashed lines stand for W L '- W_L^{\\prime -} and W R '- W_R^{\\prime -} respectively, and the two straight dotted lines present the luminosity of 300 and 1000fb -1 1000~fb^{-1}."
],
[
"$W^\\prime \\rightarrow q\\bar{q}^{\\prime }$ channel for {{formula:613c9032-0963-4adf-9bc1-5976c1613493}} production",
"According to Fig.",
"REF , the search of $W^\\prime $ is in favor of $W^\\prime \\rightarrow q\\bar{q}^{\\prime }$ decay modes for the large branch fraction.",
"Then we focus on the following process $pp\\rightarrow tW^{\\prime -} \\rightarrow bW^+ W^{\\prime -}\\rightarrow bl^{+}+jj+\\displaystyle {\\lnot }E_T,$ where the charged lepton is from the associated top quark.",
"The differential distributions with the transverse momentum of the three jets and charged lepton are shown in Fig.",
"REF (a) and (b) as well as the missing transverse energy.",
"The first two jets with the largest transverse momentum mostly derive from $W^\\prime $ so that two peaks appear nearby 1.5 TeV which is half of the $W^\\prime $ mass.",
"The normalized differential distributions with $\\Delta R$ which are displayed in Fig.",
"REF (c) for $pp\\rightarrow tW^{\\prime -}\\rightarrow bl^{+}+jj+\\displaystyle {\\lnot }E_T$ process is broader than that in processes (REF ) and (REF ).",
"Figure: (a) The normalized differential distributions with the transverse momentum of the jets (j 1 ,j 2 ,j 3 j_1,j_2,j_3)with p Tj 1 >p Tj 2 >p Tj 3 p_{Tj_1}>p_{Tj_2}>p_{Tj_3} for M=3M=3 TeV in the process of pp→tW L '- →l + +3jets+¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow l^{+}+3jets+\\displaystyle {\\lnot }E_T at s=33\\sqrt{s}=33 TeV.",
"(b) The same as (a) but for the charged lepton (solid line) and transverse missing energy ¬E T \\displaystyle {\\lnot }E_T (dashed line).",
"(c) The minimal angular separation distributions between jets (solid line) and that between jets and the charged lepton (dashed line).We use the same methods as in Sec.",
"III A to obtain the momentum of neutrino and reconstruct the intermediate resonances.",
"The following relation is adopted, $&&{\\bf p}_{\\nu T}=-({\\bf p}_{lT}+\\sum _{j=1}^{3}{\\bf p}_{jT}),\\nonumber \\\\\\ &&m_{W}^{2}=(p_{\\nu }+p_{l})^{2},\\nonumber \\\\\\ &&M_{jl\\nu }^{2}=(p_{l}^{2}+p_{\\nu }+p_{j})^{2},$ with j named $j_a$ refers to any one of the three jets which makes $M_{jl\\nu }$ be closest to the top quark mass and $j_{b(c)}$ for the left two.",
"Considering the unlike transverse momentum distributions of jet with the center of mass energy at 14 and 33 TeV, we employ the basic cuts as Cut $B_{1}$ : $&{\\rm For~~ 14~~ TeV}\\left\\lbrace \\begin{array}{ll}&p_{j_{1}T}>200~{\\rm GeV},~~~~p_{j_{2}T}>100~{\\rm GeV},~~~~p_{j_{3}T}>20~~{\\rm GeV},\\nonumber \\\\&p_{lT}>20~~{\\rm GeV},~~~~~\\displaystyle {\\lnot }E_T>20~{\\rm GeV},\\nonumber \\\\&|\\eta _{l}|<2.5,~~~~|\\eta _{j}|<2.5,~~~~~\\Delta R_{jj(lj)}>0.4,\\nonumber \\\\&|M_{j_{a}l\\nu }-m_{t}|\\le 30~{\\rm GeV}.\\end{array} \\right.&\\\\&{\\rm For~~ 33~~ TeV}\\left\\lbrace \\begin{array}{ll}&p_{j_{1}T}>550~{\\rm GeV},~~~~p_{j_{2}T}>550~{\\rm GeV},~~~~p_{j_{3}T}>20~~{\\rm GeV},\\nonumber \\\\&p_{lT}>20~~{\\rm GeV},~~~~~\\displaystyle {\\lnot }E_T>20~{\\rm GeV},\\nonumber \\\\&|\\eta _{l}|<2.5,~~~~|\\eta _{j}|<2.5,~~~~~\\Delta R_{jj(lj)}>0.4,\\nonumber \\\\&|M_{j_{a}l\\nu }-m_{t}|\\le 30~{\\rm GeV}.\\end{array} \\right.&$ Once the jet derived from top quark is confirmed by the reconstruction of top quark, the two remaining jets are absolutely from the heavy charged gauge boson $W^\\prime $ .",
"As presented in Fig.",
"REF , the resonance peak is obvious around the $W^\\prime $ mass in the differential distribution with $M_{j_bj_c}$ .",
"Figure: The distributions of 1/σdσ/dM jj 1/\\sigma d\\sigma /dM_{jj} with respect tothe invariant mass reconstructed by the remaining two jetsfor M=1,2,3M=1,~2,~3 TeV for the process ofpp→tW L '- →3 jets +l + +¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow 3{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T after cut B 1 B_{1} at the LHC for 33 TeV.Hence the further cut is Cut $B_{2}$ :$~~~|M_{j_bj_c}-M|\\le 10\\%\\, M $ .",
"Comparing the signal to the main SM backgrounds $Wjjj$ , $WWj$ and $WZj$ , one can find that b-tagging can help to purify the signal, so we require the jet used to reconstruct top quark to be a b-jet.",
"We show the total cross section after all the cuts for the signal process (REF ) at the LHC with 14 TeV and 33 TeV in Fig.",
"REF (a), as well as the integral luminosity as a function of $M$ assuming the significance is $3\\sigma $ in Fig.",
"REF (b).",
"The cross section is up to 0.003 (0.005) $fb$ for a mass of 2 TeV $W^{\\prime }_{L}$ ($W^{\\prime }_{R}$ ) with 14 TeV, as well as 0.118 (0.165) $fb$ with 33 TeV, while it is obviously suppressed by the strictly kinematic cuts for the $W^\\prime $ mass lighter than 2 TeV with 33 TeV.",
"The main backgrounds with the same detector signal including $tW$ , $Wjjj$ , $WWj$ and $WZj$ are simulated by MadEvent program [31].",
"One can find that the luminosity need to be up to $10^5 fb^{-1}$ if the charged bosons with $M=2$ TeV can be detected at 14 TeV.",
"However, there are about forty events can be detected for a $M=2.8$ TeV $W^\\prime $ with the luminosity of $1000 fb^{-1}$ at 33 TeV.",
"Figure: (a) The total cross section as a function ofthe charged gauge bosons mass MM at s=14\\sqrt{s}=14 and 33 TeVfor the process of pp→tW '- →3 jets +l + +¬E T pp\\rightarrow tW^{\\prime -}\\rightarrow 3{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T.",
"(b) The integral luminosity needed at the LHC for s=14\\sqrt{s}=14 and 33 TeVat S/B=3σS/\\sqrt{B}=3\\sigma sensitivity.",
"The solid and dashed lines stand for W L '- W_L^{\\prime -} and W R '- W_R^{\\prime -} respectively, and the two straight dotted lines present the luminosity of 300 and 1000fb -1 1000~fb^{-1}.Once a heavy charged boson $W^\\prime $ is discovered at the LHC, it will be imperative to determine its chiral couplings to SM fermions.",
"Accounting for different chiral couplings between $W^\\prime _Lq\\bar{q}^{\\prime }$ and $W^\\prime _Rq\\bar{q}^{\\prime }$ , we investigate the charged lepton angular distribution which depends on the chiral couplings of $W^\\prime $ to light quarks.",
"The chirality of the $W^\\prime $ coupling to the light quarks can be translated to the angular distribution of the charged lepton and a forward-backward asymmetry is defined as follows $&&\\frac{1}{\\sigma }\\,\\frac{d\\sigma }{d\\cos \\theta ^{*}}\\,=\\,\\frac{1}{2}\\,[1\\,+\\, A_{FB}\\cos \\theta ^{*}],~~~~~~A_{FB}=\\frac{\\sigma (cos\\theta ^{*}\\ge 0)-\\sigma (cos\\theta ^{*}<0)}{\\sigma (cos\\theta ^{*}\\ge 0)+\\sigma (cos\\theta ^{*}<0)}$ with $\\cos \\theta ^{*}=\\frac{{\\bf p}_{l^{+}}^{*}\\cdot {\\bf p}_{t}^{*}}{|{\\bf p}_{l^{+}}^{*}||{\\bf p}_{t}^{*}|} .$ Here ${\\bf p}_{l^{+}}^{*}$ is the 3-momentum of charged lepton in the top quark rest frame, and ${\\bf p}_{t}^{*}$ is the 3-momentum of the top quark in $tW^{\\prime -}$ center of mass frame.",
"The charged lepton angular distributions with respect to $cos\\theta ^{*}$ before and after cuts are displayed in Fig.",
"REF corresponding to the process (REF ) at 33 TeV LHC.",
"The result shows that most charged leptons moving against the direction of the top quark for the left-handed type interaction, while it leads to the inverse tendency for the right-handed type interaction.",
"Thus we can separate the hemisphere of top quark direction from the opposite hemisphere according to $\\cos \\theta ^* \\ge 0$ or $\\cos \\theta ^* < 0$ , then the forward-backward asymmetry leads to inverse sign which is listed in Table.",
"REF .",
"Although the distributions after the cuts in the $cos\\theta ^*=-1$ region are severely distorted by the acceptance cuts, due to the charged leptons moving against the top quark direction carry less energy than those in the remaining region, the forward-backward asymmetry with $-0.5<\\cos \\theta <0.5$ is also an excellent characteristic quantity to distinguish $W^{\\prime }_L$ from $W^{\\prime }_R$ .",
"Figure: The angular distributions of thecharged lepton for M=3M=3 TeV with (left) and without (right) cutsat s=33\\sqrt{s}=33 TeV for the process of pp→tW '- →3 jets +l + +¬E T pp\\rightarrow tW^{\\prime -}\\rightarrow 3{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T.",
"The solid line and dashed line stand for W L '- W^{\\prime -}_L and W R '- W^{\\prime -}_R, respectively.Table: The forward-backward asymmetry A FB A_{FB}for pp→tW '- →3 jets +l + +¬E T pp\\rightarrow tW^{\\prime -}\\rightarrow 3{\\rm jets}+l^{+}+\\displaystyle {\\lnot }E_T at the LHC with s=33\\sqrt{s}=33 TeV beforeand after the cuts."
],
[
"$W^\\prime \\rightarrow l\\nu $ channel for {{formula:c59df7b3-ef98-43fd-9a5d-d682664f3850}} production",
"The leptonic decay modes of $W^\\prime $ depend on the lepton spectrum and the flavor mixing in the given model.",
"Although the right-handed $W^\\prime $ boson can couple to a charged lepton and right-handed neutrino in some new physics models, the decay modes of $W^\\prime _R\\rightarrow l \\nu _R$ is not considered due to the mass of $\\nu _R$ larger than $W^\\prime $ or the different signal comparing to $W^\\prime _L\\rightarrow l \\nu $ .",
"In this section, we focus on the process $pp\\rightarrow tW^{\\prime -}_L \\rightarrow bW^+ W^{\\prime -}\\rightarrow bjj+l^{-}+\\displaystyle {\\lnot }E_T$ with $W^{\\prime -} \\rightarrow l^- \\nu $ decay mode, which provides the charged lepton with large transverse momentum as an excellent trigger at the LHC.",
"The distributions of $\\sigma ^{-1}d\\sigma /dP_T$ with the transverse momentum of the three jets and lepton are shown in Fig.",
"REF (a) and (b).",
"There is an obvious jacobian peak at $P_T=M/2$ in the charged lepton transverse momentum distribution.",
"In Fig.",
"REF (c), we display the corresponding distributions for $\\Delta R$ .",
"Figure: (a) The normalized differential distributions with the transverse momentum of the jets (j 1 ,j 2 ,j 3 j_1,j_2,j_3)with p Tj 1 >p Tj 2 >p Tj 3 p_{Tj_1}>p_{Tj_2}>p_{Tj_3} for M=3M=3 TeV in the process of pp→tW L '- →3 jets +l - +¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow 3{\\rm jets}+l^{-}+\\displaystyle {\\lnot }E_T at s=33\\sqrt{s}=33 TeV.",
"(b) The same as (a) but for the charged lepton (solid line) and transverse missing energy ¬E T \\displaystyle {\\lnot }E_T (dashed line).",
"(c) The minimal angular separation distributions between jets (solid line) and that between jets and the charged lepton (dashed line).We employ the basic acceptance cuts as Cut $C_{1}$ : $&{\\rm For~~ 14~~ TeV}\\left\\lbrace \\begin{array}{ll}& p_{lT}>200~{\\rm GeV},~~~~\\displaystyle {\\lnot }E_T>200~{\\rm GeV} , ~~~~p_{jT}>20~{\\rm GeV},\\nonumber \\\\&|\\eta _{l}|<2.5,~~~~|\\eta _{j}|<2.5,~~~~\\Delta R_{jj(lj)}>0.4.\\nonumber \\end{array} \\right.&\\\\&{\\rm For~~ 33~~ TeV}\\left\\lbrace \\begin{array}{ll}& p_{lT}>350~{\\rm GeV},~~~~\\displaystyle {\\lnot }E_T>350~{\\rm GeV} , ~~~~p_{jT}>20~{\\rm GeV},\\nonumber \\\\&|\\eta _{l}|<2.5,~~~~|\\eta _{j}|<2.5,~~~~\\Delta R_{jj(lj)}>0.4.\\nonumber \\end{array} \\right.&$ And we require the top quark mass be reconstructed by all of the three jets as well as $W$ boson mass reconstructed by two of the jets as the further cut Cut $C_{2}$ : $~~|M_{j_{a}j_{b}}-m_{W}|\\le 10~{\\rm GeV},~~~~|M_{j_{a}j_{b}j_{c}}-m_{t}|\\le 30~{\\rm GeV},$ where $M_{jjj}^{2}=(\\sum _{j=1}^{3}{p}_{j})^2.$ In addition, the jet which is not used to reconstruct $W$ boson is tagged as a b-jet to suppress the background processes.",
"The differential distributions $\\sigma ^{-1}d\\sigma /dH_T$ for process (REF ) are shown in Fig.",
"REF , where $H_{T}=\\sum _{j=1}^{3}{p}_{jT}+{p}_{lT}+\\displaystyle {\\lnot }E_T.$ One can see that a peak appears around the $W^\\prime $ mass which is a significant excess comparing to the SM backgrounds.",
"Figure: The distributions of 1/σdσ/dH T 1/\\sigma d\\sigma /dH_T with respect toH T H_T for M=1,2,3M=1,~2,~3 TeV for the process ofpp→tW L '- →l - +3jets+¬E T pp\\rightarrow tW_L^{\\prime -}\\rightarrow l^{-}+3jets+\\displaystyle {\\lnot }E_T after cut C 2 C_{2} at the LHC for 33 TeV.In Fig.",
"REF , we present the total cross section for the pure left-handed $W^\\prime $ with and without cuts at the LHC with 14 TeV and 33 TeV.",
"The cross section reaches 0.02 $fb$ for $M=3$ TeV with $\\sqrt{s}=33$ TeV.",
"Corresponding to the process (REF ) with final states $3{\\rm jets}+l^{-}+\\displaystyle {\\lnot }E_T$ , the dominant backgrounds from SM are $tW$ , $Wjjj$ , $Wjjj$ and $WZj$ , which are simulated by MadEvent.",
"Supposing the integral luminosity to be $300 fb^{-1}$ at 14 TeV, a $W^\\prime _L$ with mass up to 1.7 TeV can be found at $3\\sigma $ significance, which can be enlarged to 4 TeV on the condition of integral luminosity of $1000 fb^{-1}$ at the LHC with $\\sqrt{s}=33$ TeV.",
"In a more appealing seesaw version [32], [33] of the theory, as in the papers below, $W^\\prime _R$ decays into the heavy right-handed neutrino with the spectacular signatures of lepton number violation [34].",
"In this case, there have been dedicated searches and theoretical discussions [35], [36], [37].",
"Perhaps the $W^\\prime $ can be first detected at the LHC associated with the heavy right-handed neutrino.",
"Figure: (a) The total cross section as a function ofthe charged gauge bosons mass MM at s=14\\sqrt{s}=14 and 33 TeVfor the process of pp→tW L '- →3 jets +l - +¬E T pp\\rightarrow tW^{\\prime -}_L\\rightarrow 3{\\rm jets}+l^{-}+\\displaystyle {\\lnot }E_T.",
"(b) The integral luminosity needed at the LHC for s=14\\sqrt{s}=14 and 33 TeVat S/B=3σS/\\sqrt{B}=3\\sigma sensitivity.",
"The two straight dotted lines present the luminosity of 300 and 1000fb -1 1000~fb^{-1}."
],
[
"Summary",
"The observation of a new charged vector boson $W^\\prime $ is an unambiguous signal for new physics beyond the SM.",
"As a result, it is important to search for $W^\\prime $ production signal and its related phenomena via different production and decay channels at the LHC.",
"In this paper, we focus on investigating the collider signature for the $tW^\\prime $ associated production with $W^\\prime \\rightarrow tb$ , $W^\\prime \\rightarrow q \\bar{q}^{\\prime }$ and $W^\\prime \\rightarrow l \\nu $ respectively at the LHC.",
"It is found that with the most common coupling parameters and present mass constraints for $W^\\prime $ , due to the limited cross section, it is difficult to observe $W^\\prime $ production signal via $tW^\\prime $ associated production at 14 TeV.",
"However, if the center of mass energy of the LHC is updated to 33 TeV or even higher, a distinct signal for $W^\\prime $ production can be observed via $W^\\prime \\rightarrow q \\bar{q}^{\\prime }$ and $W^\\prime \\rightarrow l \\nu $ after adopting proper kinematic cuts, e.g., large transverse momentum cut to the jets/leptons, etc.",
"Once the $tW^\\prime $ production is observed at the LHC, it will become important to study the interactions between $W^\\prime $ and fermions.",
"For this aim, as an example, we analyze the angular distribution of the charged lepton for $pp \\rightarrow tW^{\\prime -} \\rightarrow bW^+ W^{\\prime -} \\rightarrow bl^+ + jj + \\displaystyle {\\lnot }E_T$ process and the related forward-backward asymmetry induced by top quark spin.",
"Our results show that the charged lepton angular distribution is related to the chiral couplings of $W^\\prime $ to fermions and the forward-backward asymmetry depending on this angular distribution.",
"These observables can be used to distinguish $W^\\prime _L$ from $W^\\prime _R$ .",
"If the LHC can served as a discovery machine for the new charged gauge boson $W^\\prime $ , our work will be useful to search for its production signal and to explore its properties."
],
[
"Acknowledgments",
"We would like to thank Profs.",
"Shi-Yuan Li and Shou-Shan Bao for their helpful discussions and comments.",
"This work is supported in part by National Science Foundation of China (NSFC) under grant Nos.11305075, 11325525, 11275114, 10935012 and 11375200.",
"Li is also supported in part by Natural Science Foundation of Shandong Province under grant No.ZR2013AQ006."
]
] | 1403.0347 |
[
[
"Comment on \"Tuning the Magnetic Dimensionality by Charge Ordering in the\n Molecular TMTTF Salts\""
],
[
"Abstract Yoshimi et al.",
"[arXiv:1110.3573] have attempted to explain the pressure(P)-dependent behavior of Fabre salts which exhibit charge order (CO), antiferromagnetic (AFM), and spin-Peierls (SP) phases.",
"Experiments find two AFM phases, AFM1 at large P and AFM2 at small P. Yoshimi et al.",
"suggest that there also exist two distinct zero-temperature SP phases, SP1 and SP2.",
"Here we point out that the occurrence of two distinct SP phases contradicts experiments, and is found because of unrealistic model parameters."
],
[
"Comment on “Tuning the Magnetic Dimensionality by Charge Ordering in the Molecular TMTTF Salts” Yoshimi et al.",
"[1] have attempted to explain the pressure(P)-dependent behavior of Fabre salts which exhibit charge order (CO), antiferromagnetic (AFM), and spin-Peierls (SP) phases.",
"Experiments find two AFM phases [2], [3], AFM$_1$ at large P and AFM$_2$ at small P. Yoshimi et al.",
"suggest that there also exist two distinct zero-temperature SP phases, SP$_1$ and SP$_2$ .",
"Here we point out that the occurrence of two distinct SP phases contradicts experiments [2], [3], and is found in [Yoshimi12a] because of unrealistic model parameters.",
"Experiments [2], [3] emphasize co-operative interaction between the ferroelectric charge order (FCO) and AFM$_2$ phases.",
"In the experimental phase diagram [2], [3] $T_{CO}$ and the Néel temperature in the AFM$_2$ phase both decrease with $P$ .",
"Thus charge occupancies in the FCO and AFM$_2$ phases are likely the same.",
"In contrast, P increases [2], [3] the SP transition temperature, indicating that FCO and SP$_2$ phases compete.",
"No CO was detected for $P>$ 0.5 GPa in (TMTTF)$_2$ SbF$_6$ [2], [3], in the $P$ region where the SP$_2$ phase occurs at lower temperature.",
"It is then unlikely that SP$_2$ and FCO coexist at zero temperature.",
"The hopping parameters used by the authors in their model calculations are realistic.",
"Their choice of Coulomb interactions is however unrealistic.",
"The onsite Coulomb interaction assumed, $U/t_{a2}$ =4, is too small—in the purely electronic one dimensional model no CO occurs for this $U$ [4], [5].",
"The assumed intersite Coulomb interactions $V_b$ = 0 and $V_q=V_a$ , are also unrealistic.",
"Given the lattice geometry (see Fig.",
"1) it is highly unlikely that $V_b \\ll V_q$ , and with large interchain separation $V_q=V_a$ is equally unrealistic.",
"$4U8$ and $V_b$ $\\simeq $ $V_q\\ll V_a$ is more appropriate.",
"We repeated the calculations with more realistic $V_a = V$ , $V_b = V_q= 0$ , and $4\\le U\\le 8$ .",
"For these parameters, the intra-dimer charge structure factor ($C_{-}({\\bf q})$ in [1]) peaks at several q values, indicating comparable energies for both FCO and the checkerboard pattern CO, in agreement with experiments [6].",
"Peaks in Figure: (color online) 8×\\times 2 phase diagram for U=6U=6, V a =VV_a=V,and K 2 =1K_2=1.",
"The inset shows the lattice structure assumed by.",
"As K 1 K_1 increases the size of the FCO+SP phaseshrinks.",
"Other points do not significantly change with K 1 K_1.$S_\\pm ({\\bf q})$ remain at the same q values as in Fig.",
"2 of [1].",
"We conclude that the $V_{ij}$ assumed in [Yoshimi12a] is not required to explain coexisting FCO/AFM order in the AFM$_2$ state.",
"We also repeated (see Fig.",
"1) the 8$\\times $ 2 calculations with these parameters.",
"We have three main observations: (i) for $V_a = V$ , $V_b= V_q = 0$ , we find a phase diagram similar to that in [1], but with FCO entering at larger $V$ as expected [4], [5].",
"The choice $V_q=V$ , $V_b=0$ is also not required to realize the FCO phase; FCO can be stabilized by antiferromagnetic superexchange along the $t_b$ bonds; (ii) As $U$ increases the FCO+SP phase narrows; (iii) For both these and the parameters assumed in [1], the width of the FCO+SP phase is directly proportional to the strength of the inter-site electron phonon coupling (larger $K_1$ gives weaker coupling).",
"Unconditional transitions in the thermodynamic limit occur in the limit of 0$^+$ phonon coupling.",
"Importantly, point (iii) was not discussed in [1], and together with (ii) suggests that in the thermodynamic limit the FCO+2DAFM and DM+SP phases may share a common border.",
"To understand the phase diagram one must consider thermodynamics.",
"For large Coulomb interactions the free energy is dominated by spin excitations.",
"We have previously shown that the same DM+SP ground state can have two kinds of soliton spin excitations, (i) with local CO, or (ii) with uniform charge but local bond distortion [7].",
"In this picture, to the left of the line bisecting the SP phase [2] soliton excitations with local CO dominate at finite T; to the right occur excitations with uniform site charges.",
"A unique SP ground state is expected at all pressures between AFM$_1$ and AFM$_2$ .",
"We acknowledge support from the Department of Energy grant DE-FG02-06ER46315.",
"A. B.",
"Ward$^1$ , R. T. Clay$^1$ , and S. Mazumdar$^2$ $^1$ Department of Physics & Astronomy HPC$^2$ Center for Computational Sciences Mississippi State University Mississippi State, MS 39762-5167 $^2$ Department of Physics University of Arizona Tucson, AZ 85721 $^1$ Department of Physics & Astronomy HPC$^2$ Center for Computational Sciences Mississippi State University Mississippi State, MS 39762-5167 $^2$ Department of Physics University of Arizona Tucson, AZ 85721"
]
] | 1403.0120 |
[
[
"A New Framework for the Performance Analysis of Wireless Communications\n under Hoyt (Nakagami-q) Fading"
],
[
"Abstract We present a novel relationship between the distribution of circular and non-circular complex Gaussian random variables.",
"Specifically, we show that the distribution of the squared norm of a non-circular complex Gaussian random variable, usually referred to as squared Hoyt distribution, can be constructed from a conditional exponential distribution.",
"From this fundamental connection we introduce a new approach, the Hoyt transform method, that allows to analyze the performance of a wireless link under Hoyt (Nakagami-q) fading in a very simple way.",
"We illustrate that many performance metrics for Hoyt fading can be calculated by leveraging well-known results for Rayleigh fading and only performing a finite-range integral.",
"We use this technique to obtain novel results for some information and communication-theoretic metrics in Hoyt fading channels."
],
[
"Introduction",
"The characterization of the distribution of a complex Gaussian random variable (RV) is arguably one of the most relevant problems in engineering and statistics.",
"In the contexts of information and communication theory, the distribution of the norm of the complex Gaussian random variable $Z=X+jY$ (where $X$ and $Y$ are jointly Gaussian) finds application in many problems such as signal detection, noise characterization, or wireless fading channel modeling, just to name a few.",
"In the literature, the most general caseDepending on the specific context, different authors have pursued the characterization of the envelope $E=\\sqrt{X^2+Y^2}$ or the squared envelope $R=E^2$ .",
"However, both distributions are related by a simple transformation and therefore have a similar form.",
"Throughout this paper, we will focus on the distribution of $R$ , which model the instantaneous power of a complex Gaussian signal.",
"of $X$ and $Y$ having different mean and variance was addressed by Beckmann [1], as a generalization of the previous results obtained by Rice [2] and Hoyt [3], and recently revisited in [4], [5].",
"In this general situation the chief distribution functions (pdf and cdf) of ${R=X^2+Y^2}$ have complicated forms.",
"In the specific case of $X$ and $Y$ being independent Gaussian RVs with zero mean and arbitrary variance, or equivalently being correlated Gaussian RVs with zero mean and equal variances, the RV $Z$ is said to be a zero mean non-circular (or improper) Gaussian RV [6].",
"This occurs in many practical scenarios, such as in the detection of non-stationary complex random signals [7], or in the characterization of multipath fading [8].",
"In this latter situation, the Hoyt [3] or Nakagami-$q$ fading [8] distribution is used to model short-term variations of radio signals resulting from the addition of scattered waves which can be described as a complex Gaussian RV where the in-phase and quadrature components have zero mean and different variances, or equivalently, where the in-phase and quadrature components are correlated.",
"This distribution is commonly used to model signal fading due to strong ionospheric scintillation in satellite communications [9] or in general those fading conditions more severe than Rayleigh, and it includes both Rayleigh fading and one-sided Gaussian fading as special cases.",
"Furthermore, it was shown in [10] that the second order statistics of Hoyt fading best fit measurement data in mobile satellite channels with heavy shadowing.",
"One of the main advantages of Rayleigh fading, which is perhaps the most popular model for the random fluctuations of the signal amplitude when transmitted through a wireless link when there is no direct line-of-sight (LOS) between the transmit and receive ends, relies on its comparatively simple analytic manipulation, as the received signal-to-noise ratio (SNR) is exponentially distributed.",
"Conversely, the received SNR in Hoyt fading has a much more complicated form and sophisticated special functions are required to characterize the pdf or cdf [11] of a squared Hoyt RV.",
"Both Rayleigh and Hoyt fading have been extensively investigated in the last few decades [12]; however, while the derivation of information and communication-theoretic performance metrics such as channel capacity [13] and outage probability (OP) [14] is usually tractable mathematically for the Rayleigh case, it is way more complicated to analyze the very same scenario when Hoyt fading is assumed.",
"Dozens of papers have been published in the last years with the aim of analyzing very diverse scenarios where Hoyt fading is considered, for the sake of extending already known results for Rayleigh fading to this more general situation [15], [16], [17], [11], [18].",
"However, despite the relationship that can be inferred between both distributions, existing analyses in the literature for Hoyt fading do not exploit this connection and usually require tedious and complicated derivations.",
"To the best of our knowledge, there is no standard procedure that takes advantage of the relationship between both distributions, and therefore the calculations for Hoyt fading must be done from scratch.",
"In this paper, we present a novel connection between the distribution of the squared norm of a non-circular complex Gaussian RV and its circular counterpart.",
"In other words, we introduce a useful relationship between the squared Hoyt distribution and the exponential distribution, which greatly simplifies the analysis of the former.",
"By exploiting the fact that the cdf of a squared Hoyt distributed random variable is a weighted Rice $Ie$ -function, we demonstrate that the squared Hoyt distribution can be constructed from a conditional exponential distribution.",
"This connection has important relevance in practice: Since most communication-theoretic metrics are computed with a linear operation over the SNR distribution, we show that performance results for Hoyt fading channels can be readily obtained by leveraging previously known results for Rayleigh fading, and computing a very simple finite-range integral.",
"This general procedure will be denoted as the Hoyt transform method.",
"The main takeaway is that there is no need to redo any calculation in order to analyze the performance of communication systems in Hoyt fading, if there are available results for the simpler Rayleigh case.",
"Instead, the application of the Hoyt transform yields the desired performance result in a direct way.",
"In some cases, the Hoyt transform has analytical solution and hence the expressions for Hoyt fading are of similar complexity to those obtained for Rayleigh scenarios.",
"Otherwise, the results for Hoyt fading have the form of a finite-range integral with constant integration limits, over the performance metric of interest for the Rayleigh case.",
"Integrals of this form are very usual in communications, including proper-integral forms for the Gaussian $Q$ -function [19], the Marcum $Q$ -function [20] or the Pawula $F$ -function [21], or those obtained with the application of the moment generating function (MGF) approach to the calculation of error probability [22].",
"Therefore, the numerical computation of the Hoyt transform introduced in this paper is simpler than other alternatives that require the evaluation of infinite series or inverse Laplace transforms.",
"As an additional advantage, our new approach also permits to obtain upper and lower bounds of different performance metrics in a simple way.",
"Using this general procedure, we provide novel analytical results for three scenarios of interest in information and communication theory: We analyze the Shannon capacity of adaptive transmission techniques in Hoyt fading channels, thus extending the results given in [13].",
"Thanks to the Hoyt transform method, we can calculate the asymptotic capacity in the low and high-SNR regimes in closed-form.",
"We show that the asymptotic capacity loss per bandwidth unit in the high-SNR regime is up to 1.83 bps/Hz compared to the AWGN case, and up to 1 bps/Hz when compared to the Rayleigh case.",
"We investigate the physical layer security of a wireless link in the presence of an eavesdropper, where both the desired and wiretap links are affected by Hoyt fading.",
"Known analytical results are available for different scenarios such as Rayleigh [23], Nakagami-$m$ [24], Rician [25], or Two-Wave with Diffuse Power [26] fading models.",
"However, to the best of our knowledge, there are no results in the literature for the physical layer security in Hoyt fading channels.",
"We evaluate evaluate the OP of a Hoyt-faded wireless link affected by arbitrarily distributed co-channel interference and background noise.",
"Specifically, we show that the OP in this general scenario will be given in terms of a Hoyt transform of the MGF of the aggregate interference, admitting a very simple evaluation even for very general fading distributions such as the $\\eta $ -$\\mu $ or $\\kappa $ -$\\mu $ models [27].",
"The rest of this paper is organized as follows.",
"In Section , some preliminary definitions are introduced and the Rice $Ie$ -function is reviewed.",
"Then, the main mathematical contributions are presented in Section : the connection between the squared Hoyt and the exponential distributions, and its application to define the Hoyt transform method to the performance analysis in Hoyt fading channels.",
"This approach is used in Sections to to obtain analytical results in the aforementioned scenarios.",
"Numerical results are presented in Section , whereas the main conclusions are outlined in Section .",
"Let $Z=X+jY$ be a zero-mean non-circular complex Gaussian RV, where $X$ and $Y$ are independent jointly Gaussian RVs with variances $\\sigma _x^2$ and $\\sigma _y^2$ .",
"Then, the random variable ${R=X^2+Y^2}$ is said to follow the squared Hoyt distribution, and its pdf is given by $ f_R(x)= \\frac{1+q^2}{2q \\overline{\\gamma }} \\exp \\left[-\\frac{(1+q^2)^2 x}{4q^2 \\overline{\\gamma }}\\right] I_0 \\left(\\frac{(1-q^4) x}{4q^2 \\overline{\\gamma }}\\right),$ where $I_0(\\cdot )$ is the modified Bessel function of the first kind and zero order, $\\overline{\\gamma }=\\mathbb {E}\\lbrace R\\rbrace $ and $q=\\sigma _y/\\sigma _x$ , assuming without loss of generality that $\\sigma _y\\le \\sigma _x$ .",
"Therefore, we have $q\\in [0,1]$ .",
"If $q=1$ , then $Z$ is a zero-mean circular complex Gaussian RV and therefore $R$ is exponentially distributed, with pdf $f_R(x)= \\frac{1}{\\overline{\\gamma }}e^{-x/\\overline{\\gamma }}.$"
],
[
"The Rice $Ie$ -function",
"Let $k$ and $x$ be non-negative real numbers with $0 \\le k \\le 1$ , the Rice $Ie$ -function is defined as [2] $ Ie(k,x)=\\int _0^x e^{-t}I_0(kt) dt.$ The Rice $Ie$ -function admits different infinite series representations [28], [29], and it is not considered a tabulated function, in the sense that it is not included as a built-in function in standard mathematical software packages such as Matlab or Mathematica.",
"However, after the appropriate change of notation this function can be written in compact form, as [30] $ Ie(k,x)=\\frac{1}{\\sqrt{1-k^2}} \\left[Q(\\sqrt{ax},\\sqrt{bx})-Q(\\sqrt{bx},\\sqrt{ax})\\right]$ or equivalently, $ Ie(k,x)=\\frac{1}{\\sqrt{1-k^2}} \\left[2Q(\\sqrt{ax},\\sqrt{bx})-e^{-x}I_0(kx)-1\\right],$ where $a=1+\\sqrt{1-k^2}$ , $b=1-\\sqrt{1-k^2}$ and $ Q(\\alpha ,\\beta )=\\int _\\beta ^\\infty t e^{-\\frac{t^2+\\alpha ^2}{2}}I_0(\\alpha t) dt$ is the first order Marcum $Q$ -function.",
"Since both the modified Bessel function $I_0$ and the Marcum $Q$ -function are tabulated functions, (REF ) and (REF ) can be easily computed.",
"However, subsequent manipulations of these expressions are generally complicated and in many situations it may be preferable to express the Rice $Ie$ -function in integral form.",
"Replacing $I_0(\\cdot )$ in (REF ) by its integral representation, namely [31] $ I_0(z)=\\frac{1}{\\pi } \\int _0^\\pi e^{z\\cos \\theta } d\\theta ,$ after some manipulation we can write [28] $ Ie(k,x)=\\frac{1}{\\sqrt{1-k^2}}- \\frac{1}{\\pi } \\int _0^\\pi \\frac{e^{-x(1-k\\cos \\theta )}}{1-k\\cos \\theta } d\\theta ,$ which has important advantages with respect to (REF ), as the integration limits do not depend on the arguments of the defined function, and the integrand is given in terms of elementary functions.",
"However, for reasons that will become clear in the next Section, a much more convenient representation of the Rice $Ie$ -function for the purpose of this work is the one provided in the following proposition.",
"Proposition 1 The Rice $Ie$ -function can be written in integral form as $ Ie(k,x)=\\frac{1}{\\sqrt{1-k^2}}\\left[1 - \\frac{1}{\\pi } \\int _0^\\pi \\exp \\left(-x \\frac{{1-k^2}}{1-k\\cos \\theta }\\right) d\\theta \\right].$ Although not specifically stated in this form, this result follows from the identities provided by Pawula in [30] [32].",
"In particular, from [32] the following identity holds: $ W\\int _0^\\pi \\frac{e^{-(U-V\\cos \\theta )}}{U-V\\cos \\theta } d\\theta =\\int _0^\\pi \\exp \\left(- \\frac{{W^2}}{U-V\\cos \\theta }\\right) d\\theta ,$ where $W=\\sqrt{U^2-V^2}$ .",
"By identifying $U=x$ and $V/U=k$ , and with the help of (REF ), the desired expression is obtained."
],
[
"Main Results",
"We now introduce the main mathematical contributions of this work, which are given in a set of lemmas and corollaries.",
"Lemma 1 Let $R|\\theta $ be an exponentially distributed random variable, conditioned on $\\theta $ , with pdf $ f_{R|\\theta }(x)= \\frac{1}{\\gamma (\\theta ,q)} e^{-x/\\gamma (\\theta ,q)},$ where $\\theta $ is a random variable uniformly distributed between 0 and $\\pi $ , and $ \\gamma (\\theta ,q) \\triangleq \\overline{\\gamma }\\left(1-\\frac{1-q^2}{1+q^2}\\cos \\theta \\right)=\\mathbb {E}\\left\\lbrace R|\\theta \\right\\rbrace .$ Then, the unconditional random variable $R$ , with pdf $ f_R(x)= \\frac{1}{\\pi } \\int _0^\\pi \\frac{1}{\\gamma (\\theta ,q)}e^{-x/\\gamma (\\theta ,q)}d\\theta ,$ follows a squared Hoyt distribution with average $\\mathbb {E}\\left\\lbrace R\\right\\rbrace =\\overline{\\gamma }$ and parameter $q$ , i.e., (REF ) is an alternative expression for the pdf given in (REF ).",
"The cdf of $R$ will be given by $ F_R(x)=1 - \\frac{1}{\\pi } \\int _0^\\pi e^{-x/\\gamma (\\theta ,q)} d\\theta .$ The cdf of the $R$ can be calculated as $ F_R(x)= \\int _0^x \\frac{1+q^2}{2q \\overline{\\gamma }}\\exp \\left[-\\frac{(1+q^2)^2 t}{4q^2 \\overline{\\gamma }}\\right]I_0 \\left(\\frac{(1-q^4) t}{4q^2 \\overline{\\gamma }}\\right) dt,$ which can be written using the definition of the Rice $Ie$ -function in (REF ) as $ F_R(x)= \\frac{2q}{1+q^2}Ie \\left( \\frac{1-q^2}{1+q^2} ,\\frac{(1+q^2)^2}{4q^2 \\overline{\\gamma }} x\\right).$ Using the alternative definition for the Rice $Ie$ -function in (REF ), the cdf of the SNR can be written after some algebraic manipulation as in (REF ).",
"Finally, by taking the derivative of (REF ), the desired pdf in (REF ) is obtained.",
"By comparing (REF ) with the pdf of an exponential distribution, Lemma REF states that a squared Hoyt RV can be viewed as a finite-range integral of a exponentially distributed RV with continuously varying averages.",
"Note that the factor that multiplies $\\cos \\theta $ in (REF ) coincides with the squared third eccentricity $\\epsilon $ of the ellipse represented by the underlying non-circular complex Gaussian random variable of the Hoyt distribution [33], i.e.",
"$\\epsilon =\\frac{1-q^2}{1+q^2}$ .",
"A direct application of Lemma REF in a communication-theoretic context follows: Any performance metric in Hoyt fading channels that can be obtained by averaging over the SNR pdf (e.g.",
"outage probability, channel capacity, error probability) can be calculated from existing results for Rayleigh fading, by performing a finite-range integral.",
"In this situation, $\\bar{\\gamma }$ represents the average SNR and $R$ (or $\\gamma $ , indistinctly) denotes the instantaneous SNR.",
"Since most performance metrics of interest for Rayleigh fading in the literature are usually given in closed-form, the proposed approach allows for easily extending the results to Hoyt fading in a very simple manner.",
"This is formally stated in the following lemma, where the Hoyt transform is introduced.",
"Lemma 2 Let $h(\\gamma )$ be a performance metric depending on the instantaneous SNR $R$ , and let $\\overline{h}_R(\\overline{\\gamma })$ be the metric in Rayleigh fading with average SNR $\\overline{\\gamma }$ obtained by averaging over an interval of the pdf of the SNR, i.e., $ \\overline{h}_R(\\overline{\\gamma })= \\int _a^bh(x) \\frac{1}{\\overline{\\gamma }}e^{-x/\\overline{\\gamma }}dx,$ with $0 \\le a < b \\le \\infty $ .",
"Then, the performance metric in Hoyt fading channels with average SNR $\\overline{\\gamma }$ , denoted as $\\overline{h}_H(\\overline{\\gamma })$ , can be calculated as $ \\overline{h}_H(\\overline{\\gamma })= \\frac{1}{\\pi } \\int _0^\\pi \\overline{h}_R(\\gamma (\\theta ,q))d\\theta =\\mathcal {H}\\left\\lbrace \\overline{h}_R(\\overline{\\gamma });q\\right\\rbrace .$ where $\\mathcal {H}\\left\\lbrace \\cdot ;q\\right\\rbrace $ is the Hoyt transform operation.",
"The metric $\\overline{h}_H(\\overline{\\gamma })$ is obtained as $ \\overline{h}_H(\\overline{\\gamma })= \\int _a^bh(x) f(x)dx.$ where $f(x)$ is the pdf of a squared Hoyt random variable given in (REF ).",
"Thus, we can write $ \\overline{h}_H(\\overline{\\gamma })= \\int _a^bh(x) \\frac{1}{\\pi } \\int _0^\\pi \\frac{1}{\\gamma (\\theta ,q)}e^{-x/\\gamma (\\theta ,q)}d\\theta dx,$ and reversing the order of integrationA sufficient condition for the double integral to be reversible is that $h(x)$ is a nonnegative continuous function, which is the case of most performance metrics of interest, such as channel capacity, symbol error rate, outage probability, etc.",
"yields $ \\overline{h}_H(\\overline{\\gamma })= \\frac{1}{\\pi } \\int _0^\\pi \\left[\\int _a^bh(x)\\frac{1}{\\gamma (\\theta ,q)}e^{-x/\\gamma (\\theta ,q)}dx\\right]d\\theta .$ By recognizing that the integral between brackets is actually $\\overline{h}_R(\\gamma (\\theta ,q))$ , (REF ) is finally obtained.",
"We see that Lemma REF provides a very simple and direct way to analyze the performance of communication systems operating in Hoyt fading channels.",
"In fact, some interesting dualities with the popular MGF approach to the error-rate performance analysis of digital communication systems over fading channels [22] can be inferred: In the reference work by Simon and Alouini, error-rate expressions are obtained “in the form of a single integral with finite limits and an integrand composed of elementary functions, thus readily enabling numerical evaluation”; in our work, the Hoyt transform also facilitates the derivation of expressions of the same form, with the integrand being now directly the performance metric obtained in the Rayleigh case.",
"However, while the MGF approach is applicable to obtain a specific performance metric (error-rate) in general fading channels; the Hoyt transform approach is applicable to obtain general performance metrics in a specific fading channel (Hoyt).",
"An interesting consequence of Lemma REF is the following corollary.",
"Corollary 1 The MGF of a squared Hoyt random variable of average $\\overline{\\gamma }$ and shape parameter $q$ can be written as $ \\phi (s)=\\frac{1}{\\pi } \\int _0^\\pi \\frac{1}{1-\\gamma (\\theta ,q)s}d\\theta .$ This result follows directly from Lemma REF and the fact that the MGF of an exponentially distributed random variable of average $\\overline{\\gamma }$ is given by $(1-\\overline{\\gamma }s)^{-1}$ .",
"This corollary provides an alternative demonstration of the integral representation of the pdf of a squared Hoyt random variable given in (REF ).",
"Indeed, because of the way it has been constructed, it is clear that (REF ) is the MGF of a random variable which pdf is given by (REF ).",
"On the other hand, the integral in (REF ) can be solved in closed-form, using [31], yielding $ \\phi (s)=\\left[1-2\\overline{\\gamma }s+\\frac{q^2(2\\overline{\\gamma }s)^2}{(1+q^2)^2} \\right]^{-1/2},$ which is the well-known MGF of a squared Hoyt random variable [12].",
"Therefore, from the uniqueness theorem of the MGF, (REF ) and (REF ) are actually the same pdf.",
"Another benefit of the Hoyt transform method relies in the fact that the calculations are based on an integration involving a bounded trigonometric function; hence, this permits to find simple upper and lower bounds of the performance metrics.",
"These bounds can be found by taking into account that symbol error rate performance metrics are usually convex decreasing functions with respect to the SNR, whereas channel capacity metrics are typically concave increasing functions.",
"The following proposition establishes a sufficient condition to determine the monotonicity and convexity of some important average performance metric functions.",
"Proposition 2 Let $h(\\gamma )$ be a performance metric depending on the instantaneous SNR $\\gamma $ and let $\\overline{h}_R(\\overline{\\gamma })$ be defined as in Lemma 1.",
"If $h(\\gamma )$ is a decreasing convex (increasing concave) function of $\\gamma $ in $[0,\\infty )$ , then $\\overline{h}_R(\\overline{\\gamma })$ is a decreasing convex (increasing concave) function of $\\overline{\\gamma }$ .",
"If $h(\\gamma )$ is a decreasing convex function then the first and second order derivatives of $h(\\gamma )$ verify ${h^{\\prime }(\\gamma )\\le 0}$ , $h^{\\prime \\prime }(\\gamma )\\ge 0$ .",
"By a simple change of variables in (REF ), considering the interval $[0,\\infty )$ , we can write $ \\overline{h}_R(\\overline{\\gamma })= \\int _0^\\infty h(\\overline{\\gamma } x) e^{-x} dx,$ and its first and second order derivatives verify $ \\overline{h}^{\\prime }_R(\\overline{\\gamma })= \\int _0^\\infty h^{\\prime }(\\overline{\\gamma } x) x e^{-x} dx < 0,$ $ \\overline{h}^{\\prime \\prime }_R(\\overline{\\gamma })= \\int _0^\\infty h^{\\prime \\prime }(\\overline{\\gamma } x) x^2 e^{-x} dx > 0.$ Therefore, $\\overline{h}_R(\\overline{\\gamma })$ is a decreasing convex function of $\\overline{\\gamma }$ .",
"Analogously, if $h(\\gamma )$ is an increasing concave function, then the first and second order derivatives of $h(\\gamma )$ verify $h^{\\prime }(\\gamma )\\ge 0$ , $h^{\\prime \\prime }(\\gamma )\\le 0$ .",
"Thus, $\\overline{h}_R(\\overline{\\gamma })$ is an increasing concave function.",
"Now we present the aforementioned bounds in the next lemmas: Lemma 3 Let $\\overline{h}_R(\\overline{\\gamma })$ and $\\overline{h}_H(\\overline{\\gamma })$ be functions obtained by averaging a given function $h(\\gamma )$ in, respectively, Rayleigh and Hoyt fading channels, where $\\overline{\\gamma }$ is the average SNR, and let $\\overline{h}_R(\\overline{\\gamma })$ be a decreasing convex function.",
"Then, the following inequality holds: $ \\overline{h}_R(\\overline{\\gamma }) \\le \\overline{h}_H(\\overline{\\gamma })\\le \\overline{h}_R \\left(\\frac{2q^2}{1+q^2}\\overline{\\gamma }\\right),$ Lemma 4 Let $\\overline{h}_R(\\overline{\\gamma })$ and $\\overline{h}_H(\\overline{\\gamma })$ be functions obtained by averaging a given function $h(\\gamma )$ in, respectively, Rayleigh and Hoyt fading channels, where $\\overline{\\gamma }$ is the average SNR, and let $\\overline{h}_R(\\overline{\\gamma })$ be a concave increasing function.",
"Then, the following inequality holds: $ \\overline{h}_R \\left(\\frac{2q^2}{1+q^2}\\overline{\\gamma }\\right) \\le \\overline{h}_H(\\overline{\\gamma }) \\le \\overline{h}_R(\\overline{\\gamma }).$ Let us first demonstrate (REF ): As $\\overline{h}_R(\\overline{\\gamma })$ is a decreasing function of $\\overline{\\gamma }$ and the lowest value of $\\gamma (\\theta ,q)$ is obtained for $\\theta =0$ , an upper bound of $\\overline{h}_H(\\overline{\\gamma })$ can be found as $ \\begin{split}\\overline{h}_H(\\overline{\\gamma })= &\\frac{1}{\\pi } \\int _0^\\pi \\overline{h}_R(\\gamma (\\theta ,q)) d\\theta \\le \\frac{1}{\\pi } \\int _0^\\pi \\overline{h}_R(\\gamma (0,q)) d\\theta \\\\ & =\\overline{h}_R(\\gamma (0,q))=\\overline{h}_R \\left(\\frac{2q^2}{1+q^2}\\overline{\\gamma }\\right).\\end{split}$ A lower bound of $\\overline{h}_H(\\overline{\\gamma })$ can be found from Jensen's inequality and taking into account that $\\overline{h}_R(\\overline{\\gamma })$ is convex: $ \\begin{split}\\overline{h}_R(\\overline{\\gamma })= &\\overline{h}_R \\left(\\frac{1}{\\pi }\\int _0^\\pi \\gamma (\\theta ,q) d\\theta \\right) \\le \\\\ &\\frac{1}{\\pi } \\int _0^\\pi \\overline{h}_R(\\gamma (\\theta ,q)) d\\theta =\\overline{h}_H(\\overline{\\gamma })\\end{split}$ On the other hand, (REF ) can be obtained analogously when $\\overline{h}_R(\\overline{\\gamma })$ is a concave increasing function.",
"The bounds in Lemmas REF and REF state that performance in Hoyt fading, for a given average SNR, will be bounded between that of Rayleigh fading with the same average SNR and that of Rayleigh fading when the average SNR is scaled by a factor $2q^2/(1+q^2)$ .",
"Note also that the derived bounds are asymptotically exact as $q \\rightarrow 1$ .",
"We have introduced a general approach to the analysis of wireless communication systems operating under Hoyt fading.",
"In the following sections, we use this technique to derive novel results for different performance metrics of interest."
],
[
"Channel Capacity",
"The channel capacity in Rayleigh fading channels was characterized in [13] for different transmission policies.",
"Even though closed-form expressions were attained for the Rayleigh case, the channel capacity in Hoyt fading channels is much more complicated to evaluate.",
"In fact, only infinite series expressions of very complicated argument are available in the literature [15], [16], which do not facilitate the extraction of any insights.",
"Using the Hoyt transform method, we will now show how to use readily available performance results derived for Rayleigh channels to directly obtain the same performance metric in Hoyt fading.",
"The capacity per bandwidth unit using optimum rate adaptation (ORA) policy with constant transmit power is calculated as $\\frac{C_{\\text{ora}}}{B}=\\overline{C}=\\int _{0}^{\\infty }\\log _2\\left(1+\\gamma \\right)f_{\\gamma }(\\gamma )d\\gamma ,$ where $\\log $ is the natural logarithm.",
"This capacity metric obtained by averaging the Shannon capacity on a flat-fading channel using the pdf of $\\gamma $ , and has dimensions of bps/Hz.",
"For a communication system operating under Rayleigh fading with average SNR at the receiver side given by $\\overline{\\gamma }$ , a closed-form expression for this metric was obtained in [13] as $\\overline{C}^{\\text{Ray}}=\\log _2 (e) e^{1/{\\overline{\\gamma }}}E_1(1/{\\overline{\\gamma }}),$ where $E_1(\\cdot )$ is the exponential integral function.",
"Since $\\overline{C}^{\\text{Ray}}$ is computed in the form stated in Lemma REF , then the we can directly calculate this metric considering a Hoyt fading channel as $\\overline{C}^{\\text{Hoyt}}=\\frac{\\log _2 (e)}{\\pi } \\int _{0}^{\\pi } e^{1/{\\gamma (\\theta ,q)}}E_1\\left(\\frac{1}{\\gamma (\\theta ,q)}\\right)d\\theta .$ Note that (REF ) is given in terms of a finite integral over a smooth and well-behaved function.",
"Hence, it can be calculated very accurately.",
"A simple lower bound can be found as $ \\overline{C}^{\\text{Hoyt}} \\ge \\frac{1}{\\ln 2}e^{ (1+q^2) /(2q^2\\overline{\\gamma }) }E_1 \\left(\\frac{(1+q^2)}{2q^2\\overline{\\gamma }}\\right).$ We now provide asymptotic capacity results in the low-SNR and high-SNR regimes.",
"In the first situation, it is known that the asymptotic capacity in Rayleigh fading is given by [34] $\\overline{C}_{\\overline{\\gamma }\\Downarrow }\\approx \\log _2 e \\frac{d\\mathcal {M}_{\\gamma }(s)}{ds}|_{s=0} = \\log _2 e \\overline{\\gamma },$ where $\\mathcal {M}_{\\gamma }(s)$ represents the MGF of the SNR.",
"Since (REF ) is obtained through linear operations over the distribution of the SNR, we can calculate the asymptotic capacity in Hoyt fading channels in the low-SNR regime as $\\overline{C}_{\\overline{\\gamma }\\Downarrow }^{\\text{Hoyt}}\\approx \\frac{\\log _2 e}{\\pi } \\int _{0}^{\\pi }\\overline{\\gamma }(\\theta ,q)d\\theta = \\log _2 e \\overline{\\gamma },$ which is the same as in the Rayleigh case, but also the same as in the Rician case [35].",
"In the high-SNR regime, the asymptotic capacity can be expressed in the following form [36] $\\overline{C}_{\\overline{\\gamma }\\Uparrow }\\approx \\log _2 e \\cdot \\frac{\\partial }{\\partial n} \\mathbb {E}\\left[\\gamma ^n\\right]|_{n=0}.$ which is asymptotically exact.",
"After some manipulations, we can equivalently express (REF ) as $\\overline{C}_{\\overline{\\gamma }\\Uparrow }&\\approx \\log _2(e)\\cdot \\log \\overline{\\gamma }- \\mu ,$ where $\\mu $ is a constant value independent of the average SNR, but dependent on the specific channel model.",
"The AWGN case yields a value of $\\mu =0$ , which is usually taken as a reference.",
"The effect of fading causes $\\mu >0$ and therefore there is a non-nil capacity loss due to fading for a given value of $\\overline{\\gamma }$ .",
"In the case of Rayleigh fading, it is a well-known result that ${\\mu =\\log _2(e)\\cdot \\gamma _e}$ , where $\\gamma _e$ is the Euler-Mascheroni constant.",
"Therefore, for a fixed value of $\\overline{\\gamma }$ , the parameter $\\mu $ can be regarded as the capacity loss with respect to the AWGN case, being $\\mu _{Rayleigh}\\approx 0.83\\,\\text{bps/Hz}$ .",
"There has been a renewed interest in the research community on different fading models that allow for characterizing a more severe fading condition than Rayleigh fading [37], [38].",
"Different models, such as the Two-Ray [37] or Hoyt fading meet this condition.",
"Very recently, the value of $\\mu $ for the Two-Ray fading channel was derived [35], yielding a capacity loss of $\\mu _{\\text{Two-ray}}=1\\,\\text{bps/Hz}$ .",
"This shows that the Two-Ray model is indeed more detrimental than Rayleigh fading, as it provokes a larger capacity loss.",
"However, the value of $\\mu $ for Hoyt fading is unknown in the literature to the best of our knowledge.",
"Using Lemma REF in (REF ), we have $\\overline{C}_{\\overline{\\gamma }\\Uparrow }^{\\text{Hoyt}}&\\approx \\tfrac{\\log _2(e)}{\\pi }\\int _0^{\\pi } \\log \\left\\lbrace \\overline{\\gamma }\\left(1-\\tfrac{1-q^2}{1+q^2}\\cos \\theta \\right)\\right\\rbrace d\\theta - \\mu _{\\text{Rayleigh}}.$ After some straightforward manipulations, we have $\\overline{C}_{\\overline{\\gamma }\\Uparrow }^{\\text{Hoyt}}&\\approx \\log _2(e)\\log \\overline{\\gamma }- \\mu _{\\text{Rayleigh}} - \\log _2(e)\\log \\left\\lbrace \\tfrac{2(1+q^2)}{(1+q)^2}\\right\\rbrace .$ Therefore, the capacity loss in Hoyt fading with respect to the AWGN case is given by $\\mu _{\\text{Hoyt}} = \\log _2(e)\\gamma _e + \\log _2(e)\\log \\left\\lbrace \\frac{2(1+q^2)}{(1+q)^2}\\right\\rbrace .$ The second term in (REF ) can be regarded as the capacity loss with respect to the Rayleigh case, and therefore equals 0 if $q=1$ .",
"In the limit case of $q=0$ , corresponding to the one-sided Gaussian distribution, we have the larger capacity loss given by $\\mu _{\\text{Hoyt}}^{q=1} = \\log _2(e)\\gamma _e +1 \\approx 1.83 \\text{bps/Hz}.$ Therefore, the capacity loss of Hoyt fading with respect to AWGN can be as large as 1 bps/Hz more than in the Rayleigh case.",
"These results are new to the best of our knowledge."
],
[
"Problem Definition",
"We consider the problem in which two legitimate peers, say Alice and Bob, wish to communicate over a wireless link in the presence of an eavesdropper, say Eve, that observes their transmission through a different link.",
"Let us denote as $\\gamma _b$ the instantaneous SNR at the receiver for the link between Alice and Bob, and let $\\gamma _e$ be the instantaneous SNR at the eavesdropper for the wiretap link between Alice and Eve.",
"Unlike the classical setup for the Gaussian wiretap channel [39], it is known that fading provides an additional layer of security to the communication between Alice and Bob [23], [40], allowing for a secure transmission even when Eve experiences a better SNR than the legitimate receiver Bob.",
"According to the information-theoretic formulation in [23], the secrecy capacity in this scenario is defined as $C_S=\\left[C_B-C_E\\right]^+,$ where $[a]^+ \\equiv \\max \\lbrace a,0 \\rbrace $ , $C_B$ is the capacity of the main channel $C_B=\\log \\left(1+\\gamma _b\\right),$ and $C_E$ is the capacity of the eavesdropper channel $C_E=\\log \\left(1+\\gamma _e\\right).$ For the sake of simplicity, we assumed a normalized bandwidth $B=1$ in the previous capacity definitions.",
"Since channel capacity is by definition a non-negative metric, the secrecy capacity for a given realization of the fading links is therefore given by $ C_S = \\left[\\log \\left( {1 + \\gamma _b } \\right) - \\log \\left( {1 + \\gamma _e } \\right)\\right]^+.$ In [23], [40], the physical layer security of the communication between Alice and Bob in the presence of Eve was characterized in terms of several performance metrics of interest, assuming that both wireless links undergo Rayleigh fading.",
"Specifically, closed-form expressions were derived for the probability of strictly positive secrecy capacity ${{\\mathcal {P} }(C_S>0)}$ , and for the outage probability of the secrecy capacity ${{\\mathcal {P} }(C_S<R_S)}$ , where $R_S$ is defined as the threshold rate under which secure communication cannot be achieved.",
"As these expressions will be used in the forthcoming analysis, we reproduce them for the readers' convenience ${\\mathcal {P} }(C_S>0)&=\\frac{\\bar{\\gamma }_b}{\\bar{\\gamma }_b+\\bar{\\gamma }_e},\\\\{\\mathcal {P} }(C_S<R_S)&=1-\\frac{\\bar{\\gamma }_b}{\\bar{\\gamma }_b+2^{R_S}\\bar{\\gamma }_e}\\exp {\\left(-\\frac{2^{R_S}-1}{\\bar{\\gamma }_b}\\right)},$ where $\\bar{\\gamma }_b$ and $\\bar{\\gamma }_e$ are the average SNRs at Bob and Eve, respectively.",
"We note that ${{\\mathcal {P} }(C_S>0)=1-{\\mathcal {P} }(C_S<R_S)_{R_S=0}}$ ; hence, the probability of strictly positive secrecy capacity will be considered as a particular case of the secrecy outage probability."
],
[
"Secrecy Outage Probability Analysis",
"Let us consider the scenario where the wireless links experience a more severe fading than Rayleigh, say Hoyt, where $q_b$ and $q_e$ represent the Hoyt shape parameters for the desired and eavesdropper links, respectively.",
"We also define the eccentricities associated with both Hoyt distributions as $\\epsilon _b=\\frac{1-q_b^2}{1+q_b^2}$ and $\\epsilon _e=\\frac{1-q_e^2}{1+q_e^2}$ According to the Hoyt transform, the outage probability of the secrecy capacity in Hoyt fading channels is given by ${\\mathcal {P} }(C_S < R_S ) = &1 - \\frac{1}{{\\pi ^2 }}\\int _0^\\pi \\int _0^\\pi \\exp \\left( { - \\tfrac{{2^{R_S } - 1}}{{\\bar{\\gamma }_b \\left( {1 - \\epsilon _b \\cos \\theta _b } \\right)}}} \\right) \\\\& \\nonumber \\times \\tfrac{{\\bar{\\gamma }_b \\left( {1 - \\epsilon _b \\cos \\theta _b } \\right)}}{{\\bar{\\gamma }_b \\left( {1 - \\epsilon _b \\cos \\theta _b } \\right) + 2^{R_S } \\bar{\\gamma }_e \\left( {1 - \\epsilon _e \\cos \\theta _e } \\right)}}d\\theta _e d\\theta _b.$ We observe that the integral over $\\theta _e$ can be solved, and hence we obtain $P(C_S < R_S ) =& 1 - \\frac{1}{\\pi }\\int _0^\\pi \\exp \\left( { - \\tfrac{{2^{R_S } - 1}}{{\\bar{\\gamma }_b \\left( \\theta \\right)}}} \\right) \\\\& \\nonumber \\times \\tfrac{{\\bar{\\gamma }_b \\left( \\theta \\right)}}{{\\bar{\\gamma }_b \\left( \\theta \\right) + 2^{R_S } \\bar{\\gamma }_e }}\\tfrac{1}{{\\sqrt{1 - \\left( {\\frac{{\\epsilon _e 2^{R_S } \\bar{\\gamma }_e }}{{\\bar{\\gamma }_b \\left( \\theta \\right) + 2^{R_S } \\bar{\\gamma }_e }}} \\right)^2 } }}d\\theta ,$ where ${\\bar{\\gamma }_b \\left( \\theta \\right)}={\\bar{\\gamma }_b \\left( {1 - \\epsilon _b \\cos \\theta } \\right)}$ .",
"Hence, the secrecy capacity OP is given in terms of a very simple integral form.",
"This result is new in the literature to the best of our knowledge, and shows the strength and versatility of the Hoyt transform method to derive new performance metrics for Hoyt fading by leveraging existing results for Rayleigh fading.",
"Directly from (REF ), the probability of strictly positive secrecy capacity can be easily obtained as $P(C_S {\\rm { > }}0) = \\frac{1}{\\pi }\\int _0^\\pi {\\tfrac{{\\bar{\\gamma }_b \\left( \\theta \\right)}}{{\\bar{\\gamma }_b \\left( \\theta \\right) + \\bar{\\gamma }_e }}\\tfrac{1}{{\\sqrt{1 - \\left( {\\frac{{\\varepsilon _e \\bar{\\gamma }_e }}{{\\bar{\\gamma }_b \\left( \\theta \\right) + \\bar{\\gamma }_e }}} \\right)^2 } }}} d\\theta .$ Expressions (REF ) and (REF ) admit an easy manipulation, in order to extract insights on the effect of fading severity into the secrecy capacity OP.",
"One clear example arises if we assume that the eavesdropper link suffers from a more severe fading compared to the desired link: this can be achieved by setting $q_b=1$ , and seeing what is the impact of $q_e$ .",
"In this case, the integral over $\\theta $ disappears, yielding to a closed-form expression for the secrecy capacity OP, and hence $\\left.",
"{P(C_S < R_S )} \\right|_{q_b = 1} = 1 - \\frac{{\\bar{\\gamma }_b }}{{\\bar{\\gamma }_b + 2^{R_S } \\bar{\\gamma }_e }}\\frac{{\\exp \\left( { - \\frac{{2^{R_S } - 1}}{{\\bar{\\gamma }_b }}} \\right)}}{{\\sqrt{1 - \\left( {\\frac{{\\varepsilon _e 2^{R_S } \\bar{\\gamma }_e }}{{\\bar{\\gamma }_b + 2^{R_S } \\bar{\\gamma }_e }}} \\right)^2 } }}$ Comparing () and (REF ), we observe that both have similar form, and the effect of the distribution of the fading for the eavesdropper link is captured by a multiplicative term that modulates the result for the Rayleigh case.",
"Since this additional term is always larger than one, it is clear that for a fixed value of $\\bar{\\gamma }_b$ and $\\bar{\\gamma }_e$ , the secrecy capacity OP $P(C_S < R_S )$ decreases with $q_e$ .",
"This illustrates the fact that when Eve suffers from a more severe fading, then the probability of having a secure communication between Alice and Bob grows.",
"A similar conclusion can be extracted when examining the probability of strictly positive secrecy capacity in this particular scenario: $\\left.",
"{P(C_S > 0 )} \\right|_{q_b = 1} =\\frac{{\\bar{\\gamma }_b }}{{\\bar{\\gamma }_b + \\bar{\\gamma }_e }}\\frac{1}{{\\sqrt{1 - \\left( {\\frac{{\\epsilon _e \\bar{\\gamma }_e }}{{\\bar{\\gamma }_b + \\bar{\\gamma }_e }}} \\right)^2 } }}.$ Again, the effect of considering $q_e<1$ (i.e.",
"a more severe fading than Rayleigh for the eavesdropper link) causes that ${P(C_S > 0 )}$ grows as $q_e$ is reduced."
],
[
"Outage Probability in Noise-Limited Scenarios",
"The outage probability is one of the most important performance metrics in wireless communications, and it is defined as the probability that the received SNR falls below a predefined threshold $\\gamma _o$ .",
"Thus, the outage probability is given by ${P_{out}(\\gamma _o)=F(\\gamma _o)}$ , where $F(\\cdot )$ is the cdf of the received SNR.",
"Therefore, the outage probability under Hoyt fading can be written, from (REF ), as $ P_{out}(\\gamma _o)=1 - \\frac{1}{\\pi } \\int _0^\\pi e^{-\\gamma _o/\\gamma (\\theta ,q)} d\\theta ,$ which can be very efficiently computed, as the integrand varies smoothly for all possible values of parameter $q$ .",
"Conversely, the integrand of the integral representation of the outage probability given in [41] becomes sharply peaked when $q$ is close to 0.",
"It is also interesting to note that, as the outage probability in Rayleigh fading (given by $ 1- e^{ - \\gamma _o /\\overline{\\gamma } }$ ) is an increasing concave function with respect to $\\overline{\\gamma }$ , the outage probability in Hoyt fading can be bounded as $ 1- e^{ - \\gamma _o /\\overline{\\gamma } } \\le P_{out}(\\gamma _o) \\le 1-e^{ - (1+q^2) \\gamma _o /(2q^2\\overline{\\gamma }) }.$ Alternatively, the outage probability can be expressed in closed-form using (REF ) and (REF ) as $ \\begin{split} P_{out}(\\gamma _o)= & Q\\left(a(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}},b(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}}\\right)\\\\ & -Q\\left(b(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}},a(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}}\\right)\\end{split}$ or, equivalently, from (REF ) and (REF ), $ \\begin{split} P_{out}(\\gamma _o)= & 2Q\\left(a(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}},b(q)\\sqrt{\\frac{\\gamma _o}{\\overline{\\gamma }}}\\right)\\\\ & -e^{-d(q)\\gamma _o/\\overline{\\gamma }}I_0\\left(c(q){\\frac{\\gamma _o}{\\overline{\\gamma }}}\\right)-1,\\end{split}$ where $ \\nonumber \\begin{split}& a(q)=\\frac{1+q }{2q}\\sqrt{1+q^2} , \\ b(q)=\\frac{1-q }{2q}\\sqrt{1+q^2}, \\\\ &c(q)=\\frac{1-q^4 }{4q^2}, \\ \\ \\ d(q)=\\frac{(1+q^2)^2}{4q^2},\\end{split}$ which were previously derived in [11] in a slightly different way.",
"Note, however, that (REF ) is easier to compute than these closed-form expressions because the Marcum $Q$ -function is actually a two-fold integral, as the Bessel function is an integral itself.",
"Although a finite-range integral can be used to compute the Marcum $Q$ -function [12], the integrand is sharply peaked for low values of $q$ , which complicates the computation.",
"Mathematical software packages often use truncated infinite power series [42] to compute the Marcum $Q$ -function."
],
[
"Outage Probability with Co-Channel Interference",
"In many practical scenarios, the signal of interest is affected by co-channel interference.",
"In this case the outage probability can be defined as the probability that the signal-to-interference-plus-noise ratio (SINR) falls below a threshold level $\\gamma _o$ .",
"More formally, let $\\mathcal {X}$ denote the SNR of the signal of interest and $\\mathcal {Y}$ the total interference-to-noise ratio (INR).",
"The outage probability will thus be defined as $ P_{out}(\\gamma _o)=P(\\mathcal {X}<\\gamma _o(\\mathcal {Y}+1))=F_\\mathcal {X}(\\gamma _o(\\mathcal {Y}+1)),$ where $F_\\mathcal {X}(\\cdot )$ is the cdf of $\\mathcal {X}$ .",
"In scenarios where the background noise can be neglected, the signal-to-interference ratio (SIR) is typically considered instead of the SINR, which usually simplifies the analysis.",
"We derive in this section a simple expression for the outage probability for the case when the signal of interest undergoes Hoyt fading and there is an arbitrary number of interferers.",
"Our analysis will be quite general, as we assume that background noise is not necessarily neglected and each interferer undergoes an arbitrary fading.",
"The main result for this scenario is presented in the following proposition.",
"Proposition 3 Let the desired signal undergo Hoyt fading, and the interfering signals experience arbitrarily distributed fading.",
"Then, the outage probability in the presence of co-channel interference and background noise is given by $ P_{out}(\\gamma _o)= 1 -\\frac{1}{\\pi } \\int _0^\\pi e^{-\\gamma _o/\\gamma (\\theta ,q)}\\phi _\\mathcal {Y} \\left(-\\frac{\\gamma _o}{\\gamma (\\theta ,q)}\\right)d\\theta .$ where $\\phi _\\mathcal {Y}$ is the MGF of $\\mathcal {Y}$ .",
"As $\\mathcal {X}$ follows a squared Hoyt distribution, from (REF ) we have that the conditional (on $\\mathcal {Y}$ ) outage probability can be written as $ \\left.P_{out}(\\gamma _o)\\right|_{\\mathcal {Y}=y}=1 - \\frac{1}{\\pi } \\int _0^\\pi e^{-\\gamma _o(y+1)/\\gamma (\\theta ,q)} d\\theta .$ Averaging over $\\mathcal {Y}$ , the unconditional outage probability will be $ P_{out}(\\gamma _o)=1 -\\int _0^\\infty \\frac{1}{\\pi } \\int _0^\\pi e^{-\\gamma _o(y+1)/\\gamma (\\theta ,q)} d\\theta f_\\mathcal {Y}(y) dy,$ where $f_\\mathcal {Y}(\\cdot )$ denotes the pdf of $\\mathcal {Y}$ .",
"By interchanging the order of integration we can rewrite (REF ) as $ \\begin{split}P_{out}(\\gamma _o)= & 1 -\\frac{1}{\\pi } \\int _0^\\pi e^{-\\gamma _o/\\gamma (\\theta ,q)}\\\\ & \\times \\left[\\int _0^\\infty e^{-y \\gamma _o/\\gamma (\\theta ,q)}f_\\mathcal {Y}(y) dy \\right]d\\theta .\\end{split}$ By noticing that the MGF of a positive random variable $\\alpha $ is defined as $\\phi _\\alpha (s)=\\int _0^\\infty e^{st}f_\\alpha (t)dt$ , with $f_\\alpha (\\cdot )$ being the pdf of $\\alpha $ , (REF ) is obtained.",
"Proposition REF allows to analyze the outage probability in Hoyt fading channels with arbitrarily distributed co-channel interference in the presence of background noise.",
"Since the MGF is given in closed-form for the most relevant fading distributions, then (REF ) can be easily evaluated as a finite-range integral.",
"The particular case where the background noise can be neglected is presented in the next corollary: Corollary 2 When the total interference power is much higher that the background noise and the latter can be neglected, the outage probability is given by $ P_{out}(\\gamma _o)= 1 -\\frac{1}{\\pi } \\int _0^\\pi \\phi _\\mathcal {Y} \\left(-\\frac{\\gamma _o}{\\gamma (\\theta ,q)}\\right)d\\theta .$ When the background noise can be neglected, the outage probability can be defined as $ P_{out}(\\gamma _o)=P(\\mathcal {X}<\\gamma _o\\mathcal {Y})=F_\\mathcal {X}(\\gamma _o\\mathcal {Y}),$ and by following the same steps as in Proposition REF , (REF ) is obtained.",
"When $L$ independent interferers are considered, the MGF of $\\mathcal {Y}$ will be $ \\phi _\\mathcal {Y}(s)=\\prod _{1=1}^L \\phi _{\\mathcal {Y}_i}(s),$ where $\\phi _{\\mathcal {Y}_i}(\\cdot )$ is the MGF corresponding to the i-th interferer.",
"In recent years, several general fading models such as $\\eta $ -$\\mu $ or $\\kappa $ -$\\mu $ have been proposed, showing a better fit to experimental measurements than traditional fading models in many different environments [27].",
"The $\\eta $ -$\\mu $ model includes Hoyt, Nakagami-$m$ , Rayleigh and one-sided Gaussian fading as particular cases, whereas the $\\kappa $ -$\\mu $ model includes Rician, Nakagami-$m$ and Rayleigh as particular cases.",
"In spite of their generality, these models have a MGF that can be expressed in simple terms [43].",
"The MGF of $\\eta $ -$\\mu $ is given by $ \\phi (s)= \\left(\\frac{4\\mu ^2 h}{ (2(h-H)\\mu -s\\overline{\\gamma }) (2(h+H)\\mu -s\\overline{\\gamma }) }\\right)^\\mu ,$ and the distribution is defined in two different formats.",
"In format 1 we have $H=(\\eta ^{-1}-\\eta )/4$ and $h=(2+\\eta ^{-1}+\\eta )/4$ , with $0<\\eta <\\infty $ , while in format 2: $H=\\eta /(1-\\eta ^2)$ and $h=1/(1-\\eta ^2)$ , with $-1<\\eta <1$ .",
"For the $\\kappa $ -$\\mu $ fading model we have $ \\phi (s)= \\left(\\frac{\\mu (1+\\kappa )}{\\mu (1+\\kappa )-s\\overline{\\gamma }} \\right)^\\mu \\exp \\left(\\frac{\\mu ^2\\kappa (1+\\kappa )}{\\mu (1+\\kappa )-s\\overline{\\gamma }}-\\mu \\kappa \\right),$ Introducing (REF ) and/or (REF ) as factors in (REF ) we can consider many different and general interfering scenarios.",
"Note that, as we have not imposed any restriction on the interferers statistics, our outage probability expression also includes the case of correlated interferers, as long as the MGF of the total interference power is known.",
"Fortunately, there exist closed-form expressions for the MGF of the addition of correlated signal powers of traditional fading models, such as Rayleigh, Nakagami-$m$ or Rician [14], and more general fading models such as $\\eta $ -$\\mu $ [44]."
],
[
"Numerical results",
"After describing how the Hoyt transform method can be applied to obtain new results for some information and communication-theoretic performance metrics, now we discuss the main implications that arise in practical scenarios."
],
[
"Channel capacity",
"First, we study the Shannon capacity of ORA technique in Hoyt fading channels.",
"In the following set of figures, we evaluate the capacity per bandwidth unit in Hoyt fading channels using (REF ).",
"Figure: Normalized capacity vs γ ¯\\bar{\\gamma } using ORA policy, for different values of qq.",
"Markers indicate the lower bounds on capacity given by (), for q=0.5q=0.5 ('x') and q=0.9q=0.9 (squares).In Fig.",
"REF , we observe how the capacity loss due to a more severe fading is low for values of $q>0.5$ , being under $0.15$ bps/Hz in this range.",
"In fact, it is noted how the achievable performance when $q=0.9$ is practically coincident with the Rayleigh case.",
"Markers indicate the simple lower bounds obtained in (REF ).",
"We see how the lower bound becomes tighter as $q$ is increased, whereas the performance for the Rayleigh case serves as an upper bound.",
"The accuracy of the asymptotic approximations yielding from (REF ) in the high-SNR regime is evaluated in Fig.",
"REF .",
"The capacity of the AWGN and Rayleigh ($q=1$ ) cases are included as reference.",
"Figure: Capacity per bandwidth unit vs γ ¯\\overline{\\gamma } using ORA policy, for different values of qq.",
"Straight thin lines represent the asymptotic results for high-SNR.We observe that for values of $q$ close to 1, the capacity loss with respect to the Rayleigh case is almost negligible.",
"However, we see that as $q$ is reduced, the gap between the capacities in the Rayleigh and Hoyt cases grows.",
"We see how the approximations for the capacity are asymptotically exact for sufficiently large $\\bar{\\gamma }$ .",
"However, we also observe that for $q=0$ the convergence between the exact and asymptotic capacity takes place at a larger value of $\\bar{\\gamma }$ .",
"We must note that the asymptotic results obtained for Hoyt fading are also valid for communication systems using optimal power and rate adaptation (OPRA) policy, since in the high-SNR regime this policy has the same performance as ORA [13].",
"In Fig.",
"REF , we study the asymptotic capacity loss due to Hoyt fading with respect to the reference AWGN case, which is given by (REF ).",
"The capacity loss for Rayleigh and Two-Ray [35] are also included for comparison purposes.",
"Figure: Asymptotic capacity loss μ Hoyt \\mu _{\\text{Hoyt}} as a function of qq.As indicated in (REF ), the maximum capacity loss is obtained when $q=0$ .",
"This corresponds to the more severe fading condition that Hoyt fading can represent.",
"As $q\\rightarrow 1$ , we see that the capacity loss tends to the value obtained in [13], i.e.",
"approximately $0.83$ bps/Hz.",
"The comparison with the Two-Ray fading model is also interesting: we see that the capacity loss in Hoyt and Two-Ray fading is coincident for $q\\approx 0.48$ .",
"This suggests that Hoyt fading represents a more severe condition than the Two-Ray model if $q<0.48$ , when the asymptotic capacity loss is chosen as the performance metric of interest."
],
[
"Secrecy capacity",
"Now we focus on the scenario considered in Section ; specifically, we will evaluate the effect of considering that the links between Alice and Bob (and equivalently between Alice and Eve) can suffer from different fading severities, quantified by the parameters $q_b$ and $q_e$ .",
"In Fig.",
"REF , the secrecy capacity OP derived in (REF ) is represented as a function of the average SNR at Bob $\\bar{\\gamma }_b$ , for different sets of values of the Hoyt shape parameters.",
"We assume that the normalized rate threshold value used to declare an outage is $R_S=0.1$ , and an average SNR at Eve $\\bar{\\gamma }_e=15$ dB.",
"Figure: Outage probability of secrecy capacity as a function of γ ¯ b \\bar{\\gamma }_b, for different values of q e q_e and q b q_b.",
"Parameter values γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.For a given value of $q_b$ , we observe two different effects depending on the magnitude of $\\bar{\\gamma }_b$ : in the low-medium SNR region, we see how a lower value of $q_e$ (i.e.",
"a more severe fading in the eavesdropper link) makes the occurrence of a secrecy outage to be less likely.",
"Hence, ${\\mathcal {P} }(C_S<R_S)$ decreases with $q_e$ for a given $\\bar{\\gamma }_b$ ; we also note how the secrecy in this region is barely affected by the value of $q_b$ .",
"Conversely, in the large SNR region we observe how the outage secrecy probability is mainly dominated by the fading severity of the desired link $q_b$ .",
"In this region, it is the distribution of $\\gamma _b$ the dominant factor in the secure communication between Alice and Bob.",
"The probability of strictly positive secrecy capacity given in (REF ) is evaluated in Fig.",
"REF , for the same set of parameter values considered in the previous figure.",
"Figure: Probability of strictly positive secrecy capacity as a function of γ ¯ b \\bar{\\gamma }_b, for different values of q e q_e and q b q_b.",
"Parameter value γ ¯ e =15\\bar{\\gamma }_e=15 dB.",
"Solid lines only indicate q b =0.2q_b=0.2; solid lines with markers are included for q b =0.8q_b=0.8.We can extract similar conclusions with regard of the effects of the fading severity in the desired and eavesdropper links.",
"For low values of $\\bar{\\gamma }_b$ , the secure communication is mainly determined by the distribution of $\\gamma _e$ ; specifically, considering $\\bar{\\gamma }_b=5$ dB we see how ${\\mathcal {P} }(C_S>0)$ is twice larger for $q_e=0.1$ , compared to $q_e=0.5$ .",
"We also observe how a less severe fading in the desired link (i.e.",
"a larger value of $q_b$ ) leads this probability to be larger."
],
[
"Outage Probability with interference",
"Now, we consider the scenario analyzed in Section , where the OP of wireless links in Hoyt fading is investigated in the presence of arbitrarily distributed co-channel interference and background noise.",
"We consider that the interference can be distributed according to the general $\\eta $ -$\\mu $ and $\\kappa $ -$\\mu $ distributions [27], widely employed in the literature for modeling NLOS and LOS propagation, respectively.",
"For the sake of simplicity in the discussion, we consider a single interferer; however, note that the analysis introduced in Section can accommodate an arbitrary number of interferers, admitting correlation between them as well as combinations of $\\eta $ -$\\mu $ and $\\kappa $ -$\\mu $ distributions.",
"We will start considering that both co-channel interference and background noise are present, the desired link undergoes Hoyt fading, and the interference is distributed according to the $\\eta $ -$\\mu $ or $\\kappa $ -$\\mu $ distribution with arbitrary values of their parameters.",
"We use format 2 for the $\\eta $ -$\\mu $ distribution.",
"We define the average SINR in this scenario as SINR$=\\frac{\\bar{\\gamma }_d}{1+\\bar{\\gamma }_i}$ , where $\\bar{\\gamma }_d$ is the ratio between the desired signal average receive power and the noise power (i.e., the SNR), whereas $\\bar{\\gamma }_i$ accounts for the ratio between the interferer average receive signal power at the receiver and the noise power (i.e., an interference-to-noise ratio, INR).",
"Fig.",
"REF evaluates the OP for different fading conditions for the desired and interfere links.",
"We assume INR$=5$ dB, and a SINR threshold value used to declare an outage $\\gamma _{o}=0$ dB.",
"In the low SINR regime, it is possible to note the effect of the distribution of the interference in the OP: as $\\mu $ is reduced, the propagation of the interfering signal experiences a more severe fade in both types of fading and hence the OP decays.",
"Conversely, we observe that for large values of the average SINR the OP tends asymptotically to a value that is determined by the distribution of the desired link, i.e.",
"by parameter $q$ .",
"For a given value of SINR, the OP grows as $q$ is reduced (i.e., an outage is declared with more probability when the direct link experiences a more severe fade).",
"Figure: Outage probability with co-channel interference and background noise as a function of the average SINR, considering a Hoyt distributed desired link and interference distributed according to η\\eta -μ\\mu or κ\\kappa -μ\\mu distributions.",
"Parameter values are INR=5=5 dB and γ o =0\\gamma _{o}=0 dB.Now, we assume an interference-limited system, meaning that the background noise will be neglected.",
"In this scenario, we define the average SIR as the ratio between the average powers of the desired and interfering signal.",
"As in the previous case, we set the threshold value used to declare an outage as $\\gamma _{o}=0$ dB.",
"In Fig.",
"REF , the OP is represented as a function of the SIR, for different values of parameters $q$ , $\\kappa $ , $\\eta $ and $\\mu $ .",
"In the large SIR regime, it is reinforced the conclusion that the asymptotic OP is dominated by the distribution of the desired link.",
"For low values of SIR, a similar behavior as in the previous figure is observed.",
"However, it is noted a larger difference in the OP values when $\\mu $ changes; this is due to the fact that when background noise vanishes, the differences between the different distributions of the interference become more evident.",
"Figure: Outage probability with co-channel interference as a function of the average SIR, considering a Hoyt distributed desired link and interference distributed according to η\\eta -μ\\mu or κ\\kappa -μ\\mu distributions.",
"Parameter value γ o =0\\gamma _{o}=0 dB."
],
[
"Conclusions",
"We have provided a new look at the analysis of wireless communication systems in Hoyt (Nakagami-$q$ ) fading.",
"Unlike previous approaches in the literature, we have found a connection between the Rayleigh and Hoyt distributions that facilitates the analysis in the latter scenario.",
"By deriving integral expressions for the pdf and cdf of the squared Hoyt distribution, we have shown that the squared Hoyt distribution is in fact a composition of exponential distributions with continuously varying averages.",
"Using this connection, we have introduced the Hoyt transform approach as a way to obtain easy-to-compute finite-range integral expressions of different performance metrics in Hoyt fading channels, as well as simple upper and lower bounds which become asymptotically tight as $q\\rightarrow 1$ , by simply leveraging existing results for Rayleigh fading channels.",
"As a direct application, we have derived new expressions for several scenarios of interest in information and communication theory: (a) capacity analysis of adaptive transmission policies in Hoyt fading channels, (b) wireless information-theoretic security in Hoyt fading, and (c) outage probability analysis of Hoyt fading channels with arbitrarily distributed interference and background noise.",
"A further implication of the results in this paper is that there is no need to reproduce complicated calculations for Hoyt fading in the cases where tractable expressions are available for the Rayleigh case.",
"Instead, these analyses can be easily extended to Hoyt scenarios by using a straightforward finite-range integral."
],
[
"Acknowledgments",
"The work of Juan M. Romero-Jerez was supported by the Spanish Government-FEDER public Project No.",
"TEC2013-42711-R.",
"The work of F.J. Lopez-Martinez was funded by Junta de Andalucia (P11-TIC-7109), Spanish Government-FEDER (TEC2013-44442-P, COFUND2013-40259), the University of Malaga and the European Union under Marie-Curie COFUND U-mobility program (ref.",
"246550)."
]
] | 1403.0537 |
[
[
"Fossil Groups Origins III. Characterization of the sample and\n observational properties of fossil systems"
],
[
"Abstract (Abridged) Fossil systems are group- or cluster-sized objects whose luminosity is dominated by a very massive central galaxy.",
"In the current cold dark matter scenario, these objects formed hierarchically at an early epoch of the Universe and then slowly evolved until present day.",
"That is the reason why they are called {\\it fossils}.",
"We started an extensive observational program to characterize a sample of 34 fossil group candidates spanning a broad range of physical properties.",
"Deep $r-$band images were taken for each candidate and optical spectroscopic observations were obtained for $\\sim$ 1200 galaxies.",
"This new dataset was completed with SDSS DR7 archival data to obtain robust cluster membership and global properties of each fossil group candidate.",
"For each system, we recomputed the magnitude gaps between the two brightest galaxies ($\\Delta m_{12}$) and the first and fourth ranked galaxies ($\\Delta m_{14}$) within 0.5 $R_{{\\rm 200}}$.",
"We consider fossil systems those with $\\Delta m_{12} \\ge 2$ mag or $\\Delta m_{14} \\ge 2.5$ mag within the errors.",
"We find that 15 candidates turned out to be fossil systems.",
"Their observational properties agree with those of non-fossil systems.",
"Both follow the same correlations, but fossils are always extreme cases.",
"In particular, they host the brightest central galaxies and the fraction of total galaxy light enclosed in the central galaxy is larger in fossil than in non-fossil systems.",
"Finally, we confirm the existence of genuine fossil clusters.",
"Combining our results with others in the literature, we favor the merging scenario in which fossil systems formed due to mergers of $L^\\ast$ galaxies.",
"The large magnitude gap is a consequence of the extreme merger ratio within fossil systems and therefore it is an evolutionary effect.",
"Moreover, we suggest that at least one candidate in our sample could represent a transitional fossil stage."
],
[
"Introduction",
"Fossil systems are group- or cluster-sized [65], [22] objects whose luminosity is dominated by a very massive central galaxy.",
"In the current cold dark matter (CDM) scenario, these objects formed hierarchically at an early epoch of the Universe and then slowly evolved until present day.",
"That is the reason why they are called fossils.",
"The study of this particular kind of objects started two decades ago, when [76] suggested that RX-J1340.6+4018 was probably the remains of an ancient group of galaxies.",
"Later, [49] gave the first observational definition of Fossil Groups (FGs) as systems characterized by a magnitude gap larger than 2 mag in the $r-$ band between the two brightest galaxies of the system within half the virial radius.",
"Moreover, the central galaxy should be surrounded by a diffuse X-ray halo, with a luminosity of at least $L_{\\rm X} > 10^{42}$ $h_{50}^{-2}$ erg ${\\rm s^{-1}}$ , with the aim of excluding bright isolated galaxies.",
"Many optical and X-ray observational properties of FGs have been studied, but always on small samples or individual systems.",
"These properties can be grouped in: (i) properties of the intracluster hot component; (ii) properties of the galaxy population; and (iii) properties of the brightest group galaxy (hereafter BGG).",
"Referring to the hot gas component, fossil and non-fossil systems generally show a similar $L_{\\rm X}-T_{\\rm X}$ relation [53], [44].",
"Differences in scaling relations that combine both optical and X-ray properties were detected.",
"In particular, some authors found different relations in optical vs X-ray luminosity ($L_{{\\rm opt}}-L_{\\rm X}$ ), X-ray luminosity vs velocity dispersion of the clusters galaxies ($L_{\\rm X}-\\sigma _{{\\rm v}}$ ), and X-ray temperature vs velocity dispersion ($T_{\\rm X}-\\sigma _{{\\rm v}}$ ).",
"In these works, for any given L$_{{\\rm opt}}$ , FGs are more luminous and hotter in the X-rays than normal groups or clusters.",
"These differences were interpreted as a deficit formation of $L^\\ast $ galaxies in FGs [78].",
"In contrast, other authors such as [84] and [44] did not find any different relation between X-ray and optical quantities for FGs and normal groups and clusters.",
"They claimed that the previous differences were due to observational biases in the selection of FGs or inhomogeneity between the FGs and the comparison sample.",
"In addition, high $S/N$ and high resolution X-ray observations of fossil systems seem to confirm that fossil systems are formed inside high centrally concentrated dark matter (DM) halos [83], [52], with large mass-to-light ratios, which could indicate an early formation.",
"Nevertheless, most of the fossil systems do not show cooling cores [28] as normal clusters, suggesting that strong heating mechanisms, such as AGN feedback or cluster mergers, could heat the central regions of their DM halos [83], [51], [52], [66].",
"The galaxy luminosity function (hereafter LF) is a powerful tool for studying the galaxy population in clusters.",
"In the past, several works investigated the galaxy LF in fossil systems.",
"They found that the LF of these objects are well fitted by single Schechter function, but there is a large variety of values in the faint-end slope ($\\alpha $ ) of the LFs of FGs.",
"In particular, the values of $\\alpha $ measured goes from $-1.6$ to $-0.6$ [22], [52], [65], [66], [91], [5], [57].",
"Unfortunately, all these studies were performed on single FGs or very small samples, and a systematic study of LFs of statistically meaningful samples of FGs remains to be done.",
"The brightest central galaxies of fossil systems are amongst the most massive and luminous galaxies known in the Universe.",
"In fact, the luminosity and the fraction of light contained in the BGGs correlate with the magnitude gap [44].",
"Some observations [52] show that these objects are different from both isolated elliptical galaxies and central galaxies in non-fossil clusters in the sense that they have disky isophotes in the centre and their luminosity correlates with velocity dispersion, while other authors [55], [67] found no differences in isophotal shapes between fossil and non fossil central galaxies.",
"In [67] we analised deep K-band images of 20 BGGs in fossil and non-fossil systems and showed that these galaxies follow the tilted fundamental plane of normal ellipticals [11].",
"This fact suggests that BGGs are dynamically relaxed systems that suffer dissipational mergers during their formation.",
"On the other hand, they depart from both Faber-Jackson and luminosity-size relations.",
"In particular, BGGs have larger effective radii and smaller velocity dispersions than those predicted by these relations.",
"We infer that BGGs grew throughout dissipational mergers in an early stage of their evolution and then assembled the bulk of their mass through subsequent dry mergers.",
"Nevertheless, stellar population studies of BGGs in fossil systems suggest that their age, metallicity and $\\alpha $ -enhancement are similar to those of bright ellipticals field galaxies [55], [34].",
"In numerical simulations, FGs are found to be a particular case of structure formation.",
"They are supposed to be located in highly concentrated DM halos, so that they can assembly half of their DM mass at z $>$ 1.",
"Then, the FGs grow via minor mergers only, and only accrete $\\approx $ 1 galaxy from z $\\approx 1$ down to present time, while regular groups accrete about three galaxies in the same range of time [89].",
"[25] show that the mass assembled at any redshift is higher in fossil than in non-fossil systems.",
"This means that the formation time is, on average, shorter for FGs than for regular systems [30], [89], leaving to FGs enough time to merge $L^\\ast $ galaxies in one very massive central object.",
"In fact, simulations predict that the timescale for merging via dynamical friction is inversely proportional to the mass of the galaxy, thus favouring the merging of larger objects.",
"So, the dynamical friction would be responsible for the lack of $L^\\ast $ galaxies which is reflected in the requested magnitude gap of the observational definition.",
"Moreover, to enhance the high efficiency in the merging process, FGs should have particular dynamical properties, such as the location of massive satellites on orbits with low angular momentum [82].",
"So, a combination of high mass satellite and low angular momentum orbits boosts the efficiency of the merging process [18].",
"Recently, [56] demonstrate that the growth of the BCGs since z$\\sim $ 1 is mainly due to major mergers, suggesting that this could be the dominant mechanism in accreting the mass of central galaxies in cluster and thus supporting indirectly the merging scenario for fossil systems, which would differ from regular clusters only for the early time formation.",
"Nevertheless, this evolutionary picture in which fossil groups became fossils in the early Universe and then evolved undisturbed is not the only proposed scenario.",
"In the framework of the merging scenario [27] suggest that first ranked galaxies in fossil systems has the last major merger later than non-fossil ones.",
"This means that the formation of large magnitude gaps as those in nowadays fossil systems is a long term process.",
"In addition, [89] predict that the fossilness could be a transitional status of some systems.",
"Thus, some fossil systems have become non-fossil ones in recent epoch due to accretion of nearby galaxy groups.",
"An alternative formation scenario in which the magnitude gap of the systems appears at the beginning of the formation process can also explain the reported observational properties.",
"This is the so called monolithic scenario, in which fossil systems formed with a top heavy LF.",
"In this scenario, the magnitude gap is due to a primordial deficient formation of $L^\\ast $ galaxies [70].",
"All the observational results presented in the literature are limited by the small number of FGs known in the literature.",
"A more general study of fossil systems is needed in order to discriminate between these two formation scenarios.",
"For this reason, we started an extensive observational program called Fossil Groups Origins (FOGO), aimed at carrying out a large, systematic and multiwavelenght study of a sample of 34 FGs candidates identified by [81].",
"The specific goals of the program include mass and dynamics of FGs, properties of their galaxy populations, formation of the central galaxies and their connection with the intragroup medium, properties of the extended diffuse light, and agreement with old and new simulations.",
"The details of the project are resumed in the first paper of the series [5].",
"The structural properties of the BGGs in fossil and non-fossil systems were shown in the second paper [67].",
"The L$_{\\rm X}$ -L$_{{\\rm opt}}$ relation of fossil and normal systems will be presented in a forthcoming paper (Girardi et al., in prep).",
"This is the third paper of the series, devoted to the characterization of the sample.",
"In particular, we recomputed the magnitude gaps of the systems by using new spectroscopic redshift measurements.",
"These new data provide us robust cluster membership and global properties for the systems.",
"Only 15 out of 34 turned out to be fossil systems according with the standard definition [49].",
"We have also analised the relations between central galaxies in FGs and non-FGs and other quantities such as magnitude gaps and velocity dispersion of the host halo.",
"FGs follow the same relations than non-FGs, but they are extreme cases.",
"The paper is organized as follows.",
"The description of the sample is given in Section 2.",
"The available dataset is shown in Section 3.",
"Radial velocities determination are presented in Section 4.",
"The results are given in Section 5.",
"Sections 6 and 7 report the discussion and conclusions of the paper, respectively.",
"Unless otherwise stated, we give errors at the 68% confidence level.",
"Throughout this paper, we use $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{\\Lambda }=0.7$ and $\\Omega _M=0.3$"
],
[
"Description of the sample",
"The FOGO sample is based on the [81] FG candidates selected from the Sloan Digital Sky Survey Data Release 5 [2].",
"[81] selected 112.510 galaxies brighter than $r=19$ in the Luminous Red Galaxy catalog [33].",
"A cross-match with the Rosat All Sky Survey catalog [85] was performed to look for the presence of a diffuse X-ray halo of at least $10^{42}$ erg s$^{-1}$ and closer than 0$$ 5 from the position of each LRG.",
"FInally, they looked for the brightest companions of each of the remaining LRGs within a fixed radius of $0.5h_{70}^{-1}$ Mpc to satisfy the magnitude gap $\\Delta m_{12} \\ge 2$ between the two brightest galaxies of the group.",
"The final catalog comprises 34 FG candidates with some unique characteristics: the sample spans the last 5 Gyr of galaxy evolution ($0\\le z\\le 0.5$ ), it has a wide range of X-ray luminosities and therefore masses, and the absolute magnitude of the central BGG covers a large range ($-25.3\\le M_{r}\\le -21.3$ ).",
"In this work we present the analysis of the 34 systems of the sample of [81].",
"For each system we were able to compute new $\\Delta m_{12}$ and $\\Delta m_{14}$ gaps combining our deep $r-$ band images with photometric data from the SDSS DR7.",
"We measure the LOS velocity dispersion for those systems with at least 10 members within $R_{200}$ The radius $R_{{\\rm \\delta }}$ is the radius of a sphere with mass over density $\\delta $ times the critical density at the redshift of the galaxy system.",
"This subsample is formed by 24 groups with redshift obtained mainly from our own spectroscopy ($\\sim 1200$ new velocities, see Sect.",
"REF ).",
"In Fig.",
"REF we show the distribution of redshifts, X-ray luminosities and absolute magnitude of the 34 BGGs of the sample by [81] and of our subsample of 24 objects plus FGS28, for which only one member galaxy is found within $R_{200}$ .",
"The Kolmogorov-Smirnov test confirms that our subsample of 25 and the whole sample of 34 FG candidates by [81] come from the same parent distribution.",
"Hereafter, we identify each system using the notation FGS + ID, where ID is the id number in Table 1 of [81]."
],
[
"Photometric data",
"Deep images for 22 of the FG candidates were obtained using the 2.5-m Nordic Optical Telescope (NOT) at the Roque de los Muchachos Observatory (ORM, La Palma, Spain) in the period between 2008-2011.",
"We used Andalucia Faint Object Spectrograph and Camera (ALFOSC) in imaging mode with SDSS $r-$ band filter.",
"The detector was a CCD of 2048$\\times $ 2048 pixels with a plate scale of 019 pixel$^{-1}$ .",
"For other 10 candidates, images in the same band were taken at the 2.5-m Isaac Newton Telescope (INT) at the ORM in the same period using the Wide Field Camera (WFC).",
"It consists of 4 2000$\\times $ 4000 CCDs with a scale of 033 pixel$^{-1}$ .",
"All the images were obtained under photometric conditions, and the mean value of the seeing FWHM was 10.",
"Only FGS27 and FGS33 were observed under bad seeing conditions.",
"Their final combined images have a seeing FWHM $\\ge $ 2and therefore they were replaced with SDSS images.",
"For FGS09 and FGS26 it was impossible to obtain deep images due to the presence of a very bright star located close to their BGGs.",
"SDSS photometric images were also used for these two systems.",
"Data reduction was performed using standard IRAFIRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.",
"routines, correcting for bias and flat field.",
"In most of the cases, after these corrections, we detected some residual light.",
"In order to achieve the best possible flat field correction, we obtained a super-flat field using a combination of the scientific images.",
"Then we corrected once again the scientific images with such a new super-flat field [5].",
"The images were combined and calibrated using SDSS data of the not saturated stars available in the field of view.",
"The typical RMS error of the calibration is 0.08 mag."
],
[
"Spectroscopic data",
"We used the SDSS DR6 [3] photometric redshifts ($z_{{\\rm phot}}$ ) to select reliable targets for multi-object spectroscopy (MOS).",
"For each FG candidate we downloaded a catalog with all galaxies brighter than $m_r = 22$ within a radius of 30 arcmin around each BGG.",
"The $m_r$ value represents the completeness limit of the photometric catalog of the SDSS, the selected radius is larger than a virial radius for all our FG candidates.",
"Then, we considered as possible targets only galaxies with photometric redshift within the range of $\\Delta z_{{\\rm phot}} \\pm 0.15$ from the spectroscopic redshift of the BGG.",
"This value was chosen because the typical error on photometric redshift in the SDSS DR6 is about 0.1.",
"Finally, we visually selected the targets trying to maximize the number of slits per mask.",
"We also put 60 slits on galaxies with spectroscopic redshift in the SDSS DR6 for a comparison.",
"We observed a total of 51 masks with on average 30 slits per mask.",
"We obtained 1227 spectra with a $S/N \\ge 5$ , which is enough to measure the line-of-sight (LOS) velocity of the galaxies.",
"MOS observations were performed under International Time Program (ITP) of the ORM in the period between 2008-2010.",
"Additional observations were done under one Italian and two Spanish Time Allocation Comitee (TAC) runs between 2008-2012.",
"The data were taken at the 3.5-m Telescopio Nazionale Galileo (TNG) telescope using Device Optimized for the LOw RESolution (DOLORES) in the MOS mode.",
"The instrument has a CCD of $2048\\times 2048$ pixels with a pixel size of 13.5 $\\mu $ m and a 0252 pixel$^{-1}$ scale.",
"We used the LR-B Grism with a dispersion of 187 $Å$ mm$^{-1}$ , together with 16 slits.",
"This led to a final resolution of $R=365$ in the wavelength range 3000-8430 Å.",
"The typical exposure time was of 3$\\times $ 1800s per mask and the mean FWHM of the seeing was 10.",
"We performed the data reduction using standard IRAF procedures.",
"We did not correct for bias and flat field because the uncorrected spectra result less noisy than the corrected ones.",
"In particular, the measured LOS velocities are the same in both the corrected and uncorrected spectra, but the uncertainties are larger when the bias and flat field corrections are applied.",
"The cosmic rays correction was performed during the combining process of the different exposures we obtained for each mask.",
"The sky was evaluated by measuring the median value in the outer regions of each spectrum.",
"To perform the wavelength calibration we used two different lamps (Ne+He and Ne+Hg) to have arc lines in both the red and blue part of the spectrum.",
"The typical uncertainty of the wavelength calibration was $< 0.1$ $Å$ (RMS).",
"Finally, we corrected for the instrumental distortions by measuring the [OI]$\\lambda $ 5577 $Å$ sky line.",
"This is crucial since we divided each mask exposure in individual exposures of 1800s, sometimes taken in different days or runs.",
"The mean error associated to the instrumental distortions is 0.85 $Å$ (which corresponds to 45 km s$^{-1}$ ), but it can be as large as 8 $Å$ (450 km s$^{-1}$ ).",
"We corrected all the measured LOS velocities to take into account for the systematic error due to the instrumental distortions."
],
[
"Line of sight velocity measurement",
"The LOS velocities were measured using the cross-correlation technique [88] implemented in the IRAF task XCSAO of the package RVSAORVSAO was developed at the Smithsonian Astrophysical Observatory Telescope Data Center.. For each spectrum the task performs a cross-correlation with six templates [50], corresponding to different types of galaxies (E, S0, Sa, Sb, Sc, Irr).",
"The template with the highest value of the S/N of the cross-correlation peak was chosen.",
"In addition, we visually inspected all the spectra to verify the velocity determination.",
"In most of the cases the LOS velocity was obtained from the absorption lines.",
"Nevertheless, in some spectra with low $S/N$ (especially for faint objects with $m_r > 20.5$ ) the emission lines were measured with the IRAF task EMSAO to obtain the LOS velocity.",
"In Fig.",
"REF the absorption lines are detectable in the five brightest objects but not in the faintest one.",
"The latter is actually the only galaxy with $m_r > 21$ for which we measured the LOS velocity.",
"The nominal uncertainties given by the cross–correlation algorithm are known to be smaller than the true ones [59], [8], [35], [80].",
"The uncertainties obtained through the RVSAO procedure were multiplied by a factor 2, following previous analyses [9] on data acquired with the same instrumental setup and with comparable quality or our own.",
"Moreover, to be conservative, we assumed the largest between the formal uncertainty and 100 km s$^{-1}$ for the LOS velocities measured with the EMSAO procedure .",
"We adopted the weighted mean of the different determinations and the corresponding error for the galaxies with repeated measurements .",
"The RMS of this difference for 48 galaxies is 107 km s$^{-1}$ ."
],
[
"Additional line of sight velocities",
"In order to have the largest possible number of LOS velocities for the 34 FG candidates, we added all available redshifts within $R_{200}$ from the SDSS-DR7.",
"Figure REF shows the comparison between our and SDSS LOS velocity measures for the 60 galaxies for which both values are available.",
"The RMS of the difference between the two values is 84 km s$^{-1}$ , which is consistent with the results of Sect.",
"REF .",
"Finally, for FGS05 we added the LOS velocities given by [42] and obtained with the same instrumental setup and data analysis.",
"Figure: Differences in LOS velocity for galaxies with both FOGO and SDSS measurements (top panel) and for galaxies with repeated FOGO measurements (bottom panel).",
"Dotted lines represent the 1σ1\\sigma scatter of the data."
],
[
"Spectroscopic completeness",
"The completeness of the spectroscopic sample is a crucial parameter since it is used in the derivation of several quantities, such as the spectroscopic luminosity function.",
"For each magnitude bin we defined our completeness as the ratio between the number of galaxies of the 25 FG candidates for which we were able to obtain a redshift ($N_z$ ) from either the FOGO or SDSS spectroscopy and the number of targets ($N_{z_{{\\rm phot}}}$ , see Sec.",
"REF ): $C=\\frac{N_z}{N_{z_{\\rm phot}}}.$ In Fig.",
"REF we show our completeness as a function of the $r-$ band magnitude.",
"Our sample is more than 70% complete down to $m_r=17$ and more than 50% complete at $m_r=18$ .",
"In a similar way, for each magnitude bin, we defined the success rate as the ratio between the number of galaxies of the 18 FG candidates for which we are able to measure a redshift with our own spectroscopy ($N_{z_{{\\rm our}}}$ ) and the total number of observed galaxies ($N_{{\\rm obs}}$ ): $SR=\\frac{ N_{z_{\\rm our}} }{ N_{{\\rm obs}} }.$ Figure REF also shows the success rate as function of the $r-$ band magnitude.",
"Notice that we have a success rate larger than 75% for objects with $m_r < 21$ .",
"The success rate decreases abruptly $m_r > 21$ .",
"Thus, we conclude that the adopted combination of instrumental setup and exposure time is effective for measuring the redshift of galaxies with $m_r \\le 21$ .",
"Figure: Spectroscopic completeness of the 25 FG candidates with either FOGO or SDSS spectroscopy (top panel) and success rate of the 18 FG candidates with FOGO spectroscopy only (bottom panel) as a function of r-r-band magnitude."
],
[
"System membership",
"The identification of systems and the membership of individual galaxies were obtained using a two-step procedure applied to the galaxies in the region within $R_{{\\rm 200}}$ .",
"First, we used DEDICA [75], which is an adaptive kernel procedure that works under the assumption that a cluster corresponds to a local maximum in the density of galaxies.",
"Then, we adopted the likelihood ratio test [63] to assign a membership probability to each single galaxy relatively to an identified cluster.",
"According to the DEDICA procedure, each FG candidate was detected as a very significant peak (at a confidence level $>99\\%$ ) at the redshift corresponding to that of the BGG.",
"Only FGS14, FGS23, and FGS26 were detected as two close peaks (with $\\Delta v$ $<$ 1500 km s$^{-1}$ in the rest frame).",
"For each FG candidate, the redshift of the BGG, the redshift distribution of the galaxies, and the redshift peak associated to the BGG are shown in Fig.",
"REF .",
"Some structures are isolated (e.g., FGS20) while others present clear foreground (e.g, FGS07) or background (e.g., FGS15) contamination.",
"The corresponding members were then identified using the distance-velocity diagram (Fig.",
"REF ), which consists in the so called \"shifting gapper\" method [36], [39].",
"This procedure rejects galaxies within a fixed distance bin that are too far in velocity from the main body of the system.",
"The distance bin is shifted outwards out to $R_{200}$ .",
"The procedure was then iterated until the number of cluster members converged to a stable value.",
"Following [36], we used a velocity gap of 1000 km s$^{-1}$ in the cluster rest–frame and a distance bin of 0.6 Mpc (or large enough to include 10 galaxies).",
"In the case of FGS02 we slightly modified the gap value (1100 km s$^{-1}$ ) to be more conservative and avoid the rejection of a few galaxies at the edge of the system.",
"For all the systems our own spectroscopic data extend to at least 0.5 $R_{{\\rm 200}}$ , except for FGS28, FGS30, and FGS31 for which we covered 0.4 $R_{{\\rm 200}}$ .",
"The membership efficiency, defined as the fraction of confirmed members with respect to the observed targets, turns out to be 59%.",
"FGS15 seems to be a peculiar case within the subsample of 25 FG candidates.",
"It has only 13 members spanning a large range in velocity between one another (up to 6000 km s$^{-1}$ ).",
"Thus, it is not clear if either this system is very massive or it is part of a larger structure, such as a filament.",
"Another system with peculiar properties is FGS28.",
"It is the smallest group of the [81] sample.",
"It has the faintest BGG and the lowest X-ray luminosity, and it seems much more an isolated giant elliptical galaxy than a group of galaxies.",
"The peak associated to the BGG in the velocity histogram (Fig.",
"REF ) is not significant and we found only four (possible) members which are at a distance about $2\\,R_{200}$ of the group.",
"We argue that this BGG is actually a large and isolated galaxy which is embedded in a high density environment due to the presence of another cluster in the same region.",
"For both FGS28 and FGS15, we used the membership only for calculating the magnitude gaps, but not for estimating the LOS velocity dispersion and mass."
],
[
"Cluster global properties",
"In Table REF we present the general properties for each system of the [81] sample.",
"We estimated $R_{{\\rm 200}}$ from $L_{{\\rm X}}$ , which is available for each of the 34 FG candidates from the RASS Catalogs.",
"We decided to recalculate $L_{\\rm X}$ because of the discrepancies between the values reported by [81] and other measurements available in the literature for some well-studied clusters of their sample [14] For each FG candidate, we took into account the counts from the RASS Bright Source Catalog [85] or, alternatively, from the , RASS Faint Source Catalog [86] in the broad band 0.1-2.4 keV.",
"We used the total Galactic column density ($N_{\\rm H}$ ) as taken from the NASA's HEASARC $N_{\\rm H}$ toolhttp://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl and the redshift $z$ as listed by [81].",
"We used an iterative procedure based on PIMMSftp:${\\rm //legacy.gsfc.nasa.gov/software/tools/pimms4\\_3.tar.gz}$ [68].",
"Details of the procedure are available in [43].",
"With our new $L{\\rm _{x}}$ estimates, we were able to calculate $R_{{\\rm 500}}$ using the relation proposed by [16]: $R_{500}=0.753\\, {\\rm Mpc}\\, h^{-0.544}_{100} \\,E(z)^{-1}\\, L_{X,44}^{0.228}$ where $E(z)=h(z)/h_0$ and $L_{X,44}$ is the X-ray luminosity in units of $h_{70}^{-2}10^{44}$ erg s${-1}$ in the 0.1-0.24 keV band.",
"We calculated $R_{200}=1.516 \\times R_{500}$ as prescripted by [6].",
"We computed the mean LOS velocity dispersion $\\sigma _{{\\rm v}}$ by using the bi-weight estimator of the ROSTAT package [10] for systems with more than 10 members.",
"For the remaining systems we computed $\\sigma _{{\\rm v}}$ using the bi-weight estimator and shifting gapper.",
"We applied both the cosmological correction and standard correction for velocity uncertainties [24].",
"By assuming sphericity, dynamical equilibrium and that galaxy distribution traces mass distribution, we followed [40] and [41] to compute the virial mass as: $M_{\\rm v} = \\frac{3\\pi }{2G} \\sigma _v \\,R_{{\\rm pv}}-{\\rm SPT}$ where SPT is the surface pressure term correction [87], while $R_{{\\rm pv}}$ is two times the projected mean harmonic radius.",
"We could not compute $R_{{\\rm pv}}$ by using data of observed galaxies since our $z$ data do not cover the whole $R_{{\\rm 200}}$ region.",
"Therefore, we used an alternative estimate which is valid for a typical galaxy distribution in clusters [40].",
"We assumed 20% for the SPT correction, as obtained by combining data on many clusters and valid at a radius around $R_{{\\rm 200}}$ [19], [40].",
"Table: Global properties of our sample."
],
[
"L$_{\\rm X}$ -{{formula:d3764243-c325-4924-b03c-2ffe31970fb3}} relation",
"Once we obtained the luminosity in X-ray and velocity dispersion of the galaxies, we were able to evaluate the $L_{{\\rm X}}-\\sigma _{v}$ relation.",
"This relation is connected with the formation of the cluster.",
"In fact, theoretical predictions based on purely gravitational collapse lead to $L_{\\rm X} \\propto \\sigma _{v}^4$ .",
"There are several observational studies of this relation, the majority of them finding values between 4 and 5.3 for the slope [79], [31], [69], [17], [90], [60], [41], [71], [45].",
"In Fig.",
"REF we show the distribution and best-fit to our data.",
"We found a slope consistent within the errors to the theoretical predictions.",
"We derived the best-fit $L_{{\\rm X}}-\\sigma _{{\\rm v}}$ relation using the FITEXY algorithm in IDLInteractive Data Language is distributed by ITT Visual Information Solution.",
"It is available from http://www.ittvis.com/ which account for measurements uncertainties in both variables.The ROSTAT package gave us the uncertainties in the velocity dispersion measurements, while for the X-ray luminosity we used the counts uncertainties listed by RASS-BCS/FSC and computed the relative error.",
"The same relative error was assumed for $L_{\\rm X}$ and we found a median value of $\\sigma _{L{\\rm X}}\\sim 25\\%$ .",
"Our best-fitting relation is: ${\\rm log}(L_{{\\rm X}})=(33.35\\pm 0.73) + (3.72\\pm 0.26) \\,{\\rm log}(\\sigma _{{\\rm v}})$ This relation is shown in Fig.",
"REF together with the $L_{\\rm X}-\\sigma _{{\\rm v}}$ relation for the WINGS nearby cluster sample [38], [20].",
"The two relations are in good agreement with one another."
],
[
"Fossilness determination",
"A fossil system is defined as having a large gap in magnitude in the $r-$ band between the two brightest members of the system, namely larger than 2 magnitues within 0.5 $R_{200}$ .",
"We calculated the distance from the BGG and magnitude for each galaxy to verify the fossilness criterium of our sample.",
"In this way, we obtained a diagram (such as that shown in Fig.",
"REF ) that allowed us to constrain the main properties of the system, such as the magnitude gap, virial radius, cluster membership, and magnitude of the BGG.",
"The magnitudes of the galaxies were obtained from SDSS-DR7 and our own photometry (see Sec.",
"REF ).",
"We used the extinction corrected Petrosian and model $r$ -band magnitudes for all the objects in the SDSS-DR7 database.",
"In addition, we have our own photometry for 30 out of 34 galaxy systems.",
"Our photometry only covers the central regions of the clusters but it is about 2 magnitudes deeper in the $r-$ band (see Fig.",
"REF ).",
"We ran SExtractor [12] on our images in order to obtain the $r-$ band MAG-BESTMAG-BEST is an aperture magnitude enclosing the total light of the galaxy.",
"It usually coincides with MAG-AUTO, which is the best total magnitude provided for galaxies by SExtractor.",
"The latter provides MAG-ISOCOR instead of MAG-AUTO if the galaxy is at least 10% contaminated by another object.",
"magnitude.",
"Determining the magnitude of the BGGs is not straightforward.",
"In SExtractor a successful deblending strongly depends on both the angular size of the BGG and the number of its close satellites.",
"In order to circumvent this problem, we recomputed the magnitude of the BGGs using an ad hoc procedure on our images.",
"In particular, we modeled the light of the BGGs by masking its close satellites.",
"The model was done in IRAF by using the bmodel task, which adopts as input the isophotal fit of the galaxy provided by ellipse [48].",
"The magnitude of the BGGs was computed using these uncontaminated models.",
"Besides, the modeled light of the BGGs was subtracted from the original images and the magnitudes of all other galaxies were obtained by running SExtractor in the resulting images.",
"SDSS photometry suffers from both deblending and overestimation of sky levels near bright galaxies [13].",
"The final magnitude of each galaxy was obtained by averaging the available magnitudes to have a more realistic estimate of the uncertainty.",
"This was computed as the RMS of the available magnitudes.",
"Table REF presents the value of the gap between the two brightest ($\\Delta m_{12}$ ) and the first and fourth ranked galaxies ($\\Delta m_{14}$ ) within 0.5 $R_{200}$ for each system.",
"Some gaps are marked as lower limits because our spectroscopy failed to determine a redshift for some \"bright\" target galaxy (see Sec.REF for details).",
"For this reason we were not able to assign a membership to these objects.",
"According to [49] and [26], a system if fossil if $\\Delta m_{12} \\ge 2.0$ or $\\Delta m_{14} \\ge 2.5$ mag, respectively.",
"We considered as fossil systems those that satisfy at least one of the previous criteria taking into account the errors in the magnitude gaps determination.",
"More explicitly, a system is fossil if $\\Delta m_{12}+ \\epsilon _{12} \\ge 2.0$ or $\\Delta m_{14}+ \\epsilon _{14} \\ge 2.5$ , being $\\epsilon _{12}$ and $\\epsilon _{14}$ the $1 \\sigma $ uncertainties in the magnitude gaps.",
"In Table 1 we highlighted the 15 systems that follow the previous criteria.",
"Notice that the two methods find 12 and 13 fossil systems respectively, despite [26] claimed that their method is expected to find 50% more fossil systems than [49].",
"The fossil definitions take into account not only the magnitude gaps but also the virial radius of the system.",
"Thus, uncertainties in the $R_{200}$ determination reflect in uncertainties in the fossil classification.",
"Therefore, we computed the variation in the number of fossil systems taking into account a 25% uncertainty in $R_{{\\rm 200}}$ in agreement with [43].",
"The number of fossil systems is $15^{+8}_{-4}$ .",
"The upper limit gives the number of fossil systems for a 25% smaller ${\\rm R}_{200}$ .",
"Similarly, the lower limit corresponds to the number of fossil systems for a 25% larger ${\\rm R}_{200}$ .",
"Figure: Magnitude-distance diagram for the group FGS20.",
"The distance from the BGG is given as a function of the magnitude for each galaxy.",
"The black points represent all the galaxies within the FoV, grey circles are the target galaxies (see Sec.",
"for details), red stars represent the spectroscopically confirmed members, and green crosses are the spectroscopically confirmed non-members.",
"The red solid horizontal line corresponds to 0.5 R 200 R_{200}, black dashed-dotted line marks R 200 R_{200}, and black dashed line represents 0.5 Mpc which is the limit used by to define the fossilness of the group.",
"The red solid and dashed-dotted vertical lines indicate the 2-mag and 2.5-mag gaps from the BGG which determine the fossilness of the group following the criteria by and , respectively."
],
[
"Correlations with the magnitude gaps",
"In Fig.",
"REF we show $\\Delta m_{12}$ as a function of both the absolute $r-$ band magnitude of the BGG (M$_{{\\rm BGG}}$ ) and X-ray luminosity of the system.",
"Our FG candidates show mainly $\\Delta m_{12}>1$ .",
"In order to have more clusters in the range $0<\\Delta m_{12}<1$ , we included the sample of nearby ($z<0.1$ ) galaxy clusters from [4].",
"Their $\\Delta m_{12}$ was obtained from spectroscopically confirmed members within 0.5 $R_{{\\rm 200}}$ once we applied the same evolutionary and K corrections that we used for the [81] sample.",
"For both relations we computed the Spearman correlation coefficients.",
"We found a strong correlation (significance $> 3\\sigma $ ) between $\\Delta m_{12}$ and M$_{{\\rm BGG}}$ .",
"On average, the larger is $\\Delta m_{12}$ the brightest is the central objects.",
"The M$_{\\rm BGG}$ and $\\Delta m_{12}$ correlation is somewhat expected in the classical scenario of the formation of fossil systems, in which the central galaxy has grown due to merging of nearby $L^\\ast $ galaxies.",
"A similar correlation was also observed for central galaxies in other cluster samples [7].",
"On the contrary, the relationship between log(L$_{\\rm X}$ ) and $\\Delta m_{12}$ is weaker (significance $< 2\\sigma $ ).",
"We also analysed the fraction of total optical luminosity contained within the central galaxy ($L_{{\\rm BGG}}/L_{{\\rm tot}}$ ) as a function of the magnitude gaps $\\Delta m_{12}$ and $\\Delta m_{14}$ .",
"In this case, the total luminosity $L_{{\\rm tot}}$ represents the sum of the luminosities of all the galaxies with $|(g-r)-(g-r)_{BGG}| \\le 0.2$ , $M_{{\\rm r}} \\le -20.0$ , and within 0.5 $R_{200}$ .",
"We limited this analysis to systems with $z\\le 0.25$ because for farther systems we were unable to reach $M_{\\rm r}=-20.0$ .",
"Figure REF shows a clear correlation (Spearman test significance $> 3 \\sigma $ ) between $L_{{\\rm BGG}}/L_{{\\rm tot}}$ and the two magnitude gaps.",
"Fossil systems are, on average, those objects with a larger fraction of light in the BGG.",
"Nevertheless, they are characterized by a large range of $L_{{\\rm BGG}}/L_{{\\rm tot}}$ values (0.25 $< L_{{\\rm BGG}}/L_{{\\rm tot}} <$ 0.75).",
"Most of the systems with $L_{{\\rm BGG}}/L_{{\\rm tot}} >$ 0.5 are fossil ones.",
"Similar relations were also found in other fossil samples [44] and recently [43] suggest that the growth in mass of the BGGs is directly correlated with $\\Delta m_{12}$ and that this correlation is necessary to justify the BGGs over luminosity.",
"Fossil systems represent always extreme cases in the correlations.",
"However, Fig.",
"REF and Fig.REF indicate that not all the properties of the clusters depend on $\\Delta m_{12}$ .",
"Figure: Absolute rr-band magnitude of the BGG (top panel) and X-ray luminosity of the system (bottom panel) as a function of the gap in magnitude between the two brightest galaxies of the systems studied in this paper (red open circles) and in .",
"The green filled circles are the genuine fossil systems.",
"The points with a right arrow are those systems for which the magnitude gap is a lower limit."
],
[
"Correlation with the velocity dispersion",
"In Fig.",
"REF we show the correlation between the fraction of light enclosed in the BGG ($L_{{\\rm BGG}}/L_{{\\rm tot}}$ ) as function of the LOS velocity dispersion $\\sigma _{{\\rm v}}$ of the cluster for the same sample of Fig.",
"REF .",
"This is a well known correlation which was originally reported by [58].",
"They argued that this correlation, together with the correlation between the luminosity of the BGGs and the mass of the system, indicates that BGGs grow by merging galaxies.",
"In addition, they claimed that the decreasing of the BGG luminosity fraction with cluster mass indicates that the rate of luminosity growth of the BGGs is slower than the rate at which clusters acquire galaxies from the field.",
"Fig.",
"REF clearly shows that fossil systems delineate the upper envelope of the expected trend of the $L_{{\\rm BGG}}/L_{{\\rm tot}}-\\sigma _{{\\rm v}}$ relation of non-fossil systems.",
"So, for a given velocity dispersion (or mass), fossil systems have a larger fraction of light enclosed in the BGG.",
"Following Lin & Mohr, we infer that the growth rate of BGGs in fossil systems is larger than that of BGGs in non-fossil systems.",
"Notice that non-fossil systems which are located in the upper envelope of Fig.",
"REF have either large gaps in magnitude or gaps calculated as lower limits only.",
"The former are systems dominated by the BGG which are classified as non-fossil systems only due to the arbitrariness of the fossilness criteria.",
"The latter are expected to be genuine fossil systems, but for which further investigation is needed to constrain $\\Delta m_{12}$ and $\\Delta m_{14}$ .",
"Figure: Fraction of light of the BGG as function of Δm 12 \\Delta m_{12} and Δm 14 \\Delta m_{14} for the systems studied in this paper (red open circles) and in with z≤0.25z\\le 0.25.",
"The green filled circles represent the genuine fossil systems.",
"The points with a right arrow are those systems for which the magnitude gap is a lower limit.Figure: Fraction of light of the BGG as function of the LOS velocity dispersion (mass) of the system.",
"The symbols and colors are the same of Fig.",
".",
"Upward arrows indicate those systems for which the magnitude gaps (and thus the L BGG /_{{\\rm BGG}}/L tot _{{\\rm tot}}) represent lower limits."
],
[
"Differences with {{cite:d5d16e60588172b69019437ff871dec681acf78f}}",
"We carefully analysed the sample of FG candidates presented by [81].",
"As they claimed, the number of fossil systems depends critically on the parameters adopted for measuring the magnitude gaps.",
"Hence, it is very important to highlight the differences between their and our methodology.",
"First of all, [81] used a fixed radius to look for the second brightest galaxies, namely 0.5 Mpc, whereas we were able to compute the virial radius for each cluster from its X-ray luminosity.",
"The value of 0.5 $R_{200}$ that we obtained for the [81] candidates varies between 0.29 Mpc and 1.02 Mpc, with a mean value of 0.63 Mpc.",
"This means that, on average, the radius within which we looked for the second brightest galaxies is larger than the one adopted in [81].",
"Therefore, we expect that not all the candidates proposed by [81] are genuine fossil systems.",
"Moreover, the procedure to define cluster membership adopted by [81] suffers from three main problems.",
"First, a small number of spectroscopic redshifts were available in the SDSS-DR5 for the 34 FG candidates.",
"Second, [81] considered as members all the galaxies with the spectroscopic redshift in the range $z_{\\rm c} \\pm \\Delta z$ , being $z_{\\rm c}$ the cluster redshift and $\\Delta z=0.002$ .",
"It is worth noting that $\\Delta z$ was fixed for all the clusters, and did not take into account the differences in velocity dispersion (or mass) between the systems.",
"Third, [81] considered as cluster members also the galaxies without the spectroscopic redshift but with a photometric redshift in the range $z_{\\rm c} \\pm 0.035$ .",
"This photometric redshift window is too small to deal with the typical errors of 0.1 expected for SDSS-DR5 photometric redshifts.",
"In addition, they only took into account galaxies with errors smaller than 0.1 in the photometric redshift.",
"On the contrary, we were able to obtain a number of spectroscopic velocities good enough to compute the LOS velocity dispersion of the systems and to accurately define the cluster membership.",
"In particular, for 25 out of 34 systems the cluster membership was obtained using a two-step method of member selection which works both in the redshift space and projected space phase.",
"This method is much more robust than the simple $z$ -cut proposed by [81].",
"Moreover, our photometric redshift criterion for the membership adopts a much larger window of $z_c \\pm 0.15$ .",
"For all these reasons, it is not surprising that only half of the systems proposed by [81] as FG candidates turned out to be genuine fossil systems according to our criteria.",
"Finally, it is important to notice that there are 12 systems in the sample that are not fossils but for which the magnitude gaps we calculated are lower limits only.",
"This means that, in principle, there could be other genuine fossil systems in the sample, which could be identified by completing the spectroscopic survey."
],
[
"Formation scenarios for fossil systems",
"There are two models that are mainly used in the literature to explain the formation scenarios for fossil systems.",
"The first and widely accepted one [30], [82], [89], [26] is that fossil systems were assembled at a higher redshift than regular clusters and, due to particularly favorable conditions, they had enough time to merge $L^\\ast $ galaxies in a single giant BGG.",
"This process results in the observed gap in magnitude which is a consequence of the system evolution.",
"The second scenario [70] suggests that the BGG was created in a single monolithic collapse and that the gap in magnitude is present from the beginning.",
"The two formation processes are called merging and failed group scenario, respectively.",
"The observational properties of our systems are the following: Fossil systems are observed at any LOS velocity dispersion (mass) of the host DM halo (Table REF ).",
"Indeed, we have both fossil groups and fossil clusters.",
"Systems with larger $\\Delta m_{12}$ have brighter BGGs (Fig.",
"REF , top panel).",
"Systems with larger $\\Delta m_{12}$ or $\\Delta m_{14}$ have a larger fraction of total galaxy light contained within the BGG (Fig.",
"REF ).",
"Fossil systems follow the same $L_{{\\rm BGG}}/L_{{\\rm tot}}-\\sigma _{{\\rm v}}$ relation of non-fossil systems.",
"However, fossil systems are extreme cases of this relation and show larger $L_{{\\rm BGG}}/L_{{\\rm tot}}$ for a fixed LOS velocity dispersion (Fig.",
"REF ).",
"Fossil systems lie on the same $L_{{\\rm X}}-\\sigma _{{\\rm v}}$ relation of non-fossil systems.",
"All these facts can be interpreted in the terms of the merging scenario: the fossil systems are the result of massive merger episodes in the early Universe due to the fact that their galaxies follow more low angular momentum orbits than galaxies in non-fossil systems.",
"In this case, the observed magnitude gap is an indication of the evolutionary state of the system.",
"Thus, the systems with larger magnitude gaps are expected to have brighter central galaxies and larger fraction of light enclosed in their BGGs.",
"The growth of the BGG in fossil systems is larger than in non-fossil ones, but it is expected to follow the same rules of normal clusters [58].",
"The properties we observed in fossil systems can also be explained by the failed group scenario.",
"Nevertheless, there are other properties that seem to disfavor this scenario.",
"For example, the monolithic formation of elliptical galaxies predicts strong metallicity gradients whereas the stellar population gradients are erased by mergers.",
"In this context, [34] found flat age and metallicity gradients for a sample of central galaxies in fossil systems,which are not compatible with the failed group scenario.",
"Moreover, the bend observed at high masses in the scaling relations of early-type galaxies also suggests that fossil systems were formed by mergers [11].",
"In particular, for the BGGs of our 34 FG candidates a two-phase merger scenario was proposed to explain their scaling relations.",
"Indeed, [67] claimed that these objects went through dissipational mergers in an early stage of their evolution and assembled the bulk of their mass through subsequent dry mergers.",
"This process seems similar to the one proposed by [27], in which the BGGs in fossil systems have undergo their last major merger later than in non-fossil systems."
],
[
"Transitional fossil systems",
"The correlation between the gap in magnitude and absolute magnitude of the BGG suggests that the scenario that suggests the existence of transitional fossil systems [89] can not be applied to those systems with the brightest BGGs.",
"In fact, the probability that two systems with such a bright central galaxy would merge is negligible.",
"For this reason, we expect that current fossil systems hosting the most luminous BGGs will be fossil forever.",
"Nevertheless, the [89] scenario could explain the case of FGS06.",
"The BGG has a magnitude of $M_r=-22.88$ whereas the second ranked, located at $\\sim 0.4$ $R_{200}$ , has $M_r=-22.68$ .",
"These magnitude values are typical of central galaxies in groups and clusters (see Fig.",
"REF ).",
"The third, fourth and fifth ranked galaxies have magnitudes of $M_r=-21.19$ , $-20.77$ , and $-20.18$ , respectively.",
"Moreover, FGS06 is the only non-fossil system in our sample with an abrupt change in its magnitude gap if the second ranked galaxy would not be taken into account.",
"In fact, the median value of the magnitude gap change for non-fossil systems in our sample is $0.2\\pm 0.2$ mag, whereas FGS06 would suffer a 1.6 mag change.",
"For these reasons, we suggest that FGS06 could be a good candidate for a transitional fossil systems as those described by [89]."
],
[
"Conclusions",
"We characterized the sample of 34 FG candidates proposed by [81] by using a unique collection of new optical photometric and spectroscopic data.",
"This dataset was completed with SDSS-DR7 archival data.",
"This large collection of radial velocities provided us robust cluster membership and global cluster properties for a subsample of 25 systems which were not available before.",
"The fossilness determination of the 34 FG candidates was revisited.",
"In particular, the magnitudes of the galaxies in each system were obtained by averaging three different magnitudes: Petrosian and model magnitudes from SDSS-DR7 and MAG-BEST SExtractor magnitude from our data.",
"This was done because the magnitude of the BGGs can be affected by close satellites both in the SDSS and SExtractor analyses.",
"Therefore, we computed new magnitude gaps ($\\Delta m_{12}$ and $\\Delta m_{14}$ ) within 0.5 $R_{200}$ for each system.",
"The systems with $\\Delta m_{12} \\ge 2$ or $\\Delta m_{14} \\ge 2.5$ mag within the errors were classified as fossils.",
"By applying this criterion, the total number of fossil systems in the sample is $15^{+8}_{-4}$ .",
"The uncertainties in the total number of fossil systems reflect the uncertainties in the $R_{200}$ determination.",
"Moreover, there are 12 systems for which one or both the magnitude gaps are lower limits.",
"For these systems, a more extended spectroscopic survey is needed in order to define their fossilness.",
"We derived the main observational properties of the fossil systems in our sample.",
"The fossil systems span a wide range of masses and we can confirm the existence of genuine fossil clusters in our sample.",
"In particular, five fossil systems have LOS velocity dispersions $\\sigma _{{\\rm v}} > 700$ km s$^{-1}$ , from both the $L_X$ luminosity and \"shifting gapper\" procedure.",
"Clear correlations were found between the magnitude gaps and luminosity of the BGGs.",
"In particular, the systems with larger $\\Delta m_{12}$ have brighter BGGs, and the systems with larger $\\Delta m_{12}$ or $\\Delta m_{14}$ have larger fraction of the total galaxy light in the BGGs.",
"The fossil systems also follow the same $L_{{\\rm BGG}}/L_{{\\rm tot}}-\\sigma _{{\\rm v}}$ relation of non-fossil systems.",
"Nevertheless, they are extreme cases in the studied relations.",
"In particular, the fossil systems have brighter BGGs than normal systems for any given LOS velocity dispersion (mass).",
"All these properties can be explained by the two mainly accepted proposed scenarios of formation of fossil structures and thus are not conclusive in this sense.",
"Nevertheless, we suggest that fossil systems with very bright central galaxies are not transitional phases of regular clusters and groups because, if this was the case, we should find systems with small gaps but very bright and massive central galaxies.",
"These systems are not observed because the probability that two systems with such a bright BGG would merge is negligible.",
"On the contrary, the systems with fainter BGGs possibly experienced a transitional fossil stage, which ended with the merging of another galaxy system.",
"This could be the case of FGS06.",
"The FOGO project will continue in the next future by analyzing other observational properties of fossil systems.",
"In a forthcoming paper we will focus on the LFs of fossil and normal systems.",
"This analysis will be crucial for the understanding of the formation and evolution of these galaxy aggregations, because the LFs of fossil systems in the merging scenario are expected to have a lack of $L^\\ast $ galaxies.",
"In contrast, the failed group formation scenario expects to find differences between fossil and normal systems in both the bright and faint ends of the LFs.",
"We would like to thank the anonymous referee for the useful comments which helped us to improve the paper.",
"This work was partially funded by the Spanish MICINN (grant AYA2010-21887-C04-04), and the local Canarian Government (grant ProID20100140).",
"This article is based on observations made with the Isaac Newtown Telescope, Nordic Optical Telescope, and Telescope Nazionale Galileo operated on the island of La Palma by the Isaac Newton Group, the Nordic Optical Telescope Scientific Association and the Fundaci—n Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) respectively, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof’sica de Canarias.",
"E.D.",
"gratefully acknowledges the support of the Alfred P. Sloan Foundation.",
"M.G.",
"acknowledges financial support from the MIUR PRIN/2010-2011 (J91J12000450001).",
"E.M.C.",
"is supported by Padua University (grants 60A02-1283/10,5052/11, 4807/12).",
"JIP and JVM acknowledge financial support from the Spanish MINECO under grant AYA2010-21887-C04-01, and from Junta de Andaluc’a Excellence Project PEX2011-FQM7058.",
"JMA acknowledges support from the European Research Council Starting Grant (SEDmorph; P.I.",
"V. Wild)"
]
] | 1403.0588 |
[
[
"Lagrangian immersions in the product of Lorentzian two manifold"
],
[
"Abstract For Lorentzian 2-manifolds $(\\Sigma_1,g_1)$ and $(\\Sigma_2,g_2)$ we consider the two product para-K\\\"ahler structures $(G^{\\epsilon},J,\\Omega^{\\epsilon})$ defined on the product four manifold $\\Sigma_1\\times\\Sigma_2$, with $\\epsilon=\\pm 1$.",
"We show that the metric $G^{\\epsilon}$ is locally conformally flat (resp.",
"Einstein) if and only if the Gauss curvatures $\\kappa_1,\\kappa_2$ of $g_1,g_2$, respectively, are both constants satisfying $\\kappa_1=-\\epsilon\\kappa_2$ (resp.",
"$\\kappa_1=\\epsilon\\kappa_2$).",
"We give the conditions on the Gauss curvatures for which every Lagrangian surface with parallel mean curvature vector is the product $\\gamma_1\\times\\gamma_2\\subset\\Sigma_1\\times\\Sigma_2$, where $\\gamma_1$ and $\\gamma_2$ are curves of constant curvature.",
"We study Lagrangian surfaces in the product $d{\\mathbb S}^2\\times d{\\mathbb S}^2$ with non null parallel mean curvature vector and finally, we explore the stability and Hamiltonian stability of certain minimal Lagrangian surfaces and $H$-minimal surfaces."
],
[
"Introduction",
"This article is a continuation of our previous work [8] on which we have studied minimal Lagrangian surfaces in the Kähler structures endowed in the product $\\Sigma _1\\times \\Sigma _2$ of Riemannian two manifolds.",
"Here, we consider again the product structure $\\Sigma _1\\times \\Sigma _2$ , where $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ are Lorentzian surfaces.",
"By Lorentz surface we mean a connected, orientable 2-manifold $\\Sigma $ endowed with a metric $g$ of indefinite signature.",
"Analogously with the Riemannian case, we may construct a para-Kähler structure $(g,j,\\omega )$ on a Lorentzian surface $(\\Sigma ,g)$ .",
"We recall that a para-Kähler structure is a pair $(g,j)$ defined on a manifold $M$ of even dimension and has the same properties than a Kähler one, except that $j$ is paracomplex structure rather than complex, i.e, we have $j^2=Id$ and is integrable, by meaning that the Nijenhuis tensor, $N^J(X,Y):=[X,Y]+[JX,JY]-J[JX,Y]-J[X,JY],$ vanishes.",
"Moreover, the compatibilty condition of the metric $g$ with the paracomplex structure $j$ becomes $g(j.,j.)=-g(.,.",
")$ .",
"The symplectic structure $\\omega $ can be defined by $\\omega (.,.)=g(j.,.",
")$ .",
"For Lorentzian surfaces $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ , we denote by $\\kappa _1$ and $\\kappa _2$ the Gauss curvatures of $g_1$ and $g_2$ , respectively.",
"For $\\epsilon \\in \\lbrace -1,1\\rbrace $ , we may define an almost para-Kähler structure $(G^{\\epsilon },\\Omega ^\\epsilon ,J)$ on the product $\\Sigma _1\\times \\Sigma _2$ where $G^{\\epsilon }$ is the para-Kähler metric, the endomorphism $J$ is an almost paracomplex structure and $\\Omega ^\\epsilon $ is the symplectic 2-form.",
"This structure will be described in Section and in particular we prove: Theorem 1 If $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ are Lorentzian two manifolds, the quadruples $(\\Sigma _1\\times \\Sigma _2, G^{\\epsilon },J,\\Omega ^{\\epsilon })$ are 4-dimensional para-Kähler structures.",
"Furthermore, the para-Kähler metric $G^{\\epsilon }$ is conformally flat $($ resp.",
"Einstein$)$ if and only if the Gauss curvatures $\\kappa _1$ and $\\kappa _2$ are constants with $\\kappa _1=-\\epsilon \\kappa _2$ $($ resp.",
"$\\kappa _1=\\epsilon \\kappa _2)$ .",
"In Section we study the surface theory of the para-Kähler structures constructed in Section .",
"An analogous result with Theorem 3 of [8] is the following theorem: Theorem 2 Let $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ be Lorentzian two manifolds and let $(G^{\\epsilon },J,\\Omega ^{\\epsilon })$ be the para-Kähler product structures on $\\Sigma _1\\times \\Sigma _2$ constructed in Section .",
"Assume that one of the following holds: (i) The metrics $g_1$ and $g_2$ are both non-flat almost everywhere and away from flat points we have $\\epsilon \\kappa _1\\kappa _2<0$ .",
"(ii) Only one of the metrics $g_1$ and $g_2$ is flat while the other is non-flat almost everywhere.",
"Then every $\\Omega ^{\\epsilon }$ -Lagrangian surface with parallel mean curvature vector is locally the product $\\gamma _1\\times \\gamma _2$ , where each curve $\\gamma _i\\subset \\Sigma _i$ has constant curvature.",
"Note 1 Note that the Theorem 3 of [8] holds true for Lagrangian immersions with parallel mean curvature vector in $(\\Sigma _1\\times \\Sigma _2, G^{\\epsilon },J,\\Omega ^{\\epsilon })$ , where $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ are Riemannian two manifolds.",
"We show that Theorem REF is no longer true when $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ are both flat.",
"In particular, we construct minimal Lagrangian immersions in the paracomplex Euclidean space ${\\mathbb {D}}^2$ , endowed with the pseudo-Hermitian product structure, such that they are not a product of straight lines in ${\\mathbb {D}}$ (Proposition REF ).",
"If ${\\mbox{d}}{\\mathbb {S}}^2$ denotes the anti-De Sitter 2-space, the Theorem REF tells us that $\\Omega ^-$ -Lagrangian surfaces in ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ with parallel mean curvature vector are locally the product of curves in ${\\mbox{d}}{\\mathbb {S}}^2$ with costant curvature.",
"The following theorem proves an analogue result with Theorem 1 in [6]: Theorem 3 Every $\\Omega ^+$ -Lagrangian immersions in ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ , with non null parallel mean curvature vector, is locally the product $(\\gamma _1,\\gamma _2)$ of curves in ${\\mbox{d}}{\\mathbb {S}}^2$ with costant curvatures $k_{1},k_2$ , respectively such that $k_1^2+k_2^2>0$ .",
"Minimality is the first order condition for a submanifold to be volume-extremizing in its homology class.",
"Minimal submanifolds that are local extremizers of the volume are called stable minimal submanifolds.",
"The stability of a minimal submanifold is determined by the monotonicity of the second variation of the volume functional.",
"The second order condition for a minimal submanifold to be volume-extremizing was first derived by Simons [12] and then Harvey while Lawson have proven that minimal Lagrangian submanifolds of a Calabi-Yau manifold is calibrated, which implies by Stokes theorem, that are volume-extremizing [9].",
"For the stability of minimal Lagrangian surface in $(\\Sigma _1\\times \\Sigma _2, G^{\\epsilon },J,\\Omega ^{\\epsilon })$ we prove the following: Theorem 4 Assume that the Gauss curvatures $\\kappa _1$ and $\\kappa _2$ satisfy the conditions of Theorem REF .",
"(i) If $\\kappa _1$ and $\\kappa _2$ are both nonpositive (nonnegative), then every $G^+$ -minimal Lagrangian surface $\\phi \\times \\psi $ where the geodesics $\\phi ,\\psi $ are spacelike (timelike), is stable.",
"(ii) If $\\kappa _1$ is nonpositive and $\\kappa _2$ is nonnegative, then every $G^-$ -minimal Lagrangian surface $\\phi \\times \\psi $ where the geodesics $\\phi $ is spacelike and $\\psi $ is timelike, is stable.",
"A Lagrangian submanifold $\\Sigma $ of a (para-) Kähler manifold is said to be Hamiltonian minimal (or $H$ -minimal) if it is a critical point of the volume functional with respect to Hamiltonian variations.",
"A $H$ -minimal Lagrangian submanifold is characterized by the fact that its mean curvature vector is divergence-free, that is, ${\\mbox{div}}JH=0$ , where $J$ is the (para-) complex structure and $H$ is the mean curvature vector of $\\Sigma $ .",
"If the second variation of the volume functional of a $H$ -minimal submanifold is monotone for any Hamiltonian compactly supported variation, it is said to be Hamiltonian stable (or $H$ -stable).",
"In [10] and [11], the second variation formula of a $H$ -minimal submanifold has been derived in the case of a Kähler manifold, while for the pseudo-Kähler case it has been given in [4].",
"The next theorem, in Section , investigates the $H$ -stability of projected rank one Hamiltonian $G^{\\epsilon }$ -minimal surfaces in $\\Sigma _1\\times \\Sigma _2$ : Theorem 5 Let $\\Phi =(\\phi ,\\psi )$ be of projected rank one Hamiltonian $G^{\\epsilon }$ -minimal immersion in $(\\Sigma _1\\times \\Sigma _2,G^{\\epsilon })$ such that $\\epsilon _{\\phi }\\kappa _1\\le 0$ and $\\epsilon _{\\psi }\\kappa _2\\le 0$ along the curves $\\phi $ and $\\psi $ respectively.",
"Then $\\Phi $ is a local maximizer of the volume in its Hamiltonian isotopy class.",
"Acknowledgements.",
"The author would like to thank H. Anciaux, B. Guilfoyle and W. Klingenberg for their helpful and valuable suggestions and comments."
],
[
"The Product para-Kähler structure",
"Let $(\\Sigma ,g)$ be a two dimensional oriented manifold endowed with a non degenerate Lorentzian metric $g$ .",
"Then in a neighbourhood of any point there exist local isothermic coordinates $(s,t)$ , i.e., $g_{ss}=-g_{tt}$ and $g_{st}=0$ (see [2]).",
"The endomorphism $j:{\\mbox{T}}\\Sigma \\rightarrow {\\mbox{T}}\\Sigma $ defined by $j(\\partial /\\partial s)=\\partial /\\partial t$ and $j(\\partial /\\partial t)=\\partial /\\partial s$ , satisfies $j^2={\\mbox{Id}}_{T\\Sigma }$ and $g(j.,j.)=-g(.,.",
")$ .",
"It follows that $j$ defines a paracomplex structure and if we set $\\omega (\\cdot , \\cdot )=g(j_k\\cdot , \\cdot )$ , the quadruple $(\\Sigma ,g,j,\\omega )$ is a 2-dimensional para-Kähler manifold.",
"For $k=1,2$ , let $(\\Sigma _k,g_k,j_k,\\omega _k)$ be the para-Kähler structures defined as before, and consider the product structure $\\Sigma _1\\times \\Sigma _2$ .",
"The identification $X\\in {\\mbox{T}}(\\Sigma _1\\times \\Sigma _2)\\simeq (X_1,X_2)\\in {\\mbox{T}}\\Sigma _1\\oplus {\\mbox{T}}\\Sigma _2$ , gives the natural splitting ${\\mbox{T}}(\\Sigma _1\\times \\Sigma _2)={\\mbox{T}}\\Sigma _1\\oplus {\\mbox{T}}\\Sigma _2$ .",
"For $(x,y)\\in \\Sigma _1\\times \\Sigma _2$ , let $X=(X_1,X_2)$ and $Y=(Y_1,Y_2)$ be two tangent vectors in ${\\mbox{T}}_{(x,y)}(\\Sigma _1\\times \\Sigma _2)$ and define the metric $G^{\\epsilon }$ by $G^{\\epsilon }_{(x,y)}(X,Y)=g_1(X_1,Y_1)(x)+\\epsilon g_2(X_2,Y_2)(y),$ where $\\epsilon \\in \\lbrace -1,1\\rbrace $ .",
"If $\\nabla $ denotes the Levi-Civita connection with respect to the metric $G^{\\epsilon }$ , we have $\\nabla _X Y=(D^1_{X_1} Y_1,D^2_{X_2} Y_2)$ , where $D^1,D^2$ denote the Levi-Civita connections with respect to the metrics $g_1$ and $g_2$ , respectively.",
"The endomorphism $J\\in {\\mbox{End}}({\\mbox{T}}\\Sigma _1\\oplus {\\mbox{T}}\\Sigma _2)$ defined by $J=j_1\\oplus j_2$ is an almost paracomplex structure on $\\Sigma _1\\times \\Sigma _2$ , while the two-forms $\\Omega ^{\\epsilon }=\\pi _1^{\\ast }\\omega _1+\\epsilon \\pi _2^{\\ast }\\omega _2$ , are symplectic structures, with $\\pi _i:\\Sigma _1\\times \\Sigma _2\\rightarrow \\Sigma _i$ is the $i$ -th projection.",
"Proof of Theorem REF : It is clear that the Nijenhuis tensor of $J$ given by (REF ) vanishes.",
"Furthermore, $J$ and $\\Omega ^{\\epsilon }$ are compatible, i.e., $\\Omega ^{\\epsilon }(J.,J.",
")=-\\Omega ^{\\epsilon }(.,.",
")$ and thus the quadruples $(\\Sigma _1\\times \\Sigma _2, G^{\\epsilon },J,\\Omega ^{\\epsilon })$ are para-Kähler structures.",
"Let $(e_1,e_2)$ and $(v_1,v_2)$ be orthonormal frames on $\\Sigma _1$ and $\\Sigma _2$ respectively, both oriented such that $|e_1|^2=|v_1|^2=1$ and $|e_2|^2=|v_2|^2=-1$ and consider an orthonormal frame $(E_1,E_2,E_3, E_4)$ of $G^{\\epsilon }$ defined by $E_1=(e_1,v_1+v_2),\\quad E_2=(e_2,v_1+v_2),\\quad E_3=(\\epsilon (e_2-e_1),v_1),\\quad E_4=(\\epsilon (e_1-e_2),v_2).$ The Ricci curvature tensor ${\\mbox{Ric}}^{\\epsilon }$ of $G^{\\epsilon }$ gives ${\\mbox{R}ic}^{\\epsilon }(E_1,E_1)_{(x,y)}=\\kappa _1(x),\\qquad {\\mbox{R}ic}^{\\epsilon }(E_3,E_3)_{(x,y)}=\\kappa _2(y),$ and using the fact ${\\mbox{R}ic}^{\\epsilon }(J.,J.",
")=-{\\mbox{R}ic}^{\\epsilon }(.,.",
")$ , the scalar curvatute ${\\mbox{R}}^\\epsilon $ is: ${\\mbox{R}}^\\epsilon =2(\\kappa _1(x)+\\epsilon \\kappa _2(y)).$ If $G^{\\epsilon }$ is conformally flat, it is scalar flat [5] and thus, from (REF ), the Gauss curvatures $\\kappa _1, \\kappa _2$ are constants with $\\kappa _1=-\\epsilon \\kappa _2$ .",
"Conversely, assuming that $\\kappa _1=-\\epsilon \\kappa _2=c$ , where $c$ is a real constant and following the same computations with the proof of Theorem 2.2 in [8], we prove that the self-dual ${\\mbox{W}}^+$ and the anti-self-dual part ${\\mbox{W}}^-$ of the Weyl tensor vanish and therefore the metric $G^\\epsilon $ is conformally flat.",
"On the other hand, a direct computation shows that ${\\mbox{R}ic}^\\epsilon (E_i,E_j)=cG^\\epsilon (E_i,E_j)$ if and only if $\\kappa _1=\\epsilon \\kappa _2=c$ , where $c$ is a real constant and the theorem follows.",
"$\\Box $ Corollary 1 Let $(\\Sigma ,g)$ be a Lorentzian two manifold.",
"The para-Kähler metric $G^-$ (resp.",
"$G^+$) of the four dimensional Kähler manifold $\\Sigma \\times \\Sigma $ is conformally flat (resp.",
"Einstein) if and only if the metric $g$ is of constant Gaussian curvature."
],
[
"Lagrangian immersions in $\\Sigma _1\\times \\Sigma _2$",
"In this section, we study Lagrangian immersions in the product $\\Sigma _1\\times \\Sigma _2$ endowed with the para-kähler structure $(G^\\epsilon ,J,\\Omega ^\\epsilon )$ constructed in section .",
"An immersion $\\Phi :S\\rightarrow \\Sigma _1\\times \\Sigma _2$ of a surface $S$ is said to be Lagrangian if $\\Phi ^{\\ast }\\Omega ^\\epsilon $ vanishes for every point of $S$ .",
"In this case, the paracomplex structure $J:{\\mbox{T}}S\\rightarrow {\\mbox{N}}S$ is a bundle isomorphism between the tangent bundle ${\\mbox{T}}S$ and the normal bundle ${\\mbox{N}}S$ .",
"It is well known that a Lagrangian immersion of a pseudo-Riemannian Kähler manifold is indefinite if the Kähler metric is indefinite.",
"Althought the signature of the para-Kähler metric is always neutral, a Lagrangian immersion can be either Riemannian or indefinite.",
"If $\\pi _i$ are the projections of $\\Sigma _1\\times \\Sigma _2$ onto $\\Sigma _i$ , $i=1,2$ , we denote by $\\phi $ and $\\psi $ the mappings $\\pi _1\\circ \\Phi $ and $\\pi _2\\circ \\Phi $ , respectively, and we write $\\Phi =(\\phi ,\\psi )$ .",
"Definition 1 The immersion $\\Phi =(\\phi ,\\psi ): S\\rightarrow \\Sigma _1\\times \\Sigma _2$ is said to be of projected rank zero at a point $p\\in S$ if either ${\\mbox{r}ank}(\\phi (p))=0$ or ${\\mbox{r}ank}(\\psi (p))=0$ .",
"$\\Phi $ is of projected rank one at $p$ if either ${\\mbox{r}ank}(\\phi (p))=1$ or ${\\mbox{r}ank}(\\psi (p))=1$ .",
"Finally, $\\Phi $ is of projected rank two at $p$ if ${\\mbox{r}ank}(\\phi (p))={\\mbox{r}ank}(\\psi (p))=2$ .",
"Note that the fact that $\\Phi $ is an immersion, implies that $\\Phi $ is locally either of projected rank zero, one or two.",
"Let $\\Phi =(\\phi ,\\psi )$ be of projected rank zero immersion in $\\Sigma _1\\times \\Sigma _2$ .",
"Assuming, without loss of generality, that ${\\mbox{r}ank}(\\phi )=0$ , the map $\\phi $ is locally a constant function and the map $\\psi $ is a local diffeomorphism.",
"Following the same argument with Proposition 3.2 in [8], we show that there are no Lagrangian immersions in $\\Sigma _1\\times \\Sigma _2$ of projected rank zero.",
"In order to discuss Lagrangian surfaces of projected rank one, we need to extend the definition of Cornu spirals for a pseudo-Riemannian two manifold.",
"Definition 2 Let $(\\Sigma ,g)$ be a pseudo-Riemannian two manifold.",
"A non-null regular curve $\\gamma $ of $\\Sigma $ is called a Cornu spiral of parameter $\\lambda $ if its curvature $\\kappa _{\\gamma }$ is a linear function of its arclength parameter such that $\\kappa _{\\gamma }(s)=\\lambda s+\\mu $ , where $s$ is the arclength and $\\lambda ,\\mu $ are real constants.",
"Proposition 1 Let $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ be Lorentzian surfaces and $\\Phi $ be a nondegenerate $\\Omega ^\\epsilon $ -Lagrangian surface of projected rank one.",
"Then, 1) $\\Phi $ can be locally parametrised by $\\Phi :S\\rightarrow \\Sigma _1\\times \\Sigma _2:(s,t)\\mapsto (\\phi (s),\\psi (t))$ , where $\\phi $ and $\\psi $ are respectively non-null regular curves in $\\Sigma _1$ and $\\Sigma _1$ with $s,t$ being their arclength parameters, 2) the induced metric $\\Phi ^{\\ast }G^{\\epsilon }$ is flat, 3) $\\Phi $ is $G^{\\epsilon }$ -minimal if and only if the curves $\\phi $ and $\\psi $ are geodesics, 4) $\\Phi $ is $H$ -minimal if and only if $\\phi $ and $\\psi $ are Cornu spirals of parameters $\\lambda _{\\phi }$ and $\\lambda _{\\psi }$ , respectively, such that $\\epsilon _{\\phi }\\lambda _{\\phi }+\\epsilon \\epsilon _{\\psi }\\lambda _{\\psi }=0,$ where $\\epsilon _{\\phi }=g_1(\\phi ^{\\prime },\\phi ^{\\prime })$ and $\\epsilon _{\\psi }=g_2(\\psi ^{\\prime },\\psi ^{\\prime })$ .",
"Let $\\Phi =(\\phi ,\\psi ):S\\rightarrow \\Sigma _1\\times \\Sigma _2$ be of projected rank one Lagrangian immersion.",
"Then either $\\phi $ or $\\psi $ is of rank one.",
"Assume, without loss of generality, that $\\phi $ is of rank one.",
"The nondegeneracy of $\\omega _2$ implies that $\\psi $ is of rank one and thus $S$ is locally parametrised by $\\Phi :U\\subset S\\rightarrow \\Sigma _1\\times \\Sigma _2:(s,t)\\mapsto (\\phi (s),\\psi (t))$ , where $\\phi $ and $\\psi $ are regular curves in $\\Sigma _1$ and $\\Sigma _2$ , respectively.",
"We denote by $s,t$ the arclength parameters of $\\phi $ and $\\psi $ , respectively, such that $g_1(\\phi ^{\\prime },\\phi ^{\\prime })=\\epsilon _{\\phi }$ and $g_s(\\psi ^{\\prime },\\psi ^{\\prime })=\\epsilon _{\\psi }$ , where $\\epsilon _{\\phi },\\epsilon _{\\psi }\\in \\lbrace -1,1\\rbrace $ .",
"The Frénet equations give $D^1_{\\phi ^{\\prime }}\\phi ^{\\prime }=k_{\\phi }j_1\\phi ^{\\prime }\\qquad D^2_{\\psi ^{\\prime }}\\psi ^{\\prime }=k_{\\psi }j_2\\psi ^{\\prime },$ where $k_{\\phi }$ and $k_{\\psi }$ are the curvatures of $\\phi $ and $\\psi $ , respectively.",
"Moreover, $\\Phi _s=(\\phi ^{\\prime },0)$ and $\\Phi _t=(0,\\psi ^{\\prime })$ and thus $\\nabla _{\\Phi _s}\\Phi _s=(k_{\\phi }j_1\\phi ^{\\prime },0),\\qquad \\nabla _{\\Phi _t}\\Phi _t=(0,k_{\\psi }j_2\\psi ^{\\prime }),\\qquad \\nabla _{\\Phi _t}\\Phi _s=(0,0).$ The immersion $\\Phi $ is flat since the first fundamental form $G^{\\epsilon }_{ij}=G^{\\epsilon }(\\partial _i\\Phi ,\\partial _j\\Phi )$ is given by $G_{ss}=\\epsilon _{\\phi },\\; G_{tt}=\\epsilon \\epsilon _{\\psi }$ , and $G_{st}=0$ .",
"The second fundamental form $h^{\\epsilon }$ of $\\Phi $ is completely determined by the following tri-symmetric tensor $h^{\\epsilon }(X,Y,Z):=G^{\\epsilon }(h^{\\epsilon }(X,Y),JZ)=\\Omega ^{\\epsilon }(X,\\nabla _Y Z).$ We then have $h^{\\epsilon }_{sst}=h^{\\epsilon }_{stt}=0,\\; h_{sss}^{\\epsilon }=-\\epsilon _{\\phi }k_{\\phi } $ and $h_{ttt}^{\\epsilon }=-\\epsilon \\epsilon _{\\psi } k_{\\psi }$ .",
"Denoting by $H^{\\epsilon }$ the mean curvature of $\\Phi $ we have $2H^{\\epsilon }=\\epsilon _{\\phi }k_{\\phi }J\\Phi _s+\\epsilon \\epsilon _{\\psi }k_{\\psi }J\\Phi _t,$ and we can see easily that the Lagrangian immersion $\\Phi $ is $G^{\\epsilon }$ -minimal if and only if the curves $\\phi $ and $\\psi $ are geodesics.",
"Moreover, if $\\Phi $ is a $G^{\\epsilon }$ -minimal Lagrangian it is totally geodesic, since the second fundamental form vanishes identically.",
"The condition (REF ) for Hamiltonian $G^\\epsilon $ -minimal Lagrangian surfaces of projected rank one, is given by the fact that $\\mbox{div}^{\\epsilon }(2JH^{\\epsilon })=\\epsilon _{\\phi }\\frac{D}{ds}k_{\\phi }(s)+\\epsilon \\epsilon _{\\psi } \\frac{D}{dt}k_{\\psi }(t),$ and the Proposition follows.",
"We now prove our next theorem: Proof of Theorem REF : Let $\\Phi =(\\phi ,\\psi ):S\\rightarrow \\Sigma _1\\times \\Sigma _2$ be a $\\Omega ^{\\epsilon }$ -Lagrangian immersion of projected rank two.",
"Then the mappings $\\phi :S\\rightarrow \\Sigma _1$ and $\\psi :S\\rightarrow \\Sigma _2$ are both local diffeomorphisms.",
"The Lagrangian condition yields $\\phi ^{\\ast }\\omega _1=-\\epsilon \\psi ^{\\ast }\\omega _2.$ Without loss of generality we consider an orthonormal frame $(e_1,e_2)$ of $\\Phi ^{\\ast }G^{\\epsilon }$ such that, $G^{\\epsilon }(d\\Phi (e_1),d\\Phi (e_1))=\\epsilon _1 G^{\\epsilon }(d\\Phi (e_2),d\\Phi (e_2))=1,\\qquad G^{\\epsilon }(d\\Phi (e_1),d\\Phi (e_2))=0.$ Note that for $\\epsilon _1=1$ the induced metric $\\Phi ^{\\ast }G^{\\epsilon }$ is Riemannian while for $\\epsilon _1=-1$ the induced metric $\\Phi ^{\\ast }G^{\\epsilon }$ is Lorentzian.",
"Let $(s_1,s_2)$ and $(v_1,v_2)$ be oriented orthonormal frames of $(\\Sigma _1,g_1)$ and $(\\Sigma _2,g_2)$ , respectively, such that $|s_1|^2=-|s_2|^2=1$ and $|v_1|^2=-|v_2|^2=1$ with $j_1s_1=s_2$ and $j_2v_1=v_2$ .",
"Then there exist smooth functions $\\lambda _1,\\lambda _2,\\mu _1,\\mu _2$ on $\\Sigma _1$ and $\\bar{\\lambda }_1,\\bar{\\lambda }_2,\\bar{\\mu }_1,\\bar{\\mu }_2$ on $\\Sigma _2$ such that $d\\phi (e_1)=\\lambda _1 s_1+\\lambda _2 s_2,\\;\\; d\\phi (e_2)=\\mu _1 s_1+\\mu _2 s_2,\\;\\;d\\psi (e_1)=\\bar{\\lambda }_1 v_1+\\bar{\\lambda }_2 v_2,\\;\\; d\\psi (e_2)=\\bar{\\mu }_1 v_1+\\bar{\\mu }_2 v_2.$ Using the Lagrangian condition (REF ), we have $(\\lambda _1\\mu _2-\\lambda _2\\mu _1)(\\phi (p))=-\\epsilon (\\bar{\\lambda }_1\\bar{\\mu }_2-\\bar{\\lambda }_2\\bar{\\mu }_1)(\\psi (p)),\\quad \\forall \\;\\; p\\in S.$ Moreover, the assumption that $\\Phi $ is of projected rank two, implies that $\\lambda _1\\mu _2-\\lambda _2\\mu _1\\ne 0$ for every $p\\in S$ .",
"For the mean curvature vector $H^{\\epsilon }$ of the immersion $\\Phi $ , consider the one form $a_{H^{\\epsilon }}$ defined by $a_{H^{\\epsilon }}=G^{\\epsilon }(JH^{\\epsilon },\\cdot )$ .",
"Since $\\Phi $ is a Lagrangian in a para-Kähler 4-manifold $da_{H^{\\epsilon }}=-\\Phi ^{\\ast }\\rho ^{\\epsilon },$ where $\\rho ^{\\epsilon }$ is the Ricci form of $G^{\\epsilon }$ .",
"The fact that $\\Phi $ has parallel mean curvature vector implies that the one form $a_{H^{\\epsilon }}$ is closed and thus, $(\\mu _1\\lambda _2-\\mu _2\\lambda _1)\\Big [\\Big (\\lambda _1^2-\\lambda _2^2+\\epsilon _1(\\mu _1^2-\\mu _2^2)\\Big )\\kappa _1-\\Big (\\bar{\\lambda }_1^2-\\bar{\\lambda }_2^2+\\epsilon _1(\\bar{\\mu }_1^2-\\bar{\\mu }_2^2)\\Big )\\kappa _2\\Big )\\Big ]=0.$ Hence, $\\Big (\\lambda _1^2-\\lambda _2^2+\\epsilon _1(\\mu _1^2-\\mu _2^2)\\Big )\\kappa _1=\\Big (\\bar{\\lambda }_1^2-\\bar{\\lambda }_2^2+\\epsilon _1(\\bar{\\mu }_1^2-\\bar{\\mu }_2^2)\\Big )\\kappa _2.$ Following the same argument with the proof of Theorem 3.5 in [8], we show that $\\lambda _1^2-\\lambda _2^2+\\epsilon _1(\\mu _1^2-\\mu _2^2)=1, \\quad \\mbox{and}\\quad \\bar{\\lambda }_1^2-\\bar{\\lambda }_2^2+\\epsilon _1(\\bar{\\mu }_1^2-\\bar{\\mu }_2^2)=\\epsilon ,$ and the relation (REF ) becomes $\\kappa _1(\\phi (p))=\\epsilon \\kappa _2(\\psi (p)),\\qquad \\mbox{for}\\;\\mbox{every}\\;\\; p\\in S,$ which implies that the metrics $g_1$ and $g_2$ can satisfy neither condition (i) nor condition (ii) of the statement and the theorem follows.",
"$\\Box $ We immediately obtain the following corollary: Corollary 2 Let $(\\Sigma ,g)$ be a non-flat Lorentzian two manifold.",
"Then every $G^{-}$ -minimal Lagrangian surface immersed in $\\Sigma \\times \\Sigma $ is of projected rank one and consequently the product of two geodesics of $(\\Sigma ,g)$ .",
"Example 1 Consider the real space ${\\mathbb {R}}^3$ , endowed with the pseudo-Riemannian metric $\\left<.,.\\right>_p=-\\sum _{i=1}^p dx_i^2+\\sum _{i=p+1}^3 dx_i^2$ .",
"We define the de Sitter 2-space ${\\mbox{d}}{\\mathbb {S}}_a^2$ and the anti de Sitter 2-space ${\\mbox{A}d}{\\mathbb {S}}^2_a$ of radius $a>0$ , by ${\\mbox{d}}{\\mathbb {S}}_a^2:=\\lbrace x\\in {\\mathbb {R}}^3|\\; \\left<x,x\\right>_1=a^2\\rbrace $ and ${\\mbox{A}d}{\\mathbb {S}}^2_a:=\\lbrace x\\in {\\mathbb {R}}^3|\\; \\left<x,x\\right>_2=a^2\\rbrace $ .",
"Note that ${\\mbox{d}}{\\mathbb {S}}_a^2$ and ${\\mbox{A}d}{\\mathbb {S}}^2_a$ are anti-isometric and therefore we only use the De Sitter 2-space ${\\mbox{d}}{\\mathbb {S}}_a^2$ .",
"Moreover, the Gauss curvature is constant with $\\kappa ({\\mbox{d}}{\\mathbb {S}}_a^2)=a^{-1}$ .",
"For positives $a,b$ with $a\\ne b$ , the Theorem REF implies that every $\\Omega ^\\epsilon $ -Lagrangian surface with parallel mean curvature in ${\\mbox{d}}{\\mathbb {S}}_a^2\\times {\\mbox{d}}{\\mathbb {S}}_b^2$ is locally the product of geodesics $\\gamma _1\\times \\gamma _2$ .",
"Example 2 The space $L^{-}({\\mbox{A}d}{\\mathbb {S}}^3)$ of oriented timelike geodesics in the anti-De Sitter 3-space ${\\mbox{A}d}{\\mathbb {S}}^3$ is diffeomorphic to the product ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ .",
"The para-Kähler metric $G^\\epsilon $ is invariant under the natural action of the isometry group ${\\mbox{I}so}({\\mbox{A}d}{\\mathbb {S}}^3,g)$ (see [1] and [3]).",
"It is clear by Corollary REF that every $G^{-}$ -minimal Lagrangian surface immersed in $L^{-}({\\mbox{A}d}{\\mathbb {S}}^3)$ is locally the product of two geodesics in ${\\mbox{d}}{\\mathbb {S}}^2$ .",
"The set of para-complex numbers ${\\mathbb {D}}$ is defined to be the two-dimensional real vector space ${\\mathbb {R}}^2$ endowed with the commutative algebra structure whose product rule is $(x_1,y_1)\\cdot (x_2,y_2)=(x_1x_2+y_1y_2,x_1y_2+x_2y_1).$ The number $(0,1)$ , whose square is $(1,0)$ , will be denoted by $\\tau .$ It is convenient to use the following notation: $(x,y) \\simeq z =x + \\tau y .$ In particular, one has the same conjugation operator than in ${\\mathbb {C}}$ , i.e., $\\overline{x+ \\tau y } = x - \\tau y$ with corresponding square norm $| z |^2 := z .",
"\\bar{z} = x^2 -y^2$ .",
"In other words, the metric associated to $|.|^2$ is the Minkowski metric $dx^2 - dy^2.$ On the Cartesian product ${\\mathbb {D}}^2$ with para-complex coordinates $(z_1,z_2)$ , we define the canonical para-Kähler structure $(J,G^\\epsilon )$ by $J(z_1,z_2):=(\\tau z_1,\\tau z_2),\\qquad G^\\epsilon :=dz_1 d\\bar{z}_1+\\epsilon dz_2 d\\bar{z}_2.$ We now give a similar result of Dong [7] for Lagrangian graphs in ${\\mathbb {C}}^n$ , to show that there are $G^-$ -minimal Lagrangian surface of rank two in $\\Sigma \\times \\Sigma $ when $(\\Sigma ,g)$ is a flat Lorentzian surface.",
"Proposition 2 Let $U$ be some open subset of ${\\mathbb {R}}^2$ , $u$ a smooth, real-valued function defined on $U$ and $\\Phi $ be the graph of the gradient i.e.",
"the immersion $\\Phi :U\\rightarrow {\\mathbb {D}}^2:x\\mapsto x+\\tau \\nabla u(x)$ , where $\\nabla u$ is the gradient of $u$ with respect to the Minkowski metric $g$ of ${\\mathbb {R}}^2$ .",
"Then $\\Phi $ is $G^-$ -minimal Lagrangian if and only if $u$ is a harmonic function with respect to the metric $g$ .",
"If $x=(x_1,x_2)$ , the first derivatives of the immersion $\\Phi (x_1,x_2)=(x_1+\\tau u_{x_1},x_2-\\tau u_{x_2})$ are $\\Phi _{x_1}=(1+\\tau u_{x_1x_1},-\\tau u_{x_1x_2}),\\qquad \\Phi _{x_2}=(\\tau u_{x_1x_2},1-\\tau u_{x_2x_2}).$ The symplectic structure is $\\Omega ^{-}(.,.)=G^{-}(J.,.",
")$ .",
"Then $\\Omega ^{-}(\\Phi _{x_1},\\Phi _{x_2})=0$ and therefore $\\Phi $ is a Lagrangian immersion.",
"The Lagrangian angle is $\\beta &=&{\\mbox{a}rg}(1-u_{x_1x_1}u_{x_2x_2}+u_{x_1x_2}^2+\\tau (u_{x_1x_1}-u_{x_2x_2})).\\nonumber $ Then $\\Phi $ is $G^{-}$ -minimal if and only if $u_{x_1x_1}-u_{x_2x_2}=0$ and the proposition follows.",
"The immersion $\\Phi $ in the Proposition REF is of rank two at the open subset $\\lbrace (x_1,x_2)\\in U|\\; u_{x_1x_2}\\ne 0\\rbrace $ .",
"We denote by $d{\\mathbb {S}}^2$ the De Sitter 2-space of radius one.",
"Then we prove our next result: Proof of Theorem REF : If ${\\mathbb {R}}^{1,2}$ denotes the Lorentzian space $({\\mathbb {R}}^3,\\left<,\\right>_1)$ , we define the Lorentzian cross product $\\otimes $ in ${\\mathbb {R}}^{1,2}$ by $u\\otimes v:={\\mbox{I}}_{1,2}\\cdot (u\\times v),$ where $u\\times v$ is the standard cross product in ${\\mathbb {R}}^3$ and ${\\mbox{I}}_{1,2}={\\mbox{d}iag}(-1,1,1)$ .",
"For $u,v,w\\in {\\mathbb {R}}^3$ we have $\\left<u\\otimes v,u\\otimes w\\right>_1=-\\left<u,u\\right>_1\\left<v,w\\right>_1+\\left<u,v\\right>_1\\left<u,w\\right>_1.$ The paracomplex structure $j$ on ${\\mbox{d}}{\\mathbb {S}}^2$ is given by $j_x(v):=x\\otimes v$ , where $v\\in {\\mathbb {R}}^3$ is such that $\\left<x,v\\right>_1=0$ .",
"It can be verified easily that $x\\otimes (x\\otimes v)=v$ .",
"If $h$ denotes the second fundamental form of the inclusion map $i:{\\mbox{d}}{\\mathbb {S}}^2\\hookrightarrow {\\mathbb {R}}^3$ and $u,v\\in {\\mbox{T}}_x{\\mbox{d}}{\\mathbb {S}}^2$ , we have $h_x(u,v)=-\\left<u,v\\right>_1x$ .",
"We consider the para-Kähler structure $(G^+,\\Omega ^+,J)$ of the product ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ and denote by $\\tilde{h}$ the second fundamental form of ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ into ${\\mathbb {R}}^3\\times {\\mathbb {R}}^3$ .",
"Thus, $\\tilde{h}_{(x,y)}(U,V)=(-\\left<u_1,v_1\\right>_1x,-\\left<u_2,v_2\\right>_1y),$ where $U=(u_1,u_2), V=(v_1,v_2)\\in {\\mbox{T}}_{(x,y)}({\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2)$ .",
"From (REF ) we obtain $\\tilde{h}_{(x,y)}(JU,JV)=-\\tilde{h}_{(x,y)}(U,V).$ For an orthonormal frame $(v_1,v_2)$ of ${\\mbox{d}}{\\mathbb {S}}^2$ , oriented such that $|v_1|^2=-|v_2|^2=1$ , we consider the following oriented orthonormal frame $(E_1,E_2=JE_1,E_3,E_4=JE_3)$ of $G^+$ defined by $E_1=(v_1,v_1+v_2),\\qquad E_3=(-v_1+v_2,v_1),$ and we prove that the mean curvature $\\tilde{H}$ of the inclusion map of ${\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ into ${\\mathbb {R}}^3\\times {\\mathbb {R}}^3$ is $2\\tilde{H}_{(x,y)}=-(x,y).$ Let $\\Phi =(\\phi ,\\psi ):S\\rightarrow {\\mbox{d}}{\\mathbb {S}}^2\\times {\\mbox{d}}{\\mathbb {S}}^2$ be a $\\Omega ^+$ - Lagrangian immersion with non null parallel mean curvature vector $H$ .",
"Following similar arguments with Theorem 1 of [6], consider an orthonormal frame $(e_1,e_2)$ with respect to the induced metric such that $|e_1|^2=\\epsilon _1|e_2|^2=1$ .",
"Then the equation (REF ) becomes, $|d\\phi (e_1)|^2+\\epsilon _1 |d\\phi (e_2)|^2=|d\\psi (e_1)|^2+\\epsilon _1 |d\\psi (e_2)|^2=1.$ By the proof of Theorem REF , we have $(|d\\phi e_1|^2-\\epsilon _1|d\\phi e_2|^2)^2+4\\epsilon _1g_1(d\\phi e_1,d\\phi e_2)^2=1+4\\epsilon _1 C^2,$ where $C:=\\lambda _2\\mu _1-\\lambda _1\\mu _2=\\bar{\\lambda }_1\\bar{\\mu }_2-\\bar{\\lambda }_2\\bar{\\mu }_1$ and is called the associated Jacobian of the Lagrangian immersion $\\Phi $ (see [6] and [13] for definitions and further details).",
"Note that the vanishing of the associated Jacobian is equivalent with the fact that the Lagrangian immersion $\\Phi $ is of projected rank one.",
"From (REF ) and (REF ), the mean curvature vector $\\bar{H}$ of $S$ in ${\\mathbb {R}}^3\\times {\\mathbb {R}}^3$ is $\\bar{H}= H-\\frac{1}{2}\\Phi .$ Since $\\nabla ^{\\bot }H=0$ together with the Lagrangian condition, implies that $\\nabla JH=0$ , where $\\nabla ^{\\bot }$ and $\\nabla $ denote the normal and tangential part of the Levi-Civita connection of $G^+$ .",
"From Theorem REF , we know that $G^+$ is Einstein and so there exists locally a function $\\beta $ on $S$ such that $JH=\\nabla \\beta $ .",
"Thus, using the Boschner formula, $\\frac{1}{2}\\Delta |\\nabla \\beta |^2={\\mbox{R}ic}(\\nabla \\beta ,\\nabla \\beta )+g(\\nabla \\beta ,\\nabla \\Delta \\beta )+g(\\nabla ^2\\beta ,\\nabla ^2\\beta ),$ we have that the induced metric $g$ is flat.",
"Furthermore, the normal curvature of $\\Phi $ vanishes.",
"It is important to mention that the Boschner formula holds also for pseudo-Riemannian metrics [4].",
"Let $(x,y)$ be isothermal local coordinates of $g$ , i.e., $g=e^{2u}(dx^2+\\epsilon _1 dy^2)$ and let $z=x+iy$ .",
"Note that for $\\epsilon _1=1$ , the variable $z$ is a local holomorphic coordinate, while for $\\epsilon _1=-1$ , the variable $z$ is a local paraholomorphic coordinate.",
"A brief computation gives, $G^+(\\Phi _z,\\Phi _z)=g_1(\\phi _z,\\phi _z)+g_2(\\psi _z,\\psi _z)=0$ From (REF ) we have that $\\Phi _{z\\bar{z}}=e^{2u}\\Big (H-\\Phi /2\\Big )/2$ , and $\\Phi _{zz}=2u_z\\Phi _z-G^+(H,J\\Phi _z)J\\Phi _z-2G^+(\\Phi _{zz},J\\Phi _z)J\\Phi _{\\bar{z}}-\\frac{1}{2}G^+(\\Phi _z,\\hat{\\Phi }_z)\\hat{\\Phi },$ where $\\hat{\\Phi }=(\\phi ,-\\psi )$ .",
"A direct computation implies, $G^+(J\\Phi _{\\bar{z}},\\hat{\\Phi }_z)=i\\epsilon _1 e^{2u}C,$ and thus, $\\hat{\\Phi }_z=2G^+(\\Phi _z,\\hat{\\Phi }_z)\\Phi _{\\bar{z}}-2i\\epsilon _1 CJ\\Phi _z.$ From (REF ), it follows that $|G^+(\\Phi _z,\\hat{\\Phi }_z)|^2=4|g_1(\\phi _z,\\phi _z)|^2=4|g_2(\\psi _z,\\psi _z)|^2=\\frac{1+4\\epsilon _1 C^2}{4}.$ It is not hard for one to obtain, $|G^+(H,J\\Phi _z)|^2=-\\frac{|H|^2}{4}.$ Since the normal curvature of $\\Phi $ vanishes, the Ricci equation yields, $|G^+(\\Phi _{zz},J\\Phi _z)|^2=-\\frac{e^{6u}(|H|^2+4C^2)}{16}.$ We now use the Gauss equation to get $C^2=2e^{-4u}(|\\Phi _{zz}|^2-|\\Phi _{z\\bar{z}}|^2)$ , which implies, $\\frac{C^2e^{4u}}{2}=2|u_z|^2e^{2u}-\\frac{e^{2u}|G^+(H,J\\Phi _z)|^2}{2}-2e^{-2u}|G^+(\\Phi _{zz},J\\Phi _z)|^2$ $\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad +\\frac{|G^+(\\Phi _z,\\hat{\\Phi }_z)|^2}{2}-\\frac{e^{4u} (|H|^2+1/2)}{4},$ and from (REF ), (REF ) and (REF ) we finally obtain $C^2=-4\\epsilon _1|u_z|^2e^{-2u}$ .",
"Since $g$ is flat, it is possible to choose local coordinates $(x,y)$ such that $g=dx^2+\\epsilon _1 dy^2$ , that is, the function $u$ is constant and therefore $C=0$ .",
"Then $\\Phi $ is of projected rank one and in particular, it is locally a product of curves with constant curvature such that they cannot be both geodesics.",
"$\\Box $"
],
[
"($H$ -) Stability of ({{formula:1efc8597-0d07-49f1-a944-417fd15a2123}} -) minimal Lagrangian surfaces",
"For the stability of $G^\\epsilon $ -minimal Lagrangian surfaces in the para-Kähler structure $(\\Sigma _1\\times \\Sigma _2,G^\\epsilon ,J,\\Omega ^\\epsilon )$ , we prove the following theorem: Proof of Theorem REF : Let $\\Phi :S\\rightarrow \\Sigma _1\\times \\Sigma _2$ be a $G^{\\epsilon }$ -minimal Lagrangian surface.",
"By assumption, $\\Phi $ is of projected rank one and therefore it is locally the product $\\gamma _1\\times \\gamma _2$ of non-null geodesics parametrised by $\\Phi (s,t)=(\\phi (s),\\psi (t))$ .",
"If $(S_t)$ is a normal variation of $S$ with velocity $X\\in {\\mbox{N}}S$ , the second variation formula is, $\\delta ^2 V(S)(X)=\\int _{S} \\Big (G^{\\epsilon }(\\nabla ^{\\bot }X,\\nabla ^{\\bot }X)-G^{\\epsilon }(A_X,A_X)+G^{\\epsilon }(R^{\\bot }(X),X)\\Big )dV,$ where, $A_X$ is the shape operator and $R^{\\bot }(X):={\\mbox{T}r}\\Big ((Y_1,Y_2)\\mapsto R(Y_1,X) Y_2)\\Big )$ .",
"For $X=X^1J\\Phi _s+X^2J\\Phi _t$ , we have, $G^{\\epsilon }(\\nabla ^{\\bot }X,\\nabla ^{\\bot }X)= -(X^1_s)^2-(X^2_t)^2-\\epsilon \\epsilon _{\\phi }\\epsilon _{\\psi }\\big ((X^2_s)^2+(X^1_t)^2\\big )$ .",
"Furthermore, $G^{\\epsilon }(A_X,A_X)=0$ and a brief computation gives, $G^{\\epsilon }(R^{\\bot }(X),X)=\\epsilon _{\\phi }(X^1)^2\\kappa (g_1)+\\epsilon _{\\psi }(X^2)^2\\kappa (g_2).$ The metric $G^{\\epsilon }$ is of neutral signature and therefore a necessary condition for a minimal surface to be stable is that the induced metric $\\Phi ^{\\ast }G^\\epsilon $ must be Riemannian [2].",
"This implies that $\\epsilon _{\\phi }=\\epsilon \\epsilon _{\\psi }$ and hence the second variation formula becomes $\\delta ^2 V(S)(X)=\\int _{S}-(X^1_s)^2-(X^2_s)^2-(X^1_t)^2-(X^2_t)^2+\\epsilon _{\\phi }(X^1)^2\\kappa (g_1)+\\epsilon _{\\psi }(X^2)^2\\kappa (g_2),$ and the theorem follows.",
"$\\Box $ Example 3 We give the following examples of $G^\\epsilon $ -minimal Lagrangian surfaces that are stable: Table: NO_CAPTIONWe now study the $H$ -stability of a $H$ -minimal surface in the para-Kähler structure $(\\Sigma _1\\times \\Sigma _2,G^\\epsilon ,\\Omega ^\\epsilon ,J)$ .",
"We recall that the $H$ -stability of a $H$ -minimal surface $S$ in a pseudo-Riemannian manifold $(M, G)$ is given by the monotonicity of the second variation formula of the volume $V(S)$ under Hamiltonian deformations (see [4] and [10]).",
"For a smooth compactly supported function $u\\in C^{\\infty }_{c}(S)$ the second variation $\\delta ^2 V(S)(X)$ formula in the direction of the Hamiltonian vector field $X=J\\nabla u$ is: $\\delta ^2 V(S)(X)=\\int _{S} -(\\Delta u)^2 + {\\mbox{R}ic}^G(\\nabla u,\\nabla u)+2G(h(\\nabla u,\\nabla u),nH)+G^2(nH,J\\nabla u),$ where $h$ is the second fundamental form of $S$ , ${\\mbox{R}ic}^G$ is the Ricci curvature tensor of the metric $G$ , and $\\Delta $ with $\\nabla $ denote the Laplacian and gradient, respectively, with respect to the metric $G$ induced on $S$ .",
"For the Hamiltonian stability of projected rank one Hamiltonian $G^{\\epsilon }$ -minimal surfaces we prove the following theorem: Proof of Theorem REF : Let $\\Phi =(\\phi ,\\psi ):S\\rightarrow \\Sigma _1\\times \\Sigma _2$ be of projected rank one Hamiltonian $G^{\\epsilon }$ -minimal immersion and let $(s,t)$ be the corresponded arclengths of $\\phi $ and $\\psi $ , respectively.",
"After a brief computation, the second variation formula for the volume functional with respect to the Hamiltonian vector field $X=J\\nabla u$ given by (REF ), becomes $\\delta ^2 V(S)(X)=\\int _{S} -(\\epsilon _{\\phi }u_{ss}+\\epsilon \\epsilon _{\\psi } u_{tt})^2+\\epsilon _{\\phi }u_s^2\\kappa _1+\\epsilon _{\\psi }u_t^2\\kappa _2-(\\epsilon _{\\phi }u_s k_{\\phi }-\\epsilon \\epsilon _{\\psi } u_t k_{\\psi })^2,$ where $\\kappa _1,\\kappa _2$ are the Gauss curvatures of $g_1$ and $g_2$ , respetively and the theorem follows.",
"$\\Box $ We also have the next Proposition: Corollary 3 Let $(\\Sigma ,g)$ be a Riemannian two manifold of positive (negative) Gaussian curvature.",
"Assume that every $G^-$ -minimal Lagrangian surface $\\Phi =(\\phi ,\\psi )$ immersed in $\\Sigma \\times \\Sigma $ is a pair of timelike (spacelike) geodesics in $\\Sigma $ .",
"Then $\\Phi $ is $H$ -stable.",
"Example 4 We know from [3] that every $G^-$ -minimal Lagrangian immersion in ${\\mbox{A}d}{\\mathbb {S}}^3$ is the Gauss map of a equidistant tube along a geodesic $\\gamma $ in ${\\mbox{A}d}{\\mathbb {S}}^3$ and following the example REF it must be locally parametrised as the product of geodesics in ${\\mbox{d}}{\\mathbb {S}}^2$ .",
"In this example, we are going to see exactly how can we obtain this product of geodesics.",
"Assume that $\\gamma $ is a spacelike geodesic.",
"Since the metric $G^-$ is invariant under the natural action of the isometry group of ${\\mbox{A}d}{\\mathbb {S}}^3$ , we may assume that $\\gamma $ is parametrised by $\\gamma (s)=(0,0,\\cos s,\\sin s)$ .",
"In this case the tube over $\\gamma $ with constant distance $d>0$ is parametrised by, $f:{\\mathbb {S}}^1\\times {\\mathbb {S}}^1\\rightarrow {\\mbox{A}d}{\\mathbb {S}}^3:(s,t)\\mapsto (\\sinh d\\cos t,\\sinh d\\sin t,\\cosh d\\cos s,\\cosh d\\sin s).$ The normal vector field is $N(s,t)=(\\cosh d\\cos t,\\cosh d\\sin t,\\sinh d\\cos s,\\sinh d\\sin s)$ , and thus $(f,v_1:=f_s/|f_s|,v_2:=f_t/|f_t|,N)\\in SO(2,2)$ .",
"It is also known by [3], that $ L^-({\\mbox{A}d}{\\mathbb {S}}^3)$ is identified with the Grassmannian ${\\mbox{G}r}^-(2,4):=\\lbrace x\\wedge y\\in \\Lambda ^2({\\mathbb {R}}^3):\\; y\\in T_x{\\mbox{A}d}{\\mathbb {S}}^3,\\; \\left<y,y\\right>_2=-1\\rbrace $ .",
"For an oriented orthonormal frame $\\lbrace e_1,e_2,e_3,e_4\\rbrace $ of ${\\mathbb {R}}^{2,2}:=({\\mathbb {R}}^4,\\left<.,.\\right>_2)$ such that $-|e_1|=-|e_2|=|e_3|=|e_4|=1$ we define the subspaces ${\\mathbb {R}}_{\\pm }^{1,2}$ of $\\Lambda ^2({\\mathbb {R}}^3)$ generated by the vectors $E^1_{\\pm }=(e_1\\wedge e_2\\pm e_3\\wedge e_4)/\\sqrt{2},\\quad E^2_{\\pm }=(e_1\\wedge e_3\\pm e_4\\wedge e_2)/\\sqrt{2},\\quad E^3_{\\pm }=(e_1\\wedge e_4\\pm e_2\\wedge e_3)/\\sqrt{2}.$ We define the de Sitter 2-spaces by ${\\mbox{d}}{\\mathbb {S}}_\\pm ^2:=\\lbrace x=x_1E^1_{\\pm }+x_2E^2_{\\pm }+x_3E^3_{\\pm }\\in {\\mathbb {R}}_{\\pm }^{1,2}|\\; \\left<x,x\\right>_2=-1\\rbrace .$ If $u_1\\wedge u_2\\in {\\mbox{G}r}^-(2,4)$ take $u_3,u_4$ such that $(u_1,u_2,u_3,u_4)\\in SO(2,2)$ .",
"The map ${\\mbox{G}r}^-(2,4)\\rightarrow {\\mbox{d}}{\\mathbb {S}}_-^2\\times {\\mbox{d}}{\\mathbb {S}}_+^2:u_1\\wedge u_2\\mapsto ((u_1\\wedge u_2+u_3\\wedge u_4)/\\sqrt{2},(u_1\\wedge u_2-u_3\\wedge u_4)/\\sqrt{2}),$ is a diffeomorphism.",
"The Gauss map $\\bar{f}=f\\wedge N\\in L^-({\\mbox{A}d}{\\mathbb {S}}^3)$ is identified to the following pair $(\\phi ,\\psi )\\in {\\mbox{d}}{\\mathbb {S}}^2_-\\times {\\mbox{d}}{\\mathbb {S}}^2_+$ , given by $\\phi (s,t)=(f\\wedge N+v_1\\wedge v_2)/\\sqrt{2},\\quad \\psi (s,t)=(f\\wedge N-v_1\\wedge v_2)/\\sqrt{2}.$ By setting $u=t-s$ and $v=t+s$ , we observe that $\\phi $ and $\\psi $ are the following geodesics of ${\\mbox{d}}{\\mathbb {S}}^2_{\\pm }$ : $\\phi (u)=-\\cos u E_-^2+\\sin u E^3_-,\\quad \\psi (v)=-\\cos (v) E_+^2-\\sin (v) E^3_+.$ Firthermore, $\\phi $ and $\\psi $ are timelike geodesics and therefore $\\bar{f}$ is a (unstable) $H$ -stable minimal Lagrangian torus in $L^{-}(\\mathop {\\mbox{A}d}{\\mathbb {S}}^3)$ .",
"For the case where $\\gamma $ is a timelike geodesic, a similar argument shows that $\\bar{f}$ is $H$ -unstable.",
"We emphasize here that by using different methods in [4], it was first proven that the Gauss map $\\bar{f}$ is $H$ -stable."
]
] | 1403.0305 |
[
[
"Giant thermoelectric effect in graphene-based topological insulators\n with nanopores"
],
[
"Abstract Designing thermoelectric materials with high figure of merit $ZT=S^2 G T/\\kappa$ requires fulfilling three often irreconcilable conditions, i.e., the high electrical conductance $G$, small thermal conductance $\\kappa$ and high Seebeck coefficient $S$.",
"Nanostructuring is one of the promising ways to achieve this goal as it can substantially suppress lattice contribution to $\\kappa$.",
"However, it may also unfavorably influence the electronic transport in an uncontrollable way.",
"Here we theoretically demonstrate that this issue can be ideally solved by fabricating graphene nanoribbons with heavy adatoms and nanopores.",
"These systems, acting as a two-dimensional topological insulator with robust helical edge states carrying electrical current, yield a highly optimized power factor $S^2G$ per helical conducting channel.",
"Concurrently, their array of nanopores impedes the lattice thermal conduction through the bulk.",
"Using quantum transport simulations coupled with first-principles electronic and phononic band structure calculations, the thermoelectric figure of merit is found to reach its maximum $ZT \\simeq 3$ at $T \\simeq 40$ K. This paves a way to design high-$ZT$ materials by exploiting the nontrivial topology of electronic states through nanostructuring."
],
[
"Methods",
"The electronic subsystem of GNR + heavy-adatoms is described by the tight-binding Hamiltonian of Kane-Mele type [5] with a single $p_z$ orbital per site of the honeycomb lattice $\\hat{H} & = & - t \\sum _{\\langle \\mathbf {m}\\mathbf {n} \\rangle ,\\sigma } \\hat{c}_{\\mathbf {m}\\sigma }^\\dagger \\hat{c}_{\\mathbf {n}\\sigma } \\nonumber \\\\\\mbox{} && + i\\lambda _\\mathrm {SO} \\sum _\\mathcal {P} \\sum _{\\langle \\langle \\mathbf {m}\\mathbf {n} \\rangle \\rangle \\in \\mathcal {P},\\sigma ,\\sigma ^\\prime } \\nu _\\mathbf {nm} \\hat{c}_{\\mathbf {m}\\sigma }^\\dagger \\hat{s}^z_{\\sigma \\sigma ^\\prime } \\hat{c}_{\\mathbf {n}\\sigma ^\\prime }.$ Here the operator $\\hat{c}_{\\mathbf {m}\\sigma }^\\dagger $ ($\\hat{c}_{\\mathbf {m}\\sigma }$ ) creates (annihilates) electron on site $\\mathbf {m}$ of the lattice in spin state $\\sigma $ and $\\hat{s}^z$ is the Pauli matrix.",
"The nearest-neighbor hopping $t=2.7$ eV in the first term in Eq.",
"(REF ) sets the unit of energy scale.",
"The spin-dependent hopping in the second term, where $\\nu _\\mathbf {mn} = 1$ for moving counterclockwise around the hexagon and $\\nu _\\mathbf {mn} = -1$ otherwise, acts between next-nearest neighbor sites of only those hexagons $\\mathcal {P}$ of the honeycomb lattice which host indium adatoms.",
"The strength of such SOC, which can be viewed as locally enhanced version of the tiny intrinsic SOC in pristine graphene [18], is parameterized by $\\lambda _\\mathrm {SO}$ .",
"This minimal effective model in Eq.",
"(REF ) is sufficient [16] to fit—using $\\lambda _\\mathrm {SO}=0.0037t$—the low-energy spectrum obtained from first-principles calculations for $4 \\times 4$ graphene supercell with two indium adatoms using VASP package [28], which gives $E_G \\approx 11.5$ meV.",
"The electron-core interactions are described by the projector augmented wave (PAW) method [38], [39], and we use Perdew-Burke-Ernzerhof (PBE) [40] parametrization of the generalized gradient approximation (GGA) for the exchange-correlation functional.",
"The cutoff energies for the plane wave basis set used to expand the Kohn-Sham orbitals are 500 eV for all calculations.",
"A $11 \\times 11 \\times 1$ $k$ -point mesh within Monkhorst-Pack scheme is used for the Brillouin zone (BZ) integration.",
"Structural relaxations and total energy calculations are performed ensuring that the Hellmann-Feynman forces acting on ions are less than $0.005$ eV/Å.",
"Starting from the matrix representation $\\mathbf {H}$ of the Hamiltonian in Eq.",
"(REF ), we compute the electronic retarded GF [12], $\\mathbf {G}(E) = [E - \\mathbf {H} - {\\Sigma }_L(E) - {\\Sigma }_R(E)]^{-1}$, where ${\\Sigma }_{L,R}$ are the self-energies introduced by the semi-infinite ideal (without disorder, adatoms or nanopores) ZGNR leads assumed to be attached to 2D TI wire in Figs.",
"REF (b) or REF (c).",
"The retarded GF and the level broadening matrices ${\\Gamma }_{L,R} (E) = i[{\\Sigma }_{L,R}(E)-{\\Sigma }_{L,R}^\\dagger (E)]$ allow us to obtain the electronic zero-bias transmission function, $T_{\\rm el}(E) = {\\rm Tr} \\left\\lbrace {\\Gamma }_R (E) {\\bf G}(E) {\\Gamma }_L (E) {\\bf G}^\\dagger (E) \\right\\rbrace $ , which determines electronic transport quantities through Eq. ().",
"The phononic band structure plotted in Figs.",
"REF (a) and REF (b) was computed via first-principles methodology using combined VASP [28] and Phonopy packages [37].",
"The details of VASP calculations are the same as delineated above (except that we use $3 \\times 3 \\times 1$ $k$ -point mesh), but here we start from $4 \\times 4$ graphene supercell hosting one indium adatom and then enlarged it to $8 \\times 8$ supercell in order to capture accurately force constants between a range of neighboring carbon atoms or carbon atoms and indium adatoms.",
"Figures REF (a) and REF (b) demonstrate appearance of new low energy bands due to the presence of indium adatoms.",
"Although the effect of SOC on phononic band structures can be profound for materials containing heavy elements, especially on surfaces and in thin films (as exemplified by the recent calculations on Bi$_2$ Te$_3$ and Bi$_2$ Te$_3$ [41]), the inclusion of SOC in Fig.",
"REF (b) generates only a small difference.",
"To construct the empirical up to fourth-nearest neighbors force constant model, we varied and optimized the FCs to fit as closely as possible the phononic dispersions plotted in Figs.",
"REF (a) and REF (b).",
"Using the FC matrix ${\\bf K}$ , the diagonal matrix ${\\bf M}$ containing atomic masses, and self-energies ${\\Pi }_{L,R}$ of the semi-infinite ideal ZGNR leads [obtained using FCs extracted from the dotted line in Fig.",
"REF (a)], we compute the phononic version of the retarded GF [13], ${\\bf D}(\\omega )=[\\omega ^2 {\\bf M} - {\\bf K} - {\\Pi }_L(\\omega ) - {\\Pi }_R(\\omega )]^{-1}$.",
"This, together with the level broadening matrices ${\\Lambda }_{L,R}(\\omega )=i[{\\Pi }_{L,R}(\\omega ) - {\\Pi }_{L,R}^\\dagger (\\omega )]$, gives the phononic transmission function, $T_{\\rm ph}(\\omega ) = {\\rm Tr} \\left\\lbrace {\\Lambda }_R (\\omega ) {\\bf D}(\\omega ) {\\Lambda }_L (E) {\\bf D}^\\dagger (\\omega ) \\right\\rbrace $, which determines $\\kappa _\\mathrm {ph}$ through Eq.",
"(REF ).",
"The significant difference between $\\kappa _\\mathrm {ph}$ for GNRs with nanopores but neglecting heavy adatoms [dashed line in Figs.",
"REF (e) and REF (f)] and $\\kappa _\\mathrm {ph}$ when heavy adatoms and the corresponding SOC are included [solid line in Figs.",
"REF (e) and REF (f)] confirms the necessity for the procedure delineated above.",
"We note that the values of $\\kappa _\\mathrm {ph}$ [solid lines in Figs.",
"REF (e) and REF (f)] based on FCs extracted from the phononic band structure of bulk graphene with heavy adatoms in Fig.",
"REF (b) are most likely overestimated—more precise FCs would require computationally very expensive procedure which considers a large number of atoms confined within the nanoribbon geometry and in the presence of nanopores [13], [35]."
],
[
"Acknowledgments",
"P.-H.C. and B.K.N.",
"were supported by US NSF under Grant No.",
"ECCS 1202069.",
"M.S.B.",
"and N.N.",
"were supported by Grant-in-Aids for Scientific Research (21244053) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, Strategic International Cooperative Program (Joint Research Type) from Japan Science and Technology Agency, and also by Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program)."
]
] | 1403.0239 |
[
[
"Zeta Function Regularization of Photon Polarization Tensor for a\n Magnetized Vacuum"
],
[
"Abstract In this paper, we have developed a systematic technique to regularize double summations of Landau levels and analytically evaluated the photon vacuum polarization at an external magnetic field.",
"The final results are described by Lerch transcendent $\\Phi(z,s,v)$ or its $z$-derivation.",
"We have found that the tensor of vacuum polarization is split into not only longitudinal and transverse parts but also another mixture component.",
"We have obtained a complete expression of the magnetized photon vacuum polarization at any kinematic regime and any strength of magnetic field for the first time.",
"In the weak $B$-fields, after canceling out a logarithmic counter term, all three scalar functions are limited to the usual photon polarization tensor without turning on magnetic field.",
"In the strong $B$-fields, the calculations under Lowest Landau Level approximation are only valid at the region $M^2\\gg q_{\\shortparallel}^2$, but not correct while $q_{\\shortparallel}^2\\gg M^2$, where, an imaginary part has been missed.",
"It reminds us, a recalculation of the gap equation under a full consideration of all Landau Levels is necessary in the next future."
],
[
"Introduction",
"It has been a long time ago that Schwinger studied the physical problems of fermions moving in a constant magnetic field [1].",
"Working in Landau and symmetric gauges, he obtained the exact fermion propagator written in an integral representative.",
"In 1990, Chodos et al.",
"demonstrated that the propagator is equivalent to a summation representative by decomposing over Landau poles [2].",
"The summation of Landau levels clearly indicates that the related calculations of magnetized matter are more complicated than usual ones.",
"Especially, if the infinite series are not convergent, it would be hard to extract the finite results by general regularization methods, such as Pauli-Villars, dimensional regularization, and so on.",
"Basically, Quantum Electrodynamics (QED) vacuum is modified after turning on an external electromagnetic field and then several striking phenomena are created, such as photon decay into an electron-positron pair via Schwinger mechanism [1], [3], [4]; vacuum birefringence of a photon [5], [6]; photon splitting and so on [7], [8].",
"As a fundamental information of magnetized vacuum, photon vacuum polarization tensor [9], [10], [11], [12], [13] is expressed in terms of a double summation of infinite series with respect to two Landau levels occupied by virtual charged particles.",
"Therefore, even though the tensor structure of one-loop diagram is plain to carry out, the subsequent calculations are particularly complicate to approach.",
"Before, most works were focusing on the strong filed limit where an assumption of Lowest Landau Level (LLL) has been applied [10], [14].",
"However, it is not clear how solid of LLL approximation is and what kind of invalid physics would be caused.",
"Without a complete description of vacuum polarization, it also limits us to explore other possible non-trivial phenomena in response to the external electromagnetic fields.",
"The regularization and renormalization procedures are essential ingredients of quantum field theory.",
"Among these methods, a powerful technique named as zeta function regularization is always greatly expected when encountering infinite sums [15], [16].",
"For instance, the thermal radiation [17] and Casimir effect [18] have been computed by the zeta function directly.",
"After our investigation, we confirm that the zeta function technique can be utilized to address the photon vacuum polarization tensor $\\Pi ^{\\mu \\nu }$ in the presence of $B$ , as well.",
"Remarkably, we find that the Lerch transcendent [19], [20], belonging to zeta family, is very efficient to handle the double infinite series of Landau levels.",
"After regularizing the divergent series, it gives us a finite value of $\\Pi ^{\\mu \\nu }$ despite of the double summations.",
"Recently, an extremely strong magnetic field of the strength of $10^{18-20}\\textmd {G}$ , equivalent to the order of $m_{\\pi }^{2}$ , has been realized in non-central heavy-ion collisions at the Relativistic Heavy Ion Collider (RHIC) or the Large Hadron Collider (LHC) [21], [22].",
"It draws a lot of interests to explore the theoretical problems of Quantum Chromodynamics (QCD) vacuum and strong-interaction matter under hadron-scaled magnetic fields.",
"Hence, the purposes of this paper are double folds.",
"Firstly, we are going to develop a systematic technique to extract finite results after double summing infinite Landau levels in the existence of external magnetic fields.",
"We obtain a complete expression of the magnetized photon vacuum polarization at any kinematic regime and any magnitude of $B$ .",
"Secondly, after a full theoretical formula has been established, we wish our work can be served as a standard routine and toolbox to involve in future researches in QCD$\\times $ QED physics.",
"The paper is organized as follows.",
"We demonstrate the procedure on how to derive the basic form of $\\Pi ^{\\mu \\nu }$ at external $B$ in section .",
"Dimension regularization in longitudinal space also is explained here.",
"In section , we present how powerful of the zeta regularization method in extracting finite physical results from infinite series.",
"We examine the limiting behaviors of $\\Pi ^{\\mu \\nu }$ and physical explanations at different kinematics regimes in section .",
"We end up with the summary and future applications in section .",
"In appendices, we provide supplemental materials of Lerch transcendent and some useful mathematic formula."
],
[
"Vacuum Polarizations",
"The decomposed fermion propagator ${S}(k)$ is written as [14]: ${S}(k)=i\\exp \\left(2\\hat{k}_{\\perp }^{2}\\right)\\sum _{n=0}^{\\infty }\\frac{(-1)^{n}{D}_{n}(eB,k)}{k_{0}^{2}-k_{3}^{2}-M^{2}-2neB},$ where $&&{D}_{n}(eB,k)=2({k}_{}+M)\\mathcal {O}^{-}L_{n}(-4\\hat{k}_{\\perp }^{2})\\nonumber \\\\&&-2({k}_{}+M)\\mathcal {O}^{+}L_{n-1}(-4\\hat{k}_{\\perp }^{2})-4{k}_{\\perp }L_{n-1}^{1}(-4\\hat{k}_{\\perp }^{2})$ with $\\hat{k}_{\\perp }=k_{\\perp }^{2}/(2eB)$ .",
"$L_{n}^{a}(\\xi )$ are the generalized Laguerre polynomials, where $L_{n-i}^{a}(\\xi )=0$ if $n<i$ .",
"$\\mathcal {O}^{\\pm }=(1\\pm \\mathrm {i}\\gamma ^{1}\\gamma ^{2})/2$ are the projecting operators corresponding to the spin state of charged particle paralleling or anti-paralleling to the direction of external field $B$ .",
"They satisfy following commutation relations: $\\mathcal {O}^{\\pm }\\gamma ^{\\mu }\\mathcal {O}^{\\pm }=\\mathcal {O}^{\\pm }\\gamma _{}$ and $\\mathcal {O}^{\\pm }\\gamma ^{\\mu }\\mathcal {O}^{\\mp }=\\mathcal {O}^{\\pm }\\gamma _{\\perp }$ .",
"Here, the metric convention $g^{\\mu \\nu }$ is decomposed into two orthogonal subspaces $g^{\\mu \\nu }_{}=\\mathrm {diag}(1,0,0,-1)$ and $g^{\\mu \\nu }_{\\perp }=\\mathrm {diag}(0,-1,-1,0)$ .",
"Similar decompositions are adopted for four dimensional momentum $k^{\\mu }=k^{\\mu }_{}+k^{\\mu }_{\\perp },~k^{\\mu }_{}=(k^{0},0,0,k^{3}),~k^{\\mu }_{\\perp }=(0,k^{1},k^{2},0)$ and Dirac matrices $\\gamma ^{\\mu }_{}=(\\gamma ^{0},0,0,\\gamma ^{3}),~\\gamma ^{\\mu }_{\\perp }=(0,\\gamma ^{1},\\gamma ^{2},0)$ .",
"To calculate $\\Pi ^{\\mu \\nu }=-\\mathrm {i}e^2\\hbox{Tr}[{S}(k)\\gamma ^{\\mu }{S}(p)\\gamma ^{\\nu }]$ , where $p=k+q$ and $e$ is the electrical charge.",
"The general Feynman parameter for the denominator factor has been introduced [23].",
"Besides, Schwinger observed that all propagators may be rewritten as Gaussian integrals by using a so called proper time representation [24].",
"After combining them, the denominator factor is shown as: $\\frac{1}{ab}=\\int _{0}^{1}\\mathrm {d}x\\int _{0}^{\\infty }\\mathrm {d}\\tau ~\\tau \\exp \\left[\\left(xa+(1-x)b\\right)\\tau \\right]$ where $a=k_{}^2-M^{2}-2neB+i\\epsilon $ and $b=(k_{}+q_{})^2-M^{2}-2meB+i\\epsilon $ in our work.",
"$\\tau $ is the variable of proper time.",
"Before step on, we normalize the momentum to dimensionless, where $\\hat{q}^{2}=q^{2}/(2eB)$ , $\\hat{M}^{2}=M^{2}/(2eB)$ , and so on.",
"It is common known to shift $k$ to $k-(1-x)q$ to simplify the later calculations.",
"However, we only shift $\\hat{k}_{}$ to $\\hat{k}_{}-(1-x)\\hat{q}_{}$ as usual, but shift transverse momentum $\\hat{{k}}_{\\perp }$ to $\\hat{{k}}_{\\perp }-\\beta \\hat{q}_{\\perp }/(\\alpha +\\beta )$ .",
"The reason will show later by the explicit expression of Eq.",
"(REF ).",
"Also, the detailed meaning of notation $\\alpha ,\\beta $ are described in Eq.",
"(REF ).",
"Now, the tensor structure of vacuum polarization $I^{\\mu \\nu }$ becomes $I^{\\mu \\nu }&=&2\\hbox{Tr}\\left[{k}_{}^{1-x}\\gamma ^{\\mu }_{}{k}_{}^{x}\\gamma ^{\\nu }\\right]\\left({L}_{n}{L}_{m}+{L}_{n-1}{L}_{m-1}\\right)\\nonumber \\\\&&-2\\hbox{Tr}\\left[{k}_{}^{1-x}\\gamma ^{\\mu }_{\\perp }{k}_{}^{x}\\gamma ^{\\nu }\\right]\\left({L}_{n}{L}_{m-1}+{L}_{n-1}{L}_{m}\\right)\\nonumber \\\\&&-4\\hbox{Tr}\\left[{k}_{}^{1-x}\\gamma ^{\\mu }{k}_{\\perp }^{\\alpha }\\gamma ^{\\nu }\\right]\\left({L}_{n}-{L}_{n-1}\\right){L}^{1}_{m-1}\\nonumber \\\\&&-4\\hbox{Tr}\\left[{k}_{\\perp }^{\\beta }\\gamma ^{\\mu }{k}_{}^{x}\\gamma ^{\\nu }\\right]{L}^{1}_{n-1}\\left({L}_{m}-{L}_{m-1}\\right)\\nonumber \\\\&&+16\\hbox{Tr}\\left[{k}_{\\perp }^{\\beta }\\gamma ^{\\mu }{k}_{\\perp }^{\\alpha }\\gamma ^{\\nu }\\right]{L}^{1}_{n-1}{L}^{1}_{m-1},$ where ${k}_{}^{x}={k}_{}+x{q}_{}+M,&&{k}_{}^{1-x}={k}_{}-(1-x){q}_{}+M,\\nonumber \\\\{k}_{\\perp }^{\\alpha }={k}_{\\perp }+\\frac{\\alpha }{\\alpha +\\beta }{q}_{\\perp },&&{k}_{\\perp }^{\\beta }={k}_{\\perp }-\\frac{\\beta }{\\alpha +\\beta }{q}_{\\perp }.$ The augments of $L^{\\alpha ,\\beta }_{n,m}$ have been abbreviated.",
"Therefore, $\\Pi ^{\\mu \\nu }=\\int \\mathrm {d}\\Gamma ~I^{\\mu \\nu }\\mathrm {e}^{-\\left[\\hat{M}^{2}-\\eta \\hat{q}_{}^{2}+nx+m(1-x)-\\hat{k}_{}^{2}-i\\epsilon \\right]\\tau },$ where $\\eta =x(1-x)$ and $\\mathrm {d}\\Gamma &=&-\\mathrm {i}e^{2}\\sum _{n=0}^{\\infty }\\sum _{m=0}^{\\infty }(-1)^{n+m}\\int _{0}^{1}\\mathrm {d}x\\int _{0}^{\\infty }\\mathrm {d}\\tau ~\\tau \\nonumber \\\\&&\\cdot \\int \\frac{\\mathrm {d}^{2-\\epsilon }\\hat{k}_{}}{(2\\pi )^{2}}\\int \\frac{\\mathrm {d}^{2}\\hat{k}_{\\perp }}{(2\\pi )^{2}}\\exp \\left[2\\hat{k}_{\\perp }^{2}+2\\hat{p}_{\\perp }^{2}\\right].$ Under the help of generating function of Laguerre polynomials [20]: $\\sum _{n=0}^{\\infty }t^{n}L_{n-i}^{a}(\\xi )=\\frac{t^{i}}{(1-t)^{a+1}}\\exp \\left[\\frac{-t\\xi }{1-t}\\right]$ for $|t|<1$ , we are able to evaluate the summation of Landau level by a direct manner.",
"We have $&&\\exp \\left[2\\hat{k}_{\\perp }^{2}+2\\hat{p}_{\\perp }^{2}\\right]\\sum _{n=0}^{\\infty }\\sum _{m=0}^{\\infty }(-1)^{(n+m)}z^{\\frac{n}{2}}z^{\\frac{m}{2}}\\nonumber \\\\&&\\cdot \\exp \\left[-(nx+m(1-x))\\tau \\right]L_{n-i}^{a}(-4\\hat{k}_{\\perp }\\mathbf {}^{2})L_{m-j}^{b}(-4\\hat{p}_{\\perp }^{2})\\nonumber \\\\&&=\\frac{t_{1}^{i}t_{2}^{j}}{(1-t_{1})^{a+1}(1-t_{2})^{b+1}}\\exp \\left[\\frac{\\alpha \\beta }{\\alpha +\\beta }\\hat{q}_{\\perp }^{2}\\right]\\nonumber \\\\&&\\cdot \\exp \\left[(\\alpha +\\beta )\\left(\\hat{k}_{\\perp }+\\frac{\\beta }{\\alpha +\\beta }\\hat{q}_{\\perp }\\right)^{2}\\right]$ where the regulator $z=\\mathrm {e}^{-\\epsilon _{\\perp }}$ (for $\\epsilon _{\\perp }\\rightarrow 0$ ) has been included in each summation.",
"The last exponential term explains the unusual shifting of transverse momentum which is early taken in Eq.",
"(REF ).",
"Here, $t_{1}=-z^{\\frac{1}{2}}\\mathrm {e}^{-x\\tau }$ , $t_{2}=-z^{\\frac{1}{2}}\\mathrm {e}^{-(1-x)\\tau }$ , and $\\alpha =\\frac{2(1+t_{1})}{1-t_{1}},\\qquad \\beta =\\frac{2(1+t_{2})}{1-t_{2}}.$ In principle, another term $J^{\\mu \\nu }&=&-4\\mathrm {i}\\left\\lbrace \\hbox{Tr}\\left[{k}_{}^{1-x}\\gamma ^{1}\\gamma ^{2}\\gamma ^{\\mu }{k}_{\\perp }^{\\alpha }\\gamma ^{\\nu }\\right]\\left({L}_{n}+{L}_{n-1}\\right){L}^{1}_{m-1}\\right.\\nonumber \\\\&&\\left.+\\hbox{Tr}\\left[{k}_{\\perp }^{\\beta }\\gamma ^{\\mu }{k}_{}^{x}\\gamma ^{1}\\gamma ^{2}\\gamma ^{\\nu }\\right]{L}^{1}_{n-1}\\left({L}_{m}+{L}_{m-1}\\right)\\right\\rbrace $ should appear in $\\Pi ^{\\mu \\nu }$ .",
"But it definitively vanishes because of the symmetry between $x\\leftrightarrow 1-x$ .",
"In other words, hold by Ward identity."
],
[
"Lerch Transcendent Regularization",
"After summation, it is remarkable that we acquire such a simple term, which takes the form of $\\Pi ^{\\mu \\nu }=4\\int \\mathrm {d}\\Gamma _{E}~\\frac{\\tau \\mathrm {e}^{-v_{0}\\tau }\\mathcal {I}^{\\mu \\nu }}{(1-t_{1})(1-t_{2})}\\exp \\left[h(x,\\tau )\\hat{q}_{\\perp }^{2}\\right],$ where $v_{0}=\\hat{M}^{2}-\\eta \\hat{q}_{}^{2}-\\mathrm {i}\\epsilon $ and $h(x,\\tau )=\\alpha \\beta /(\\alpha +\\beta )$ .",
"The integral volume space after performing wick rotation is $\\mathrm {d}\\Gamma _{E}=e^{2}\\int _{0}^{1}\\mathrm {d}x\\int _{0}^{\\infty }\\mathrm {d}\\tau \\int \\frac{\\mathrm {d}\\Gamma _{k_{}}}{(2\\pi )^{2}}\\int \\frac{\\mathrm {d}\\Gamma _{k_{\\perp }}}{(2\\pi )^{2}},$ where $\\int \\mathrm {d}\\Gamma _{k_{}}&=&\\int \\mathrm {d}^{2-\\epsilon }\\hat{k}_{}\\exp \\left[-\\hat{k}_{}^{2}\\tau \\right],\\nonumber \\\\\\int \\mathrm {d}\\Gamma _{k_{\\perp }}&=&\\int \\mathrm {d}^{2}\\hat{k}_{\\perp }\\exp \\left[-(\\alpha +\\beta )\\hat{k}_{\\perp }^{2}\\right].$ Note here $k^{0}=\\mathrm {i}k_{E}^{0}$ and $k^2=-k_{E}^{2}$ .",
"In the appendix , we detailed explore the value of $h(x,\\tau )$ .",
"We find that it locates in the interval $[0,1]$ for $x\\in [0,1]$ and $\\tau \\in (0,\\infty )$ .",
"The feature of Gaussian integrals is that its dominant contribution is coming from the point where the argument of the exponential is stationary [25], [26].",
"Therefore, from the view of mathematics, when $\\hat{q}_{\\perp }^{2}\\rightarrow 0$ , the mainly contributed interval of $h(x,\\tau )$ is at its maximum value with $\\tau \\gg 1$ , i.e.",
"$h(x,\\tau )=1$ .",
"As $\\hat{q}_{\\perp }^{2}\\rightarrow \\infty $ , $h(x,\\tau )$ is at its minimum region $\\eta \\tau $ to produce stationary points with $\\tau \\ll 1$ .",
"From the view of physics, since the sphere space symmetry is almost not broken for $B\\rightarrow 0$ , $h(x,\\tau )$ is closing to the coefficient of $\\hat{q}_{}^2$ , i.e., $\\eta \\tau $ .",
"On the other hand, the transverse spaces are decoupling from the whole system as the strength of magnetic field increasing, but the longitudinal space is left.",
"All in all, after considering the magnitude of $\\hat{q}_{\\perp }^{2}$ , the suitable approximation of $h(x,\\tau )$ can be applied as $h(x,\\tau )=\\theta (\\tau )\\operatorname{sech}\\frac{q_{\\perp }^{2}}{2eB}+\\eta \\tau \\tanh \\frac{q_{\\perp }^{2}}{2eB}.$ Then, we rewrite Eq.",
"(REF ) as $\\Pi ^{\\mu \\nu }=4C\\int \\mathrm {d}\\Gamma _{E}~\\frac{\\tau \\mathrm {e}^{-v\\tau }\\mathcal {I}^{\\mu \\nu }}{(1-t_{1})(1-t_{2})},$ where $C=\\exp \\left[\\hat{q}_{\\perp }^{2}\\operatorname{sech}\\hat{q}_{\\perp }^{2}\\right]$ and $v=\\frac{M^2-\\eta q_{}^2-\\eta q_{\\perp }^{2}\\tanh \\frac{q_{\\perp }^{2}}{2eB}}{2eB}-\\mathrm {i}\\epsilon .$ Following Eq.",
"(REF ), one obtains the full description of the tensor $\\mathcal {I}^{\\mu \\nu }&=&2\\left[-\\frac{\\epsilon }{2}g_{}^{\\mu \\nu }k_{}^{2}-\\eta (2q_{}^{\\mu }q_{}^{\\nu }-g_{}^{\\mu \\nu }q_{}^{2})+g_{}^{\\mu \\nu }M^{2}\\right]\\nonumber \\\\&&\\cdot \\left(1+t_{1}t_{2}\\right)-2g_{\\perp }^{\\mu \\nu }\\left[k_{}^{2}+\\eta q_{}^{2}+M^{2}\\right](t_{1}+t_{2})\\nonumber \\\\&&+4\\left(q_{}^{\\mu }q_{\\perp }^{\\nu }+q_{\\perp }^{\\mu }q_{}^{\\nu }\\right)\\left[\\frac{\\alpha (1-x)(1-t_{1})t_{2}}{(\\alpha +\\beta )(1-t_{2})}+\\mathrm {S.T.",
"}\\right]\\nonumber \\\\&&+16\\left[(\\alpha +\\beta )g_{}^{\\mu \\nu }k_{\\perp }^{2}-\\frac{\\alpha \\beta }{\\alpha +\\beta }(2q_{\\perp }^{\\mu }q_{\\perp }^{\\nu }-g^{\\mu \\nu }q_{\\perp }^{2})\\right]\\nonumber \\\\&&\\cdot \\frac{t_{1}t_{2}}{(1-t_{1})(1-t_{2})(\\alpha +\\beta )},$ where S.T.",
"denotes the related symmetric terms by exchanging $x$ and $1-x$ .",
"Both the functions of longitudinal and transverse momentum are at a simple Gaussian form in the integrand.",
"Carrying out the integration, one gets $\\pi ^{\\mu \\nu }=\\frac{1}{4\\pi ^{2}}\\int _{0}^{\\infty }\\mathrm {d}\\tau \\frac{\\tau ^{\\frac{\\epsilon }{2}}\\mathrm {e}^{-v\\tau }}{1-z\\mathrm {e}^{-\\tau }}\\left\\langle \\mathcal {I}^{\\mu \\nu }(\\tau ^{s})\\right\\rangle ,$ where $\\left\\langle \\mathcal {I}^{\\mu \\nu }(\\tau ^{s})\\right\\rangle =\\tau ^{1-\\frac{\\epsilon }{2}}(\\alpha +\\beta )\\int \\mathrm {d}\\Gamma _{k_{}}\\mathrm {d}\\Gamma _{k_{\\perp }}~\\mathcal {I}^{\\mu \\nu }(\\tau ^{s}).$ The denominator in Eq.",
"(REF ) is determined by algebra $\\frac{1}{(1-t_{1})(1-t_{2})(\\alpha +\\beta )}=\\frac{1}{4(1-z\\mathrm {e}^{-\\tau })}.$ The result of integrating Eq.",
"(REF ) is exactly proportional to $\\Gamma (s+\\frac{\\epsilon }{2})\\Phi (z,s+\\frac{\\epsilon }{2},v)$ [19], [20], where $\\Phi (z,s+\\frac{\\epsilon }{2},v)$ is the Lerch transcendent, see the details in appendix .",
"Thus, it is straightforward for us to write down the final expression of $\\pi ^{\\mu \\nu }=\\pi ^{\\mu \\nu }_{a}+\\pi ^{\\mu \\nu }_{b}+\\pi ^{\\mu \\nu }_{c}$ , which are $\\pi ^{\\mu \\nu }_{a}&=&-\\frac{\\epsilon }{4}g_{}^{\\mu \\nu }(2eB)\\Gamma \\left(\\frac{\\epsilon }{2}\\right)\\left(\\Phi (z,\\frac{\\epsilon }{2},v)+\\Phi (z,\\frac{\\epsilon }{2},v+1)\\right)\\nonumber \\\\&&-\\frac{1}{2}\\left[\\eta \\left(2q_{}^{\\mu }q_{}^{\\nu }-g_{}^{\\mu \\nu }q_{}^{2}\\right)-g_{}^{\\mu \\nu }M^{2}\\right]\\nonumber \\\\&&\\cdot \\left(\\Phi (z,1+\\frac{\\epsilon }{2},v)+\\Phi (z,1+\\frac{\\epsilon }{2},v+1)\\right)\\nonumber \\\\&&+g_{}^{\\mu \\nu }(2eB)D_{z}\\Phi (z,1+\\frac{\\epsilon }{2},v),$ $\\pi ^{\\mu \\nu }_{b}&=&\\frac{1}{2}g_{\\perp }^{\\mu \\nu }(2eB)D_{z}\\Phi (z,1+\\frac{\\epsilon }{2},v+x)+\\mathrm {S.T.",
"}\\\\ &&+\\frac{1}{2}g_{\\perp }^{\\mu \\nu }\\left(\\eta q_{}^{2}+M^{2}\\right)\\Phi (z,1+\\frac{\\epsilon }{2},v+x)+\\mathrm {S.T.",
"}\\\\ &&-\\frac{1}{2}\\left(q_{}^{\\mu }q_{\\perp }^{\\nu }+q_{\\perp }^{\\mu }q_{}^{\\nu }\\right)(1-x)\\left(D_{z}\\Phi (z,1+\\frac{\\epsilon }{2},v-x)\\right.",
"\\nonumber \\\\&& \\left.-D_{z}\\Phi (z,1+\\frac{\\epsilon }{2},v+x)\\right)+\\mathrm {S.T.",
"},$ $\\pi ^{\\mu \\nu }_{c}&=&-\\left(2q_{\\perp }^{\\mu }q_{\\perp }^{\\nu }-g^{\\mu \\nu }q_{\\perp }^{2}\\right)\\left(\\operatorname{sech}\\hat{q}_{\\perp }^2 D_{z}\\Phi (z,1+\\frac{\\epsilon }{2},v)\\right.\\nonumber \\\\&& \\left.+\\eta \\tanh \\hat{q}_{\\perp }^2 D_{z}\\Phi (z,2+\\frac{\\epsilon }{2},v)\\right).$ Here, not only $\\Phi $ is going to divergent for $s$ being non-positive integer at $z\\rightarrow 1$ , but also $\\Gamma (s+\\frac{\\epsilon }{2})$ .",
"When $s=0$ , the unpleasant divergence of Gamma function is originated from the UV-divergence with $k_{}^{2}$ in the numerator.",
"Loosely speaking, the dimension regularization in the longitudinal space should reduce such quadratic divergence to a logarithmic one, and then $s$ has to be greater than zero while evaluating Eq.",
"(REF ).",
"To cure the divergent $\\Gamma (\\frac{\\epsilon }{2})$ , we employ the formula $\\frac{1}{\\tau }\\simeq \\frac{z\\mathrm {e}^{-\\tau }}{1-z\\mathrm {e}^{-\\tau }}\\bigg |_{\\tau \\rightarrow 0, z\\rightarrow 1}$ to increase the power of $\\tau $ and therefore get the associated term in Eq.",
"(REF ).",
"We also adopt the approximate form of $h(x,\\tau )$ to get Eq.",
"(REF ).",
"We separate $\\Pi ^{\\mu \\nu }$ into three parts, in the form of $\\Pi ^{\\mu \\nu }=\\frac{e^2 C}{4\\pi ^{2}}\\int _{0}^{1}\\pi ^{\\mu \\nu }~\\mathrm {d}x =\\frac{e^2 C}{4\\pi ^{2}}P^{\\mu \\nu }_{i}(q^2)\\pi _{i},$ where $\\pi _{i}=\\pi _{},\\pi _{\\perp },\\pi _{m}$ correspond to longitudinal, transverse and mixture scalar function, respectively.",
"The projecting operators are $P_{}^{\\mu \\nu }(q^{2})=q_{}^{\\mu }q_{}^{\\nu }-g_{}^{\\mu \\nu }q_{}^{2},~P_{\\perp }^{\\mu \\nu }(q^{2})=q_{\\perp }^{\\mu }q_{\\perp }^{\\nu }-g_{\\perp }^{\\mu \\nu }q_{\\perp }^{2}$ and $P_{m}^{\\mu \\nu }(q^{2})=q^{\\mu }q^{\\nu }-g^{\\mu \\nu }q^{2}-P_{}^{\\mu \\nu }(q^{2})-P_{\\perp }^{\\mu \\nu }(q^{2}).$ Led by the explicit form of (REF ), we get $\\pi _{}&=&-\\int _{0}^{1}\\eta \\left[\\Phi (z,1,v)+\\Phi (z,1,v+1)\\right]\\mathrm {d}x\\nonumber \\\\&=&\\int _{0}^{1}\\eta \\left[\\psi (v)+\\psi (v+1)\\right]\\mathrm {d}x$ where the logarithmic divergence has been removed, see Eq.",
"(REF ).",
"$\\psi $ is di-gamma function, which is the logarithmic derivative of the Gamma function, $\\psi (v)=\\partial _{v}\\ln \\Gamma (v)$ .",
"The detailed limiting behaviors of function $\\Phi $ are discussed in appendix .",
"Meanwhile, from the summation representative of Lerch transcendent [19], [20], we obtain $D_{z}\\Phi (z,s,v)=\\Phi (z,s-1,v+1)-v\\Phi (z,s,v+1).$ Hence, the transverse scalar self-energy function can be derived as: $\\pi _{\\perp }&=&2\\int _{0}^{1}\\left(\\eta v\\tanh \\hat{q}_{\\perp }^2\\zeta (2,v+1)-\\operatorname{sech}\\hat{q}_{\\perp }^2\\zeta (0,v+1)\\right.",
"\\nonumber \\\\&& \\left.+\\psi (v+1)\\left(\\eta \\tanh \\hat{q}_{\\perp }^2-v\\operatorname{sech}\\hat{q}_{\\perp }^2\\right)\\right)\\mathrm {d}x$ via formula (REF ).",
"The mixture scalar function is described as $\\pi _{m}=\\int _{0}^{1}\\eta \\left(\\psi (v+x)+\\psi (v+1-x)\\right)\\mathrm {d}x.$ The full tensor structure of $P_{m}^{\\mu \\nu }$ is combined by several components, including $\\pi ^{\\mu \\nu }_{b}$ and a partial of Eq.",
"(REF ) and Eq.",
"(REF ).",
"Their associated scalar functions are closed but not exactly equivalent in our current calculation, shown by (REF ), (), (), and so on.",
"As a gauge invariant tensor, such slight differences should not exist.",
"But, they are inevitable since we have to simplify $h(x,\\tau )$ to access a final answer.",
"Here, we pick up the typical and dominant parts among them as our results of $\\pi _{m}$ , given in Eq.",
"(REF )."
],
[
"Discussion of Results",
"Di-Gamma function $\\psi (v)=-1/v+\\psi (v+1)$ has a simple pole at $v=0$ with residue $-1$ [19], [20].",
"It leads that $\\Pi ^{\\mu \\nu }$ is not a purely real valued function but contains imaginary part when the variable of $\\psi $ becoming negative.",
"Since the maximum value of $\\eta =x(1-x)$ is at the point $x=1/2$ , the threshold condition is $q_{}^2=4M^2$ for strong magnetic fields, where transverse momenta have been decoupled.",
"The reason that the threshold condition is not sensitive to the magnitude of magnetic fields is because the virtual pair gains energy from the external momentum of photon and becomes real via photon decay.",
"One should distinguish it with the pair production in strong electric fields via Schwinger effect, i.e.",
"virtual pair gains energy from the external electric field [16].",
"Strong Magnetic Field Limit For $eB\\gg q_{\\perp }^2$ , one has $C=\\exp \\left[\\hat{q}_{\\perp }^2\\right]$ and $v=\\hat{M}^{2}-\\eta \\hat{q}_{}^{2}$ .",
"Plus, both $\\hat{q}_{}^2$ and $\\hat{M}^2$ are not greater than 1 for $eB\\gg q_{}^{2}$ and $eB\\gg M^2$ .",
"The maximal value of $\\lfloor \\hat{q}_{}^{2}/4-\\hat{M}^2\\rfloor $ (floor function) is equal to zero.",
"Therefore, among three scalar functions, only $\\pi _{}$ contains imaginary part in the small distance regime where $q_{}^2>4M^2$ , which is ${\\mathrm {Im}}\\,\\pi _{}=\\frac{-4\\pi eBM^2}{q_{}^2\\sqrt{\\Delta _{}}},$ where $\\Delta _{}={q_{}^4-4q_{}^2M^2}$ .",
"Since here we have $|v|\\ll 1$ , $\\psi $ can be expanded as [20]: $\\psi (v+1)=-\\gamma +v\\zeta _{2}+\\mathcal {O}(v^2).$ $\\gamma \\approx 0.577216$ is the Euler-Gamma constant and $\\zeta _{2}=\\pi ^{2}/6$ .",
"It results in ${\\mathrm {Re}}\\,\\pi _{}=\\left\\lbrace \\begin{array}{ll}\\pi _{0}-\\frac{8eBM^2\\mathrm {arctanh}\\frac{q_{}^2}{\\sqrt{\\Delta _{}}}}{q_{}^2\\sqrt{\\Delta _{}}}, & \\hbox{\\small {$q_{}^2> 4M^2$};} \\\\\\pi _{0}+\\frac{8eBM^2\\arctan \\frac{q_{}^2}{\\sqrt{-\\Delta _{}}}}{q_{}^2\\sqrt{-\\Delta _{}}}, & \\hbox{\\small {$q_{}^2\\le 4M^2$},}\\end{array}\\right.$ where $\\pi _{0}=-2eB/q_{}^2-\\gamma /3$ .",
"Furthermore, one deduces that ${\\mathrm {Re}}\\,\\pi _{}=\\pi _{0}$ for $q_{}^2\\gg M^2$ and $\\pi _{}=eB/(2M^2)-\\gamma /3$ for $q_{}^2\\ll M^2$ .",
"$\\pi _{\\perp }$ and $\\pi _{m}$ are purely real valued functions at this limit.",
"One gets that $\\pi _{m}=-4\\ln A,~~\\pi _{\\perp }=\\mathcal {O}\\left(\\frac{\\Lambda ^2}{eB}\\right),$ where $A\\approx 1.28243$ is the Glaisher-Kinkelin constant and $\\Lambda ^{2}\\sim M^2$ and/or $q^2$ .",
"Evidently, one gets $\\pi _{}\\gg \\pi _{m}\\gg \\pi _{\\perp }$ at strong $B$ limit, which demonstrates the decoupling of transverse spaces.",
"Medium Regime Di-Gamma function $\\psi $ is meromorphic with simple poles at not only zero point but also $v=-1,-2,...$ with residues $-1$ .",
"Let $\\tilde{q}^2=q_{}^{2}+q_{\\perp }^{2}\\tanh \\hat{q}_{\\perp }^{2}$ , one finds that only $\\pi _{}$ becomes complex while $\\tilde{q}^2>4M^2$ but $\\tilde{q}^2\\le 2eB$ .",
"The imaginary component of $\\pi _{m}$ arise while $\\tilde{q}^2>2M^2+2eB+2\\sqrt{M^4+2eBM^2}$ .",
"All three scalar functions are characterized by an imaginary part while $\\tilde{q}^2>4(M^2+2eB)$ .",
"Thus, if $n=\\lfloor \\tilde{q}^2/(8eB)-\\hat{M}^2\\rfloor $ is a finite positive integer, $\\psi $ can be transformed through the following series [20] $\\psi (v)=\\psi (v+n+1)-\\sum _{k=0}^{n}\\frac{1}{v+k}.$ Then, one has ${\\mathrm {Im}}\\,{\\pi }_{}=\\sum _{k=0}^{n}\\frac{-8\\pi eBM^2_{k}}{\\tilde{q}^2\\sqrt{\\Delta _{k}}}+\\frac{4\\pi eBM^{2}}{\\tilde{q}^2\\sqrt{\\Delta _{0}}},$ where $M^2_{k}=M^2+2keB$ and $\\Delta _{k}=\\tilde{q}^4-4\\tilde{q}^2M^2_{k}$ .",
"The physical explanation of $n$ is the highest Landau level which possibly occupied by the created real fermion-anti-fermion pair.",
"The transverse function takes the form of: ${\\mathrm {Im}}\\,\\pi _{\\perp }=\\sum _{k=1}^{n}\\frac{-8\\pi eBM^2_{k}}{\\tilde{q}^2\\sqrt{\\Delta _{k}}}\\tanh \\hat{q}_{\\perp }^2+\\frac{8\\pi keB}{\\sqrt{\\Delta _{k}}}\\operatorname{sech}\\hat{q}_{\\perp }^2.$ Besides, let $n=\\lfloor \\tilde{q}^{2}/(8eB)+eB/(2\\tilde{q}^2)-\\hat{M}^2-1/2\\rfloor $ , one has ${\\mathrm {Im}}\\,\\pi _{m}=\\sum _{k=0}^{n}\\frac{-8\\pi eB\\left(\\tilde{q}^2M^{2}_{k+\\frac{1}{2}}-2e^2B^2\\right)}{\\tilde{q}^4\\sqrt{\\Theta _{k}}},$ where $\\Theta _{k}=\\tilde{q}^4+4e^2B^2-4\\tilde{q}^2M^{2}_{k+\\frac{1}{2}}$ .",
"Integrating analytically with respect to $x$ is not a plain task.",
"However, the numerical evaluation of a complete result of $\\Pi ^{\\mu \\nu }$ in any kinematic regime and any strength of $B$ -field is not computationally expensive.",
"We leave it in the future work.",
"An interesting algebra given here is that $\\int _{0}^{n}\\frac{2eBM^2_{k}}{q^2\\sqrt{\\Delta _{k}}}\\mathrm {d}k=\\frac{1}{12}\\sqrt{1-\\frac{4M^2}{q^2}}\\left(1+\\frac{2M^2}{q^2}\\right).$ Weak Magnetic Field Limit The fermion propagator at a constant magnetic field reduces to a free one if $B\\rightarrow 0$ .",
"It indicates that the three scaler functions have to become uniform and close to the usual results without $B$ -fields.",
"For $eB\\ll q^2$ and $eB\\ll M^2$ , $C=1$ and $\\tilde{q}^2=q^2$ .",
"Especially, $v$ is possible going to either $+\\infty $ or $-\\infty $ .",
"At these regimes, $\\psi (v)$ is logarithmically divergent [19].",
"A counter term of course has to be introduced to regularize results.",
"According to the properties of $\\psi $ , one has [19] $\\psi (v)=\\frac{1}{m}\\sum _{k=0}^{m-1}\\psi \\left(\\frac{v+k}{m}\\right)+\\ln m.$ Let $m=\\lceil \\hat{M}^{2}\\rceil $ (ceiling function).",
"The regularized $\\hat{\\psi }(v)$ is defined by $\\hat{\\psi }(v)=\\psi (v)-\\ln m$ .",
"The counter term $\\ln m$ is zero at strong magnetic field limit, i.e., $m=1$ for $M^2\\ll eB$ .",
"Another important property of counter term is that it is a purely real valued function and should not affect the imaginary part of $\\Pi ^{\\mu \\nu }$ .",
"Through ${\\mathrm {Im}}\\,\\psi (v)={\\mathrm {Im}}\\,\\ln v$ at $v\\rightarrow -\\infty $ , one can easily derive the imaginary part of the complex scalar functions for $q^2\\gg M^{2}$ : ${\\mathrm {Im}}\\,\\pi _{}=\\pi _{\\mathbb {I}}+\\frac{4\\pi eBM^2}{q^2\\sqrt{\\Delta }};~~{\\mathrm {Im}}\\,\\pi _{\\perp }=\\pi _{\\mathbb {I}}+\\frac{8\\pi eBM^2}{q^2\\sqrt{\\Delta }}.$ Note that here $\\pi _{\\mathbb {I}}=\\frac{-\\pi }{3}\\sqrt{1-\\frac{4M^2}{q^2}}\\left(1+\\frac{2M^2}{q^2}\\right).$ The results of Eq.",
"(REF ) are exactly the same as the ones if performed via early formula (REF ).",
"Besides, ${\\mathrm {Im}}\\,\\pi _{m}=\\frac{-\\pi }{3}\\frac{\\sqrt{\\Theta _{0}}}{q^2}\\left(1+\\frac{4e^2B^2}{q^4}+\\frac{2M^2+2eB}{q^2}\\right).$ Obviously, when $|v|\\rightarrow \\infty $ , the differences among three scalar functions are at the order of $eB/\\Lambda ^{2}$ , examined by above results.",
"Thus, we only provide the real part of the scalar function $\\pi _{}$ at the weak magnetic field limit, which is ${\\mathrm {Re}}\\,\\hat{\\pi }_{}=\\frac{1}{m}\\left(\\sum _{k=0}^{m-1}+\\sum _{k=1}^{m}\\right)\\int _{0}^{1}\\psi \\left(\\frac{M^2_{k}-\\eta q^2}{M^2}\\right) \\mathrm {d}x.$ The limiting behaviors are in the forms of: ${\\mathrm {Re}}\\,\\hat{\\pi }_{}|_{q^2\\gg M^2}=\\frac{1}{3}\\left(\\ln \\frac{q^2}{M^2}-\\frac{5}{3}\\right)+\\frac{2M^2}{q^2};$ $\\hat{\\pi }_{}|_{q^2\\ll M^2}=\\frac{-\\gamma }{3}+\\frac{\\zeta _{2}}{3}\\left(\\frac{1}{2}-\\frac{q^2}{5M^2}\\right).$"
],
[
"Conclusions and Outlooks",
"In this paper, we evaluated the photon vacuum polarization tensor in the presence of a homogeneous, purely magnetic field in a systematic framework.",
"The quadratic UV divergence of longitudinal momentum is cured by dimension regularization procedures.",
"We developed the zeta function technique to deal with the infinite series and then subtract the finite results.",
"Therefore, beyond LLL approximation, we obtained a full description of vacuum polarization tensor in response to all the Landau levels at any field strength of $B$ for the first time.",
"The final answers are characterized by the di-Gamma function $\\psi (v)$ and zeta functions.",
"Remarkably, we caught an intuitive understanding of the imaginary part of the photon polarization beyond the threshold $\\tilde{q}^2=4M^2$ in our context.",
"That is, the singular behavior of $\\psi (v)$ near each non-positive integer $-k$ describes the probability which a photon decays into a real fermion-anti-fermion pair at the given Landau level state $k$ .",
"The allowed quantum states are determined by the magnitude of $n=\\lfloor \\tilde{q}^{2}/(8eB)-\\hat{M}^2\\rfloor $ , with $k=0,1,...,n$ .",
"Such kind of pair creation process was observed by Hattori and Itakura [11].",
"Their work, and other similar ones [12], [13], used an alternative method for calculating the magnetized vacuum polarization tensor, which is based on an integration of double proper times.",
"Below the threshold, $\\tilde{q}^{2}<4M^{2}$ , the vacuum polarization tensor is a purely real valued function.",
"In the specified physical parameter regime, strong field limit, we analog our answers with the early ones under LLL approximation.",
"We confirmed that LLL approximation is valid in the region where $M^2\\gg q_{}^2$ but not correct in the region $q_{}^2\\gg M^2$ [10], [14], where the imaginary part has been missed.",
"We then focused on another interesting kinematics regime, weak field limit.",
"We found out the correct counter term and demonstrated that $\\Pi ^{\\mu \\nu }$ is limited to the usual vacuum polarization at $B\\rightarrow 0$ .",
"As showed by above sections, the integration with respect to the proper time is carried out by Lerch transcendent $\\Phi (z,s,v)$ straightforwardly.",
"It also should be realized as a performance of double summation with respect to two charged virtual particles.",
"Our current work for the first time brings a complete and surprising expression to the photon vacuum polarization at a constant external magnetic field.",
"Our approach is based upon a consistent applying of Lerch transcendent regularization.",
"The final description is rendered in a simple language of Lerch transcendent or its $z$ -derivation.",
"The whole procedure is elegant and mathematically rigorous.",
"Therefore, it allows us to enlarge the scope of this technique.",
"In other words, Lerch transcendent can be served as a fundamental tool to sum the infinite series with respect to the Landau levels.",
"It has opened a new window to investigate a wide range of phenomena in all physical regimes.",
"Especially, we expect our results will play an important role in studying the observations in the heavy ion collisions at RHIC and LHC experiments.",
"In the next paper, we will extend our studies to finite temperatures.",
"In particular, we will compute the gap equation via the complete photon vacuum polarization tensor.",
"It had been studied under the assumption which the dynamical mass of fermion is determined within the LLL approximation.",
"One should recalculate this essential ingredient based on our presented results."
],
[
"Lerch Transcendent",
"The integral representative of Lerch transcendent $\\Phi (z,s,v)$ is $\\Phi (z,s,v)=\\frac{1}{\\Gamma (s)}\\int _{0}^{\\infty }\\frac{\\tau ^{s-1}\\exp (-v\\tau )}{1-z\\mathrm {e}^{-\\tau }}~\\mathrm {d}\\tau ,$ for $|z|\\le 1, z\\ne 1, {\\mathrm {Re}}\\,s>0$ or $z=1, {\\mathrm {Re}}\\,s>1$ .",
"While as the summation representative is given by $\\Phi (z,s,v)=\\sum _{n=0}^{\\infty }z^{n}(n+v)^{-s}.$ for $|z|<1, v\\ne 0,-1,-2,...$ [19], [20].",
"Rendered by the integral representative, its derivation with respect to $z$ is: $D_{z}\\Phi (z,s,v)=\\frac{1}{\\Gamma (s)}\\int _{0}^{\\infty }\\frac{z\\mathrm {e}^{-\\tau }\\tau ^{s-1}\\exp (-v\\tau )}{(1-z\\mathrm {e}^{-\\tau })^{2}}~\\mathrm {d}\\tau ,$ where $D_{z}=z\\partial _{z}$ .",
"Besides, $\\Phi (1,s,v)=\\zeta (s,v)$ , where $\\zeta $ is the Riemann's Zeta Function.",
"$\\zeta (s,v)$ has a meromorphic continuation in the $s$ plane, its only singularity in $\\mathbf {C}$ being a simple pole at $s = 1$ with residue 1.",
"Therefore, we get the limiting behaviors of the Lerch transcendent as below [19]: $\\lim _{z\\rightarrow 1}\\Phi (z,1,v)=-\\log (1-z)-\\psi (v)$ for s=1.",
"When ${\\mathrm {Re}}\\,(s)<1$ , $\\lim _{z\\rightarrow 1}\\Phi (z,s,v)=\\frac{\\Gamma (1-s)}{(1-z)^{1-s}}+\\zeta (s,v).$ Another important identity of $\\Phi $ is $\\Phi (z,s,v)=z\\Phi (z,s,v+1)+\\frac{1}{v^{s}}.$"
],
[
"$h(x,\\tau )$",
"Remind you that $t_{1}=-z^{\\frac{1}{2}}\\mathrm {e}^{-x\\tau }$ , $t_{2}=-z^{\\frac{1}{2}}\\mathrm {e}^{-(1-x)\\tau }$ and $h(x,\\tau )=\\frac{\\alpha \\beta }{\\alpha +\\beta }=\\frac{1+t_{1}+t_{2}+t_{1}t_{2}}{1-t_{1}t_{2}}.$ Because the $n$ -th Bernoulli polynomials $B_{n}(x)$ represent the coefficients of $\\tau ^{n-1}/n!$ in the expansion of the generating function [20]: $\\frac{\\mathrm {e}^{x\\tau }}{\\mathrm {e}^{\\tau }-1}=\\sum _{n=0}^{\\infty }B_{n}(x)\\frac{\\tau ^{n-1}}{n!",
"},$ Eq.",
"(REF ) is deduced to $h(x,\\tau )\\big |_{z\\rightarrow 1}&=&2\\sum _{m=1}^{\\infty }(B_{2m}(0)-B_{2m}(x))\\frac{\\tau ^{2m-1}}{(2m)!",
"}\\nonumber \\\\&=& \\eta \\tau +\\mathcal {O}(\\tau ^{3}),$ for $\\tau <1$ .",
"However, for $\\tau >1$ , we have to use another formula, which $h(x,\\tau )\\big |_{z\\rightarrow 1}&\\le & \\frac{1-2\\sqrt{t_{1}t_{2}}+t_{1}t_{2}}{1-t_{1}t_{2}}=\\tanh \\frac{\\tau }{4}.$ Generally, $\\tanh \\frac{\\tau }{4}$ can be replaced by step fucntion $\\theta (\\tau )$ and its maximum value is equal to 1.",
"Acknowledgement.— We thank K. Hattori for pointing out a severe mistake.",
"We also thank I. Shovkovy for useful discussions.",
"This work is supported by the NSFC under Grant No.",
"11275213, DFG and NSFC (CRC 110), CAS key project KJCX2-EW-N01 and Youth Innovation Promotion Association of CAS.",
"This Project of J. C. is supported by China Postdoctoral Science Foundation Grant No.",
"2013M530732."
]
] | 1403.0442 |
[
[
"The Density and Mass of Unshocked Ejecta in Cassiopeia A through Low\n Frequency Radio Absorption"
],
[
"Abstract Characterizing the ejecta in young supernova remnants is a requisite step towards a better understanding of stellar evolution.",
"In Cassiopeia A the density and total mass remaining in the unshocked ejecta are important parameters for modeling its explosion and subsequent evolution.",
"Low frequency (<100 MHz) radio observations of sufficient angular resolution offer a unique probe of unshocked ejecta revealed via free-free absorption against the synchrotron emitting shell.",
"We have used the Very Large Array plus Pie Town Link extension to probe this cool, ionized absorber at 9 arcseconds and 18.5 arcseconds resolution at 74 MHz.",
"Together with higher frequency data we estimate an electron density of 4.2 electrons per cubic centimeters and a total mass of 0.39 Solar masses with uncertainties of a factor of about 2.",
"This is a significant improvement over the 100 electrons per cubic centimeter upper limit offered by infrared [S III] line ratios from the Spitzer Space Telescope.",
"Our estimates are sensitive to a number of factors including temperature and geometry.",
"However using reasonable values for each, our unshocked mass estimate agrees with predictions from dynamical models.",
"We also consider the presence, or absence, of cold iron- and carbon-rich ejecta and how these affect our calculations.",
"Finally we reconcile the intrinsic absorption from unshocked ejecta with the turnover in Cas A's integrated spectrum documented decades ago at much lower frequencies.",
"These and other recent observations below 100 MHz confirm that spatially resolved thermal absorption, when extended to lower frequencies and higher resolution, will offer a powerful new tool for low frequency astrophysics."
],
[
"General Background on Cas A",
"Cassiopeia A (Cas A; 3C 461, G111.7-2.1) is the 2nd-youngest-known supernova remnant (SNR) and, at a distance of 3.4 kpc [43], it lies just beyond the Perseus Arm of the Galaxy.",
"With the discovery of light echoes from the explosion, we now know that Cas A resulted from a type IIb explosion [35].",
"Cas A is one of the strongest synchrotron radio-emitting objects in the sky and has been observed extensively with the Very Large Array (VLA) since its commissioning in 1980.",
"The morphology of Cas A is quite complex with structure distributed over a variety of spatial scales.",
"The terminology used to describe some of these structures was coined in some of the earliest papers describing the optical and resolved radio images [53], [47].",
"The most prominent feature is the almost circular “Bright Ring” at a radius of $\\approx 100$ which is generally regarded as marking the location of ejecta that have interacted with the reverse shock [39].",
"A fainter “plateau” of radio emission is seen out to a radius of $\\approx 150$ .",
"To the northeast, where the shell becomes broken, is the “jet” and extending in the opposite direction to the southwest is the counter-jet.",
"The jet and counter-jet do not represent outflow in the classical sense.",
"Instead they describe locations where the fastest-moving ejecta are observed well beyond the plateau and farthest from the explosion center [22].",
"To the southeast, iron-rich ejecta extend beyond the Bright Ring and into the plateau.",
"The jets and extended iron-rich structure are likely the result of an asymmetric explosion of the progenitor [26], [28].",
"The light echo data also indicate an asymmetric explosion [45]."
],
[
"Unshocked Ejecta",
"In addition to the shocked ejecta described above, there is a class of ejecta still interior to the reverse shock in Cas A.",
"These “unshocked ejecta” were discovered via absorption of low frequency ($<$ 100 MHz) radio emission [32] and are also seen to radiate in the infrared in the emission lines of [O4], [S3], [S4], and [Si2] [13].",
"The term “unshocked” is somewhat of a misnomer because all of the ejecta were originally shocked by the passage of the blast wave through the star.",
"However, the ejecta cooled during the subsequent expansion of the SNR.",
"What we consider to be shocked ejecta today are those ejecta that have crossed through the reverse shock with the term “unshocked ejecta” referring to those ejecta that are still interior to the reverse shock.",
"Thus the simple cartoon of Cas A's structure is that of cold ejecta in the interior of a roughly spherical shell composed of shocked gas that radiates strongly in multiple bands.",
"At low frequencies, the radio emission from the far side of the shocked shell is absorbed by the cold, unshocked ejecta in the interior.",
"In §REF we provide a more thorough description of the geometry assumed for our analysis.",
"The infrared emission from the unshocked ejecta in Cas A occurs because they have been photoionized.",
"An analysis of the unshocked ejecta observed in the supernova remnant SN1006, based on calculations of photoionization cross-sections and Bethe parameters, showed that the unshocked ejecta are photoionized by two primary sources [19].",
"Ambient ultraviolet starlight is all that is necessary to photoionize Si1 and Fe1 since their ionization potentials are below the Lyman limit.",
"For ions with ionization potentials above the Lyman limit, the ultraviolet and soft X-ray radiation field of the shocked ejecta in SN1006 is such that species up to O3 (54.9 eV), Si4 (45.1 eV), and Fe4 (54.8 eV) can be photoionized.",
"A similar analysis was performed for Cas A assuming photoionization equilibrium, abundances appropriate for a core-collapse SNR, a thermal bremsstrahlung spectrum for the shocked ejecta, and utilizing a simple one-dimensional hydrodynamic model to follow Cas A's evolution [14].",
"This simplified model predicts that [Si2] (34.8) and [O4] (25.9) should be the dominant infrared lines in the unshocked ejecta, directly in line with observations [13].",
"In addition, [14] predicts strong [O3] (88.4), which we will show in §REF is present in spectra from the Infrared Space Observatory [52], [12].",
"Cas A was spectrally mapped with the Spitzer Space Telescope and Doppler shifts were measured which allowed a 3D mapping of the ejecta distribution, including the unshocked component [11], [29].",
"We now know based on the Spitzer data that the ejecta are organized into a “thick disk” structure, tilted at $\\sim 70$ from the line-of-sight, providing further evidence that the explosion, or subsequent evolution of the SNR prior to the reverse shock encounter, must have been asymmetric.",
"An upper limit of 100 cm$^{-1}$ was determined for the electron density of the unshocked ejecta based on infrared [S3] line ratios, but the actual density is likely much lower [14], [49].",
"The absorption seen in the low frequency radio observations provides a means to probe the density and mass of the unshocked ejecta because the free-free optical depth ($\\tau _{\\nu }$ ) is related to emission measure and thus density.",
"[32] attempted to determine the total mass of the unshocked ejecta, but due to using a $\\tau _{\\nu }$ appropriate for a hydrogenic gas and a temperature that was too high, arrived at 19M$_{}$ , which is unreasonably large considering that the total ejecta mass is likely only 2-4 M$_{}$ [28].",
"Given the role that the ejecta play in the evolution of SNRs, it is important to provide an accurate census of the total mass present, thus prompting a new look at the low frequency absorption analysis of [32]."
],
[
"Low Frequencies on the Legacy VLA",
"The upgraded Karl G. Jansky Very Large Array (VLA) primarily accesses frequencies above 1 GHz through its broadband Cassegrain focus systems.",
"Its predecessor, hereafter the “legacy” VLA, also accessed two relatively narrow bands below 1 GHz through its primary focus systems [33], [31].",
"These included the “P band” and “4 band” systems operating at 330 MHz (1990-2009) and 74 MHz (1998-2009), respectively.",
"Both systems provided sub-arcminute resolution imaging and were widely used over their lifetimes.",
"These systems were removed during the VLA upgrade and have been only recently replaced with a new “Low Band” receiving system [10].",
"The first call for proposals using the 330 MHz band of this new system was issued by NRAO in February, 2013."
],
[
"Pie Town Link",
"The legacy 74 MHz and 330 MHz VLA systems achieved their maximum angular resolution of $\\sim 20$ and $\\sim 6$ in the A configuration (maximum baseline $\\sim $ 36 km), respectively.",
"An 8-antenna prototype of the 74 MHz system was used to observe Cas A [32] early on, adding to the body of VLA work extending from 330 MHz to higher frequencies.",
"As the resolution was still relatively poor compared to shorter wavelengths, NRL and NRAO added a 74 MHz feed system to the Pie Town antennaThe Pie Town antenna already had a permanent 330 MHz feed as part of the VLBA., utilizing an optical fiber link connecting the innermost Very Large Baseline Array (VLBA) antenna to the legacy VLAThe optical fiber link was experimental and is no longer active.. With a maximum baseline of 73 km, this improved the angular resolution at 74 MHz and 330 MHz by a factor of two to approximately 9$$ and 3$$ respectively.",
"Cas A was thereafter reobserved with the full (27-antenna) 74 MHz and the 330 MHz legacy systems using the Pie Town link capability in August 2003."
],
[
"This Paper",
"In this paper, we report on legacy VLA observations in 1997-1998 at frequencies of 5 GHz, 1.4 GHz, 330 MHz, and 74 MHz, and on the Pie Town link, observations at frequencies of 330 MHz and 74 MHz in 2003.",
"We use our derived images to determine the degree of free-free absorption present, applying the appropriate formula for when the composition of the gas is not dominated by hydrogen.",
"Finally, we calculate the density and mass of the unshocked ejecta and discuss the implications of our results."
],
[
"1997-1998 and 2003 Legacy VLA Observations and Calibration",
"Observations were made with the legacy VLA in 1997 and 1998 using all four configurations from the most extended A-configuration with a maximum antenna separation of 36.4 km to the most compact D-configuration with a shortest separation of 35 m. These are summarized in Table REF .",
"Data were taken at 4410.0, 4640.0, 4985.0, and 5085.0 MHz (hereafter called 5 GHz or C band), at 1285.0, 1365.1, 1464.9, and 1665.0 MHz (hereafter called 1.4 GHz or L band), at 327.5 and 333.0 MHz (hereafter called 330 MHz or P band), and at 73.8 MHz (hereafter called 74 MHz or 4 band).",
"The 5 GHz and 1.4 GHz data were taken in continuum mode with pairs of frequencies observed simultaneously.",
"The purpose of observing at multiple 5 GHz and 1.4 GHz bands is to improve sensitivity while avoiding bandwidth smearing.",
"The 330 MHz and 74 MHz data were taken in spectral line mode.",
"Note that the D-configuration data taken at 330 MHz and 74 MHz had insufficient resolution to be useful for this paper and were not utilized.",
"In order to increase the spatial resolution of the lowest frequency images, further observations at 74 MHz and 330 MHz were taken with the legacy VLA in A-configuration and utilizing the Pie Town (PT) link (hereafter called A+PT).",
"We were allocated an 18-hour observing block in Aug 2003 with 12.2 hours on Cas A and the remainder of the time on calibrators and other sources of interest at low frequencies.",
"The 2003 observations are also summarized in Table REF .",
"lcccc Cas A 1997-1998 and 2003 Legacy VLA Observationsa 0.46 On-Source Time Bandwidthc per Band Date Configuration Bandb (MHz) (minutes) 1997 May B 4 1.5625 33 P 3.125 67 L 12.5 241 C 12.5 293 1997 Sep C 4 1.5625 24 P 3.125 24 L 12.5 144 C 12.5 142 1997 Nov, Dec D 4 1.5625 14 P 3.125 14 L 12.5 84 C 12.5 123 1998 Mar A 4 1.5625 61 P 3.125 61 L 6.25 464 C 6.25 467 2003 Aug A+PT 4 1.5625 732 P 3.125 732 aOriginal proposal codes were AR378 (1997-1998) and AD480 (2003).",
"bThese are telescope time on source at each band.",
"For C and L band observations, this is the combined time for both frequency pairs since pairs of frequencies are observed simultaneously.",
"cThese bandwidths are for each frequency observed.",
"Standard calibration procedures at 1.4 and 5 GHz were used for the data as described in the AIPS Cookbookhttp://www.aips.nrao.edu/cook.html.",
"Primary flux calibration was based on the source 3C 48, the polarization calibrator was 3C 138, and the phase calibrator was J2355+4950.",
"After initial calibration, multiple passes of self-calibration and imaging were performed on the Cas A data to improve the antenna phase and gain solutions.",
"The observations at 330 MHz and 74 MHz were performed in spectral line mode due to the presence of narrow-band radio frequency interference (RFI).",
"Images of Cygnus A at 330 MHz and 74 MHz, after normalization to the [4] absolute flux density scale, were used for both flux and bandpass calibration.",
"RFI was flagged out and the data were combined into a single channel at each band.",
"Iterative cycling between self calibration and imaging was then used to improve the phase and gain solutions for Cas A and to derive the final images.",
"74 MHz legacy VLA data, especially for the B, A, and A+PT link configurations, require corrections for ionospheric phase variations that can be rapid and severe over baselines longer than several kilometers [31].",
"Fortunately, Cas A so completely dominates the visibility phase measurements that straightforward self-calibration, yielding one phase correction per antenna, is perfectly capable of measuring the ionospheric effects.",
"Moreover, the signal-to-noise is sufficiently high that the fluctuations can be tracked and corrected at the native sampling rate of 6.7 seconds.",
"Standard 74 MHz and 330 MHz data reduction also typically requires wide-field, multi-faceted imaging of spectral line data, to address non-coplanar baseline imaging and bandwidth smearing, respectively [31].",
"Fortunately, Cas A is of small angular size (only $\\sim 300$ across) such that there is no need for implementing angular-dependent gain solutions nor dealing with the non-coplanar array.",
"Furthermore, Cas A is so strong that the contribution of other sources in the primary beams at both frequencies is negligible, and hence neither technique was required, greatly simplifying the data reduction."
],
[
"Total Intensity Images",
"cccrrrrrr Final Image Statistics 5 1 $u$ -$v$ range $\\theta _{LAS}$ a FOVb Noise Dynamicc $S_{\\nu }$ d $\\delta S_{\\nu }$ e Band Epoch (k$\\lambda $ ) (arcsec) (arcmin) (mJy bm$^{-1}$ ) Range (Jy) (Jy) 9c25 Resolution P 2003 0.7-81 170 150 5.5 40 6217 124 L 1997/8 0.5-81 300 30 1.8 193 2204 44 C 1997/8 0.5-81 300 7.5 3.2 194 809 16 9c9 Resolution 4 2003 0.7-18 170 600 190 368 18951 379 P 2003 0.7-18 170 150 26 135 6217 124 9c185 Resolution 4 1997/8 0.12-9 800 600 387 513 18555 371 P 1997/8 0.5-9 300 150 94 267 6185 124 (0.12-9) (5859)f L 1997/8 0.12-9 800 30 21 249 2179 44 aThe largest angular scale visible to the array.",
"Note that Cas A is 300$$ (5$$ ) in diameter.",
"bThe field of view (FOV), or primary beam, of the array set by the single dish aperture and the observing band.",
"At 5 GHz, the FOV is only 1.5 times the size of Cas A. cdynamic range=(image maximum)/(3-$\\sigma $ ) dIntegrated flux densities are reported at the frequencies of 4.64 GHz, 1.285 GHz, 330 MHz, and 73.8 MHz.",
"eBased on a 2% uncertainty in the absolute flux calibration [42].",
"fIntegrated flux density for 330 MHz image using 0.12 k$\\lambda $ as the minimum $u$ -$v$ distance to show the magnitude of the Van Vleck bias.",
"We present in this section sets of “matched” images at 25, 9, and 185 resolution.",
"The individual configuration data at each band were concatenated and images made with matched beam sizes.",
"Minimum and maximum spatial frequency ($u$ -$v$ ) ranges were used to match spatial sampling with the exception of the 25 and 185 330-MHz images which have a different minimum $u$ -$v$ cutoff than the other images due to the native minimum $u$ -$v$ distance for the 2003 A+PT data and to avoid the Van Vleck bias for the 1997-1998 data, as discussed below.",
"The AIPS maximum-entropy deconvolution routine VTESS was used to restore the total intensity images.",
"The default image for VTESS was a 5 GHz image of appropriate spatial resolution.",
"Using the same default image in VTESS for all of the bands essentially forces the images to look as much alike as possible, with any resulting differences being due to the requirements of the data.",
"Initial zero-spacing flux guesses were supplied to VTESS based on the 5 GHz integrated flux density and Cas A's average spectral index of -0.77.",
"The “negative flux” option was used so that VTESS was not required to get within 5% of the flux estimate.",
"The standard correction for primary beam attenuation was applied to all of the final images.",
"Since there were four images at 1.4 GHz and four images at 5 GHz, the images for each band needed to be normalized in flux density and averaged together.",
"While we could have concatenated the final calibrated $u$ -$v$ datasets and then performed the imaging, we chose to normalize the images directly.",
"There are two primary reasons for this choice.",
"First, the slightly different primary beam correction at each band introduces a small, but not insignificant error.",
"Second, it is generally not a good idea to force the $u$ -$v$ data to the same flux density, and grid on one plane for objects, such as Cas A, that have significant spectral gradients because each $u$ -$v$ cell has a slightly different visibility amplitude at each band.",
"Furthermore, since the individual 5 GHz and 1.4 GHz bands are close together, the relative calibration uncertainty is large enough that Cas A's well-known average spectral index cannot be used to scale between them.",
"This necessitates doing the concatenation and normalization in the imaging plane.",
"We determined the average scaling factor between each set of 5 GHz and 1.4 GHz images using the radial surface brightness profiles extracted from a series of 20 wedges.",
"The 1.4 GHz images were scaled to the 1.285 GHz image and the 5 GHz images were scaled to the 4.64 GHz image.",
"In Table REF , we summarize the integrated flux densities, noise figures, and $u$ -$v$ ranges for the final images and in Figure REF we plot the integrated radio spectrum of Cas A.",
"The uncertainty in the integrated flux density at each band is calculated based on a 2% uncertainty in the absolute flux scale of the VLA [42].",
"Figure: Integrated flux density measurements at each of the four observingbands and using the uu-vv ranges indicated in Table .",
"The errorin flux density is 2%, which is much smaller than the plotting symbols.The filled triangles indicate the flux densities of the 1997-1998 1.4 GHz and330 MHz images using their native minimum uu-vv distances to demonstrate themagnitude of the Van Vleck bias in each.",
"The dotted line represents aspectral index of -0.77.At 330 MHz, legacy VLA observations of Cas A suffer from the Van Vleck bias [55] which depends very strongly and non-linearly on the correlated flux density and also affects phase since the degradation operates independently on the real and imaginary portions of the visibility separately.",
"Thus, short antenna spacings report a reduced flux density which skews the final reconstructed flux density of the image by about 6% as shown in Table REF and Figure REF .",
"An effect this large would create a significant bias in spectral index and absorption measurements.",
"For the A+PT 330 MHz data taken in 2003, the minimum $u$ -$v$ distance is 0.7 k$\\lambda $ , which is long enough to avoid the Van Vleck bias.",
"In order to avoid this bias for the 1997-1998 330 MHz image, which has data from shorter baselines, we impose a 0.5 k$\\lambda $ minimum to the $u$ -$v$ range.",
"This limit happens to correspond to the minimum baseline for the 5 GHz data in D-configuration and results in a completely negligible Van Vleck bias [50].",
"Unfortunately, variations in spatial sampling between images can affect spectral index measurements.",
"Different deconvolution methods provide different abilities to “fill in” the holes in the $u$ -$v$ sampling.",
"Strict overlap or matching of $u$ -$v$ coverage is not required to get a good image, especially for an exceptionally bright source such as Cas A.",
"Furthermore, good deconvolution algorithms are able to recover the short-spacing flux, especially when they permit a total flux estimate and allow a default image, as VTESS does.",
"The Van Vleck bias also should affect the 1.4 GHz images based on the correlation coefficient [50], however we note only a 1% difference in the total flux density when using 0.5 k$\\lambda $ vs. 0.12 k$\\lambda $ as the minimum of the $u$ -$v$ range (0.12 k$\\lambda $ corresponds to the minimum baseline for the 1.4 GHz data in D-configuration.)",
"The 1% flux density difference at 1.4 GHz is small enough that other factors besides the Van Vleck bias might be accountable.",
"The integrated spectrum of Cas A plotted in Figure REF shows that the spectral index ($\\alpha _{\\nu _1}^{\\nu _2}\\equiv \\log (S_1/S_2)/\\log (\\nu _1/\\nu _2)$ ) is constant from 5 GHz up to 330 MHz ($\\alpha _{330}^{1.4}=-0.76\\pm 0.01$ and $\\alpha _{1.4}^{5}=-0.77\\pm 0.01$ ) when using a minimum $u$ -$v$ cutoff for the 330 MHz image.",
"From 330 MHz to 74 MHz, the integrated spectrum flattens slightly ($\\alpha _{74}^{330}=-0.74\\pm 0.01$ ).",
"The uncertainties in spectral index are calculated based on a formal propagation of the uncertainty in integrated flux density, and thus the spectral flattening at 74 MHz is statistically significant.",
"The spectral index between 5 GHz and 330 MHz is the same as that reported by [4] for epoch 1980 but the spectral index between 330 MHz and 74 MHz is steeper than the 1980 result.",
"[4] derived a frequency-dependent secular decrease in flux density which would flatten the spectral index over time.",
"Newer low frequency observations have shown that there is no frequency dependence to the secular decrease and the overall rate is about 8% yr$^{-1}$ [44], [23].",
"The integrated flux densities we report in Table REF at 5 GHz, 1.4 GHz, and 330 MHz are only $\\sim $ 3% less than the 1980 values from [4] and the 74 MHz integrated flux densities are largely consistent with those reported by [23].",
"There is no significant change in integrated flux density between 1997-1998 and 2003.",
"Given the differences in aperture coverage, constraints imposed during image reconstruction, and flux standards applied for the different data sets, as well as the short-term variability observed in Cas A [3], [23], our flux densities are in line with expectations.",
"Thus we argue that there is no significant curvature in Cas A's integrated spectrum between 5 GHz and 330 MHz but there is a significant flattening of the integrated spectrum at lower frequencies.",
"We did not concatenate the 1997-1998 and 2003 data sets in order to make images containing data from both epochs because of two complicating factors.",
"The first is the non-negligible proper-motion induced shift, due to expansion, across the epochs, and the second are the variations in the brightness of compact features over time.",
"[3], for example, found that two-thirds of the compact radio features in Cas A have brightness changes in the range of -2.2% yr$^{-1}$ to +5.2% yr$^{-1}$ and that the compact features were on average brightening at the rate of 1.6% yr$^{-1}$ .",
"At low spatial resolution, the proper motions and small-scale brightness changes are mitigated, but then there is no reason to combine the A+PT data with the lower resolution data except to improve signal-to-noise, which as we will demonstrate below, was not necessary for the analysis at 185 resolution.",
"Figure: 5 GHz (top) and 1.4 GHz (middle) images of Cas A from 1997-1998 and330 MHz image (bottom) from 2003.",
"The resolution is 25 and the fluxdensity scales are in Jy bm -1 ^{-1}.",
"The uu-vv ranges used to construct theseimages and the noise and integrated flux densities are reported inTable ."
],
[
"25 Resolution Images",
"Figure REF shows the final 1997-1998 5 GHz and 1.4 GHz images and the 2003 330 MHz image at 25 resolution.",
"With the improved resolution using the Pie Town link, we can see that the 330 MHz image captures the same large- and small-scale features as at higher frequencies, thus supporting the constant spectral index found from 5 GHz to 330 MHz.",
"The dynamic range of the 330 MHz image is on par with the 1.4 GHz image and noticeably better than the 5 GHz image.",
"The relatively poor dynamic range of the 5 GHz image is partly because the primary beam of the VLA at 5 GHz is only 1.5 times larger than Cas A, as indicated in Table REF .",
"The high quality of the 330 MHz image demonstrates how well the AIPS task FLGIT excises RFI and the power of self-calibration for dealing with ionospheric effects."
],
[
"9 Resolution Images",
"Figure REF shows the final 2003 330 MHz and 74 MHz images at 9 resolution, which is the best resolution achieved with the A+PT link at 74 MHz.",
"The dynamic range on the 330 MHz image is about 3 times better than for the 74 MHz image.",
"Clumpy structure is observed on a variety of spatial scales down to the resolution limit.",
"Comparing to the 25 resolution images in Figure REF , it is clear that the same synchrotron features are responsible for the emission at 74 MHz.",
"The bottom panel of Figure REF is a plot of the angle-averaged radial surface brightness profiles of the 330 MHz and 74 MHz images.",
"The profiles have been normalized to their peak values.",
"The emission in the center of the 74 MHz image is noticeably fainter than expected based on the appearance of the 330 MHz image and therefore contributes to the flattening of the integrated spectrum from 330 MHz to 74 MHz observed in Figure REF .",
"Figure: Top and middle: 2003 A+PT configuration images of Cas A at 330 MHzand 74 MHz.",
"The resolution is 9.",
"The flux density scales are inJy bm -1 ^{-1}.",
"The uu-vv ranges used to construct these images and the noiseand integrated flux densities are reported in Table .",
"Bottom:Angle-averaged radial surface brightness profiles of the 330 MHz and 74 MHzimages normalized to their peak values."
],
[
"185 Resolution Images",
"Figure REF shows the final 1.4 GHz, 330 MHz and 74 MHz images at 185 resolution from the 1997-1998 observations.",
"This represents the best resolution attained with the legacy VLA in A-configuration at 74 MHz.",
"The dynamic range on the 1.4 GHz image is about 2 times better than for the 330 MHz and 74 MHz images.",
"The 74 MHz image at this lower resolution has a dynamic range about twice that at higher resolution using the A+PT link.",
"The higher dynamic range allows us to better probe the large-scale diffuse emission from Cas A.",
"The absorption of emission from the center of Cas A is, again, clearly observed.",
"Figure: 1.4 GHz (top left), 330 MHz (top right), and 74 MHz (bottom left)images of Cas A from 1997-1998.",
"The resolution is 185 and the fluxdensity scales are in Jy bm -1 ^{-1}.",
"The uu-vv ranges used to constructthese images and the noise and integrated flux densities are reported inTable .",
"The angle-averaged radial surface brightness profiles ofthe 330 MHz and 74 MHz normalized to their peak values are shown at bottomright."
],
[
"Spectral Index at 9 Resolution",
"The 2003 330 MHz and 74 MHz images at 9 resolution were combined to form the spectral index image shown in Figure REF .",
"Only spectral index data from regions greater than 10-$\\sigma $ on the 330 MHz image were retained.",
"The range of spectral indices at 9 resolution is $\\alpha _{74}^{330}\\approx $ -0.95 to -0.35, with the flattest spectrum corresponding to the bright clump in the center of Cas A.",
"[2] report an average $\\alpha _{1.4}^{5}\\approx $ -0.78 for the collection of clumps in that location, thus a significant spectral flattening has occurred.",
"The abundance of clumps with flat (blue) spectral indices in the center are not necessarily an indication of absorption.",
"Many of the central clumps are known to have flat spectral indices based on images at higher frequencies [2].",
"In order to assess the degree of absorption in Cas A, we need to look at the difference in spectral index from high frequencies to low frequencies.",
"Since we did not take higher frequency data with the 2003 observations, and the proper motion and small-scale brightness variations across epochs would be evident at 9 resolution, we must work with our lower resolution images."
],
[
"Spectral Index at 185 Resolution",
"The 1997-1998 74 MHz, 330 MHz, and 1.4 GHz images at 185 resolution were combined to form the spectral index images presented in Figure REF .",
"Only spectral index data from regions greater than 10-$\\sigma $ on the 330 MHz image were retained.",
"These images show similar patterns in spectral index as those presented by [32] at 25 resolution but at higher signal-to-noise for the images tied to 74 MHz because of the full complement of antennas available.",
"As with the higher resolution spectral index image in Figure REF , the central features in the $\\alpha _{74}^{330}$ image are notably flatter than the Bright Ring.",
"One distinct advantage at lower spatial resolution is that the signal-to-noise is high enough to probe the spectral index of the diffuse emission as well as the clumped emission.",
"The range of spectral indices between 330 MHz and 1.4 GHz (Figure REF left) is $\\alpha _{330}^{1.4}\\approx $ -0.85 to -0.65 and is typical of that observed at higher frequencies [2].",
"The range of spectral indices between 74 MHz and 330 MHz (Figure REF right) is $\\alpha _{74}^{330}\\approx $ -0.95 to -0.5, which is slightly larger than that between 330 MHz and 1.4 GHz.",
"Both images in Figure REF are plotted using the same spectral index scale in order to better demonstrate the spectral flattening effect in the central regions of the $\\alpha _{74}^{330}$ image.",
"At 185 resolution, the central region is not as flat as at 9 resolution.",
"This could be because the lower resolution smooths out large variations and also because of the lower dynamic range of the 9 resolution image.",
"Figure: Spectral index image between 74 MHz and 330 MHz made from the 2003data set.",
"Only spectral index data from regions greater than 10-σ\\sigma onthe 330 MHz image are shown.",
"The 330 MHz image is used as a brightnesschannel to illuminate the spectral index colors.",
"The resolution is 9.As noted by [32], the spectral indices for features on the Bright Ring are about the same from high frequencies to low frequencies.",
"At the center of Cas A, however, there is a significant flattening of the spectral index between 330 MHz and 74 MHz.",
"Given that the spectral flattening is confined to the center of the 74 MHz image, there is likely an absorbing medium in or near Cas A that affects the observed low frequency emission but not the observed high frequency emission."
],
[
"Spectral Index Difference ($\\Delta \\alpha $ )",
"The spectral index variations observed at high frequencies reflect the different energy spectra of accelerated electron populations.",
"These different energy spectra may be due to the acceleration at the forward and reverse shocks and to intrinsic breaks in the underlying electron energy spectrum [2].",
"The spectral index variations observed at low frequencies are due to both the accelerated electron populations and effects from free-free absorption.",
"In order to isolate spectral changes due only to free-free absorption, we assume that there is no curvature in the emitted synchrotron spectrum at low frequencies.",
"Thus, when we subtract the two spectral index images, the resulting image, $\\Delta \\alpha =\\alpha _{74}^{330}-\\alpha _{330}^{1.4}$ , shows only variations due to free-free absorption.",
"Our $\\Delta \\alpha $ image is shown in Figure REF with the same signal-to-noise cutoff as the spectral index images.",
"The 330 MHz image was again used to illuminate the colors.",
"Typical $\\Delta \\alpha $ values for regions with good S/N ratio vary from -0.1 to 0.25 where positive $\\Delta \\alpha $ indicates absorption and statistical variations due to noise are of order $\\pm $ 0.01 to 0.02.",
"Most of the free-free absorption is confined to the middle of Cas A with typical $\\Delta \\alpha $ values of 0.1-0.25.",
"This is the same absorption morphology noted by [32] who found a peak $\\Delta \\alpha $ =0.3 with their 25 resolution images.",
"Figure: Spectral index images between 330 MHz and 1.4 GHz (top) and between74 MHz and 330 MHz (bottom) made from the 1997-1998 data set.",
"Only spectralindex data from regions greater than 10-σ\\sigma on the 330 MHz image areshown.",
"The 330 MHz image is used as a brightness channel to illuminate thespectral index colors.",
"The resolution is 185."
],
[
"Comparison with Infrared Emission from Unshocked Ejecta",
"One of the most exciting discoveries with Spitzer was the presence of emission from unshocked ejecta in Cas A.",
"This emission is most prominent in the lines of [O4] and [Si2] and less prominent for [S3] and [S4] [13].",
"The density and temperature conditions derived for this material based on the infrared line ratios confirms that it is indeed cold, low density, unshocked, photoionized ejecta [49].",
"In order to confirm the hypothesis of [32] that the low frequency free-free absorption seen in Cas A is due to unshocked ejecta, we compare the $\\Delta \\alpha $ image to the infrared [Si2] image.",
"As shown in the top left panel of Figure REF , strong free-free absorption ($\\Delta \\alpha 0.1$ ) corresponds well to bright [Si2] emission ($5\\times 10^{-7}$ W m$^{-2}$ sr$^{-1}$ ).",
"However, there are bright [Si2] regions that have low or no associated absorption.",
"Figure: Top left: Δα\\Delta \\alpha contours plotted over infrared [Si2]emission in greyscale at a spatial resolution of 185.Positive Δα\\Delta \\alpha values (solid contours) indicate free-freeabsorption.",
"The contour levels correspond to Δα\\Delta \\alpha values of:-0.05, 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, and 0.30 and the signal-to-noisecutoff for the contours is 10-σ\\sigma on the 330 MHz image.",
"Top right: Sliceprofiles taken at the location indicated on the inset of the Δα\\Delta \\alpha image.",
"The yy-axis indicates the value of Δα\\Delta \\alpha and the 74 MHz and[Si2] profiles have been scaled to show correlations between featuresin the images.",
"The dotted line indicates the “average” Δα\\Delta \\alpha chosen for our analysis.",
"Bottom left: Same as top left except with blankingapplied to exclude shocked ejecta.",
"Bottom right: Δα\\Delta \\alpha vs.[Si2] surface brightness from only unshocked ejecta.",
"The solid lineindicates the running mean of the Δα\\Delta \\alpha values and shows a generalincrease of absorption with [Si2] surface brightness.There is not a one-to-one correlation between free-free absorption and [Si2] emission for three primary reasons.",
"First, while the [Si2] image is a good representative of the emission observed from the unshocked ejecta in the Spitzer data, it also contains emission from shocked ejecta.",
"As the ejecta at large radius cross the reverse shock, become heated, and contribute to the Bright Ring emission, the free-free absorption drops drastically due to the temperature dependence.",
"We can filter the [Si2] image using the infrared [Ar2] image to blank those regions corresponding to shocked ejecta.",
"The remaining [Si2] emission, shown in the bottom left panel of Figure REF , is dominated by unshocked ejecta and, when plotted against $\\Delta \\alpha $ (bottom right panel of Figure REF ), shows a good correlation to the low frequency absorption.",
"The second reason for scatter in the correlation between the free-free absorption and [Si2] emission is that the [Si2] image provides no information about other unshocked components that may be present but not detected, such as higher ionization states of Si- and O-rich ejecta or the presence of Fe-rich ejecta, for example.",
"Finally, the $\\Delta \\alpha $ image is a function of both the column density of the absorbing medium and the distribution of radio emitting regions along the line of sight.",
"The presence of clumped radio emission biases the $\\Delta \\alpha $ calculation to higher or lower values depending on whether the clump is on the far or near side of the unshocked ejecta in Cas A.",
"The greater the brightness disparity between the clump and the diffuse background, the stronger the bias.",
"We show this case in the top right panel of Figure REF where we plot slice profiles for $\\Delta \\alpha $ and 74 MHz and [Si2] surface brightnesses.",
"The peak $\\Delta \\alpha $ occurs at the location of a particularly bright emission clump, but has no obvious correlation to any distinct [Si2] features.",
"We therefore conclude that the free-free absorption is clearly correlated to the observed emission from the unshocked ejecta and is a tracer of this material as presumed by [32].",
"Since free-free absorption depends on the temperature and density of the absorbing medium, our low frequency radio data provide a means of determining the conditions in the unshocked ejecta.",
"When coupled with assumptions about geometry, we can also calculate the total mass of the observed unshocked ejecta.",
"We carry out these calculations in the following sections."
],
[
"Free-Free Optical Depth",
"Our use of thermal absorption to probe the properties of the unshocked ejecta in Cas A relies on simplifying assumptions.",
"Our measurement of $\\Delta \\alpha $ , discussed above, can yield an equivalent free-free optical depth, but relies on our knowledge of the line-of-sight brightness distribution of the nonthermal, illuminating radiation from Cas A's shell.",
"The high spatial resolution images in Figure REF show that Cas A consists of both clumpy and diffuse components.",
"We assume the smooth emission has radial symmetry, consistent with shell decomposition models [16], [18], so that equal contributions come from the front and back side of the shell along any line of sight.",
"As discussed in §, the presence of clumps will bias our measurement of $\\Delta \\alpha $ .",
"However, it is almost impossible to avoid clumped emission in Cas A, as Figure REF shows.",
"We could try using a higher frequency image to account for the clumps, but without a priori information about which clumps are in front of or behind the absorbing medium, we would introduce more uncertainties into the analysis.",
"Therefore, we choose to select a representative absorption figure that effectively averages out the contamination from clumps.",
"In the top right panel of Figure REF , we show the absorption profile taken along the slice indicated in the inset.",
"The peaks and valleys of the central absorption ($\\Delta \\alpha $ ) are correlated with 74 MHz surface brightness variations, as expected.",
"Typical $\\Delta \\alpha $ values in the center of Cas A range from 0.1-0.25 so we choose by eye $\\Delta \\alpha $ =0.15 as a compromise between high and low absorption towards the center of Cas A.",
"Since this is a somewhat arbitrary choice, we will investigate in §REF how the final mass and density estimates for the unshocked ejecta depend on the value of $\\Delta \\alpha $ .",
"We also assume that the cold ejecta lie only in the interior of Cas A, and therefore only affect the 74 MHz emission from the back side of the shell, again consistent with three-dimensional models [11].",
"We further assume the 330 MHz emission is completely unabsorbed – consistent with our spectral index measurements from 330 MHz to 5 GHz.",
"Based on the above picture, we derive in Appendix A the free-free optical depth $\\tau $ as a function of $\\Delta \\alpha $ : $\\tau = -\\ln [2 (10^{a \\Delta \\alpha } -0.5)]$ where $a=\\log (74/330)$ .",
"In this formulation, spectral index is defined as $\\alpha _{74}^{330}=\\frac{\\log (S_{74}/S_{330})}{\\log (74/330)}$ and therefore is negative for a synchrotron spectrum.",
"Using $\\Delta \\alpha $ =0.15 results in a $\\tau $ of about 0.51."
],
[
"Emission Measure of the Unshocked Ejecta",
"The free-free absorption $\\tau $ can be characterized by the following formula [46]: $&\\tau _{\\nu }=&3.014\\times 10^{4} \\left(\\frac{Z^2}{f}\\right)\\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^{-2}\\left(\\frac{T_e}{\\mathrm {K}}\\right)^{-3/2}\\left(\\frac{EM}{\\mathrm {pc\\,cm}^{-6}}\\right) \\nonumber \\\\& & \\times \\ln \\left[49.55 Z^{-1}\\left(\\frac{T_e}{\\mathrm {K}}\\right)^{3/2} \\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^{-1}\\right]$ where $Z$ is the average atomic number of the ions dominating the cold ejecta, $f$ is the electron to ion ratio, $\\nu $ is the radio frequency in MHz at which the absorption is observed, $T_e$ is the temperature in K of the ionized absorbing medium, and $EM$ is the emission measure in units of pc cm$^{-6}$ .",
"Emission measure is defined as: $EM \\equiv \\int n_e^2 dl$ where $n_e$ is the electron density of the gas in units of cm$^{-3}$ and $dl$ is the path length along the line-of-sight measured in parsecs.",
"Inverting Equation REF to solve for $EM$ yields: $EM=\\frac{3.318\\times 10^{-5} \\left(\\frac{\\tau _{\\nu } f}{Z^2}\\right)\\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^2 \\left(\\frac{T_e}{\\mathrm {K}}\\right)^{3/2}}{\\ln \\left[49.55\\,Z^{-1}\\left(\\frac{T_e}{\\mathrm {K}}\\right)^{3/2} \\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^{-1}\\right]}.$"
],
[
"Composition of the Absorbing Medium",
"To interpret $EM$ , we first need to place realistic constraints on both $Z$ and $f$ .",
"Based on our observed correlation between $\\Delta \\alpha $ and infrared [Si2] emission, and the emission from unshocked ejecta in the infrared lines of [O4], [S3], and [S4] observed with Spitzer [13], [14], we assume the absorption is dominated by Si- and O-rich ejecta, including Si, S, Ar, Ca, Mg, Ne, and O.",
"To determine the relative abundances of these species in Cas A, we rely on X-ray derived abundances indicating that the ejecta in Cas A are O-dominated [56] and, based on stellar composition models, are typically more than a million times more abundant than hydrogen [58].",
"Based on the abundances quoted in [56] from XMM-Newton observations of Cas A we adopt a weighted average $Z$ of 8.34.",
"Spectra from Spitzer and ISO [52], [12] indicate that both [O3] and [O4] are observed from the unshocked ejecta.",
"While models of unshocked ejecta in Cas A predict higher ionization states to be present, the material responsible for the observed infrared emission and most of the thermal absorption is in these lower ionization states [14].",
"Therefore, we adopt an electron/ion ratio, $f$ , of 2.5.",
"Figure: ISO spectrum of the center of Cas A taken with the LWS instrumentin 1996.",
"The dotted line marks the rest wavelength of [O3].",
"Usingthe rest wavelength of 88.36 μ\\mu m for [O3] results in Dopplervelocities of the fitted Gaussian components as indicated.",
"These velocitiescorrespond to the known velocity structure of the unshocked ejecta.",
"We note that [Fe2], with a rest wavelength of87.38 μ\\mu m, also falls into this same wavelength range, but the data arecompletely consistent with the [O3] origin given the known velocitystructure.At the observing frequency of 73.8 MHz, the last parameter to consider is $T_e$ .",
"[32] assumed, rather arbitrarily, $T_e$ =1000 K in their calculation.",
"A much better estimate of the temperature in the unshocked ejecta comes from the analysis of the infrared emission lines.",
"In order to produce strong [O4] and [Si2] lines but weak or absent [S4] and [Ar3] lines in the Spitzer data, $T_e$ must be $\\sim $ 100-500 K [14].",
"Therefore, we will adopt $T_e$ =300 K. Parameterized in terms of our values for $\\tau _{\\nu }$ , $f$ , $Z$ , $T_e$ , and $\\nu $ , the emission measure becomes: $EM=\\frac{17.21\\left(\\frac{\\tau _{\\nu }}{0.51}\\right)\\left(\\frac{f}{2.5}\\right)\\left(\\frac{Z}{8.34}\\right)^{-2}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/2}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)^2}{\\ln \\left[418.3\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)^{-1}\\right]}.$"
],
[
"Geometry of Cas A",
"In order to estimate density from the emission measure and then estimate the total mass in the unshocked ejecta, we must assume a geometry for the absorbing medium.",
"[11] and [29] showed that the unshocked ejecta, as traced by [O4] and [Si2] emission, are concentrated onto two thick sheets interior to the reverse shock – one front and one rear.",
"The two sheets are separated by a much lower density region, where only a little emission is seen.",
"One might expect unshocked C-rich and Fe-rich ejecta at various locations interior to the reverse shock as well based on stratification models of the ejecta [13], however these species are not observed from the unshocked ejecta either in the Spitzer IRS spectra or the ISO LWS spectra, which could indicate that they are of lower density than the O-rich or Si-rich material, or simply absent [14].",
"Figure: Simplified drawing of the geometry of the shocked and unshockedmaterial in Cas A.",
"The shaded, circumferential ring represents the regionbounded on its interior and exterior by the reverse shock and the blast wave,respectively.",
"It consists of smooth (blue shaded) and clumped (blue stars)synchrotron emitting components.",
"The two orange ellipsoids representthermally absorbing, unshocked ejecta emitting infrared lines of Si and O. Cand Fe, depicted as lying exterior and interior to the Si and O emittingmaterial, respectively, have not been detected but might be present at lowerdensities.A simplified drawing of the geometry involved is shown in Figure REF .",
"As explained in §REF , the emission from shocked material in Cas A can be modeled as a spherically symmetric shell consisting of smooth and clumped components.",
"The clumpy shocked material may be ejecta, circumstellar material, or filaments associated with the forward or reverse shock.",
"The unshocked ejecta are confined to two thick sheets in the interior.",
"The high spectral resolution Spitzer mapping shows that the sheets are composed of thin filaments (15 or 0.03 pc thick) with several filaments along any given line-of-sight [29].",
"Therefore, we estimate the combined thickness of the front and back sheets to be of order 10$$ which translates to $L=$ 0.16 pc at the distance to Cas A of 3.4 kpc [11], [43].",
"The sheets intersect the reverse shock at a radius of about 90$$ (or 1.48 pc), so that the total volume of absorbing material is $V=\\pi R^2 L =$ 1.1 pc$^3$ .",
"The geometry can be modified further by the use of a clumping factor.",
"We know from the optical images of shocked ejecta in Cas A that there is a great deal of structure on small scales with typical knot sizes between 02 and 04 [17].",
"It is not known how much of this clumping was present prior to reverse shock passage.",
"The effect of clumping is to reduce the total volume of material while at the same time increasing the density of the material.",
"In order to determine the effect of clumping on our calculations, we simply modify each of our radius or thickness measures by the clumping factor ($CL$ or $CR$ ) so that when $C=1$ there is no clumping and when $C<1$ there is some degree of clumping.",
"Our default value of $C$ is to assume no further clumping beyond the filamentary structures identified by [29]."
],
[
"Electron Density and Mass",
"If we assume that the density is constant, then emission measure defined in Equation REF simplifies to $EM=n_e^2CL$ where $L$ is the combined thickness of the unshocked ejecta sheets.",
"In Appendix C we show electron density fully parameterized in terms of $\\tau _{\\nu }$ , $f$ , $Z$ , $T_e$ , $\\nu $ , $C$ , and $L$ , however the variables in the denominator are suppressed by both a natural logarithm and a square root.",
"Therefore we show here a simplified version of the parameterized formula in order to show how electron density depends on these various quantities: $n_e \\approx & 4.23 \\left(\\frac{\\tau _{\\nu }}{0.51}\\right)^{1/2}\\left(\\frac{f}{2.5}\\right)^{1/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/4} \\nonumber \\\\& \\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)C^{-1/2}\\left(\\frac{L}{0.16\\,\\mathrm {pc}}\\right)^{-1/2}.$ Thus, our estimate of electron density in the unshocked ejecta is 4.2 cm$^{-3}$ .",
"The total mass of unshocked ejecta is simply $M=\\rho V$ where $\\rho $ is the mass density of the ions, $\\rho =Zm_pn_e/f$ , and $m_p$ is proton mass.",
"If we substitute Equation REF for $n_e$ , assume the two thick sheet geometry, and parameterize mass in terms of $\\tau _{\\nu }$ , $f$ , $Z$ , $T_e$ , $\\nu $ , $C$ , $R$ , and $L$ then, as shown in Appendix C, the dependence on $Z$ becomes minuscule as the only surviving $Z$ -term is mitigated by the natural logarithm and square root in the denominator.",
"As we did for electron density, we show here a simplified parameterized equation for the total mass of the unshocked ejecta expressed in solar masses: $M \\approx & 0.39 \\left(\\frac{\\tau _{\\nu }}{0.51}\\right)^{1/2}\\left(\\frac{f}{2.5}\\right)^{-1/2}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/4} \\nonumber \\\\& \\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)C^{5/2}\\left(\\frac{R}{1.48\\,\\mathrm {pc}}\\right)^2\\left(\\frac{L}{0.16\\,\\mathrm {pc}}\\right)^{1/2}.$ We therefore estimate a total mass in unshocked ejecta of 0.39 M$_{}$ which is significantly improved compared to the 19 M$_{}$ estimated by [32]."
],
[
"Effect of Varying Parameters",
"We now delve into the effect our choices of $\\Delta \\alpha $ , $C$ , $T_e$ , $f$ , $Z$ , and geometry have on the final electron density and mass estimates.",
"We start with our measurement of $\\Delta \\alpha $ since the values are constrained by the radio observations.",
"In Figure REF we plot Equations REF and REF as functions of $\\Delta \\alpha $ (via $\\tau $ ).",
"If $\\Delta \\alpha $ is varied from 0.1 to 0.3, typical of the $\\Delta \\alpha $ values found near the middle of Cas A, the electron density and mass estimates change by a factor of two (3.3 to 6.6 cm$^{-3}$ and 0.31 to 0.62 M$_{}$ , respectively).",
"Even if “very strong” absorption ($\\Delta \\alpha $ =0.4) were observed uniformly across the center of Cas A, the maximum mass estimate would still be below 1 M$_{}$ .",
"Also shown in Figure REF are Equations REF and REF plotted as functions of clumping.",
"Recall that $C=1$ indicates no clumping, while smaller values of $C$ increase the amount of clumping in the ejecta.",
"As clumping increases, the electron density increases and the total mass estimate decreases.",
"For all modest clumping estimates, the electron density remains below 10 cm$^{-3}$ and the total mass remains below 1 M$_{}$ .",
"Figure: Plots of Equation (top) and Equation (bottom) as functions of Δα\\Delta \\alpha (via τ\\tau ) and clumping factor.For each curve, one of the variables are held constant as indicated.",
"Each ofthe xx-axes is independent with Δα\\Delta \\alpha indicated on the bottomxx-axis and clumping (CC) indicated on the top xx-axis in each plot.Figure: Plots of Equation (top) and Equation (bottom) as functions of T e T_e, ff, and ZZ.",
"For each curve, two of thevariables are held constant as indicated.",
"Each of the xx-axes isindependent with temperature indicated on the bottom xx-axis and atomicnumber (ZZ) and electron/ion ratio (ff) indicated on the top xx-axis ineach plot.The values of $T_e$ , $f$ , and $Z$ depend on emission models and observations that may be affected by components that are missing because we have no probe for them, such as higher ionization states of Si [14].",
"To illustrate the effect of varying these quantities, we plot Equations REF and REF in Figure REF .",
"For each of the plotted curves, two of the values are held constant as indicated.",
"All reasonable estimates of $T_e$ , $f$ , and $Z$ require that the electron density be less than 10 cm$^{-3}$ .",
"While the total unshocked ejecta mass is much more strongly influenced by temperature and electron/ion ratio than atomic number, plausible variations in all three variables constrain the mass below 1 M$_{}$ .",
"The final effect we would like to address is that of the geometry.",
"While the two thick sheet geometry for the unshocked ejecta is supported by the morphology of the infrared [Si2] and [O4] emission, and the correlation between the $\\Delta \\alpha $ map and [Si2] is quite good, this is not a one-to-one correlation.",
"Other species possibly in different locations along the line-of-sight might be responsible for some of the observed thermal absorption.",
"Therefore, we consider here a spherical geometry in which the unshocked ejecta are modeled as a filled sphere of absorbing material interior to the bright radio-emitting ring.",
"Since the emission measure derives from the $\\Delta \\alpha $ measurement, it does not change, however the path length through the absorbing medium does.",
"If we use $L=2.96$ pc through the center of Cas A so that $R=L/2=1.48$ pc as before, and we assume that $n_e$ is constant, then the electron density from Equations REF and REF becomes 0.98 cm$^{-3}$ and the total mass of the unshocked ejecta becomes 1.12 M$_{}$ as derived in Equation REF .",
"A clumping factor would increase the electron density and decrease the total mass estimate."
],
[
"Improvement over Previous Mass Estimate",
"As mentioned previously, [32] estimated a total mass of 19 M$_{}$ for the unshocked ejecta.",
"The dominant reason for this high mass value is that the free-free optical depth equation used by [32] was for a hydrogenic gas, rather than an oxygenic gas, as appropriate for a core-collapse SNR.",
"In addition, [32] made a number of assumptions about other model parameters that significantly affected their final mass values.",
"Specifically, their temperature was 1000 K compared to our value of 300 K. They used their maximum $\\Delta \\alpha $ of 0.3 compared to our more modest estimate of 0.15.",
"They correctly assumed the ejecta were dominated by oxygen, but incorrectly assumed that the oxygen was singly ionized.",
"[32] also assumed a spherical ejecta distribution rather than our two-thick-sheet morphology.",
"Thus, the confluence of parameter values chosen, and using an incorrect formula for free-free optical depth, resulted in a very high mass estimate in the earlier work."
],
[
"Comparison with Theoretical Expectations for Density and Mass",
"The density and temperature conditions in the unshocked ejecta and the total mass of this material play a key role in many aspects of supernova remnant physics.",
"[37] and [27], [28] have incorporated results from their analysis of the million-second observation of Cas A with Chandra into hydrodynamic simulations of supernova remnant evolution that account for factors such as expansion into a stellar wind and acceleration of the reverse shock.",
"Their models, designed to match the dynamics of Cas A, result in a range of total and shocked ejecta masses, which are then partitioned into the various elemental species based on their derived abundances from the X-ray data.",
"Their most recent prediction for the total mass of ejecta that are currently unshocked is 0.18-0.3 M$_{}$ , depending on the model chosen.",
"Which species dominate the composition of the unshocked ejecta is debatable since the explosion is known to have been asymmetric [45].",
"[28] argue that there is little Fe left in the unshocked ejecta based on several factors including the lack of infrared Fe emission from the unshocked ejecta [49], the location of the X-ray-emitting Fe-rich ejecta at the same or larger radius than the Si-rich ejecta [26] and the carbon atmosphere of the neutron star [25].",
"Our estimate of 0.39 M$_{}$ for the unshocked ejecta is near the upper limit predicted by [28], however reasonable variations of our parameters, such as increasing the clumping factor, can lead to lower values of total unshocked ejecta mass in line with the prediction.",
"We can also compare the density of the unshocked ejecta to the hydrodynamic model of [28].",
"Their ejecta model assumes a uniform density core surrounded by a power-law envelope.",
"The density of the core as a function of time is given by: $\\rho _{\\mathrm {core}}=\\left(\\frac{M_{ej}}{v_{\\mathrm {core}}^3}\\right)\\left(\\frac{3}{4 \\pi }\\right)\\left(\\frac{n-3}{n}\\right)t^{-3}$ where $M_{ej}$ is the total mass of the ejecta and the factor $(n-3)/n$ gives the fraction of the ejecta mass in the core [37], [28], [51].",
"$v_{\\mathrm {core}}$ is the velocity of the boundary between the core and the power-law envelope.",
"For an $n=10$ power-law envelope and $M_{ej}=$ 3 M$_{}$ , [28] report $v_{\\mathrm {core}}\\approx 9000$ km s$^{-1}$ .",
"At an age of 330 years, $\\rho _{\\mathrm {core}}\\approx 1\\times 10^{-24}$ g cm$^{-3}$ .",
"Our “filled sphere” geometry provides a convenient comparison because it, too, assumes a uniform density ejecta distribution throughout the interior of the SNR with no clumping.",
"For $n_e=0.98$ cm$^{-3}$ , $f$ =2.5, and $Z$ =8.34, we derive $\\rho _{\\mathrm {sph}}=5.5\\times 10^{-24}$ g cm$^{-3}$ , which is only a factor of 5 larger than [28].",
"The ejecta in Cas A are distributed in a complex fashion and the unshocked ejecta show signs of filamentary behavior in the high spectral resolution Spitzer data [29].",
"For any given emission measure, if the material is tied up in clumps, the final density estimate will be higher and the final mass estimate would be lower than when the material is distributed uniformly along the line-of-sight.",
"In order to match our mass estimate with that of [28], this would argue for the unshocked ejecta being primarily distributed in the two-shell geometry.",
"Any unshocked C-rich or Fe-rich ejecta must then be of such low density that it makes little contribution to the observed thermal absorption – or one or both of these components are missing entirely.",
"Our calculations favor a low density and low temperature environment in the unshocked ejecta consistent with the analysis of the Spitzer infrared data [14].",
"Temperatures $\\lesssim $ 500 K are not unexpected given the rapid expansion of the SNR, which could conceivably cool the ejecta to $\\sim $ 10s of Kelvin, and the fact that the infrared flux densities infer unshocked dust temperatures of about 35 K [41].",
"Expectations for the density of the unshocked ejecta are not clear.",
"While hydrodynamic models can predict an average density, as in Equation REF , the clumping of the ejecta renders the average somewhat meaningless.",
"If the unshocked ejecta we detect via thermal absorption and mid-infrared emission eventually radiate strongly in the X-rays upon shock heating, then our density estimate is in line with that expectation.",
"The electron densities observed in the X-ray-emitting (shocked) ejecta are $\\sim $ 10s cm$^{-3}$ which would be consistent with the standard strong shock compression ratio of 4.",
"On the other hand, much more dense (1000s cm$^{-3}$ ) shocked ejecta are observed optically [9] indicating that the unshocked ejecta must have a much denser, clumped component in order to reproduce the observed range of shocked ejecta densities [39].",
"A high degree of clumping might arise due to the instabilities associated with the explosion reverse shock which forms inside of the star in the first few hours after the explosion begins [24], [30].",
"Similarly, turbulence and strong clumping are associated with expanding Fe-Ni bubbles in the inner ejecta [38], [5].",
"The expansion of Fe-Ni bubbles naturally renders the innermost Fe ejecta of very low density, consistent with the lack of detected infrared Fe emission in the unshocked ejecta."
],
[
"Comparison to SN1006",
"The only other confirmed detection of unshocked ejecta in an SNR is that of SN1006 where high-velocity absorption features were observed in ultraviolet lines of Fe2 and Si2 [60], [20].",
"High resolution spectra showed no detectable unshocked Si3 or Si4 and thus the bulk of the unshocked Si is accounted for with Si2 absorption [57], [21].",
"The total Si mass calculated from the derived column densities (0.25 M$_{}$ ) is roughly comparable with the Si mass predicted from some white dwarf carbon deflagration models (0.16 M$_{}$ ) and the derived Si mass from X-ray data (0.2 M$_{}$ ) [20].",
"Accounting for the unshocked Fe in SN1006 has proven to be more difficult.",
"[20] argue that the unshocked Fe absorption should be dominated by Fe2 because the photoionization cross section is five times that of Si2.",
"However, the total mass in Fe2 (0.029 M$_{}$ ) is less than 1/10 of that expected from carbon deflagration models (0.3 M$_{}$ ) [19], [20].",
"The absence of Fe2 absorption along some lines-of-sight indicate that there is likely no large-scale mixing or overturning of Fe ejecta, so the bulk of the Fe must still be interior to the reverse shock [57].",
"There is the possibility that the Fe is tied up in higher ionization states [19], but then the lack of Si3 and Si4 absorption from the unshocked ejecta is puzzling [20].",
"Therefore, it seems that some of the unshocked ejecta in SN1006 are still unaccounted for.",
"In Cas A, there is a similar “problem” between the strong emission from [Si2] and the lack of any detectable infrared Fe emission from the unshocked ejecta in the Spitzer IRS bands or the ISO LWS spectra.",
"At the density we infer for the unshocked ejecta, [Si2] should be much stronger than [Fe2] and [Fe3], however [Fe5] should be comparable to the [Si2] [14], which is not observed [49].",
"As explained in §REF , the Fe discrepancy in Cas A may be resolved if the bulk of the Fe ejecta have already passed through the reverse shock and the remaining interior Fe is of lower density than the unshocked Si- and O-rich ejecta.",
"However, we need a better knowledge of Cas A's progenitor and the expected Fe yields for a Type IIb explosion of such a star in order to determine if all of the Fe ejecta in Cas A are truly accounted for.",
"One other comparison we can make to SN1006 is of the density in the unshocked ejecta.",
"[20] estimate $\\rho _{\\mathrm {FeII}}\\sim 4\\times 10^{-27}$ g cm$^{-3}$ at the center of their unshocked ejecta profile with an estimate of about half that figure for the current unshocked mass density of Si2.",
"If we scale the mass density of the (O-rich) unshocked ejecta in Cas A by 27 to account for the difference in age of the two SNRs and thus the dilution of the density due to expansion, then $\\rho _{\\mathrm {sph}}^{1000}=2\\times 10^{-25}$ g cm$^{-3}$ .",
"Therefore the observed unshocked ejecta in Cas A are about 100 times more dense than in SN1006.",
"Given that Cas A is the result of a Type IIb explosion and SN1006 resulted from a Type Ia explosion, stark differences in ejecta density are to be expected due to differing stellar compositions, explosion mechanisms and resulting nucleosynthesis, initial ejecta density profiles, instabilities in the ejecta that may lead to clumping, and circumstellar environments.",
"The question is are the unshocked ejecta in Type Ia SNRs systematically less dense than in core-collapse remnants?",
"The extension of low frequency radio analysis to other SNRs could be used to test this hypothesis.",
"Furthermore, since the absorption is frequency dependent, working at frequencies lower than 74 MHz will allow the detection of lower density ejecta in Cas A and other SNRs."
],
[
"Implications for Lower Frequency Spectrum",
"[23] revisited the low frequency, integrated spectrum of Cas A utilizing legacy VLA and Long Wavelength Development Array 74 MHz measurements.",
"We consider here how the thermal absorption discussed in this paper relates to the turnover in the integrated spectrum of Cas A at much lower frequencies, as originally documented by [4].",
"Given the current limited angular resolution of observations towards Cas A below 74 MHz, our discussion is necessarily limited.",
"Examination of the [4] Figure 1a indicates the integrated spectrum of Cas A begins to turn over below 38 MHz.",
"The [4] modeled spectrum (epoch 1965) which did not fit for a turnover, predicts $\\sim $ 93,000 Jy at 10 MHz.",
"The lowest reliable flux density measurement at that frequency is the Penticton 10 MHz flux density of 28,000 Jy (epoch 1965.9, i.e.",
"contemporaneous, [6]), which is a factor of 3.3 below the expected value.",
"(This corresponds to a 10 MHz free-free optical depth by a uniform external absorber of $\\sim $ 1.2, an oversimplification since some of the absorption must be internal.)",
"As described in Section REF , an average free-free optical depth for the unshocked ejecta at 74 MHz is $\\sim $ 0.5, which rapidly inflates to $\\gg $ 1 at 10 MHz, i.e.",
"completely opaque.",
"If the absorbing ejecta blocked $\\sim $ 30% of the back surface of Cas A's nonthermal emission, $\\sim $ 85% of Cas A's 10 MHz emission, or $\\sim $ 79,000 Jy, should escape the remnant.",
"We can account for the remaining deficit by an intervening thermal absorber with $\\tau _{10\\,\\mathrm {MHz}} \\sim -\\ln (28,000/79,000) \\sim 1$ .",
"Thus, a simplistic scenario has $\\sim $ 79,000 Jy of Cas A's 10 MHz emission emerging from the remnant after being attenuated by $\\sim $ 15% from the unshocked ejecta.",
"Thereafter the emerging emission is further attenuated by an intervening ionized component of the ISM with 10 MHz optical depth $\\sim $ 1.",
"The distribution of low density gas in the interstellar medium was constrained by [34], using the observed, patchy thermal absorption inferred from the low frequency, integrated spectra of Galactic SNRs.",
"(Several of those systems have since been resolved with legacy VLA data [36], [7].)",
"The measured optical depths at 30.9 MHz typically ranged from 0.1-1, though some remnants showed no turnovers while others were more heavily absorbed.",
"The nature of the absorption was consistent with higher frequency recombination line observations that postulated extended H2 region envelopes (EHEs) associated with normal H2 regions [1].",
"The inferred ISM 10 MHz free-free optical depth unity discussed above, and required to further attenuate Cas A's emerging emission, scales to $\\sim $ 0.1 at 30.9 MHz.",
"Thus the low frequency turnover in the integrated spectrum of Cas A is qualitatively consistent with a combination of intrinsic absorption from unshocked ejecta and extrinsic absorption by EHEs located along the line of sight.",
"In retrospect the [4] low frequency turnover would be conspicuous in its absence based on what we now know about ionized gas in both the ISM and Cas A."
],
[
"Future Work",
"The radio detection of thermally absorbing, unshocked ejecta inside a young Galactic SNR by the legacy VLA 74 MHz system only scratches the surface of possibilities for emerging low frequency systems with greater capabilities.",
"With the transition to the fully digital electronics and new correlator of the VLA, the narrowband 74 and 330 MHz legacy systems have been replaced with an improved, broad-band “Low Band” receiving system [10].",
"The new, single receiver system brings significant improvements over its two narrow band predecessors, and will become available to the VLA user community in 2014.",
"For broad band systems probing below 74 MHz, e.g.",
"LOFAR Low Band [54], or LWA, the frequency dependence of thermal absorption significantly enhances the effect.",
"Gradually tuning to lower frequencies as the absorption grows can break the optical depth degeneracy between temperature and emission measure, and deconvolve the radial superposition of nonthermal emitting and thermal absorbing regions.",
"Lower frequency observations with sufficient angular resolution (e.g.",
"LOFAR Low Band) can test our qualitative assessment of the relative contributions of intrinsic and extrinsic thermal absorption contributing to the observed turnover in Cas A's integrated spectrum below 38 MHz.",
"It is also important to constrain the emission at frequencies above 74 MHz just prior to the onset of absorption, as offered e.g.",
"by LOFAR High Band, MWA, the GMRT, and VLA Low Band.",
"[23] also confirmed shorter scale (5-10 yr) temporal variations at 74 MHz consistent with previous reports in the literature for low frequencies [15], [59].",
"These have generally been attributed to the interaction of the forward shock with an inhomogeneous CSM or ISM.",
"However some variation must also arise from inside Cas A, as the reverse shock advances into the unshocked ejecta and affects its opacity.",
"Together with the secular decrease due to adiabatic expansion [48], the overall temporal behavior is necessarily complex, and requires higher resolution, broad band low frequency measurements to understand.",
"We suggest that Cas A's low frequency flux density should be monitored routinely across a range of lower frequencies; for instruments such as LOFAR or LWA this would require only a few minutes of observations every few weeks or months and would allow a much better determination of the range of timescales over which such variations occur."
],
[
"Conclusions",
"We have imaged Cas A from 5 GHz to 74 MHz in all four configurations of the Legacy VLA with follow-up observations at 74 and 330 MHz with the legacy VLA+PT link.",
"Our spatially resolved spectral index maps confirm the interior spectral flattening measured earlier, but at higher signal-to-noise and resolution.",
"Comparison with Spitzer infrared spectra confirms the earlier hypothesis that the spectral flattening is due to thermal absorption by cool, unshocked ejecta photoionized by X-ray radiation from Cas A's reverse shock.",
"We use the spectral flattening to measure the free-free optical depth.",
"Next, using priors of electron temperature, atomic number, and electron to ion ratios, we derive an emission measure from the measured optical depth.",
"With an assumed geometry, informed from three-dimensional modeling based on higher frequency studies, we use the emission measure to place constraints on both the density and total mass of the unshocked ejecta.",
"We consider modest, physically plausible variations in both our priors and the assumed geometry, and find that the effect on the total mass is relatively modest, varying by a factor of about two.",
"Furthermore, our derived total mass is consistent with recent model predictions [28].",
"After accounting for the relative ages of Cas A and SN1006, our derived mass density is much higher than found in SN1006, not unexpected since Cas A (Type IIb) and SN1006 (Type Ia) emerged from two fundamentally different supernova explosion types.",
"However, if there is a systematic difference in unshocked ejecta density for core collapse vs.",
"Type Ia SNRs, low frequency radio data can be used to test this hypothesis.",
"Finally, we consider the contribution of the intrinsic thermal absorption to the known turnover of Cas A's integrated spectrum at much lower frequencies.",
"We find that the intrinsic thermal absorption from the unshocked ejecta, combined with extrinsic absorption from a known, patchy distribution of low density ISM gas, are completely consistent with the low frequency turnover.",
"The promise of the emerging instruments is expanding the population of SNRs, young and old, that can be probed for intrinsic and extrinsic thermal absorption and shock acceleration variations beyond pathologically bright sources like Cas A.",
"More generally, the seemingly ubiquitous detection of resolved thermal absorption by the 74 MHz legacy VLA against the Galactic background [40] and towards, discrete non thermal sources [36], [7], [8] confirms the phenomena will continue to emerge as a powerful tool for low frequency astrophysics.",
"The VLA is operated by the National Radio Astronomy Observatory, which is a facility of the National Science Foundation, operated under cooperative agreement by Associated Universities, Inc. All sub-GHz systems on the VLA have been developed cooperatively between the National Radio Astronomy Observatory and the Naval Research Laboratory.",
"Basic research in radio astronomy at the Naval Research Laboratory is supported by 6.1 base funding.",
"Partial funding for this research at West Virginia Wesleyan College was provided by Chandra Grant GO0-11089X and by the NASA-West Virginia Space Grant Consortium.",
"Facility: VLA"
],
[
"Derivation of $\\tau -\\Delta \\alpha $ Relation",
"The definition of spectral index between 74 and 330 MHz is $\\alpha _{74}^{330}=\\frac{\\log (S_{74}/S_{330})}{\\log (74/330)},$ where $\\alpha $ is negative for nonthermal emission.",
"We also define $a=\\log \\left(\\frac{74}{330}\\right)$ .",
"Assuming no curvature in the emitted synchrotron spectrum, we expect ($e$ ) the spatially resolved spectral index $\\alpha $ to remain unchanged from above 330 MHz to between 330 and 74 MHz, defining $\\alpha _e$ .",
"$\\alpha _o$ is then defined as the observed ($o$ ) spectral index due to absorption.",
"The difference between observed and expected spectral index is then: $\\Delta \\alpha =\\alpha _o-\\alpha _e$ ($\\Delta \\alpha $ is positive), so that: $\\Delta \\alpha = \\frac{\\log (S_{o74}/S_{330}) - \\log (S_{e74}/S_{330})}{a} = \\frac{\\log (S_{o74}/S_{e74})}{a}$ To obtain observed and expected 74 MHz flux density, we assume equal synchrotron emission from the front and back sides ($S_f=S_b$ ) of Cas A: $S_e=S_f + S_b = 2S_f\\\\S_o=S_f+S_f e^{-\\tau } = (1+e^{-\\tau })S_f\\\\\\frac{S_o}{S_e}=\\frac{1+e^{-\\tau }}{2}$ Substituting for $\\frac{S_o}{S_e}$ we have: $a \\Delta \\alpha = \\log \\left(\\frac{S_{o74}}{S_{e74}}\\right)=\\log \\left(\\frac{1+e^{-\\tau }}{2}\\right)\\\\2 (10^{a \\Delta \\alpha }) = 1 + e^{-\\tau }\\\\\\tau = -\\ln [2 (10^{a \\Delta \\alpha } -0.5)]$ We note that Equation REF is slightly different than that published in [32] due to a typographical error."
],
[
"Calculation of Free-Free Optical Depth $\\tau $",
"In this section, we follow the derivation in chapter 9 of [46].",
"The definition of free-free optical depth ($\\tau $ ) is: $\\tau _{\\nu }=-\\int \\kappa _{\\nu } dl$ where $\\kappa _{\\nu } = \\frac{4 Z^2 e^6}{3 c} \\frac{n_i n_e}{\\nu ^2} \\frac{1}{\\sqrt{2 \\pi (m k T_e)^3}} \\left<g_{ff}\\right>$ is the absorption coefficient and the Gaunt factor is $\\left<g_{ff}\\right> = \\ln \\left[\\left(\\frac{2 k T_e}{\\gamma m_e}\\right)^{3/2} \\frac{m_e}{\\pi \\gamma Z e^2 \\nu }\\right].$ $Ze$ is the ion charge, $n_i$ and $n_e$ are number density of ions and electrons, respectively.",
"$c$ is the speed of light, $\\nu $ is the observation frequency, $m_e$ is the electron mass, $k$ is Boltzmann's constant, $\\gamma =1.781$ , and $T_e$ is the effective temperature of the gas which must be $>20$ K. For classic H2 regions one normally assumes a hydrogenic gas where $Z=1$ and $n_e=n_i$ , but that is not the case here [58].",
"Let $f=n_e/n_i$ and use emission measure $=EM \\equiv \\int n_e^2\\,dl$ and express $T_e$ in K, $\\nu $ in MHz, and $EM$ in pc cm$^{-6}$ so that: $ \\tau _{\\nu }=3.014\\times 10^{4} \\left(\\frac{Z^2}{f}\\right)\\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^{-2}\\left(\\frac{T_e}{\\mathrm {K}}\\right)^{-3/2}\\left(\\frac{EM}{\\mathrm {pc\\,cm}^{-6}}\\right)\\ln \\left[49.55 Z^{-1}\\left(\\frac{T_e}{\\mathrm {K}}\\right)^{3/2} \\left(\\frac{\\nu }{\\mathrm {MHz}}\\right)^{-1}\\right].$"
],
[
"Full Parameterized Equations for Electron Density and Total Mass",
"The electron density follows from Equation REF and our preferred geometry as $n_e=\\sqrt{EM/CL}$ .",
"Substituting Equation REF into this expression yields: $n_e = \\frac{10.4 \\left(\\frac{\\tau _{\\nu }}{0.51}\\right)^{1/2}\\left(\\frac{f}{2.5}\\right)^{1/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/4}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)C^{-1/2}\\left(\\frac{L}{0.16\\,\\mathrm {pc}}\\right)^{-1/2}}{\\left\\lbrace \\ln \\left[418.3\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)^{-1}\\right]\\right\\rbrace ^{1/2}}$ where $n_e$ is in units of cm$^{-3}$ .",
"The total mass in unshocked ejecta is calculated from $M=(Zm_pn_e/f)(\\pi C^3 R^2 L)$ , where $m_p$ is proton mass.",
"Substituting Equation REF for $n_e$ and absorbing constant terms results in the following equation for total mass in solar masses: $M = \\frac{0.97 \\left(\\frac{\\tau _{\\nu }}{0.51}\\right)^{1/2}\\left(\\frac{f}{2.5}\\right)^{-1/2}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/4}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)C^{5/2}\\left(\\frac{R}{1.48\\,\\mathrm {pc}}\\right)^2\\left(\\frac{L}{0.16\\,\\mathrm {pc}}\\right)^{1/2}}{\\left\\lbrace \\ln \\left[418.3\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)^{-1}\\right]\\right\\rbrace ^{1/2}}.$ If a spherical geometry is assumed, then $M=(Zm_pn_e/f)(\\frac{4}{3}\\pi C^3 R^3)$ which becomes: $M = \\frac{2.73 \\left(\\frac{\\tau _{\\nu }}{0.51}\\right)^{1/2}\\left(\\frac{f}{2.5}\\right)^{-1/2}\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/4}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)C^{5/2}\\left(\\frac{R}{1.48\\,\\mathrm {pc}}\\right)^3\\left(\\frac{L}{2.96\\,\\mathrm {pc}}\\right)^{-1/2}}{\\left\\lbrace \\ln \\left[418.3\\left(\\frac{T_e}{300\\,\\mathrm {K}}\\right)^{3/2}\\left(\\frac{Z}{8.34}\\right)^{-1}\\left(\\frac{\\nu }{73.8\\,\\mathrm {MHz}}\\right)^{-1}\\right]\\right\\rbrace ^{1/2}}.$"
]
] | 1403.0032 |
[
[
"Vlasov equation and $N$-body dynamics - How central is particle dynamics\n to our understanding of plasmas?"
],
[
"Abstract Difficulties in founding microscopically the Vlasov equation for Coulomb-interacting particles are recalled for both the statistical approach (BBGKY hierarchy and Liouville equation on phase space) and the dynamical approach (single empirical measure on one-particle $(\\mathbf{r},\\mathbf{v})$-space).",
"The role of particle trajectories (characteristics) in the analysis of the partial differential Vlasov--Poisson system is stressed.",
"Starting from many-body dynamics, a direct derivation of both Debye shielding and collective behaviour is sketched."
],
[
"Introduction",
"Plasmas are many-body systems, in which the long-range Coulomb interactions play the leading role.",
"Because of the long range coupling, each particle interacts with many other particles, so that it seems natural to attempt describing this medium in a mean-field limit.",
"This approach has proved successful as it provides a theoretical frame for computing various phenomena, in quantitative agreement with physical observation – for natural and for laboratory plasmas.",
"However, because the Coulomb field diverges strongly near its source, a complete derivation of the Vlasov–Poisson equations from first principles in Coulomb systems is yet to be constructed (see [24], [26] and references therein).",
"We recently tried to recover some vlasovian results starting from the $N$ -body picture, and taking the $N \\rightarrow \\infty $ limit after solving approximately the dynamics [15].",
"While this yields an alternative derivation for some Vlasov-based results, it does not provide a direct validation of the Vlasov–Poisson system in the continuum limit – actually, it bypasses the Vlasov picture by leading also easily to the analysis of collisions, and shows moreover how Debye shielding alters the Poisson description of the interactions [16].",
"In this paper, we discuss some of the respective merits of the partial differential equation viewpoint inherent to the Vlasov equation and of the ordinary differential equation viewpoint of many-body Coulomb force dynamics.",
"We hope this discussion will help the reader to grasp how delicate foundational issues the Vlasov equation involves.",
"While our selection of topics is bound to remain far from exhaustive, we hope it will refresh the reader's view of the Vlasov equation.",
"Specifically, Sect.",
"focuses on the meaning of the entity obeying the Vlasov equation, namely the distribution function.",
"Section comments on the role of particle trajectories in the theory of Vlasov–Poisson systems.",
"Section sketches a specific instance where attention to trajectories provides better insight into well-known vlasovian phenomena – this is the only section where a specific microscopic model (actually, jellium) is considered."
],
[
"Status of the distribution function and its evolution",
"Major merits of the Vlasov equation include a rather successful application to the modeling of hot plasmas, enabling physicists to use powerfulThe standard toolbox of the plasma physicist would be dramatically damaged by the removal of partial differential equations and the associated techniques – even the most “elementary” ones like separation of variables.",
"techniques of partial differential equations to address physical phenomena, contributions to progress in functional analysisThis progress makes the modern theory of partial differential equations incommensurately more powerful, benefiting to many other fields.",
"motivated by challenges it raises, stressing an interesting stage (in order to formulate a physical problem, one needs not merely to write its equations but first to choose the mathematical entities called to play) for describing the physics.",
"Issue REF indicates how much needed is a comprehensive foundation for this fruitful model.",
"We address issue REF (and indirectly issue REF ) in Sect. .",
"Let us stress in the current section the conceptually crucial issue REF : the central quantity in the classical theory of the Vlasov equation is a non-negative continuous function $f$ on Boltzmann's $\\mu $ -space [38], $f(\\mathbf {r},\\mathbf {v},t)$ , which in the simple Coulomb interaction case obeys a closed evolution equationThe Poisson equation is not an evolution equation : it simply provides an explicit relation expressing the electric field in terms of the current, instantaneous $f$ – in this sense, the electric field does not describe additional degrees of freedom, but is a mere subsidiary entity slaved to the particles.."
],
[
"Pedestrian approach",
"Classical, continuously differentiable distribution functions $f(\\mathbf {r},\\mathbf {v},t)$ are often interpreted in terms of some averages.",
"Since the particles are completely described by their positions and velocities, $(\\mathbf {r}_j(t),\\mathbf {v}_j(t))$ , one first associates with these data a “spiky” distribution, involving Dirac distributions on $(\\mathbf {r},\\mathbf {v})$ space.",
"This is too wild an object for simple calculations, and one would gladly smoothe it.",
"Moreover, as the field acting on particles varies in space, distinct particles are subject to different accelerations, and their motions may significantly separate with time.",
"Yet, while particles drift apart from each other, other particles may come around them in such a way that the overall distribution does not seem to change much : recall a cloud made of many tiny drops, or the large-scale hydrodynamics of air made of many molecules.",
"Simultaneously, the field generated by the particles varies wildly on microscopic scales, so that one expects particle motions to be hardly accessible to the theory on these scales.",
"Therefore, one would also gladly limit the particle motion analysis to large scales, over which the Coulomb field might appear smoother.",
"A first smoothing procedure is simply to replace the Coulomb interaction (with point sources) by a regularized, or mollified [36], interaction, where the source of the Coulomb field is a sphere with finite charge density centered on the point particle.",
"The force on a particle is then also computed by summing the electric field over the sphere.",
"This is done in particle-based numerical schemes, and if one keeps a fixed regularizing form function, one can even obtain the Vlasov equation from the $N$ -body model rigorously (see e.g.",
"[36]).",
"However, the derivation of the Vlasov equation depends crucially on the size of the mollifying sphere.Physically, one expects the classical model to fail anyway at describing Coulomb-interacting particles at very short distances.",
"Indeed, the “classical radius of the electron” (at which scale the point particle model should break down) is far smaller than its Compton length, so that quantum effects are expected to imply a different modeling.",
"It may thus be considered physically sensible to model the $N$ -body plasma with a “quantum-mollified” Coulomb interaction (even for large, finite $N$ ).",
"This will seem even more harmless since the quantum description of two-body interaction [6] leads to a scattering cross section identical to the classical Rutherford cross section if particles bear charges with equal signs – moreover, the same expression applies to particles with opposite signs, for which the classical picture allows for arbitrarily close approaches.",
"Besides, physicists most often consider plasmas where typical binary collisions involve closest approaches much further away than the Compton length, and molecular dynamics simulations simply integrate the Coulomb force for short range while smoothing the interaction for longer range [9].",
"Yet the mathematical issue is important, for it should provide insight into more complicated problems too.",
"A second procedure to formalize the “cloud” picture is to deem irrelevant some subtleties of particle motion.",
"Rather, one pays attention to the evolution of the particles currently in a “mesoscopic” domain $\\Delta U_\\mathbf {r}$ (with size $| \\Delta U_\\mathbf {r} | = \\Delta x \\Delta y \\Delta z$ ), with velocities in a similar range $\\Delta U_{\\mathbf {v}}$ (with $| \\Delta U_\\mathbf {v} | = \\Delta v_x \\Delta v_y \\Delta v_z$ ).",
"The distribution function is then used to compute the “coarse-grained” distribution $| \\Delta U_\\mathbf {r} \\times \\Delta U_\\mathbf {v} |^{-1} \\int _{\\Delta U_\\mathbf {r} \\times \\Delta U_\\mathbf {v}}f(\\mathbf {r},\\mathbf {v},t) \\, {\\mathrm {d}}^3 \\mathbf {r} \\, {\\mathrm {d}}^3 \\mathbf {v}$ , where the range and domain of interest are large enough to contain so many particles (say $Q \\gg 1$ ) that their number would fluctuate moderately with time (say on the smaller scale $Q^{1/2}$ if these fluctuations follow a central-limit type of scaling, or $Q^{2/3}$ for a surface-vs-volume scaling in position space, or $Q^{5/6}$ for a boundary-vs-volume scaling in $(\\mathbf {r},\\mathbf {v})$ -space).",
"To extract a smooth function $f$ from this fluctuating picture, a third procedure consists in introducing an “ensemble” of realizations of the plasma, see e.g.",
"[1], [23], [21].",
"The function $f$ in the Vlasov equation is then viewed as the average of individual spiky distributions.",
"As the force acting on a particle is due to the field generated by all other particles, this force is expressed as an integral over the distribution of those source particles, from which the target particle is excluded.",
"This leads to computing a field $\\mathbf {E}(\\mathbf {r}_1,t)$ from an integral over the two-particle joint distribution $f^{(2)} (\\mathbf {r}_1, \\mathbf {v}_1, \\mathbf {r}_2, \\mathbf {v}_2, t)$ , in the spirit of derivations of the Boltzmann equation in gas theory.Until 2013, rigorous derivations of the Boltzmann equation were limited to short times (on the order of the mean free path) or small initial data (ultimately expanding in vacuum without collision).",
"This is a topic of hard and active research, see [20], [40] and references therein.",
"Because of the long range nature of the Coulomb interaction, one expects particles to be “almost” independent and $f^{(2)}$ to almost factorize – otherwise, one should solve an evolution equation for $f^{(2)}$ where the source involves $f^{(3)}$ , etc...",
"The resulting set of equations, viz.",
"the BBGKY hierarchy, stands at the core of the statistical approach below.Note that, in the $N$ -body case, giving the spiky distribution amounts to locating all particle, and the joint distribution for pairs is uniquely determined by these data.",
"In contrast, given a smooth distribution $f^{(1)}$ on ${\\mathbb {R}}^6$ , there exist infinitely many joint distributions $f^{(2)}$ on $R^{12}$ such that $\\int f^{(2)} (\\mathbf {r}_1, \\mathbf {v}_1, \\mathbf {r}_2, \\mathbf {v}_2, t)\\, {\\mathrm {d}}^3 \\mathbf {r}_2 {\\mathrm {d}}^3 \\mathbf {v}_2= f^{(1)} (\\mathbf {r}_1, \\mathbf {v}_1, t)$ along with $f^{(2)} (1,2) = f^{(2)} (2,1)$ .",
"The factorization $f^{(2)} (1,2) = f^{(1)}(1) f^{(1)}(2)$ may hold at initial time but is generally not preserved by the time evolution.",
"This “propagation of (initial) chaos” (or “Kac property”) is an important, non-trivial issue, for understanding the $N \\rightarrow \\infty $ limit [31].",
"Let us stress again that the usual conceptual setting for these derivations involves probabilistic averages over ensembles to generate smooth functions.",
"Yet a physical plasma is a single realisation of the possible plasmas considered in an ensemble.",
"The particles in it do not respond to the average field generated by the ensemble, and each particle follows a single, regular enough, trajectory (which is not a diffusion process).",
"An additional difficulty met with continuous distribution functions $f$ solving the Vlasov equation is the absence of H-theorem.",
"Indeed, the Vlasov equation preserves all functionals of the form $\\int _{{\\mathbb {R}}^6} G(f(\\mathbf {r}, \\mathbf {v})) {\\mathrm {d}}^3 \\mathbf {r} {\\mathrm {d}}^3 \\mathbf {v}$ , and evolutions towards a kind of equilibrium can only lead to the formation of finer and finer filaments rippling the surface representing $f$ over $(\\mathbf {r}, \\mathbf {v})$ space.This is also viewed from the fact that the Vlasov equation for hamiltonian dynamics “shuffles” (without altering their volumes) the respective domains over which $f$ takes its various values, i.e.",
"at any time $t$ the set $\\lbrace (\\mathbf {r}, \\mathbf {v}) : a < f(\\mathbf {r}, \\mathbf {v}, t) \\le b \\rbrace $ has the same 6-dimensional volume (see [21] for this shuffling image).",
"For the one-dimensional models, with $(\\mathbf {r}, \\mathbf {v}) = (x,v)$ , volume preservation reduces to area preservation, which makes hamiltonian dynamics so different from dissipative ones.",
"Recall also how mixing occurs in non-diffusive, viscosity-free incompressible fluid flows.",
"When such ripples become finer than a typical interparticle distance in the $N$ -body system, they lose physical significance [18].",
"However, the BBGKY evolution equations do not incorporate such a destruction of unphysical ripples.",
"In numerical simulations, this filamentation is a delicate issue.",
"On the one hand, modeling accurately the partial differential equation requires increasing computational power as filamentation proceeds.",
"On the other hand, the numerical smoothing due to various interpolation schemes is not granted to reproduce the physical smoothing of the distribution function due to phenomena not included in the smooth Vlasov model, such as finite-$N$ effects, perturbing interactions, etc.",
"To conclude, there is a single Vlasov equation, but there are various views of the distribution function whose evolution it is meant to describe.",
"For a given plasma physics problem, which of these views, if any, should be considered ?",
"This issue is usually overlooked, and the outcome of the vlasovian calculation is deemed relevant."
],
[
"Technical approach",
"The statistical approach to deriving the Vlasov equation (see e.g.",
"Appendix A in [23]) starts from $N$ -body dynamics, introduces the high-dimensional phase space $\\Gamma = {\\mathbb {R}}^{6N}$ , and considers the Liouville equation for a statistical measure $\\mathfrak {f}$ on $\\Gamma $ .Given an initial data $z \\in \\Gamma $ , the $N$ -body equation of motion $\\dot{z} = g(z)$ generates the evolution of the $6N$ degrees of freedom so that $z(t) = T_{t,0} (z_0)$ for initial data $z(0) = z_0$ .",
"A statistical measure on $\\Gamma $ is a distribution $\\mathfrak {f}$ , and the Liouville equation $\\partial _t \\mathfrak {f} + \\partial _z \\cdot (g \\mathfrak {f}) = 0$ transports an initial measure $\\mathfrak {f}_0$ for $z_0$ into an evolved measure $\\mathfrak {f}_t = \\mathfrak {f}_0 \\circ T_{0,t}$ (just by finding the preimage at time 0 of the state at time $t$ ).",
"For instance, for free motion, $z = (\\mathbf {r}_1, \\mathbf {v}_1, \\ldots \\mathbf {r}_N, \\mathbf {v}_N)$ , $g(z) = (\\mathbf {v}_1, \\mathbf {0}, \\ldots \\mathbf {v}_N, \\mathbf {0})$ so that $T_{t,0} z = (\\mathbf {r}_1 + \\mathbf {v}_1 t, \\mathbf {v}_1,\\ldots \\mathbf {r}_N + \\mathbf {v}_N t, \\mathbf {v}_N)$ while $\\mathfrak {f}(z, t)= \\mathfrak {f}(\\mathbf {r}_1 - \\mathbf {v}_1 t, \\mathbf {v}_1, \\ldots \\mathbf {r}_N - \\mathbf {v}_N t, \\mathbf {v}_N, 0)$ .",
"This statistical measure may be interpreted as the probability distribution of, say, ${\\mathcal {N}}$ “replicas” (${\\mathcal {N}}\\gg 1$ ) of the $N$ -body system, and the symmetrized one-particle marginalThat is, $f(\\mathbf {r}, \\mathbf {v})= N^{-1} \\sum _{j=1}^N \\int _\\Gamma \\delta ( \\mathbf {r} - \\mathbf {r}_j) \\delta (\\mathbf {v} - \\mathbf {v}_j) \\ \\mathfrak {f}(\\mathbf {r}_1, \\mathbf {v}_1, \\ldots \\mathbf {r}_N, \\mathbf {v}_N, t)\\, \\prod _{l = 1}^N {\\mathrm {d}}^3 \\mathbf {r}_l {\\mathrm {d}}^3 \\mathbf {v}_l$ : integration with the Dirac distribution generates the one-particle distribution for particle $j$ , and the sum symmetrizes over all particles.",
"of $\\mathfrak {f}$ obeys the first equation in the BBGKY hierarchy.",
"One then identifies this symmetrized marginal with a one-particle measure on the “molecular” $\\mu $ -spaceThis space would be the phase space of a particle, if the force field acting on it were known (though possibly time-dependent).",
"However, for the many-body problem, particles interact, so that the phase space is $\\Gamma = \\otimes _{j=1}^N {\\mathbb {R}}^6$ while $\\mu $ -space is just ${\\mathbb {R}}^6$ (see notes 72 and 118 in [12] for a historical gas theory analogue)., and one expresses the force field generating the evolution by some integrals of the two-particle marginal of $\\mathfrak {f}$ .",
"For the Vlasov–Poisson system, one then requires that the electric field in the Vlasov equation solves the Poisson equation where the source is the average of the individual particle distributions generated by all replicas.The source of $\\mathbf {E}$ is not the $N^{-1} \\sum _{j=1}^N \\delta (\\mathbf {r} - \\mathbf {r}_j)$ given by a single plasma realisation, but the density $\\int _{{\\mathbb {R}}^3} f(\\mathbf {r},\\mathbf {v}) {\\mathrm {d}}^3 \\mathbf {v}$ obtained from an average over ${\\mathcal {N}}$ replicas.",
"In this approach, the evolution of $f$ is thus subsidiary to the evolution of many replicas, each of which evolves independently of the ${\\mathcal {N}}-1$ other replicas, and the $f$ of interest in the Vlasov equation is then an average over ${\\mathcal {N}}$ replicas in the limit ${\\mathcal {N}}\\rightarrow \\infty $ .",
"But how can (${\\mathcal {N}}-1$ ) thought-experimental replicas drive the evolution of the single physically realized $N$ -body system ?",
"How should the field acting on a given particle in the physically observed plasma be the by-product of an average over ${\\mathcal {N}}$ Gedanken plasmas ?",
"The dynamical approach to the Vlasov equation [43], [26] starts with an actual system of $N$ bodies, interacting by instant-action-at-a-distance or via dynamical fields (see e.g.",
"[19] for a simple example).",
"For undistinguishable particles, the data of $N$ points in $\\mu $ -space, namely a setNot a sequence where labels do matter !",
"$M = \\lbrace (\\mathbf {r}_1, \\mathbf {v}_1), \\ldots , (\\mathbf {r}_N, \\mathbf {v}_N)\\rbrace $ (thanks to Coulomb repulsion, no particles can be at the same position, hence this set counts exactly $N$ points), is equivalent to the counting measure ${\\mathrm {d}}\\mu ^{(N)}= \\sum _{j=1}^N \\delta (\\mathbf {r}- \\mathbf {r}_j) \\delta (\\mathbf {v} - \\mathbf {v}_j) \\,{\\mathrm {d}}^3 \\mathbf {r}\\,{\\mathrm {d}}^3 \\mathbf {v}$ on ${\\mathbb {R}}^6$ .",
"When particles move with time, the measure also evolves, and for finite $N$ the evolution of $\\mu ^{(N)}$ provides all information on the motion of all particles, as particles cannot swap their identity (exchanging two particle labels would require their trajectories to be discontinuous, or to meet at a same position with the same velocity – which is ruled out by the dynamics).",
"The measure $\\mu ^{(N)}$ determines the force field exactly as the $N$ particle data do, and this field generates the vector field according to which $\\mu ^{(N)}$ is transported.",
"In order to handle the limit $N \\rightarrow \\infty $ , it is convenient to consider the normalized empirical measure $N^{-1} \\mu ^{(N)}$ , which is non-negative and verifies $N^{-1} \\mu ^{(N)}({\\mathbb {R}}^6) = 1$ .",
"Indeed, this makes $\\mu ^{(N)}/N$ formally akin to a probability measure, and indeed one may interpret $N^{-1} \\mu ^{(N)}({\\mathcal {A}})$ as the fraction of the plasma particles which are in some subset ${\\mathcal {A}}\\subset {\\mathbb {R}}^6$ , or as the probability that a particle with randomly picked labelOne shall not confuse this “label” randomness, essentially related to particle undistinguishability in a single realization of the plasma, with the random choice of a replica out of ${\\mathcal {N}}$ possible realizations of the plasma.",
"be in ${\\mathcal {A}}$ .",
"The limit $N \\rightarrow \\infty $ makes sense formally for non-negative normalized measures, and indeed this space of measures can be equipped with various kinds of distances generating physically reasonable topologies [43], [31] to give an operational meaning to the notation $\\lim _{N \\rightarrow \\infty }$ .",
"One such distance was considered by Kolmogorov and Smirnov to test the likelihood for sample random data to follow a given law, and it is used in convergence theorems for distributions [11].",
"On the contrary, the limit $N \\rightarrow \\infty $ is ill-defined in the phase-space approach involved in the Liouville equation, on which the BBGKY hierarchy relies.",
"The very space in which the phase point (representing the $N$ -particle system) evolves varies with $N$ : its dimension increases like $N$ .",
"Therefore, phase space $\\Gamma $ is simply not the good stage for performing the limit.",
"The derivation of the Vlasov equation in the (dynamical) measure approach is rather short (actually, shorter than the BBGKY-hierarchy based derivation) and both conceptually and physically clear, provided the interaction is not too singular (with the statistical approach providing no better derivation in the singular case either).",
"Key ingredients are the existence and uniqueness of solutions to individual particles' equations of motion in a given, regular enough, force field, say $\\mathbf {E}(\\mathbf {r},t)$ (with e.g.",
"Lipschitz regularity, i.e.",
"${\\Vert \\mathbf {E}(\\mathbf {r},t) - \\mathbf {E}(\\mathbf {r}^{\\prime },t) \\Vert } \\le K(t) {\\Vert \\mathbf {r} - \\mathbf {r}^{\\prime } \\Vert }$ ), by dualityThis is just using the fact that $\\mu (T_{t,0} (y) \\in {\\mathcal {A}}) = \\mu (y \\in T_{0,t} \\, {\\mathcal {A}})$ , because the space of measures $\\mu $ is dual to the space of observables $q$ , in the linear algebraic interpretation of the formula $\\int _{{\\mathbb {R}}^6} q(y) \\, {\\mathrm {d}}\\mu (y)$ [2].",
"and because the single-particle dynamics in the given field is measure-preserving in the “molecular” $\\mu $ -space, the existence and uniqueness of evolution of measures in the same given, regular enough, force field $\\mathbf {E}(\\mathbf {r},t)$ , regularity of the force field generated by arbitrary measures (which fails for Coulomb field with point sources), a self-consistency argument to formulate the Vlasov equation coupled with the fields as a fixed point problem in a suitable (Banach) function space – much like the standard technique used for solving iteratively ordinary differential equations.",
"The measure approach reaches further than just proving that the limit $N \\rightarrow \\infty $ commutes with time evolution.",
"It actually shows that, given initial measures $\\mu _1(0)$ and $\\mu _2(0)$ on $(\\mathbf {r},\\mathbf {v})$ -space, which need not be empirical nor absolutely continuous, the Vlasov equation defines unique time-dependent measures $\\mu _1(t)$ and $\\mu _2(t)$ and that the distance between the evolved measures is controlled by their initial distance and a constant $C$ depending on $\\mu _1(0)$ and $\\mu _2(0)$ , $\\mathrm {dist}(\\mu _1(t), \\mu _2(t))\\le C^{\\prime } \\, \\mathrm {dist}(\\mu _1(0), \\mu _2(0)) \\, {\\mathrm {e}}^{C {| t |}} .$ Here $C^{\\prime }$ may occur for technical reasons, and $C$ is an upper estimate for the largest Lyapunov exponent, controlling the rate at which particles separate in the fields $\\mathbf {E}_1$ and $\\mathbf {E}_2$ generated by the matter distributions described by $\\mu _1$ and $\\mu _2$ (see [19], [14] for a smooth interaction example).",
"This estimate implies that, to ensure $\\mathrm {dist}(\\mu ^{(N)}(t)/N, \\mu _f(t)) \\sim \\delta $ with $\\mathrm {dist}(\\mu ^{(N)}(0)/N, \\mu _f(0)) \\sim N^{-c}$ (with $c = 1/2$ or 1, say), one needs an initial accuracy corresponding to some $N \\sim (C^{\\prime } / \\delta )^{1/c} \\exp (C {| t |} /c)$ , which is too demanding when the times of interest exceed a few Lyapunov e-folding times.",
"As physicists are more interested in estimating specific observables, say local current densities or electric field energy density, estimates like (REF ) provide upper bounds on the errors due to approximating $\\mu _1(0)$ with $\\mu _2(0)$ .",
"However, actual discordances between the values of specific observables are generally much smaller – for instance, if $\\mu _1(0)$ and $\\mu _2(0)$ describe two stationary solutions, the observables are constants of the motion."
],
[
"Vlasov–Poisson evolution",
"While the dynamical foundation of the Vlasov equation, in the previous section, stressed the motion of particles, we must stress that some important results on the Vlasov–Poisson system make no use of particle trajectories.",
"In this section, we first indicate how the smooth solutions to the Vlasov–Poisson system may involve non-trajectorial concepts, and then comment on some aspects of the system which may, or need to, involve trajectories."
],
[
"Without trajectories",
"As a partial differential equations system, the Vlasov–Poisson model is well-posed and much studied (see e.g.",
"[41]).",
"Theorems on solutions existence rely on some regularity in initial data, typically $f$ has compact support and bounded, continuous derivatives $\\partial _\\mathbf {r} f$ , $\\partial _\\mathbf {v} f$ .",
"Proofs involve estimates for the spatial density $\\rho (\\mathbf {r}, t) = \\int _{{\\mathbb {R}}^3} f(\\mathbf {r},\\mathbf {v},t) \\, {\\mathrm {d}}^3 \\mathbf {v}$ in various $L^p$ norms, along with estimates for the electric field $\\mathbf {E}$ (whose $L^2$ norm is proportional to the total potential energy) and its gradients, which involve norms of $\\partial _\\mathbf {r} \\rho $ .",
"Moments of $f$ (and of course of $\\rho $ ), as well as the upper bound on particle velocity $P(t) = \\sup \\lbrace {| \\mathbf {v} |}~: f(\\mathbf {r},\\mathbf {v},t) > 0 \\textrm { for some } \\mathbf {r} \\rbrace $ , play a role in the analysis of the Vlasov–Poisson system.",
"It may occur that gradients of $f$ get steep, due to filamentation in $\\mu $ -space.",
"Yet, gradients of $\\rho $ or of $\\bar{f}(\\mathbf {v},t)~:= \\int _{{\\mathbb {R}}^3} f(\\mathbf {r},\\mathbf {v},t) \\, {\\mathrm {d}}^3 \\mathbf {r}$ may be better controlled, and even decay in some sense to zero, as e.g.",
"in the nonlinear theory of Landau damping [34], [35], [44], [4].",
"The analysis of continuous solutions to the Vlasov–Poisson system does not rely only on the more usual physical quantities like $\\rho $ or $\\mathbf {j}(\\mathbf {r},t) = \\int _{{\\mathbb {R}}^3} \\mathbf {v} f(\\mathbf {r},t) \\, {\\mathrm {d}}^3 \\mathbf {v}$ , and on standard global invariants like total momentum or energy.",
"It also takes advantage of Casimir invariants, which may be expressed in terms of the functions $C_a [f] (t)~:=\\int _{{\\mathbb {R}}^6} 1(f(\\mathbf {r},\\mathbf {v},t) > a) \\, {\\mathrm {d}}^3 \\mathbf {r} \\, {\\mathrm {d}}^3 \\mathbf {v}$ where $1(A) = 1$ if $A$ is true and $1(A) = 0$ otherwise.",
"Note that $C_a$ is a decreasing function of $a$ , with $C_a = + \\infty $ for any $a < 0$ , and $C_0$ being the measure of the support of $f$ .",
"These functions $C_a$ measure the level sets of $f$ and provide a tool for calculating other functionals like ${\\Vert f \\Vert }_{L^1} = \\int _0^{\\infty } C_a[f] \\, {\\mathrm {d}}a$ , ${\\Vert f \\Vert }_{L^p}^p = p \\int _0^{\\infty } a^{p-1} \\,ÊC_a[f] \\, {\\mathrm {d}}a$ , or $\\int _{{\\mathbb {R}}^6} f(\\mathbf {r},\\mathbf {v}) \\ln (f(\\mathbf {r},\\mathbf {v})/c) {\\mathrm {d}}^3 \\mathbf {r} {\\mathrm {d}}^3 \\mathbf {v}= \\int _0^{\\infty } (1 + \\ln (a/c)) C_a[f] {\\mathrm {d}}a$ .",
"The Vlasov equation preserves all Casimir invariants, and a physical interpretation for these conservation laws is that vlasovian evolution must be invariant under the group of particle relabelings.",
"Such a group is discrete for the $N$ -body system (and then $C_a$ is undefined), but it is continous for smooth $f$ , and one expects it to generate integral invariants following Noether's theorem.",
"Adding to the total energy a suitably chosen Casimir invariant enabled proving the stability of some spherically symmetric equilibria of the gravitational Vlasov–Poisson system [33].",
"In numerical simulations and in simple analytic models, the case where $C_a$ is a simple step function occurs for waterbag distributions [5], equal to $h$ on a domain ${\\mathcal {A}}(t)$ with measure $1/h$ , and vanishing outside ${\\mathcal {A}}(t)$ (then $C_a = 0$ for $a \\ge h$ and $C_a = 1/h$ for $0 \\le a < h$ )."
],
[
"With trajectories",
"Particle trajectories are instrumental in understanding the Vlasov equation because this equation transports the distribution $f$ along the characteristics of the Vlasov operator.",
"Good control on particle trajectories is crucial e.g.",
"to proofs of the existence of solutions globally in time [39], [22], [42] (using the Lipschitz regularity of the electric field generated by the distribution function) and to the proof of Landau damping in the nonlinear regime [34], [35], [44], [4].",
"Estimates on individual particle trajectories in small field limits are obtained for the fields generated by the Vlasov–Poisson dynamics, and provide further insight in the latter, using estimates for velocity moments of the distribution function [30], [37].",
"Estimates on particle trajectory crossings are also used to construct Lyapunov functionals for the evolution from “small” initial data in order to assess the decay to spatial uniformity and zero electric field in long times, and the “asymptotic completeness”This issue is whether the nonlinear Vlasov flow can or cannot be approximated by the free streaming flow in the limit $t \\rightarrow \\infty $ .",
"of dynamics [7], [8].",
"Incidentally, Mouhot and Villani also stress the stabilizing role of plasma echoes (a particle effect, associated with bunching) for the nonlinear theory of Landau damping.",
"However, characteristics (i.e.",
"trajectories) are not considered individually ; rather, smoothness of $f$ is important, and nonlinear particle motion does not break it for finite time – while mixing in $(\\mathbf {r},\\mathbf {v})$ space generates small scale oscillations (ripples) which are homogenized in the long run.",
"On the other hand, particle trajectories may be ignored when considering stationary solutions, see e.g.",
"the orbital stability problem solved by Lemou, Méhats and Raphaël (see [33] for a review).",
"Nor do they appear explicitly in the discussion of Bernstein-Greene-Kruskal modes and trapping in space-periodic Vlasov–Poisson system [29].",
"Note that all these statements refer to the Vlasov–Poisson system as a partial-differential deterministic evolution equation system, in which any particle is following a trajectory defined by the Vlasov equation characteristics.",
"It is not analysed with probabilistic methods."
],
[
"Direct many-body dynamics",
"A finite-$N$ approach always stresses trajectories.",
"Analogues of the stability theorems for Landau damping in the nonlinear regime would be a Kolmogorov–Arnol'd–Moser theorem [4] for global-in-time stability, or a Nekhoroshev-type theorem for stability over exponentially long times (say $\\exp (a N^b)$ for some $a, b > 0$ ).",
"This is indeed a challenging target for mathematical analysis (see the recent tutorial [44]).",
"Interestingly, (partial-differential-equations-based) Landau damping theorems involve periodic boundary conditions, so that the electrons may be viewed as immersed in a neutralizing background (the “jellium” model) – while results on asymptotic completeness involve small initial data in an infinite space.",
"For periodic jellium, with finitely many bodies, interacting through repulsive Coulomb force, no two particles can be simultaneously at the same point.",
"The equations of motion are well-posed, and solutions to the initial value problem (Cauchy problem) exist for all times.For actual plasmas, with ions and electrons, Coulomb attraction permits that particles come arbitrarily close, with potential energy diverging to $-\\infty $ while kinetic energy diverges to $+\\infty $ .",
"This makes their $N$ -body description even more difficult.",
"The similar difficulty in celestial mechanics is well known [17].",
"The continuum limit, where $N \\rightarrow \\infty $ along with $\\mu ^{(N)}/N$ approaching an absolutely continuous measure with a smooth density $f$ , is however difficult because the limit imposes that some particles come arbitrarily close to each other, and the characteristic time scales for their relative motion become arbitrarily short.",
"The short-range singularity of the Coulomb potential is too strong for the usual rigorous derivations of the Vlasov equation to apply.",
"Kiessling [26] exposes a new approach along with a review of the state of the art.",
"Without relying on the Vlasov description, one can use the $N$ -body description directly to understand some fundamental physical effect in plasmas.",
"This issue was recently revisited [15] for $N$ electrons (with mass $m_{\\mathrm {e}}$ and charge $-e$ ) in a cube with side $L$ with periodic boundary conditions (the latter play the role of a neutralizing background).",
"For large $N$ , when the particle positions are not too far from a spatially uniform distribution, and velocities are close to a distribution $f_0(\\mathbf {v})$ , one separates the motion of a particle into a ballistic approximation, $\\mathbf {r}_l^{(0)}(t) = \\mathbf {r}_{l0} + \\mathbf {v}_{l0} t$ , and a correction $\\delta \\mathbf {r}_j(t) = \\mathbf {r}_j(t) - \\mathbf {r}_l^{(0)}(t)$ due to the interactions.",
"Perturbative analysisSee appendix B in [15] for the outline of a nonlinear version.",
"shows that the dominant contribution to their interaction is not given by the full Coulomb potential (with Fourier components $ \\tilde{\\varphi }(\\mathbf {m},t)= - \\sum _{j = 1}^N e / (\\epsilon _0 {\\Vert \\mathbf {k}_\\mathbf {m} \\Vert }^2)\\exp (- {\\mathrm {i}}\\mathbf {k}_{\\mathbf {m}} \\cdot \\mathbf {r}_j(t))$ for $\\mathbf {k}_\\mathbf {m} = 2\\pi \\mathbf {m}/L$ with $\\mathbf {m} \\in {\\mathbb {Z}}^3 \\setminus \\lbrace \\mathbf {0}\\rbrace $ ), but only the Debye-shielded potential $\\delta \\Phi (\\mathbf {r},t)= \\sum _j \\delta \\Phi (\\mathbf {r} - \\mathbf {r}_j(0) - \\dot{\\mathbf {r}}_j(0) t,\\dot{\\mathbf {r}}_j(0))$ , where $\\delta \\Phi (\\mathbf {r},\\mathbf {v})= - \\frac{e}{L^3 \\epsilon _0} \\sum _{{\\mathbf {m}} \\ne {\\mathbf {0}}}\\frac{\\exp ({\\mathrm {i}}\\mathbf {k}_{\\mathbf {m}} \\cdot \\mathbf {r})}{ {\\Vert \\mathbf {k}_{\\mathbf {m}} \\Vert }^2 \\,\\epsilon (\\mathbf {m},\\mathbf {k}_{\\mathbf {m}} \\cdot \\mathbf {v} + {\\mathrm {i}}\\varepsilon ) }$ and $\\epsilon $ approachesThe ${\\mathrm {i}}\\varepsilon $ prescription stems from inverting the Laplace transform as the integral in $\\epsilon $ is undefined for real-valued $\\omega = \\mathbf {k}_{\\mathbf {m}} \\cdot \\mathbf {v}$ .",
"As this diverging $\\epsilon $ occurs in the denominator in (REF ), however, the finite $N$ version does not need the ${\\mathrm {i}}\\varepsilon $ .",
"A similar discussion occurs in the analysis of discrete analogues of van Kampen-Case modes [14].",
"the usual dielectric function $\\epsilon (\\mathbf {m},\\omega )= 1 - e^2 / (L^3 m_{\\mathrm {e}}\\epsilon _0)\\int (\\omega -\\mathbf {k}_{\\mathbf {m}} \\cdot \\mathbf {v})^{-2} f_0(\\mathbf {v})\\, {\\mathrm {d}}^3 \\mathbf {v}$ in the $N \\rightarrow \\infty $ limit.",
"While Debye shielding of probes or test charges is well known in the kinetic theory of plasmas, this result brings forward the direct appearance of Debye shielded Coulomb potential in the dynamics – which is not usually stressed in classical Vlasov–Poisson theory.",
"Moreover, the faster decay of the shielded force enables one to compute transport phenomena in terms of particle “binary collisions” over the whole range of impact parameters [16], while the classical collisional models were constructed separately for large and small impact parameters.",
"In the same calculations, the collective response of particles to any one of them also “dresses” the latter, so that the effective description of the plasma deviates simultaneously from both the Vlasov equation (stressing now $N$ bodies) and from the Poisson equation (through shielding and dressing).",
"The potential (REF ) lends itself easily to Laplace analysis in time [15].",
"In the $N \\rightarrow \\infty $ limit, it then also approaches formally (outside its sources) the potential given by the standard expression involving the Landau contour calculation, for both the unstable (growing wave) case and the damping case.",
"The twist in this derivation of Landau damping in the linear regime is that it involves not the full Coulomb interaction but its (perturbatively constructed) shielded version.",
"This comes as a surprise, for screening generates a “short range” effective model ; yet the shielded potential is still Coulomb-singular at its source, so that the standard derivations of the Vlasov equation do not apply better than for the full Coulomb interaction."
],
[
"Final remarks",
"Let us first recall that finite-$N$ dynamics may always depart from smooth Vlasov solutions on the characteristic time scales of instabilities, as noted in eq.",
"(REF ).",
"Apart form this obvious example, it is known that for interactions smoother than Coulomb's, finite-$N$ distributions can indeed remain quasi-stationary for extremely long times (see [28] for a recent review), even if the finite-$N$ model does not admit corresponding periodic solutions [13].",
"Moreover, the large, finite $N$ regime of near-equilibrium distributions, as occurs in the physical problem motivating research on orbital stability, is of interest for the Newton gravitation interaction [33], where it also branches to the elegant special solutions to the $N$ -body dynamics with symmetries known as “choreographies” [32].",
"Similar results in the Coulomb case relate to the motion of cold ion clouds or non-neutral plasmas [25], [10].",
"As physicists have since long understood that finite-$N$ systems are never completely described by classical Vlasov–Poisson theory, long-time evolutions of plasmas are often described using other kinetic models in the $N \\rightarrow \\infty $ limit.",
"Rigorous derivations of such models, like the Landau equation or the Balescu–Lenard–Guernsey equation, are a challenge for this century (see [27] and the truly short ch.",
"6 in [43]).",
"Actually, corrections to the Vlasov model often go by the name of “collisions”, and we noted in Sect.",
"how surprising they may be even in their simpler instances [15], [16].",
"Collisions do not necessarily account for the graininess effects due to finite $N$ [18].",
"It is a pleasure for YE to acknowledge stimulating discussions with G. Belmont, F. Califano, C. Krafft, G. Manfredi, Ph.",
"Morrison, F. Pegoraro, F. Valentini and participants to Vlasovia 2013 in Nancy.",
"The authors are grateful to L. Couëdel for a critical reading of the manuscript, and to their colleagues in Marseilles for fruitful comments."
]
] | 1403.0056 |
[
[
"Multi-Objective Resource Allocation for Secure Communication in\n Cognitive Radio Networks with Wireless Information and Power Transfer"
],
[
"Abstract In this paper, we study resource allocation for multiuser multiple-input single-output secondary communication systems with multiple system design objectives.",
"We consider cognitive radio networks where the secondary receivers are able to harvest energy from the radio frequency when they are idle.",
"The secondary system provides simultaneous wireless power and secure information transfer to the secondary receivers.",
"We propose a multi-objective optimization framework for the design of a Pareto optimal resource allocation algorithm based on the weighted Tchebycheff approach.",
"In particular, the algorithm design incorporates three important system objectives: total transmit power minimization, energy harvesting efficiency maximization, and interference power leakage-to-transmit power ratio minimization.",
"The proposed framework takes into account a quality of service requirement regarding communication secrecy in the secondary system and the imperfection of the channel state information of potential eavesdroppers (idle secondary receivers and primary receivers) at the secondary transmitter.",
"The adopted multi-objective optimization problem is non-convex and is recast as a convex optimization problem via semidefinite programming (SDP) relaxation.",
"It is shown that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP relaxed problem.",
"Besides, two suboptimal resource allocation schemes for the case when the solution of the dual problem is unavailable for constructing the optimal solution are proposed.",
"Numerical results not only demonstrate the close-to-optimal performance of the proposed suboptimal schemes, but also unveil an interesting trade-off between the considered conflicting system design objectives."
],
[
"Introduction",
"The explosive growth of the demand for ubiquitous, secure, and high data rate wireless communication services has led to a tremendous solicitation of limited radio resources such as bandwidth and energy.",
"In practice, fixed spectrum allocation has been implemented for resource sharing in traditional wireless communication systems.",
"Although interference can be avoided by assigning different wireless services to different licensed frequency bands, such a fixed spectrum allocation strategy may result in spectrum under utilization.",
"In fact, the Federal Communications Commission (FCC) has reported that 70 percent of the allocated spectrum in the United States is not fully utilized, cf.",
"[2].",
"As a result, cognitive radio (CR) has emerged as one of the most promising solutions to improve spectrum efficiency [3].",
"In particular, CR enables a secondary system to access the spectrum of a primary system as long as the interference from the secondary system does not severely degrade the quality of service (QoS) of the primary system [2]–[11].",
"CR is not only applicable to traditional cellular networks, but also has the potential to improve the performance of wireless sensor networks [5], [6].",
"In [7] and [8], cooperative spectrum sensing and the sensing-throughput trade-off were studied for single antenna systems, respectively.",
"In [9], joint beamforming and power control was studied for transmit power minimization in multiple-transmit-antenna CR downlink systems.",
"In [10] and [11], by taking into account the imperfectness of channel state information, robust beamforming designs were proposed for CR networks with single and multiple secondary users, respectively.",
"Furthermore, a detailed performance analysis of transmit antenna selection in multiple-antenna networks was presented in [12] for multi-relay networks and then extended to CR relay networks in [13].",
"However, since the transmit precoding strategies in [12] and [13] are not optimized, they do not fully exploit the available degrees of freedom in the network for maximizing the system performance.",
"Although the current spectrum scarcity may be partially overcome by CR technology, wireless communication devices, such as wireless sensors, are often powered by batteries with limited energy storage capacity.",
"This constitutes another major bottleneck for providing communication services and extending the lifetime of networks.",
"On the other hand, energy harvesting is envisioned to provide a perpetual energy source to facilitate self-sustainability of power-constrained communication devices [14]–[24].",
"In addition to conventional renewable energy sources such as biomass, wind, and solar, wireless power transfer has emerged as a new option for prolonging the lifetime of battery-powered wireless devices.",
"Specifically, the transmitter can transfer energy to the receivers via electromagnetic waves in radio frequency (RF).",
"Nowadays, energy harvesting circuits are able to harvest microwatt to milliwatt of power over the range of several meters for a transmit power of 1 Watt and a carrier frequency of less than 1 GHz [14].",
"Thus, RF energy can be a viable energy source for devices with low-power consumption, e.g.",
"wireless sensors [16], [17].",
"The integration of RF energy harvesting capabilities into communication systems provides the possibility of simultaneous wireless information and power transfer (SWIPT) [16]–[24].",
"As a result, in addition to the traditional QoS constraints such as communication reliability, efficient energy transfer is expected to play an important role as a new QoS requirement.",
"This new requirement introduces a paradigm shift in the design of both resource allocation algorithms and transceiver signal processing.",
"In [19] and [20], the fundamental trade-off between the maximum achievable data rate and energy transfer was studied for a noisy single-user communication channel and a pair of noisy coupled-inductor circuits, respectively.",
"Then, in [21], the authors extended the trade-off study to a two-user multiple-antenna transceiver system.",
"In [22], the authors proposed separated receivers for SWIPT to facilitate low-complexity receiver design; these receivers can be built by using off-the-shelf components.",
"In [23], different resource allocation algorithms were designed for broadband far field wireless systems with SWIPT.",
"In [24], the authors showed that the energy efficiency of a communication system can be improved by RF energy harvesting at the receivers.",
"Nevertheless, resource allocation algorithms maximizing the energy harvesting efficiency of SWIPT CR systems have not been reported in the literature yet.",
"Besides, two conflicting system design objectives arise naturally for a CR network providing SWIPT service to the secondary receivers in practice.",
"On the one hand, the secondary transmitter should transmit with high power to facilitate energy transfer to the energy harvesting receivers.",
"On the other hand, the secondary transmitter should transmit with low power to cause minimal interference at the primary receivers.",
"Thus, considering these conflicting system design objectives, the single objective resource allocation algorithms proposed in [19]–[24] may not be applicable in SWIPT CR networks.",
"Furthermore, transmitting with high signal power may also cause substantial information leakage and high vulnerability to eavesdropping.",
"Recently, physical (PHY) layer security has attracted much attention in the research community for preventing eavesdropping [25]–[34].",
"In [25], the authors proposed a beamforming scheme for maximization of the energy efficiency of secure communication systems.",
"In [26] and [27], the spatial degrees of freedom offered by multiple antennas were used to degrade the channel of the eavesdroppers deliberately via artificial noise transmission.",
"Thereby, communication secrecy was guaranteed at the expense of allocating a large portion of the transmit power to artificial noise generation.",
"In [28], the authors addressed the power allocation problem in CR secondary systems with PHY layer security provisioning.",
"However, the resource allocation algorithm designs in [25]–[28] cannot be directly extended to the case of of RF energy harvesting due to the differences in the underlying system models.",
"On the other hand, [1] and [30] studied different resource allocation algorithms for providing secure communication in systems with separated information and energy harvesting receivers.",
"Yet, the assumption of having perfect channel state information (CSI) of the energy harvesting receivers in [1] and [30] may be too optimistic if the energy harvesting receivers do not interact with the transmitter periodically.",
"In [29], the case where the transmitter has only imperfect CSI of the energy harvesting receivers was considered and a robust beamforming design was proposed to minimize the total transmit power of a system with simultaneous energy and secure information transfer.",
"In [31], the authors studied resource allocation algorithm design for secure information and renewable green energy transfer to mobile receivers in distributed antenna communication systems.",
"In [32] and [33], beamforming algorithm design and secrecy outage capacity was studied for multiple-antenna potential eavesdropper and passive eavesdroppers, respectively.",
"However, the beamforming algorithms developed in [29]–[32] may not be applicable in CR networks.",
"Furthermore, in [34], the secrecy outage probability of CR networks was investigated in the presence of a passive eavesdropper.",
"Form the above discussions, we conclude that for CR communication systems providing simultaneous wireless energy transfer and secure communication services, conflicting system design objectives such as total transmit power minimization, energy harvesting efficiency maximization, and interference power leakage-to-transmit power ratio minimization play an important role for resource allocation.",
"However, the problem formulations in [19]–[34] focus on a single system design objective and cannot be used to study the trade-off between the aforementioned conflicting design goals.",
"In this paper, we address the above issues and the contributions of the paper are summarized as follows: Different from our previous work in [29], in this paper, we propose a new non-convex multi-objective optimization problem with the aim to jointly minimize the total transmit power, maximize the energy harvesting efficiency, and minimize the interference power leakage-to-transmit power ratio for CR networks with SWIPT.",
"The problem formulation takes into account the imperfectness of the CSI of potential eavesdroppers (idle secondary receivers) and primary receivers in secondary multiuser multiple-input single-output (MISO) systems with RF energy harvesting receivers.",
"The solution of the optimization problem leads to a set of Pareto optimal resource allocation policies.",
"The considered non-convex optimization problem is recast as a convex optimization problem via semidefinite programming (SDP) relaxation.",
"We show that the global optimal solution of the original problem can be constructed by exploiting both the primal and the dual optimal solutions of the SDP relaxed problem.",
"The obtained solution structure is also applicable to the multi-objective optimization of the total harvested power, the interference power leakage, and the total transmit power.",
"Two suboptimal resource allocation schemes are proposed for the case when the solution of the dual problem of the SDP relaxed problem is unavailable for construction of the optimal solution.",
"Our results unveil a non-trivial trade-off between the considered system design objectives which can be summarized as follows: (1) A resource allocation policy minimizing the total transmit power also leads to a low total interference power leakage in general; (2) energy harvesting efficiency maximization and transmit power minimization are conflicting system design objectives; (3) the maximum energy harvesting efficiency is achieved at the expense of high interference power leakage and high transmit power."
],
[
"System Model",
"In this section, we first introduce the notation used in this paper.",
"Then, we present the adopted CR downlink channel model for secure communication with SWIPT."
],
[
"Notation",
"We use boldface capital and lower case letters to denote matrices and vectors, respectively.",
"For a square-matrix $\\mathbf {S}$ , $\\operatorname{\\mathrm {Tr}}(\\mathbf {S})$ denotes the trace of matrix $\\mathbf {S}$ .",
"$\\mathbf {S}\\succ \\mathbf {0}$ and $\\mathbf {S}\\succeq \\mathbf {0}$ indicate that $\\mathbf {S}$ is a positive definite and a positive semidefinite matrix, respectively.",
"$(\\mathbf {S})^H$ and $\\operatorname{\\mathrm {Rank}}(\\mathbf {S})$ denote the conjugate transpose and the rank of matrix $\\mathbf {S}$ , respectively.",
"$\\mathbf {I}_{N}$ denotes an $N\\times N$ identity matrix.",
"$\\mathbb {C}^{N\\times M}$ and $\\mathbb {R}^{N\\times M}$ denote the space of $N\\times M$ matrices with complex and real entries, respectively.",
"$\\mathbb {H}^N$ represents the set of all $N$ -by-$N$ complex Hermitian matrices.",
"$\\vert \\cdot \\vert $ and $\\Vert \\cdot \\Vert $ denote the absolute value of a complex scalar and the Euclidean norm of a matrix/vector, respectively.",
"$\\operatorname{\\mathrm {diag}}(x_1, \\cdots , x_K)$ denotes a diagonal matrix with the diagonal elements given by $\\lbrace x_1, \\cdots , x_K\\rbrace $ .",
"$\\mathrm {Re}(\\cdot )$ extracts the real part of a complex-valued input.",
"The distribution of a circularly symmetric complex Gaussian (CSCG) vector with mean vector $\\mathbf {x}$ and covariance matrix $\\mathbf {\\Sigma }$ is denoted by ${\\cal CN}(\\mathbf {x},\\mathbf {\\Sigma })$ , and $\\sim $ means “distributed as\".",
"${\\cal E}\\lbrace \\cdot \\rbrace $ represents statistical expectation.",
"For a real valued continuous function $f(\\cdot )$ , $\\nabla _{\\mathbf {X}} f(\\mathbf {X})$ denotes the gradient of $f(\\cdot )$ with respect to matrix $\\mathbf {X}$ .",
"$[x]^+$ stands for $\\max \\lbrace 0,x\\rbrace $ ."
],
[
"Downlink Channel Model",
"We consider a CR secondary network for short distance downlink communication.",
"There are one secondary transmitter equipped with $N_{\\mathrm {T}}>1$ antennas, $K$ secondary receivers, one primary transmitterWe note that the considered system model can be extended to include multiple primary transmitters at the expense of a more involved notation., and $J$ primary receivers.",
"The primary transmitter, primary receivers, and secondary receivers are single-antenna devices that share the same spectrum, cf.",
"Figure REF .",
"We assume $N_{\\mathrm {T}}>J$ to enable efficient communication in the CR secondary network.",
"The secondary transmitter provides SWIPT services to the secondary receivers while the primary transmitter provides broadcast services to the primary receivers.",
"In practice, the CR secondary operator may rent spectrum from the primary operator under the condition that the interference leakage from the secondary system to the primary system is properly controlled.",
"We assume that the secondary receivers are ultra-low power devices, such as wireless sensorsThe power consumption of typical sensor micro-controllers, such as the Texas Instruments micro-controller: MSP430F2274 [35], is in the order of microwatt in the idle mode.",
"As a result, wireless power transfer is a viable option for the energy supply of wireless sensors., which either harvest energy or decode information from the received radio signals in each time instant, but are not able to perform both concurrently due to hardware limitations [22], [24].",
"In each scheduling slot, the secondary transmitter not only conveys information to a given secondary receiver, but also transfers energyWe adopt the normalized energy unit Joule-per-second in this paper.",
"Therefore, the terms “power\" and “energy\" are used interchangeably.",
"to the remaining $K-1$ idle secondary receivers to extend their lifetimes.",
"Figure: A CR network where K=3K=3 secondary receivers (1 active and 2 idle receivers) share the same spectrum with J=2J=2 primary receivers.",
"The secondary transmitter conveys information and transfers power(/energy) to the KK secondary receivers simultaneously.",
"The red dotted ellipsoids illustrate the dual use of artificial noise for providing security and facilitating efficient energy transfer to the secondary receivers.We note that only one secondary receiver is selected for information transfer to reduce the multiple access interference leakage to the primary receivers [28].",
"On the other hand, the information signal of the desired secondary receiver is overheard by both the $K-1$ idle secondary receivers and the $J$ primary receivers.",
"Hence, if the idle secondary receivers and the primary receivers are malicious, they may eavesdrop the signal of the selected secondary receiver, which has to be taken into account for resource allocation design to provide communication secrecy in the secondary network.",
"Thus, for guaranteeing communication security, the secondary transmitter has to employ a resource allocation algorithm that accounts for this unfavourable scenario and treat both idle secondary receivers and primary receivers as potential eavesdroppers.",
"We assume a frequency flat slow fading channel.",
"The received signals at the desired secondary receiver, idle secondary receiver $k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , and primary receiver $j\\in \\lbrace 1,\\ldots ,J\\rbrace $ are given by, respectively, $\\hspace*{-11.38109pt}y\\hspace*{-8.53581pt}&=&\\hspace*{-8.53581pt}\\mathbf {h}^{H} \\mathbf {x}+q \\sqrt{P^{\\mathrm {PU}}} d+z, \\\\\\hspace*{-11.38109pt}y_{k}^\\mathrm {Idle}\\hspace*{-8.53581pt}&=&\\hspace*{-8.53581pt}\\mathbf {g}_{k}^{H} \\mathbf {x}+ f_k\\sqrt{P^{\\mathrm {PU}}}d+ z_k ,\\, \\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace ,\\,\\mbox{and}\\\\\\hspace*{-11.38109pt}y_{j}^{\\mathrm {PU}}\\hspace*{-8.53581pt}&=&\\hspace*{-8.53581pt}\\mathbf {l}_{j}^{H} \\mathbf {x}+t_j\\sqrt{P^{\\mathrm {PU}}} d + z_{\\mathrm {PU}_j} ,\\,\\, \\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace .$ Here, $\\mathbf {x}\\in \\mathbb {C}^{ N_T \\times 1}$ denotes the symbol vector transmitted by the secondary transmitter.",
"$\\mathbf {h}^{H}\\in \\mathbb {C}^{1\\times N_T}$ , $\\mathbf {g}_{k}^{H} \\in \\mathbb {C}^{1\\times N_T}$ , and $\\mathbf {l}^{H}_j\\in \\mathbb {C}^{1\\times N_T}$ are the channel vectors between the secondary transmitter and the desired secondary receiver, idle receiver (potential eavesdropper) $k$ , and primary receiver (potential eavesdropper) $j$ , respectively.",
"$P^{\\mathrm {PU}}$ and $d\\in \\mathbb {C}^{1\\times 1}$ are the transmit power of the primary transmitter and the information signal intended for the primary receivers, respectively.",
"$q \\in \\mathbb {C}^{1\\times 1}$ , $f_k \\in \\mathbb {C}^{1\\times 1}$ , and $t_j \\in \\mathbb {C}^{1\\times 1}$ are the communication channels between the primary transmitter and desired secondary receiver, idle secondary receiver $k$ , and primary receiver $j$ , respectively.",
"$z_{\\mathrm {PU}_j}$ includes the joint effects of the thermal noise and the signal processing noise at primary receiver $j$ and is modelled as additive white Gaussian noise (AWGN) with zero mean and varianceWe assume that the noise characteristics are identical for all primary receivers due to similar hardware architectures.",
"$\\sigma _{\\mathrm {PU}}^2$ .",
"$z$ and $z_k$ include the joint effects of thermal noise and signal processing noise at the desired secondary receiver and idle secondary receiver $k$ , respectively, and are modeled as AWGN.",
"Besides, the equivalent noises at the desired and idle secondary receivers, which capture the joint effect of the received interference from the primary transmitter, i.e., $q \\sqrt{P^{\\mathrm {PU}}} d$ and $f_k\\sqrt{P^{\\mathrm {PU}}}d$ , thermal noise, and signal processing noise, are also modeled as AWGN with zero mean and variances $\\sigma _{\\mathrm {z}}^2$ and $\\sigma _{\\mathrm {z}_k}^2$ , respectively.",
"Remark 1 In this paper, we assume that the primary network is a legacy system and the primary transmitter does not actively participate in transmit power control.",
"Furthermore, we assume that the primary transmitter transmits a Gaussian signal and we focus on quasi-static fading channels such that all channel gains remain constant within the coherence time of the secondary system.",
"These assumptions justify modelling the interference from the primary transmitter to the secondary receivers as additive white Gaussian noise with different powers for different secondary receivers.",
"This model has been commonly adopted in the literature for resource allocation algorithm design [10], [28], [36].",
"To guarantee secure communication and to facilitate an efficient power transfer in the secondary system, artificial noise is generated at the secondary transmitter and is transmitted concurrently with the information signal.",
"In particular, the transmit signal vector $\\mathbf {x}=\\underbrace{\\mathbf {w}s}_{\\mbox{desired signal}}+\\underbrace{\\mathbf {v}}_{\\mbox{artificial noise}}$ is adopted at the secondary transmitter, where $s\\in \\mathbb {C}^{1\\times 1}$ and $\\mathbf {w}\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ are the information bearing signal for the desired receiver and the corresponding beamforming vector, respectively.",
"We assume without loss of generality that ${\\cal E}\\lbrace \\vert s\\vert ^2\\rbrace =1$ .",
"$\\mathbf {v}\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ is the artificial noise vector generated by the secondary transmitter to combat the potential eavesdroppers.",
"Specifically, $\\mathbf {v}$ is modeled as a complex Gaussian random vector with mean $\\mathbf {0}$ and covariance matrix $\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}}, \\mathbf {V}\\succeq \\mathbf {0}$ .",
"We note that $\\mathbf {w}$ and $\\mathbf {V}$ have to be optimized such that the transmit signal of the secondary transmitter does not interfere severely with the primary users."
],
[
"Resource Allocation Problem Formulation",
"In this section, we define different quality of service (QoS) measures for the secondary CR network for providing wireless power transfer and secure communication to the secondary receivers while protecting the primary receivers.",
"Then, we formulate three resource allocation problems reflecting three different system design objectives.",
"For convenience, we define the following matrices: $\\mathbf {H}=\\mathbf {h}\\mathbf {h}^H$ , $\\mathbf {G}_k=\\mathbf {g}_k\\mathbf {g}_k^H, k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , and $\\mathbf {L}_j=\\mathbf {l}_j\\mathbf {l}_j^H, j\\in \\lbrace 1,\\ldots ,J\\rbrace $ ."
],
[
"System Achievable Rate and Secrecy Rate",
"Given perfect CSI at the receiver, the achievable rate (bit/s/Hz) between the secondary transmitter and the desired secondary receiver is given by $C=\\log _2\\Big (1+\\Gamma \\Big )\\,\\,\\,\\,\\mbox{and}\\,\\,\\,\\,\\Gamma =\\frac{\\mathbf {w}^H\\mathbf {H}\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\mathbf {V})+\\sigma _{\\mathrm {z}}^2} ,$ where $\\Gamma $ is the received signal-to-interference-plus-noise ratio (SINR) at the desired secondary receiver.",
"On the other hand, the achievable rates between the secondary transmitter and idle secondary receiver $k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ and primary receiver $j\\in \\lbrace 1,\\ldots ,J\\rbrace $ are given by $\\hspace*{-8.53581pt}C^{\\mathrm {Idle}}_k\\hspace*{-8.53581pt}&=&\\hspace*{-8.53581pt}\\log _2\\Big (1+\\Gamma ^{\\mathrm {Idle}}_k\\Big ),\\,\\,\\,\\,\\,\\Gamma ^{\\mathrm {Idle}}_k= \\frac{\\mathbf {w}^H\\mathbf {G}_k\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k\\mathbf {V})+\\sigma _{\\mathrm {z}_k}^2},\\,\\mbox{and} \\\\\\hspace*{-8.53581pt}C^{\\mathrm {PU}}_j\\hspace*{-8.53581pt}&=&\\hspace*{-8.53581pt}\\log _2\\Big (1+\\Gamma ^{\\mathrm {PU}}_j\\Big ),\\,\\,\\,\\,\\,\\Gamma ^{\\mathrm {PU}}_j= \\frac{\\mathbf {w}^H\\mathbf {L}_j\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_k\\mathbf {V})+\\sigma _{\\mathrm {PU}}^2}, $ respectively, where $\\Gamma ^{\\mathrm {Idle}}_k$ and $\\Gamma ^{\\mathrm {PU}}_j$ are the received SINRs at idle secondary receiver $k$ and primary receiver $j$ , respectively.",
"Since both the idle secondary receivers and the primary receivers are potential eavesdroppers, the maximum achievable secrecy rate between the secondary transmitter and the desired receiver is given by $C_\\mathrm {sec}=\\Big [C - \\underset{\\underset{j\\in \\lbrace 1,\\ldots ,J\\rbrace }{k\\in \\lbrace 1,\\ldots ,K-1\\rbrace }}{\\max } \\lbrace C^{\\mathrm {Idle}}_k,C^{\\mathrm {PU}}_j\\rbrace \\Big ]^+.$ In the literature, the secrecy rate, i.e., (REF ), is commonly adopted as a QoS requirement for system design to ensure secure communication [26], [27].",
"In particular, $C_{\\mathrm {sec}}$ quantifies the maximum achievable data rate at which a transmitter can reliably send secret information to the intended receiver such that the eavesdroppers are unable to decode the received signal [37] even if the eavesdroppers have unbounded computational capabilityWe note that, in practice, the malicious secondary idle receivers and primary receivers do not have to decode the eavesdropped information in real time.",
"They can act as information collectors to sample the received signals and store them for future decoding by other energy unlimited and powerful computational devices.."
],
[
"Energy Harvesting Efficiency",
"In the considered CR system, the secondary receivers harvest energy from the RF when they are idle to extend their lifetimesIn fact, nowadays many sensors are equipped with hybrid energy harvesters for harvesting energy from different energy sources such as solar and thermal-energy [38], [39].",
"Thus, the harvested energy from the radio frequency may be used as a supplement for supporting the energy consumption of the secondary receivers..",
"The energy harvesting efficiency plays an important role in the system design of such secondary networks and has to be considered in the problem formulation.",
"To this end, we define the energy harvesting efficiency in the secondary system as the ratio of the total power harvested at the idle secondary receivers and the total power radiated by the secondary transmitter.",
"The total amount of energy harvested by the $K-1$ idle secondary receivers is modeled as $\\mathrm {HP}(\\mathbf {w},\\mathbf {V})=\\sum _{k=1}^{K-1}\\eta _k\\Big (\\mathbf {w}^H\\mathbf {G}_k\\mathbf {w}+\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k\\mathbf {V})\\Big ),$ where $\\eta _k$ is a constant, $0\\le \\eta _k\\le 1,\\forall k$ , which represents the RF energy conversion efficiency of idle secondary receiver $k$ in converting the received radio signal to electrical energy.",
"We note that the power received at the secondary receivers from the primary transmitter and the AWGN power are neglected in (REF ) as we focus on the worst-case scenario for robust energy harvesting system design.",
"On the other hand, the power radiated by the transmitter can be expressed as $\\mathrm {TP}(\\mathbf {w},\\mathbf {V})=\\Vert \\mathbf {w}\\Vert ^2+\\operatorname{\\mathrm {Tr}}(\\mathbf {V}).$ Thus, the energy harvesting efficiency of the considered secondary CR system is given by $\\eta _{\\mathrm {eff}}(\\mathbf {w},\\mathbf {V})=\\frac{\\mathrm {HP}(\\mathbf {w},\\mathbf {V})}{\\mathrm {TP}(\\mathbf {w},\\mathbf {V})}.$"
],
[
"Interference Power Leakage-to-Transmit Power Ratio",
"In the considered CR network, the secondary receivers and the primary receivers share the same spectrum resource.",
"However, the primary receivers are licensed users and thus the secondary transmitter is required to ensure the QoS of the primary receivers via a careful resource allocation design.",
"Strong interference may impair the primary network when the secondary transmitter increases its transmit power for providing SWIPT services to the secondary receivers.",
"As a result, the interference power leakage-to-transmit power ratio (IPTR) is an important performance measure for designing the secondary CR network and should be captured in the resource allocation algorithm design.",
"To this end, we first define the total interference power received by the $J$ primary receivers as $\\mathrm {IP}(\\mathbf {w},\\mathbf {V})=\\sum _{j=1}^{J}\\Big (\\mathbf {w}^H\\mathbf {L}_j\\mathbf {w}+\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j\\mathbf {V})\\Big ).$ Thus, the IPTR of the considered secondary CR network is defined as $\\mathrm {IP}_{\\mathrm {ratio}}(\\mathbf {w},\\mathbf {V})=\\frac{\\mathrm {IP}(\\mathbf {w},\\mathbf {V})}{\\mathrm {TP}(\\mathbf {w},\\mathbf {V})}.$"
],
[
"Channel State Information",
"In this paper, we focus on a Time Division Duplex (TDD) communication system with slowly time-varying channels.",
"In practice, handshakingThe legitimate receivers can either take turns to send the handshaking signals or transmit simultaneously with orthogonal pilot sequences.",
"is performed between the secondary transmitter and the secondary receivers at the beginning of each scheduling slot.",
"This allows the secondary transmitter to obtain the statuses and the QoS requirements of the secondary receivers.",
"As a result, by exploiting the channel reciprocity, the downlink CSI of the secondary transmitter to the secondary receivers can be obtained by measuring the uplink training sequences embedded in the handshaking signals.",
"Thus, we assume that the secondary-transmitter-to-secondary-receiver fading gains, $\\mathbf {h}$ and $\\mathbf {g}_{k},\\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , can be reliably estimated at the secondary transmitter at the beginning of each scheduling slot with negligible estimation error.",
"Then, during the transmission, the desired secondary receiver is required to send positive acknowledgement (ACK) packets to inform the secondary transmitter of successful reception of the information packets.",
"Hence, the transmitter is able to update the CSI estimate of the desired receiver frequently via the training sequences in each ACK packet.",
"Therefore, perfect CSI for the secondary-transmitter-to-desired-secondary-receiver link, i.e., $\\mathbf {h}$ , is assumed over the entire transmission period.",
"However, the remaining $K-1$ secondary receivers are idle and there is no interaction between them and the secondary transmitter after handshaking.",
"As a result, the CSI of the idle secondary receivers becomes outdated during transmission.",
"To capture the impact of the CSI imperfection and to isolate specific channel estimation methods from the resource allocation algorithm design, we adopt a deterministic model [40]–[43] for the resulting CSI uncertainty.",
"In particular, the CSI of the link between the secondary transmitter and idle secondary receiver $k$ is modeled as $\\mathbf {g}_k&=&\\mathbf {\\hat{g}}_k + \\Delta \\mathbf {g}_k,\\, \\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace , \\mbox{ and}\\\\{\\Omega }_k&\\triangleq & \\Big \\lbrace \\Delta \\mathbf {g}_k\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1} :\\Delta \\mathbf {g}_k^H \\Delta \\mathbf {g}_k \\le \\varepsilon _k^2\\Big \\rbrace ,\\forall k,$ where $\\mathbf {\\hat{g}}_k\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ is the CSI estimate available at the secondary transmitter at the beginning of a scheduling slot and $ \\Delta \\mathbf {g}_k$ represents the unknown channel uncertainty due to the time varying nature of the channel during transmission.",
"The continuous set ${\\Omega }_k$ in () defines a Euclidean sphere and contains all possible channel uncertainties.",
"Specifically, the radius $\\varepsilon _k$ represents the size of the sphere and defines the uncertainty region of the CSI of idle secondary receiver (potential eavesdropper) $k$ .",
"In practice, the value of $\\varepsilon _k^2$ depends on the coherence time of the associated channel and the duration of transmission.",
"Furthermore, to capture the imperfectness of the CSI of the primary receiver channels at the secondary transmitter, we adopt the same CSI error model as for the idle secondary receivers.",
"In fact, the primary receivers are not directly interacting with the secondary transmitter.",
"Besides, the primary receivers may be silent for non-negligible periods of time due to bursty data communication.",
"As a result, the CSI of the primary receivers can be obtained only occasionally at the secondary transmitter when the primary receivers communicate with a primary transmitter.",
"Hence, we model the CSI of the link between the secondary transmitter and primary receiver $j$ as $\\mathbf {l}_j&=&\\mathbf {\\hat{l}}_j + \\Delta \\mathbf {l}_j,\\, \\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace , \\mbox{ and}\\\\{\\Psi }_j&\\triangleq & \\Big \\lbrace \\Delta \\mathbf {l}_j\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1} :\\Delta \\mathbf {l}_j^H \\Delta \\mathbf {l}_j \\le \\upsilon _j^2\\Big \\rbrace ,\\forall j,$ where $\\mathbf {\\hat{l}}_j$ is the estimate of the channel of primary receiver $j$ at the secondary transmitter and $ \\Delta \\mathbf {l}_j$ denotes the associated channel uncertainty.",
"${\\Psi }_j$ and $\\upsilon _j^2$ in () define the continuous set of all possible channel uncertainties and the size of the uncertainty region of the estimated CSI of primary receiver $j$ , respectively.",
"We note that, in practice, the channel estimation qualities for primary receivers and secondary receivers at the secondary transmitter may be different which leads to different values for $\\varepsilon _k$ and $\\upsilon _j$ ."
],
[
"Optimization Problem Formulations",
"We first propose three problem formulations for single-objective system design for secure communication in the secondary CR network.",
"In particular, each single-objective problem formulation considers one aspect of the system design.",
"Then, we consider the three system design objectives jointly under the framework of multi-objective optimization.",
"In particular, the adopted multi-objective optimization enables the design of a set of Pareto optimal resource allocation policies.",
"The first problem formulation aims at maximizing the energy harvesting efficiency while providing secure communication in the secondary CR network.",
"The problem formulation is as follows: Problem 1 Energy Harvesting Efficiency Maximization: $&&\\hspace*{0.0pt} \\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {maximize}}}\\,\\, \\,\\, \\min _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}\\,\\, \\eta _{\\mathrm {eff}}(\\mathbf {w},\\mathbf {V})\\nonumber \\\\\\mbox{s.t.}",
"&&\\hspace*{-17.07164pt}\\mbox{C1: }\\frac{\\mathbf {w}^H\\mathbf {H}\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\mathbf {V})+\\sigma _\\mathrm {z}^2} \\ge \\Gamma _{\\mathrm {req}}, \\\\&&\\hspace*{-17.07164pt}\\mbox{C2: }\\hspace*{-5.69054pt}\\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}\\frac{\\mathbf {w}^H\\mathbf {G}_k\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k\\mathbf {V})\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt}\\sigma _{\\mathrm {z}_k}^2}\\hspace*{-1.42262pt} \\le \\hspace*{-1.42262pt} \\Gamma _{\\mathrm {tol}_k},\\forall k\\hspace*{-1.42262pt}\\in \\hspace*{-1.42262pt}\\lbrace 1,\\ldots ,K-1\\rbrace ,\\\\&&\\hspace*{-17.07164pt}\\mbox{C3: }\\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}\\,\\,\\frac{\\mathbf {w}^H\\mathbf {L}_j\\mathbf {w}}{\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j\\mathbf {V})+\\sigma _{\\mathrm {PU}}^2} \\hspace*{-1.42262pt}\\le \\hspace*{-1.42262pt} \\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}},\\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace ,\\\\&&\\hspace*{-17.07164pt}\\mbox{C4: } \\Vert \\mathbf {w}\\Vert ^2 +\\operatorname{\\mathrm {Tr}}(\\mathbf {V})\\le P_{\\max }, \\quad \\quad \\mbox{C5:}\\,\\, \\mathbf {V}\\succeq \\mathbf {0}.$ The system objective in (REF ) is to maximize the worst case energy harvesting efficiency of the system for channel estimation errors $\\Delta \\mathbf {g}_k$ belonging to set $\\Omega _k$ .",
"Constant $\\Gamma _{\\mathrm {req}}$ in C1 specifies the minimum required received SINR of the desired secondary receiver for information decoding.",
"$\\Gamma _{\\mathrm {tol}_k},\\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , and $\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}},\\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace $ , in C2 and C3, respectively, are given system parameters which denote the maximum tolerable received SINRs at the potential eavesdroppers in the secondary network and the primary network, respectively.",
"In practice, depending on the considered application, the system operator chooses the values of $\\Gamma _{\\mathrm {req}}$ , $\\Gamma _{\\mathrm {tol}_k},\\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , and $\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}},\\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace $ , such that $\\Gamma _{\\mathrm {req}}\\gg \\Gamma _{\\mathrm {tol}_k}>0$ and $\\Gamma _{\\mathrm {req}}\\gg \\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}>0$ .",
"In other words, the secrecy rate of the system is bounded below by $C_\\mathrm {sec}\\ge \\log _2(1+\\Gamma _{\\mathrm {req}})-\\log _2(1+\\underset{k,j}{\\max }\\lbrace \\Gamma _{\\mathrm {tol}_k},\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}\\rbrace )> 0$ .",
"We note that although $\\Gamma _{\\mathrm {req}}$ , $\\Gamma _{\\mathrm {tol}_k}$ , and $\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}$ in C1, C2, and C3, respectively, are not optimization variables in this paper, a balance between secrecy rate and system achievable rate can be struck by varying their values.",
"$P_{\\max }$ in C4 specifies the maximum transmit power in the power amplifier of the analog front-end of the secondary transmitter.",
"C5 and $\\mathbf {V}\\in \\mathbb {H}^{N_\\mathrm {T}}$ are imposed since covariance matrix $\\mathbf {V}$ has to be a positive semidefinite Hermitian matrix.",
"To facilitate the presentation and without loss of generality, we rewrite Problem 1 in (REF ) in the equivalent form [44]: $&&\\hspace*{-56.9055pt} \\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\, \\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}\\,\\, -\\eta _{\\mathrm {eff}}(\\mathbf {w},\\mathbf {V})\\nonumber \\\\\\hspace*{17.07164pt}\\mbox{s.t.}",
"&&\\hspace*{-8.53581pt}\\mbox{C1 -- C5}.$ The second system design objective is the minimization of the total transmit power of the secondary transmitter and can be mathematically formulated as: Problem 2 Total Transmit Power Minimization: $&&\\hspace*{-56.9055pt} \\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\, \\mathrm {TP}(\\mathbf {w},\\mathbf {V})\\nonumber \\\\\\hspace*{8.53581pt}\\mbox{s.t.}",
"&&\\hspace*{-8.53581pt}\\mbox{C1 -- C5}.$ Problem REF yields the minimum total transmit power of the secondary transmitter while ensuring that the QoS requirement on secure communication is satisfied.",
"We note that Problem REF does not take into account the energy harvesting capability of the idle secondary receivers and focuses only on the requirement of secure communication via constraints C1, C2, and C3.",
"Besides, although transmit power minimization has been studied in the literature in different contexts [1], [45], [46], combining Problem 2 with the new Problems 1 and 3 (see below) offers new insights for the design of CR networks providing secure wireless information and power transfer to secondary receivers.",
"The third system design objective concerns the minimization of the worst case IPTR while providing secure communication in the secondary CR network.",
"The problem formulation is given as: Problem 3 Interference Power Leakage-to-Transmit Power Ratio Minimization: $&&\\hspace*{-56.9055pt}\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\,\\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}\\,\\,\\mathrm {IP}_{\\mathrm {ratio}}(\\mathbf {w},\\mathbf {V})\\nonumber \\\\\\hspace*{8.53581pt}\\mbox{s.t.}",
"&&\\hspace*{-8.53581pt}\\mbox{C1 -- C5}.$ Remark 2 In (REF ) and (REF ), the maximization of the energy harvesting efficiency and the minimization of the IPTR are chosen as design objectives, respectively.",
"Alternative design objectives are the maximization of the total harvested power, $\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {maximize}}}\\,\\, \\underset{{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}}{\\min } \\mathrm {HP}(\\mathbf {w},\\mathbf {V})$ , and the minimization of the total interference power leakage, $\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\,\\underset{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}{\\max }\\mathrm {IP}(\\mathbf {w},\\mathbf {V})$ .",
"We will show later that the maximization of the energy harvesting efficiency in (REF ) and the minimization of the IPTR in (REF ) subsume the total harvested power maximization and the total interference power leakage minimization as special cases, respectively.",
"Please refer to Remark REF for the solution of the total interference power leakage minimization and total harvested power maximization problems.",
"Remark 3 In fact, the optimization problem in (REF ) can be extended to the minimization of the maximum received interference leakage per primary receiver.",
"However, such problem formulation does not facilitate the study of the trade-off between interference leakage, energy harvesting, and total transmit power as the system performance is always limited by those primary users which have strong channels with respect to the secondary transmitter.",
"In practice, the system design objectives in Problems 1–3 are all desirable for the system operators of secondary CR networks in providing simultaneous power and secure information transfer.",
"Yet, theses objectives are usually conflicting with each other and each objective focuses on only one aspect of the system.",
"In the literature, multi-objective optimization has been proposed for studying the trade-off between conflicting system design objectives via the concept of Pareto optimality.",
"For facilitating the following exposition, we denote the objective function and the optimal objective value for problem formulation $p\\in \\lbrace 1,2,3\\rbrace $ as $F_{p}(\\mathbf {w},\\mathbf {V})$ and $F_p^*$ , respectively.",
"We define a resource allocation policy which is Pareto optimal as: Definition [47]: A resource allocation policy, $\\lbrace \\mathbf {w,V}\\rbrace $ , is Pareto optimal if and only if there does not exist another policy, $\\lbrace \\mathbf {w^{\\prime },V^{\\prime }}\\rbrace $ , such that $ F_i(\\mathbf {w}^{\\prime },\\mathbf {V}^{\\prime })\\le F_i(\\mathbf {w},\\mathbf {V}), \\forall i\\in \\lbrace 1,2,3\\rbrace $ , and $ F_j(\\mathbf {w}^{\\prime },\\mathbf {V}^{\\prime })< F_j(\\mathbf {w},\\mathbf {V})$ for at least one index $j\\in \\lbrace 1,2,3\\rbrace $ .",
"The set of all Pareto optimal resource allocation polices is called the Pareto frontier or the Pareto optimal set.",
"In this paper, we adopt the weighted Tchebycheff method [47] for investigating the trade-off between objective functions 1, 2, and 3.",
"In particular, the weighted Tchebycheff method can provide the complete Pareto optimal set despite the non-convexity (if any) of the considered problemsIn the literature, different scalarization methods have been proposed for achieving the points of the complete Pareto set for multi-objective optimization [47], [48].",
"However, the weighted Tchebycheff method requires a lower computational complexity compared to other methods such as the weighted product method and the exponentially weighted criterion.",
"; it provides a necessary condition for Pareto optimality.",
"The complete Pareto optimal set can be achieved by solving the following multi-objective problem: Problem 4 Multi-Objective Optimization – Weighted Tchebycheff Method: $&&\\hspace*{-71.13188pt}\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\max _{p\\in \\lbrace \\,1,\\,2,\\,3\\rbrace }\\,\\, \\Bigg \\lbrace \\lambda _p \\Big (\\frac{F_p\\big (\\mathbf {w},\\mathbf {V}\\big )-F_p^*}{\\vert F_p^*\\vert }\\Big )\\Bigg \\rbrace \\nonumber \\\\\\mbox{s.t.}",
"&&\\hspace*{8.53581pt}\\mbox{C1 -- C5},$ In fact, by varying the values of $\\lambda _p$ , Problem 4 yields the complete Pareto optimal set [47], [48].",
"Besides, Problem REF is a generalization of Problems 1, 2, and 3.",
"In particular, Problem REF is equivalentHere, “equivalent\" means that the considered problems share the same optimal resource allocation solution(s).",
"to Problem $p$ when $\\lambda _p=1$ and $\\lambda _i=0, \\forall i\\ne p$ .",
"For instance, if the secondary energy harvesting receivers do not require wireless power transfer from the secondary transmitter, without loss of generality, we can set $\\lambda _1=0$ in Problem 4 to study the tradeoff between the remaining two system design objectives.",
"In addition, this commonly adopted approach also provides a non-dimensional objective function, i.e., the unit of the objective function is normalized.",
"Remark 4 Finding the Pareto optimal set of the multi-objective optimization problem provides a set of Pareto optimal resource allocation policies.",
"Then, depending on the preference of the system operator, a proper resource allocation policy can be selected from the set for implementation.",
"We note that the resource allocation algorithm in [29] cannot be directly applied to the problems considered in this paper since it was designed for single-objective optimization, namely for the for minimization of the total transmit power.",
"Remark 5 Another possible problem formulation for the considered system model is to move some of the objective functions in (REF ), (REF ), and (REF ) to the set of constraints and constrain each of them by some constant.",
"Then, by varying the constants, trade-offs between different objectives can be struck.",
"However, in general, such a problem formulation does not reveal the Pareto optimal set due to the non-convexity of the problem."
],
[
"Solution of the Optimization Problems",
"The optimization problems in (REF ), (REF ), and (REF ) are non-convex with respect to the optimization variables.",
"In particular, the non-convexity arises from objective function 1, objective function 3, and constraint C1.",
"In order to obtain tractable solutions for the problems, we recast Problems 1, 2, 3, and 4 as convex optimization problems by semidefinite programming (SDP) relaxation [49], [50] and study the tightness of the adopted relaxation in this section."
],
[
"Semidefinite Programming Relaxation",
"To facilitate the SDP relaxation, we define $\\mathbf {W}=\\mathbf {w}\\mathbf {w}^H,\\, \\mathbf {W}=\\frac{\\overline{\\mathbf {W}}}{\\xi }, \\mathbf {V}=\\frac{\\overline{\\mathbf {V}}}{\\xi },\\, \\xi =\\frac{1}{\\operatorname{\\mathrm {Tr}}({\\mathbf {W}})+\\operatorname{\\mathrm {Tr}}({\\mathbf {V}})},$ and rewrite Problems 1 – 4 in terms of new optimization variables $\\overline{\\mathbf {W}}$ , $\\overline{\\mathbf {V}}$ , and $\\xi $ .",
"Transformed Problem 1 Energy Harvesting Efficiency Maximization: $&&\\hspace*{-28.45274pt}\\underset{\\overline{\\mathbf {V}},\\overline{\\mathbf {W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\xi }{\\operatorname{\\mathrm {minimize}}}\\,\\, \\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}-\\sum _{k=1}^{K-1}\\eta _k\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))\\nonumber \\\\\\mbox{s.t.}",
"&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C1}}$:} \\frac{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\overline{\\mathbf {W}})}{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\overline{\\mathbf {V}})+\\sigma _\\mathrm {z}^2\\xi } \\ge \\Gamma _{\\mathrm {req}}, \\\\&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C2}}$: }\\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}\\frac{\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k\\overline{\\mathbf {W}})}{\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k\\overline{\\mathbf {V}})+\\sigma _{\\mathrm {z}_k}^2\\xi } \\le \\Gamma _{\\mathrm {tol}_k},\\forall k,\\\\&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C3}}$: }\\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}\\frac{\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j\\overline{\\mathbf {W}})}{\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j\\overline{\\mathbf {V}})+\\sigma _{\\mathrm {PU}}^2\\xi } \\le \\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}},\\forall j,\\\\&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C4}}$: }\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {W}}) +\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {V}})\\le P_{\\max }\\xi , \\\\&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C5}}$:}\\,\\, \\mathbf {\\overline{W}},\\mathbf {\\overline{V}}\\succeq \\mathbf {0},\\hspace*{14.22636pt}\\mbox{$\\overline{\\mbox{C6}}$:}\\,\\, \\xi \\ge 0, \\\\&&\\hspace*{-14.22636pt}\\mbox{$\\overline{\\mbox{C7}}$:}\\,\\, \\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {W}})+\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {V}})= 1,\\,\\,\\mbox{$\\overline{\\mbox{C8}}$:}\\,\\, \\operatorname{\\mathrm {Rank}}(\\mathbf {\\overline{W}})=1,$ where $\\overline{\\mathbf {W}}\\succeq \\mathbf {0}$ , $\\overline{\\mathbf {W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}$ , and $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}})=1$ in (REF ) are imposed to guarantee that $\\overline{\\mathbf {W}}=\\xi \\mathbf {w}\\mathbf {w}^H$ after optimizing $\\mathbf {\\overline{W}}$ .",
"Transformed Problem 2 Total Transmit Power Minimization: $ \\underset{\\overline{\\mathbf {V}},\\mathbf {\\overline{W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}, \\xi }{\\operatorname{\\mathrm {minimize}}}&&\\,\\, \\frac{1}{\\xi }\\nonumber \\\\&&\\hspace*{-56.9055pt}\\mbox{s.t.",
"}\\, \\mbox{$\\overline{\\mbox{C1}}$ -- $\\overline{\\mbox{C8}}$}.$ Transformed Problem 3 Interference Power Leakage-to-Transmit Power Ratio Minimization: $ \\underset{\\overline{\\mathbf {V}},\\mathbf {\\overline{W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}, \\xi }{\\operatorname{\\mathrm {minimize}}}&&\\,\\, \\hspace*{-14.22636pt}\\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}\\sum _{j=1}^{J}\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))\\nonumber \\\\&&\\hspace*{0.0pt}\\mbox{s.t.",
"}\\, \\mbox{$\\overline{\\mbox{C1}}$ -- $\\overline{\\mbox{C8}}$}.$ Transformed Problem 4 Multi-Objective Optimization: $&&\\hspace*{28.45274pt}\\underset{\\overline{\\mathbf {V}},\\mathbf {\\overline{W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}, \\xi ,\\tau }{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\tau \\nonumber \\\\&&\\hspace*{22.76228pt}\\mbox{s.t.",
"}\\,\\mbox{$\\overline{\\mbox{C1}}$ -- $\\overline{\\mbox{C8}}$},\\\\&&\\hspace*{-34.14322pt}\\mbox{$\\overline{\\mbox{C9}}$a: }\\frac{\\lambda _1}{\\vert F_1^*\\vert } (\\overline{F_1}-F_1^*)\\le \\tau ,\\,\\,\\,\\mbox{$\\overline{\\mbox{C9}}$b: }\\frac{\\lambda _2}{\\vert F_2^*\\vert } (\\overline{F_2}-F_2^*)\\le \\tau ,\\\\&&\\hspace*{-34.14322pt}\\mbox{$\\overline{\\mbox{C9}}$c: }\\frac{\\lambda _3}{\\vert F_3^*\\vert } (\\overline{F_3}-F_3^*)\\le \\tau ,$ where $\\overline{F_1}=\\underset{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}{\\min }-\\sum _{k=1}^{K-1}\\varepsilon _k\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))$ , $\\overline{F_2}= \\frac{1}{\\xi }$ , $\\overline{F_3}= \\underset{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}{\\max }\\sum _{j=1}^{J}\\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))$ , $\\tau $ is an auxiliary optimization variable, and (REF ) is the epigraph representation [44] of (REF ).",
"Proposition 1 The above transformed Problems (REF )–(REF ) are equivalent to the original problems in (REF )–(REF ), respectively.",
"Specifically, we can recover the solutions of the original problems from the solutions of the transformed problems based on (REF ).",
"Please refer to Appendix A.",
"Since transformed Problem 4 is a generalization of transformed Problems 1, 2, and 3, we focus on the methodology for solving transformed ProblemIn studying the solution structure of transformed Problem 4, we assume that the optimal objective values of transformed Problems 1–3 are given constants, i.e., $F^*_p,\\forall p\\in \\lbrace 1,2,3\\rbrace $ , are known.",
"Once the structure of the optimal resource allocation scheme of transformed Problem 4 is obtained, it can be exploited to obtain the optimal solution of transformed Problems 1–3.",
"4.",
"In practice, the considered problems may be infeasible when the channels are in unfavourable conditions and/or the QoS requirements are too stringent.",
"However, in the sequel, for studying the trade-off between different system design objectives and the design of different resource allocation schemes, we assume that the problem is always feasibleWe note that multiple optimal solutions may exist for the considered problems and the proposed optimal resource allocation scheme is able to find at least one of the global optimal solutions.. First, we address constraints $\\mbox{$\\overline{\\mbox{C2}}$}$ , $\\mbox{$\\overline{\\mbox{C3}}$}$ , and $\\mbox{$\\overline{\\mbox{C9}}$}$ .",
"We note that although these constraints are convex with respect to the optimization variables, they are semi-infinite constraints which are generally intractable.",
"For facilitating the design of a tractable resource allocation algorithm, we introduce two auxiliary optimization variables $E_{k}^{\\mathrm {SU}}$ and $I_{j}^{\\mathrm {PU}}$ and rewrite transformed Problem 4 in (REF ) as $&&\\underset{\\mathbf {\\overline{W}},\\overline{\\mathbf {V}}\\in \\mathbb {H}^{N_{\\mathrm {T}}},I_{j}^{\\mathrm {PU}},E_{k}^{\\mathrm {SU}}, \\xi ,\\tau ,}{\\operatorname{\\mathrm {minimize}}}\\,\\,\\,\\tau \\nonumber \\\\&&\\mbox{s.t.",
"}\\,\\mbox{$\\overline{\\mbox{C1}}$ -- $\\overline{\\mbox{C8}}$},\\\\&&\\hspace*{-17.07182pt}\\mbox{$\\overline{\\mbox{C9}}$a: }\\lambda _1 \\Big (\\sum _{k=1}^{K-1}E_{k}^{\\mathrm {SU}}-F_1^*\\Big )\\le \\tau \\vert F_1^*\\vert ,\\\\&&\\hspace*{-17.07182pt}\\mbox{$\\overline{\\mbox{C9}}$b: }\\lambda _2(\\overline{F_2}-F_2^*)\\le \\tau \\vert F_2^*\\vert , \\\\&&\\hspace*{-17.07182pt}\\mbox{$\\overline{\\mbox{C9}}$c: }\\lambda _3 \\Big (\\sum _{j=1}^{J}I_{j}^{\\mathrm {PU}}-F_3^*\\Big )\\le \\tau \\vert F_3^*\\vert ,\\\\&&\\hspace*{-17.07182pt}\\mbox{$\\overline{\\mbox{C10}}$: }E_{k}^{\\mathrm {SU}}\\ge \\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}- \\eta _k\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}})),\\forall k,\\\\&&\\hspace*{-17.07182pt}\\mbox{$\\overline{\\mbox{C11}}$: }I_{j}^{\\mathrm {PU}}\\ge \\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j} \\operatorname{\\mathrm {Tr}}(\\mathbf {L}_j(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}})),\\forall j.$ In fact, the introduced auxiliary variables $E_{k}^{\\mathrm {SU}}$ and $I_{j}^{\\mathrm {PU}}$ decouple the original two nested semi-infinite constraints into two semi-infinite constraints and two affine constraints, i.e., $\\mbox{$\\overline{\\mbox{C10}}$}$ , $\\mbox{$\\overline{\\mbox{C11}}$}$ and $\\mbox{$\\overline{\\mbox{C9}}$a}$ , $\\mbox{$\\overline{\\mbox{C9}}$c}$ , respectively.",
"It can be verified that (REF ) is equivalent to (REF ), i.e., constraints $\\mbox{$\\overline{\\mbox{C10}}$}$ and $\\mbox{$\\overline{\\mbox{C11}}$}$ are satisfied with equality for the optimal solution.",
"Next, we transform constraints $\\mbox{$\\overline{\\mbox{C2}}$}$ , $\\mbox{$\\overline{\\mbox{C3}}$}$ , $\\mbox{$\\overline{\\mbox{C10}}$}$ , and $\\mbox{$\\overline{\\mbox{C11}}$}$ into linear matrix inequalities (LMIs) using the following lemma: Lemma 1 (S-Procedure [44]) Let a function $f_m(\\mathbf {x}),m\\in \\lbrace 1,2\\rbrace ,\\mathbf {x}\\in \\mathbb {C}^{N\\times 1},$ be defined as $f_m(\\mathbf {x})=\\mathbf {x}^H\\mathbf {A}_m\\mathbf {x}+2 \\mathrm {Re} \\lbrace \\mathbf {b}_m^H\\mathbf {x}\\rbrace +c_m,$ where $\\mathbf {A}_m\\in \\mathbb {H}^N$ , $\\mathbf {b}_m\\in \\mathbb {C}^{N\\times 1}$ , and $c_m\\in \\mathbb {R}$ .",
"Then, the implication $f_1(\\mathbf {x})\\le 0\\Rightarrow f_2(\\mathbf {x})\\le 0$ holds if and only if there exists a $\\delta \\ge 0$ such that $\\delta \\begin{bmatrix}\\mathbf {A}_1 & \\mathbf {b}_1 \\\\\\mathbf {b}_1^H & c_1 \\\\\\end{bmatrix} -\\begin{bmatrix}\\mathbf {A}_2 & \\mathbf {b}_2 \\\\\\mathbf {b}_2^H & c_2 \\\\\\end{bmatrix} \\succeq \\mathbf {0},$ provided that there exists a point $\\mathbf {\\hat{x}}$ such that $f_k(\\mathbf {\\hat{x}})<0$ .",
"Now, we apply Lemma 1 to constraint $\\mbox{$\\overline{\\mbox{C2}}$}$ .",
"In particular, we substitute $\\mathbf {g}_k=\\mathbf {\\hat{g}}_k +\\Delta \\mathbf {g}_k$ into constraint $\\mbox{$\\overline{\\mbox{C2}}$}$ .",
"Therefore, the implication, $&&\\hspace*{-1.42262pt} \\Delta \\mathbf {g}_k^H \\Delta \\mathbf {g}_k\\hspace*{-2.84526pt}\\le \\hspace*{-2.84526pt} \\varepsilon _k^2\\\\\\Rightarrow \\,\\hspace*{-8.53581pt}&&\\hspace*{-5.12149pt} \\mbox{$\\overline{\\mbox{C2}}$: }0\\hspace*{-2.84526pt}\\ge \\hspace*{-2.84526pt} \\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k} \\Delta \\mathbf {g}_k^H\\big (\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}}-\\overline{\\mathbf {V}}\\big )\\Delta \\mathbf {g}_k\\\\&& \\hspace*{-19.34787pt}+2\\mathrm {Re}\\Big \\lbrace \\mathbf {\\hat{g}}_k^H\\big (\\hspace*{-1.42262pt}\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}}\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\hspace*{-1.42262pt}\\big )\\Delta \\mathbf { g}_k\\Big \\rbrace +\\mathbf {\\hat{g}}_k^H\\big (\\hspace*{-1.42262pt}\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}}\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\hspace*{-1.42262pt}\\big )\\mathbf {\\hat{g}}_k\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\xi \\sigma _{\\mathrm {z}_k}^2,\\forall k,$ holds if and only if there exists a $\\delta _k\\ge 0$ such that the following LMI constraint holds: $\\mbox{$\\overline{\\mbox{C2}}$: } &&\\hspace*{-14.22636pt}\\mathbf {S}_{\\mathrm {\\overline{C2}}_k}(\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}, \\xi ,\\delta _k)\\\\ =&&\\hspace*{-17.07164pt}\\begin{bmatrix}\\delta _k\\mathbf {I}_{N_{\\mathrm {T}}}+\\overline{\\mathbf {V}}-\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}} & \\hspace*{-17.07164pt} (\\overline{\\mathbf {V}}-\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}})\\mathbf {\\hat{g}}_k \\\\\\hspace*{-5.69054pt}\\mathbf {\\hat{g}}_k^H (\\overline{\\mathbf {V}}-\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}}) & \\hspace*{-17.07164pt} -\\delta _k\\varepsilon _k^2 +\\xi \\sigma _{\\mathrm {z}_k}^2+ \\mathbf {\\hat{g}}_k^H (\\overline{\\mathbf {V}}-\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_k}}) \\mathbf {\\hat{g}}_k \\\\\\end{bmatrix}\\\\=&&\\hspace*{-17.07164pt}\\begin{bmatrix}\\delta _k\\mathbf {I}_{N_{\\mathrm {T}}}+\\overline{\\mathbf {V}} & \\overline{\\mathbf {V}}\\mathbf {\\hat{g}}_k \\\\\\mathbf {\\hat{g}}_k^H \\overline{\\mathbf {V}} & -\\delta _k\\varepsilon _k^2 +\\xi \\sigma _{\\mathrm {z}_k}^2+ \\mathbf {\\hat{g}}_k^H \\overline{\\mathbf {V}} \\mathbf {\\hat{g}}_k \\\\\\end{bmatrix} \\\\- &&\\hspace*{-17.07164pt} \\frac{1}{\\Gamma _{\\mathrm {tol}_k}} \\mathbf {U}_{\\mathbf {g}_k}^H\\overline{\\mathbf {W}}\\mathbf {U}_{\\mathbf {g}_k}\\succeq \\mathbf {0}, \\forall k,$ for $\\delta _k\\ge 0, k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ where $\\mathbf {U}_{\\mathbf {g}_k}=\\Big [\\mathbf {I}_{N_{\\mathrm {T}}}\\quad \\mathbf {\\hat{g}}_k\\Big ]$ .",
"Similarly, we rewrite constraints $\\mbox{$\\overline{\\mbox{C3}}$}$ , $\\mbox{$\\overline{\\mbox{C10}}$}$ , and $\\mbox{$\\overline{\\mbox{C11}}$}$ in the form of (REF ) which leads to $\\mbox{$\\overline{\\mbox{C3}}$: }&&\\hspace*{-17.07164pt}0\\hspace*{-2.84526pt}\\ge \\hspace*{-2.84526pt} \\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j} \\Delta \\mathbf {l}_j^H\\big (\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}}\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\big )\\Delta \\mathbf {l}_j\\\\&&\\hspace*{-42.67912pt}+2\\mathrm {Re}\\Big \\lbrace \\hspace*{-1.42262pt}\\mathbf {\\hat{l}}_j^H\\big (\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}}\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\big )\\Delta \\mathbf { l}_j\\hspace*{-1.42262pt}\\Big \\rbrace \\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\mathbf {\\hat{l}}_j^H\\big (\\frac{\\overline{\\mathbf {W}}}{\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}}\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\big )\\mathbf {\\hat{l}}_j\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\xi \\sigma _{\\mathrm {PU}}^2,\\forall k,\\\\\\mbox{$\\overline{\\mbox{C10}}$: }&&\\hspace*{-17.07164pt}0\\hspace*{-2.84526pt}\\ge \\hspace*{-2.84526pt} \\max _{\\Delta \\mathbf {g}_k\\in {\\Omega }_k} -\\eta _k\\Big \\lbrace \\Delta \\mathbf {g}_k^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\Delta \\mathbf {g}_k\\\\&&\\hspace*{-42.67912pt}+2\\mathrm {Re}\\Big \\lbrace \\mathbf {\\hat{g}}_k^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\Delta \\mathbf { g}_k\\Big \\rbrace \\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\mathbf {\\hat{g}}_k^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\mathbf {\\hat{g}}_k\\Big \\rbrace \\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}E_{k}^{\\mathrm {SU}},\\forall k,\\,\\mbox{and}\\\\\\mbox{$\\overline{\\mbox{C11}}$: }&&\\hspace*{-17.07164pt}0\\hspace*{-2.84526pt}\\ge \\hspace*{-2.84526pt} \\max _{\\Delta \\mathbf {l}_j\\in {\\Psi }_j} \\Delta \\mathbf {l}_j^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\Delta \\mathbf {l}_j\\\\ &&\\hspace*{-42.67912pt}+ 2\\mathrm {Re}\\Big \\lbrace \\mathbf {\\hat{l}}_j^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\Delta \\mathbf { l}_j\\Big \\rbrace \\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\mathbf {\\hat{l}}_j^H\\big (\\overline{\\mathbf {W}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt} \\overline{\\mathbf {V}}\\big )\\mathbf {\\hat{l}}_j\\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}I_{j}^{\\mathrm {PU}},\\forall j,$ respectively.",
"Figure: NO_CAPTIONBy using Lemma 1, constraint $\\mbox{$\\overline{\\mbox{C3}}$}$ , $\\mbox{$\\overline{\\mbox{C10}}$}$ , and $\\mbox{$\\overline{\\mbox{C11}}$}$ can be equivalently written as $&&\\mbox{$\\overline{\\mbox{C3}}$: } \\mathbf {S}_{\\mathrm {\\overline{C3}}_j}(\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}, \\xi ,\\gamma _j)\\\\\\hspace*{-7.11317pt}&=&\\hspace*{-7.11317pt}\\begin{bmatrix}\\gamma _j\\mathbf {I}_{N_{\\mathrm {T}}}+\\overline{\\mathbf {V}} & \\overline{\\mathbf {V}}\\mathbf {\\hat{l}}_j \\\\\\mathbf {\\hat{l}}_j^H \\overline{\\mathbf {V}} & \\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\gamma _j\\upsilon _j^2 +\\xi \\sigma _{\\mathrm {PU}}^2+ \\mathbf {\\hat{l}}_j^H \\overline{\\mathbf {V}} \\mathbf {\\hat{l}}_j \\\\\\end{bmatrix}\\\\\\hspace*{-1.42262pt}&-&\\hspace*{-1.42262pt}\\frac{ \\mathbf {U}_{\\mathbf {l}_j}^H\\overline{\\mathbf {W}}\\mathbf {U}_{\\mathbf {l}_j}}{\\Gamma _{\\mathrm {tol}_j}^\\mathrm {PU}} \\hspace*{-1.42262pt}\\succeq \\hspace*{-1.42262pt}\\mathbf {0}, \\forall j,\\\\&&\\mbox{$\\overline{\\mbox{C10}}$: }\\mathbf {S}_{\\mathrm {\\overline{C10}}_k}(\\overline{\\mathbf {W}},\\overline{\\mathbf {V}},E_{k}^{\\mathrm {SU}}, \\varphi _k)\\\\\\hspace*{-7.11317pt}&=&\\hspace*{-7.11317pt}\\begin{bmatrix}\\varphi _k\\mathbf {I}_{N_{\\mathrm {T}}}\\hspace*{-1.42262pt}+\\hspace*{-1.42262pt}\\overline{\\mathbf {V}}& \\hspace*{-1.42262pt}\\overline{\\mathbf {V}}\\mathbf {\\hat{g}}_k \\\\ \\mathbf {\\hat{g}}_h^H \\overline{\\mathbf {V}}& \\hspace*{-1.42262pt}-\\hspace*{-1.42262pt}\\varphi _k\\varepsilon _k^2 \\hspace*{-1.42262pt}+\\hspace*{-1.42262pt}\\frac{E_{k}^{\\mathrm {SU}}}{\\eta _k}\\hspace*{-1.42262pt} +\\hspace*{-1.42262pt} \\mathbf {\\hat{g}}_k^H \\overline{\\mathbf {V}} \\mathbf {\\hat{g}}_k \\\\\\end{bmatrix}\\\\\\hspace*{-1.42262pt}&+&\\hspace*{-1.42262pt} \\mathbf {U}_{\\mathbf {g}_k}^H\\overline{\\mathbf {W}}\\mathbf {U}_{\\mathbf {g}_k}\\hspace*{-1.42262pt}\\succeq \\hspace*{-1.42262pt}\\mathbf {0}, \\forall k,\\\\&&\\mbox{$\\overline{\\mbox{C11}}$: }\\mathbf {S}_{\\mathrm {\\overline{C11}}_j}(\\overline{\\mathbf {W}},\\overline{\\mathbf {V}},I_{j}^{\\mathrm {PU}},\\omega _j)\\\\\\hspace*{-7.11317pt}&=&\\hspace*{-7.11317pt}\\begin{bmatrix}\\omega _j\\mathbf {I}_{N_{\\mathrm {T}}}-\\overline{\\mathbf {V}}& -\\overline{\\mathbf {V}}\\mathbf {\\hat{l}}_j \\\\-\\mathbf {\\hat{l}}_j^H \\overline{\\mathbf {V}}& -\\omega _j\\upsilon _j^2 +I_{j}^{\\mathrm {PU}} - \\mathbf {\\hat{l}}_j^H \\overline{\\mathbf {V}} \\mathbf {\\hat{l}}_j \\\\\\end{bmatrix}\\\\\\hspace*{-1.42262pt}&-& \\hspace*{-1.42262pt} \\mathbf {U}_{\\mathbf {l}_j}^H\\overline{\\mathbf {W}}\\mathbf {U}_{\\mathbf {l}_j}\\hspace*{-1.42262pt}\\succeq \\hspace*{-1.42262pt} \\mathbf {0}, \\forall j,$ respectively, with $\\mathbf {U}_{\\mathbf {l}_j}=\\Big [\\mathbf {I}_{N_{\\mathrm {T}}}\\quad \\mathbf {\\hat{l}}_j\\Big ]$ and new auxiliary optimization variables $\\gamma _j\\ge 0, j\\in \\lbrace 1,\\ldots ,J\\rbrace ,$ $\\varphi _k\\ge 0, k\\in \\lbrace 1,\\ldots ,K-1\\rbrace ,$ and $\\omega _j\\ge 0, j\\in \\lbrace 1,\\ldots ,J\\rbrace $ .",
"We note that now constraints $\\mbox{$\\overline{\\mbox{C2}}$}, \\mbox{$\\overline{\\mbox{C3}}$}, \\mbox{$\\overline{\\mbox{C10}}$}$ , and $\\mbox{$\\overline{\\mbox{C11}}$}$ involve only a finite number of convex constraints which facilitates an efficient resource allocation algorithm design.",
"As a result, we obtain the following equivalent optimization problem on the top of this page in (REF ), where $\\mathbf {\\Theta }\\triangleq \\lbrace \\mathbf {I}^{\\mathrm {PU}},\\mathbf {E}^{\\mathrm {SU}}, \\xi ,\\tau ,\\gamma ,\\delta ,\\varphi ,\\omega ,\\overline{\\mathbf {V}}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {\\overline{W}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}\\rbrace $ denotes the set of optimization variables after transformation; $\\mathbf {I}^{\\mathrm {PU}}$ and $\\mathbf {E}^{\\mathrm {SU}}$ are auxiliary variable vectors with elements ${I}_j^{\\mathrm {PU}},\\forall j\\in \\lbrace 1,\\ldots ,J\\rbrace ,$ and ${E}_k^{\\mathrm {SU}},\\forall k\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , respectively; $\\delta ,\\gamma ,\\varphi $ , and $\\omega $ are auxiliary optimization variable vectors with elements $\\delta _k,\\gamma _j,\\varphi _k$ , and $\\omega _j\\ge 0$ connected to the constraints in (REF )–(), respectively.",
"The remaining non-convexity of problem (REF ) is due to the combinatorial rank constraint in $\\mbox{$\\overline{\\mbox{C8}}$}$ on the beamforming matrix $\\overline{\\mathbf {W}}$ .",
"In fact, by relaxing constraint $\\mbox{$\\overline{\\mbox{C8}}$: }\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}})=1$ , i.e., removing it from (REF ), the considered problem is a convex SDP and can be solved efficiently by numerical solvers such as SeDuMi [51] and CVX [52].",
"Besides, if the obtained solution for the relaxed SDP problem admits a rank-one matrix $\\overline{\\mathbf {W}}$ , i.e., $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}})=1$ , then it is the optimal solution of the original problem.",
"In general, the adopted SDP relaxation may not yield a rank-one solution and the result of the relaxed problem serves as a performance upper bound for the original problem.",
"Nevertheless, in the following, we show that there always exists an optimal solution for the relaxed problem with $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\overline{W}}) = 1$ .",
"In particular, the optimal solution of the relaxed version of (REF ) with $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\overline{W}}) = 1$ can be obtained from the solution of the dual problem of the SDP relaxed version of (REF ).",
"In other words, we can obtain the global optimal solutions of non-convex Problems 1, 2, 3, and 4.",
"Furthermore, we propose two suboptimal resource allocation schemes which do not require the solution of the dual problem of the SDP relaxed problem."
],
[
"Optimality Condition for SDP Relaxation",
"In this subsection, we first reveal the tightness of the proposed SDP relaxation.",
"The existence of a rank-one solution matrix $\\overline{\\mathbf {W}}$ for the relaxed SDP version of transformed Problem 4 is summarized in the following theorem which is based on [30]We note that [30] was designed for a communication system with SWIPT for the case of perfect CSI and single objective optimization.",
"The application of the results of [30] to the scenarios considered in this paper is only possible after performing the steps and transformations introduced in Section ii@ to Section iv@-A.. Theorem 1 Suppose the optimal solution for the SDP relaxed version of (REF ) is denoted as $\\mathbf {\\Lambda }^*\\triangleq \\lbrace \\mathbf {I}^{\\mathrm {PU}*},\\mathbf {E}^{\\mathrm {SU}*}, \\xi ^*,\\tau ^*,\\gamma ^*, \\delta ^*,\\varphi ^*,\\omega ^*,\\overline{\\mathbf {V}}^*,\\mathbf {\\overline{W}}^*\\rbrace $ and $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}}^*)>1$ .",
"Then, there exists a feasible solution for the SDP relaxed version of (REF ), denoted as $\\mathbf {\\widetilde{\\Lambda }}\\triangleq \\lbrace \\mathbf {\\widetilde{I}}^{\\mathrm {PU}},\\mathbf {\\widetilde{E}}^{\\mathrm {SU}}, \\widetilde{\\xi },\\widetilde{\\tau },\\widetilde{\\gamma },\\widetilde{\\delta },\\widetilde{\\varphi },\\widetilde{\\omega },$ ${\\mathbf {\\widetilde{V}}},\\mathbf {{ \\widetilde{W}}}\\rbrace $ , which not only achieves the same objective value as $\\mathbf {\\Lambda }^*$ , but also admits a rank-one matrix $\\mathbf {\\widetilde{W}}$ , i.e., $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\widetilde{W}})=1$ .",
"The solution $\\mathbf {\\widetilde{\\Lambda }}$ can be constructed exploiting $\\mathbf {\\Lambda }^*$ and the solution of the dual problem of the relaxed version of (REF ).",
"Please refer to Appendix B for the proof of Theorem REF and the method for constructing the optimal solution.",
"Since there always exists an achievable optimal solution with a rank-one beamforming matrix $\\mathbf {\\widetilde{W}}$ , the global optimum of (REF ) can be obtained despite the SDP relaxation.",
"By utilizing Theorem 1, we specify the optimal solution of transformed Problems 1–3 in the following corollary.",
"Figure: NO_CAPTIONCorollary 1 Transformed Problems 1–3 can be solved optimally by applying SDP relaxation and the solution of each problem can be obtained with the method provided in the proof of Theorem 1.",
"In particular, Problem $p$ can be solved by solving Problem REF with $\\lambda _p=1$ , $\\lambda _i=0, \\forall i\\ne p$ , $i\\in \\lbrace 1,2,3\\rbrace $ , and setting $F_p^*,\\forall p\\in \\lbrace 1,2,3\\rbrace ,$ to any non-negative and finite constant$F_p^*,\\forall p\\in \\lbrace 1,2,3\\rbrace $ , are considered to be given constants in Problem 4 for studying the trade-offs between objective functions 1, 2, and 3.",
"Setting $F_p^*=c$ where $0<c<\\infty $ is a constant in Problem 4 is used for recovering the solution of Problems 1, 2, and 3.",
"However, this does not imply that the optimal value of Problem $p$ is equal to $c$ ..",
"Remark 6 The computational complexity of the proposed optimal algorithm with respect to the numbers of secondary users $K$ , the number of primary users $J$ , and transmit antennas $N_{\\mathrm {T}}$ at the secondary transmitter can be characterized as $&&\\operatorname{\\cal O}\\Bigg (\\Big (\\sqrt{2N_{\\mathrm {T}}}\\log \\big (\\frac{1}{\\kappa }\\big )\\Big )\\Big ((2K+2J)(2N_{\\mathrm {T}})^3\\\\&&+(2N_{\\mathrm {T}})^2(2K+2J)^2+(2K+2J)^3\\Big )\\Bigg )$ for a given solution accuracy $\\kappa >0$ , where $\\operatorname{\\cal O}(\\cdot )$ is the big-O notation.",
"We note that polynomial time computational complexity algorithms are considered to be fast algorithms in the literature [53] and are desirable for real time implementation.",
"Besides, the computational complexity can be further reduced by adopting a tailor made interior point method [54], [55].",
"Also, we would like to emphasize that in practice, $\\lambda _1,\\lambda _2,$ and $\\lambda _3$ are given parameters and thus we only need to compute one point of the trade-off region."
],
[
"Suboptimal Resource Allocation Schemes",
"As discussed in Appendix B, constructing the optimal solution $\\mathbf {\\widetilde{\\Lambda }}$ with $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\widetilde{W}})=1$ requires the solution of the dual problem of problem (REF ) as the Lagrange multiplier matrix $\\mathbf {Y}^*$ is needed in (REF ).",
"Nevertheless, $\\mathbf {Y}^*$ may not be provided by some numerical solvers and thus the construction of a rank-one solution matrix $\\mathbf {\\widetilde{W}}$ may not be possible.",
"In the following, we propose two suboptimal resource allocation schemes based on the solution of the primal problem of the relaxed version of (REF ) which do not require knowledge of $\\mathbf {Y}^*$ when $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}}^*)>1$ ."
],
[
"Suboptimal Resource Allocation Scheme 1",
"The first proposed suboptimal resource allocation scheme is a hybrid scheme and is based on the solution of the relaxed version of (REF ).",
"We first solve (REF ) by SDP relaxation.",
"If the solution admits a rank-one $\\overline{\\mathbf {W}}$ , then the global optimal solution of (REF ) is obtained.",
"Otherwise, we construct a suboptimal beamforming matrix $ \\overline{\\mathbf { W}}_{\\mathrm {sub}}$ .",
"Suppose $\\overline{\\mathbf {W}}$ is a matrix with rank $N$ .",
"Then $\\overline{\\mathbf {W}}$ can be written as $\\overline{\\mathbf {W}}=\\sum _{t=1}^N \\vartheta _t \\mathbf {e}_t\\mathbf {e}^H_t $ , where $\\vartheta _t$ and $\\mathbf {e}_t\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ are the descending eigenvalues, i.e., $\\vartheta _1\\ge \\vartheta _2\\ge ,\\ldots ,\\ge \\vartheta _t,\\ldots ,\\ge \\vartheta _N$ , and eigenvectors associated with $\\overline{\\mathbf {W}}$ , respectively.",
"Now, we introduce the suboptimal beamforming vector $\\overline{\\mathbf { w}}_{\\mathrm {sub}}=\\mathbf {e}_1$ such that $\\overline{\\mathbf { W}}_{\\mathrm {sub}} = \\overline{\\mathbf { w}}_{\\mathrm {sub}}\\overline{\\mathbf { w}}_{\\mathrm {sub}}^H$ .",
"Then, we define a scalar optimization variable $P_b$ which controls the power of the suboptimal beamforming matrix.",
"As a result, a new optimization problem is then given by (REF ) on the top of this page, where $\\mathbf {\\Theta }_{\\mathrm {sub}}\\triangleq \\lbrace P_b,\\mathbf {I}^{\\mathrm {PU}},\\mathbf {E}^{\\mathrm {SU}}, \\xi ,\\tau ,\\gamma ,\\delta ,\\varphi ,\\omega ,\\overline{\\mathbf {V}}\\in \\mathbb {H}^{N_{\\mathrm {T}}}\\rbrace $ is the new set of optimization variables for suboptimal resource allocation scheme 1.",
"The problem formulation in (REF ) is jointly convex with respect to the optimization variables and can be solved by using efficient numerical solvers.",
"Besides, the solution of (REF ) satisfies the constraints of (REF ), thus the solution of (REF ) serves as a suboptimal solution for (REF ) since the beamforming matrix $\\overline{\\mathbf { W}}_{\\mathrm {sub}}$ is fixed which leads to reduced degrees of freedom for resource allocation."
],
[
"Suboptimal Resource Allocation Scheme 2",
"The second proposed suboptimal resource allocation scheme is also a hybrid scheme.",
"It adopts a similar approach to solve the problem as suboptimal resource allocation scheme 1, except for the choice of the suboptimal beamforming matrix $\\overline{\\mathbf { W}}_{\\mathrm {sub}}$ when $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}}^*)>1$ .",
"Here, the choice of beamforming matrix $\\overline{\\mathbf { W}}_{\\mathrm {sub}}$ is based on the rank-one Gaussian randomization scheme [56].",
"Specifically, we calculate the eigenvalue decomposition of $\\overline{\\mathbf {W}}^*=\\mathbf {U}\\mathbf {\\Sigma }\\mathbf {U}^H$ , where $\\mathbf {U}=\\Big [\\mathbf {e}_1\\ldots \\mathbf {e}_N\\Big ]$ and $\\mathbf {\\Sigma }=\\operatorname{\\mathrm {diag}}\\big ({\\vartheta _1},\\ldots ,{\\vartheta _N}\\big )$ are an $N_\\mathrm {T}\\times N_\\mathrm {T}$ unitary matrix and a diagonal matrix, respectively.",
"Then, we adopt the suboptimal beamforming vector $\\overline{\\mathbf { w}}_{\\mathrm {sub}}=\\mathbf {U}\\mathbf {\\Sigma }^{1/2}\\mathbf {r}, \\overline{\\mathbf { W}}_{\\mathrm {sub}}=P_b\\overline{\\mathbf { w}}_{\\mathrm {sub}}\\overline{\\mathbf { w}}_{\\mathrm {sub}}^H$ , where $\\mathbf {r}\\in {\\mathbb {C}}^{N_{\\mathrm {T}}\\times 1}$ and $\\mathbf {r}\\sim {\\cal CN}(\\mathbf {0}, \\mathbf {I}_{N_{\\mathrm {T}}})$ .",
"Subsequently, we follow the same approach as in (REF ) for optimizing $\\mathbf {\\Theta }_{\\mathrm {sub}}$ and obtain a suboptimal rank-one solution $P_b \\overline{\\mathbf { W}}_{\\mathrm {sub}}$ .",
"We note that suboptimal resource allocation scheme 2 provides flexibility for trading computational complexity and system performance which is not offered by scheme 1.",
"In fact, by executing scheme 2 repeatedly for different Gaussian distributed random vectors $\\mathbf {r}$ , the performance of scheme 2 can be improved by selecting the best $\\overline{\\mathbf { w}}_{\\mathrm {sub}}=\\mathbf {U}\\mathbf {\\Sigma }^{1/2}\\mathbf {r}$ over different trials.",
"Remark 7 We note that the solution of the total received interference power minimization ($\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {minimize}}}\\,\\underset{\\Delta \\mathbf {l}_j\\in {\\Psi }_j}{\\max }$ $\\mathrm {IP}(\\mathbf {w},\\mathbf {V})$ ) and total harvested power maximization ($\\underset{\\mathbf {V}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\mathbf {w}}{\\operatorname{\\mathrm {maximize}}}\\, \\underset{{\\Delta \\mathbf {g}_k\\in {\\Omega }_k}}{\\min } \\mathrm {HP}(\\mathbf {w},\\mathbf {V})$ ) problems can be obtained by applying Corollary 1 and solving Problem 4 after setting $\\zeta =1$ and removing constraint $\\mbox{$\\overline{\\mbox{C7}}$}$ ."
],
[
"Results",
"We evaluate the system performance of the proposed resource allocation schemes using simulations.",
"The important simulation parameters are summarized in Table REF .",
"A reference distance of 2 meters for the path loss model is selected.",
"There are $K$ receivers uniformly distributed between the reference distance and the maximum service distance of 20 meters in the secondary network.",
"Besides, we assume that the primary transmitter is 40 meters away from the secondary transmitter.",
"In particular, there are $J$ primary receivers uniformly distributed between 20 meters and 40 from the secondary transmitter, cf.",
"Figure REF .",
"Because of path loss and channel fading, different secondary receivers experience different interference powers from the primary transmitterFrom Table REF , we observe that the secondary transmitter, which needs to provide both information and energy, has a higher maximum transmit power budget compared to the primary transmitter which only provides information signals to the receivers in its networks.",
"In fact, by exploiting the extra degrees of freedom offered by the multiple transmit antennas, the secondary transmitter can transmit a high power to the secondary receivers and cause a minimal interference to the primary network.",
"On the contrary, the primary transmitter is equipped with a single antenna only and has to transmit with a relatively small power to avoid harmful interference.. To facilitate the presentation, in the sequel, we define the normalized maximum channel estimation errors of primary receiver $j$ and idle secondary receiver $k$ as $\\sigma _{\\mathrm {PU}_j}^2=\\frac{\\upsilon ^2_j}{\\Vert \\mathbf {l}_j\\Vert ^2}$ and $\\sigma _{\\mathrm {SU}_k}^2=\\frac{\\varepsilon ^2_k}{\\Vert \\mathbf {g}_k\\Vert ^2}$ , respectively, with $\\sigma _{\\mathrm {PU}_a}^2=\\sigma _{\\mathrm {PU}_b}^2,\\forall a, b\\in \\lbrace 1,\\ldots ,J\\rbrace $ , for all primary receivers and $\\sigma _{\\mathrm {SU}_c}^2=\\sigma _{\\mathrm {SU}_d}^2,\\forall c, d\\in \\lbrace 1,\\ldots ,K-1\\rbrace $ , for all secondary receivers, respectively.",
"Unless specified otherwise, we assume normalized maximum channel estimation errors of idle secondary receiver $k$ and primary receiver $j$ of $\\sigma _{\\mathrm {SU}_k}^2=0.01,\\sigma _{\\mathrm {PU}_j}^2=0.05,\\forall k,j$ .",
"Besides, we study the trade-off between the different objective functions via the solution of Problem 4 for two cases.",
"In particular, in Case I, we study the trade-off between the objective functions for total harvested power maximization, total interference power leakage minimization, and total transmit power minimization, cf.",
"Remark REF and Remark REF ; in Case II, we study the trade-off between the objective functions for energy harvesting efficiency maximization, average IPTR minimization, and average total transmit power minimization.",
"The average system performance shown in the following sections is obtained by averaging over different realizations of both path loss and multipath fading.",
"Figure: CR SWIPT network simulation topology.Table: System parameters."
],
[
"Trade-off Regions for Case I and Case II",
"Figures REF and REF depict the trade-off regions for the system objectives for Case I and Case II achieved by the proposed optimal resource allocation scheme, respectively.",
"There are one active secondary receiver, $K-1=3$ idle secondary receivers, and $J=2$ primary receivers.",
"The trade-off regions in Figures REF and REF are obtained by solving Problem 4 via varying the values of $0\\le \\lambda _p\\le 1,\\forall p\\in \\lbrace 1,2,3\\rbrace $ , uniformly for a step size of $0.04$ such that $\\sum _p \\lambda _p=1$ .",
"We use asterisk markers to denote the trade-off region achieved by the considered resource allocation scheme and colored circles to represent the Pareto frontier [47].",
"Figure: System objective trade-off region for Case II.For the trade-off region for Case I in Figure REF , it can be observed that although the system design objectives of total transmit power minimization and total interference power leakage minimization do not share the same optimal solution (a single point which is the minimum of both objective functions), these two objectives are partially aligned with each other.",
"Specifically, a large portion of the trade-off region is concentrated at the bottom of the figure.",
"In other words, a resource allocation policy which minimizes the total transmit power can also reduce the total interference power leakage effectively and vice versa.",
"On the contrary, the objective of total harvested power maximization conflicts with the other two objective functions.",
"In particular, in order to maximize the total harvested power, the secondary transmitter has to transmit with full power in every time instant despite the imperfection of the CSI.",
"The associated resource allocation policy with full power transmission corresponds to the top corner point in Figure REF .",
"Besides, if the secondary transmitter employs a large transmit power, a high average total interference power leakage at the primary receivers will result.",
"Furthermore, the total harvested power in the system is in the order of milliwatt which is sufficient to charge the sensor type idle secondary receivers, despite the existence of the primary receivers, cf.",
"footnote REF .",
"For the trade-off region for Case II in Figure REF , it can be seen that a significant portion of the trade-off region is concentrated near the bottom and the remaining parts spread over the entire space of the figure.",
"The fact that the trade-off region is condensed near the bottom indicates that resource allocation policies which minimize the total transmit power can also reduce the IPTR to a certain extent and vice versa.",
"However, there also exist resource allocation policies that incur a high transmit power while achieving a low IPTR, i.e., the points located near an average total transmit power of 30 dBm and average IPTR $=0.1\\%$ .",
"This can be explained by the fact that the objective functions for energy harvesting efficiency maximization and IPTR minimization are invariant to a simultaneous positive scaling of both $\\mathbf {W}$ and $\\mathbf {V},$ e.g.",
"$\\frac{\\mathrm {HP}(c\\mathbf {W},c\\mathbf {V})}{\\mathrm {TP}(c\\mathbf {W},c\\mathbf {V})}=\\frac{\\mathrm {HP}(\\mathbf {W},\\mathbf {V})}{\\mathrm {TP}(\\mathbf {W},\\mathbf {V})} $ and $\\frac{\\mathrm {IP}(c\\mathbf {W},c\\mathbf {V})}{\\mathrm {TP}(c\\mathbf {W},c\\mathbf {V})}=\\frac{\\mathrm {IP}(\\mathbf {W},\\mathbf {V})}{\\mathrm {TP}(\\mathbf {W},\\mathbf {V})}$ for $c>0$ .",
"As a result, if total transmit power minimization is not a system design objective in Problem 4, i.e., $\\lambda _2=0$ , optimal solutions of Problem 4 in the trade-off region may exist such that the secondary transmitter transmits with a high power while still satisfying all constraints.",
"On the other hand, to achieve the maximum energy harvesting efficiency in the secondary network, i.e., the top corner point in Figure REF , the secondary transmitter has to transmit with maximum power which leads to a high average IPTR.",
"Figure: System objectives trade-off region for Case II.For comparison, in Figure REF , we plot the trade-off regions achieved by a baseline resource allocation scheme for Case I and Case II in Figure 3.",
"For the baseline scheme, we adopt maximum ratio transmission (MRT) with respect to the desired secondary receiver for information beamforming matrix ${\\mathbf {W}}$ .",
"In other words, the beamforming direction of matrix ${\\mathbf {W}}$ is fixed and it has a rank-one structure.",
"Then, we optimize the artificial noise covariance matrix $\\mathbf {{V}}$ and the power of ${\\mathbf {W}}$ in Problem 4 via varying the values of $0\\le \\lambda _p\\le 1,\\forall p\\in \\lbrace 1,2,3\\rbrace $ .",
"We note that the proposed baseline scheme requires the same amount of CSI as the proposed optimal scheme.",
"However, the baseline scheme does not fully exploit the available CSI for resource allocation optimization and, as a result, the required computational complexity is reduced roughly by half compared to the proposed optimal scheme.",
"As can be observed from Figures REF , REF , REF , and REF , the baseline scheme is effective in maximizing the energy harvesting efficiency and the total harvested power in the high transmit power regime and is able to approach the optimal trade-off region achieved by the proposed optimal SDP based resource allocation scheme.",
"This can be explained by the fact that both the optimal scheme and the baseline scheme optimize the covariance matrix of the artificial noise which contributes most of the power transferred to the idle secondary receivers.",
"In particular, by exploiting the spatial degrees of freedom offered by the multiple antennas, multiple narrow energy beams can be created via the proposed optimization framework for transfer of the artificial noise.",
"The narrow energy beams help in focusing energy on the idle secondary receivers which increases the energy transfer efficiency.",
"Nevertheless, when the total transmit power budget of the secondary transmitter is small, the baseline scheme may not be able to satisfy the QoS constraints which leads to a smaller trade-off region compared to the proposed optimal scheme.",
"Figure: Average energy harvesting efficiency."
],
[
"Average Total Harvested Power and Average Energy Harvesting Efficiency",
"Figures REF and REF depict the average total harvested power and the average energy harvesting efficiency of the secondary system versus the average total transmit power for different numbers of secondary receivers, $K$ , respectively.",
"The curves in Figures REF and REF are obtained for Case I and Case II, respectively.",
"Specifically, for each case, we solve Problem 4 for $\\lambda _3=0$ and $0\\le \\lambda _p\\le 1,\\forall p\\in \\lbrace 1,2\\rbrace $ , where the values of $\\lambda _p,p\\in \\lbrace 1,2\\rbrace ,$ are uniformly varied for a step size of $0.01$ such that $\\sum _p \\lambda _p=1$ .",
"It can be observed from Figures REF and REF that the average total harvested power and the average energy harvesting efficiency are monotonically increasing functions with respect to the total transmit power.",
"In other words, total harvested power/energy harvesting efficiency maximization and total transmit power minimization are conflicting system design objectives.",
"Also, the proposed optimal scheme outperforms the baseline scheme.",
"In particular, the proposed optimal scheme fully utilizes the available CSI and provides a larger trade-off region, e.g.",
"$2.7$ dB less transmit power in the considered scenario compared to the baseline scheme in Figure REF .",
"Besides, the two proposed suboptimal schemes perform close to the trade-off region achieved by the optimal SDP resource allocation scheme.",
"Furthermore, all trade-off curves are shifted in the upper-right direction if the number of secondary receivers is increased.",
"This is due to the fact that for a larger number of secondary users, there are more idle secondary receivers in the system harvesting the power radiated by the transmitter which improves the energy harvesting efficiency and the total harvested power.",
"Also, having additional idle secondary receivers means that there are more potential eavesdroppers in the system.",
"Thus, more artificial noise generation is required for neutralizing information leakage.",
"We note that in all the considered scenarios, the proposed resource allocation schemes are able to guarantee the minimum secrecy data rate requirement of $C_\\mathrm {sec}\\ge 5.6582$ bit/s/Hz despite the imperfectness of the CSI.",
"Figure: Average energy harvesting efficiency.Figures REF and REF show the average total harvested power and the average energy harvesting efficiency of the secondary system versus the average total interference power leakage and the average IPTR, respectively, for different numbers of desired secondary receivers, $K$ .",
"The curves in Figures REF and REF are obtained by solving Problem 4 for $\\lambda _2=0$ and varying the values of $0\\le \\lambda _p\\le 1,\\forall p\\in \\lbrace 1,3\\rbrace $ , uniformly for a step size of $0.01$ such that $\\sum _p \\lambda _p=1$ for Case I and Case II, respectively.",
"The average total harvested power and the average energy harvesting efficiency increase with increasing average total interference power leakage and increasing average IPTR, respectively.",
"This observation indicates that total harvested power maximization and energy harvesting efficiency maximization are conflicting with total interference power leakage minimization and IPTR minimization, respectively.",
"Besides, the two proposed suboptimal schemes perform close to the trade-off curve achieved by the optimal resource allocation scheme.",
"Furthermore, all the trade-off curves are shifted in the upper-right direction as the number of secondary receivers is increased.",
"In fact, there are more potential eavesdroppers in the system when the number of idle secondary receivers increases.",
"Thus, more artificial noise has to be radiated by the secondary transmitter for guaranteeing communication security which leads to a higher IPTR and a higher interference power leakage.",
"On the other hand, the baseline scheme achieves a smaller trade-off region in both Figures REF and REF compared to the proposed optimal and suboptimal schemes.",
"This performance gap reveals the importance of the optimization of beamforming matrix $\\overline{\\mathbf {W}}$ for minimizing the total interference power leakage and the IPTR.",
"Figure: Average IPTR."
],
[
"Average Total Interference Power Leakage and Average IPTR",
"Figures REF and REF depict the average total interference power leakage and the average IPTR of the secondary system versus the average total transmit power for different numbers of secondary receivers, $K$ , respectively.",
"The curves in Figures REF and REF are obtained by solving Problem 4 for Case I and Case II, respectively, by setting $\\lambda _1=0$ and varying the values of $0\\le \\lambda _p\\le 1,\\forall p\\in \\lbrace 2,3\\rbrace $ , uniformly for a step size of $0.01$ such that $\\sum _p \\lambda _p=1$ .",
"Interestingly, we observe from Figures REF and REF that a higher transmit power may not correspond to a stronger interference leakage to the primary system or a higher IPTR, if the degrees of freedom offered by the multiple antennas are properly exploited.",
"Furthermore, the baseline scheme achieves a significantly worse trade-off compared to the proposed optimal and suboptimal schemes, e.g.",
"10 dB more interference leakage in Figure REF .",
"Also, a resource allocation policy that minimizes the total transmit power can only minimize the total interference power leakage simultaneously to a certain extent or vice versa in general.",
"For minimizing the total interference power leakage, the secondary transmitter sacrifices some degrees of freedom to reduce the received strengths of both information signal and artificial noise at the primary receivers.",
"Thus, fewer degrees of freedom are available for providing reliable and secure communication to the secondary receivers such that a higher transmit power is required.",
"In fact, in the proposed optimal scheme, both the beamforming matrix $\\mathbf {\\overline{W}}$ and the artificial noise covariance matrix $\\mathbf {\\overline{V}}$ are jointly optimized for performing resource allocation based on the CSI of all receivers.",
"In contrast, in the baseline scheme, the direction of the beamforming matrix is fixed which leads to fewer degrees of freedom for resource allocation.",
"Thus, the baseline scheme performs worse than the proposed schemes.",
"On the other hand, for a given required average interference leakage power or average IPTR, increasing the number of secondary receivers induces a higher transmit power in both cases.",
"Indeed, constraint C2 on communication secrecy becomes more stringent for an increasing number of secondary receivers.",
"In other words, it leads to a smaller feasible solution set for resource allocation optimization.",
"As a result, the efficiency of the resource allocation schemes in jointly optimizing the multiple objective functions decreases for a larger number of secondary receivers $K$ ."
],
[
"Conclusions",
"In this paper, we studied the resource allocation algorithm design for CR secondary networks with simultaneous wireless power transfer and secure communication based on a multi-objective optimization framework.",
"We focused on three system design objectives: transmit power minimization, energy harvesting efficiency maximization, and IPTR minimization.",
"Besides, the proposed multi-objective problem formulation includes total harvested power maximization and interference power leakage minimization as special cases.",
"In addition, our problem formulation takes into account the imperfectness of the CSI of the idle secondary receivers and the primary receivers at the secondary transmitter.",
"By utilizing the primal and dual optimal solutions of the SDP relaxed problem, the global optimal solution of the original problem can be constructed.",
"Furthermore, two suboptimal resource allocation schemes were proposed for the case when the solution of the dual problem is unavailable.",
"Simulation results illustrated the performance gains of the proposed schemes compared to a baseline scheme, and unveiled the trade-off between the considered system design objectives: (1) A resource allocation policy minimizing the total transmit power also leads to a low total interference power leakage in general; (2) energy harvesting efficiency maximization and transmit power minimization are conflicting system design objectives; (3) maximum energy harvesting efficiency is achieved at the expense of high interference power leakage and high transmit power."
],
[
"Proof of Proposition 1",
"The proof is based on the Charnes-Cooper transformation [27], [58].",
"By applying the change of variables in (REF ) to (REF ), Problem 1 in (REF ) can be equivalently transformed to $&&\\hspace*{-28.45274pt} \\underset{\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\xi }{\\operatorname{\\mathrm {minimize}}}\\,\\, \\,\\, \\frac{-\\sum _{k=1}^{K-1}\\eta _k\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))}{\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {W}})+\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {V}})}\\\\\\mbox{s.t.}",
"&&\\hspace*{0.0pt}\\mbox{$\\overline{\\mbox{C1}}$} -\\mbox{$\\overline{\\mbox{C5}}$},\\,\\, \\mbox{$\\overline{\\mbox{C6}}$:}\\,\\, \\xi > 0,\\mbox{$\\overline{\\mbox{C7}}$:}\\,\\, \\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {W}})+\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {V}})= 1,\\mbox{$\\overline{\\mbox{C8}}$}.$ Now, we show that (REF ) is equivalent to $&&\\hspace*{-34.1433pt} \\underset{\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}\\in \\mathbb {H}^{N_{\\mathrm {T}}},\\xi }{\\operatorname{\\mathrm {minimize}}}\\,\\, \\,\\, -\\sum _{k=1}^{K-1}\\eta _k\\operatorname{\\mathrm {Tr}}(\\mathbf {G}_k(\\overline{\\mathbf {W}}+\\overline{\\mathbf {V}}))\\\\\\mbox{s.t.}",
"&&\\hspace*{0.0pt}\\mbox{$\\overline{\\mbox{C1}}$} -\\mbox{$\\overline{\\mbox{C5}}$},\\,\\,\\mbox{$\\overline{\\mbox{C6}}$:}\\,\\, \\xi \\ge 0,\\,\\, \\mbox{$\\overline{\\mbox{C7}}$:}\\,\\, \\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {W}})+\\operatorname{\\mathrm {Tr}}(\\overline{\\mathbf {V}})= 1,\\mbox{ $\\overline{\\mbox{C8}}$}.\\nonumber $ We denote the optimal solution of (REF ) as $(\\overline{\\mathbf {W}}^*,\\overline{\\mathbf {V}}^*,\\xi ^*)$ .",
"If $\\xi ^*=0$ , then $\\overline{\\mathbf {W}}=\\overline{\\mathbf {V}}=\\mathbf {0}$ according to $\\mbox{$\\overline{\\mbox{C3}}$}$ .",
"Yet, this solution cannot satisfy $\\mbox{$\\overline{\\mbox{C1}}$}$ for $\\Gamma _{\\mathrm {req}}>0$ .",
"As a result, without loss of generality and optimality, constraint $\\xi >0$ can be replaced by $\\xi \\ge 0$ .",
"The equivalence between transformed Problems 2, 3, and 4 and their original problem formulations can be proved by following a similar approach as above."
],
[
"Proof of Theorem 1",
"The proof is divided into two parts.",
"In the first part, we investigate the structure of the optimal solution $\\mathbf {\\overline{W}}^*$ of the relaxed version of problem (REF ).",
"Then, in the second part, we propose a method to construct a solution $\\mathbf {\\widetilde{\\Lambda }}\\triangleq \\lbrace \\mathbf {\\widetilde{I}}^{\\mathrm {PU}},\\mathbf {\\widetilde{E}}^{\\mathrm {SU}}, \\widetilde{\\xi },\\widetilde{\\tau },\\widetilde{\\gamma },\\widetilde{\\delta },\\widetilde{\\varphi },\\widetilde{\\omega },{\\mathbf {\\widetilde{V}}},\\mathbf {{ \\widetilde{W}}}\\rbrace $ that achieves the same objective value as $\\mathbf {\\Lambda }^*\\triangleq \\lbrace \\mathbf {I}^{\\mathrm {PU}*},\\mathbf {E}^{\\mathrm {SU}*}, \\xi ^*,\\tau ^*,$ $\\gamma ^*, \\delta ^*,\\varphi ^*,\\omega ^*,\\overline{\\mathbf {V}}^*,\\mathbf {\\overline{W}}^*\\rbrace $ but admits a rank-one $\\mathbf {\\widetilde{W}}$ .",
"It can be shown that the relaxed version of problem (REF ) is jointly convex with respect to the optimization variables and satisfies Slater's constraint qualification.",
"As a result, the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient conditions [44] for the optimal solution of the relaxed version of problem (REF ).",
"The Lagrangian function of the relaxed version of problem (REF ) is given by ${\\cal L}&=&\\hspace*{-8.53581pt} \\operatorname{\\mathrm {Tr}}\\Big (\\big (\\mathbf {I}_{N_{\\mathrm {T}}}(\\alpha +\\mu )-\\mathbf {Y}-\\beta \\mathbf {H}\\big )\\overline{\\mathbf {W}}\\Big )\\\\&-&\\hspace*{-8.53581pt} \\sum _{k=1}^{K-1}\\operatorname{\\mathrm {Tr}}\\Big ( \\mathbf {S}_{\\mathrm {\\overline{C2}}_k}\\big (\\overline{\\mathbf {W}},\\overline{\\mathbf {V}},\\delta _k\\big )\\mathbf {D}_{\\mathrm {\\overline{C2}}_k}\\Big ) \\\\&-&\\hspace*{-8.53581pt}\\sum _{j=1}^{J}\\operatorname{\\mathrm {Tr}}\\Big ( \\mathbf {S}_{\\mathrm {\\overline{C3}}_j}\\big (\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}, \\xi ,\\gamma _j\\big ) \\mathbf {D}_{\\mathrm {\\overline{C3}}_j}\\Big )\\\\&-& \\hspace*{-8.53581pt}\\sum _{j=1}^{J}\\operatorname{\\mathrm {Tr}}\\Big ( \\mathbf {S}_{\\mathrm {\\overline{C11}}_j}\\big (\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}, \\xi ,\\omega _j\\big )\\mathbf {D}_{\\mathrm {\\overline{C11}}_j}\\Big )\\\\\\hspace*{-14.22636pt}&-&\\hspace*{-8.53581pt}\\sum _{k=1}^{K-1}\\operatorname{\\mathrm {Tr}}\\Big ( \\mathbf {S}_{\\mathrm {\\overline{C10}}_k}\\big (\\overline{\\mathbf {W}},\\overline{\\mathbf {V}}, \\xi ,\\varphi _k\\big ) \\mathbf {D}_{\\mathrm {\\overline{C10}}_k}\\Big )+\\Omega ,$ where $\\Omega $ denotes the collection of the terms that only involve variables that are not relevant for the proof.",
"$\\beta ,\\alpha \\ge 0$ , and $\\mu $ are the Lagrange multipliers associated with constraints $\\mbox{$\\overline{\\mbox{C1}}$, $\\overline{\\mbox{C4}}$}$ , and $\\mbox{$\\overline{\\mbox{C7}}$}$ , respectively.",
"Matrix $\\mathbf {Y}\\succeq \\mathbf {0}$ is the Lagrange multiplier matrix for the semidefinite constraint on matrix $\\overline{\\mathbf {W}}$ in $\\mbox{$\\overline{\\mbox{C4}}$}$ .",
"$\\mathbf {D}_{\\mathrm {\\overline{C2}}_k}\\succeq \\mathbf {0},\\forall k\\in \\lbrace 1,\\,\\ldots ,\\,K-1\\rbrace ,$ and $\\mathbf {D}_{\\mathrm {\\overline{C3}}_j}\\succeq \\mathbf {0},\\forall j\\in \\lbrace 1,\\,\\ldots ,\\,J\\rbrace $ , are the Lagrange multiplier matrices for the maximum tolerable SINRs of the idle secondary receivers and the primary receivers in $\\mbox{$\\overline{\\mbox{C2}}$}$ and $\\mbox{$\\overline{\\mbox{C3}}$}$ , respectively.",
"$\\mathbf {D}_{\\mathrm {\\overline{C10}}_k}\\succeq \\mathbf {0},\\forall k\\in \\lbrace 1,\\,\\ldots ,\\,K-1\\rbrace ,$ and $\\mathbf {D}_{\\mathrm {\\overline{C11}}_j}\\succeq \\mathbf {0},\\forall j\\in \\lbrace 1,\\,\\ldots ,\\,J\\rbrace $ , are the Lagrange multiplier matrices associated with constraints $\\mbox{$\\overline{\\mbox{C10}}$}$ and $\\mbox{$\\overline{\\mbox{C11}}$}$ , respectively.",
"In the following, we focus on the KKT conditions related to the optimal $\\mathbf {\\overline{W}}^*$ : $\\mathbf {Y}^*,\\mathbf {D}^*_{\\mathrm {\\overline{C2}}_k},\\mathbf {D}^*_{\\mathrm {\\overline{C3}}_j},\\mathbf {D}^*_{\\mathrm {\\overline{C10}}_k},\\mathbf {D}^*_{\\mathrm {\\overline{C11}}_j}\\hspace*{-5.69054pt}&\\succeq &\\hspace*{-5.69054pt} \\mathbf {0},\\alpha ^*,\\beta ^*\\ge 0,\\mu ^*,\\\\\\mathbf {Y^*\\overline{\\mathbf {W}}^*}\\hspace*{-5.69054pt}&=&\\hspace*{-5.69054pt}\\mathbf {0}, \\\\\\nabla _{\\overline{\\mathbf {W}}^*}{\\cal L}\\hspace*{-5.69054pt}&=&\\hspace*{-5.69054pt}\\mathbf {0}, $ where $\\mathbf {Y}^*,\\mathbf {D}^*_{\\mathrm {\\overline{C2}}_k},\\mathbf {D}^*_{\\mathrm {\\overline{C3}}_j},\\mathbf {D}^*_{\\mathrm {\\overline{C10}}_k},\\mathbf {D}^*_{\\mathrm {\\overline{C11}}_j},\\mu ^*,\\beta ^*$ , and $\\alpha ^*$ are the optimal Lagrange multipliers for the dual problem of (REF ).",
"From the complementary slackness condition in (), we observe that the columns of $\\overline{\\mathbf {W}}^*$ are required to lie in the null space of $\\mathbf {Y}^*$ for $\\overline{\\mathbf {W}}^*\\ne \\mathbf {0}$ .",
"Thus, we study the composition of $\\mathbf {Y}^*$ to obtain the structure of $\\overline{\\mathbf {W}}^*$ .",
"The KKT condition in () can be expressed as $&&\\mathbf {Y}^*+\\beta ^*\\mathbf {H}\\\\&=&\\mathbf {I}_{N_{\\mathrm {T}}}(\\mu ^*+\\alpha ^*)+\\sum _{k=1}^{K-1} \\mathbf {U}_{\\mathbf {g}_k}\\Big (\\frac{\\mathbf {D}^*_{\\mathrm {\\overline{C2}}_k}}{\\Gamma _{\\mathrm {tol}_k}}-\\mathbf {D}_{\\mathrm {\\overline{C10}}_k}^*\\Big )\\mathbf {U}_{\\mathbf {g}_k}^H \\\\&+&\\sum _{j=1}^J\\mathbf {U}_{\\mathbf {l}_j}\\Big (\\frac{\\mathbf {D}^*_{\\mathrm {\\overline{C3}}_j}}{\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}}+\\mathbf {D}_{\\mathrm {\\overline{C11}}_j}^*\\Big )\\mathbf {U}_{\\mathbf {l}_j}^H.$ For notational simplicity, we define $\\mathbf {A}^*&=&\\mathbf {I}_{N_{\\mathrm {T}}}(\\mu ^*+\\alpha ^*)+\\sum _{k=1}^{K-1} \\mathbf {U}_{\\mathbf {g}_k}\\Big (\\frac{\\mathbf {D}^*_{\\mathrm {\\overline{C2}}_k}}{\\Gamma _{\\mathrm {tol}_k}}-\\mathbf {D}_{\\mathrm {\\overline{C10}}_k}^*\\Big )\\mathbf {U}_{\\mathbf {g}_k}^H \\\\ &+&\\sum _{j=1}^J\\mathbf {U}_{\\mathbf {l}_j}\\Big (\\frac{\\mathbf {D}^*_{\\mathrm {\\overline{C3}}_j}}{\\Gamma _{\\mathrm {tol}_j}^{\\mathrm {PU}}}+\\mathbf {D}_{\\mathrm {\\overline{C11}}_j}^*\\Big )\\mathbf {U}_{\\mathbf {l}_j}^H.$ Besides, there exists at least one optimal solution with $\\beta ^*>0$ , i.e., constraint $\\mbox{$\\overline{\\mbox{C1}}$}$ is satisfied with equality.",
"Suppose that for the optimal solution, constraint $\\mbox{$\\overline{\\mbox{C1}}$}$ is satisfied with strict inequality, i.e., $\\frac{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\overline{\\mathbf {W}}^*)}{\\operatorname{\\mathrm {Tr}}(\\mathbf {H}\\overline{\\mathbf {V}}^*)+\\sigma _\\mathrm {z}^2\\xi ^*} > \\Gamma _{\\mathrm {req}}$ .",
"Then, we can replace $\\xi ^*$ with $\\overline{\\xi }^*=\\xi ^* c$ for $c>1$ such that $\\mbox{$\\overline{\\mbox{C1}}$}$ is satisfied with equality.",
"We note that the new solution $\\overline{\\xi }^*$ not only satisfies all the constraints, but also provides a larger feasible solution set for minimizing $\\tau $ , cf.",
"constraints $\\mbox{$\\overline{\\mbox{C4}}$}$ and $\\mbox{$\\overline{\\mbox{C9}}$b}$ .",
"As a result, there always exist at least one optimal solution such that constraint $\\mbox{$\\overline{\\mbox{C1}}$}$ is satisfied with equality.",
"In order to obtain the optimal solution in practice, we can replace the inequality “$\\ge $ \"with equality “$=$ \" in $\\mbox{$\\overline{\\mbox{C2}}$}$ without loss of optimality.",
"From (REF ) and (REF ), we can express the Lagrange multiplier matrix $\\mathbf {Y}^*$ as $\\mathbf {Y}^*=\\mathbf {A}^*-\\beta ^*\\mathbf {H},$ where $\\beta ^*\\mathbf {H}$ is a rank-one matrix since $\\beta ^*>0$ .",
"Without loss of generality, we define $r=\\operatorname{\\mathrm {Rank}}(\\mathbf {A}^*)$ and the orthonormal basis of the null space of $\\mathbf {A}^*$ as $\\mathbf {\\mathbf {\\Upsilon }}\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times (N_{\\mathrm {T}}-r)}$ such that $\\mathbf {A}^*\\mathbf {\\Upsilon }=\\mathbf {0}$ and $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\Upsilon })=N_{\\mathrm {T}}-r$ .",
"Let ${\\phi }_t\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ , $1\\le t\\le N_{\\mathrm {T}}-r$ , denote the $t$ -th column of $\\mathbf {\\Upsilon }$ .",
"By exploiting [30], it can be shown that $\\mathbf {H}\\mathbf {\\Upsilon }=\\mathbf {0}$ and we can express the optimal solution of $\\overline{\\mathbf {W}}^*$ as $\\overline{\\mathbf {W}}^*=\\sum _{t=1}^{N_{\\mathrm {T}}-r} \\psi _t {\\phi }_t {\\phi }_t^H + \\underbrace{f\\mathbf {u}\\mathbf {u}^H}_{\\mbox{Rank-one}},$ where $\\psi _t\\ge 0, \\forall t\\in \\lbrace 1,\\ldots ,N_{\\mathrm {T}}-r\\rbrace ,$ and $f>0$ are positive scalars and $\\mathbf {u}\\in \\mathbb {C}^{N_{\\mathrm {T}}\\times 1}$ , $\\Vert \\mathbf {u}\\Vert =1$ , satisfies $\\mathbf {u}^H\\mathbf {\\Upsilon }=\\mathbf {0}$ .",
"In the second part of the proof, for $\\operatorname{\\mathrm {Rank}}(\\overline{\\mathbf {W}}^*)>1$ , we construct another solution $\\mathbf {\\widetilde{\\Lambda }}\\triangleq \\lbrace \\mathbf {\\widetilde{I}}^{\\mathrm {PU}},\\mathbf {\\widetilde{E}}^{\\mathrm {SU}}, \\widetilde{\\xi },\\widetilde{\\tau },$ $\\widetilde{\\gamma },\\widetilde{\\delta },\\widetilde{\\varphi },\\widetilde{\\omega },{\\mathbf {\\widetilde{V}}},\\mathbf {{ \\widetilde{W}}}\\rbrace $ based on (REF ).",
"Let $\\mathbf {\\widetilde{W}}\\hspace*{-5.69054pt}&=&\\hspace*{-5.69054pt}f\\mathbf {u}\\mathbf {u}^H=\\overline{\\mathbf {W}}^*-\\sum _{t=1}^{N_{\\mathrm {T}}-r} \\psi _t {\\phi }_t {\\phi }_t^H,\\\\\\mathbf {\\widetilde{V}}&=&\\overline{\\mathbf { V}}^*+\\sum _{t=1}^{N_{\\mathrm {T}}-r} \\psi _t {\\phi }_t {\\phi }_t^H,\\\\\\mathbf {\\widetilde{I}}^{\\mathrm {PU}}\\hspace*{-5.69054pt}&=&\\hspace*{-5.69054pt}\\mathbf { I}^{\\mathrm {PU}*},\\,\\,\\mathbf {\\widetilde{E}}^{\\mathrm {SU}}=\\mathbf { E}^{\\mathrm {SU}*},\\,\\, \\widetilde{\\xi }=\\xi ^*,\\,\\,\\\\\\widetilde{\\tau }\\hspace*{-5.69054pt}&=&\\hspace*{-5.69054pt}\\tau ^*,\\,\\,\\widetilde{\\gamma }=\\gamma ^*,\\,\\,\\widetilde{\\delta }=\\delta ^*,\\,\\,\\widetilde{\\varphi }=\\widetilde{\\varphi }^*,\\,\\,\\widetilde{\\omega }=\\omega ^*.$ Then, we substitute the constructed solution $\\mathbf {\\widetilde{\\Lambda }}$ into the objective function and the constraints in (REF ) which yields the equations in (REF ) on the top of next page.",
"Figure: NO_CAPTIONIt can be seen from (REF ) that the constructed solution set $\\mathbf {\\widetilde{\\Lambda }}$ achieves the same optimal value as the optimal solution $\\mathbf {\\Lambda }^*$ while satisfying all the constraints.",
"Thus, $\\mathbf {\\widetilde{\\Lambda }}$ is also an optimal solution of (REF ).",
"Besides, the constructed beamforming matrix $\\mathbf {\\widetilde{W}}$ is a rank-one matrix, i.e., $\\operatorname{\\mathrm {Rank}}(\\mathbf {\\widetilde{W}})=1$ .",
"On the other hand, we can obtain the values of $f$ and $\\psi _t$ in (REF ) by substituting the variables in (REF ) into the relaxed version of (REF ) and solving the resulting convex optimization problem for $f$ and $\\psi _t$ ."
]
] | 1403.0054 |
[
[
"Quantum Random State Generation with Predefined Entanglement Constraint"
],
[
"Abstract Entanglement plays an important role in quantum communication, algorithms, and error correction.",
"Schmidt coefficients are correlated to the eigenvalues of the reduced density matrix.",
"These eigenvalues are used in Von Neumann entropy to quantify the amount of the bipartite entanglement.",
"In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states.",
"We also show that it is possible to adjust the entanglement between subsystems by changing the quantum gates corresponding to the Schmidt coefficients.",
"In this manner, random quantum states with predefined bipartite entanglement amounts can be generated using random Schmidt basis.",
"This provides a technique for generating equivalent quantum states for given weighted graph states, which are very useful in the study of entanglement, quantum computing, and quantum error correction."
],
[
"Introduction",
"In quantum information, a quantum state encodes information and is used in the design of algorithms.",
"Random numbers and random matrix theory [1] play important roles in various applications, ranging from wireless communications [2] to determining physical properties of a quantum system [3].",
"Consequently, generating random quantum states is important in quantum communication and information.",
"For instance, unique random states can be used to design quantum bills (money) [4].",
"Statistical properties of random quantum states show that random quantum states generated within some restricted set of states can still be effectively random [5].",
"A quantum state, defined as a vector in Hilbert space, contains all the accessible-measurable information about the system [6].",
"Entanglement is one of the quantum mechanical accessible phenomena used to build efficient quantum algorithms.",
"For a given multi-qubit state, a graph state [7], [8], [9] can be used to specify the graph-based representation of the entanglement between qubits.",
"A graph state is comprised of vertices and edges, where the vertices correspond to qubits and edges represent entanglement between qubits.",
"Non-local properties [10], [11] and entanglement characterizations [12] of graph states have been studied, and their use has been demonstrated for different applications in quantum error correction [13], quantum communication, and one-way quantum computation[14], among others (Hein et al.",
"[15] present an excellent review of the applications of graph states).",
"Realization of graph states has been experimentally demonstrated for six photons [16].",
"As a generalization of graph states, weighted graph states include weights on each edge, quantifying the amount of the entanglement.",
"Weighted graph states are shown to be useful in the study of bipartite entanglement in spin chains [17] and many-body quantum states [18].",
"Based on a weighted graph state representation of certain classes of multi-particle entangled states, a variational method [19], [20] is proposed for arbitrary spin and infinite-dimensional systems.",
"These representations are also used in error correction schemes in one-way quantum computing [21], and in many other applications (please refer to Hein at al.[15]).",
"In this paper, we map the Schmidt basis and the associated coefficients to quantum circuits to generate random quantum states.",
"We show that for state generation, by using quantum gates corresponding to Schmidt coefficients, the amount of the bipartite entanglement between subsystems can be controlled.",
"Therefore, we show that if quantum gates corresponding to the Schmidt basis are chosen randomly, one can generate random states with bipartite entanglement amounts predefined by the gates implementing the coefficients.",
"This provides a way to tune the entanglement between subsystems in a generated state, which can be used to generate certain type of weighted graph states on quantum computers.",
"This can be used in the utilization and characterization of entanglement [22] in quantum communication, cryptography, and cluster state computation [7].",
"In addition, our method can be used to simulate entanglement distribution of particular quantum systems in quantum computing.",
"An example of this is in the simulation of the entanglement distribution in light-harvesting complexes[23], [24] to investigate energy transfer and efficiency.",
"Given Hilbert spaces $H_A$ and $H_B$ of dimension $d_A$ and $d_B$ , and a quantum state $\\left|\\psi \\right\\rangle \\in H_{AB}=H_A\\otimes H_B$ , the Schmidt decomposition is defined as: $\\left|\\psi \\right\\rangle =\\sum _i^{min(d_A,d_B)}s_i\\left|u_i\\right\\rangle \\left|v_i\\right\\rangle ,$ where $s_i$ s are the Schmidt coefficients, and $\\left|u_i\\right\\rangle $ and $\\left|v_i\\right\\rangle $ are the state vectors that form the Schmidt bases in $H_A$ and $H_B$ , respectively.",
"The reduced density matrix for system $A$ or $B$ can be found from the Schmidt decomposition as follows: $\\rho _A = \\sum _i s_i^2 \\left|u_i\\right\\rangle \\left\\langle u_i\\right|.$ The above expression shows that the coefficients of the Schmidt decomposition are related to the eigenvalues of the reduced density matrix."
],
[
"Von Neumann Entropy",
"For a given density matrix $\\rho $ , the von Neumann Entropy is defined as: $S(\\rho )=-Tr(\\rho \\ln {\\rho }),$ where the notation $Tr$ describes the trace of a matrix.",
"If the density matrix $\\rho $ with eigenvalues $\\lambda _j$ and associated eigenvectors $\\left|j\\right\\rangle $ has the eigenvalue decomposition $\\rho =\\sum _j\\lambda _j\\left|j\\right\\rangle \\left\\langle j\\right|$ , then the entropy can be defined as: $S(\\rho )=-\\sum _j\\lambda _j\\ln {\\lambda _j}$ For pure states, we can use the Schmidt coefficients in the von Neumann Entropy to quantify the bipartite entanglement between systems $A$ and $B$ as: $S(\\rho _A)=S(\\rho _B)=-\\sum _j^{min(d_A,d_B)}s_j^2\\ln {s_j^2},$ where $\\rho _A$ and $\\rho _B$ are the density matrices for the systems $A$ and $B$ , respectively."
],
[
"Random State Generation With Predetermined Entanglement",
"Since the Schmidt coefficients are important in determining entanglement, by suitably mapping the Schmidt decomposition to a circuit design, we can control entanglement.",
"Please note that in this paper, for simplicity, we will only consider the real space for the circuit designs, but they can be generalized to complex space."
],
[
"2-qubit Case",
"The Schmidt decomposition for $H=H_1\\otimes H_2$ , and $H_1$ and $H_2\\in \\mathcal {R}^{\\otimes ^1}$ is as follows: $\\left|\\psi \\right\\rangle =\\sum _{i=1}^{2}s_i\\left|u_i\\right\\rangle \\left|v_i\\right\\rangle .$ The circuit in Fig.REF can be used generate any general $\\left|\\psi \\right\\rangle $ state for two qubits, where the entanglement defined by the quantum gate $R$ whose elements are determined from the Schmidt coefficients.",
"In the figure $X$ is the quantum $NOT$ gate, $U$ and $V$ are the Schmidt basis, and the elements of $R$ are the Schmidt coefficients determining the entanglement: $R=\\left(\\begin{matrix}s_1 &-s_2\\\\s_2 &s_2\\end{matrix} \\right).$ Choosing the elements $s_1$ and $s_2$ , which are the Schmidt coefficients, and random Schmidt basis $U$ and $V$ , one can also create a two-qubit random state with predetermined entanglement."
],
[
"Generalization to $n$ qubits",
"We can generalize the idea to an $n$ -qubit system: The Kronecker tensor product of the Schmidt bases $U$ and $V$ can be written in matrix form as: $U \\otimes V =\\left[\\begin{matrix}u_{\\bullet 1}\\otimes v_{\\bullet 1}& \\dots &u_{\\bullet 1}\\otimes v_{\\bullet k}&u_{\\bullet 2}\\otimes v_{\\bullet 1}& \\dots & u_{\\bullet 2}\\otimes v_{\\bullet k}& \\dots &u_{\\bullet k}\\otimes v_{\\bullet 1}& \\dots & u_{\\bullet k}\\otimes v_{\\bullet k}\\end{matrix}\\right],$ where $u_{\\bullet i}$ and $v_{\\bullet j}$ represent the $i$ th and $j$ th column of $U$ and $V$ , respectively, and $k$ represents the number of columns.",
"In the Schmidt decomposition of a vector $\\left|\\psi \\right\\rangle $ : $\\left|\\psi \\right\\rangle = \\sum _{i=1}^k s_i u_{\\bullet i} \\otimes v_{\\bullet i},$ the Schmidt coefficients $s_1 \\dots s_k$ are related to the columns: $1, (k+2),(2k+3), \\dots , (k^2)$ , respectively.",
"Therefore, if we have an input state $\\left|\\varphi \\right\\rangle $ to $(U\\otimes V)$ in the following form: $\\left|\\varphi \\right\\rangle =\\left[{\\begin{matrix}s_1\\\\0\\\\\\vdots \\\\0\\\\s_2\\\\0\\\\\\vdots \\\\0\\\\s_3\\\\0\\\\\\vdots \\\\0\\\\s_k\\end{matrix}}\\right],$ then $(U\\otimes V) \\left|\\varphi \\right\\rangle =\\left|\\psi \\right\\rangle = \\sum _{i=1}^ks_i u_{\\bullet i} \\otimes v_{\\bullet i}$ .",
"If we assume the initial input to the circuit is $\\left|\\mathbf {0}\\right\\rangle $ , then the first column of the matrix representation of the circuit defines the output.",
"Therefore, to generate $\\left|\\varphi \\right\\rangle $ , first we construct the Schmidt coefficients in the first column of the matrix $S$ of dimension $min(d_A, d_B)$ .",
"If $S$ is on the first subsystem, then the global unitary operator is $(S\\otimes I)$ with the first column: $\\left[{\\begin{matrix}s_1\\\\0\\\\\\vdots \\\\0\\\\s_2\\\\0\\\\\\vdots \\\\0\\\\s_3\\\\0\\\\\\vdots \\\\0\\\\s_k\\\\0\\\\ \\vdots \\\\ 0\\end{matrix}}\\right]$ To get the Schmidt coefficients to the rows $1, (k+2),(2k+3), \\dots , (k^2)$ as in Eq.",
"(REF ), we apply a permutation matrix $P$ to switch the rows and columns: $P(S\\otimes I)P$ .",
"Therefore, the final circuit can be represented by the matrix vector product as: $\\left|\\psi \\right\\rangle =(U\\otimes V )P(S\\otimes I)P\\left|\\mathbf {0}\\right\\rangle .$ In the corresponding circuit design, $U$ and $V$ are defined as the operators on the first and the second subsystems, respectively.",
"$S$ , whose first column is the Schmidt coefficients, is considered on the first subsystem.",
"Since the operator $P$ is a permutation matrix, it can be implemented by a combination of controlled $NOT$ ($CNOT$ ) gates."
],
[
"4-qubit Case",
"As an example, consider a 4-qubit system, where the subsystems are composed of two qubits: $H_{12}$ for the first and second qubits and $H_{34}$ for the third and fourth qubits.",
"Thus, in the Schmidt decomposition, there are four coefficients: $s_1,s_2,s_3,$ and $s_4$ .",
"The circuit in Fig.REF generates any quantum state that has the Schmidt coefficients implemented by $S$ and Schmidt bases implemented by $U$ and $V$ .",
"For the implementation of $S$ given in Fig.REF , we follow the idea first presented in ref.",
"[25]: The coefficients are divided into two unit vectors as $1/k_1\\left[{\\begin{matrix} s_1 \\\\ s2\\end{matrix}}\\right]$ and $1/k_2\\left[{\\begin{matrix} s_3 \\\\ s4\\end{matrix}}\\right]$ , with normalization constants $1/k_1$ and $1/k_2$ .",
"Then, the rotation gates in Fig.REF are defined as: $R_1=\\frac{1}{k_1}\\left(\\begin{matrix}s_1 &-s_2\\\\s_2& s_1\\end{matrix}\\right),\\ R_2=\\frac{1}{k_2}\\left(\\begin{matrix}s_3 &-s_4\\\\s_4& s_3\\end{matrix}\\right),$ and $R_3=\\left(\\begin{matrix}k_1& -k_2\\\\k_2 &k_1\\end{matrix} \\right).$ $S$ is constructed using the above rotation gates as: $S=\\left(\\begin{matrix}R_1& I\\\\ I&R_2\\end{matrix}\\right)(R_3 \\otimes I).$ The first column of $S$ consists of the Schmidt coefficients, which is shown in Fig.REF .",
"Figure: Quantum circuit for 4 qubits which is found by following the Schmidt decomposition and can generate any quantum state of dimension 16.",
"In the circuit UU and VV are the Schmidt basis and S implements the Schmidt coefficients controlling the entanglement between H 12 H_{12} and H 34 H_{34}.Figure: Quantum circuit that has the matrix representation whose first column is the Schmidt coefficients."
],
[
"Bipartite Entanglement Control for $n$ qubits",
"We now show that we can sequentially combine the Schmidt decomposition circuits for two qubits to control the entanglement between various parts of the system with the rest of the system in the random state: e.g., for a 5 qubit system, controlling entanglement between $H_1$ and $H_{2345}$ and $H_{12}$ and $H_{345}$ ."
],
[
"Connecting three qubits linearly",
"We start with the initial state $\\left|\\psi _0\\right\\rangle =\\left|000\\right\\rangle $ .",
"If we assume the first two qubits are entangled by the circuit in Fig.REF , where the Schmidt basis is chosen to be identity: $V_1=U_1=I$ , then we get the following: $\\left|\\psi _1\\right\\rangle = s_1\\left|0\\right\\rangle \\left|0\\right\\rangle \\left|0\\right\\rangle + s_2\\left|1\\right\\rangle \\left|1\\right\\rangle \\left|0\\right\\rangle ,$ where $s_1$ and $s_2$ are Schmidt coefficients.",
"To also entangle the 2nd and 3rd qubits, we apply the same Schmidt circuit to these qubits: First, the $CNOT$ gate is applied: $\\begin{split}\\left|\\psi _2\\right\\rangle = &s_1\\left|0\\right\\rangle CNOT(\\left|0\\right\\rangle \\left|0\\right\\rangle ) + s_2\\left|1\\right\\rangle CNOT(\\left|1\\right\\rangle \\left|0\\right\\rangle )\\\\=& s_1\\left|0\\right\\rangle \\left|0\\right\\rangle \\left|0\\right\\rangle + s_2\\left|1\\right\\rangle \\left|1\\right\\rangle \\left|1\\right\\rangle .\\end{split}$ Then, we apply the rotation gate $R$ , which has the Schmidt coefficients, $k_1$ and $k_2$ , as elements: $R_2=\\left(\\begin{matrix}k_1 &-k_2\\\\k_2 &k_1\\end{matrix} \\right)$ This generates the following state: $\\begin{split}\\left|\\psi _3\\right\\rangle = & s_1\\left|0\\right\\rangle R_2\\left|0\\right\\rangle \\left|0\\right\\rangle + s_2\\left|1\\right\\rangle R_2\\left|1\\right\\rangle \\left|1\\right\\rangle \\\\= & s_1\\left|0\\right\\rangle (k_1\\left|0\\right\\rangle + k_2\\left|1\\right\\rangle )\\left|0\\right\\rangle + s_2\\left|1\\right\\rangle (-k_2\\left|0\\right\\rangle + k_1\\left|1\\right\\rangle )\\left|1\\right\\rangle \\\\= &s_1k_1\\left|000\\right\\rangle + s_1k_2\\left|010\\right\\rangle -s_2k_2\\left|101\\right\\rangle + s_2k_1\\left|111\\right\\rangle \\\\\\end{split}$ After the second $CNOT$ , the final state becomes: $\\left|\\psi _4\\right\\rangle = s_1k_1\\left|000\\right\\rangle + s_1k_2\\left|011\\right\\rangle -s_2k_2\\left|101\\right\\rangle + s_2k_1\\left|110\\right\\rangle \\\\$ Since $U_2$ and $V_2$ are the local operators, they do not change the entanglement.",
"Therefore, the entanglement between $H_{12}$ and $H_{3}$ , and the entanglement between $H_1$ and $H_{23}$ can be found from $\\left|\\psi _4\\right\\rangle $ .",
"The von Neumann entropy $S(\\rho _3)= S(\\rho _{12})$ defines the entanglement between the systems $H_{12}$ and$H_{3}$ .",
"For the entropy, the reduced density matrix $\\rho _3$ can be found as follows: $\\begin{split}\\rho _3 = &Tr_{H_{12}}(\\left|\\psi _4\\right\\rangle \\left\\langle \\psi _4\\right|)\\\\= & (s_1k_1)^2\\left|0\\right\\rangle \\left\\langle 0\\right|+ (s_1k_2)^2\\left|1\\right\\rangle \\left\\langle 1\\right| +(s_2k_2)^2\\left|1\\right\\rangle \\left\\langle 1\\right|\\\\+ &(s_2k_1)^2\\left|0\\right\\rangle \\left\\langle 0\\right|\\\\= & \\left((s_1k_1)^2 + (s_2k_1)^2 \\right)\\left|0\\right\\rangle \\left\\langle 0\\right|+ \\left( (s_1k_2)^2 + (s_2k_2)^2\\right)\\left|1\\right\\rangle \\left\\langle 1\\right| \\\\= &(k_1)^2 \\left|0\\right\\rangle \\left\\langle 0\\right| + (k_2)^2\\left|1\\right\\rangle \\left\\langle 1\\right|\\end{split}$ Since $S(\\rho _3)=(k_1)^2\\ln {(k_1)^2}+(k_2)^2\\ln {(k_2)^2}$ , which is determined solely from the Schmidt coefficients, the entanglement is controlled as expected.",
"The entanglement between $H_{12}$ and $H_3$ is also defined as $S(\\rho _{1})=S(\\rho _{23})$ .",
"Here, the reduced density matrix $\\rho _1$ can be obtained as follows: $\\begin{split}\\rho _1= &Tr_{H_{23}}(\\left|\\psi _4\\right\\rangle \\left\\langle \\psi _4\\right|)\\\\= & \\left((s_1k_1)^2 + (s_1k_2)^2 \\right)\\left|0\\right\\rangle \\left\\langle 0\\right|+ \\left( (s_2k_2)^2 + (s_2k_1)^2\\right)\\left|1\\right\\rangle \\left\\langle 1\\right| \\\\= &(s_1)^2 \\left|0\\right\\rangle \\left\\langle 0\\right| + (s_2)^2\\left|1\\right\\rangle \\left\\langle 1\\right|\\end{split}$ Finally, we find $S(\\rho _1)=(s_1)^2\\ln {(s_1)^2}+(s_2)^2\\ln {(s_2)^2}$ .",
"Eq.",
"(REF ) and Eq.",
"(REF ) prove that we can use the Schmidt circuit sequentially to achieve desired entanglement between two disentangled subsystems."
],
[
"Definition of a Graph State",
"For a given multi-qubit state, a graph state is an instance of the graph-based representation of the entanglement between qubits[15].",
"They are used in determining the capacity of quantum channels and quantum error correction.",
"We use weighted graph states, where vertices represent qubits (a vertex can also be a subsystem), and an edge between vertices $v_i$ and $v_j$ determines the bipartite entanglement between subsystems $i$ and $j$ .",
"If a graph state is acyclic, i.e.",
"there is only one edge connecting two subsystems, successively using the Schmidt circuit in Fig.REF and controlling the Schmidt coefficients, as done for three qubits above, the desired entanglement between each subsystems can be derived.",
"Example circuits are given in Fig.REF and Fig.REF for linear (path) and star graphs with five qubits.",
"A linear graph or path graph consists of vertices and edges that can be drawn as a single straight line where there are two terminal vertices of degree 1 at the beginning and at the end of the line and the remaining vertices are in the middle and have degree 2.",
"A star graph with $n$ vertices have one vertex of degree $n-1$ and all the other vertices have degree 1.",
"When a star graph is drawn, as its name suggests, it forms a star where the vertex having degree $n-1$ is located in the middle.",
"Similar circuit designs can be generated for different graphs in the same manner.",
"Figure: Quantum circuit that can generate random state for 5 qubits with the entanglement amount between qubits defined on the linear graph.Figure: Quantum circuit that can generate random state for 5 qubits with the entanglement amount between qubits defined on the star graph."
],
[
"Numerical Results",
"The statistical properties of the entanglement of a large bipartite quantum system have been analyzed by Facchi et.",
"al [26].",
"Pasquale et.",
"al [27] have investigated the statistical distribution of the Schmidt coefficients to obtain characterization of the statistical features of the bipartite entanglement of a large quantum system in a pure state.",
"In addition, the behavior of bipartite entanglement at the fixed von Neumann entropy has been recently studied in ref.[28].",
"Here, we now show the degree of the randomness of the output states, generated by the circuits described in the previous section, through a given probability distribution of the generated states.",
"Let $G(m, n)$ be an $m\\times n$ matrix of independent and identically distributed standard normal real random variables.",
"The distribution of the matrices is defined as [1], [29]: $\\frac{1}{(2\\pi )^{\\beta mn/2}}e^{-\\frac{1}{2}||G||_F^2},$ where $||G||_F=\\sqrt{Tr(G^*G)}$ is the Frobenius norm of the matrix $G$ , and $\\beta $ takes values based on considering real matrices($\\beta =1$ ), the complexes ($\\beta =2$ ), or the quaternions $(\\beta =4)$ .",
"In $MATLAB$ , we use the function $G=randn(m,n)$ to generate matrices with the above Gaussian distribution with $\\beta =1$ .",
"Starting with a normally distributed matrix and taking QR or singular value decomposition of the matrix generates random orthogonal matrices distributed according to Haar measure.",
"One can also generate standard random orthogonal matrices with the same distribution by using successive plane rotations with random angle values generated according to Gaussian distribution [30].",
"Therefore, for quantum states generated using the Schmidt decomposition by choosing random Schmidt coefficients and basis (or the corresponding quantum gates), the distribution of the overlaps or the angle values between these states is expected to be Gaussian.",
"This is also numerically shown in Fig.REF for 1000 random quantum states (All histograms in the figures are drawn by using 1000 number of states.)",
"generated using random Schmidt basis and coefficients for an eight-qubit star graph state.",
"Figure: Angles between the generated quantum states for an eight-qubit star graph:Both the Schmidt coefficients and basis are chosen randomly for each state.When we use the same Schmidt coefficients but different random bases to generate random quantum states; if the sizes of the subsystems are greater than one qubit, the distribution of the histogram of the generated quantum states are still Gaussian.",
"This is numerically shown in Fig.REF for a four-qubit system composed of two-qubit subsystems, $H_{12}$ and $H_{34}$ .",
"We draw the distribution of the angles between 1000 random quantum states which has the same Schmidt coefficients but different bases: the comparison of the histograms in Fig.REF and Fig.REF shows that the distributions are very similar when the entanglement between subsystem is high and low.",
"Therefore, if the sizes of the subsystems are greater than one qubit, the amount of the bipartite entanglement between subsystems does not affect the distribution, which is Gaussian.",
"Figure: Histograms of the angles between the generated quantum states for four qubits,where the Schmidt coefficients are fixed to certain values and the amountof the entanglement is high in (a) and low in (b).However, if the size of one of the subsystems is one qubit and the amount of the entanglement is fixed, examples are shown in Fig.REF for an eight-qubit linear graph and in Fig.REF for an eight-qubit star graph, then the distribution of the angles between the generated quantum states are affected by the amount of the bipartite entanglement between these subsystems.",
"As an example in Fig.REF for a two-qubit system, two different random set of Schmidt coefficients are used to generate two group of 1000 quantum random states (the states in the same group have the same Schmidt coefficients): the entanglement is high for the first group and low for the second group.",
"While the histogram of the first group shown in Fig.REF looks more uniform-like, the histogram for the second group shown in Fig.REF is more Gaussian-like.",
"This indicates that the distribution changes by the amount of the entanglement and is Gaussian when the entanglement is low; however, becomes more uniform-like when the amount of the entanglement is increased.",
"This appears more clearly in Fig.REF , where the average bipartite entanglement ($\\overline{S}$ ) between subsystems of a five-qubit star graph changes for each different group of 1000 random states.",
"As shown in Fig.REF , the distribution becomes more uniform-like when the average entanglement ($\\overline{S}$ ) is increased.",
"Please also note that the reason for using five qubits is to perform the computer simulations faster while keeping the system size large enough.",
"In addition, we show the histograms of star graphs of different number of qubits with different average bipartite entanglements in Fig.REF , which also supports Fig.REF and the above argument.",
"Figure: Histogram of the angles between the generated random quantum states for eight qubits where the Schmidt coefficients are fixed and the corresponding bipartite entanglements are given as weights of the edges on the graph.Figure: Histogram of the angles between the generated random quantum states for eight qubits where the Schmidt coefficients are fixed and the corresponding bipartite entanglements are given as weights of the edges on the graph.Figure: Histograms of the angles between the generated quantum states for two qubits,where the Schmidt coefficients are fixed to a certain value and the amountof the entanglement is high in (a) and low in (b).Figure: S ¯=0.9440\\overline{S}=0.9440Figure: 10-qubit: S ¯=0.6766\\overline{S}=0.6766"
],
[
"Conclusion",
"In this paper, we map the Schmidt decomposition for a general quantum state into quantum circuits, which can be used to generate random quantum states.",
"We show that in random state generation, the entanglement amount between subsystems can be controlled by using quantum gates implementing the desired Schmidt coefficients.",
"We also show that one can combine the Schmidt circuits sequentially to generate an equivalent quantum state for an acyclic weighted graph state in which vertices and edges correspond to subsystems and the bipartite entanglement between subsystems, respectively, and also the amount of entanglement is given by the weights of the edges.",
"Our method can be used in different applications and protocols relying on entanglement.",
"In the simulation of quantum systems, one can use the method to create an instance of the desired system.",
"In addition, decoherence effects the quality of the entanglement and generally cause errors in computations.",
"A similar idea can be used in quantum error correction to correct an imperfect bipartite entanglement, and so a quantum channel."
]
] | 1403.0270 |
[
[
"Universality and Borel Summability of Arbitrary Quartic Tensor Models"
],
[
"Abstract We extend the study of \\emph{melonic} quartic tensor models to models with arbitrary quartic interactions.",
"This extension requires a new version of the loop vertex expansion using several species of intermediate fields and iterated Cauchy-Schwarz inequalities.",
"Borel summability is proven, uniformly as the tensor size $N$ becomes large.",
"Every cumulant is written as a sum of explicitly calculated terms plus a remainder, suppressed in $1/N$.",
"Together with the existence of the large $N$ limit of the second cumulant, this proves that the corresponding sequence of probability measures is uniformly bounded and obeys the tensorial universality theorem."
],
[
"Introduction",
"Matrix models [1], [2] allow to study in two dimensions critical phenomena on a random geometry and the quantization of gravity coupled with conformal matter.",
"Tensor models [3] are their generalization to higher dimensions.",
"Colored [4] and invariant [5] tensor models support a $1/N$ expansion [6], [7], [8].",
"This analytical tool is crucial for establishing their large $N$ and double scaling limit [9], [10], [11] as well as their application to the study of critical phenomena in random geometries [12], [13], [14], [15], [16], [17] and possibly to quantization of gravity [18] in higher dimensions.",
"Already for quartic interactions, tensor models have a very rich structure: whereas there exists a unique quartic invariant one can build out of a matrix, $\\mathrm {Tr}\\left[ MM^{*}MM^{*}\\right]$ , there are numerous possible quartic invariants for tensors, as the indices of four tensors can be contracted in many different ways.",
"The mathematical study of tensor models as probability measures has recently been started [19], with the rigorous non-perturbative construction of a specific quartic tensor model completed in [20] using the Loop Vertex Expansion (LVE) [21], [22], [23].",
"While this model is symmetric over all the colors, hence is a genuine tensor model, and not just a model for a rectangular matrix, it includes only the simplest possible quartic interactions, called melonic.",
"In particular, for this melonic model, a factorization property holds which is crucial for establishing the results in [20].",
"The factorization fails as soon as one considers a slightly more general model: even adding just some non melonic quartic invariants spoils it.",
"This brings the natural question: are the results of [20] generic, or are they the result of a lucky accident?",
"In this paper we prove that these non perturbative results and the tensorial universality theorem of [19] hold for any quartic tensor model.",
"However, as the factorization fails, we need to use an entirely different technique to establish them.",
"This paper is organized as follows.",
"Section introduces the framework of tensor models with an arbitrary quartic interactions and presents our theorems.",
"The rest of the sections contains the proofs."
],
[
"The model and the main results",
"In this section we introduce the notations and state our main results.",
"The proofs of these results are presented starting with the next section."
],
[
"Generalities",
"Let us consider a Hermitian inner product space $V$ of dimension $N$ and $\\lbrace e_n| n=1,\\dots N\\rbrace $ an orthonormal basis in $V$ .",
"The dual of $V$ , $V^{\\vee }$ is identified with the complex conjugate $\\bar{V}$ via the conjugate linear isomorphism $z \\rightarrow z^{\\vee }(\\cdot )=\\langle z, \\cdot \\rangle \\; .$ We denote $e^n \\equiv e_n^{\\vee } = \\langle e_n , \\cdot \\rangle $ the basis dual to $e_n $ .",
"Then $\\Bigl (\\sum _n z^n e_n \\Bigr )^{\\vee }(\\cdot ) = \\sum _n \\overline{ z^n }\\langle e_n, \\cdot \\rangle = \\sum _{n} (z^{\\vee })_n e^n (\\cdot )\\Rightarrow (z^{\\vee })_n = \\overline{ z^n } \\; .$ A covariant tensor of rank $D$ is a multilinear form ${\\bf T}: V^{\\otimes D} \\rightarrow \\mathbb {C}$ .",
"We denote its components in the tensor product basis by $T_{n^1\\dots n^D} \\equiv {\\bf T} (e_{n^1},\\dots , e_{n^D}) \\; , \\qquad {\\bf T} = \\sum _{n^1,\\dots n^D} T_{n^1\\dots n^D} \\; \\;e^{n^1} \\otimes \\dots \\otimes e^{n^D} \\; .$ A priori $T_{n^1\\dots n^D}$ has no symmetry properties, hence its indices have a well defined position.",
"We call the position of an index its color, and we denote ${\\cal D}$ the set of colors $\\lbrace 1,\\dots D\\rbrace $ .",
"A tensor can be seen as a multilinear map between vector spaces.",
"There are in fact as many choices as there are subsets ${\\cal C}\\subset {\\cal D}$ : for any such subset the tensor is a multilinear map ${\\bf T}: V^{\\otimes {\\cal C}} \\rightarrow \\bar{V}^{ \\otimes {\\cal D}\\setminus {\\cal C}}$ : ${\\bf T}(z^{(c)}, c \\in {\\cal C}) = \\sum _{n^c, c\\in {\\cal C}} T_{n^1\\dots n^D} \\prod _{c\\in {\\cal C}} [z^{(c)}]^{n^c} \\; .$ We denote $n^{{\\cal C}}= (n^c,c\\in {\\cal C})$ the indices with colors in ${\\cal C}$ .",
"The complementary indices are then denoted $n^{{\\cal D}\\setminus {\\cal C}} =( n^c,c\\notin {\\cal C})$ .",
"In this notation the set of all the indices of the tensor should be denoted $n^{{\\cal D}}=(n^1,\\dots n^D)$ .",
"We will use whenever possible the shorthand notation $n\\equiv n^{\\cal D}$ .",
"The matrix elements of the linear map (in the appropriate tensor product basis) are $T_{ n^{{\\cal D}\\setminus {\\cal C}} n^{{\\cal C}} } \\equiv T_n \\equiv T_{n^1\\dots n^D} \\; \\text{ with } n^c \\in n^{{\\cal C}} \\cup n^{{\\cal D}\\setminus {\\cal C}} \\;,\\;\\; \\forall c \\; .$ As we deal with complex inner product spaces, the dual tensor $ {\\bf T}^{\\vee }$ is defined by $ {\\bf T}^{\\vee } \\left( (z^{(1)})^{\\vee }, \\dots (z^{(D)})^{\\vee } \\right) \\equiv \\overline{ {\\bf T} \\left( z^{(1)}, \\dots z^{(D)} \\right) } \\; .$ Taking into account that $\\sum _{n^{{\\cal D}}} \\overline{T_{n^1\\dots n^D}} \\; \\overline{ (z^{(1)} )^{n^1} } \\dots \\overline{ (z^{(D)} )^{n^D} }= \\sum _{n^{{\\cal D}}} \\overline{T_{n^1\\dots n^D}} \\; ( z^{(1)\\vee })_{n^1} \\dots (z^{(D)\\vee })_{n^D} \\;,$ we obtain the following expressions for the dual tensor and its components $ {\\bf T}^{\\vee } = \\sum _{n^1, \\dots n^D} \\overline{T_{n_1\\dots n_D}}\\; \\; e_{n^1} \\otimes \\dots e_{n^D} \\; , \\;\\;({\\bf T}^{\\vee })^{n^1\\dots n^D} = \\overline{T_{n_1\\dots n_D}} \\; .$ The dual tensor is a conjugated multilinear map ${\\bf T}^{\\vee } : \\bar{V}^{\\otimes {\\cal D}\\setminus {\\cal C}} \\rightarrow V^{\\otimes {\\cal C}} $ with matrix elements $( T^{\\vee })^{ n^{{\\cal C}} n^{{\\cal D}\\setminus {\\cal C}} } \\equiv \\overline{ T_{ n^1\\dots n^D} } \\; \\text{ with }n^c \\in n^{{\\cal C}} \\cup n^{{\\cal D}\\setminus {\\cal C}} \\; ,\\;\\; \\forall c \\; .$ From now on we denote $ \\bar{T}_{n_1\\dots n_D} \\equiv \\overline{T_{n_1\\dots n_D}} $ , we write all the indices in subscript, and we denote the contravariant indices with a bar.",
"Indices are always understood to be listed in increasing order of their colors.",
"We denote $ \\delta _{n^{{\\cal C}} \\bar{n}^{{\\cal C}}} = \\prod _{c\\in {\\cal C}} \\delta _{n^c \\bar{n}^c} $ and $\\mathrm {Tr}_{{\\cal C}}$ the partial trace over the indices $n^c, c\\in {\\cal C}$ ."
],
[
"Trace invariants and tensor models",
"Under unitary base change, covariant tensors transform under the tensor product of $D$ fundamental representations of $U(N)$ : the group acts independantly on each index of the tensor.",
"For $U^{(1)}...U^{(D)}\\in U(N)$ , ${\\bf T} \\rightarrow \\left( U^{(1)} \\otimes ... \\otimes U^{(D)} \\right){\\bf T} ,\\qquad {\\bf T}^{\\vee } \\rightarrow {\\bf T}^{\\vee } \\left( U^{(1)*} \\otimes ... \\otimes U^{(D)*} \\right).$ In components, it writes, $T_{a^{{\\cal D}}}\\rightarrow \\sum _{m^{{\\cal D}}} U^{(1)}_{a^1 m^1}...U^{(D)}_{a^D m^D}\\ T_{m^{{\\cal D}}},\\qquad \\bar{T}_{\\bar{a}^{{\\cal D}}}\\rightarrow \\sum _{m^{{\\cal D}}} \\bar{U}^{(1)}_{\\bar{a}^1 \\bar{m}^1}... \\bar{U}^{(D)}_{\\bar{a}^D \\bar{m}^D}\\ \\bar{T}_{\\bar{m}^{{\\cal D}}}.$ A trace invariant is a invariant quantity under the action of the external tensor product of $D$ independant copies of the unitary group $U(N)$ which is built by contracting indices of a product of tensor entries.",
"The tensor and its dual can be composed as linear maps to yield a map from $V^{\\otimes {\\cal C}}$ to $V^{\\otimes {\\cal C}}$ $[ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{\\bar{n}^{ {\\cal C}} n^{ {\\cal C}} }= \\sum _{n^{{\\cal D}\\setminus {\\cal C}} , \\bar{n}^{{\\cal D}\\setminus {\\cal C}} }\\bar{T}_{ \\bar{n}^1\\dots \\bar{n}^D}\\delta _{\\bar{n}^{{\\cal D}\\setminus {\\cal C}} n^{{\\cal D}\\setminus {\\cal C}} } T_{ n^1\\dots n^D} \\; .$ The unique quadratic trace invariant is the (scalar) Hermitian pairing of ${\\bf T}^{\\vee }$ and ${\\bf T}$ which writes: ${\\bf T}^{\\vee } \\cdot _{{\\cal D}} {\\bf T} = \\sum _{n^{{\\cal D}} \\bar{n}^{{\\cal D}}}\\bar{T}_{ \\bar{n}^1\\dots \\bar{n}^D} \\delta _{ \\bar{n}^{{\\cal D}} n^{{\\cal D}}} T_{ n^1\\dots n^D} \\; ,$ A connected quartic trace invariant $V_{\\mathcal {C}}$ for ${\\bf T}$ is specified by a subset of indices ${\\cal C}\\subset {\\cal D}$ : $V_{{\\cal C}}({\\bf T}^{\\vee },{\\bf T} ) = \\mathrm {Tr}_{{\\cal C}} \\Big [ \\left[ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} \\right] \\cdot _{{\\cal C}}\\left[{\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} \\right] \\Big ] \\;,$ where we denoted $\\cdot _{{\\cal C}}$ the product of operators from $V^{\\otimes {\\cal C}}$ to $V^{\\otimes {\\cal C}}$ .",
"In components this invariant writes: $\\sum _{n, \\bar{n}, m, \\bar{m}}\\left( \\bar{T}_{\\bar{n}}\\ \\delta _{\\bar{n}^{{\\cal D}\\setminus {\\cal C}} n^{{\\cal D}\\setminus {\\cal C}} } \\ T_{n} \\right) \\ \\delta _{n^{{\\cal C}}\\bar{m}^{{\\cal C}}} \\delta _{ \\bar{n}^{{\\cal C}} m^{{\\cal C}}}\\ \\left( \\bar{T}_{\\bar{m}}\\ \\delta _{\\bar{m}^{{\\cal D}\\setminus {\\cal C}} m^{{\\cal D}\\setminus {\\cal C}} } \\ T_{m} \\right) \\; .$ A generic quartic tensor model is then the (invariant) perturbed Gaussian measure for a random tensor: $d\\mu = \\Bigl ( \\prod _n N^{D-1} \\frac{ d \\bar{T}_{{n}} dT_n }{2 \\imath \\pi } \\Bigr ) \\,e^{ -N^{D-1} \\big ( {\\bf T}^{\\vee } \\cdot _{{\\cal D}} {\\bf T}+\\lambda \\sum _{\\mathcal {C} \\in \\mathcal {Q}} V_{\\mathcal {C}}({\\bf T}^{\\vee },{ \\bf T} ) \\big ) } \\;,$ where $\\mathcal {Q}$ is some set of ${\\cal C}$ s. From now, we will denote $\\mathcal {N}_\\mathcal {Q} = |\\mathcal {Q}|$ , the cardinal of $\\mathcal {Q}$ .",
"The melonic models previously treated in the literature [20] are obtained by restricting to $\\mathcal {Q}=\\lbrace \\mathcal {C},|\\mathcal {C}|=1 \\rbrace $ .",
"Considering arbitrary $\\mathcal {Q}$ s has important consequences.",
"One of the features of the melonic model of [20] is that, in the loop vertex expansion, the amplitude of graphs factors in contributions associated to the faces (technically this is done by introducing Schwinger parameters on the resolvents) and one immediately recovers the appropriate scaling with $N$ .",
"This does not hold in the general model.",
"Recovering the appropriate scaling in $N$ requires a new technique in this more general case relying on iterated Cauchy-Schwarz inequalities which we will present below.",
"The moment-generating function of the measure $d\\mu $ is defined as : $& Z(J, \\bar{J})=\\int d\\mu \\;\\; e^{ \\sum _{n} T_{n} \\bar{J}_{n} + \\sum _{\\bar{n}}\\bar{T}_{\\bar{n} }J_{\\bar{n} }} \\;,$ and its cumulants are thus written : $& \\kappa (T_{n_1}\\bar{T}_{\\bar{n}_1}...T_{n_k}\\bar{T}_{\\bar{n}_k})=\\frac{\\partial ^{(2k)} \\Bigl ( \\ln Z(J,\\bar{J}) \\Bigr ) }{\\partial \\bar{J}_{n_1}\\partial J_{\\bar{n}_1}...\\partial \\bar{J}_{n_k}\\partial J_{\\bar{n}_k}} \\Bigg {\\vert }_{J =\\bar{J} =0}.$"
],
[
"Gaussian measure and universality",
"The Gaussian measure of covariance $\\sigma ^2$ for a random tensor is: $d\\mu _{G} = \\Bigl ( \\prod _n \\frac{N^{D-1}}{\\sigma ^2} \\frac{ d \\bar{T}_{{n}} dT_n }{2 \\imath \\pi } \\Bigr ) \\,e^{ -\\sigma ^{-2}N^{D-1} \\ {\\bf T}^{\\vee } \\cdot _{{\\cal D}} {\\bf T} }\\;,$ For any trace invariant $\\mathrm {B}({\\bf T}^{\\vee },{ \\bf T})$ made of $k$ covariant and $k$ contravariant tensors, there are two non-negative integers, $\\Omega (\\mathrm {B})$ ans $R(\\mathrm {B})$ , such that the large $N$ limit of the Gaussian expectation follows [19] $\\lim _{N\\rightarrow \\infty }N^{\\Omega (\\mathrm {B})-1}\\mu _G \\left(\\mathrm {B}({\\bf T}^{\\vee },{ \\bf T})\\right)=R(\\mathrm {B})\\ \\sigma ^{2k}.$ $\\Omega (\\mathrm {B})$ is called the convergence order of B.",
"Definition 1 A random tensor ${ \\bf T}$ with the probability measure $\\mu $ converges in distribution to the distributional limit of a Gaussian tensor of covariance $\\sigma ^2$ if the large $N$ limit of the expectation of any trace invariant $\\mathrm {B}$ equals the limit of the Gaussian expectation of the invariant.",
"$\\lim _{N\\rightarrow \\infty }N^{\\Omega (\\mathrm {B})-1}\\mu \\left(\\mathrm {B}({\\bf T}^{\\vee },{ \\bf T})\\right)=R(\\mathrm {B})\\ \\sigma ^{2k}.$ Definition 2 A random tensor ${\\bf T}$ with the probability measure $\\mu $ is trace invariant if its cumulants are linear combinaisons of trace invariant operators.",
"$\\kappa (T_{n_1}\\bar{T}_{\\bar{n}_1}...T_{n_k}\\bar{T}_{\\bar{n}_k})=\\sum _{\\pi , \\bar{\\pi }}\\sum _{\\rho _{{\\cal D}} }\\mathfrak {K}(\\rho _{{\\cal D}})\\prod _{d=1}^k\\prod _{c=1}^D\\delta _{n_{\\pi (d)}^c \\bar{n}_{\\rho _c\\bar{\\pi }(d)}^c} \\;,$ where $\\rho _{{\\cal D}}=(\\rho _1, \\dots \\rho _D)$ runs over $D$ -uples of permutations of $k$ elements, $\\pi $ and $\\bar{\\pi }$ are permutations over $k$ elements, and $\\mathfrak {K}(\\rho _{{\\cal D}})$ depends only on $\\rho _{{\\cal D}}$ , $\\lambda $ and $N$ .",
"We denote $\\mathfrak {C}(\\rho _{{\\cal D}})$ the number of connected components of the graph associated to $\\rho _{{\\cal D}}$ .",
"(We represent $\\rho _{\\cal D}$ as a $D$ -colored bipartite graph with $k$ black and $k$ white $D$ -valent vertices, each set being indexed form 1 to $k$ , and the white vertex $l$ is connected to the black vertex $\\rho _c (l)$ by an edge of color $c$ .",
"Then, $\\mathfrak {C}(\\rho _{{\\cal D}})$ is the number of connected components of this graph).",
"Definition 3 A trace invariant probability measure $\\mu $ is properly uniformly bounded at large $N$ if $\\lim _{n\\rightarrow \\infty } N^{D-1} \\mathfrak {K}(\\lbrace 1\\rbrace _{\\mathcal {D}}) < \\infty ,$ with $\\lbrace 1\\rbrace _{\\mathcal {D}}$ being the $D$ -uples of trivial permutations over $k=1$ element, and if $N^{ 2(D-1) k - D +\\mathfrak {C}(\\rho _{{\\cal D}}) } \\mathfrak {K}(\\rho _{\\mathcal {D}}) < K(\\rho _{\\mathcal {D}}),\\ \\forall k\\ne 1,\\ \\forall N,$ for some constant $K(\\rho _{\\mathcal {D}})$ .",
"The main result of this paper is to prove that for a perturbation parameter $\\lambda \\in [\\, 0, \\frac{1}{8\\mathcal {N}_{\\mathcal {Q}}})$ , quartic tensor models are properly bounded trace invariant probability distributions, and thus obey the second universality theorem in [19].",
"Theorem.",
"(Universality) For large $N$ , a random tensor whose probability distribution $\\mu $ is trace invariant and properly uniformly bounded converges in distribution to a Gaussian tensor of covariance $\\sigma ^2=\\lim _{N\\rightarrow \\infty }\\ N^{D-1} \\mathfrak {K}(\\,\\lbrace 1\\rbrace _{\\mathcal {D}}\\,)$ ."
],
[
"BKAR Formula",
"In order to compute the cumulants of $\\mu $ we need to compute the logarithm of the generating function $Z(J,\\bar{J})$ .",
"This can be done by using at first a replica trick, then the Brydges-Kennedy-Abdesselam-Rivasseau (BKAR) forest formula [24], [25], [26] interpolating the Gaussian measure between the replicas, and finally expressing the logarithm as a sum over trees.",
"Let $X$ be a complex vector, for any function $V(X,\\bar{X})$ , denoting $d\\mu _C$ the Gaussian measure of covariance $C$ , $\\int d\\mu _C \\;\\; X_a \\bar{X}_b = C_{ab} \\; ,$ we have: $& \\mathrm {ln}\\int e^{V(X,\\bar{X})}d\\mu \\quad =\\quad \\sum _{v\\ge 1}\\frac{1}{v!}",
"\\sum _{T_v}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}\\nonumber \\\\&\\times \\quad \\left[\\prod _{l_{ij}\\in T_v}\\left(\\sum _{a,b}\\left(\\frac{\\partial }{\\partial X_{a}^{i}}C_{ab}\\frac{\\partial }{\\partial \\bar{X}_{b}^{j}} + \\frac{\\partial }{\\partial X_{a}^{j}}C_{ab}\\frac{\\partial }{\\partial \\bar{X}_{b}^{i}}\\right)\\right)\\right]\\prod _{i=1}^v V\\left(X^i,\\bar{X}^i\\right),$ where $T_v$ runs over combinatorial trees with $v$ vertices labelled from 1 to $v$ , $l_{ij}$ denotes the edge of the tree connecting the vertices $i$ and $j$ and $u_{ij}$ is an interpolation parameter running from 0 to 1.",
"The $X^i$ and $\\bar{X}^i$ are random complex vectors associated with the vertices $i$ of the tree and distributed with the interpolated Gaussian measure $d\\mu _{T_v,u}$ defined as: $\\int d\\mu _{T_v, u} \\;\\;\\; X^i_{a}\\bar{X}^j_{b}= w_{ij}C_{ab} \\;,$ where, denoting $\\mathcal {P}_{ij}$ is the unique path in $T_v$ joining the vertices $i$ and $j$ , $w_{ij} = {\\left\\lbrace \\begin{array}{ll}1 \\qquad & i=j \\\\\\min _{ l_{km}\\in \\mathcal {P}_{ij} }\\lbrace u_{km}\\rbrace \\quad & i\\ne j\\end{array}\\right.}",
"\\; .$ The matrix $w_{ij}$ is positive, hence the measure $d\\mu _{T_v,u} $ is well defined.",
"Remark that the integral with measure $ d\\mu _{T_v, u}$ can be rewritten as: $& \\int d\\mu _{T_v, u} \\; \\; F(X,\\bar{X}) =\\crcr & \\quad = \\Big [e^{ \\sum _{i,j} w_{ij} \\Bigl [ \\sum _{ab} C_{ab}\\left( \\frac{\\partial }{\\partial X_{a}^{i} } \\frac{\\partial }{\\partial \\bar{X}_{b}^{j} } \\right) \\Bigr ] } F(X, \\bar{X})\\Big ]_{X=0} \\; ,$ where the matrix $w_{ij}$ is symmetric, namely $w_{ij} = w_{ji}$ ."
],
[
"Summary of the results",
"We denote $\\mathcal {N}_{\\mathcal {Q}}=|\\mathcal {Q}|$ the number of distinct interaction terms in a given model (hence for melonic models $\\mathcal {N}_{\\mathcal {Q}}=D$ ).",
"Note that ${\\cal C}$ and ${\\cal D}\\setminus {\\cal C}$ play a complementary role in a trace invariant.",
"Given an invariant, there are two natural choices of ${\\cal C}$ : ${\\cal C}$ are the colors shared by two tensors such that the color 1 never belongs to ${\\cal C}$ , $1\\notin {\\cal C}$ .",
"${\\cal C}$ are the colors shared by two tensors such that $|{\\cal C}|\\le |{\\cal D}\\setminus {\\cal C}|$ , that is $|{\\cal C}|\\le D/2$ .",
"The Loop Vertex Expansion.",
"A first set of results of this paper concerns the loop vertex expansion (introduced in [21], [22]) of the cumulants of the measure in eq.",
"(REF ).",
"Let us denote $\\sigma ^{\\cal {C}}_{a^{\\cal C} b^{\\cal C}}$ a $N^{|\\cal {C}|} \\times N^{|\\cal {C}|}$ matrix with line and column indices of colors in $\\cal {C}$ , $a^{\\cal {C}} = (a^c | c\\in \\cal {C})$ , and let us denote $\\mathbf {1}^{{\\cal D}\\setminus {\\cal C}}$ the identity matrix on the indices of colors ${\\cal D}\\setminus {\\cal C}$ .",
"We define $& A(\\sigma )=\\sqrt{\\frac{\\lambda }{ N^{D-1}}}\\left( \\sum _{\\mathcal {C}\\in \\cal {Q}}\\mathbf {1}^{{\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{\\mathcal {C}}-\\sigma ^{{\\mathcal {C}}*}) \\right) \\; ,\\crcr & \\big [A(\\sigma ) \\big ]_{nm} = \\sqrt{\\frac{\\lambda }{ N^{D-1}}}\\sum _{\\mathcal {C}\\in \\cal {Q}} \\delta _{n^{{\\cal D}\\setminus {\\cal C}} m^{{\\cal D}\\setminus {\\cal C}}} \\;\\;(\\sigma ^{\\mathcal {C}}-\\sigma ^{{\\mathcal {C}}*})_{n^{\\cal {C}} m^{\\cal {C}} } \\; .$ Note that, as $A(\\sigma )$ is anti Hermitian, $ [ \\mathbf {1}^{{\\cal D}} + A(\\sigma ) ]^{-1}$ is well defined for all $\\lambda \\in \\mathbb {C} \\setminus (-\\infty ,0)$ .",
"The loop vertex expansion of the generating function of the moments of $\\mu $ is: Lemma 1 The generating function of the moments of $\\mu $ (i.e.",
"the partition function of the quartic model) with a set of interactions $\\mathcal {Q}$ is: $&Z(J,\\bar{J})=\\int \\prod _{\\mathcal {C} \\in \\mathcal {Q}}\\prod _{a^{\\cal {C}}b^{\\cal {C}}}\\frac{d\\sigma _{a^{\\cal {C}}b^{\\cal {C}}}^{\\mathcal {C}}d\\bar{\\sigma }_{a^{\\cal {C}}b^{\\cal {C}}}^{\\mathcal {C}}}{2\\imath \\pi }\\;\\; e^{-\\ \\sum _{ \\mathcal {C} \\in \\cal {Q} }\\mathrm {Tr}_{{\\cal C}} \\big [\\sigma ^{{\\mathcal {C}}*}\\sigma ^{\\mathcal {C}}\\big ] \\ }\\crcr & \\qquad \\times e^{-\\ \\mathrm {Tr}_{{\\cal D}} \\big [ \\ln \\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma )\\right) \\big ]+ \\frac{1}{N^{D-1}} \\sum _{nm } \\bar{J}_{n}\\left[\\frac{1}{ \\mathbf {1}^{{\\cal D}} + A(\\sigma ) }\\right]_{nm}J_{m} } \\; .$ This lemma is proven in subsection REF .",
"The next theorem establishes the loop vertex expansion of the cumulants of $\\mu $ .",
"This expansion relies on the BKAR formula adapted to tensor models, which requires adding a number of twists.",
"The cumulants of $\\mu $ are expressed as sums over plane trees with marked vertices and colored edges.",
"Plane trees have a well defined ordering at the vertices, and a mark is a specified starting point for this ordering.",
"The edges of the trees are colored by subsets of colors ${\\cal C}$ .",
"We have here the first important difference between the melonic model of [20] and the general case presented here: in the former case the edges of the trees had a unique color, while now they can carry several.",
"Let us denote $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }$ a plane tree with $v$ vertices, labelled $1,2\\dots v$ , and whose vertices $\\lbrace i_d\\rbrace , d=1\\dots k$ are marked.",
"We denote ${\\mathcal {C}}(l_{ij})$ the colors of the tree edge $l_{ij}$ , and $T_v$ the combinatorial tree associated to $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }$ .",
"The $\\sigma ^{\\cal C},\\bar{\\sigma }^{\\cal C}$ fields are now replicated over the vertices, $\\sigma ^{{\\cal C}\\ i}$ and $\\bar{\\sigma }^{{\\cal C} \\ i}$ and the interpolated Gaussian measure $d\\mu _{T_v,u}(\\sigma )$ is degenerated over the colorings ${\\cal C}$ : $e^{\\sum _{ij}w_{ij} \\left[ \\sum _{\\mathcal {C}; a^{\\cal C} b^{\\cal C} }\\left(\\frac{\\partial }{\\partial \\sigma _{a^{\\cal C}b^{\\cal C}}^{ {\\mathcal {C} \\ i } } }\\frac{\\partial }{\\partial \\bar{\\sigma }_{a^{\\cal C}b^{\\cal C}}^{{\\mathcal {C} \\ j }}}\\right) \\right] } \\; .$ The contribution of each tree is a certain contraction of resolvent operators and external source terms $J$ and $\\bar{J}$ .",
"The resolvents are defined as: $R(\\sigma )= \\left[ {\\bf 1}^{{\\cal D}} + A(\\sigma ) \\right]^{-1} \\; .$ We adopt the following graphical representation.",
"We represent every vertex of the plane tree as a fat vertex, having $D$ interior strands, corresponding to the $D$ indices of the resolvent.",
"The strands are labelled 1 to $D$ from the most interior strand to the exterior one.",
"Plane trees have a well defined notion of corners which are pieces of the vertices comprised between two consecutive halflines.",
"Every corner of the vertex is the represented as $D$ parallel strands crossed by a vertical line, as in Figure REF on the left.",
"Figure: Graphical representation of a resolvent (left) and a J m J ¯ n J_m\\bar{J}_n term (right) for rank D=3D=3 tensors.The marks correspond to $J$ and $\\bar{J}$ sources.",
"We either represent them as $D$ parallel strands crossed by with a wiggly line (as in Figure REF on the right) or as a pair of caps gluing together the $D$ strands as in Figure REF .",
"The caps (pictured as dashed in Figure REF ) represent a $J$ and a $\\bar{J}$ source.",
"Both $J$ and $\\bar{J}$ have $D$ indices which are contracted with resolvent indices.",
"This is pictured by the fact that the dashed strands hook to solid strands in Figure REF .",
"Figure: Detailed representation of the external sources JJ and J ¯\\bar{J}.A plane tree is then a set of edges connecting such vertices.",
"Every edge will transmit the strands corresponding to the indices ${\\cal C}$ from one vertex to the other, and will connect on the same vertex the indices in ${\\cal D}\\setminus {\\cal C}$ .",
"Hence edges have multiple strands, as in Figure REF .",
"Figure: An example of plane tree with 1 marked and 3 regular vertices, and edges of color𝒞={3}{\\cal C}= \\lbrace 3 \\rbrace , {2}\\lbrace 2\\rbrace and {2,3}\\lbrace 2,3\\rbrace .",
"Resolvents and JJ ¯J\\bar{J} marks are represented as in Fig.",
"and stuck to their respective vertices.The number of corners of the vertex $i$ , denoted $res(i)$ , is equal to the degree of the vertex if it is not marked and it is equal to the degree of the vertex plus one if it is.",
"We label the corners of the vertex $i$ by $p$ in the order they are encountered when turning clockwise around the vertex.",
"To the $p$ 'th corner of the vertex $i$ we associate a resolvent $R(\\sigma ^i)_{n_{i,p} m_{i,p}}$ .",
"To every edge of a tree we associate a contraction of the indices of the four resolvents corresponding to the four corners incident to the edge.",
"If the edge $l_{ij}$ is incident to the corners $q$ and $q+1$ of the vertex $i$ , and $p$ and $p+1$ of the vertex $j$ , the contraction associated to the edge is: $\\delta _{ {\\cal D}}^{l_{ij},{\\mathcal {C}}(l_{ij})}=\\left( \\delta _{m^{{\\cal D}\\setminus {\\cal C}}_{i,q} n^{{\\cal D}\\setminus {\\cal C}}_{i,q+1} } \\right)\\delta _{ m^{{\\cal C}}_{i,q} n^{{\\cal C}}_{j,p+1} } \\delta _{ m^{{\\cal C}}_{j,p} n^{{\\cal C}}_{i,q+1} }\\left( \\delta _{m^{{\\cal D}\\setminus {\\cal C}}_{j,p} n^{{\\cal D}\\setminus {\\cal C}}_{j,p+1} } \\right) \\; .$ Theorem 1 The measure $\\mu $ is trace invariant, and its cumulants are given by: $\\kappa (T_{n_1}\\bar{T}_{\\bar{n}_1}...T_{n_k}\\bar{T}_{\\bar{n}_k})=\\sum _{\\pi , \\bar{\\pi }}\\sum _{\\rho _{{\\cal D}} }\\mathfrak {K}(\\rho _{{\\cal D}})\\prod _{d=1}^k\\prod _{c=1}^D\\delta _{n_{\\pi (d)}^c \\bar{n}_{\\rho _c\\bar{\\pi }(d)}^c} \\;,$ where $\\rho _{{\\cal D}}=(\\rho _1, \\dots \\rho _D)$ runs over $D$ -uples of permutations of $k$ elements, $\\pi $ and $\\bar{\\pi }$ are permutations over $k$ elements.",
"With the notation introduced above, $\\mathfrak {K}(\\rho _{{\\cal D}})$ is: $\\mathfrak {K}(\\rho _{{\\cal D}}) = &\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}} \\left( \\prod _{c=1}^{D}\\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right)\\crcr &\\quad \\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m}\\\\&\\quad \\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{ n_{i,p} m_{i,p}}\\right)\\left(\\prod _{l\\in \\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C} }(l) \\rbrace } } \\delta _{ {\\cal D}}^{l,{\\mathcal {C}}(l)}\\right) \\crcr & \\quad \\times \\left( \\prod _{d=1}^k \\prod _{c=1}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\; , \\nonumber $ where $\\tau _{{\\cal D}}=(\\tau _1, \\dots \\tau _D)$ runs over $D$ -uples of permutations of $k$ elements and $ \\mathrm {Wg}(\\pi ,N)$ is the Weingarten function [27], [28].",
"This theorem will be proved in the section .",
"We see from eq.",
"(REF ) that the cumulants are linear combinations of trace invariant operators [19] (pairwise identifications of the indices of $T$ and $\\bar{T}$ ) of the type: $\\prod _{d=1}^k \\prod _{c=1}^D\\delta _{n_{ d }^c \\bar{n}_{\\rho _c ( d)}^c} \\; .$ Such an operator is specified by $D$ permutations $(\\rho _1,\\dots \\rho _D)$ .",
"It is canonically represented as an edge colored graph [20].",
"The graph is obtained as follows: we draw a black and a white vertex for every $d$ and, for all $d$ and $c$ , we connect the black vertex $d$ to the white vertex $\\rho _c(d)$ by an edge of color $c$ .",
"The $\\pi $ and $\\bar{\\pi }$ permutations in eq.",
"(REF ) are just trivial relabellings of the vertices.",
"The $\\prod _{d=1}^k \\delta _{n_d^c \\, m_{\\rho _c(d)}^c}\\ \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}$ factors in eq.",
"(REF ) and eq.",
"(REF ) represent new contractions between the indices of the resolvents $R(\\sigma )$ and the $J$ and $\\bar{J}$ tensors.",
"We call these contractions external strands.",
"The permutations $\\tau $ encode contractions exclusively between the indices of resolvents, while the permutations $\\rho $ encode contractions exclusively between the indices of $J$ and $\\bar{J}$ .",
"We represent each such contraction as a ribbon edge, like in Figure REF .",
"For convenience we inserted a third category of strands (represented as dotted in Figure REF ) in between the solid and the dashed strands.",
"Figure: Detailed representation of the vertex with external strands.The permutations $\\tau _c$ are represented as ribbon edges having a solid strand and a dotted strand, while the $\\rho _c$ permutations have a dashed and a dotted strand.",
"The cycles of the permutation $\\tau _c\\rho _c^{-1}$ are the closed circuits made of dotted strands.",
"A typical example of a tree with external strands is presented in Figure REF .",
"Figure: Tree with external strands.The graph of each trace invariant in $J$ and $\\bar{J}$ in the expansion (REF ) is immediately read off our graphical representation: it is the graph associated to the permutation $\\rho _{{\\cal D}}$ , hence it is the graph made by the dashed strands.",
"If one splits the interactions such that $1\\notin {\\cal C}$ , which (as already mentioned) is always possible, the color 1 factors completely.",
"The cumulants rewrite then in terms of: reduced resolvents: $\\mathcal {R}(\\sigma )=\\left[\\mathbf {1}^{{\\cal D}\\setminus \\lbrace 1\\rbrace }+\\sqrt{\\frac{\\lambda }{ N^{D-1}}}\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}}\\otimes (\\sigma ^{\\mathcal {C}}-\\sigma ^{{\\mathcal {C}}*}) \\right)\\right]^{-1} \\;,$ which are operators on a vector space of dimension $N^{D-1}$ corresponding to the indices $\\mathbf {n}\\equiv (n^2 \\dots n^D)$ of colors different form 1. reduced edge contractions: $& \\delta _{ {\\cal D}\\setminus \\lbrace 1\\rbrace }^{l_{ij},{\\mathcal {C}}(l_{ij})}= \\\\&\\; =\\left( \\delta _{m^{{\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}}_{i,q} n^{{\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}}_{i,q+1} } \\right)\\delta _{ m^{{\\cal C}}_{i,q} n^{{\\cal C}}_{j,p+1} } \\delta _{ m^{{\\cal C}}_{j,p} n^{{\\cal C}}_{i,q+1} }\\left( \\delta _{m^{{\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}}_{j,p} n^{{\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}}_{j,p+1} } \\right) \\; .",
"\\nonumber $ Corollary 1 The cumulants of the measure $\\mu $ are alternatively given by: $&\\kappa (T_{n_1}\\bar{T}_{\\bar{n}_1}...T_{n_k}\\bar{T}_{\\bar{n}_k}) = \\crcr & \\quad =\\sum _{\\pi , \\bar{\\pi }}\\sum _{ \\rho _{{\\cal D}\\setminus \\lbrace 1\\rbrace } }\\mathfrak {K}( \\rho _{{\\cal D}\\setminus \\lbrace 1\\rbrace } )\\prod _{d=1}^k \\delta _{n_{\\pi (d)}^1\\bar{n}_{\\bar{\\pi }(d)}^1}\\prod _{c=2}^D\\delta _{n_{\\pi (d)}^c \\bar{n}_{\\rho _c\\bar{\\pi }(d)}^c} \\;,$ where $ \\rho _{{\\cal D}\\setminus \\lbrace 1\\rbrace }=(\\rho _2...\\rho _D)$ runs over collections of $D-1$ permutations over $k$ elements, $\\pi $ and $\\bar{\\pi }$ are permutations over $k$ elements, and $\\mathfrak {K}( \\rho _{{\\cal D}\\setminus \\lbrace 1\\rbrace } )$ is: $\\mathfrak {K}( \\rho _{{\\cal D}\\setminus \\lbrace 1\\rbrace })= &\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1)+k-v}} \\sum _{\\tau _{{\\cal D}\\setminus \\lbrace 1\\rbrace } }\\left( \\prod _{c\\ne 1} \\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right)\\crcr &\\quad \\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m}\\\\&\\quad \\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} \\mathcal {R}(\\sigma ^i)_{\\mathbf {n}_{i,p}\\mathbf {m}_{i,p}}\\right)\\left(\\prod _{l\\in \\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C} }(l) \\rbrace } } \\delta _{ {\\cal D}\\setminus \\lbrace 1\\rbrace }^{l,{\\mathcal {C}}(l)}\\right) \\crcr & \\quad \\times \\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\; , \\nonumber $ where $\\tau _{{\\cal D}\\setminus \\lbrace 1\\rbrace } = (\\tau _2,\\dots \\tau _{D}) $ runs over $D$ -uples of permutations of $k$ elements.",
"Eq.",
"(REF ) is not just a trivial evaluation of eq.",
"(REF ): both the scaling with $N$ and the number of sums in the first lines of the two equations differ.",
"In fact this equation represents a very different repackaging of the terms.",
"The advantage of this second representation resides in the fact that the expansion in eq.",
"(REF ) is written in terms of trace invariants which are trivial on the color 1: the permutation associated to the indices of color 1 is the identity permutation $e(d)=d$ .",
"If we represent such invariants as edge color graphs, the edges of color 1 always connect the white and the black vertex with the same $d$ .",
"Mixed Expansion.",
"The LVE expansion of the cumulants can be refined to an expansion in trees decorated by loop edges.",
"The loop edges are of the same nature as the tree edges: they have colors ${\\cal C}$ , they connect vertices and they are adjacent to corners.",
"Like tree edges, loop edges represent identifications of indices of the adjacent resolvents.",
"We represent them as edges with solid strands.",
"A tree ${\\cal T}$ decorated by loop edges ${\\cal L}$ (and external strands $\\tau _{{\\cal D}}$ ) is a graph.",
"All its edges have strands, and all its vertices are fat.",
"We call the closed circuits of solid and dotted strands faces.",
"The faces have a color $c\\in {\\cal D}$ .",
"There are three categories of faces: faces made of solid strands which do not reach any mark.",
"We call them internal faces and denote their number $F_{\\rm int}({\\cal T},{\\cal L})$ .",
"faces made of solid strands which reach at least a mark.",
"When reaching a mark the face follows the permutation $\\tau $ .",
"We call such faces external and denote their number $F_{\\rm ext}({\\cal T},{\\cal L},\\tau _{{\\cal D}})$ .",
"faces made of the dotted strands.",
"We call them $\\tau \\rho ^{-1}$ -faces as they track the permutation $\\tau \\rho ^{-1}$ .",
"The number of $\\tau \\rho ^{-1}$ -faces is the number of cycles of the permutation $\\tau \\rho ^{-1}$ , which we denote $C(\\tau \\rho ^{-1})$ .",
"They appear explicitly in the Weingarten function.",
"The mixed expansion of the cumulants is Theorem 2 The cumulants of $\\mu $ write: $&\\mathfrak {K}(\\rho _{{\\cal D}})=\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}} \\left( \\prod _{c = 1}^D\\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right) \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\crcr &\\times \\Bigg {[} \\sum _{q=0}^L \\left(\\frac{-2\\lambda }{N^{D-1}}\\right)^q \\frac{1}{q!}",
"\\sum _{{\\cal L}, |{\\cal L}| = q}N^{F_{\\rm int} ({\\cal T},{\\cal L})+F_{ \\rm ext } ({\\cal T},{\\cal L},\\tau _{ {\\cal D}} ) } \\crcr & \\qquad \\qquad \\qquad \\qquad \\times \\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right) \\prod _{l\\in {\\cal L}} w_{i(l)j(l)}\\crcr & \\qquad + \\left(\\frac{-2\\lambda }{N^{D-1}}\\right)^{L+1} \\frac{1}{L!}",
"\\sum _{ {\\cal L}, |{\\cal L}| = L+1 } \\crcr & \\qquad \\qquad \\qquad \\qquad \\times \\int _0^1 dt \\; (1-t)^L \\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\left( \\prod _{l\\in {\\cal L}} w_{i(l)j(l)} \\right) \\crcr & \\qquad \\times \\int d\\mu _{T_v,u}(\\sigma ) \\sum _{m,n} \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right)\\crcr &\\qquad \\times \\left(\\prod _{l\\in {\\cal L}}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)} \\right) \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\Bigg {]}\\; .$ This theorem is proved in section .",
"Absolute convergence and bounds.",
"From now on we only study the case $|{\\cal C}|\\le D/2$ , namely we always separate an interaction with the intermediate matrix field of minimal size.",
"This is of course always possible.",
"We will study the analyticity properties and the scaling with $N$ of the cumulants starting from the mixed expansion in eq.",
"(REF ).",
"Theorem 3 The series in (REF ) is absolutely convergent for $\\lambda \\in [0,\\frac{1}{8\\mathcal {N}_{\\mathcal {Q}}})$ .",
"In this domain the cumulants obey the bound $| \\mathfrak {K}(\\rho _{{\\cal D}}) | \\le N^{D - 2(D-1) k - \\mathfrak {C}(\\rho _{{\\cal D}}) } K(\\lambda ) \\; ,$ for some $K$ depending only on $\\lambda $ , and the rescaled second cumulant $N^{D-1}\\mathfrak {K}({1}_{\\mathcal {D}})$ admits a finite limit at large $N$ .",
"Hence the measure $\\mu $ is properly uniformly bounded.",
"This theorem will be proved in section .",
"This shows that the measure $\\mu $ obeys the universality theorem.",
"Uniform Borel summability.",
"We subsequently establish the uniform Borel summability of the cumulants at the origin.",
"Theorem 4 The cumulants can be analytically continued for complex $\\lambda =r e^{i\\phi }$ with $r<\\frac{1}{8\\mathcal {N}_{\\mathcal {Q}}}\\left(\\mathrm {cos}\\frac{\\phi }{2}\\right)^2$ .",
"In this domain they obey the bound: $| \\mathfrak {K}(\\rho _{{\\cal D}}) | \\le N^{D - 2(D-1) k - \\mathfrak {C}(\\rho _{{\\cal D}}) }K\\left( \\frac{|\\lambda |}{ \\left(\\cos \\phi /2 \\right)^2 } \\right) \\;,$ and are Borel summable in $\\lambda $ uniformly in $N$ .",
"This theorem will be proved in section .",
"The $1/N$ expansion.",
"Furthermore, the mixed expansion in eq.",
"(REF ) is the non perturbative $1/N$ expansion of the cumulants in the following sense Corollary 2 The rest term in the mixed expansion is analytic in the domain $r<\\frac{1}{8\\mathcal {N}_{\\mathcal {Q}}}\\left(\\mathrm {cos}\\frac{\\phi }{2}\\right)^2$ and in this domain it admits the bound: $&\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}} \\left( \\prod _{c = 1}^D\\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right) \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\crcr &\\qquad \\times \\left(\\frac{-2\\lambda }{N^{D-1}}\\right)^{L+1} \\frac{1}{L!}",
"\\sum _{ {\\cal L}, |{\\cal L}| = L+1 } \\int _0^1 dt \\; (1-t)^L \\crcr & \\qquad \\times \\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\left( \\prod _{l\\in {\\cal L}} w_{i(l)j(l)} \\right) \\crcr & \\qquad \\times \\int d\\mu _{T_v,u}(\\sigma ) \\sum _{m,n} \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right)\\crcr &\\qquad \\times \\left(\\prod _{l\\in {\\cal L}}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)} \\right) \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\Bigg {]} \\crcr & \\le N^{D - (D-1)k - (L+1) \\left(\\frac{D}{2}-1\\right) }\\left( \\frac{|\\lambda |}{ \\left(\\cos \\phi /2 \\right)^2 } \\right)^{L+k}K^{\\prime }\\left( \\frac{|\\lambda |}{ \\left(\\cos \\phi /2 \\right)^2 } \\right)\\; .$ for some bounded function $K^{\\prime }$ ."
],
[
"Loop Vertex Expansion",
"In this section we prove Lemma REF , Theorem REF and Corollary REF ."
],
[
"Intermediate field representation",
"The Hubbard Stratonovich intermediate field representation relies on the observation that, for any complex numbers $Z_1,Z_2$ , $\\int \\frac{d\\bar{z} dz}{2 \\imath \\pi } \\; e^{-z\\bar{z} - z Z_1 + \\bar{z} Z_2} = e^{-Z_1Z_2} \\; .$ We will now apply this formula for the quartic interaction terms.",
"We have: $& e^{-N^{D-1}\\lambda \\mathrm {Tr}_{{\\cal C}} \\left[ [ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ] \\cdot _{{\\cal C}}[ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ] \\right] } = \\crcr & \\; \\; = \\prod _{ n^{{\\cal C}} \\bar{n}^{{\\cal C}} m^{{\\cal C}} \\bar{m}^{{\\cal C}} }e^{-N^{D-1}\\lambda [ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} }\\delta _{n^{{\\cal C}} \\bar{m}^{{\\cal C}}} \\delta _{ m^{{\\cal C}} \\bar{n}^{{\\cal C}}}[ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ \\bar{m}^{{\\cal C}} m^{{\\cal C}} } }\\crcr &\\; \\; = \\prod _{ n^{{\\cal C}} \\bar{n}^{{\\cal C}} }e^{-N^{D-1}\\lambda [ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} }[ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ n^{{\\cal C}} \\bar{n}^{{\\cal C}} } }\\crcr &= \\prod _{ n^{{\\cal C}} \\bar{n}^{{\\cal C}} } \\Bigg (\\int \\frac{d\\sigma _{n^{\\cal C} \\bar{n}^{\\cal C}}^{\\mathcal {C}}d\\bar{\\sigma }_{n^{\\cal C} \\bar{n}^{\\cal C}}^{\\mathcal {C}}}{2\\imath \\pi }\\quad e^{- \\sigma _{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} }^{{\\cal C}} \\bar{\\sigma }_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} }^{{\\cal C}} }\\crcr & \\qquad \\times e^{-\\sqrt{\\lambda N^{D-1} } [ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} } \\sigma ^{{\\cal C}}_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} } }\\;\\;e^{ \\sqrt{\\lambda N^{D-1} } [ {\\bf T}^{\\vee } \\cdot _{{\\cal D}\\setminus {\\cal C}} {\\bf T} ]_{ n^{{\\cal C}} \\bar{n}^{{\\cal C}} } \\bar{\\sigma }^{{\\cal C}}_{ \\bar{n}^{{\\cal C}} n^{{\\cal C}} } } \\Bigg )\\crcr &= \\int \\left(\\prod _{a^{{\\cal C}}b^{{\\cal C}}} \\frac{d\\sigma _{a^{{\\cal C}}b^{{\\cal C}}}^{\\mathcal {C}}d\\bar{\\sigma }_{a^{{\\cal C}}b^{{\\cal C}}}^{\\mathcal {C}}}{2 \\imath \\pi }\\right)e^{- \\mathrm {Tr}_{{\\cal C}} \\left[ \\sigma ^{{\\cal C}*} \\sigma ^{{\\cal C}} \\right] } \\crcr & \\qquad \\qquad \\times e^{ - \\sqrt{\\lambda N^{D-1}}\\sum _{n \\bar{n}} \\bar{T}_{\\bar{n}} \\left[ \\mathbf {1}^{{\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{\\mathcal {C}}-\\sigma ^{{\\mathcal {C}}*}) \\right]_{\\bar{n} n} T_{ n } } \\; .$ The generating function is then: $& Z(J, \\bar{J}) = \\crcr & =\\int \\left(\\prod _{n} N^{D-1}\\frac{dT_{n} d \\bar{T}_{n } }{2\\imath \\pi }\\right)\\left(\\prod _{\\mathcal {C}}\\prod _{a^{{\\cal C}}b^{{\\cal C}}} \\frac{d\\sigma _{ a^{{\\cal C}}b^{{\\cal C}} }^{\\mathcal {C}}d\\bar{\\sigma }_{ a^{{\\cal C}}b^{{\\cal C}} }^{\\mathcal {C}}}{2\\pi }\\right)e^{-\\sum _{\\mathcal {C}}\\mathrm {Tr}\\sigma ^{\\mathcal {C}}\\sigma ^{{\\mathcal {C}}*}} \\crcr &\\;\\; \\times e^{ -N^{D-1}\\sum _{n \\bar{n}} \\bar{T}_{\\bar{n} } \\left[ {\\mathbf {1}}^{{\\cal D}} \\ +\\ \\sqrt{\\frac{\\lambda }{ N^{D-1}}}\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{\\mathcal {C}}-\\sigma ^{{\\mathcal {C}}*}) \\right)\\right]_{\\bar{n}; n}T_{n} } \\crcr & \\;\\; \\times e^{ \\sum _{n} T_{n} \\bar{J}_{n} + \\sum _{\\bar{n}}\\bar{T}_{\\bar{n} }J_{\\bar{n} }} \\; .$ The integral over $T$ and $\\bar{T}$ is now Gaussian, and a direct computation leads to eq.",
"(REF )."
],
[
"Forest formula",
"To simplify notations we sometimes drop the superscript ${\\cal C}$ on the (multi) indices of $\\sigma ^{{\\cal C}}$ .",
"According to our equation (REF ), the logarithm of $Z(J,\\bar{J})$ is: $\\mathrm {ln} Z(J,\\bar{J})&=\\sum _{v\\ge 1}\\frac{1}{v!}",
"\\sum _{T_v}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma )\\crcr &\\times \\left[\\prod _{l_{ij}\\in T_v}\\left(\\sum _{{\\cal C},ab }\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}+ \\frac{\\partial }{\\partial \\sigma _{ab}^{j\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\right)\\right]\\crcr &\\times \\prod _{i=1}^v \\left(-\\mathrm {Tr}\\ \\mathrm {ln}\\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma ^i)\\right)+N^{1-D}\\bar{J}_n\\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma ^i)\\right)^{-1}_{nm}J_m\\right),$ where $T_v$ are combinatorial trees with $v$ vertices and the interpolated Gaussian measure $d\\mu _{T_v, u}$ is degenerated over ${\\cal C}$ : $\\int F(\\sigma ) \\; d\\mu _{T_v, u} = \\left[e^{\\sum _{i,j}w_{ij}\\sum _{ {\\cal C},ab }\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}} }}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}\\right)}F(\\sigma )\\right]_{\\sigma =0} \\; ,$ and $w_{ij}$ is defined in equation (REF ).",
"Expanding the product over $i$ we get: $&\\mathrm {ln} Z(J,\\bar{J})=\\sum _{v\\ge 1}\\frac{1}{v!}",
"\\sum _{T_v}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma )\\crcr & \\;\\; \\times \\left[\\prod _{l_{ij}\\in T_v}\\left(\\sum _{ {\\cal C},ab }\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}+ \\frac{\\partial }{\\partial \\sigma _{ab}^{j\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\right)\\right]\\sum _{k=1}^v\\ \\sum _{i_1<...< i_k} \\crcr & \\;\\; \\times \\prod _{d=1}^k\\ N^{1-D}\\bar{J}_n\\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma ^{i_d})\\right)^{-1}_{nm}J_m\\prod _{i\\ne i_1..i_k}-\\mathrm {Tr}\\ \\mathrm {ln}\\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma ^i)\\right) \\;.$ The logarithm of $Z(J,\\bar{J})$ is then a sum over trees $T_{v,k,\\lbrace i_d\\rbrace }$ with $k$ marked vertices (the $J\\bar{J}$ vertices) and $v-k$ regular vertices.",
"The sum over ${\\mathcal {C}}$ gives us a sum over trees with colored edges, each coloring corresponding to a set ${\\mathcal {C}}\\in \\mathcal {Q}$ .",
"Before taking into account the action of the derivatives, to each marked vertex $i_d$ of the tree $T_{v,k,\\lbrace i_d\\rbrace }$ is associated a resolvent operator $R(\\sigma ^{i_d})=\\left(\\mathbf {1}^{{\\cal D}} + A(\\sigma ^{i_d}) \\right)^{-1}$ (and a pair $J, \\bar{J}$ ), and to each unmarked vertex $i$ is associated a $ -\\mathrm {Tr}\\ \\mathrm {ln}\\left(\\mathbf {1} + A(\\sigma ^i)\\right) $ factor.",
"We now have to evaluate the action of the derivatives: $& \\frac{\\partial }{\\partial \\sigma ^{i \\ {\\cal C}}_{a^{{\\cal C}} b^{{\\cal C}} }} \\left[ R(\\sigma ^{i}) \\right]_{nm} = \\crcr & =\\frac{\\partial }{\\partial \\sigma ^{i \\ {\\cal C}}_{a^{{\\cal C}} b^{{\\cal C}} }} \\left[ \\sum _{q=0}^{\\infty } \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right)^q\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{i \\ {\\cal C}}-\\sigma ^{ i \\ {\\cal C}*}) \\right)^q \\right]_{nm} \\crcr & = \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right) \\sum _{a^{{\\cal D}\\setminus {\\cal C}} b^{{\\cal D}\\setminus {\\cal C}} }\\sum _{q_1,q_2=0}^{\\infty } \\crcr & \\; \\times \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right)^{q_1}\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{i \\ {\\cal C}}-\\sigma ^{ i \\ {\\cal C}*}) \\right)^{q_1}_{n a} \\crcr & \\; \\times \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right)^{q_2}\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{i \\ {\\cal C}}-\\sigma ^{ i \\ {\\cal C}*}) \\right)^{q_2}_{b m}\\delta _{a^{{\\cal D}\\setminus {\\cal C}} b^{{\\cal D}\\setminus {\\cal C}} } \\crcr & = \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right) \\sum _{ a^{{\\cal D}\\setminus {\\cal C}} b^{{\\cal D}\\setminus {\\cal C}} }\\left[ R(\\sigma ^{i}) \\right]_{n a } \\delta _{a^{{\\cal D}\\setminus {\\cal C}} b^{{\\cal D}\\setminus {\\cal C}} } \\left[ R(\\sigma ^{i}) \\right]_{b m} \\crcr & \\frac{\\partial }{\\partial \\sigma ^{i \\ {\\cal C}}_{a^{{\\cal C}} b^{{\\cal C}} }} \\left( - \\mathrm {Tr}\\ln [ \\left(\\mathbf {1} + A(\\sigma ^i)\\right) ] \\right) = \\crcr & = \\frac{\\partial }{\\partial \\sigma ^{i \\ {\\cal C}}_{a^{{\\cal C}} b^{{\\cal C}} }} \\left[ \\sum _{q=1}^{\\infty }\\frac{(-1)^q}{q}\\left( \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right)^q \\mathrm {Tr}\\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{i \\ {\\cal C}}-\\sigma ^{ i \\ {\\cal C}*}) \\right)^q\\right] \\crcr & = \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right) \\sum _{q=0}^{\\infty } \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right)^q \\crcr & \\;\\; \\times \\sum _{a^{{\\cal D}\\setminus {\\cal D}} b^{{\\cal D}\\setminus {\\cal C}}} \\delta _{a^{{\\cal D}\\setminus {\\cal D}} b^{{\\cal D}\\setminus {\\cal C}} }\\left[ \\left( \\sum _{\\mathcal {C}}\\mathbf {1}^{ {\\cal D}\\setminus {\\cal C}}\\otimes (\\sigma ^{i \\ {\\cal C}}-\\sigma ^{ i \\ {\\cal C}*}) \\right)^q \\right]_{ba} \\crcr & = \\left( - \\sqrt{\\frac{\\lambda }{N^{D-1}}} \\right) \\sum _{a^{{\\cal D}\\setminus {\\cal D}} b^{{\\cal D}\\setminus {\\cal C}}} \\delta _{a^{{\\cal D}\\setminus {\\cal D}} b^{{\\cal D}\\setminus {\\cal C}} }\\left[ R(\\sigma ^{i}) \\right]_{b a } \\; ,$ and similarly for $\\bar{\\sigma }$ .",
"A couple of derivative operators (i.e.",
"an edge) adds a resolvent on each vertex it acts on.",
"On each vertex, marked or not, acts at least one derivative operator, thus we obtain at least a resolvent per vertex.",
"Each vertex is then a partial trace of this resolvents (and a pair $J,\\bar{J}$ if it is marked).",
"The action of the derivatives acting on a vertex $(p+1)$ -th edge hooked to a vertex is the sum of the $p$ positions one can add a resolvent into the partial trace of eq.",
"(REF ), and induces a well defined ordering of the resolvents at a vertex.",
"The sum in (REF ) becomes thus a sum over plane trees, with well defined orderings of the half edges at every vertex, with resolvents associated to the corners.",
"We have thus expressed $\\mathrm {ln} Z$ as a sum over plane trees with colored edges and marked vertices $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }$ .",
"The contribution of a tree is a product of resolvents and $J\\bar{J}$ with indices contracted in a certain pattern.",
"To the edge $l_{ij}$ connecting the vertices $i$ and $j$ , incident at the corners $q$ and $q+1$ of the vertex $i$ and $p$ and $p+1$ of the vertex $j$ , corresponds the contraction: $\\delta _{{\\cal D}}^{l_{ij},{\\mathcal {C}}(l_{ij})}=\\left( \\delta _{m^{{\\cal D}\\setminus {\\cal C}}_{i,q} n^{{\\cal D}\\setminus {\\cal C}}_{i,q+1} } \\right)\\delta _{ m^{{\\cal C}}_{i,q} n^{{\\cal C}}_{j,p+1} } \\delta _{ m^{{\\cal C}}_{j,p} n^{{\\cal C}}_{i,q+1} }\\left( \\delta _{m^{{\\cal D}\\setminus {\\cal C}}_{j,p} n^{{\\cal D}\\setminus {\\cal C}}_{j,p+1} } \\right) \\; .$ Collecting everything and taking into account that each edge is the sum of two terms we obtain: $&\\mathrm {ln} Z(J,\\bar{J})=\\sum _{v\\ge 1}\\frac{1}{v!",
"}\\sum _{k=1}^v\\ \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m} \\crcr & \\qquad \\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{n_{i,p}m_{i,p}}\\right)\\left(\\prod _{l\\in T_v}\\frac{-2\\lambda }{N^{D-1}} \\ \\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right) \\crcr & \\qquad \\times \\left(\\prod _{d=1}^k\\ N^{1-D}\\bar{J}_{n_{i_d,q+1}}J_{m_{i_d,q}}\\right) \\; ,$ where res$(i)$ , the number of resolvents associated with the vertex $i$ , equals its degree for unmarked vertices, and its degree plus one for marked vertices, in the last line $q$ denotes the position of the $J\\bar{J}$ mark on the vertex $i_d$ , and all the indices $n$ and $m$ are summed.",
"If we always chose ${\\cal C}$ such that $1\\notin {\\cal C}$ , then $A(\\sigma )$ is always trivial on the index of color 1, $A(\\sigma ) = 1^{\\lbrace 1\\rbrace } \\otimes \\Bigl ( \\sqrt{ \\frac{\\lambda }{N^{D-1} } } \\sum _{{\\cal C}} {\\bf 1}^{{\\cal D}\\setminus \\lbrace 1\\rbrace \\setminus {\\cal C}} \\otimes (\\sigma ^{{\\cal C}}- \\sigma ^{{\\cal C}*})\\Bigr ) \\;,$ and the same holds for $R(\\sigma )$ , that is $R(\\sigma )=\\mathbf {1}^{\\lbrace 1\\rbrace }\\otimes \\mathcal {R}(\\sigma )$ with $\\mathcal {R}(\\sigma )$ defined in eq.",
"(REF ).",
"Also, $\\delta ^{l,{\\cal C}(l)}$ writes as $\\delta _{{\\cal D}}^{l,{\\cal C}(l)}=\\delta _{m_{i,q}^1n_{i,q+1}^1}\\delta _{m_{j,p}^1n_{j,p+1}^1}\\delta _{{\\cal D}\\setminus \\lbrace 1\\rbrace }^{l,{\\cal C}(l)}$ with $\\delta _{{\\cal D}\\setminus \\lbrace 1\\rbrace }^{l, {\\cal C}(l)} $ defined in eq.",
"(REF ).",
"The trace over the index of color 1 can be explicitly evaluated.",
"Denoting $\\mathbf {n} =n^{{\\cal D}\\setminus \\lbrace 1\\rbrace } = (n^2,\\dots n^D)$ , we get $ &\\mathrm {ln} Z(J,\\bar{J})=\\sum _{v\\ge 1}\\frac{1}{v!",
"}\\sum _{k=1}^v\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1)+k-v}} \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }} \\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right) \\\\& \\;\\times \\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m} \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}\\setminus \\lbrace 1\\rbrace }^{l,{\\mathcal {C}}(l)}\\right)\\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} \\mathcal {R}(\\sigma ^i)_{\\mathbf {n}_{i,p}\\mathbf {m}_{i,p}}\\right) \\crcr & \\; \\times \\left(\\prod _{d=1}^k \\bar{J}_{n_{i_d,q+1}}J_{m_{i_d,q}} \\delta _{ n^1_{i_d,q+1} m^1_{i_d,q} } \\right)\\nonumber \\;.$ The contribution of the strands of color 1 has been completely factored out.",
"The only trace of the color 1 still subsisting is the contraction of the indices of colors 1 between the source tensors $J$ and $\\bar{J}$ on the same vertex $i_d$ in the last line of the equation above."
],
[
"Cumulants",
"The cumulants are computed by evaluating the derivatives of eq.",
"(REF ) with respect to $J$ and $\\bar{J}$ .",
"However, in its present form, eq.",
"(REF ) obscures the invariance properties of the cumulants under unitary transformations.",
"To identify the appropriate invariant structure we use a trick introduced in [20].",
"Let us consider a set of unitary operators $U^c\\in U(N), c\\in {\\cal D}$ .",
"The Gaussian measure is invariant under the change of variables, of unit Jacobian, $\\sigma ^{i\\ {\\cal C}}\\rightarrow ( \\otimes _{c\\in {\\cal C}} U^ {c*}) \\ \\sigma ^{i\\ {\\mathcal {C}}} \\ ( \\otimes _{c\\in {\\cal C}} U^ {c }) \\; .$ Using $\\mathbf {1}=U^{c*} U^c$ , under this change of variables the resolvent changes like $& R(\\sigma )\\rightarrow \\ \\left( \\bigotimes _{c\\in {\\cal D}} U^{c*} \\right) \\ R(\\sigma )\\ \\left( \\bigotimes _{c\\in {\\cal D}} U^{c } \\right) \\; .$ If two resolvent indices are contracted together (through an edge contraction $\\delta ^{l, {\\cal C}(l) }_{{\\cal D}}$ ) an $U^c$ and $U^{c *}$ are multiplied together and drop out.",
"Thus, the only surviving $U^c$ 's are those from indices contracted with the $J, \\bar{J}$ source terms: $&\\mathrm {ln} Z(J,\\bar{J})= \\sum _{v\\ge 1}\\frac{1}{v!",
"}\\sum _{k=1}^v\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1)}} \\crcr &\\qquad \\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m} \\crcr &\\qquad \\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{ { n}_{i,p} { m}_{i,p}}\\right)\\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right) \\crcr & \\qquad \\times \\prod _{d=1}^k \\left( \\bar{J}_{n_d}J_{m_d}\\prod _{c=1}^D U^{c*}_{n_d^c n^c_{i_d,q+1}}U^c_{m_{i_d,q}^cm^c_d}\\right).\\nonumber $ The unitary operators can be added to our graphical representation of Figure REF by inserting a piece of a dotted strand in between the dashed strands representing the indices of $J$ and the solid strands representing the indices of the resolvent, as in Figure REF .",
"Figure: The UU and U * U^{*} transformations.Now, as $\\int _{U(N)}dU=1$ , we have the trivial equality $\\mathrm {ln} Z(J,\\bar{J})=\\int _{U(N)}...\\int _{U(N)}\\mathrm {ln} Z(J,\\bar{J})\\ dU^1 \\dots dU^D \\; ,$ and, for each value of $U^c$ , we use the previous change of variables.",
"The integral over $U(N)$ can be explicitly performed [27], [28]: $& \\int _{U(N)}dU^c \\; \\prod _{d=1}^k U^{c*}_{n_d^c n^c_{i_d,q+1}} U^c_{m_{i_d,q}^c m^c_d} \\crcr & \\qquad \\qquad =\\sum _{\\rho _c,\\tau _c} \\mathrm {Wg}(N,\\tau _c \\rho _c^{-1}) \\prod _{d=1}^k \\delta _{n_d^c \\, m_{\\rho _c(d)}^c}\\ \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c},$ where $\\rho _c$ and $\\tau _c$ run over all the permutations of $k$ elements.",
"The functions $\\mathrm {Wg}(N,\\pi )$ , introduced in [29], are known as the Weingarten functions.",
"They depend only of the cycle structure of the permutation $\\pi $ [27], [28], and if $\\pi $ has $q$ cycles of lengths $d_1,\\dots d_q$ , then [27], [28]: $& \\mathrm {Wg}(N,(1)) = \\frac{1}{N} \\; , \\qquad \\mathrm {Wg}(N,\\pi ) = \\prod _{j=1}^q V_{d_j} + O(N^{k-2n-2}) \\; ,\\crcr & V_{d} = N^{1-2d} (-1)^{d-1} \\frac{1}{d} \\binom{2d-2}{d-1} +O(N^{-1-2d}) \\; .$ The $\\prod _{d=1}^k \\delta _{n_d^c \\, m_{\\rho _c(d)}^c}\\ \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}$ factors in eq.",
"(REF ) representing new contractions between the indices of the resolvents $R(\\sigma )$ and the $J$ and $\\bar{J}$ tensors become the external strands.",
"The logarithm of $Z$ writes $&\\mathrm {ln} Z(J,\\bar{J})=\\sum _{v\\ge 1}\\frac{1}{v!",
"}\\sum _{k=1}^v\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }}\\sum _{\\rho _{{\\cal D}},\\tau _{{\\cal D}}} \\left( \\prod _{c=1}^D \\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right) \\\\& \\; \\times \\sum _{n_d,m_d}\\left[ \\prod _{d=1}^k\\left( \\bar{J}_{n_d}J_{m_d}\\prod _{c=1}^D \\delta _{n_d^c \\, m_{\\rho _c(d)}^c}\\right) \\right] \\crcr & \\;\\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }} \\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m}\\crcr &\\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{ n_{i,p} m_{i,p}}\\right)\\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=1}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\nonumber \\;.$ In this form $\\mathrm {ln} Z(J,\\bar{J})$ is explicitly a sum of trace invariants made of $J$ and $\\bar{J}$ tensors, and the graph of each such trace invariant is the graph made by the dashed strands.",
"The coefficient of a trace invariant is a sum over trees (decorated by external strands $\\tau _{{\\cal D}}$ ).",
"The cumulants are computed by taking the partial derivatives of eq.",
"(REF ) with respect to the external sources, $\\kappa (T_{n_1}\\bar{T}_{\\bar{n}_1}...T_{n_k}\\bar{T}_{\\bar{n}_k})&=\\frac{\\partial ^{2k}}{\\partial \\bar{J}_{n_1}\\partial J_{\\bar{n}_1} \\dots \\partial \\bar{J}_{n_k}\\partial J_{\\bar{n}_k}} \\mathrm {ln} Z(J,\\bar{J})\\Big {\\vert }_{J =\\bar{J} =0}\\nonumber \\\\&=\\sum _{\\pi , \\bar{\\pi }}\\sum _{\\rho _{{\\cal D}}}\\mathfrak {K}(\\rho _{{\\cal D}})\\prod _{d=1}^k\\prod _{c=1}^D\\delta _{n_{\\pi (d)}^c \\bar{n}_{\\rho _c\\bar{\\pi }(d)}^c},$ with $&\\mathfrak {K}(\\rho _{{\\cal D}})=\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}} \\left( \\prod _{c = 1}^D \\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right) \\crcr &\\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m} \\\\&\\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{ n_{i,p} m_{i,p}}\\right)\\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=1}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\nonumber \\; ,$ which achieves the proof of Theorem REF .",
"For corollary REF , one follows the same steps with minor adaptations, starting from eq.",
"(REF ).",
"Thus the unitary operators $U^c$ act only on the colors $2 \\dots D$ , the reduced resolvent changes like $& \\mathcal {R}(\\sigma )\\rightarrow \\ \\left( \\bigotimes _{c\\in {\\cal D}\\setminus \\lbrace 1\\rbrace } U^{c*} \\right) \\ \\mathcal {R}(\\sigma )\\ \\left( \\bigotimes _{c\\in {\\cal D}\\setminus \\lbrace 1\\rbrace } U^{c } \\right) \\; ,$ and so on.",
"The main difference comes from the fact that the strand 1 appears only as a direct contraction of the $J$ and $\\bar{J}$ corresponding to the same mark, but does not appear anymore among the indices of the resolvents.",
"In particular this leads to a novel graphical representation, in which the solid strands of color 1 have been erased, there are no $\\tau _1$ or $\\rho _1$ ribbon edges, and the $J$ and $\\bar{J}$ of a mark are connected by a dashed strand of color 1.",
"We call such a vertex reduced and we picture it like in Figure REF .",
"Figure: Detailed representation of the reduced vertex with external strands."
],
[
"The mixed expansion",
"Let us go back to eq.",
"(REF ), $&\\mathfrak {K}(\\rho _{{\\cal D}})=\\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{(-2\\lambda )^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}} \\left( \\prod _{c = 1}^D \\mathrm {Wg}(N,\\tau _c \\rho _c^{-1})\\right) \\crcr &\\times \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{n,m} \\\\&\\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R(\\sigma ^i)_{ n_{i,p} m_{i,p}}\\right)\\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=1}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\nonumber \\; ,$ and refine a term in this sum by Taylor expanding up to an order $L$ using the formula $f(\\sqrt{\\lambda })=& \\sum _{q=0}^{L}\\frac{1}{q!",
"}\\left[\\frac{d^q}{dt^q}f(\\sqrt{t\\lambda })\\right]_{t=0}\\ \\crcr & +\\ \\frac{1}{L !",
"}\\int _0^1 dt \\; (1-t)^{L}\\frac{d^{L+1} }{dt^{L+1} } \\left( f(\\sqrt{t\\lambda }) \\right)$ on the contributions of the trees $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }$ , using $\\frac{d}{dt} R(\\sqrt{t} \\sigma )_{ n m}=\\frac{1}{2t} \\left( \\sum _{\\mathcal {C}}\\left( \\sigma _{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}+ \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\right) R( \\sqrt{t} \\sigma )_{n m} \\; ,$ where we dropped the superscript on the indices of $\\sigma ^{{\\cal C}}$ , and integrating by parts $&\\frac{d}{dt}\\int d\\mu _{T_v,u}(\\sigma )\\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}}\\right)=\\frac{1}{2t}\\int d\\mu _{T_v,u}(\\sigma )\\crcr &\\qquad \\times \\sum _{i=1}^v\\sum _{\\mathcal {C}} \\left(\\sigma _{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}+ \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right) \\crcr &=\\frac{1}{2t}\\Bigg {[}e^{\\sum _{i,j}w_{ij}\\sum _{\\mathcal {C}}\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}} \\right) } \\crcr & \\qquad \\times \\sum _{i,\\,{\\mathcal {C}}}\\left(\\sigma _{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}+ \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\left(\\prod _{i,\\,p} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right) \\Bigg {]}_{\\sigma =0} \\crcr &=\\Bigg {[}e^{\\sum _{i,j}w_{ij}\\sum _{\\mathcal {C}}\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}\\right)} \\crcr & \\quad \\times \\sum _{i,j,\\,{\\mathcal {C}}}\\frac{w_{ij}}{2t}\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}} + \\frac{\\partial }{\\partial \\sigma _{ab}^{j\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}}\\right)\\left(\\prod _{i,\\,p} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right) \\Bigg {]}_{\\sigma =0} \\crcr & = \\Bigg {[}e^{\\sum _{i,j}w_{ij}\\sum _{\\mathcal {C}}\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}\\right)} \\crcr & \\quad \\times \\sum _{i,j,\\,{\\mathcal {C}}}\\frac{w_{ij}}{t}\\left(\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}}\\right)\\left(\\prod _{i,\\,p} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right) \\Bigg {]}_{\\sigma =0}\\; .$ The sum over $i,j$ and ${\\cal C}$ is a sum over all the ways of adding a loop edge to the graph $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }$ .",
"Evaluating the derivatives with respect to $\\sigma ^{{\\cal C}}$ and $\\bar{\\sigma }^{{\\cal C}}$ we see that the loop edge gives the same kind of colored contraction $\\delta ^{l,{\\mathcal {C}}}_{{\\cal D}}$ as a tree edge.",
"Furthermore, each loop edge brings a factor $\\frac{-2t\\lambda }{N^{D-1}}$ (hence the $t$ 's cancel), because the matrix $w_{ij}$ is symmetric and the same loop edge $ij$ is generated by two terms: $\\frac{\\partial }{\\partial \\sigma _{ab}^{i\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{j\\ {\\mathcal {C}}}} $ and $\\frac{\\partial }{\\partial \\sigma _{ab}^{j\\ {\\mathcal {C}}}}\\frac{\\partial }{\\partial \\bar{\\sigma }_{ab}^{i\\ {\\mathcal {C}}}} $ .",
"Repeating this process $L$ times gives a sum of $\\frac{(2v+k-3+2L)!}{(2v+k-3)!",
"}$ terms labeled by trees $\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l) \\rbrace }$ decorated with $L$ colored, labelled loop edges forming the set ${\\cal L}$ , $&\\frac{d^q}{dt^q}\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma )\\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right) \\crcr & \\qquad \\times \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right)\\nonumber \\\\&=\\left(\\frac{(-2\\lambda )}{N^{D-1}}\\right)^q \\sum _{ {\\cal L}, |{\\cal L}|=q }\\int _0^1 \\left(\\prod _{l_{ij}\\in T_v}du_{ij}\\right)\\int d\\mu _{T_v,u}(\\sigma ) \\left(\\prod _{l\\in {\\cal L}}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)} w_{i(l)j(l)}\\right)\\nonumber \\\\&\\qquad \\times \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right)\\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right).$ Taking into account that $R(0) = {\\bf 1}^{{\\cal D}}$ proves the theorem because the first $L$ terms of the Taylor expansion up to order $L$ can be evaluated explicitly: one obtains a free sum for each of the internal and external faces of the graph, and theorem REF follows."
],
[
"Absolute convergence",
"In order to establish the absolute convergence of the series in eq.",
"(REF ), we need to establish a bound on an individual term.",
"The explicit terms (consisting in trees with up to $L$ loops) and the rest term (trees with $L+1$ loops) are bounded by very different methods, explained in the next two subsections."
],
[
"Bounds on the explicit terms",
"The global scaling in $N$ of the term associated to the tree ${\\cal T}$ decorated with $q$ loop edges ${\\cal L}$ and external strands $\\tau _{{\\cal D}}$ in eq.",
"(REF ) is $\\frac{1}{N^{(D-1)(k+v-1) }} \\frac{1}{N^{q(D-1)}} \\; \\left( \\prod _{c =1}^D\\frac{1}{N^{2k}} N^{C(\\tau _c \\rho _c^{-1})} \\right) N^{F_{\\rm int} ({\\cal T},{\\cal L}) + F_{ {\\rm ext} } ({\\cal T},{\\cal L},\\tau _{{\\cal D}}) } \\;,$ where we used the asymptotic behavior (REF ) of the Weingarten functions.",
"We thus need to bound the number of faces (internal or external) of the tree ${\\cal T}$ decorated by the loop edges ${\\cal L}$ .",
"Recall that $\\mathfrak {C}(\\rho _{{\\cal D}}) $ denotes the number of connected components of the graph associated to the permutations $\\rho _{{\\cal D}}$ .",
"We denote (naturally) $F_{\\rm int} ({\\cal T})$ and $F_{ {\\rm ext} } ({\\cal T},\\tau _{{\\cal D}})$ the numbers of internal and external faces of the tree ${\\cal T}$ itself (with external strands $\\tau _{{\\cal D}}$ ) with no loop edges.",
"Lemma 2 We have the following bounds: $F_{\\rm int} ({\\cal T},{\\cal L}) + F_{ {\\rm ext} } ({\\cal T},{\\cal L},\\tau _{{\\cal D}})\\le F_{\\rm int} ({\\cal T}) + F_{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}}) + q \\frac{D}{2} \\; ,$ $\\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F^{{\\cal D}}_{\\rm int}({\\cal T}) + F^{{\\cal D}}_{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}}) \\le (D+1) k + D + (D-1) v - \\mathfrak {C}(\\rho _{{\\cal D}}) \\; .$ Before proving these statements, let us comment on a subtle point: this bound holds for trees having at most a mark per vertex.",
"In the next section we will use the Cauchy Schwarz inequalities which lead to vertices having several marks in order to bound the rest term.",
"The scaling with $N$ of such graphs obeys a weaker bound and, in order to establish theorem REF we will need to push the expansion up to a relatively high (but finite) number of loops.",
"Proof: Equation (REF ) is trivial, taking into account that for all loop edges $|{\\cal C}|\\le D/2$ and adding an loop edge on a graph can at most divide $|{\\cal C}|$ faces into two.",
"The proof of eq.",
"(REF ) is somewhat more involved.",
"It is done by an iterative procedure consisting in deleting at each step a leaf (univalent vertex) of the tree together with the tree edge it is hooked to and tracking the evolution of $ \\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F_{\\rm int}({\\cal T}) + F_{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}}) + \\mathfrak {C}(\\rho _{{\\cal D}}) $ .",
"Chose the univalent vertex $i$ , connected to the rest of the tree by an edge of colors ${\\cal C}$ .",
"There exists an unique vertex in the tree to which $i$ is hooked, called its ancestor.",
"The deletion is defined as follows: if $i$ has no mark, the deletion consists in erasing all the faces with color in ${\\cal D}\\setminus {\\cal C}$ containing $i$ and reconnecting the faces with color in ${\\cal C}$ passing through $i$ directly on its ancestor.",
"if $i$ has a mark, then it is one of the marked vertices $i_1,\\dots i_k$ .",
"Say $i$ is $i_d$ .",
"The deletion proceeds in two steps.",
"If $\\tau _c (d) \\ne d $ then we replace it by the permutation $\\tilde{\\tau }_c$ defined as: ${\\left\\lbrace \\begin{array}{ll}\\tilde{\\tau }_c (d) = d \\; , \\\\\\tilde{\\tau }_c \\left( \\tau _c^{-1} (d)\\right) = \\tau _c(d) \\; , \\\\\\tilde{\\tau }_c (d^{\\prime }) = \\tau _c(d^{\\prime }) \\; , \\quad \\forall d^{\\prime } \\ne d, \\tau _c^{-1}(d) .\\end{array}\\right.",
"}$ Graphically this comes to cutting the ribbon edges representing the permutations $\\tau _c$ incident at $i_d$ and reconnecting them the other way around.",
"Similarly, if $\\rho _c (d) \\ne d $ then we replace it by the permutation $\\tilde{\\rho }_c$ defined as: ${\\left\\lbrace \\begin{array}{ll}\\tilde{\\rho }_c (d) = d \\; , \\\\\\tilde{\\rho }_c \\left( \\rho _c^{-1} (d)\\right) = \\rho _c(d) \\; , \\\\\\tilde{\\rho }_c (d^{\\prime }) = \\rho _c(d^{\\prime }) \\; , \\quad \\forall d^{\\prime } \\ne d, \\rho _c^{-1}(d) .\\end{array}\\right.",
"}$ Graphically this comes to cutting the ribbon edges representing the permutations $\\rho _c$ incident at $i_d$ and reconnecting them the other way around.",
"For a vertex $i_d$ such that $\\tau _c(d) = d$ and $\\rho _c(d) = d$ for all colors, the deletion consists in reconnecting the solid strands of color in ${\\cal C}$ on its ancestor and deleting all the other strands (solid, dashed and dotted).",
"For a vertex $i_d$ (represented in Figure REF ) such that for all colors $\\tau _c(d)\\ne d$ and $\\rho _c(d)\\ne d$ , the deletion leads to the drawing in Figure REF .",
"Figure: Deletion of a vertex with external sources.Let us denote $ \\tau _c^r$ , $\\rho _c^r$ and ${\\cal T}^r$ the permutations and the tree obtained after having erased one vertex.",
"If the vertex had a mark and the deletion was done in two steps, at the intermediary step we have the same tree ${\\cal T}$ , but the permutations $\\tilde{\\tau }_{{\\cal D}}$ and $\\tilde{\\rho }_{{\\cal D}}$ .",
"If $i$ has no mark, the permutations $\\tau $ and $\\rho $ are unaffected by the deletion.",
"As we erase at most $|{\\cal D}\\setminus {\\cal C}|\\le D-1$ faces, we have $& \\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F_{\\rm int}({\\cal T}) + F_{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}}) + \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\crcr & \\qquad \\le \\sum _{c = 1}^D C(\\tau ^r_c [\\rho _c^r]^{-1}) + F_{\\rm int}({\\cal T}^r) + F _{ {\\rm ext} }({\\cal T}^r,\\tau _{{\\cal D}}^r) + \\mathfrak {C}(\\rho ^r_{{\\cal D}}) +D-1 \\; .$ and the graph obtained after deletion has one fewer vertex.",
"We now analyze the evolution of this quantity in the case when $i$ has a mark, $i=i_d$ and we go through the intermediary graph: The number of connected components of the graph of the permutations $\\rho _{{\\cal D}}$.",
"We always create a connected component with a black and a white vertex $d$ and edges $\\tilde{\\rho }_c(d) =d$ .",
"The other connected components of the graph associated to $\\rho _{{\\cal D}}$ change as follows: if $\\rho _c(d) \\ne d$ for all $c \\in {\\cal D}$ then there are two cases: the black and white vertices $d$ belong to the same connected component.",
"Then by going to $\\tilde{\\rho }_{{\\cal D}}$ this components either survives or it is split into several connected components.",
"Thus $ \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}}) -1 $ .",
"the black and white vertices $d$ belong to two distinct connected components.",
"Then the two are either merged into an unique component, or split into several.",
"We always have $ \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}})$ .",
"if there exists $c\\in {\\cal D}$ such that $ \\rho _c(d) = d$ .",
"Then the black and white vertex $d$ belong to the same connected component and this component either survives or it is split into several by going to $\\tilde{\\rho }_{{\\cal D}}$ .",
"Hence in this case we always have $ \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}}) -1 $ .",
"To summarize: ${\\left\\lbrace \\begin{array}{ll}\\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}}) -1 \\text{ if } \\exists c\\;,\\;\\; \\rho _c(d) = d \\; , \\\\\\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}}) \\text{ otherwise} \\; .\\end{array}\\right.",
"}$ The number of faces of color $c\\in {\\cal C}$ .",
"This number is: constant if $\\tau _c(d)=d$ .",
"can at most decrease by 1 if $\\tau _c(d) \\ne d$ .",
"The number of faces of color $c\\in {\\cal D}\\setminus {\\cal C}$ is: constant if $\\tau _c(d)=d$ .",
"increases by 1 if $\\tau _c(d) \\ne d$ .",
"Denoting $s^{{\\cal C}} \\le |{\\cal C}|$ the number of colors in ${\\cal C}$ such that $\\tau _c(d) \\ne d$ , and $s^{{\\cal D}\\setminus {\\cal C}}$ the number of colors in ${\\cal D}\\setminus {\\cal C}$ such that $\\tau _c(d) \\ne d$ , we get: $F _{\\rm int}({\\cal T}) + F _{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}} ) \\le F_{\\rm int} ({\\cal T}) + F_{ {\\rm ext} } ({\\cal T},\\tilde{\\tau }_{{\\cal D}}) + s^{{\\cal C}} - s^{{\\cal D}\\setminus {\\cal C}} \\; .$ The number of faces $\\tau _c\\rho _c^{-1}$ .",
"This number changes as follows: if $\\tau _c(d) = d$ , if $\\rho _c(d) = d$ , $C(\\tau _c \\rho _c^{-1})$ is constant.",
"if $\\rho _c(d) \\ne d$ , $C(\\tau _c \\rho _c^{-1})$ increases by 1. if $\\tau _c(d) \\ne d$ , if $\\rho _c(d) = d$ , $C(\\tau _c \\rho _c^{-1})$ increases by 1 if $\\rho _c(d) \\ne d $ , $C(\\tau _c \\rho _c^{-1})$ can not decrease.",
"Hence: ${\\left\\lbrace \\begin{array}{ll}\\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) \\le \\sum _{c = 1}^D C(\\tilde{\\tau }_c \\tilde{\\rho }_c^{-1}) -1\\text{ if } \\exists c {\\left\\lbrace \\begin{array}{ll}\\tau _c(d) = d \\\\ \\text{ and } \\\\ \\rho _c(d) \\ne d\\end{array}\\right.}",
"\\; , \\\\\\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) \\le \\sum _{c = 1}^D C(\\tilde{\\tau }_c \\tilde{\\rho }_c^{-1}) \\text{ otherwise} \\; .\\end{array}\\right.",
"}$ Remark that $s^{{\\cal C}} -s^{{\\cal D}\\setminus {\\cal C}} \\le |{\\cal C}| -1$ unless both $s^{{\\cal C}} = |{\\cal C}|$ and $s^{{\\cal D}\\setminus {\\cal C}}=0$ .",
"However in this case $\\tau _c(d) = d$ for all $c\\in {\\cal D}\\setminus {\\cal C}$ , and either $ \\rho _c(d) \\ne d $ or $\\rho _c(d) = d$ hence in all cases we obtain the bound: $& \\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F _{\\rm int}({\\cal T}) + F _{ {\\rm ext} } ({\\cal T},\\tau _{{\\cal D}}) + \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\crcr & \\qquad \\le \\sum _{c = 1}^D C(\\tilde{\\tau }_c \\tilde{\\rho }_c^{-1}) + F _{\\rm int}({\\cal T}) + F_{ {\\rm ext} } ({\\cal T},\\tilde{\\tau }_{{\\cal D}})+ \\mathfrak {C}(\\tilde{\\rho }_{{\\cal D}}) + |{\\cal C}| -1 \\; .$ Finally, erasing a vertex $i_d$ with $\\tau _c(d) = d$ and $\\rho _c(d)=d$ for all $c$ we obtain $& \\mathfrak {C}( \\tilde{\\rho }_{{\\cal D}}) = \\mathfrak {C}( \\rho ^r_{{\\cal D}}) +1 \\qquad C(\\tilde{\\tau }_c \\tilde{\\rho }_c^{-1}) = 1 + C(\\tau ^r_c [\\rho ^r_c]^{-1}) \\crcr & \\tilde{F}_{\\rm int} ({\\cal T}) + F_{ {\\rm ext} } ({\\cal T},\\tilde{\\tau }_{{\\cal D}}) = F _{\\rm int}({\\cal T}^r) + F_{ {\\rm ext} } ({\\cal T}^r,\\tau ^r_{{\\cal D}}) + |{\\cal D}\\setminus {\\cal C}| \\; ,$ hence $& \\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F_{\\rm int}({\\cal T}) + F _{ {\\rm ext} } ({\\cal T},\\tau _{{\\cal D}}) + \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\sum _{c=1}^D C(\\tau ^r_c [\\rho ^r_c]^{-1})\\crcr & \\qquad + F _{\\rm int}({\\cal T}^r) + F_{ {\\rm ext} } ({\\cal T}^r,\\tau ^r_{{\\cal D}}) + \\mathfrak {C}( \\rho ^r_{{\\cal D}}) +D + |{\\cal D}\\setminus {\\cal C}| + |{\\cal C}| \\; ,$ and the number of vertices and of marks both go down by 1.",
"Iterating up to the last vertex, we either end up with a vertex with no mark or with a vertex with a mark, and all $\\rho _c (d) = d ,\\tau _c (d) =d$ .",
"Counting the faces and the number of connected components of $\\rho _{{\\cal D}}$ of the two possible end graphs we obtain: $& \\sum _{c = 1}^D C(\\tau _c \\rho _c^{-1}) + F _{\\rm int}({\\cal T}) + F _{ {\\rm ext} }({\\cal T},\\tau _{{\\cal D}}) + \\mathfrak {C}(\\rho _{{\\cal D}}) \\le \\crcr & \\qquad \\le {\\left\\lbrace \\begin{array}{ll}k(D+1) + (D-1)(v-1) + D \\; , \\\\(k-1)(D+1) + (D-1) (v-1) + 2D +1 \\; ,\\end{array}\\right.",
"}$ which proves eq.",
"(REF ).",
"The main difficulty in the above proof is to obtain the $-1$ on the right hand side of (REF ).",
"This relies crucially on the observation that in a tree with a mark per vertex, when $s^{{\\cal D}\\setminus {\\cal C}}=0$ , then the strands with colors $c \\in {\\cal D}\\setminus {\\cal C}$ are such that $\\tau _c(d) = d$ .",
"This is fine as long as we have a unique mark per vertex, however if we have several marks, the proof fails.",
"$\\Box $"
],
[
"Bounds on the rest term",
"In order to establish a bound on the rest terms in equation (REF ) we will use the technique introduced in [23] and [30] of iterated Cauchy-Schwarz inequalities: $| \\mathinner {\\langle {A}|} R\\otimes R^{\\prime }\\otimes \\mathbf {1}^{\\otimes p}\\mathinner {|{B}\\rangle } | \\; \\le \\; \\Vert R\\Vert \\Vert R^{\\prime }\\Vert \\sqrt{\\mathinner {\\langle {A|A}\\rangle }}\\sqrt{\\mathinner {\\langle {B|B}\\rangle }} \\; .$ Lemma 3 We have the bound: $& \\Bigg {|}\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{m,n} \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right)\\left(\\prod _{l\\in {\\cal L}}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)} \\right)\\crcr &\\qquad \\times \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\Bigg {|} \\le \\crcr & \\le N^{D + Dk + (D-1)(v-1)+ |{\\cal L}| \\frac{D}{2}}\\; .$ Proof: The rest term is a contraction of resolvents placed around the vertices in a pattern encoded in a tree with loop edges and external strands.",
"On any tree, one can choose a resolvent and then count the resolvents following the clockwise contour walk of the tree, indexing them from $R_1$ to $R_{2n}$ (or $R_{2n+1}$ ).",
"Choosing $R_1$ and $R_{n+1}$ as the $R$ and $R^{\\prime }$ of formula (REF ), the vector $A$ is made of all the resolvents from $R_2$ to $R_n$ and the contractions between them.",
"The vector $B$ is made of all the resolvents from $R_{n+2}$ to $R_{2n}$ (or $R_{2n+1}$ ) and the contractions between them.",
"The contractions of indices of the resolvents $R_2$ to $R_n$ with indices of the resolvents $R_{n+2}$ to $R_{2n}$ (or $R_{2n+1}$ ), which can exist due to the $\\tau $ external strands or due to the loop edges, is encoded in the $\\mathbf {1}^{\\otimes p}$ operator.",
"If an index of $R_1$ is directly contracted with an index of $R_{n+1}$ , the latter contributes a Kronecker delta to the vector $A$ or $B$ .",
"We forget the dashed and dotted strands, as they do not contribute to the quantity on the left hand side of equation (REF ).",
"We represented such a splitting in Figure REF .",
"Figure: Splitting the graph in two part in order to apply the Cauchy-Schwartz inequality.We apply the formula (REF ) and, because the norm of the resolvent is bounded by 1, $\\Vert R(\\sigma )\\Vert \\le 1$ , $| \\mathinner {\\langle {A}|} R_1\\otimes R_{n+1}\\otimes \\mathbf {1}^{\\otimes p}\\mathinner {|{B}\\rangle } | \\; \\le \\; \\sqrt{\\mathinner {\\langle {A|A}\\rangle }\\mathinner {\\langle {B|B}\\rangle }}.$ The resolvents $R_1$ and $R_{n+1}$ were on the vertices of the tree and the previous splitting in $A$ and $B$ corresponds to cutting through the interior of the tree (the cut is represented in bold in Figure REF ).",
"Any contraction strand between the part $A$ and the part $B$ (due to the external strands $\\tau $ or to the loop edges) is also cut.",
"The scalar products $\\mathinner {\\langle {A|A}\\rangle }$ and $\\mathinner {\\langle {B|B}\\rangle }$ are half graphs merged with their mirror symmetric with respect to the splitting line.",
"They also have the structure of trees with loop edges and external $\\tau $ strands.",
"However, in contrast with the original graph, they can have several marks on the same vertex.",
"This happens whenever the resolvent $R_1$ (or $R_n$ ) belongs to a marked vertex.",
"This is ultimately the reason for which the bound on the rest term we establish below is weaker than the bound on the explicit terms of lemma REF .",
"As the resolvents $R_1$ and $R_{n+1}$ have been taken out, the scars (i.e.",
"the corners where $R_1$ and $R_{n+1}$ were connecting on $A$ and $B$ ) are now just direct identifications of indices.",
"That is, in the graphs merged with their mirror symmetric, two resolvents have been set to the identity ${\\bf 1}^{{\\cal D}}$ operator.",
"The corresponding corners will be represented as just $D$ parallel strands, with no vertical line, like in Figure REF .",
"Figure: The scalar product graphs 〈A|A〉\\mathinner {\\langle {A|A}\\rangle } (top) and 〈B|B〉\\mathinner {\\langle {B|B}\\rangle } (bottom).Let us denote $v^A$ , $k^A$ , ${\\cal T}^{A}$ , ${\\cal L}^{A}$ , $\\tau _{{\\cal D}}^A$ the number of vertices, marked vertices, the tree, the loop edges and the permutations of external strands corresponding to the graph $\\mathinner {\\langle {A|A}\\rangle }$ and similarly for $B$ .",
"We have the following (in)equalities: the number of vertices doubles $ 2v = v^A + v^B \\; .$ the number of marked vertices doubles $ 2 k = k^A + k^B \\; .$ the number of loop edges doubles $ |{\\cal L}| = |{\\cal L}^A| + |{\\cal L}^B| \\; .",
"$ the number of faces (closed solid strands) at least doubles $2F_{\\rm int}({\\cal T}, {\\cal L}) + 2F_{\\rm ext} ({\\cal T},{\\cal L},\\tau _{{\\cal D}}) & \\le F_{\\rm int}({\\cal T}^A, {\\cal L}^A) + F_{\\rm ext} ({\\cal T}^A,{\\cal L}^A,\\tau ^A_{{\\cal D}}) \\crcr &+ F_{\\rm int}({\\cal T}^B, {\\cal L}^B) + F_{\\rm ext} ({\\cal T}^B,{\\cal L},\\tau ^B_{{\\cal D}}) \\; .$ This is because a face is either untouched by the splitting line, or it is cut in two pieces (and in both cases it leads to two faces in the mirrored graphs), or it is cut in at least four pieces in which case it leads to at least four faces in the mirrored graphs.",
"Each mirrored graph has an even number of resolvents and contributes to the bound by the square root of its amplitude.",
"We can now iterate the process, each time eliminating two resolvents, until there are no resolvents left on any graph.",
"If we start with $2n$ resolvents we obtain $2^n$ final graphs.",
"If we start with $2n+1$ resolvents, we add an iteration in which all but one of the resolvents are already set to the identity operator.",
"The $2^n$ (resp.",
"$2^{n+1}$ ) final graphs we obtain are made solely of faces which represent traces of identity, hence each such face brings a factor $N$ .",
"We denote for, $q=1,\\dots 2^n$ (or $2^{n+1}$ ) by $k^q$ , $v^q$ , ${\\cal T}^q$ , ${\\cal L}^q$ the numbers of marked vertices, vertices, etc.",
"of the final graphs.",
"As before, adding a loop edge with color ${\\cal C}$ to a final graph can at most divide $|{\\cal C}|$ of its faces, hence $F_{\\rm int}({\\cal T}^q, {\\cal L}^q) + F_{\\rm ext}({\\cal T}^q, {\\cal L}^q,\\tau _{{\\cal D}}^q) \\le F_{\\rm int}({\\cal T}^q) + F_{\\rm ext}({\\cal T}^q,\\tau _{{\\cal D}}^q) + |{\\cal L}^q|\\frac{D}{2} \\;.$ A tree ${\\cal T}^q$ with multiple marks on the same vertex has exactly $2Dk^q$ ends of strands connected by the permutation $\\tau ^q$ , hence it has at most $Dk^q$ external faces.",
"The number of internal faces of $T^q$ is at most $1 + (D-1) v^q $ .",
"If we start with $2n$ resolvents we get the bound $& \\Bigg {|}\\int d\\mu _{T_v,u}(\\sigma ) \\sum _{m,n} \\left(\\prod _{i=1}^v\\prod _{p=1}^{\\mathrm {res}(i)} R( \\sqrt{t}\\sigma ^i)_{n_{i,p} m_{i,p}} \\right)\\left(\\prod _{l\\in {\\cal L}}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)} \\right)\\crcr &\\qquad \\times \\left(\\prod _{l\\in T_v}\\delta _{{\\cal D}}^{l,{\\mathcal {C}}(l)}\\right)\\left( \\prod _{d=1}^k \\prod _{c=2}^D \\delta _{n^c_{i_d,q+1}\\, m_{i_{\\tau _c(d)},q}^c}\\right) \\Bigg {|} \\le \\crcr & \\le N^{\\frac{1}{2^n} \\sum _{q=1}^{2^n} \\left( F_{\\rm int}({\\cal T}^q, {\\cal L}^q) + F_{\\rm ext}({\\cal T}^q, {\\cal L}^q,\\tau _{{\\cal D}}^q) \\right) }\\le N^{\\frac{1}{2^n} \\sum _{q=1}^{2^n} \\left( Dk^q + 1 + (D-1)v^q + |{\\cal L}^q|\\frac{D}{2} \\right) } \\crcr & = N^{D + Dk + (D-1)(v-1)+ |{\\cal L}| \\frac{D}{2}}$ and the same holds if we start with $2n+1$ resolvents.",
"$\\Box $ We are now ready to prove theorem REF .",
"Taking absolute values in eq.",
"(REF ), using eq.",
"(REF ) and using the lemmas REF and REF we find: $&|\\mathfrak {K}(\\rho _{{\\cal D}})|\\le \\sum _{v\\ge k}\\frac{1}{v!",
"}\\ \\frac{ |2\\lambda |^{v-1}}{N^{(D-1)(k+v-1) }} \\sum _{\\tau _{{\\cal D}}}\\left( \\prod _{c = 1}^D \\frac{2^{2k}}{N^{2k}} N^{C(\\tau _c \\rho _c^{-1})} \\right) \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\crcr &\\times \\Bigg {[} \\sum _{q=0}^L \\left(\\frac{ | 2\\lambda | }{N^{D-1}}\\right)^q \\frac{1}{q!}",
"\\sum _{{\\cal L}, |{\\cal L}| = q}N^{F_{\\rm int} ({\\cal T},{\\cal L})+F_{ \\rm ext } ({\\cal T},{\\cal L},\\tau _{ {\\cal D}} ) } \\crcr & \\qquad + \\left(\\frac{|2\\lambda |}{N^{D-1}}\\right)^{L+1} \\frac{1}{L!}",
"\\sum _{ {\\cal L}, |{\\cal L}| = L+1 }N^{D + Dk + (D-1)(v-1)+ |{\\cal L}| \\frac{D}{2}} \\Bigg {]} \\le \\crcr & \\le 2^{2Dk} k!^D \\sum _{v\\ge k}\\frac{1}{v!}",
"\\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }} \\crcr & \\qquad \\Bigg {[} \\sum _{q=0}^L \\frac{ | 2\\lambda | ^{v-1+q} }{q!}",
"\\sum _{{\\cal L}, |{\\cal L}| = q} N^{D - 2(D-1)k - \\mathfrak {C}(\\rho _{{\\cal D}})- q\\left(\\frac{D}{2}-1 \\right) } \\crcr & \\qquad \\; + \\frac{ | 2\\lambda | ^{v+L} }{L!}",
"\\sum _{{\\cal L}, |{\\cal L}| = L+1} N^{D - (D-1)k - (L+1) \\left(\\frac{D}{2}-1 \\right) } \\Bigr ] \\; .$ As $ \\mathfrak {C}(\\rho _{{\\cal D}}) \\le k $ , choosing $L \\ge \\frac{Dk}{D/2-1}-1$ ensures that $D - (D-1)k - (L+1) \\left(\\frac{D}{2}-1 \\right) \\le D - 2(D-1)k - \\mathfrak {C}(\\rho _{{\\cal D}}) \\; ,$ and $|\\mathfrak {K}(\\rho _{{\\cal D}})| & \\le N^{D - 2(D-1)k - \\mathfrak {C}(\\rho _{{\\cal D}}) } \\crcr & \\times 2^{2Dk} k!^D\\sum _{v\\ge k}\\frac{1}{v!}",
"\\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}\\sum _{q=0}^{L+1} \\frac{ | 2\\lambda | ^{v-1+q} }{q!}",
"\\frac{(2v+k-3+2q)!}{(2v+k-3)!}",
"\\; .$ Taking into account that $& \\sum _{\\mathcal {T}_{v,\\lbrace i_d\\rbrace ,\\lbrace {\\mathcal {C}}(l)\\rbrace }}1 \\crcr &= (\\mathcal {N}_{\\mathcal {Q}})^{v-1}\\sum _{d_1 \\dots d_v\\ge 1 }^{\\sum d_i=2(v-1)}\\left(\\frac{(v-2)!",
"}{\\prod _i (d_i-1)!}",
"\\prod _i (d_i-1)!\\ \\times \\sum _{i_1,\\dots i_k}^{i_k\\ne i_{k^{\\prime }}}d_{i_1}...d_{i_k}\\right) \\crcr &=(\\mathcal {N}_{\\mathcal {Q}})^{v-1}\\frac{v!(2v+k-3)!}{k!(v-k)!(v+k-1)!",
"}\\; ,$ with $(\\mathcal {N}_{\\mathcal {Q}})^{v-1}$ the number of edge coloring, $d_i$ the degree of the vertex $i$ , $\\frac{(v-2)!",
"}{\\prod _i (d_i-1)!",
"}$ the number of trees with fixed degrees and $\\prod _i (d_i-1)!$ the number of associated plane trees, and $\\prod _{d=1}^k d_{i_d}$ the number of ways to put marks on the vertices $i_d$ , we obtain $|\\mathfrak {K}(\\rho _{{\\cal D}})| & \\le N^{D - 2(D-1)k - \\mathfrak {C}(\\rho _{{\\cal D}}) } \\crcr & \\times 2^{2Dk} k!^{D-1} \\sum _{q=0}^{L+1} \\sum _{v\\ge k} |2\\mathcal {N}_{\\mathcal {Q}}\\lambda |^{v+q-1}\\frac{(2v+k-3+2q)!}{q!",
"(v-k)!(v+k-1)!}",
"\\; ,$ and the sum over $v$ converges for $|8\\mathcal {N}_{\\mathcal {Q}}\\lambda |<1$ ."
],
[
"Second cumulant",
"For $k=1$ , the rescaled second cumulant can be written $N^{D-1} \\mathfrak {K}(\\lbrace 1\\rbrace _{\\mathcal {D}})= \\sum _{v\\ge 1} a_v(N,\\lambda ) ,$ where $a_v$ contains all terms corresponding to graphs with $v$ vertices and is bounded by $|a_v(N,\\lambda )| \\le 2^{2Dk} k!^{D-1} \\sum _{q=0}^{L+1} |2\\mathcal {N}_{\\mathcal {Q}}\\lambda |^{v+q-1}\\frac{(2v-2+2q)!}{q!",
"v!",
"(v-1)!",
"}.$ This bound does not depend on $N$ and assures the uniform convergence of the series.",
"$A_v$ is composed of the amplitude of trees, trees decorated with loops and rest terms.",
"Choosing $L>\\frac{Dk}{D/2-1}-1$ ensures the rest terms to be dominated by $1/N$ .",
"A tree decorated with $q$ loops being dominated by $N^{-q(D/2-1)}$ , the large $N$ limit of $a_v$ is given by the amplitude of trees.",
"Moreover, we have $F_{\\rm int}(\\mathcal {T})+F_{\\rm ext}(\\mathcal {T},\\lbrace 1\\rbrace _{\\mathcal {D}})=(D-1)(v-1)+D$ if and only if $\\mathcal {T}$ is composed only of edges with $|\\mathcal {C}(l)|=1$ .",
"Thus, the large $N$ limit of the rescaled second cumulant is finite, and is given by the sum over all trees with edges carrying only one color.",
"If we denote $\\mathcal {N}_1$ the number of interaction terms with $|\\mathcal {C}|=1$ , we have $&\\lim _{N\\rightarrow \\infty } N^{D-1} \\mathfrak {K}(\\lbrace 1\\rbrace _{\\mathcal {D}}) \\ =\\ \\sum _{v\\ge 1} \\frac{1}{v!}",
"\\sum _{\\mathcal {T}_{v,i_1,\\lbrace \\mathcal {C}(l)\\rbrace }^{|\\mathcal {C}(l)|=1}}1 \\crcr &= \\sum _{v\\ge 1} (-2\\mathcal {N}_1\\lambda )^{v-1} \\frac{(2v-2)!}{(v-1)!v!}",
"=\\frac{-1+\\sqrt{1+8\\mathcal {N}_1\\lambda }}{4\\mathcal {N}_1\\lambda } .$ This, together with the bound in Eq.",
"REF , establishes theorem REF , shows that the measure is properly uniformly bounded and thus obeys the universality theorem."
],
[
"Uniform Borel summability",
"A function $f(\\lambda , N)$ is said to be Borel summable in $\\lambda $ uniformly in $N$ if $f$ is analytic in $\\lambda $ in a disk $\\mathrm {Re}\\frac{1}{\\lambda }>\\frac{1}{R}$ with $R>0$ independent on $N$ and admits a Taylor expansion $f(\\lambda ,N)=\\sum _{k<r} A_k(N)\\lambda ^k \\ +\\ R_{r}(\\lambda ,N) \\; , \\;\\; |R_{r}(\\lambda ,N)|\\le r!",
"\\; a^r |\\lambda |^r K(N) .$ for some $a$ independent of $N$ ."
],
[
"Analyticity",
"To establish the convergence of the series in eq.",
"(REF ) in the domain in the complex plane $\\lambda =r e^{i\\phi }$ , $\\phi \\in (-\\pi ,\\pi )$ defined by $|\\lambda |< \\frac{1}{8(\\mathcal {N}_{\\mathcal {Q}})} \\left( \\cos \\frac{\\phi }{2} \\right)^2 $ it is enough to follow step by step the proof of theorem REF and note that the norm of the resolvent is bounded by $\\Vert R(\\sigma )\\Vert \\le \\frac{1}{\\mathrm {cos}\\frac{\\phi }{2}} \\; .$ The iterated Cauchy-Schwarz inequalities go through, and it is easy to see that the norm of each resolvent contributes to the power 1 to the amplitude of the graph.",
"The total number of resolvents of a graph with $v$ vertices and $k$ marks is $2(v-1)+k$ .",
"Therefore each term of the overall bound in eq.",
"REF must be multiplied by $\\left(\\frac{1}{\\mathrm {cos}\\frac{\\phi }{2}}\\right)^{2(v-1)+k}$ , which proves the convergence and eq.",
"(REF ).",
"The convergence domain $|\\lambda |<\\frac{1}{8\\mathcal {N}_{\\mathcal {Q}}}\\left(\\mathrm {cos}\\frac{\\phi }{2}\\right)^2 \\; ,$ contains a disk $\\mathrm {Re}\\frac{1}{\\lambda }>\\frac{1}{R}$ .",
"In this domain the cumulants eq.",
"(REF ) are analytic as the resolvents themselves $R(\\sigma ^i)_{n_{i,p} m_{i,p}}$ are, which can be proved by verifying the Cauchy-Riemann equation, $r\\frac{\\partial }{\\partial r} R(\\sigma ^i)_{n_{i,p} m_{i,p}}=-\\frac{1}{2} \\left[ A(\\sigma ^i) R(\\sigma ^i)\\right]_{n_{i,p} m_{i,p}} = i\\frac{\\partial }{\\partial \\phi } R(\\sigma ^i)_{n_{i,p} m^{{\\cal D}}_{i,p}} \\; .$"
],
[
"Taylor expansion",
"The Taylor expansion in $\\lambda $ of the cumulants up to order $r$ is obtained by using the mixed expansion in theorem REF , but choosing the order $L$ up to which we develop the loop edges to depend on the number of vertices $v$ of the tree $L = \\max (0, r - v)$ .",
"For $v\\ge r+1$ we do not develop any loop edges.",
"Using the same bounds leading up to eq.",
"(REF ), and noting that the scaling with $N$ is always bounded by $N^D$ , the rest term is bounded by $|R_{r}(\\lambda ,N)| \\le & N^{D} 2^{2Dk} k!^{D-1} \\crcr & \\sum _{v\\ge k} \\Bigg {[}|2\\mathcal {N}_{\\mathcal {Q}}\\lambda |^{v+q-1} \\frac{(2v+k-3+2q)!}{q!",
"(v-k)!(v+k-1)!}",
"\\Bigg {]}_{q= \\max (0,r+1-v)}\\;,$ hence up to irrelevant overall factors the rest term is bounded by $& \\sum _{v\\ge r+1} |2\\mathcal {N}_{\\mathcal {Q}}\\lambda |^{v -1} \\frac{(2v+k-3)!}{(v-k)!(v+k-1)!}",
"\\crcr & \\quad \\quad +\\sum _{v=k}^{r+1} |2\\mathcal {N}_{\\mathcal {Q}}\\lambda |^{r}\\frac{\\left[2v+k-3+2(r+1-v)\\right]!}{(r+1-v)!",
"(v-k)!(v+k-1)!}",
"\\; .$ While the first term above is bounded by $|\\lambda |^r$ times some constant for $\\lambda $ small enough, the second one is bounded only as : $|\\lambda |^r \\frac{(2r+k-1)!}{(r-k)!}",
"\\le (2k-1)!",
"3^{2r+k-1} \\;\\;\\; r!",
"|\\lambda |^r \\; .$ Corollary REF is straightforward from the previous bounds."
]
] | 1403.0170 |
[
[
"Dynamics and the Godbillon-Vey Class of C^1 Foliations"
],
[
"Abstract Let F be a codimension-one, C^2-foliation on a manifold M without boundary.",
"In this work we show that if the Godbillon--Vey class GV(F) \\in H^3(M) is non-zero, then F has a hyperbolic resilient leaf.",
"Our approach is based on methods of C^1-dynamical systems, and does not use the classification theory of C^2-foliations.",
"We first prove that for a codimension--one C^1-foliation with non-trivial Godbillon measure, the set of infinitesimally expanding points E(F) has positive Lebesgue measure.",
"We then prove that if E(F) has positive measure for a C^1-foliation F, then F must have a hyperbolic resilient leaf, and hence its geometric entropy must be positive.",
"The proof of this uses a pseudogroup version of the Pliss Lemma.",
"The theorem then follows, as a C^2-foliation with non-zero Godbillon-Vey class has non-trivial Godbillon measure.",
"These results apply for both the case when M is compact, and when M is an open manifold."
],
[
"Introduction",
"Godbillon and Vey introduced in [26] the invariant $GV({\\mathcal {F}}) \\in H^3(M; {\\mathbb {R}})$ named after them, which is defined for a codimension-one $C^2$ -foliation ${\\mathcal {F}}$ of a manifold $M$ without boundary.",
"While the definition of the Godbillon-Vey class is elementary, understanding its relations to the geometric and dynamical properties of the foliation ${\\mathcal {F}}$ remains an open problem.",
"In the paper [72] by Thurston, where he showed that the Godbillon-Vey class can assume a continuous range of values for foliations of closed 3-manifolds, he also included Figure 1, which illustrated the concept of “hellical wobble”, which he suggested gives a relation between the value of this class and the Riemannian geometry of the foliation.",
"This geometric relation was made precise in a work by Reinhart and Wood [68].",
"More recently, Langevin and Walczak in [52], [76], [77] gave further insights into the geometric meaning of the Godbillon-Vey invariant for smooth foliations of closed 3-manifolds, in terms of the conformal geometry of the leaves of the foliation.",
"The Godbillon-Vey class appears in a surprising variety of contexts, such as the Connes-Moscovici work on the cyclic cohomology of Hopf algebras [13], [15], [14] which interprets the class in non-commutative geometry setting.",
"The works of Leichtnam and Piazza [54] and Moriyoshi and Natsume [58] gave interpretations of the value of the Godbillon-Vey class in terms of the spectral flow of leafwise Dirac operators for smooth foliations.",
"The problem considered in this work was first posed in papers of Moussu and Pelletier [59] and Sullivan [71], where they conjectured that a foliation ${\\mathcal {F}}$ with $GV({\\mathcal {F}}) \\ne 0$ must have leaves of exponential growth.",
"The support for this conjecture at that time was principally a collection of examples, and some developing intuition for the dynamical properties of foliations.",
"The geometry of the helical wobble phenomenon is related to geometric properties of contact flows, such as for the geodesic flow of a compact surface with negative curvature.",
"The weak stable foliations for such flows have all leaves of exponential growth, and often have non-zero Godbillon-Vey classes [72], [62], [68], [41], [27].",
"Moreover, the work of Thurston in [72] implies that for any positive real number $\\alpha $ there exist a $C^2$ -foliation of codimension-one on a compact oriented 3-manifold, whose Godbillon-Vey class is $\\alpha $ times the top dimension integral cohomology class.",
"These various results suggest that a geometric interpretation of $GV({\\mathcal {F}})$ might be related to dynamical invariants such as “entropy”, whose values are not limited to a discrete subset of ${\\mathbb {R}}$ .",
"Given a choice of a complete, relatively compact, 1-dimensional transversal ${\\mathfrak {X}}\\subset M$ to ${\\mathcal {F}}$ , the transverse parallel transport along paths in the leaves defines local homeomorphisms of ${\\mathfrak {X}}$ , which yields a 1-dimensional pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ as recalled in Section REF .",
"The study of the properties of foliation pseudogroups has been a central theme of foliation theory since the works of Reeb and Haefliger in the 1950's [66], [67], [28], [29].",
"The geometric entropy $ of a $ C1$-foliation $ F$ was introduced byGhys, Langevin and Walczak \\cite {GLW1988}, and can be formulated in terms of the pseudogroup $ GF$ associated to the foliation.",
"The geometric entropy is a measure of the dynamical complexity of the action of $ GF$ on $ X$, and is one of the most important dynamical invariants of $ C1$-foliations.",
"The Godbillon-Vey class $ GV(F)$ vanishes for all the known examples of foliations for which $ 0$, and the problem was posedto relate the non-vanishing of the geometric entropy $ of a codimension-one $C^2$ -foliation ${\\mathcal {F}}$ with the non-vanishing of its Godbillon-Vey class.",
"Duminy showed in the unpublished papers [18], [19] that for a $C^2$ -foliation of codimension one, $GV({\\mathcal {F}}) \\ne 0$ implies there are leaves of exponential growth.",
"(See the account of Duminy's results in Cantwell and Conlon [12], and [10].)",
"Duminy's proof began by assuming that a $C^2$ -foliation ${\\mathcal {F}}$ has no resilient leaves, or equivalently resilient orbits for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ as in Definition REF .",
"Then by the Poincaré-Bendixson theory for codimension–one, $C^2$ -foliations [12], [33], Duminy showed that the Godbillon-Vey class of ${\\mathcal {F}}$ must vanish.",
"Thus, if $GV({\\mathcal {F}}) \\ne 0$ then ${\\mathcal {F}}$ must have at least one resilient leaf.",
"If a codimension–one foliation has a resilient leaf, then by an easy argument it follows that ${\\mathcal {F}}$ has an uncountable set of leaves with exponential growth.",
"Duminy's proof is “non-constructive” and does not directly show how a non-trivial value of the Godbillon-Vey class results in resilient leaves for the foliation.",
"One of the points of this present work is to give a direct demonstration of this relation, which we show using techniques of ergodic theory for $C^1$ -foliations.",
"In the work [24], Theéorème 6.1 states that for a codimension-one, $C^2$ -foliation ${\\mathcal {F}}$ , if 0 then ${\\mathcal {F}}$ must have a resilient leaf.",
"Candel and Conlon gave a proof of this result in [9] for the special case where the foliation is the suspension of a group action on a circle, but were unable to extend the proof to the general case asserted in [24].",
"Combining these results, one concludes that for a $C^2$ -foliation ${\\mathcal {F}}$ , if the geometric entropy 0 then ${\\mathcal {F}}$ has no resilient leaves, and thus $GV({\\mathcal {F}}) = 0$ .",
"This result suggests the problem of giving a direct proof of this conclusion.",
"The development of an ergodic theory approach to the study of the secondary classes began with the work of Heitsch and Hurder [36], which was inspired by Duminy's work [18], [19].",
"A key idea introduced in [38], [39], was to use techniques from the Oseledets theory of cocycles to study the relation between foliation dynamics, and the values of the secondary classes of foliations.",
"In this paper, we use methods from the ergodic theory of $C^1$ -foliations to show that for a $C^2$ -foliation ${\\mathcal {F}}$ , the assumption $GV({\\mathcal {F}}) \\ne 0$ implies that the foliation ${\\mathcal {F}}$ has resilient leaves, and thus 0.",
"An important aspect of our proof, is that the subtle techniques of the Poincaré-Bendixson theory of $C^2$ -foliations are avoided, and the conclusion that there exists resilient leaves follows from straightforward techniques of dynamical systems.",
"The work of Duminy [18] reformulated the study of the Godbillon-Vey class for $C^2$ -foliations in terms of the “Godbillon measure”, which for a $C^1$ -foliation ${\\mathcal {F}}$ of a compact manifold $M$ , is a linear functional defined on the Borel $\\sigma $ -algebra ${\\mathcal {B}}({\\mathcal {F}})$ formed from the leaf-saturated Borel subsets of $M$ , and by extension this measure is defined on the saturated measurable subsets of $M$ .",
"These ideas are introduced and discussed in the papers [12], [18], [19], [36], [38], [39], and recalled in Section below.",
"Here is our main result, as formulated in these terms: THEOREM 1.1 If ${\\mathcal {F}}$ is a codimension–one, $C^{1}$ -foliation with non-trivial Godbillon measure $G_{{\\mathcal {F}}}$ , then ${\\mathcal {F}}$ has a hyperbolic resilient leaf.",
"In the course of our proof of this result, resilient orbits of the action of the pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ are explicitly constructed using a version of the Ping-Pong Lemma, first introduced by Klein in his study of subgroups of Kleinian groups [16], and which is discussed in Section REF .",
"For $C^{2}$ -foliations, the Godbillon-Vey class is obtained by evaluating the Godbillon measure on the “Vey class” $[v({\\mathcal {F}})] | E$ localized to a set $E \\in {\\mathcal {B}}({\\mathcal {F}})$ .",
"Only the definition of the class $[v({\\mathcal {F}})] | E$ requires that ${\\mathcal {F}}$ be $C^2$ .",
"Thus, for a $C^2$ -foliation ${\\mathcal {F}}$ , $GV({\\mathcal {F}}) \\ne 0$ implies that $G_{{\\mathcal {F}}} \\ne 0$ , and we deduce: COROLLARY 1.2 If ${\\mathcal {F}}$ is a codimension–one, $C^2$ -foliation with non-trivial Godbillon-Vey class $GV({\\mathcal {F}}) \\in H^3(M; {\\mathbb {R}})$ , then ${\\mathcal {F}}$ has a hyperbolic resilient leaf, and thus the entropy 0.",
"We next discuss the strategy of the proof of Theorem REF .",
"A key idea in dynamical systems of flows is to consider the points for which the dynamics is “infinitesimally exponentially expansive” over long orbit segments, which corresponds to points with positive Lyapunov exponent [2], [5], [61].",
"The analog for pseudogroup dynamics is to introduce the set of points in the transversal ${\\mathcal {X}}$ for which there are arbitrarily long words in ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ for which the norm of their transverse derivative matrix is exponentially growing with respect to the word norm on the pseudogroup.",
"We introduce in Section , the ${\\mathcal {F}}$ -saturated set ${\\rm E}^+({\\mathcal {F}})$ of points in $M$ where the transverse derivative cocycle for ${\\mathcal {F}}$ has positive exponent.",
"A point $x \\in {\\rm E}^+({\\mathcal {F}})\\cap {\\mathcal {X}}$ if and only if there is a sequence of holonomy maps such that the norms of their derivatives at $x$ grow exponentially fast as a function of “word length” in the foliation pseudogroup, and ${\\rm E}^+({\\mathcal {F}})$ is the leaf saturation of this set.",
"The set ${\\rm E}^+({\\mathcal {F}})$ is a fundamental construction for a $C^1$ -foliation.",
"For example, a key step in the proof of the generalized Moussu–Pelletier–Sullivan conjecture in [38] was to show that for a foliation ${\\mathcal {F}}$ with almost all leaves of subexponential growth, the Lebesgue measure $|{\\rm E}^+({\\mathcal {F}})| = 0$ .",
"Here, we show in Theorem REF that if a measurable, ${\\mathcal {F}}$ -saturated subset $B \\subset M$ is disjoint from ${\\rm E}^+({\\mathcal {F}})$ , then the Godbillon measure must vanish on $B$ .",
"The second step in the proof of Theorem REF is to show that for each point $x \\in {\\rm E}^+({\\mathcal {F}})$ , the holonomy of ${\\mathcal {F}}$ has a uniform exponential estimate along the orbit of $x$ for its transverse expansion along arbitrarily long words in the holonomy pseudogroup.",
"This follows from Proposition REF , which is pseudogroup version of what is called the “Pliss Lemma” in the literature for non-uniform dynamics [64], [55], [5].",
"If ${\\rm E}^+({\\mathcal {F}})$ has positive measure, it is then straightforward to construct resilient orbits for the action of ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ on ${\\mathcal {X}}$ , as done in the proof of Proposition REF .",
"The proof of Theorem REF then follows by combining Theorem REF , Proposition REF and Proposition REF .",
"The proofs of Propositions REF and REF are the most technical aspects of this paper.",
"One important issue that arises in the study of pseudogroup dynamical systems, is that the domain of a holonomy map in the pseudogroup may depend upon the “length” of the leafwise path used to define it, so that composing maps in the pseudogroup often results in a contraction of the domain of definition for the resulting map.",
"This is a key difference between the study of dynamics of a group acting on the circle, and that of a pseudogroup associated to a general codimension–one foliation.",
"One of the key steps in the proof of Proposition REF is to show uniform estimates on the length of the domains of compositions.",
"The proof uses these estimates to produce an abundance of holonomy pseudogroup maps with hyperbolic fixed–points.",
"We point out one application of Proposition REF , which complements the main result of [42].",
"THEOREM 1.3 Let ${\\mathcal {F}}$ be a $C^1$ -foliation of codimension-one such that no leaf of ${\\mathcal {F}}$ has a closed loop with hyperbolic transverse holonomy, then the hyperbolic set ${\\rm E}^+({\\mathcal {F}})$ is empty.",
"Finally, the extension of the methods for closed manifolds to the case of open manifolds requires only a minor modification in the definition of the Godbillon measure, as discussed in Section .",
"For codimension–one foliations, it is elementary that the existence of a resilient leaf implies 0.",
"The converse, that 0 implies there is a resilient leaf, was proved in [24] for $C^2$ -foliations, and proved in [43] for $C^1$ -foliations.",
"Let ${\\rm ``HRL({\\mathcal {F}})\"} $ denote the property that ${\\mathcal {F}}$ has a hyperbolic resilient leaf.",
"Let $|E|$ denote the Lebesgue measure of a measurable subset $E \\subset M$ .",
"The results of this paper are summarized by the following implications: THEOREM 1.4 Let ${\\mathcal {F}}$ be a codimension–one, $C^1$ -foliation of a manifold $M$ .",
"Then $g_{{\\mathcal {F}}}\\ne 0 \\Longrightarrow |{\\rm E}^+({\\mathcal {F}})| > 0 \\Longrightarrow {\\rm ``HRL({\\mathcal {F}})\"} \\Longleftrightarrow 0 ~ .$ The collaboration of the authors in Spring 1999 leading to this work was made possible by the support of the first author by the Université of Bourgogne, Dijon.",
"This support is gratefully acknowledged.",
"This manuscript is a revised version of a preprint dated March 27, 2004 and submitted for publication [45].",
"The statements of the results, and the ideas for their proofs, have not changed in the intervening period, but the revised manuscript reorganizes the proofs in Sections and , includes updated references, and incorporates the suggestions and edits given by the referee of that manuscript."
],
[
"Foliation Basics",
"In this section, we introduce some standard notions and results of foliation geometry and dynamics.",
"Complete details and further discussions are provided by the texts [8], [9], [25], [34], [75].",
"We assume that $M$ is a closed oriented smooth Riemannian $m$ -manifold, ${\\mathcal {F}}$ is a $C^r$ -foliation of codimension–1 with oriented normal bundle, for $r \\ge 1$ , and that the leaves of ${\\mathcal {F}}$ are smoothly immersed submanifolds of dimension $n \\ge 2$ , where $m = n+1$ .",
"This is sometimes referred to as a $C^{\\infty ,r}$ -foliation, where the holonomy transition maps are $C^r$ , typically for either $r =1$ or $r=2$ ."
],
[
"Regular Foliation Atlas",
"A $C^{\\infty ,r}$ -foliation atlas on $M$ , for $r \\ge 1$ , is a finite collection $\\lbrace (U_{\\alpha },\\phi _{\\alpha }) \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ such that: ${\\mathcal {U}}= \\lbrace U_{\\alpha } \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ is an open covering of $M$ .",
"$\\phi _{\\alpha } : U_{\\alpha } \\rightarrow (-1,1)^m$ is a $C^{\\infty ,r}$ –coordinate chart; that is, for $(u,w) \\in (-1,1)^n \\times (-1,1)$ , the map $\\phi _{\\alpha }^{-1}(u,w)$ is $C^{\\infty }$ in the “leaf” variable $u$ and $C^r$ in the “transverse” variable $w$ .",
"Each chart $\\phi _{\\alpha }$ is transversally oriented.",
"Given $x \\in U_{\\alpha } \\cap U_{\\beta }$ with $\\phi _{\\alpha }(x) = (u,w)$ , for the change-of-coordinates map $(u^{\\prime },w^{\\prime }) = \\phi _{\\beta } \\circ \\phi _{\\alpha }^{-1}(u,w)$ , the value of $w^{\\prime }$ is locally constant with respect to $u$ .",
"Figure: Overlapping foliation chartsThe collection of sets ${\\mathcal {V}}_{{\\mathcal {F}}} \\equiv \\left\\lbrace V_{\\alpha , w} = \\phi _{\\alpha }^{-1} (V \\times \\lbrace w\\rbrace ) \\mid V \\subset (-1,1)^n ~, ~ w \\in (-1,1) ~ , ~ \\alpha \\in {\\mathcal {A}}\\right\\rbrace $ form a subbasis for the “fine topology” on $M$ .",
"For $x \\in M$ , let $L_x \\subset M$ denote the connected component of this fine topology containing $x$ .",
"Then $L_x$ is path connected, and is called the leaf of ${\\mathcal {F}}$ containing $x$ .",
"Without loss of generality, we can assume that the coordinates are positively oriented, mapping the positive orientation for the normal bundle to $T{\\mathcal {F}}$ to the positive orientation on $(-1,1)$ .",
"Note that each leaf $L$ is a smooth, injectively immersed manifold in $M$ .",
"The Riemannian metric on $TM$ restricts to a smooth metric on each leaf.",
"The path-length metric $d_{{\\mathcal {F}}}$ on a leaf $L$ is defined by $d_{{\\mathcal {F}}}(x,y) = \\inf \\left\\lbrace \\Vert \\gamma \\Vert \\mid \\gamma \\colon [0,1] \\rightarrow L ~ {\\rm is } ~ C^1~, ~ \\gamma (0) = x ~, ~ \\gamma (1) = y \\right\\rbrace , $ where $\\Vert \\gamma \\Vert $ denotes the path length of the $C^1$ -curve $\\gamma (t)$ .",
"If $x,y \\in M$ are not on the same leaf, then set $d_{{\\mathcal {F}}}(x,y) = \\infty $ .",
"It was noted by Plante [63] that for each $x \\in M$ , the leaf $L_x$ containing the point $x$ , with the induced Riemannian metric from $TM$ is a complete Riemannian manifold with bounded geometry, that depends continuously on $x$ .",
"In particular, bounded geometry implies that for each $x \\in M$ , there is a leafwise exponential map $\\exp ^{{\\mathcal {F}}}_x \\colon T_x{\\mathcal {F}}\\rightarrow L_x$ which is a surjection, and the composition $\\iota \\circ \\exp ^{{\\mathcal {F}}}_x \\colon T_x{\\mathcal {F}}\\rightarrow L_x \\subset M$ depends continuously on $x$ in the compact-open topology.",
"We next recall the notion of a regular covering, or what is sometimes called a nice covering in the literature (see [9], or [34].)",
"For a regular foliation covering, the intersections of the coverings of leaves by the plaques of the charts have nice metric properties.",
"We first recall a standard fact from Riemannian geometry, as it applies to the leaves of ${\\mathcal {F}}$ .",
"For each $x \\in M$ and $r > 0$ , let ${\\overline{B}}_{{\\mathcal {F}}}(x, r) = \\lbrace y \\in L_x \\mid d_{{\\mathcal {F}}}(x,y) \\le r\\rbrace $ denote the closed ball of radius $r$ in the leaf containing $x$ .",
"The Gauss Lemma implies that there exists $\\lambda _x > 0$ such that ${\\overline{B}}_{{\\mathcal {F}}}(x, \\lambda _x)$ is a strongly convex subset for the metric $d_{{\\mathcal {F}}}$ .",
"That is, for any pair of points $y,y^{\\prime } \\in {\\overline{B}}_{{\\mathcal {F}}}(x, \\lambda _x)$ there is a unique shortest geodesic segment in $L_x$ joining $y$ and $y^{\\prime }$ and it is contained in ${\\overline{B}}_{{\\mathcal {F}}}(x, \\lambda _x)$ (cf.",
"[3], [17]).",
"Then for all $0 < \\lambda < \\lambda _x$ , the disk ${\\overline{B}}_{{\\mathcal {F}}}(x, \\lambda )$ is also strongly convex.",
"The compactness of $M$ and the continuous dependence of the Christoffel symbols for a Riemannian metric in the $C^2$ -topology on sections of bundles over $M$ yields: LEMMA 2.1 There exists ${\\lambda _{\\mathcal {F}}}> 0$ such that for all $x \\in M$ , ${\\overline{B}}_{{\\mathcal {F}}}(x, {\\lambda _{\\mathcal {F}}})$ is strongly convex.",
"If ${\\mathcal {F}}$ is defined by a flow without periodic points, so that every leaf is diffeomorphic to ${\\mathbb {R}}$ , then the entire leaf is strongly convex, so ${\\lambda _{\\mathcal {F}}}> 0$ can be chosen arbitrarily.",
"For a foliation with leaves of dimension $n > 1$ , the constant ${\\lambda _{\\mathcal {F}}}$ must be less than the injectivity radius for each of the leaves.",
"Let $d_{M} \\colon M \\times M \\rightarrow [0,\\infty )$ denote the path-length metric on $M$ .",
"For $x \\in M$ and ${\\epsilon }> 0$ , let $B_{M}(x, {\\epsilon }) = \\lbrace y \\in M \\mid d_{M}(x, y) < {\\epsilon }\\rbrace $ be the open ball of radius ${\\epsilon }$ about $x$ , and let ${\\overline{B}}_{M}(x, {\\epsilon }) = \\lbrace y \\in M \\mid d_{M}(x, y) \\le {\\epsilon }\\rbrace $ denote its closure.",
"Then as above, there exists ${\\lambda _{M}}> 0$ such that ${\\overline{B}}_M(x, \\lambda )$ is a strongly convex ball in $M$ for all $0 < \\lambda \\le {\\lambda _{\\mathcal {F}}}$ .",
"We use these estimates on the local geometry of $M$ and the leaves of ${\\mathcal {F}}$ to construct a refinement of the given covering of $M$ by foliations charts, which have uniform regularity properties.",
"Let ${\\epsilon _{{\\mathcal {U}}}}> 0$ be a Lebesgue number for the given covering ${\\mathcal {U}}$ of $M$ .",
"Then for each $x \\in M$ , there exists $\\alpha _x \\in {\\mathcal {A}}$ be such that $x \\in B_M(x, {\\epsilon _{{\\mathcal {U}}}}) \\subset U_{\\alpha _x}$ .",
"It follows that for each $x \\in M$ , there exists $0 < \\delta _x \\le {\\lambda _{\\mathcal {F}}}$ such that ${\\overline{B}}_{{\\mathcal {F}}}(x, \\delta _x) \\subset B_M(x, {\\epsilon _{{\\mathcal {U}}}})$ .",
"Let $(u_x, w_x) = \\phi _{\\alpha }(x)$ , and note that $\\phi _{\\alpha }({\\overline{B}}_{{\\mathcal {F}}}(x, \\delta _x)) \\subset (-1,1)^n \\times \\lbrace w_x\\rbrace $ .",
"Then there exists ${\\epsilon }_x > 0$ so that for each $w \\in (w_x - {\\epsilon }_x , w_x + {\\epsilon }_x)$ and $x_w = \\phi _{\\alpha }^{-1}(u_x, w)$ we have $\\displaystyle {\\overline{B}}_{{\\mathcal {F}}}(x_w, \\delta _x) \\subset B_M(x, {\\epsilon _{{\\mathcal {U}}}}) \\subset U_{\\alpha _x}$ is a leafwise convex subset.",
"Define $U_x$ and ${\\widetilde{U}}_x $ to be unions of leafwise strongly convex disks, $U_x = \\bigcup _{w \\in (w_x - {\\epsilon }_x/2 , w_x + {\\epsilon }_x/2)} \\, {\\overline{B}}_{{\\mathcal {F}}}(x_w, \\delta _x/2) \\quad ; \\quad {\\widetilde{U}}_x = \\bigcup _{w \\in (w_x - {\\epsilon }_x , w_x + {\\epsilon }_x)} \\, {\\overline{B}}_{{\\mathcal {F}}}(x_w, \\delta _x)$ so then $U_x \\subset {\\widetilde{U}}_x \\subset B_M(x, {\\epsilon _{{\\mathcal {U}}}}) \\subset U_{\\alpha _x}$ .",
"The restriction $\\phi _{\\alpha _x} \\colon {\\widetilde{U}}_x \\rightarrow (-1,1)^{n+1}$ is then a foliation chart, though the image is not onto.",
"Note that for each $x^{\\prime } \\in \\phi _{\\alpha _x}^{-1}(w_x - {\\epsilon }_x , w_x + {\\epsilon }_x)$ , the chart $\\phi _{\\alpha _x}$ defines a framing of the tangent bundle $T_{x^{\\prime }}L_{x^{\\prime }}$ and this framing depends $C^r$ on the parameter $x^{\\prime }$ , so we can then use the Gram-Schmidt process to obtain a $C^r$ -family of orthonormal frames as well.",
"Then using the inverse of the leafwise exponential map and affine rescaling, we obtain foliation charts ${\\widetilde{\\varphi }}_{\\alpha _x} & \\colon & {\\widetilde{U}}_x \\rightarrow (-\\delta _x,\\delta _x)^{n} \\times (w_x - {\\epsilon }_x , w_x + {\\epsilon }_x) \\cong (-2,2)^n \\times (-2,2)\\\\{\\varphi }_{\\alpha _x} & \\colon & U_x \\rightarrow (-\\delta _x/2,\\delta _x/2)^{n} \\times (w_x - {\\epsilon }_x/2 , w_x + {\\epsilon }_x/2) \\cong (-1,1)^n \\times (-1,1)$ where ${\\varphi }_{\\alpha _x}$ is the restriction of ${\\widetilde{\\varphi }}_{\\alpha _x}$ .",
"Observe that ${\\widetilde{\\varphi }}_{\\alpha _x}(x) = (\\vec{0},0) \\in (-1,1)^n \\times (-1,1)$ for each $x$ .",
"The collection of open sets $ \\lbrace U_x \\mid x \\in M \\rbrace $ forms an open cover of the compact space $M$ , so there exists a finite subcover “centered” at the points $\\lbrace x_1, \\ldots , x_{\\nu }\\rbrace \\subset M$ .",
"Set $\\delta ^{{\\mathcal {F}}}_{{\\mathcal {U}}}= \\min \\lbrace \\delta _{x_1}/2 , \\ldots , \\delta _{x_{\\nu }}/2\\rbrace ~ \\le ~ {\\lambda _{\\mathcal {F}}}/2 ~.$ This covering by foliation coordinate charts will be fixed and used throughout.",
"To simplify notation, for $1 \\le \\alpha \\le \\nu $ , set $U_{\\alpha } = U_{x_{\\alpha }}$ , ${\\widetilde{U}}_{\\alpha } = {\\widetilde{U}}_{x_{\\alpha }}$ , $\\displaystyle {\\varphi }_{\\alpha } = {\\varphi }_{x_{\\alpha }}$ , $\\displaystyle {\\widetilde{\\varphi }}_{\\alpha } = {\\widetilde{\\varphi }}_{x_{\\alpha }}$ , and ${\\mathcal {U}}= \\lbrace U_{1}, \\ldots , U_{\\nu }\\rbrace $ .",
"The resulting collection $\\displaystyle \\lbrace {\\varphi }_{\\alpha } \\colon U_{\\alpha } \\rightarrow (-1,1)^n \\times (-1,1) \\mid 1 \\le \\alpha \\le \\nu \\rbrace $ is a regular covering of $M$ by foliation charts, in the sense used in [9] or [34].",
"For each $1 \\le \\alpha \\le \\nu $ , define ${\\mathcal {X}}_{\\alpha } \\equiv (-1,1) \\cong \\lbrace \\vec{0}\\rbrace \\times (-1,1)$ and ${\\widetilde{{\\mathcal {X}}}}_{\\alpha } \\equiv (-2,2) \\cong \\lbrace \\vec{0}\\rbrace \\times (-2,2)$ .",
"The extended chart ${\\widetilde{\\varphi }}_{\\alpha }$ defines $C^r$ –embeddings $t_{\\alpha } \\colon {\\mathcal {X}}_{\\alpha } \\rightarrow U_{\\alpha } \\quad , \\quad {\\widetilde{t}}_{\\alpha } \\colon {\\widetilde{{\\mathcal {X}}}}_{\\alpha } \\rightarrow {\\widetilde{U}}_{\\alpha } ~ .$ Let ${\\mathfrak {X}}_{\\alpha } = \\tau _{\\alpha }({\\mathcal {X}}_{\\alpha })$ and ${\\widetilde{{\\mathfrak {X}}}}_{\\alpha } = {\\widetilde{t}}_{\\alpha }({\\widetilde{{\\mathcal {X}}}}_{\\alpha })$ denote the images of these maps.",
"For $n \\ge 3$ , we can assume without loss of generality that the submanifolds ${\\widetilde{{\\mathfrak {X}}}}_{\\alpha }$ and ${\\widetilde{{\\mathfrak {X}}}}_{\\beta }$ are disjoint, for $\\alpha \\ne \\beta $ .",
"Consider ${\\mathcal {X}}_{\\alpha }$ and ${\\mathcal {X}}_{\\beta }$ as disjoint spaces for $\\alpha \\ne \\beta $ , and similarly for ${\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ and ${\\widetilde{{\\mathcal {X}}}}_{\\beta }$ .",
"Introduce the disjoint unions of these spaces, as denoted by ${\\mathcal {X}}= \\bigcup _{1 \\le \\alpha \\le \\nu } {\\mathcal {X}}_{\\alpha } \\quad & \\subset & \\quad {\\widetilde{{\\mathcal {X}}}}= \\bigcup _{1 \\le \\alpha \\le \\nu } {\\widetilde{{\\mathcal {X}}}}_{\\alpha } ~ , \\\\{\\mathfrak {X}}= \\bigcup _{1 \\le \\alpha \\le \\nu } {\\mathfrak {X}}_{\\alpha } \\quad & \\subset & \\quad {\\widetilde{{\\mathfrak {X}}}}= \\bigcup _{1 \\le \\alpha \\le \\nu } {\\widetilde{{\\mathfrak {X}}}}_{\\alpha } ~ , $ Note that ${\\mathfrak {X}}$ is a complete transversal for ${\\mathcal {F}}$ , as the submanifold ${\\mathfrak {X}}$ is transverse to the leaves of ${\\mathcal {F}}$ , and every leaf of ${\\mathcal {F}}$ intersects ${\\mathfrak {X}}$ .",
"The same is true for ${\\widetilde{{\\mathfrak {X}}}}$ .",
"Let $\\tau \\colon {\\mathcal {X}}\\rightarrow {\\mathfrak {X}}\\subset M$ denote the map defined by the coordinate chart embeddings $\\tau _{\\alpha }$ , and similarly define ${\\widetilde{t}}\\colon {\\widetilde{{\\mathcal {X}}}}\\rightarrow {\\widetilde{{\\mathfrak {X}}}}\\subset M$ using the maps ${\\widetilde{t}}_{\\alpha }$ .",
"Let each ${\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ have the metric ${\\bf d}_{{\\mathcal {X}}}$ induced from the Euclidean metric on ${\\mathbb {R}}$ , where ${\\bf d}_{{\\mathcal {X}}}(x,y) = |x - y|$ for $x, y \\in {\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ .",
"Extend this to a metric on ${\\mathcal {X}}$ by setting ${\\bf d}_{{\\mathcal {X}}}(x,y) = \\infty $ for $x \\in {\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ , $y \\in {\\widetilde{{\\mathcal {X}}}}_{\\beta }$ with $\\alpha \\ne \\beta $ .",
"Let each ${\\widetilde{{\\mathfrak {X}}}}_{\\alpha }$ have the Riemannian metric induced from the Riemannian metric on $M$ , and let ${\\bf d}_{{\\mathfrak {X}}}$ denote the resulting path-length metric on ${\\mathfrak {X}}_{\\alpha }$ .",
"As before, extend this to a metric on ${\\mathfrak {X}}$ by setting ${\\bf d}_{{\\mathfrak {X}}}(x,y) = \\infty $ for $x \\in {\\widetilde{{\\mathfrak {X}}}}_{\\alpha }$ , $y \\in {\\widetilde{{\\mathfrak {X}}}}_{\\beta }$ with $\\alpha \\ne \\beta $ .",
"Given $r > 0 $ and $x \\in {\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ let $\\displaystyle {\\bf B}_{{\\widetilde{{\\mathfrak {X}}}}}(x,r) = \\lbrace y \\in {\\widetilde{{\\mathfrak {X}}}}_{\\alpha } \\mid {\\bf d}_{{\\mathfrak {X}}}(x,y) < r \\rbrace $ .",
"Introduce a notation which will be convenient for later work.",
"Given a point $x \\in {\\widetilde{{\\mathfrak {X}}}}_{\\alpha }$ and $\\delta _1, \\delta _2 > 0$ , let $ [x-\\delta _1, x +\\delta _2] \\subset {\\widetilde{{\\mathfrak {X}}}}_{\\alpha } $ be the connected closed subset bounded below by the point $x-\\delta _1$ satisfying by ${\\bf d}_{{\\mathfrak {X}}}(x,x-\\delta _1) = \\delta _1$ and $[x-\\delta _1,x]$ is an oriented interval in ${\\mathfrak {X}}_{\\alpha }$ .",
"The set $[x-\\delta _1, x +\\delta _2]$ is bounded above by the point $x+\\delta _2$ satisfying by ${\\bf d}_{{\\mathfrak {X}}}(x,x+\\delta _2) = \\delta _2$ and $[x, x+\\delta _1]$ is an oriented interval in ${\\mathfrak {X}}_{\\alpha }$ .",
"For each $1 \\le \\alpha \\le \\nu $ , let $\\pi _{\\alpha } \\equiv \\pi _t \\circ {\\varphi }_{\\alpha } \\colon U_{\\alpha } \\rightarrow {\\mathcal {X}}_{\\alpha }$ be the composition of the coordinate map ${\\varphi }_{\\alpha }$ with the projection $\\pi _t \\colon {\\mathbb {R}}^{n+1} = {\\mathbb {R}}^n \\times {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$ .",
"For each $w \\in {\\mathcal {X}}_{\\alpha }$ , the preimage ${\\mathcal {P}}_{\\alpha }(w) = \\pi _{\\alpha }^{-1} \\subset U_{\\alpha }$ is called a plaque of the chart ${\\varphi }_{\\alpha }$ .",
"For $x \\in U_{\\alpha }$ we use the notation $ {\\mathcal {P}}_{\\alpha }(x) = {\\mathcal {P}}_{\\alpha }({\\varphi }_{\\alpha }(x))$ to denote the plaque of the chart ${\\varphi }_{\\alpha }$ containing $x$ .",
"Note that $ {\\mathcal {P}}_{\\alpha }(x)$ is the connected component of the intersection of the leaf $L_x$ of ${\\mathcal {F}}$ through $x$ with the set $U_{\\alpha }$ .",
"Then the collection of all plaques for the foliation atlas is indexed by ${\\mathcal {X}}$ .",
"The maps ${\\widetilde{\\pi }}_{\\alpha } \\equiv \\pi _t \\circ {\\widetilde{\\varphi }}_{\\alpha } \\colon {\\widetilde{U}}_{\\alpha } \\rightarrow {\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ are defined analogously, with corresponding plaques ${\\widetilde{{\\mathcal {P}}}}_{\\alpha }(w)$ .",
"For $x \\in {\\widetilde{U}}_{\\alpha }$ , the plaque of the chart ${\\widetilde{\\varphi }}_{\\alpha }$ containing $x$ is denoted by ${\\widetilde{{\\mathcal {P}}}}_{\\alpha }(x) \\subset {\\widetilde{U}}_{\\alpha }$ .",
"Note that each plaque ${\\mathcal {P}}_{\\alpha }(x)$ is strongly convex in the leafwise metric, so if the intersection of two plaques $\\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ is non-empty, then it is a strongly convex subset.",
"In particular, the intersection ${\\mathcal {P}}_{\\alpha }(x) \\cap {\\mathcal {P}}_{\\beta }(y)$ is connected.",
"Thus, each plaque ${\\mathcal {P}}_{\\alpha }(x)$ intersects either zero or one plaque in $U_{\\beta }$ .",
"The same observations are also true for the extended plaques ${\\widetilde{{\\mathcal {P}}}}_{\\alpha }(x)$ ."
],
[
"Holonomy Pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$",
"A pair of indices $(\\alpha , \\beta )$ is admissible if $U_{\\alpha } \\cap U_{\\beta } \\ne \\emptyset $ .",
"For each admissible pair $(\\alpha , \\beta )$ define ${\\mathcal {X}}_{\\alpha \\beta } & = & \\lbrace x \\in {\\mathcal {X}}_{\\alpha } \\mbox{ such that } {{\\mathcal {P}}}_{\\alpha }(x) \\cap U_{\\beta } \\ne \\emptyset \\rbrace , \\\\{\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta } & = & \\lbrace x \\in {\\widetilde{{\\mathcal {X}}}}_{\\alpha } \\mbox{ such that } {{\\widetilde{{\\mathcal {P}}}}}_{\\alpha }(x) \\cap {\\widetilde{U}}_{\\beta } \\ne \\emptyset \\rbrace ~ .",
"$ Then there is a well-defined transition function ${{\\bf h}}_{\\beta \\alpha } \\colon {\\mathcal {X}}_{\\alpha \\beta } \\rightarrow {\\mathcal {X}}_{\\beta \\alpha }$ , which for $x \\in {\\mathcal {X}}_{\\alpha \\beta } $ is given by $ {\\bf h}_{\\beta \\alpha }(x) = y \\mbox{ where } {{\\mathcal {P}}}_{\\alpha }(x) \\cap {{\\mathcal {P}}}_{\\beta }(y) \\ne \\emptyset ~ .",
"$ Note that $\\displaystyle {\\bf h}_{\\alpha \\alpha } \\colon {\\mathcal {X}}_{\\alpha } \\rightarrow {\\mathcal {X}}_{\\alpha }$ is the identity map for each $\\alpha \\in {\\mathcal {A}}$ .",
"The holonomy pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ associated to the regular foliation atlas for ${\\mathcal {F}}$ is the pseudogroup with object space ${\\mathcal {X}}$ , and transformations generated by compositions of the local transformations $\\displaystyle \\lbrace {\\bf h}_{\\beta \\alpha } \\mid (\\alpha ,\\beta ) \\mbox{ admissible} \\rbrace $ .",
"The $C^{\\infty ,r}$ –hypothesis on the coordinate charts implies that each map ${\\bf h}_{\\beta \\alpha }$ is $C^r$ .",
"Moreover, the hypothesis (2) on regular foliation charts implies that each ${\\bf h}_{\\beta \\alpha }$ admits an extension to a $C^r$ -map $\\displaystyle \\widetilde{\\bf h}_{\\beta \\alpha } \\colon {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta } \\rightarrow {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ defined in a similar fashion.",
"The number of admissible pairs is finite, so there exists a uniform estimate on the sizes of the domains of these extensions.",
"We note the following consequence of these observations.",
"LEMMA 2.2 There exists ${\\epsilon }_0 > 0$ so that for every admissible pair $(\\alpha , \\beta )$ and $x \\in {\\mathcal {X}}_{\\alpha \\beta }$ then $[x-{\\epsilon }_0, x+ {\\epsilon }_0] \\subset {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ .",
"That is, if $x \\in {\\mathcal {X}}_{\\alpha }$ is in the domain of ${\\bf h}_{\\beta \\alpha }$ then $[x-{\\epsilon }_0, x+ {\\epsilon }_0]$ is in the domain of $\\widetilde{\\bf h}_{\\beta \\alpha }$ .",
"For $0 < \\delta < {\\epsilon }_0$ we introduce the closed subsets of ${\\widetilde{{\\mathcal {X}}}}$ ${\\mathcal {X}}[\\delta ] & = & \\lbrace y \\in {\\widetilde{{\\mathcal {X}}}}\\mid \\exists ~ x \\in \\overline{{\\mathcal {X}}} , ~ {\\bf d}_{{\\mathcal {X}}}(x,y) \\le \\delta \\rbrace \\\\{\\mathcal {X}}_{\\alpha \\beta }[\\delta ] & = & \\lbrace y \\in {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta } \\mid \\exists ~ x \\in \\overline{{\\mathcal {X}}_{\\alpha \\beta }} , ~ {\\bf d}_{{\\mathcal {X}}}(x,y) \\le \\delta \\rbrace ~ .",
"$ Thus, the maps ${{\\bf h}_{\\beta \\alpha }}$ are uniformly $C^r$ on ${\\mathcal {X}}_{\\alpha \\beta }[\\delta ]$ for $\\delta < {\\epsilon }_0$ .",
"Composition of elements in ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ will be defined via “plaque chains”.",
"Given $x,y \\in {\\mathcal {X}}$ corresponding to points on the same leaf, a plaque chain ${\\mathcal {P}}$ of length $k$ between $x$ and $y$ is a collection of plaques $ {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace , $ where $x_0 = x$ , $x_k = y$ and for each $0 \\le i < k$ we have $\\displaystyle {\\mathcal {P}}_{\\alpha _i}(x_i) \\cap {\\mathcal {P}}_{\\alpha _{i+1}}(x_{i+1}) \\ne \\emptyset $ .",
"We write $\\Vert {\\mathcal {P}}\\Vert = k$ .",
"A plaque chain ${\\mathcal {P}}$ also defines an “extended” plaque chain for the charts $\\lbrace ({\\widetilde{U}}_{\\alpha }, \\widetilde{\\phi }_{\\alpha })\\rbrace $ , ${\\widetilde{{\\mathcal {P}}}}= \\lbrace {\\widetilde{{\\mathcal {P}}}}_{\\alpha _1}(x_0), \\ldots , {\\widetilde{{\\mathcal {P}}}}_{\\alpha _k}(x_k) \\rbrace ~.",
"$ We say two plaque chains $ {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace \\mbox{ and } {\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\beta _0}(y_0), \\ldots , {\\mathcal {P}}_{\\beta _{\\ell }}(y_{\\ell }) \\rbrace $ are composable if $x_k = y_0$ , hence $\\alpha _k = \\beta _0$ and ${\\mathcal {P}}_{\\alpha _k}(x_k) = {\\mathcal {P}}_{\\beta _0}(y_0))$ .",
"Their composition is defined by ${\\mathcal {Q}}\\circ {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k), {\\mathcal {P}}_{\\beta _1}(y_1), \\ldots , {\\mathcal {P}}_{\\beta _{\\ell }}(y_{\\ell })\\rbrace ~ .$ The holonomy transformation defined by a plaque chain is the local diffeomorphism $ {\\bf h}_{{\\mathcal {P}}} = {\\bf h}_{\\alpha _k \\alpha _{k-1}} \\circ \\cdots \\circ {\\bf h}_{\\alpha _1 \\alpha _0} $ whose domain ${\\mathcal {D}}_{{\\mathcal {P}}} \\subset {\\mathcal {X}}_{\\alpha _0}$ contains $x_0$ .",
"Note that ${\\mathcal {D}}_{{\\mathcal {P}}} $ is the largest connected open subset of ${\\mathcal {X}}_{\\alpha _0}$ containing $x_0$ on which $\\displaystyle {\\bf h}_{\\alpha _{\\ell } \\alpha _{\\ell -1}} \\circ \\cdots \\circ {\\bf h}_{\\alpha _1 \\alpha _0}$ is defined for all $0 < \\ell \\le k$ .",
"The dependence of the domain of ${\\bf h}_{{\\mathcal {P}}}$ on the plaque chain ${\\mathcal {P}}$ is a subtle issue, yet is at the heart of the technical difficulties arising in the study of foliation pseudogroups.",
"Let $\\widetilde{\\bf h}_{{\\widetilde{{\\mathcal {P}}}}}$ be the holonomy associated to the chain $ {\\widetilde{{\\mathcal {P}}}}$ , with domain ${\\widetilde{{\\mathcal {D}}}}_{{\\widetilde{{\\mathcal {P}}}}} \\subset {\\widetilde{{\\mathcal {X}}}}_{\\alpha _0}$ the largest maximal open subset containing $x_0$ on which $\\displaystyle \\widetilde{\\bf h}_{\\alpha _{\\ell } \\alpha _{\\ell - 1}} \\circ \\cdots \\circ \\widetilde{\\bf h}_{\\alpha _1 \\alpha _0}$ is defined for all $1 < \\ell \\le k$ .",
"By the extension property of a regular atlas, the closure $\\overline{{\\mathcal {D}}_{{\\mathcal {P}}}} \\subset {\\widetilde{{\\mathcal {D}}}}_{{\\widetilde{{\\mathcal {P}}}}}$ and $\\widetilde{\\bf h}_{{\\widetilde{{\\mathcal {P}}}}}$ is an extension of ${\\bf h}_{{\\mathcal {P}}}$ .",
"Given a plaque chain $\\displaystyle {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace $ and a point $\\displaystyle y \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ , there is a “parallel” plaque chain denoted $\\displaystyle {\\mathcal {P}}(y) = \\lbrace {\\mathcal {P}}_{\\alpha _0}(y), \\ldots , {\\mathcal {P}}_{\\alpha _k}(y_k) \\rbrace $ where ${\\bf h}_{{\\mathcal {P}}}(y) = y_k$ .",
"For $x \\in {\\mathcal {X}}$ , let ${{\\mathcal {G}}}_{{\\mathcal {F}}}(x) = \\lbrace y = {\\bf h}_{{\\mathcal {P}}}(x) \\in {\\mathcal {X}}\\mid ~ {\\mathcal {P}}~ {\\rm a ~ plaque ~ chain~ for~ which} ~x \\in {\\mathcal {D}}_{{\\mathcal {P}}}\\rbrace $ denote the orbit of $x$ under the action of the pseudogroup.",
"If $L_{\\xi } \\subset M$ denotes the leaf containing $\\xi \\in U_{\\alpha }$ with $\\pi _{\\alpha }(\\xi ) =x \\in {\\mathcal {X}}_{\\alpha }$ , then $\\tau ({{\\mathcal {G}}}_{{\\mathcal {F}}}(x)) = L_{\\xi } \\cap {\\mathfrak {X}}$ ."
],
[
"The derivative cocycle",
"Given a plaque chain $\\displaystyle {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace $ from $x = x_0$ to $y = x_k$ , the derivative ${\\bf h}^{\\prime }_{{\\mathcal {P}}}(x)$ is defined using the identifications ${\\mathcal {X}}_{\\alpha } = (-1,1)$ for $1 \\le \\alpha \\le \\nu $ .",
"Note that the assumption that the foliation charts are transversally orientation preserving implies that ${\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) > 0$ for all plaque chains ${\\mathcal {P}}$ and $x \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ .",
"Given composable plaque chains ${\\mathcal {P}}$ and ${\\mathcal {Q}}$ , with $x = x_0, y = x_k = y_0, z = y_{\\ell }$ the chain rule implies ${\\bf h}^{\\prime }_{{\\mathcal {Q}}\\circ {\\mathcal {P}}}(x) = {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) ~ .$ Define the map $\\displaystyle D{\\bf h}\\colon {{\\mathcal {G}}}_{{\\mathcal {F}}}\\rightarrow {\\mathbb {R}}$ by $\\displaystyle D{\\bf h}({\\mathcal {P}},y) = {\\bf h}^{\\prime }_{{\\mathcal {P}}(y)}(y)$ , which is called the derivative cocycle for the foliation pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ acting on ${\\mathcal {X}}$ .",
"The function $\\ln \\lbrace D{\\bf h}({\\mathcal {P}},y) \\rbrace \\colon {{\\mathcal {G}}}_{{\\mathcal {F}}}\\rightarrow {\\mathbb {R}}$ is the additive derivative cocycle, or sometimes the modular cocycle for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ ."
],
[
"Resilient Leaves and Ping-Pong Games",
"A plaque chain $\\displaystyle {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace $ is closed if $x_0 = x_k$ .",
"A closed plaque chain ${\\mathcal {P}}$ defines a local diffeomorphism $\\displaystyle {\\bf h}_{{\\mathcal {P}}} \\colon {\\mathcal {D}}_{{\\mathcal {P}}} \\rightarrow {\\mathcal {X}}_{\\alpha _0}$ with $\\displaystyle {\\bf h}_{{\\mathcal {P}}}(x) = x$ , where $x = x_0 \\in {\\mathcal {X}}_{\\alpha _0}$ .",
"A point $y \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ is said to be asymptotic by iterates of ${\\bf h}_{{\\mathcal {P}}}$ to $x$ , if $\\displaystyle {\\bf h}_{{\\mathcal {P}}}^{\\ell }(y) \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ for all $\\ell > 0$ (where $\\displaystyle {\\bf h}_{{\\mathcal {P}}}^{\\ell }$ denotes the composition of $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ with itself $\\ell $ times), and $\\displaystyle \\lim _{\\ell \\rightarrow \\infty } {\\bf h}_{{\\mathcal {P}}}^{\\ell }(y) = x $ .",
"The map $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ is said to be a contraction at $x$ if there is some $\\delta > 0$ so that every $\\displaystyle y \\in {\\bf B}_{{\\mathcal {X}}}(x,\\delta )$ is asymptotic to $x$ .",
"The map $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ is said to be a hyperbolic contraction at $x$ if $0 < {\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) < 1$ .",
"In this case, there exists ${\\epsilon }> 0$ and $0 < \\lambda < 1$ so that $\\displaystyle {\\bf h}^{\\prime }_{{\\mathcal {P}}}(y) < \\lambda $ for all $ y \\in {\\bf B}_{{\\mathcal {X}}}(x,{\\epsilon })$ .",
"Hence, every point of ${\\bf B}_{{\\mathcal {X}}}(x,{\\epsilon })$ is asymptotic to $x$ , and there exists $0 < \\delta < {\\epsilon }$ so that the image of the closed $\\delta $ –ball about $x$ satisfies $ {\\bf h}_{{\\mathcal {P}}}(\\overline{{\\bf B}_{{\\mathcal {X}}}(x,\\delta )}) \\subset {\\bf B}_{{\\mathcal {X}}}(x,\\delta ) ~ .",
"$ DEFINITION 2.3 We say $x \\in {\\mathcal {X}}$ is a hyperbolic resilient point for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ if there exists a closed plaque chain $\\displaystyle {\\mathcal {P}}$ such that $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ is a hyperbolic contraction at $x = x_0$ a point $y \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ which is asymptotic to $x$ (and $y \\ne x$ ) a plaque chain ${\\mathcal {R}}$ from $x$ to $y$ .",
"Figure REF below illustrates this concept, where the closed plaque chain $\\displaystyle {\\mathcal {P}}$ is represented by a path which defines it, and likewise for the plaque chain ${\\mathcal {R}}$ from $x$ to $y$ .",
"Note that the terminal point $y$ is contained in the domain of the contraction $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ defined by ${\\mathcal {P}}$ .",
"Figure: Resilient leaf with contracting holonomy along loop 𝒫{\\mathcal {P}}The “ping-pong lemma” is a key technique for the study of 1-dimensional dynamics, which was used by Klein in his study of subgroups of Kleinian groups [16].",
"For a pseudogroup, this has the form: DEFINITION 2.4 The action of the groupoid ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ on ${\\mathcal {X}}$ has a “ping-pong game” if there exists $x,y \\in {\\mathcal {X}}_{\\alpha }$ with $x \\ne y$ and a closed plaque chain $\\displaystyle {\\mathcal {P}}$ such that $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ is a contraction at $x = x_0$ a closed plaque chain $\\displaystyle {\\mathcal {Q}}$ such that $\\displaystyle {\\bf h}_{{\\mathcal {Q}}}$ is a contraction at $y = y_0$ $y \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ is asymptotic to $x$ by $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ and $x\\in {\\mathcal {D}}_{{\\mathcal {Q}}}$ is asymptotic to $y$ by $\\displaystyle {\\bf h}_{{\\mathcal {Q}}}$ We say that the ping-pong game is hyperbolic if the maps $\\displaystyle {\\bf h}_{{\\mathcal {P}}}$ and $\\displaystyle {\\bf h}_{{\\mathcal {Q}}}$ are hyperbolic contractions.",
"Figure REF below illustrates the ping-pong dynamics, where the closed plaque chain $\\displaystyle {\\mathcal {P}}$ is represented by a path which defines it, and likewise for the plaque chain ${\\mathcal {Q}}$ .",
"Figure: Closed paths 𝒫{\\mathcal {P}} and 𝒬{\\mathcal {Q}} with contracting holonomy generate a ping-pong gameThese two notions are closely related as follows; for example, see [24] for a more detailed discussion.",
"PROPOSITION 2.5 ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ has a “ping-pong game” if and only if it has a resilient point, and has a “hyperbolic ping-pong game” if and only if it has a hyperbolic resilient point."
],
[
"The Godbillon-Vey invariant",
"In this section, we recall the definition of the Godbillon-Vey class, the basic ideas of the Godbillon measure, and how it is used to estimate the values of the Godbillon-Vey invariant.",
"The textbook by Candel and Conlon [10] gives a detailed discussion of the Godbillon-Vey class and its properties.",
"Proposition REF is the key result of this section that we will use to relate the Godbillon-Vey invariant to the dynamics of the foliation."
],
[
"The Godbillon-Vey class",
"The Godbillon-Vey class is well-defined for $C^2$ -foliations, and the Godbillon measure for $C^1$ -foliations.",
"However, giving these definitions for $C^{r}$ -foliations, for $r = 1$ or 2, adds a layer of notational complexity which obscures the basic ideas of the constructions.",
"Thus, for clarity of the exposition, we assume throughout this section that ${\\mathcal {F}}$ is a $C^{\\infty }$ -foliation, and leave to the reader the technical modifications to the arguments which are required to show the analogous results for $C^r$ -foliations, or the reader may consults the works [18], [36], [38].",
"Assume that $M$ has a Riemannian metric, and that ${\\mathcal {F}}$ is a $C^{\\infty }$ -foliation of codimension one, so that $M$ admits a covering by smooth regular foliation charts.",
"Thus, the normal bundle $Q \\rightarrow M$ to $T{\\mathcal {F}}$ can be identified with the orthogonal space to the tangential distribution $T{\\mathcal {F}}$ .",
"We may assume without loss of generality that $M$ is connected, and that both the tangent bundle $TM$ and the normal bundle $Q$ are oriented, as the dynamical properties to be studied are preserved after passing to a finite covering of $M$ .",
"Thus, $T{\\mathcal {F}}$ is defined as the kernel of a non-vanishing 1-form $\\omega $ on $M$ .",
"As the distribution $T{\\mathcal {F}}$ is integrable, the Froebenius Theorem implies that $d\\omega \\wedge \\omega = 0$ .",
"This property is used to define the Godbillon-Vey class, as described below.",
"It is also well-known from the foliation literature that integrability allows for the construction of various differential graded algebras which are derived from the de Rham complex $\\Omega ^*(M)$ of $M$ , each of which which reflect aspects of the geometry of ${\\mathcal {F}}$ .",
"These complexes and their properties are fundamental for the definition of the Godbillon measure, following the works [18], [36].",
"We first recall a basic construction that is used throughout the following discussions.",
"Let $\\omega $ be a non-vanishing 1-form on $M$ whose kernel equals $T{\\mathcal {F}}$ , and $\\vec{v}$ a vector field on $M$ such that $\\omega (\\vec{v}) =1$ .",
"The integrability of the tangential distribution $T{\\mathcal {F}}$ implies that $d\\omega \\wedge \\omega = 0$ .",
"Hence, there exists a 1-form $\\alpha $ with $d\\omega = \\omega \\wedge \\alpha $ .",
"The choice of the 1-form $\\alpha $ is not canonical, and introduce a procedure for choosing a representative for $\\alpha $ .",
"Set $\\eta = \\iota (\\vec{v}) d\\omega $ , and note that $\\eta (\\vec{v}) = 0$ .",
"Then for any choice of $\\alpha $ such that $d\\omega = \\omega \\wedge \\alpha $ , let $\\vec{u}$ be tangent to ${\\mathcal {F}}$ , then we have $\\eta (\\vec{u}) = (\\iota (\\vec{v}) d\\omega ) (\\vec{u}) = d\\omega (\\vec{v}, \\vec{u}) = (\\omega \\wedge \\alpha ) ( \\vec{v}, \\vec{u}) = \\alpha (\\vec{u})$ as $\\omega (\\vec{v}) = 1$ and $\\omega (\\vec{u}) = 0$ by definition.",
"Thus, for any 1-form $\\alpha $ such that $d\\omega = \\omega \\wedge \\alpha $ and any leaf $L$ of ${\\mathcal {F}}$ , we have that their restrictions satisfy $\\alpha |L = \\eta |L$ .",
"We introduce the following notation.",
"DEFINITION 3.1 Let $\\omega $ be a non-vanishing 1-form on $M$ whose kernel equals $T{\\mathcal {F}}$ , and $\\vec{v}$ a vector field on $M$ such that $\\omega (\\vec{v}) =1$ .",
"Define $D^{\\vec{v}}\\omega = \\iota (\\vec{v}) \\, d\\omega $ .",
"Then with this definition, we have $d\\omega = \\omega \\wedge D^{\\vec{v}}\\omega $ .",
"The restricted leafwise 1-form $D^{\\vec{v}}\\omega |{\\mathcal {F}}\\colon T{\\mathcal {F}}\\rightarrow {\\mathbb {R}}$ is known in the literature as the Reeb form, and first appeared in the works of Reeb [66], [67], and was even implicitly introduced by Poincaré [65].",
"The leafwise form $D^{\\vec{v}}\\omega |{\\mathcal {F}}$ has an interpretation as the gradient of the Radon-Nikodýn derivative along leaves for the “transverse measure” for ${\\mathcal {F}}$ defined by the 1-form $\\omega $ , as discussed in [23], [36], [38].",
"Furthermore, this is the idea behind the relation between the dynamics of ${\\mathcal {F}}$ with the flow-of-weights for the von Neumann algebra associated to ${\\mathcal {F}}$ , as discussed by Connes in [14].",
"Now set $\\eta = D^{\\vec{v}}\\omega $ , and calculate $0 = d(d\\omega ) = d( \\omega \\wedge \\eta ) = d\\omega \\wedge \\eta - \\omega \\wedge d\\eta = \\omega \\wedge \\eta \\wedge \\eta - \\omega \\wedge d\\eta = - \\omega \\wedge d\\eta ~.$ shows that $d\\eta \\wedge \\omega =0$ , and hence the 2-form $d\\eta $ is a multiple of $\\omega $ .",
"Then calculate $d(\\eta \\wedge d\\eta ) = d\\eta \\wedge d\\eta = 0$ as $\\omega \\wedge \\omega = 0$ , so that $\\eta \\wedge d\\eta $ is a closed 3-form.",
"Throughout this work, $H^*(M)$ will denote the de Rham cohomology groups of $M$ .",
"THEOREM 3.2 (Godbillon and Vey, [26]) The cohomology class $GV({\\mathcal {F}}) = [\\eta \\wedge d\\eta ] \\in H^3(M)$ is independent of the choice of the 1-forms $\\omega $ and $\\eta $ .",
"Moreover, the Godbillon-Vey class $GV({\\mathcal {F}})$ is an invariant of the foliated concordance class of ${\\mathcal {F}}$ , as noted for example in Thurston [72] and Lawson [53].",
"The definition of the Godbillon-Vey class in Theorem REF reveals very little about the relation of this cohomology class with the dynamics of the foliation ${\\mathcal {F}}$ .",
"In the case where the leaves of ${\\mathcal {F}}$ are defined by a smooth fibration $M \\rightarrow {\\mathbb {S}}^1$ , the defining 1-form $\\omega $ for ${\\mathcal {F}}$ can be chosen to be a closed form, and it is then immediate from the definition that $GV({\\mathcal {F}}) =0$ .",
"For codimension-one foliations with slightly more dynamical complexity, the proof that $GV({\\mathcal {F}}) =0$ using the definition above becomes far more involved.",
"For example, Herman showed in [37] that a foliation defined by the suspension of an action of the abelian group ${\\mathbb {Z}}^2$ on the circle must have $GV({\\mathcal {F}}) =0$ .",
"The proof used an averaging process to obtain a sequence of defining 1-forms $\\omega _n$ for which the corresponding 1-forms $D^{\\vec{v}}\\omega _n \\rightarrow 0$ .",
"Subsequently, $GV({\\mathcal {F}}) = 0$ was shown for the progressively more general classes of foliations without holonomy by Morita and Tsuboi [57], for foliations almost without holonomy by T. Mizutani, S. Morita and T. Tsuboi [56], and the case of foliations which admit SRH decompositions by Nishimori [60] and Tsuchiya [74].",
"The breakthrough idea of Duminy, which can be seen first in his paper with Sergiescu [20], and further developed in the unpublished work [18], is to introduce the notion of the Godbillon functional, and its strategy to separate the roles of the forms $\\eta $ and $d\\eta $ in the definition of $GV({\\mathcal {F}})$ , and then the study of how the contribution from the form $\\eta $ is related to the dynamical properties of ${\\mathcal {F}}$ .",
"The definition of the Godbillon functional requires considering cohomology classes defined by the forms $\\eta $ and $d\\eta $ in their “largest possible” natural contexts.",
"Introduce the spaces, for $p \\ge 1$ , $A^p(M,{\\mathcal {F}}) \\equiv \\lbrace \\xi = \\omega \\wedge \\beta \\mid \\beta \\in \\Omega ^{p-1}(M) \\rbrace \\subset \\Omega ^*(M) .$ The space $A^p(M,{\\mathcal {F}}) $ can alternately be defined as the space of $p$ -forms on $M$ which vanish when restricted to each leaf of ${\\mathcal {F}}$ .",
"Note that the identity $\\omega \\wedge \\omega = 0$ , implies that the product of forms in $A^p(M,{\\mathcal {F}})$ and $A^p(M,{\\mathcal {F}})$ always vanishes, so the sum of these spaces is a subalgebra $A^*(M,{\\mathcal {F}}) \\subset \\Omega ^*(M)$ .",
"We next show that this is a differential subalgebra.",
"The identity $d\\omega = \\omega \\wedge \\eta $ implies that $A^*(M,{\\mathcal {F}})$ is closed under exterior differentiation.",
"More precisely, let $\\xi = \\omega \\wedge \\beta \\in A^k(M,{\\mathcal {F}}) $ for $k \\ge 1$ , then $d\\xi = d(\\omega \\wedge \\beta ) = d\\omega \\wedge \\beta - \\omega \\wedge d\\beta = (\\omega \\wedge \\eta ) \\wedge \\beta - \\omega \\wedge d\\beta = \\omega \\wedge (\\eta - d\\beta ) \\in A^{k+1}(M,{\\mathcal {F}}) .$ Thus, $A^*(M,{\\mathcal {F}})$ is a differential graded algebra.",
"Let $H^{*}(M,{\\mathcal {F}})$ denote the cohomology of the differential graded complex $\\lbrace A^*(M,{\\mathcal {F}}), d \\rbrace )$ .",
"For a closed form $\\xi \\in A^k(M,{\\mathcal {F}})$ , let $[\\xi ]_{{\\mathcal {F}}} \\in H^{k}(M,{\\mathcal {F}})$ denote its cohomology class.",
"The calculation (REF ) shows that differential on the complex $A^k(M,{\\mathcal {F}}) $ is “twisted” by the 1-form $\\eta $ .",
"Twisted cohomology also arises in the study of the dynamics of Anosov flows by Fried in [22], and it would be interesting to understand if (partial) results analogous to those in [22] can be obtained from the study of the cohomology spaces $H^*(M,{\\mathcal {F}})$ .",
"The inclusion of the ideal $ A^*(M,{\\mathcal {F}}) \\subset \\Omega ^*(M)$ induces a map on cohomology $H^{*}(M,{\\mathcal {F}}) \\rightarrow H^{*}(M)$ .",
"In general, the induced map need not be injective, and the calculation of the cohomology groups $\\displaystyle H^{*}(M,{\\mathcal {F}})$ is often an intractable problem [21].",
"However, it is the fact that $\\displaystyle H^{*}(M,{\\mathcal {F}})$ is the domain of the Godbillon operators which makes them important.",
"We next discuss the linear functionals defined on these spaces.",
"First, we make explicit a property implied by the discussions above.",
"Let $\\eta $ be any choice of a 1-form satisfying $d\\omega = \\omega \\wedge \\eta $ .",
"Let $\\xi \\in A^k(M,{\\mathcal {F}})$ with $k \\ge q$ so that $\\xi = \\omega \\wedge \\beta $ for some $(k-q)$ -form $\\beta \\in \\Omega ^{k-q}(M)$ .",
"Then by the calculation (REF ), the product $\\eta \\wedge \\xi = \\eta \\wedge \\omega \\wedge \\beta $ depends only on the leafwise restriction of the form $\\eta $ .",
"Thus, $\\eta \\wedge \\xi $ is independent of the choice of $\\eta $ which satisfies $d\\omega = \\omega \\wedge \\eta $ , and in particular, it equals $D^{\\vec{v}}\\omega \\wedge \\xi $ where $\\vec{v}$ is a vector field on $M$ such that $\\omega (\\vec{v}) =1$ .",
"Again, let $\\eta $ be any choice of a 1-form satisfying $d\\omega = \\omega \\wedge \\eta $ .",
"Recall from (REF ) that the closed 2-form $d\\eta $ is in the ideal generated by $\\omega $ , so $d\\eta \\in A^2(M,{\\mathcal {F}})$ .",
"Duminy observed in [18] (see also [10],[36]) that the class $[d\\eta ]_{{\\mathcal {F}}} \\in H^{2}(M,{\\mathcal {F}})$ is independent of the choice of the 1-form $\\eta $ , and so is an invariant of ${\\mathcal {F}}$ , which he called the Vey class of ${\\mathcal {F}}$ .",
"The 2-form $d\\eta $ has some properties analogous to those of a symplectic form on $M$ , especially in the geometric interpretation of the Godbillon-Vey invariant as “helical wobble” [52], [68], [72], but the analogy is very loose.",
"The geometric meaning of the class $[d\\eta ]_{{\\mathcal {F}}}$ remains obscure, although as noted below, $[d\\eta ]_{{\\mathcal {F}}} = 0$ implies that $GV({\\mathcal {F}}) = 0$ ."
],
[
"The Godbillon operator",
"Let a defining 1-form $\\omega $ be given, and a vector field on $M$ such that $\\omega (\\vec{v}) =1$ , and set $\\eta = D^{\\vec{v}}\\omega = \\iota (\\vec{v}) d\\omega $ .",
"Given a closed form $\\xi \\in A^{p}(M,{\\mathcal {F}})$ , the product $\\eta \\wedge \\xi \\in A^{p+1}(M,{\\mathcal {F}})$ is closed, as $d(\\eta \\wedge \\xi ) = d\\eta \\wedge \\xi = \\omega \\wedge \\eta \\wedge \\omega = 0$ .",
"Moreover, if $\\xi = d\\beta $ for some form $\\beta \\in A^{p-1}(M,{\\mathcal {F}})$ , then $\\eta \\wedge \\beta \\in A^{p}(M,{\\mathcal {F}})$ and $d(-\\eta \\wedge \\beta ) = -(d\\eta ) \\wedge \\beta + \\eta \\wedge d\\beta = \\eta \\wedge \\xi ~ .$ Thus, given $[\\xi ]_{{\\mathcal {F}}} \\in H^{p+1}(M, {\\mathcal {F}})$ we obtain a well-defined class $g([\\xi ]_{{\\mathcal {F}}}) = [\\eta \\wedge \\xi ]_{{\\mathcal {F}}} \\in H^{p+1}(M, {\\mathcal {F}})$ .",
"It follows that there is a well-defined composition $g \\colon H^{p}(M,{\\mathcal {F}}) \\rightarrow H^{p+1}(M,{\\mathcal {F}}) \\rightarrow H^{p+1}(M) ~ , ~~ g([\\zeta ]_{{\\mathcal {F}}}) = [\\eta \\wedge \\zeta ]$ which is called the Godbillon operator.",
"It was shown above that the 2-form $d\\eta $ is a multiple of $\\omega $ , and is clearly a closed form, so it defines a cohomology class $[d\\eta ]_{{\\mathcal {F}}} \\in H^{2}(M,{\\mathcal {F}})$ .",
"Then we have $g([d\\eta ]_{{\\mathcal {F}}}) = [\\eta \\wedge d\\eta ] = GV({\\mathcal {F}}) \\in H^3(M)$ .",
"That is, “Godbillon(Vey) = Godbillon-Vey”.",
"If $M$ is a closed 3-manifold with fundamental class $[M]$ , then evaluating $GV({\\mathcal {F}})$ on $[M]$ yields a real number, the real Godbillon-Vey invariant of ${\\mathcal {F}}$ : $ \\langle GV({\\mathcal {F}}), [M] \\rangle = \\int _M ~ \\eta \\wedge d\\eta ~ .$ If $M$ is an open 3-manifold, then $H^3(M) = 0$ so that $GV({\\mathcal {F}}) = 0$ in this case.",
"However, the class $GV({\\mathcal {F}})$ need not vanish in the case when $M$ is open and $M$ has dimension $m > 3$ .",
"In this case, it is necessary to introduce cohomology with compact supports, in order to obtain a real-valued invariants from the class $GV({\\mathcal {F}})$ .",
"Now let $\\Omega _c^*(M) \\subset \\Omega ^*(M)$ denote the differential subalgebra of forms with compact support.",
"The cohomology of this ideal is denoted by $H_c^*(M)$ which is called with the de Rham cohomology with compact supports of $M$ .",
"Let $A_c^*(M, {\\mathcal {F}}) \\subset \\Omega _c^*(M)$ denote the differential ideal consisting of forms in $A^*(M, {\\mathcal {F}})$ with compact support.",
"Its cohomology groups are denoted by $H_c^*(M, {\\mathcal {F}})$ , and these groups are called the foliated cohomology with compact supports.",
"Given a closed form $\\zeta \\in A^{p}(M, {\\mathcal {F}})$ , let $\\xi \\in \\Omega _c^{k}(M)$ be a closed form with compact support, then the product $\\zeta \\wedge \\xi \\in A^{k+p}(M,{\\mathcal {F}})$ is again closed with compact support.",
"If either form is the boundary of a form with compact supports, then $\\psi \\wedge \\xi $ is also the boundary of a compact form.",
"Thus, there is a well-defined pairing $ H^{p}(M,{\\mathcal {F}}) \\times H_c^k(M) \\rightarrow H_c^{k + p}(M, {\\mathcal {F}}) ~ .$ In particular, given a class $[\\xi ] \\in H_c^{m-3}(M)$ represented by a smooth closed form $\\xi \\in \\Omega _c^{m-3}(M)$ , then the pairing $[d\\eta ]_{{\\mathcal {F}}} \\cup [\\xi ] = [d\\eta \\wedge \\xi ]_{{\\mathcal {F}}} \\in H_c^{m-1}(M,{\\mathcal {F}})$ is well-defined.",
"Recall that the manifold $M$ is assumed to be oriented and connected, so by Poincaré duality the pairing $H^p(M) \\otimes H_c^{m-p}(M) \\rightarrow H_c^m(M) \\cong {\\mathbb {R}}$ is non-degenerate for $0 \\le p < m$ .",
"In particular, the value of the class $ [\\eta \\wedge d\\eta ] \\in H^3(M)$ is determined by its pairings with classes in $H_c^{m-3}(M)$ .",
"This is the idea behind the next concept, which is the basis for the Godbillon measure.",
"The Godbillon operator in (REF ) applied to a class in $H_c^{m-1}(M,{\\mathcal {F}})$ yields a closed $m$ -form with compact support on $M$ , which can be integrated over the fundamental class to obtain a real number.",
"This composition yields a linear functional denoted by $G \\colon H_c^{m-1}(M,{\\mathcal {F}})\\rightarrow {\\mathbb {R}}, \\;\\; ~~ \\;\\; G([\\zeta ]_{{\\mathcal {F}}}) = \\langle [\\eta \\wedge \\zeta ], [M] \\rangle = \\int _M \\eta \\wedge \\zeta ~ .$ Note that we use the notation “$g$ ” for the Godbillon operator between cohomology groups, and the notation “$G$ ” for the linear functional on the cohomology group $H_c^{m-1}(M,{\\mathcal {F}})$ .",
"With these preliminary preparations, we have the basic result: PROPOSITION 3.3 (Duminy, [18]) The value of the Godbillon-Vey class $GV({\\mathcal {F}}) \\in H^3(M)$ is determined by the Godbillon operator $G$ in (REF ).",
"In particular, if $G \\equiv 0$ then $GV({\\mathcal {F}}) = 0$ .",
"For the case when the dimension $m =3$ and $M$ is compact, this follows by applying the linear functional $G$ to the class $[d\\eta ]_{{\\mathcal {F}}} \\in H^{2}(M, {\\mathcal {F}}) = H_c^{2}(M, {\\mathcal {F}})$ .",
"For $m > 3$ , then by Poincaré duality, the value of $GV({\\mathcal {F}}) \\in H^3(M)$ is determined by pairing the 3-form $\\eta \\wedge d\\eta $ with closed forms $\\xi \\in \\Omega _c^{m-3}(M)$ , followed by integration, to obtain $\\langle GV({\\mathcal {F}}) \\cup [\\xi ], [M] \\rangle = \\int _M ~ (\\eta \\wedge d\\eta ) \\wedge \\xi .$ Note that $[d\\eta \\wedge \\xi ]_{{\\mathcal {F}}} \\in H_c^{m-1}(M,{\\mathcal {F}})$ , so that $\\langle GV({\\mathcal {F}}) \\cup [\\xi ], [M] \\rangle = G([d\\eta \\wedge \\xi ]_{{\\mathcal {F}}})$ .",
"The claim follows.",
"This elementary observation by Duminy implies that $GV({\\mathcal {F}}) = 0$ if $G$ is the trivial functional.",
"The strategy to proving that $GV({\\mathcal {F}}) = 0$ is thus, to obtain dynamical properties of a foliation which suffice to show that the linear functional $G$ vanishes."
],
[
"The Godbillon measure",
"The linear functional $G$ possesses special properties that were hinted at in the literature preceding Duminy's work (see the survey [44] for a fuller discussion of the ideas leading up to Duminy's work.)",
"In particular, Duminy showed that the integrand in (REF ) which defines the operator $G$ can be restricted to saturated Borel subsets, to obtain well-defined real invariants.",
"This observation was systematically generalized in the work [36], to show that $G$ extends to a generalized measure on the Lebesgue measurable saturated subsets of $M$ .",
"Moreover, the values of the measure can be calculated using measurable cocycle data, as discussed in [38].",
"The extension to measurable data allows the introduction of techniques of ergodic theory.",
"We discuss the definition of the Godbillon measure and its properties in more detail below.",
"A set $B \\subset M$ is ${\\mathcal {F}}$ –saturated if for all $x \\in B$ , the leaf $L_x$ through $x$ is contained in $B$ .",
"Let ${\\mathcal {B}}({\\mathcal {F}})$ denote the $\\Sigma $ -algebra of Lebesgue measurable ${\\mathcal {F}}$ –saturated subsets of $M$ .",
"THEOREM 3.4 [18], [36] For each $B \\in {\\mathcal {B}}({\\mathcal {F}})$ , there is a well-defined linear functional $G_{{\\mathcal {F}}}(B) \\colon H_c^{m-1}(M,{\\mathcal {F}}) \\rightarrow {\\mathbb {R}}\\quad , \\quad G_{{\\mathcal {F}}}(B)([\\zeta ]_{{\\mathcal {F}}}) = \\int _B \\eta \\wedge \\zeta $ where $\\zeta \\in A_c^{m-1}(M,{\\mathcal {F}})$ is closed.",
"Note that if $B$ has Lebesgue measure zero, then $G_{{\\mathcal {F}}}(B) =0$ .",
"Moreover, the correspondence $ B \\mapsto G_{{\\mathcal {F}}}(B) \\in {\\rm Hom_{cont}}(H_c^{m-1}(M,{\\mathcal {F}}), {\\mathbb {R}})$ is a countably additive measure on ${\\mathcal {B}}({\\mathcal {F}})$ , called the Godbillon measure.",
"Part of the claim of Theorem REF is that the linear functional (REF ) is independent of the choice of the smooth 1-form $\\omega $ defining ${\\mathcal {F}}$ .",
"Much more is true, as described below.",
"The key idea, introduced in [36], is to consider the space of leafwise forms on ${\\mathcal {F}}$ which are leafwise smooth, but need only be measurable as functions on $M$ .",
"Introduce the graded differential algebra $\\Omega ^*({\\mathcal {F}})$ consisting of leafwise forms.",
"That is, for $k \\ge 0$ , the space $\\Omega ^k({\\mathcal {F}})$ consists of sections of the dual to the $k$ -$th$ exterior power of the leafwise tangent bundle $T{\\mathcal {F}}$ .",
"That is, a form $\\xi \\in \\Omega ^k({\\mathcal {F}})$ is defined, for each $x \\in M$ , on a $k$ -tuple $(\\vec{v}_1, \\ldots , \\vec{v}_k)$ of vectors in the tangent space $T_x{\\mathcal {F}}$ to the leaf $L_x$ containing $x$ .",
"Moreover, we require that for any leaf $L$ of ${\\mathcal {F}}$ , the restriction $\\xi |L$ to $L$ is a smooth form.",
"There is a leafwise exterior differential $D_{{\\mathcal {F}}}\\colon \\Omega ^k({\\mathcal {F}}) \\rightarrow \\Omega ^{k+1}({\\mathcal {F}}) \\quad , \\quad D_{{\\mathcal {F}}}(\\xi ) = d(\\xi | L).$ For $\\xi \\in \\Omega ^k({\\mathcal {F}})$ , the definition of $D_{{\\mathcal {F}}}(\\xi ) \\in \\Omega ^{k+1}({\\mathcal {F}})$ is as follows.",
"For each leaf $L$ of ${\\mathcal {F}}$ , the restriction $\\xi | L$ is a smooth $k$ -form on $L$ , so there is a well-defined exterior differential $d(\\xi | L)$ .",
"The collection of leafwise forms $\\lbrace d(\\xi | L) \\in \\Omega ^{k+1}(L) \\mid L \\subset M\\rbrace $ defines the class $D_{{\\mathcal {F}}}(\\xi ) \\in \\Omega ^{k+1}({\\mathcal {F}})$ .",
"The cohomology groups of the graded differential algebra $\\lbrace \\Omega ^*({\\mathcal {F}}), D_{{\\mathcal {F}}}\\rbrace $ are called the foliated cohomology of ${\\mathcal {F}}$ .",
"A key observation in the definition of the exterior differential in (REF ) is that it does not require any regularity for the transverse behavior of the leafwise forms.",
"Thus, one can consider the subcomplex $\\Omega _{\\infty }^*({\\mathcal {F}}) \\subset \\Omega ^*({\\mathcal {F}})$ of smooth leafwise forms and the corresponding space $H_{\\infty }^*({\\mathcal {F}})$ of smooth foliated cohomology, which was used by Heitsch in [35] to study the deformation theory of foliations.",
"If the forms are assumed to be continuous, we obtain the subcomplex $\\Omega _{c}^*({\\mathcal {F}}) \\subset \\Omega ^*({\\mathcal {F}})$ whose cohomology spaces $H_{c}^*({\\mathcal {F}})$ were studied by El Kacimi-Alaoui in [21].",
"One can also consider the subcomplex $\\Omega _{m}^*({\\mathcal {F}}) \\subset \\Omega ^*({\\mathcal {F}})$ of measurable (or bounded measurable) sections of the dual to the $k$ -$th$ exterior power of the leafwise tangent bundle $T{\\mathcal {F}}$ , then we obtain the measurable cohomology leafwise cohomology $H_{m}^*({\\mathcal {F}})$ groups used by Zimmer in [79], [80] to study the rigidity theory for measurable group actions.",
"The next result we require is formulated using the complex subcomplex $\\Omega _{m}^*({\\mathcal {F}})$ .",
"A function $f \\colon M \\rightarrow {\\mathbb {R}}$ is said to be transversally measurable if it is a measurable function, and for each leaf $L$ of ${\\mathcal {F}}$ , the restriction $f | L$ is smooth and the leafwise derivatives of $f$ are measurable functions as well.",
"Such a function $f$ is the typical element in $\\Omega _m^0({\\mathcal {F}})$ .",
"Given $f \\in \\Omega _m^0({\\mathcal {F}})$ and a form $\\xi \\in \\Omega _{c}^k({\\mathcal {F}})$ , then the product $f \\cdot \\xi \\in \\Omega _m^k({\\mathcal {F}})$ .",
"We next introduce norms on the spaces $\\Omega _m^k({\\mathcal {F}})$ .",
"For each $x \\in M$ , the Riemannian metric on $T_x M$ defines a norm on $T_x M$ , which restricts to a norm on the leafwise tangent space $T_x{\\mathcal {F}}$ .",
"The norm on the space $T_x{\\mathcal {F}}$ induces a dual norm on the cotangent bundle $T_x^* {\\mathcal {F}}$ , and also induces norms on each exterior vector space $\\Lambda ^k T_x {\\mathcal {F}}$ and on its dual $\\Omega ^k(T_x {\\mathcal {F}})$ , for all $k > 1$ .",
"We denote this norm by $\\Vert \\cdot \\Vert _x$ in each of these cases.",
"For a function $f \\in \\Omega _x({\\mathcal {F}})$ , let $\\Vert f\\Vert _x = |f(x)|$ .",
"Given a subset $B \\subset M$ , and a leafwise form $\\xi \\in \\Omega ^k({\\mathcal {F}})$ for $k \\ge 0$ , define the sup-norm over $B$ by $\\Vert \\xi \\Vert _{B} = \\sup _{x \\in B} ~ \\Vert \\xi _x\\Vert ~.$ A remarkable property of the Godbillon measure $G_{{\\mathcal {F}}}$ , as shown in Theorem 2.7 of [36], is that for $B \\in {\\mathcal {B}}$ the value of $G_{{\\mathcal {F}}}(B)$ can be calculated using a 1-form $\\omega _f = \\exp (f) \\cdot \\omega $ , where we require that $f \\in \\Omega _m({\\mathcal {F}})$ , and for $\\vec{v}$ with $\\omega _f (\\vec{v}) = 1$ and $\\eta _f = D^{\\vec{v}}(\\omega _f)$ , we have $\\Vert \\eta _f \\Vert _B < \\infty $ .",
"Then [36] shows that given a closed form with compact support $\\zeta \\in A_c^{m-1}(M, {\\mathcal {F}})$ , $G_{{\\mathcal {F}}}(B)([\\zeta ]_{{\\mathcal {F}}}) ~ = ~ \\int _B ~ \\eta _f \\wedge \\zeta ~ .$ We now recall a fundamental result, Theorem 4.3 of [38], which is a broad generalization of the ideas in the seminal work by Herman [37]: PROPOSITION 3.5 Let $B \\in {\\mathcal {B}}({\\mathcal {F}})$ .",
"Suppose there exists a sequence of transversally measurable functions $\\lbrace f_n \\mid n =1,2, \\ldots \\rbrace $ on $M$ so that the 1-forms $\\lbrace \\omega _n = \\exp (f_n) \\cdot \\omega \\mid n = 1, 2, \\ldots \\rbrace $ on $M$ satisfy $\\Vert D^{\\vec{v}_n}(\\omega _n)\\Vert _B < 1/n$ where $\\omega _n(\\vec{v}_n) =1$ .",
"Then $G_{{\\mathcal {F}}}(B) = 0$ .",
"For each $n \\ge 1$ , let $\\vec{v}_n$ be a vector field on $M$ such that $\\omega _n(\\vec{v}_n) =1$ , and set $\\eta _n = D^{\\vec{v}_n}(\\omega _n)$ .",
"Then for $[\\zeta ]_{{\\mathcal {F}}} \\in H_c^{m-1}(M,{\\mathcal {F}}) $ and each $n \\ge 1$ , we have $G_{{\\mathcal {F}}}(B)([\\zeta ]_{{\\mathcal {F}}}) = \\int _B \\; \\eta _n \\wedge \\zeta ~ .$ Estimate the norms of the integrals in (REF ): $\\left| G_{{\\mathcal {F}}}(B)([\\zeta ]_{{\\mathcal {F}}}) \\right| & = & \\lim _{n \\rightarrow \\infty } ~ \\left| \\int _B \\; \\eta _n \\wedge \\zeta \\right| \\\\& \\le & \\lim _{n \\rightarrow \\infty } ~ \\int _B \\; \\Vert \\eta _n \\Vert _B \\; \\Vert \\zeta \\Vert _B \\; dvol \\\\& \\le & \\lim _{n \\rightarrow \\infty } ~ (1/n) \\cdot \\int _B \\; \\Vert \\zeta \\Vert _B \\; dvol \\\\& = & 0 ~ .$ As this holds for all $[\\zeta ]_{{\\mathcal {F}}} \\in H_c^{m-1}(M,{\\mathcal {F}}) $ , the claim follows.",
"We note two important aspects of the proof of Proposition REF .",
"First, the $n$ -form $\\eta _n \\wedge \\zeta $ in the integrand of (REF ) depends only on the restrictions $\\eta _n | L$ for leaves $L$ of ${\\mathcal {F}}$ .",
"Thus, the pairing $\\eta _n \\wedge \\zeta $ is well-defined when ${\\mathcal {F}}$ is a $C^{\\infty ,1}$ -foliation.",
"Also, the convergence of the integral in (REF ) as $n \\rightarrow \\infty $ uses the Lebesgue dominated convergence theorem, and can be applied assuming only that the form $\\zeta \\in A_c^{m-1}(M, {\\mathcal {F}})$ is continuous.",
"In particular, for a $C^{\\infty ,2}$ -foliation the form $d\\eta $ is continuous, so the calculation above applies to multiples of this form as required for the proof of Proposition REF .",
"Proposition REF gives an effective method for showing that the Godbillon-Vey class vanishes on a set $B \\in {\\mathcal {B}}({\\mathcal {F}})$ , provided that one can construct a sequence of 1-forms $\\lbrace \\omega _n = \\exp (f_n) \\cdot \\omega \\mid n = 1, 2, \\ldots \\rbrace $ on $M$ satisfying the hypotheses of the proposition.",
"In hindsight, one can see that an analogous estimate was used in the previous works [20], [37], [56], [57], [74], [78] to show that $GV({\\mathcal {F}}) = 0$ for $C^2$ -foliations of codimension one, for foliations with various types of dynamical properties.",
"For a $C^2$ -foliation ${\\mathcal {F}}$ , Sacksteder's Theorem [70] implies that if ${\\mathcal {F}}$ has no resilient leaf, then there are no exceptional minimal sets for ${\\mathcal {F}}$ .",
"Hence, by the Poincaré-Bendixson theory, all leaves of ${\\mathcal {F}}$ either lie at finite level, or lie in “arbitrarily thin” open subsets $U \\in {\\mathcal {B}}({\\mathcal {F}})$ .",
"In his works [18], [19], Duminy used a result analogous to Proposition REF to show that $G_{{\\mathcal {F}}}(B) = 0$ , where $B$ is a union of leaves at finite level.",
"Thus, for a $C^2$ -foliation with no resilient leaves, the Godbillon measure vanishes on the union of the leaves of finite level, and also vanishes on any Borel set in their complement.",
"Thus, $GV({\\mathcal {F}}) = 0$ for a $C^2$ -foliation of codimension-one with no resilient leaves.",
"See [9], [12] for a published discussion of this proof.",
"In the next two sections, we follow a different, more direct approach to obtain this conclusion.",
"From the assumption $G_{{\\mathcal {F}}} \\ne 0$ , we conclude that the holonomy pseudogroup of a $C^{\\infty ,1}$ -foliation ${\\mathcal {F}}$ must contain resilient orbits.",
"Thus for a $C^2$ -foliation ${\\mathcal {F}}$ with $GV({\\mathcal {F}}) \\ne 0$ , we have that $G_{{\\mathcal {F}}} \\ne 0$ and hence ${\\mathcal {F}}$ must contain resilient leaves."
],
[
"Asymptotically expansive holonomy",
"In this section, we study the dynamical properties of $C^1$ -pseudogroups acting on a 1-dimensional space.",
"The main example is when there is given a codimension-one foliation ${\\mathcal {F}}$ on a compact manifold $M$ , with a regular $C^{\\infty ,1}$ -foliation atlas with associated transversal ${\\mathcal {X}}$ , and ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is its holonomy pseudogroup.",
"Then ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is generated by a finite collection of local $C^1$ -diffeomorphisms defined on open subsets of ${\\mathcal {X}}$ .",
"Recall that the charts in the foliation atlas are assumed to be transversally oriented, so for each plaque chain ${\\mathcal {P}}$ , the derivative ${\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) > 0$ for all $x \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ in its domain."
],
[
"The transverse expansion exponent function",
"We first introduce the notion of asymptotically expansive holonomy for a leaf of ${\\mathcal {F}}$ , and the associated set ${\\rm E}^+({\\mathcal {F}})$ of leaves with this property.",
"The main result of this section is that the Godbillon measure $G_{{\\mathcal {F}}}$ is supported on ${\\rm E}^+({\\mathcal {F}})$ .",
"That is, for any $B \\in {\\mathcal {B}}({\\mathcal {F}})$ , we have $G_{{\\mathcal {F}}}(B) = G_{{\\mathcal {F}}}(B \\cap {\\rm E}^+({\\mathcal {F}}))$ .",
"Hence, $G_{{\\mathcal {F}}} \\ne 0$ implies the set ${\\rm E}^+({\\mathcal {F}})$ must have positive Lebesgue measure by Theorem REF .",
"For all $x \\in {\\mathcal {X}}$ , set $\\mu _0(x) = 1$ , and and each integer $n \\ge 1$ , define the maximal n-expansion $ \\mu _n(x) = \\sup \\, \\lbrace \\, {\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) \\mid x \\in {\\mathcal {D}}_{{\\mathcal {P}}} ~ \\& ~ \\Vert {\\mathcal {P}}\\Vert \\le n\\rbrace ~ .$ The function $x \\mapsto \\mu _n(x)$ is the maximum of a finite set of continuous functions, so is a Borel function on ${\\mathcal {X}}$ , and $\\mu _n(x) \\ge 1$ as the identity transformation is the holonomy for a plaque chain of length 1.",
"LEMMA 4.1 Let $x \\in {\\mathcal {X}}$ , and let ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ be a plaque chain of length 1.",
"For the holonomy map ${\\bf h}_{{\\mathcal {Q}}}$ of this length-one plaque-chain, we have ${\\bf h}_{{\\mathcal {Q}}}(x) = y$ .",
"Then for all $n > 0$ , $ \\mu _{n-1}(x) \\le \\mu _n(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) \\le \\mu _{n+1}(x) ~ .$ Let ${\\mathcal {P}}$ be a plaque chain at $y$ with $\\Vert {\\mathcal {P}}\\Vert \\le n$ , then ${\\mathcal {P}}\\circ {\\mathcal {Q}}$ is a plaque chain at $x$ with $\\Vert {\\mathcal {P}}\\circ {\\mathcal {Q}}\\Vert \\le n+1$ , so ${\\bf h}_{{\\mathcal {P}}}^{\\prime }(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) = {\\bf h}_{{\\mathcal {P}}\\circ {\\mathcal {Q}}}^{\\prime }(x) \\le \\mu _{n+1}(x) ~ .$ As this is true for all plaque chains at $y$ with $\\Vert {\\mathcal {P}}\\Vert \\le n$ , we obtain $\\mu _n(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) \\le \\mu _{n+1}(x)$ .",
"Given a plaque chain ${\\mathcal {P}}$ at $x$ with $\\Vert {\\mathcal {P}}\\Vert \\le n-1$ , the chain ${\\mathcal {R}}= {\\mathcal {P}}\\circ {\\mathcal {Q}}^{-1}$ at $y$ has $\\Vert {\\mathcal {R}}\\Vert \\le n$ and ${\\bf h}_{{\\mathcal {P}}}^{\\prime }(x) = {\\bf h}_{{\\mathcal {R}}}^{\\prime }(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) \\le \\mu _n(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) ~ .$ As (REF ) holds for all plaque chains at $x$ with $\\Vert {\\mathcal {P}}\\Vert \\le n-1$ , we have $\\mu _{n-1}(x) \\le \\mu _n(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x)$ .",
"Define $\\lambda _n(x) =\\ln \\, (\\mu _n(x))$ , so that $\\lambda _n(x) = \\sup \\, \\lbrace \\ln ({\\bf h}^{\\prime }_{{\\mathcal {P}}}(x)) \\mid x \\in {\\mathcal {D}}_{{\\mathcal {P}}} ~ \\& ~ \\Vert {\\mathcal {P}}\\Vert \\le n\\rbrace $ .",
"Then the transverse expansion exponent at $x \\in {\\mathcal {X}}$ is defined by $ \\lambda _*(x) = \\limsup _{n \\rightarrow \\infty } \\; \\frac{ \\lambda _n(x) }{n} ~ .$ LEMMA 4.2 The transverse expansion exponent function $\\lambda _*$ is Borel measurable on ${\\mathcal {X}}$ , and constant on the orbits of ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ .",
"For each $n \\ge 1$ , the function $\\displaystyle \\lambda _n(x)/n$ is Borel, so the supremum function in (REF ) is also Borel.",
"Let $x \\in {\\mathcal {X}}$ , and let ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ be a plaque chain, then the estimate (REF ) implies that, $\\frac{ \\ln (\\mu _{n+1}(x)) }{n+1} \\ge \\frac{ \\ln ( \\mu _n(y) \\cdot {\\bf h}^{\\prime }_{\\beta \\alpha }(x)) }{n} \\cdot \\frac{n}{n+1} = \\left\\lbrace \\frac{ \\ln ( \\mu _n(y)) }{n} + \\frac{ \\ln ( {\\bf h}^{\\prime }_{\\beta \\alpha }(x)) }{n}\\right\\rbrace \\cdot \\frac{n}{n+1}$ so that $\\lambda _*(x) = \\limsup _{n \\rightarrow \\infty } \\; \\left\\lbrace \\frac{ \\ln (\\mu _{n+1}(x)) }{n+1} \\right\\rbrace \\ge \\limsup _{n \\rightarrow \\infty } \\left\\lbrace \\frac{ \\ln ( \\mu _n(y)) }{n}\\right\\rbrace = \\lambda _*(y) ~ .$ The converse inequality follows similarly.",
"Thus, $\\lambda _*(x) = \\lambda _*(y)$ if there is a plaque chain ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ .",
"The pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is generated by the holonomy defined by plaque chains of length 1, so that for each point $y \\in {{\\mathcal {G}}}_{{\\mathcal {F}}}(x)$ , there is a finite plaque chain $\\displaystyle {\\mathcal {P}}= \\lbrace {\\mathcal {P}}_{\\alpha _0}(x_0), \\ldots , {\\mathcal {P}}_{\\alpha _k}(x_k) \\rbrace $ with $x_0 = x$ and $x_k = y$ .",
"Then $\\displaystyle \\lambda _*(x_{\\ell }) = \\lambda _*(x_{\\ell + 1})$ for each $0 \\le \\ell < k$ , from which it follows that $\\lambda _*(x) = \\lambda _*(y)$ ."
],
[
"The expansion decomposition",
"We use the conclusion of Lemma REF to lift the transverse expansion exponent function $\\lambda _*$ from ${\\mathcal {X}}$ to $M$ .",
"For $\\xi \\in M$ , let $L_{\\xi }$ be the leaf containing $\\xi $ and let $x = \\pi _{\\alpha }(\\xi )$ where $\\xi \\in U_{\\alpha }$ .",
"Then by Lemma REF , the value $\\lambda _*(x)$ is independent of the choice of open set with $\\xi \\in U_{\\alpha }$ .",
"By abuse of notation, we set $\\lambda _*(\\xi ) = \\lambda _*(x)$ , which is a well-defined function on $M$ .",
"Moreover, given a leaf $L$ , set $\\displaystyle \\lambda _*(L) = \\lambda _*(\\xi )$ for some $\\xi \\in L$ , which is then well-defined as well.",
"DEFINITION 4.3 Define the ${\\mathcal {F}}$ -saturated Borel subsets of $M$ : ${\\rm E}^+({\\mathcal {F}})~ & = & ~ \\lbrace x \\in M \\mid \\lambda _*(x) > 0\\rbrace \\\\{\\rm E}^+_a({\\mathcal {F}})~ & = & ~ \\lbrace x \\in M \\mid \\lambda _*(x) > a\\rbrace , ~ {\\rm for} ~ a \\ge 0\\\\{\\rm S}({\\mathcal {F}})~ & = & ~ M - {\\rm E}^+({\\mathcal {F}})~ .$ A point $x \\in {\\rm E}^+({\\mathcal {F}})$ is said to be infinitesimally expansive.",
"The set ${\\rm E}^+({\\mathcal {F}})$ is called the hyperbolic set for ${\\mathcal {F}}$ , and is the analog for codimension-one foliations of the hyperbolic set for diffeomorphisms in Pesin theory [2], [61].",
"The set ${\\rm S}({\\mathcal {F}})$ consists of the leaves of ${\\mathcal {F}}$ for which the transverse infinitesimal holonomy has “slow growth”.",
"Both sets ${\\rm E}^+({\\mathcal {F}})$ and ${\\rm S}({\\mathcal {F}})$ are fundamental for the study of the dynamics of the foliation ${\\mathcal {F}}$ .",
"Note that if there is an holonomy map ${\\bf h}_{{\\mathcal {P}}}$ with $ x \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ , $ {\\bf h}_{{\\mathcal {P}}}(x) = x$ and $ {\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) = \\lambda > 1$ , then $x \\in {\\rm E}^+({\\mathcal {F}})$ .",
"If ${\\mathcal {P}}$ is a plaque-chain of length $k$ , then $x \\in {\\rm E}^+_a({\\mathcal {F}})$ for any $0 < a < \\ln (\\lambda )/k$ .",
"The plaque chain ${\\mathcal {P}}$ determines a closed loop $\\gamma _{{\\mathcal {P}}}$ based at $x$ in the leaf $L_x$ , and the transverse holonomy along $\\gamma _{{\\mathcal {P}}}$ is linearly expanding in some open neighborhood of $x$ .",
"Such transversally hyperbolic elements of the leaf holonomy have a fundamental role in the study of foliation dynamics, in particular in the works by Sacksteder [70], by Bonatti, Langevin and Moussu [4], and the works [40], [42].",
"However, given $x \\in {\\rm E}^+({\\mathcal {F}})$ there may not be a closed leafwise loop with infinitesimally expansive holonomy at $x$ .",
"What is always true is that there is a sequence of holonomy elements whose length tends to infinity, what has infinitesimally expansive holonomy at $x$ .",
"We make this statement precise.",
"Consider a point $x \\in {\\mathcal {X}}\\cap {\\rm E}^+_a({\\mathcal {F}})$ for $a > 0$ , and choose $\\lambda $ with $a < \\lambda < \\lambda _*(x)$ .",
"Then for all $N > 0$ , there exists $n \\ge N$ such that $ \\lambda _n(x) \\ge n \\lambda $ .",
"By the definition of $\\lambda _n(x)$ , this means there exists a plaque chain $ {\\mathcal {P}}$ with length $ \\Vert {\\mathcal {P}}\\Vert \\le n$ starting at $x$ such that ${\\bf h}^{\\prime }_{{\\mathcal {P}}}(x) \\ge \\exp \\lbrace n \\lambda \\rbrace $ .",
"By the continuity of the derivative function on ${\\mathcal {X}}$ , there exists ${\\epsilon }_n > 0$ such that on the open interval $(x -{\\epsilon }_n, x+{\\epsilon }_n) \\subset {\\mathcal {X}}$ , $ {\\bf h}^{\\prime }_{{\\mathcal {P}}}(y) \\ge \\exp \\lbrace n \\lambda /2\\rbrace ~ {\\rm for ~ all} ~ x -{\\epsilon }_n \\le y \\le x+{\\epsilon }_n ~ .$ By the Mean Value Theorem, ${\\bf h}^{\\prime }_{{\\mathcal {P}}}$ is expanding on the interval $(x -{\\epsilon }_n, x+{\\epsilon }_n)$ by a factor at least $\\exp \\lbrace n \\lambda /2\\rbrace $ .",
"Thus, the assumption $\\lambda _*(x) > \\lambda > 0$ and the definition in (REF ) implies that we can choose a sequence of plaque chains $ {\\mathcal {P}}_{\\ell }$ with lengths $ \\Vert {\\mathcal {P}}_{\\ell }\\Vert = n_{\\ell }$ starting at $x$ such that $n_{\\ell }$ is strictly increasing, and so tends to infinity, and the corresponding holonomy maps satisfy ${\\bf h}^{\\prime }_{{\\mathcal {P}}_{\\ell }}(y) \\ge \\exp \\lbrace n_{\\ell } \\lambda /2\\rbrace ~ {\\rm for ~ all} ~ x -{\\epsilon }_{n_{\\ell }} \\le y \\le x+{\\epsilon }_{n_{\\ell }} ~ .$ The constant ${\\epsilon }_{n_{\\ell }} > 0$ in (REF ) depends upon $\\ell $ , $\\lambda $ and $x$ , and is exponentially decreasing as $\\ell \\rightarrow \\infty $ .",
"It is a strong condition to have a sequence of holonomy maps as in (REF ) for elements of the holonomy pseudogroup at points $x$ , whose plaque lengths tend to infinity.",
"This is what gives the set ${\\rm E}^+({\\mathcal {F}})$ a fundamental role in the study of foliation dynamics, exactly in analog with the role of the Pesin set in smooth dynamics [2], [51], [61], [69].",
"The works [46], [47] give further study of the relation between the hyperbolic set ${\\rm E}^+({\\mathcal {F}})$ and the dynamics of the foliation.",
"In contrast, for the slow set ${\\rm S}({\\mathcal {F}})$ , the dynamics of ${\\mathcal {F}}$ on ${\\rm S}({\\mathcal {F}})$ has “less complexity”, as discussed in [47].",
"We next show that $G_{{\\mathcal {F}}}({\\rm S}({\\mathcal {F}})) =0$ , which is a measure of this lack of dynamical complexity.",
"Note that for an arbitrary saturated Borel set $B \\in {\\mathcal {B}}({\\mathcal {F}})$ , we have $G_{{\\mathcal {F}}}(B) = G_{{\\mathcal {F}}}(B \\cap {\\rm E}^+({\\mathcal {F}})) + G_{{\\mathcal {F}}}(B \\cap {\\rm S}({\\mathcal {F}}))$ so that $G_{{\\mathcal {F}}} \\ne 0$ and $G_{{\\mathcal {F}}}({\\rm S}({\\mathcal {F}})) =0$ implies the set ${\\rm E}^+({\\mathcal {F}})$ must have positive Lebesgue measure."
],
[
"A vanishing criterion",
"We use the criteria of Proposition REF to show that $G_{{\\mathcal {F}}}({\\rm S}({\\mathcal {F}})) =0$ .",
"That is, we construct a sequence of transversally measurable, non-vanishing transverse 1-forms $\\lbrace \\omega _n \\mid n = 1, 2, \\ldots \\rbrace $ on $M$ for which $\\Vert D^{\\vec{v_n}}\\omega _n\\Vert _{{\\rm S}({\\mathcal {F}})} < 1/n$ .",
"The construction of the forms $\\lbrace \\omega _n \\rbrace $ follows the method introduced in [38].",
"The first, and crucial step, is to construct an ${\\epsilon }$ –tempered cocycle (as given by (REF )) over the pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ which is cohomologous to the additive derivative cocycle, using a procedure adapted from [39].",
"This tempered cocycle is then used to produce the sequence of defining 1-forms $\\omega _n$ , using the methods of [7] and [48], [50].",
"These are the used in the proof of the following result.",
"THEOREM 4.4 For any set $B \\in {\\mathcal {B}}({\\mathcal {F}})$ , the Godbillon measure $G_{{\\mathcal {F}}}(B) = G_{{\\mathcal {F}}}(B \\cap {\\rm E}^+({\\mathcal {F}}))$ .",
"Hence, if ${\\rm E}^+({\\mathcal {F}})$ has Lebesgue measure zero, then $G_{{\\mathcal {F}}}(B) = 0$ for all $B \\in {\\mathcal {B}}({\\mathcal {F}})$ .",
"By the above remarks, it suffices to show that $G_{{\\mathcal {F}}}({\\rm S}({\\mathcal {F}})) =0$ .",
"Fix ${\\epsilon }> 0$ .",
"For $x \\in {\\mathcal {X}}\\cap {\\rm S}({\\mathcal {F}})$ , by the definition of $\\lambda _*(x) = 0$ (REF ), there exists $N_{{\\epsilon },x}$ such that $n \\ge N_{{\\epsilon },x}$ implies $ \\ln \\lbrace \\mu _n(x)\\rbrace \\le n {\\epsilon }/2$ , and hence the maximal $n$ -expansion $\\mu _n(x) \\le \\exp \\lbrace n {\\epsilon }/2\\rbrace $ .",
"For $x \\in {\\mathcal {X}}$ but $x \\notin {\\rm S}({\\mathcal {F}})$ , set $g_{{\\epsilon }}(x) = 1$ .",
"For $x \\in {\\mathcal {X}}\\cap {\\rm S}({\\mathcal {F}})$ , set $g_{{\\epsilon }}(x) = \\sum _{n=0}^{\\infty }~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _n(x) .$ For $x$ in the slow set ${\\rm S}({\\mathcal {F}})$ , the sum in (REF ) converges as the function $\\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _n(x)$ decays exponentially fast as $n \\rightarrow \\infty $ .",
"Note that while $g_{{\\epsilon }}(x) $ is finite for each $x \\in {\\mathcal {X}}$ , there need not be an upper bound for its values on ${\\mathcal {X}}\\cap {\\rm S}({\\mathcal {F}})$ .",
"Also, $g_{{\\epsilon }}$ is a Borel measurable function defined on all of ${\\mathcal {X}}$ .",
"The definition of the function $g_{{\\epsilon }}$ in (REF ) is analogous to the definition of the Lyapunov metric in Pesin theory.",
"Its role is to give a “change of gauge” with respect to which the expansion rates of the dynamical system is “normalized” for the action of ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ on ${\\mathcal {X}}$ , as made precise by Lemma REF below.",
"Let $\\vec{X} = {\\partial \\over {\\partial x}}$ be the unit-length, positively-oriented vector field on ${\\mathbb {R}}$ , let $dx$ denote the dual 1-form on ${\\mathbb {R}}$ .",
"Recall that for each $1 \\le \\alpha \\le \\nu $ , we defined ${\\mathcal {X}}_{\\alpha } \\equiv (-1,1)$ so there is an inclusion $\\iota _{\\alpha } \\colon {\\mathcal {X}}_{\\alpha } \\subset {\\mathbb {R}}$ which defines a coordinate function $x_{\\alpha } \\colon {\\mathcal {X}}_{\\alpha } \\rightarrow {\\mathbb {R}}$ .",
"Then let $dx_{\\alpha } = \\iota _{\\alpha }^*(dx)$ denote the induced 1-form on ${\\mathcal {X}}_{\\alpha }$ .",
"There is a corresponding unit vector field $\\vec{X}_{\\alpha }$ on ${\\mathcal {X}}_{\\alpha }$ , which defines the vector field $\\vec{X}$ on ${\\mathcal {X}}$ .",
"Then $dx_{\\alpha }(\\vec{X}) = 1$ on each ${\\mathcal {X}}_{\\alpha }$ .",
"For each $1 \\le \\alpha \\le \\nu $ , introduce the notation $g_{{\\epsilon }}^{\\alpha } = g_{{\\epsilon }} | {\\mathcal {X}}_{\\alpha }$ , and define the 1-form $dx_{\\alpha }^{{\\epsilon }} = g_{{\\epsilon }}^{\\alpha } \\; dx_{\\alpha }$ on ${\\mathcal {X}}_{\\alpha }$ .",
"Let $x \\in {\\mathcal {X}}$ , and let ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ be a plaque chain of length 1.",
"Then for the holonomy map ${\\bf h}_{{\\mathcal {Q}}}$ of this length-one plaque-chain, we have ${\\bf h}_{{\\mathcal {Q}}}(x) = y$ , and ${\\bf h}_{{\\mathcal {Q}}}^*(dx_{\\beta }) = {\\bf h}_{{\\mathcal {Q}}}^{\\prime } \\cdot dx_{\\alpha }$ .",
"Thus we have ${\\bf h}_{{\\mathcal {Q}}}^*(dx_{\\beta }^{{\\epsilon }}) = \\left( g_{{\\epsilon }}^{\\beta } \\circ {\\bf h}_{{\\mathcal {Q}}}\\right) \\cdot {\\bf h}_{{\\mathcal {Q}}}^{\\prime } \\cdot dx_{\\alpha }.$ The following result gives a key property of the function $g_{{\\epsilon }}$ which describes its behavior under a change of coordinates, for charts such that ${\\mathcal {P}}_{\\alpha }(x) \\cap {\\mathcal {P}}_{\\beta }(y) \\ne \\emptyset $ .",
"Recall that ${\\rm S}({\\mathcal {F}})= M - {\\rm E}^+({\\mathcal {F}})$ denotes the slow set, and let $S_{\\alpha } = \\pi _{\\alpha }({\\rm S}({\\mathcal {F}})\\cap U_{\\alpha }) \\subset {\\mathcal {X}}_{\\alpha }$ .",
"LEMMA 4.5 For $x \\in S_{\\alpha }$ and ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ , $\\exp \\lbrace -{\\epsilon }\\rbrace \\cdot g_{{\\epsilon }}^{\\alpha }(x) \\le g_{{\\epsilon }}^{\\beta }(y) \\cdot {\\bf h}^{\\prime }_{{\\mathcal {Q}}}(x) \\le \\exp \\lbrace {\\epsilon }\\rbrace \\cdot g_{{\\epsilon }}^{\\alpha }(x) ~ .$ Evaluate the expression (REF ) on the vector field $\\vec{X}$ and use the estimate (REF ), noting that ${\\bf h}_{{\\mathcal {Q}}}(x) = y$ , to obtain, $g_{{\\epsilon }}^{\\beta }(y) \\cdot {\\bf h}_{{\\mathcal {Q}}}^{\\prime }(x) & = & \\left\\lbrace \\sum _{n=0}^{\\infty } ~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _n(y)\\right\\rbrace \\; {\\bf h}_{{\\mathcal {Q}}}^{\\prime }(x) \\nonumber \\\\& \\le & \\sum _{n=0}^{\\infty } ~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _{n+1}(x) \\nonumber \\\\& < & \\exp \\lbrace {\\epsilon }\\rbrace \\cdot \\left\\lbrace \\sum _{n=1}^{\\infty } ~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _n(x) + \\mu _0(x) \\right\\rbrace \\nonumber \\\\& = & \\exp \\lbrace {\\epsilon }\\rbrace \\cdot g_{{\\epsilon }}^{\\alpha }(x) ~ .",
"$ Similarly, we have $g_{{\\epsilon }}^{\\beta }(y) \\cdot {\\bf h}_{{\\mathcal {Q}}}^{\\prime }(x) & = & \\left\\lbrace \\sum _{n=0}^{\\infty } ~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _n(y)\\right\\rbrace \\; {\\bf h}_{{\\mathcal {Q}}}^{\\prime }(x) \\nonumber \\\\& \\ge & \\sum _{n=1}^{\\infty } ~ \\exp \\lbrace - n {\\epsilon }\\rbrace \\cdot \\mu _{n-1}(x) + \\mu _0(x) \\cdot {\\bf h}_{{\\mathcal {Q}}}^{\\prime }(x) \\nonumber \\\\& \\ge & \\exp \\lbrace -{\\epsilon }\\rbrace \\cdot ~ g_{{\\epsilon }}^{\\alpha }(x) ~ .",
"$ This completes the proof of Lemma REF .",
"We next use the coordinate 1-form $dx_{\\alpha }^{{\\epsilon }}$ to define a transversally measurable 1-form $\\omega _{{\\epsilon }}$ on $M$ which defines ${\\mathcal {F}}$ .",
"The first step is to define local 1–forms $\\omega ^{\\alpha }_{{\\epsilon }}$ on the coordinate charts $U_{\\alpha }$ , then use a partition of unity to obtain the 1-form $\\omega _{{\\epsilon }}$ defined on all of $M$ .",
"Then, for appropriate choices of ${\\epsilon }$ tending to 0, we obtain 1-forms $\\lbrace \\omega _n \\rbrace $ satisfying the hypotheses of Proposition REF on ${\\rm S}({\\mathcal {F}})$ .",
"For each $1 \\le \\alpha \\le \\nu $ , use the projection $\\pi _{\\alpha } \\colon U_{\\alpha } \\rightarrow {\\mathcal {X}}_{\\alpha }$ along plaques to pull-back the form $dx_{\\alpha }$ to the closed 1-form $\\omega _{\\alpha } = \\pi _{\\alpha }^*(dx_{\\alpha })$ on $U_{\\alpha }$ .",
"Then define $\\displaystyle \\omega ^{\\alpha }_{{\\epsilon }}= \\pi _{\\alpha }^*(dx_{\\alpha }^{{\\epsilon }}) = \\left( g_{{\\epsilon }}^{\\alpha } \\circ \\pi _{\\alpha }\\right) \\cdot \\omega _{\\alpha }$ which is a transversally measurable, leafwise closed 1-form on $U_{\\alpha }$ .",
"Choose a partition of unity $\\lbrace \\rho _{\\alpha } \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ subordinate to the cover $\\lbrace U_{\\alpha } \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ of $M$ by foliation charts.",
"Then for each $1 \\le \\beta \\le \\nu $ , the 1-form $\\rho _{\\beta } \\cdot \\omega ^{\\beta }_{{\\epsilon }}$ has support contained in $U_{\\beta }$ .",
"Define the 1-form $\\omega _{{\\epsilon }} = \\sum \\, \\rho _{\\beta } \\cdot \\omega ^{\\beta }_{{\\epsilon }}$ on $M$ .",
"That is, for each $1 \\le \\alpha \\le \\nu $ , the restriction $\\omega _{{\\epsilon }}| U_{\\alpha }$ to the chart $U_{\\alpha }$ is given by $\\omega _{{\\epsilon }}|{U_{\\alpha }} = \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\rho _{\\beta }|_{U_{\\alpha }} \\cdot \\omega ^{\\beta }_{{\\epsilon }} |_{U_{\\alpha }} ~ .$ Recall that $\\vec{n}$ denotes the unit, positively-oriented vector field on $M$ orthogonal to ${\\mathcal {F}}$ , and let $\\omega $ be the 1-form on $M$ defining ${\\mathcal {F}}$ with $\\omega (\\vec{n}) =1$ .",
"Set $f_{{\\epsilon }} = \\ln (\\omega _{{\\epsilon }}(\\vec{n}))$ so that $\\omega _{{\\epsilon }} = \\exp (f_{{\\epsilon }}) \\cdot \\omega $ .",
"Then for $\\vec{v}_{{\\epsilon }} = \\exp (- f_{{\\epsilon }}) \\cdot \\vec{n}$ we have $\\omega _{{\\epsilon }}(\\vec{v}_{{\\epsilon }}) =1$ .",
"Let $\\eta _{{\\epsilon }} = D^{\\vec{v}_{{\\epsilon }}} \\omega _{{\\epsilon }} = \\iota (\\vec{v}_{{\\epsilon }}) d\\omega _{{\\epsilon }} =\\exp (- f_{{\\epsilon }}) \\cdot \\iota (\\vec{n}) d\\omega _{{\\epsilon }} ~ .$ Recall that for the definition of the Godbillon measure, we are only concerned with the restricted 1-forms $\\eta _{{\\epsilon }} | L$ for each leaf $L$ of ${\\mathcal {F}}$ .",
"That is, the integrand in (REF ) depends only on the restricted class $\\eta _{{\\epsilon }}|{\\mathcal {F}}\\in \\Omega ^1({\\mathcal {F}})$ .",
"Recall from (REF ) the definition of the leafwise differential $D_{{\\mathcal {F}}}f_{{\\epsilon }} \\equiv d(f_{{\\epsilon }}|{\\mathcal {F}}) \\in \\Omega ^1({\\mathcal {F}})$ .",
"Then using (REF ) we have the leafwise calculation in $\\Omega ^1({\\mathcal {F}})$ : $\\eta _{{\\epsilon }} |{\\mathcal {F}}& = & \\exp (- f_{{\\epsilon }}) | {\\mathcal {F}}\\cdot \\left\\lbrace \\iota (\\vec{n}) d (\\exp (f_{{\\epsilon }}) \\cdot \\omega ) \\right\\rbrace |{\\mathcal {F}}\\\\& = & \\exp (- f_{{\\epsilon }}) | {\\mathcal {F}}\\cdot \\left\\lbrace -\\exp (f_{{\\epsilon }})| {\\mathcal {F}}\\cdot D_{{\\mathcal {F}}}(f_{{\\epsilon }}) + \\exp (f_{{\\epsilon }}|{\\mathcal {F}}) \\cdot \\iota (\\vec{n}) d \\omega )|{\\mathcal {F}}\\right\\rbrace \\\\& = & - D_{{\\mathcal {F}}}(f_{{\\epsilon }}) + \\left\\lbrace \\iota (\\vec{n}) d \\omega ) \\right\\rbrace |{\\mathcal {F}}\\\\& = & \\eta |{\\mathcal {F}}- D_{{\\mathcal {F}}}(f_{{\\epsilon }}) ~ .$ Thus, the leafwise 1-forms $\\eta _{{\\epsilon }} |{\\mathcal {F}}$ and $\\eta |{\\mathcal {F}}$ differ by the leafwise exact 1-form $D_{{\\mathcal {F}}}(f_{{\\epsilon }})$ .",
"Then by the Leafwise Stokes' Theorem [36], the Godbillon measure $G_{{\\mathcal {F}}}(B)$ can be calculated using the 1-form $\\eta _{{\\epsilon }}|{\\mathcal {F}}$ restricted to $B$ .",
"We next estimate the norm $\\Vert \\eta _{{\\epsilon }}\\Vert $ .",
"Consider the 1-forms $\\displaystyle \\omega ^{\\beta }_{{\\epsilon }} |_{U_{\\alpha }}$ appearing in the expression (REF ).",
"Fix $1 \\le \\alpha \\le \\nu $ , then for $(\\alpha , \\beta )$ admissible, that is such that $U_{\\alpha } \\cap U_{\\beta } \\ne \\emptyset $ , let ${\\mathcal {Q}}= \\lbrace {\\mathcal {P}}_{\\alpha }(x), {\\mathcal {P}}_{\\beta }(y)\\rbrace $ be a plaque chain with holonomy map ${\\bf h}_{{\\mathcal {Q}}}$ .",
"Using the identity $ \\pi _{\\beta } = {\\bf h}_{{\\mathcal {Q}}} \\circ \\pi _{\\alpha }$ on $U_{\\alpha } \\cap U_{\\beta }$ and the identity (REF ), then on $U_{\\alpha } \\cap U_{\\beta }$ we have $\\omega ^{\\beta }_{{\\epsilon }} |_{U_{\\alpha } \\cap U_{\\beta }} & = & \\pi _{\\beta }^*(dx_{\\beta }^{{\\epsilon }}) |_{U_{\\alpha } \\cap U_{\\beta }} \\\\& = & \\pi _{\\alpha }^* \\circ {\\bf h}_{{\\mathcal {Q}}}^*(dx_{\\beta }^{{\\epsilon }}) |_{U_{\\alpha } \\cap U_{\\beta }}\\\\& = & \\pi _{\\alpha }^*(g_{{\\epsilon }}^{\\beta } \\circ {\\bf h}_{{\\mathcal {Q}}} \\cdot {\\bf h}_{{\\mathcal {Q}}}^{\\prime } \\cdot dx_{\\alpha }) |_{U_{\\alpha } \\cap U_{\\beta }} \\\\& = & (g_{{\\epsilon }}^{\\beta } \\circ {\\bf h}_{{\\mathcal {Q}}} \\circ \\pi _{\\alpha }) \\cdot ({\\bf h}_{{\\mathcal {Q}}}^{\\prime } \\circ \\pi _{\\alpha }) \\cdot \\pi _{\\alpha }^*(dx_{\\alpha }) |_{U_{\\alpha } \\cap U_{\\beta }} ~ .$ To simplify notation, set $k_{{\\epsilon },\\alpha \\beta }(x) = (g_{{\\epsilon }}^{\\beta } \\circ {\\bf h}_{{\\mathcal {Q}}} \\circ \\pi _{\\alpha }) \\cdot ({\\bf h}_{{\\mathcal {Q}}}^{\\prime } \\circ \\pi _{\\alpha })$ .",
"Note that $k_{{\\epsilon },\\alpha \\alpha }= g_{{\\epsilon }}^{\\alpha } \\circ \\pi _{\\alpha }$ .",
"Then in this notation, the estimate (REF ) implies for $x \\in U_{\\alpha } \\cap U_{\\beta }$ that $\\exp (-{\\epsilon }) \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) \\le k_{{\\epsilon },\\alpha \\beta }(x) \\le \\exp ({\\epsilon }) \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) ~.$ Also, each function $k_{{\\epsilon },\\alpha \\beta }$ is constant along the plaques in $U_{\\alpha } \\cap U_{\\beta }$ , so that its leafwise differential is zero; that is, $D_{{\\mathcal {F}}}k_{{\\epsilon },\\alpha \\beta } = 0$ .",
"Recall that $\\omega _{\\alpha } = \\pi _{\\alpha }^*(dx_{\\alpha })$ , so that $d\\omega _{\\alpha } = 0$ , and for $x \\in U_{\\alpha } \\cap U_{\\beta }$ we then have $\\omega ^{\\beta }_{{\\epsilon }} |_x = k_{{\\epsilon },\\alpha \\beta }(x) \\cdot \\omega _{\\alpha }|_x ~ .$ Then for $x \\in U_{\\alpha }$ and using the formulas (REF ), (REF ) and (REF ), and letting $\\vec{n}_x$ denote the value of the unit vector field $\\vec{n}$ at $x$ , we estimate $\\left\\Vert \\eta _{{\\epsilon }} |_x \\right\\Vert $ as follows: $\\left\\Vert \\eta _{{\\epsilon }} |_x \\right\\Vert = \\left\\Vert \\exp (- f_{{\\epsilon }}(x)) \\cdot \\lbrace \\iota (\\vec{n}) d\\omega _{{\\epsilon }}\\rbrace |_x \\right\\Vert & = & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\Vert \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\iota (\\vec{n}_x) d \\lbrace \\rho _{\\beta } \\cdot \\omega ^{\\beta }_{{\\epsilon }} \\rbrace |_x \\right\\Vert \\nonumber \\\\& = & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\Vert \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\iota (\\vec{n}_x) d \\lbrace \\rho _{\\beta } \\cdot k_{{\\epsilon },\\alpha \\beta } \\cdot \\omega _{\\alpha }\\rbrace |_x \\right\\Vert \\nonumber \\\\& = & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\Vert \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\cdot k_{{\\epsilon },\\alpha \\beta }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x) \\right\\Vert ~ .$ The the leafwise differential of the constant function is zero, so we have the identity $ 0 = D_{{\\mathcal {F}}}(1) = D_{{\\mathcal {F}}}(\\sum \\rho _{\\beta })= \\sum D_{{\\mathcal {F}}}\\rho _{\\beta } .$ We conclude that $0 = \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x = \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x) ~.$ Then continuing from (REF ), and using the identities (REF ) and (REF ), for $x \\in U_{\\alpha }$ we have: $\\left\\Vert \\eta _{{\\epsilon }} |_x \\right\\Vert & = & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\Vert \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\cdot \\lbrace k_{{\\epsilon },\\alpha \\beta }(x) - k_{{\\epsilon },\\alpha \\alpha }(x)\\rbrace \\cdot \\omega _{\\alpha }(\\vec{n}_x) \\right\\Vert \\nonumber \\\\& \\le & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\Vert D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\Vert \\cdot | k_{{\\epsilon },\\alpha \\beta }(x) - k_{{\\epsilon },\\alpha \\alpha }(x)| \\cdot | \\omega _{\\alpha }(\\vec{n}_x)| \\nonumber \\\\& \\le & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\lbrace \\sup _{x \\in U_{\\alpha }} \\Vert D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\Vert \\cdot |\\omega _{\\alpha }(\\vec{n}_x)| \\right\\rbrace \\cdot \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; | k_{{\\epsilon },\\alpha \\beta }(x) - k_{{\\epsilon },\\alpha \\alpha }(x)| \\nonumber \\\\& \\le & \\exp (- f_{{\\epsilon }}(x)) \\cdot \\left\\lbrace \\sup _{x \\in U_{\\alpha }} \\Vert D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\Vert \\cdot |\\omega _{\\alpha }(\\vec{n}_x)| \\right\\rbrace \\cdot \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; (\\exp ({\\epsilon }) - 1) \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) ~ .",
"$ It remains to estimate $\\exp (- f_{{\\epsilon }}(x))$ in (REF ).",
"Recall (REF ) and using (REF ) we have for $x \\in U_{\\alpha }$ that $\\exp (f_{{\\epsilon }}(x)) & = & \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\rho _{\\beta }(x) \\cdot \\omega ^{\\beta }_{{\\epsilon }}(\\vec{n}_x) \\nonumber \\\\& = & \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\rho _{\\beta }(x) \\cdot k_{{\\epsilon },\\alpha \\beta }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x) \\nonumber \\\\& \\ge & \\sum _{U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset } \\; \\rho _{\\beta }(x) \\cdot \\exp (-{\\epsilon }) \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x) \\nonumber \\\\& = & \\exp (-{\\epsilon }) \\cdot k_{{\\epsilon },\\alpha \\alpha }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x) ~ .$ Thus, we obtain the estimate $\\exp (- f_{{\\epsilon }}(x)) \\le \\exp ({\\epsilon }) \\cdot (k_{{\\epsilon },\\alpha \\alpha }(x) \\cdot \\omega _{\\alpha }(\\vec{n}_x))^{-1} .$ Then combining (REF ) and (REF ), and noting that the number of indices $\\beta $ for which $U_{\\beta } \\cap U_{\\alpha } \\ne \\emptyset $ is bounded by the cardinality $\\nu $ of the covering, we obtain $\\left\\Vert \\eta _{{\\epsilon }} |_x \\right\\Vert \\le \\left\\lbrace \\sup _{x \\in U_{\\alpha }} \\Vert D_{{\\mathcal {F}}}(\\rho _{\\beta })|_x \\Vert \\right\\rbrace \\cdot \\nu \\cdot \\exp ({\\epsilon }) (\\exp ({\\epsilon }) - 1)$ Note that the right hand side in (REF ) tends to 0 as ${\\epsilon }\\rightarrow 0$ , so that for each $n > 0$ , we can choose ${\\epsilon }_n > 0$ such that $\\Vert \\eta _{{\\epsilon }_n}\\Vert \\le 1/n$ .",
"Then set $\\omega _n = \\omega _{{\\epsilon }_n}$ , and the claim of the Theorem REF follows."
],
[
"Uniform hyperbolic expansion",
"In this section, we assume that ${\\mathcal {F}}$ is a $C^1$ -foliation with non-empty hyperbolic set ${\\rm E}^+({\\mathcal {F}})$ , and show that there exists a hyperbolic fixed-point for the holonomy pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ .",
"The proof uses a pseudogroup version of the Pliss Lemma, which is fundamental in the study of non-uniformly hyperbolic dynamics (see [1] or [5], or the original article by Pliss [64].)",
"The goal is to construct hyperbolic contractions in the holonomy pseudogroup.",
"The length of the path defining the holonomy element is not important, but rather it is important to obtain uniform estimates on the size of the domain of the hyperbolic element thus obtained, estimates which are independent of the length of the path.",
"This is a key technical point for the application of the constructions of this section in the next Section , where we construct sufficiently many contractions so that they result in the existence of a resilient orbit for the action of the holonomy pseudogroup.",
"We note that the existence of a hyperbolic contraction can also be deduced using the foliation geodesic flow methods introduced in [42], though that method does not yield estimates on the size of the domain of the hyperbolic element in the foliation pseudogroup."
],
[
"Uniform hyperbolicity and the Pliss Lemma",
"We fix a regular covering on $M$ as in Section REF , with transversals ${\\mathfrak {X}}$ and ${\\widetilde{{\\mathfrak {X}}}}$ as in (), and let ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ denote the resulting pseudogroup acting on the spaces ${\\mathcal {X}}$ and ${\\widetilde{{\\mathcal {X}}}}$ as in (REF ).",
"Recall that by Lemma REF , there exists ${\\epsilon }_0 > 0$ so that for every admissible pair $(\\alpha , \\beta )$ and $x \\in {\\mathcal {X}}_{\\alpha \\beta }$ then $[x-{\\epsilon }_0, x+ {\\epsilon }_0] \\subset {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ .",
"Recall that the space ${\\mathcal {X}}_{\\alpha \\beta }$ was defined in (REF ), and ${\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ was defined in ().",
"DEFINITION 5.1 Given $0 < {\\epsilon }_1 \\le {\\epsilon }_0$ , a constant $0 < \\delta _0 \\le {\\epsilon }_1$ is said to be a logarithmic modulus of continuity for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ with respect to ${\\epsilon }_1$ , if for $y,z \\in {\\mathcal {X}}_{\\alpha \\beta }[\\delta _0]$ with ${\\bf d}_{{\\mathcal {X}}}(y,z) \\le \\delta _0$ , then $\\left| \\log \\lbrace \\widetilde{\\bf h}_{\\beta \\alpha }^{\\prime }(y)\\rbrace - \\log \\lbrace \\widetilde{\\bf h}_{\\beta \\alpha }^{\\prime }(z)\\rbrace \\right| \\le {\\epsilon }_1 ~ .$ LEMMA 5.2 Given $0 < {\\epsilon }_1 \\le {\\epsilon }_0$ , there exists a constant $0 < \\delta _0 \\le {\\epsilon }_1$ which is a logarithmic modulus of continuity for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ with respect to ${\\epsilon }_1$ .",
"By the choice of $0 < {\\epsilon }_1 \\le {\\epsilon }_0$ , for each admissible pair $\\lbrace \\alpha , \\beta \\rbrace $ , the logarithmic derivative $\\log \\lbrace \\widetilde{\\bf h}_{\\beta \\alpha }^{\\prime }(y)\\rbrace $ is continuous on the compact subset ${\\mathcal {X}}_{\\alpha \\beta }[{\\epsilon }_1] \\subset {\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ .",
"Thus, there exists $\\delta _0(\\alpha , \\beta ) > 0$ such that (REF ) holds for this choice of $\\lbrace \\alpha , \\beta \\rbrace $ .",
"Define $\\displaystyle \\delta _0 = \\min \\lbrace \\delta _0(\\alpha , \\beta ) \\mid \\lbrace \\alpha , \\beta \\rbrace ~ {\\rm admissible}\\rbrace $ .",
"As the number of admissible pairs is finite, we have $\\delta _0 > 0$ .",
"The next result shows that if ${\\rm E}^+({\\mathcal {F}})$ is non-empty, then there are words in ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ of arbitrarily long length, along which the holonomy is “uniformly expansive”.",
"That is, there exists a constant $\\lambda _* > 0$ such that for such a word ${\\bf h}_n$ defined by a plaque chain ${\\mathcal {P}}$ of length $n$ , then ${\\bf h}_n^{\\prime }(y) \\ge \\exp \\lbrace n \\lambda _*)$ for all $\\displaystyle y \\in {\\mathcal {D}}_{{\\mathcal {P}}}$ .",
"The proof is technical, but also notable as it develops a version for pseudogroup actions of the Pliss Lemma, which is used in the study of the dynamics of partially hyperbolic diffeomorphisms, as for example in [5], [55], [64].",
"Note that Definition REF implies that the set ${\\rm E}^+({\\mathcal {F}})$ is an increasing union of the sets ${\\rm E}^+_a({\\mathcal {F}})$ for $a > 0$ , and thus given $\\xi \\in {\\rm E}^+({\\mathcal {F}})$ , there exist $a> 0$ such that $\\xi \\in {\\rm E}^+_a({\\mathcal {F}})$ .",
"We introduce a convenient notation for working with the set ${\\rm E}^+_a({\\mathcal {F}})$ .",
"For each $1 \\le \\alpha \\le \\nu $ , let ${\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}_{\\alpha } & = & \\pi _{\\alpha }({\\rm E}^+_a({\\mathcal {F}})\\cap U_{\\alpha }) \\subset {\\mathcal {X}}_{\\alpha } \\\\{\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}& = & ({\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}_{1}) \\cup \\cdots \\cup ({\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}_{\\nu }) .$ Recall that the transversals ${\\mathfrak {X}}_{\\alpha }$ and their images ${\\mathcal {X}}_{\\alpha }$ in the coordinates $U_{\\alpha }$ were defined in (REF ).",
"PROPOSITION 5.3 Let $x \\in {\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}$ for $a > 0$ , let $0 < {\\epsilon }_1 < \\min \\lbrace {\\epsilon }_0, a/100\\rbrace $ , and let $\\delta _0$ be the logarithmic modulus of continuityÊ for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ with respect to ${\\epsilon }_1$ , as chosen in Lemma REF .",
"Then for each integer $n > 0$ , there exist a point $y_n \\in {{\\mathcal {G}}}_{{\\mathcal {F}}}(x)$ , a closed interval $I_n^x \\subset {\\widetilde{{\\mathcal {X}}}}_{\\alpha }$ containing $x$ in its interior, and a holonomy map ${\\bf h}_n^x \\colon I_n^x \\rightarrow J_n^x$ such that for $y_n = {\\bf h}_n^x(x)$ , $\\displaystyle J_n^x = [y_n - \\delta _0/2, y_n+\\delta _0/2] \\subset {\\widetilde{{\\mathcal {X}}}}$ and $I_n^x = ({\\bf h}_n^x)^{-1}(J_n^x)$ , we have $({\\bf h}_n^x)^{\\prime }(z) > \\exp \\lbrace n a/2\\rbrace ~ {\\rm for ~ all} ~ z \\in I_n^x ~ .$ It follows that $\\displaystyle |I_n^x| < \\delta _0 \\, \\exp \\lbrace -n a/2\\rbrace $ .",
"This is illustrated in Figure REF .",
"Figure: Expanding holonomy map 𝐡 n x {\\bf h}_n^xFix a choice of $0 < {\\epsilon }_1 < \\min \\lbrace {\\epsilon }_0, a/100\\rbrace $ , and then choose a logarithmic modulus of continuityÊ $\\delta _0 > 0$ as in Lemma REF .",
"The set ${\\mathcal {X}}_{\\alpha \\beta }[{\\epsilon }_1]$ , as defined in () for $\\delta = {\\epsilon }_1$ , is compact, so there exists $C_0 > 0$ so that for all $(\\alpha , \\beta )$ admissible and $y \\in {\\mathcal {X}}_{\\alpha \\beta }[{\\epsilon }_1]$ , we have $\\displaystyle 1/C_0 \\le \\widetilde{\\bf h}_{\\beta \\alpha }^{\\prime }(y) \\le C_0$ .",
"From the definition of $\\lambda _*(x)$ as a $\\limsup $ in (REF ), the assumption that $\\lambda _*(x) > a$ implies that for each integer $n > 0$ , we can choose a plaque chain of length $\\ell _n \\ge n$ , given by $\\displaystyle {\\mathcal {P}}_n = \\lbrace {\\mathcal {P}}_{\\alpha _0}(z_0), \\ldots , {\\mathcal {P}}_{\\alpha _{\\ell _n}}(z_{\\ell _n}) \\rbrace $ with $z_0 = x$ , such that $\\displaystyle \\log \\lbrace {\\bf h}_{{\\mathcal {P}}_n}^{\\prime }(z_0)\\rbrace > \\ell _n \\cdot a$ .",
"Fix $n$ and the choice of the plaque chain ${\\mathcal {P}}_n$ as above.",
"For each $1 \\le j \\le \\ell _n$ let $ {\\bf h}_{\\alpha _{j},\\alpha _{j-1}}$ be the holonomy transformation defined by $\\lbrace {\\mathcal {P}}_{\\alpha _{j-1}},{\\mathcal {P}}_{\\alpha _j}\\rbrace $ , and so $ {\\bf h}_{\\alpha _{j},\\alpha _{j-1}}^{-1} = {\\bf h}_{\\alpha _{j-1},\\alpha _{j}}$ .",
"Introduce the notation ${\\widehat{\\bf h}}_{0} = Id$ , and for $1 \\le j \\le \\ell _n$ let ${\\widehat{\\bf h}}_{j} = {\\bf h}_{\\alpha _{j},\\alpha _{j-1}} \\circ \\cdots \\circ {\\bf h}_{\\alpha _{1},\\alpha _{0}}$ denote the partial composition of generators.",
"Note that $z_j = {\\widehat{\\bf h}}_{j}(z_0)$ and $z_0 = x$ , and that we have the relations $\\displaystyle {\\widehat{\\bf h}}_{j+1} = {\\bf h}_{\\alpha _{j+1},\\alpha _{j}} \\circ {\\widehat{\\bf h}}_{j}$ and $z_{j+1} = {\\widehat{\\bf h}}_{j}(z_{j})$ for $0 \\le j < \\ell _n$ .",
"For each $1 \\le j \\le \\ell _n$ , set $\\lambda _j = \\log \\lbrace {\\widehat{\\bf h}}^{\\prime }_{\\alpha _{j-1},\\alpha _{j}}(z_j)\\rbrace = - \\log \\lbrace {\\widehat{\\bf h}}^{\\prime }_{\\alpha _{j},\\alpha _{j-1}}(z_{j-1})\\rbrace ~ .$ In particular, $\\displaystyle \\log \\lbrace {\\widehat{\\bf h}}^{\\prime }_{\\ell _n}(x)\\rbrace = -(\\lambda _1 + \\cdots + \\lambda _{\\ell _n})$ .",
"Note that if $\\lambda _j < 0$ then the map ${\\widehat{\\bf h}}_{\\alpha _{j-1},\\alpha _{j}}$ is an infinitesimal contraction at $z_j$ , and ${\\widehat{\\bf h}}_{\\alpha _{j},\\alpha _{j-1}}$ is an infinitesimal expansion at $z_{j-1}$ .",
"The following algebraic definition and lemma provide the key to the analysis of the hyperbolic expansion properties of the partial compositions of the maps ${\\widehat{\\bf h}}_{j}$ .",
"DEFINITION 5.4 Let $\\lbrace \\lambda _1, \\ldots ,\\lambda _m\\rbrace $ be given, and $\\vartheta > 0$ .",
"An index $1 \\le j \\le m$ is said to be $\\vartheta $ -regular if the following sequence of partial sum estimates hold: $\\lambda _{j} + \\;\\; \\vartheta & < & 0 \\nonumber \\\\\\lambda _{j-1} + \\lambda _{j} + 2 \\vartheta & < & 0\\nonumber \\\\& \\vdots & \\\\\\lambda _{1} + \\cdots + \\lambda _{j} + j \\vartheta & < & 0 ~ .\\nonumber $ Condition (REF ) is a weaker hypothesis than assuming the uniform estimates $\\lambda _i <-\\vartheta $ for all $1 \\le i \\le j$ , but is sufficient for our purposes.",
"The next result shows that $\\vartheta $ -regular indices always exist.",
"LEMMA 5.5 Assume there are given real numbers $\\lbrace \\lambda _{1}, \\ldots , \\lambda _{m}\\rbrace $ such that $\\lambda _{1} + \\cdots + \\lambda _{m} \\le -a \\, m ~ .$ Then for any $0 < {\\epsilon }_1 < a$ , there exists an ${\\epsilon }_1$ -regular index $q_m$ , for some $1 \\le q_m \\le m$ , which satisfies $\\lambda _{1} + \\cdots + \\lambda _{q_m} ~ \\le ~ (-a + {\\epsilon }_1) \\, m ~ .$ The existence of the index $q_m$ satisfying this property is shown by contradiction.",
"We introduce the concept of an ${\\epsilon }_1$ -irregular index, for which the ${\\epsilon }_1$ -regular condition fails, and show by contradiction that not all indices can be ${\\epsilon }_1$ -irregular.",
"We say that an index $k \\le m$ is ${\\epsilon }_1$ -irregular if $\\lambda _{k} + \\cdots + \\lambda _{m} + (m - k + 1) {\\epsilon }_1 \\ge 0 ~ .$ If there is no irregular index, then observe that $q_m = m$ is an ${\\epsilon }_1$ -regular index.",
"Otherwise, suppose that there exists some index $k$ which is ${\\epsilon }_1$ -irregular.",
"The inequality (REF ) states that the index $k= 1$ is not ${\\epsilon }_1$ -irregular.",
"Let $j_m \\le m$ be the least ${\\epsilon }_1$ -irregular index, so that $\\lambda _{j_m} + \\cdots + \\lambda _{m} + (m - j_n + 1) {\\epsilon }_1 \\ge 0$ By (REF ), $j_m =1$ is is not ${\\epsilon }_1$ -irregular, so we have $2 \\le j_m \\le m$ .",
"Set $q_m = j_m -1$ , then we claim that $q_m $ is an ${\\epsilon }_1$ -regular index.",
"If not, then at least one of the inequalities in (REF ) must fail to hold.",
"That is, there is some $i \\le q_m$ with $\\lambda _{i} + \\cdots + \\lambda _{q_m} + (q_m-i +1) {\\epsilon }_1 \\ge 0 ~ .$ Add the inequalities (REF ) and (REF ), and noting that $q_m = j_m -1$ , we obtain that $i$ is also an ${\\epsilon }_1$ -irregular index.",
"As $i < j_m$ , this is contrary to the choice of $j_m$ .",
"Hence, $q_m $ is an ${\\epsilon }_1$ -regular index.",
"It remains to show that the estimate (REF ) holds.",
"As $j_m = q_m +1$ is irregular, subtract (REF ) for $k = j_m$ from (REF ) to obtain $\\lambda _{1} + \\cdots + \\lambda _{q_m} \\le -a m + (m - q_m) {\\epsilon }_1\\le (-a + {\\epsilon }_1) \\, m$ as claimed.",
"We return to considering the maps ${\\widehat{\\bf h}}_j$ defined by (REF ), and the exponents $\\lambda _j$ defined by (REF ).",
"The following result then follows directly from Lemma REF and the definitions.",
"COROLLARY 5.6 Assume that there is given $a > 0$ with $x \\in {\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}$ , a choice of integer $n > 0$ , and plaque-chain $\\displaystyle {\\mathcal {P}}_n = \\lbrace {\\mathcal {P}}_{\\alpha _0}(z_0), \\ldots , {\\mathcal {P}}_{\\alpha _{\\ell _n}}(z_{\\ell _n}) \\rbrace $ with $\\ell _n \\ge n$ , such that $\\displaystyle \\log \\lbrace {\\bf h}_{{\\mathcal {P}}_n}^{\\prime }(z_0)\\rbrace \\ge \\ell _n \\cdot a$ and $z_0 = x$ .",
"Given $0 < {\\epsilon }_1 < a$ , by Lemma REF there exists an ${\\epsilon }_1$ -regular index $q_n$ , for some $1 \\le q_n \\le \\ell _n$ chosen as in Lemma REF , such that for the map ${\\widehat{\\bf h}}_{q_n}$ defined by (REF ), ${\\widehat{\\bf h}}_{q_n}^{\\prime }(x) \\ge (a - {\\epsilon }_1) \\, \\ell _n \\ge (a - {\\epsilon }_1) \\, n~ .$ The estimate (REF ) can be interpreted as stating that “most” of the infinitesimal expansion of the map ${\\widehat{\\bf h}}_{\\ell _n}$ at $z_0$ is achieved by the action of the partial composition ${\\widehat{\\bf h}}_{q_n}$ .",
"Recall that we have a fixed choice of $0 < {\\epsilon }_1 < \\min \\lbrace {\\epsilon }_0, a/100\\rbrace $ , as given in the statement of Proposition REF , and $\\delta _0 > 0$ is chosen so that the uniform continuity estimate (REF ) in Lemma REF is satisfied.",
"Then let $1 \\le q_n \\le \\ell _n$ be the ${\\epsilon }_1$ -regular index defined in Lemma REF which satisfies (REF ).",
"We next use the ${\\epsilon }_1$ -regular condition to obtain uniform estimates on the domains for which the inverses ${\\widehat{\\bf h}}_{j}^{-1}$ are contracting, for $1 \\le j \\le q_n$ .",
"Recall that $\\displaystyle \\widetilde{\\bf h}_{\\alpha ,\\beta }$ denotes the continuous extension of the map $\\displaystyle {\\bf h}_{\\alpha ,\\beta }$ to the domain ${\\widetilde{{\\mathcal {X}}}}_{\\alpha \\beta }$ .",
"Introduce extensions ${\\bf h}_n^x$ of ${\\widehat{\\bf h}}_{q_n}$ and ${\\bf g}_n^x$ of its inverse ${\\widehat{\\bf h}}_{q_n}^{-1}$ , which are defined by ${\\bf h}_n^x & = & \\widetilde{\\bf h}_{\\alpha _{q_n},\\alpha _{q_n -1}} \\circ \\cdots \\circ \\widetilde{\\bf h}_{\\alpha _{1},\\alpha _{0}} \\\\{\\bf g}_n^x & = & \\widetilde{\\bf h}_{\\alpha _{0},\\alpha _{1}} \\circ \\cdots \\circ \\widetilde{\\bf h}_{\\alpha _{q_n-1},\\alpha _{q_n}} ~ .",
"$ Set $y_n = {\\bf h}_n^x(x) = z_{\\ell _n}$ , then by the estimate (REF ) we have $\\log \\lbrace ({\\bf g}_n^x)^{\\prime }(y_n)\\rbrace ~ = ~ \\lambda _{1} + \\cdots + \\lambda _{q_n} ~ \\le ~ (-a+{\\epsilon }_1) \\; \\ell _n < 0~.$ We next show that ${\\bf g}_n^x $ is uniformly contracting on an interval with uniform length about $y_n$ .",
"LEMMA 5.7 Set $\\delta _0^{\\prime } = \\delta _0/8$ .",
"Then the interval $J_n^x = [y_n - 4\\delta _0^{\\prime }, y_n + 4\\delta _0^{\\prime }]$ is in the domain of ${\\bf g}_n^x$ , and for all $y \\in J_n^x$ , $\\exp \\lbrace (-a - 2 {\\epsilon }_1)\\, \\ell _n\\rbrace \\le ({\\bf g}_n^x)^{\\prime }(y) \\le \\exp \\lbrace (-a + 2{\\epsilon }_1) \\, \\ell _n\\rbrace ~.$ Hence, for $I_n^x = {\\bf g}_n^x(J_n^x)$ , $|I_n^x| \\le \\delta _0 \\exp \\lbrace (-a + 2{\\epsilon }_1)\\; \\ell _n \\rbrace < \\exp \\lbrace (-a/2)\\; \\ell _n \\rbrace ~ .$ By the choice of $\\delta _0^{\\prime }$ , the uniform continuity estimate (REF ) implies that for all $y \\in J_n^x$ $ \\left| \\log \\lbrace \\widetilde{\\bf h}^{\\prime }_{\\alpha _{q_n -1},\\alpha _{q_n}}(y)\\rbrace - \\log \\lbrace \\widetilde{\\bf h}^{\\prime }_{\\alpha _{q_n - 1},\\alpha _{q_n}}(y_n)\\rbrace \\right| \\le {\\epsilon }_1 ~ .$ Thus, by the definition of $\\lambda _{q_n}$ we have that, for all $y \\in J_n^x$ , $ \\exp \\lbrace \\lambda _{q_n} -{\\epsilon }_1 \\rbrace \\le \\widetilde{\\bf h}^{\\prime }_{\\alpha _{q_n - 1},\\alpha _{q_n}}(y) \\le \\exp \\lbrace \\lambda _{q_n} + {\\epsilon }_1 \\rbrace ~ .$ The assumption that $q_n$ is ${\\epsilon }_1$ -regular implies $\\displaystyle \\lambda _{q_n}+{\\epsilon }_1 < 0$ , hence $\\exp \\lbrace \\lambda _{q_n}+ {\\epsilon }_1\\rbrace < 1$ .",
"Thus, for all $y \\in J_{n}^x$ we have ${\\bf d}_{{\\mathcal {X}}}(\\widetilde{\\bf h}_{\\alpha _{q_n -1},\\alpha _{q_n}}(y_n), \\widetilde{\\bf h}_{\\alpha _{q_n -1},\\alpha _{q_n}}(y)) \\le 4\\delta _0^{\\prime } \\exp \\lbrace \\lambda _{q_n} + {\\epsilon }_1\\rbrace < 4 \\delta _0^{\\prime } ~ .$ Now proceed by downward induction.",
"For $0 < j \\le q_n$ set ${\\bf g}_{n,j}^x = \\widetilde{\\bf h}_{\\alpha _{j-1},\\alpha _{j}} \\circ \\cdots \\circ \\widetilde{\\bf h}_{\\alpha _{q_n -1},\\alpha _{q_n}} ~, \\;\\; J_{n,j}^x = {\\bf g}_{n,j}^x(J_{n}^x) ~, \\;\\; y_{n,j} = {\\bf g}_{n,j}^x(y_n) = z_{j-1} ~ .$ Assume that for $1 < j \\le q_n$ , we are given that for all $y \\in J_{n,j}^x$ the estimates $ \\exp \\lbrace \\lambda _{j} + \\cdots + \\lambda _{q_n} - (q_n -j +1) \\, {\\epsilon }_1 \\rbrace \\le ({\\bf g}_{n,j}^x)^{\\prime }(y) \\le \\exp \\lbrace \\lambda _{j} + \\cdots + \\lambda _{q_n} + (q_n -j +1) \\, {\\epsilon }_1 \\rbrace ~,$ ${\\bf d}_{{\\mathcal {X}}}(y, y_{n,j}) \\le 4\\delta _0^{\\prime } ~ .$ The choice of $\\delta _0$ and the hypothesis (REF ) imply that for $y \\in J_{n,j}^x$ , $\\left| \\log \\lbrace \\widetilde{\\bf h}^{\\prime }_{\\alpha _{j -2},\\alpha _{j-1}}(y)\\rbrace - \\log \\lbrace \\widetilde{\\bf h}^{\\prime }_{\\alpha _{j -2},\\alpha _{j-1}}(y_{n,j})\\rbrace \\right| \\le {\\epsilon }_1 ~ .$ Recall that $z_{j-1} = y_{n,j}$ , and that $\\lambda _{j-1} = \\log \\lbrace \\widetilde{\\bf h}^{\\prime }_{\\alpha _{j -2},\\alpha _{j-1}}(y_{n,j})\\rbrace $ by (REF ), so for all $y \\in J_{n,j}^x$ we have for the inverse map $\\widetilde{\\bf h}_{\\alpha _{j -2},\\alpha _{j-1}} = \\widetilde{\\bf h}_{\\alpha _{j -1},\\alpha _{j-2}}^{-1}$ that $\\exp \\lbrace \\lambda _{j-1} -{\\epsilon }_1 \\rbrace \\le \\widetilde{\\bf h}^{\\prime }_{\\alpha _{j -2},\\alpha _{j-1}}(y) \\le \\exp \\lbrace \\lambda _{j-1} + {\\epsilon }_1 \\rbrace ~ .$ Then by the chain rule, the estimates (REF ) and the inductive hypothesis (REF ) yield the estimates $\\exp \\lbrace \\lambda _{j-1} + \\cdots + \\lambda _{q_n} - (q_n -j +2) \\, {\\epsilon }_1 \\rbrace \\le ({\\bf g}_{n,j-1}^x)^{\\prime }(y) \\le \\exp \\lbrace \\lambda _{j-1} + \\cdots + \\lambda _{q_n} + (q_n -j +2) \\, {\\epsilon }_1 \\rbrace .",
"$ Now the assumption that $q_n$ is ${\\epsilon }_1$ -regular implies $\\displaystyle \\lambda _{j-1} + \\cdots + \\lambda _{q_n} + (q_n -j +2) \\, {\\epsilon }_1 < 0$ hence $ \\exp \\lbrace \\lambda _{j-1} + \\cdots + \\lambda _{q_n} + (q_n -j +2) \\, {\\epsilon }_1\\rbrace < 1$ .",
"By the Mean Value Theorem, this yields the distance bound $\\displaystyle {\\bf d}_{{\\mathcal {X}}}(y_{n,j-1} , y) \\le 4\\delta _0^{\\prime }$ , which is the hypothesis (REF ) for $j-1$ .",
"This completes the inductive step.",
"Thus, we may take $j=1$ in inequality (REF ) and combined with the inequality (REF ), for all $y \\in J_{n}^x$ we have that $({\\bf g}_{n}^x)^{\\prime }(y) \\le \\exp \\lbrace \\lambda _{1} + \\cdots + \\lambda _{q_n} + q_n \\, {\\epsilon }_1 \\rbrace \\le \\exp \\lbrace -a \\, \\ell _n + (\\ell _n + q_n ) \\, {\\epsilon }_1 \\rbrace \\le \\exp \\lbrace (-a + 2{\\epsilon }_1) \\, \\ell _n \\rbrace ~ .$ Set $I_n^x = {\\bf g}_n^x(J_n^x)$ , then the estimate (REF ) follows by the Mean Value Theorem.",
"Since $ a - 2 {\\epsilon }_1 > a/2$ and $\\ell _n \\ge n$ , this completes the proof of Proposition REF ."
],
[
"Hyperbolic fixed-points",
"We show the existence of hyperbolic fixed-points for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ contained in the closure of ${\\rm E}^+({\\mathcal {F}})\\cap {\\mathcal {X}}$ in ${\\widetilde{{\\mathcal {X}}}}$ , with uniform estimates on the lengths of their domains of contraction.",
"PROPOSITION 5.8 Let $x \\in {\\rm E}^+_a({\\mathcal {F}})\\cap {\\mathcal {X}}$ for $a > 0$ , let $0 < {\\epsilon }_1 < \\min \\lbrace {\\epsilon }_0, a/100\\rbrace $ , and let $\\delta _0$ be chosen as in Lemma REF , and set $\\delta _0^{\\prime } = \\delta _0/8$ .",
"Given $0 < \\delta _1 < \\delta _0^{\\prime }$ and $0 < \\mu < 1$ , then there exists holonomy maps $\\phi _1, \\psi _1 \\in {{\\mathcal {G}}}_{{\\mathcal {F}}}$ , points $u_1, v_1 \\in \\overline{{\\mathcal {X}}}$ such that ${\\bf d}_{{\\mathcal {X}}}(x,v_1) < \\delta _1$ , such that we have: $\\Phi _1 = \\phi _1 \\circ \\psi _1$ has fixed point $\\Phi _1(u_1) = u_1$ ; ${\\mathcal {J}}_1 \\equiv [u_1 - \\delta _0^{\\prime }, u_1+ \\delta _0^{\\prime }]$ is contained in the domain of $\\Phi _1$ ; $\\Phi ^{\\prime }(y) < \\mu $ for all $y \\in {\\mathcal {J}}_1$ ; $\\Psi _1 = \\psi _1 \\circ \\phi _1$ has fixed point $\\Psi _1(v_1) = v_1$ ; ${\\mathcal {K}}_1 \\equiv \\psi _1({\\mathcal {J}}_1) \\subset (x - \\delta _1, x+ \\delta _1)$ .",
"The idea of the proof is to consider a sequence of maps as given by Proposition REF , for $n \\ge 1$ , and consider a subsequence of these for which the sequence of points $\\displaystyle \\lbrace y_n = {\\bf h}_n^x(x) = z_{\\ell _n} \\mid n \\ge 1\\rbrace $ cluster at a limit point.",
"We then use the estimates (REF ) on the sizes of the domains to show that the appropriate compositions of these maps are defined, and have a hyperbolic fixed point.",
"The details of this argument follow.",
"Set $\\delta _* = \\min \\lbrace 1, \\delta _0^{\\prime }/4,\\delta _1/4\\rbrace $ .",
"Then by Proposition REF , for each integer $n > 0$ , we can choose a map ${\\bf h}_n^x \\colon I_n^x \\rightarrow J_n^x$ as in (REF ), which satisfies condition (REF ).",
"Label the resulting sequence of points $y_n = {\\bf h}_n^x(x) \\in {\\mathcal {X}}$ , and the inverse maps ${\\bf g}_n^x = ({\\bf h}_n^x)^{-1}$ .",
"Let $p_n$ denote the length of the plaque chain defining ${\\bf h}_n^x$ , then $p_n$ equals the ${\\epsilon }_1$ -regular index $1 \\le q_n \\le \\ell _n$ chosen as in the proof of Corollary REF .",
"Recall that ${\\mathcal {X}}$ has compact closure in ${\\widetilde{{\\mathcal {X}}}}$ , so there exists an accumulation point $y_* \\in \\overline{{\\mathcal {X}}} \\subset {\\widetilde{{\\mathcal {X}}}}$ for the set $\\lbrace y_n \\mid n > 0\\rbrace \\subset {\\mathcal {X}}$ .",
"We can assume that ${\\bf d}_{{\\mathcal {X}}}(y_* , y_n) < \\delta _*/4$ for all $n > 0$ , first by passing to a subsequence $\\lbrace y_{n_i}\\rbrace $ which converges to $y_*$ and satisfies this metric estimate, and then reindexing the sequence.",
"Let $\\displaystyle J_n^x = [y_n - 4\\delta _0^{\\prime }, y_n+4\\delta _0^{\\prime }]$ , and set $J_* = [y_* - 3\\delta _0^{\\prime }, y_* + 3\\delta _0^{\\prime }]$ .",
"Then for all $n >0$ , we have $\\displaystyle y_n \\in (y_* -\\delta _0^{\\prime }, y_* + \\delta _0^{\\prime }) \\subset J_* \\subset J_n^x$ .",
"In particular, $y_1 \\in J_* \\subset J_1^x$ is an interior point of $J_*$ , so $x = {\\bf g}_1^x(y_1)$ is an interior point of ${\\bf g}_1^x(J_*)$ .",
"Also recall from Proposition REF , that $I_n^x = {\\bf g}_n^x(J_n^x)$ with $x \\in I_n^x$ for all $n$ , and the interval $I_n^x$ has length $|I_n^x| < \\delta _0 \\exp \\lbrace -na/2\\rbrace = 8\\delta _0^{\\prime } \\exp \\lbrace -na/2\\rbrace $ .",
"Hence, for $n$ sufficiently large, the interval $I_n^x$ is contained in the interior of ${\\bf g}_1^x(J_*)$ .",
"Without loss of generality, we again pass to a subsequence and reindex the sequence, so that we have $I_n^x \\subset {\\bf g}_1^x(J_*)$ and $\\ell _{n+1} > \\ell _{n}$ for all $n > 0$ .",
"We then have the inclusions ${\\bf g}_n^x(J_*) \\subset {\\bf g}_n^x(J_n^x) = I_n^x \\subset {\\bf g}_1^x(J_*) ~ .$ Thus, for each $n > 0$ the composition ${\\bf h}_1^x \\circ {\\bf g}_n^x \\colon J_* \\rightarrow {\\bf h}_1^x \\circ {\\bf g}_1^x(J_*) \\subset J_*$ is defined.",
"(See Figure REF .)",
"Figure: The contracting holonomy map 𝐡 1 x ∘𝐠 n x {\\bf h}_1^x \\circ {\\bf g}_n^xRecall that $p_1$ denotes the length of the plaque-chain which defines ${\\bf h}_1^x$ , and $C_0$ is the Lipschitz constant defined in the proof of Proposition REF .",
"Let $N_0$ be chosen so that for $n \\ge N_0$ we have $C_0^{p_1} \\, \\exp \\lbrace -a\\, n/2\\rbrace & < & \\min \\, \\lbrace \\mu , 1/2\\rbrace \\\\\\delta _0^{\\prime } \\, \\exp \\lbrace -a \\, n/2\\rbrace & < & \\delta _1/2 ~ .$ With the above notations, we then have: LEMMA 5.9 Fix $n \\ge N_0$ , then the map ${\\bf h}_1^x \\circ {\\bf g}_n^x$ is a hyperbolic contraction on $J_*$ with fixed-point $v_* \\in J_*$ satisfying ${\\bf d}_{{\\mathcal {X}}}(v_*, y_n) \\le \\delta _1/2$ and $({\\bf h}_1^x \\circ {\\bf g}_n^x)^{\\prime }(v_*) < \\mu $ .",
"By the choice of $C_0$ we have $({\\bf h}_1^x)^{\\prime }(y) \\le C_0^{p_1}$ for all $y$ in its domain.",
"Recall that ${\\bf g}_n^x$ is the inverse of ${\\bf h}_n^x$ which is defined by a plaque-chain of length $\\ell _n \\ge n$ , so the same holds for ${\\bf g}_n^x$ .",
"The derivative of ${\\bf g}_n^x$ satisfies the estimates (REF ) by Lemma REF , so we have $\\exp \\lbrace (-a - 2 {\\epsilon }_1)\\, \\ell _n\\rbrace \\le ({\\bf g}_n^x)^{\\prime }(y) \\le \\exp \\lbrace (-a + 2{\\epsilon }_1) \\, n\\rbrace ~.$ Thus by (REF ), for all $y \\in J_*$ the composition ${\\bf h}_1^x \\circ {\\bf g}_n^x$ satisfies $({\\bf h}_1^x \\circ {\\bf g}_n^x)^{\\prime }(y) \\le C_0^{p_1} \\, \\exp \\lbrace (-a + 2{\\epsilon }_1) \\, n\\rbrace < C_0^{p_1} \\, \\exp \\lbrace -a\\, n/2\\rbrace < \\min \\, \\lbrace \\mu , 1/2\\rbrace $ where we use that the choice of ${\\epsilon }_1 < a/100$ implies that $(-a + 2{\\epsilon }_1) < -a/2$ .",
"Thus, ${\\bf h}_1^x \\circ {\\bf g}_n^x$ is a hyperbolic contraction on $J_*$ and it follows that ${\\bf h}_1^x \\circ {\\bf g}_n^x$ has a unique fixed-point $v_* \\in J_*$ .",
"Define a sequence of points $w_{\\ell } = ({\\bf h}_1^x \\circ {\\bf g}_n^x)^{\\ell }(y_n) \\in J_*$ for $\\ell \\ge 0$ , then $\\displaystyle v_* = \\lim _{\\ell \\rightarrow \\infty }~ w_{\\ell }$ .",
"Observe that ${\\bf h}_1^x \\circ {\\bf g}_n^x(y_n) = {\\bf h}_1^x(x) = y_1$ , and recall that ${\\bf d}_{{\\mathcal {X}}}(y_* , y_n) < \\delta _*/4$ for all $n$ , hence, ${\\bf d}_{{\\mathcal {X}}}(y_1, y_n) < \\delta _*/2$ .",
"Since $w_0 = y_n$ and $w_1 = y_1$ , the estimate (REF ) implies that ${\\bf d}_{{\\mathcal {X}}}(w_{\\ell }, w_{\\ell +1}) < 2^{-\\ell } \\cdot {\\bf d}_{{\\mathcal {X}}}(w_0, w_1) < 2^{-\\ell } \\cdot \\delta _*/2 ~ .$ Summing these estimates for $\\ell \\ge 1$ , we obtain that ${\\bf d}_{{\\mathcal {X}}}(w_0 , v_*) = {\\bf d}_{{\\mathcal {X}}}(y_n , v_*) \\le \\delta _*$ so that ${\\bf d}_{{\\mathcal {X}}}(y_*, v_*) \\le {\\bf d}_{{\\mathcal {X}}}(y_*, y_n) + {\\bf d}_{{\\mathcal {X}}}(y_n, v_*) < 2 \\delta _* \\le \\delta _1/2 ~ .$ Then by (REF ) we have $\\displaystyle ({\\bf h}_1^x \\circ {\\bf g}_n^x)^{\\prime }(v_*) \\le \\mu $ , as was to be shown.",
"The conclusions of Lemma REF essentially yield the proof of Proposition REF , except that it remains to make a change of notation so the results are in the form stated in the proposition, and check that conditions (1) to (5) of Proposition REF .1 are satisfied.",
"This change of notation is done so that the conclusions are in a standard format, which will be invoked recursively in the following Section to prove there exists “ping-pong” dynamics in the holonomy pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ .",
"Choose $n \\ge N_0$ so that the hypotheses of Lemma REF are satisfied, then define $\\phi _1 = {\\bf h}_1^x$ and $\\psi _1 = {\\bf g}_n^x$ so that $\\Phi _1 = \\phi _1 \\circ \\psi _1 = {\\bf h}_1^x \\circ {\\bf g}_n^x$ , and recall that $g_1^x(J_*)$ $J_* = [y_* - 3\\delta _0^{\\prime }, y_* + 3\\delta _0^{\\prime }] \\subset J_n^x = [y_n - 4\\delta _0^{\\prime }, y_n+4\\delta _0^{\\prime }]$ for $\\delta _0^{\\prime }$ and $y_*$ as defined above.",
"Set $u_1 = v_*$ and $v_1 = {\\bf g}_n^x(v_*)$ .",
"We check that conditions (REF .1) and (REF .4) of Proposition REF are satisfied: $\\Phi _1(u_1) = \\phi _1 \\circ \\psi _1(u_1) = {\\bf h}_1^x \\circ {\\bf g}_n^x(v_*) = v_* = u_1 ~ , $ $\\Psi _1(v_1) = \\psi _1 \\circ \\phi _1(v_1) = {\\bf g}_n^x \\circ {\\bf h}_1^x({\\bf g}_n^x(v_*)) = {\\bf g}_n^x(v_*) = v_1 ~ .$ Next, for ${\\mathcal {J}}_1 = [u_1 - \\delta _0^{\\prime }, u_1 + \\delta _0^{\\prime }] = [v_* - \\delta _0^{\\prime }, v_* + \\delta _0^{\\prime }]$ as defined in (REF .2), by the estimate (REF ) we have $\\displaystyle {\\bf d}_{{\\mathcal {X}}}(y_*, v_*) < 2 \\delta _* \\le \\delta _0^{\\prime }/2$ from which it follows that ${\\mathcal {J}}_1 \\subset J_*$ .",
"Then condition (REF .3) follows from (REF ) since $u_1 = v_* \\in J_*$ .",
"Finally, to show condition (REF .5) of Proposition REF is satisfied, recall that $\\psi _1(y_n) = {\\bf g}_1^x(y_n) = x$ , that ${\\bf d}_{{\\mathcal {X}}}(y_n , v_*) < \\delta _* \\le 1$ by the proof of Lemma REF , and that $\\delta _* = \\min \\lbrace 1,\\delta _0^{\\prime }/4,\\delta _1/4\\rbrace $ .",
"Also, the estimate (REF ) combined with () and the choice of $\\delta _0^{\\prime } \\le 1$ in Definition REF yields that, for all $y \\in J_*$ $({\\bf g}_n^x)^{\\prime }(y) \\le \\exp \\lbrace (-a + 2{\\epsilon }_1) \\, \\ell _n\\rbrace < \\delta _0^{\\prime } \\cdot \\exp \\lbrace -a\\, n/2\\rbrace < \\delta _1/2 ~ .$ Thus, by the Mean Value Theorem and the estimate ${\\bf d}_{{\\mathcal {X}}}(y_n, v_*) \\le \\delta _* \\le 1$ , we have that ${\\bf d}_{{\\mathcal {X}}}(x, v_1) = {\\bf d}_{{\\mathcal {X}}}({\\bf g}_n^x(y_n), {\\bf g}_n^x(v_*)) \\le \\delta _1/2 \\cdot {\\bf d}_{{\\mathcal {X}}}(y_n, v_*) \\le \\delta _1/2 ~ .$ For any $y \\in {\\mathcal {J}}_1 = [v_* - \\delta _0^{\\prime }, v_* + \\delta _0^{\\prime }]$ we also have that ${\\bf d}_{{\\mathcal {X}}}({\\bf g}_n^x(y), v_1) = {\\bf d}_{{\\mathcal {X}}}({\\bf g}_n^x(y), {\\bf g}_n^x(v_*)) \\le \\delta _1/2 \\cdot {\\bf d}_{{\\mathcal {X}}}(y, v_*) \\le \\delta _0^{\\prime } \\delta _1/2 < \\delta _1/2 ~.$ Thus, $ {\\bf d}_{{\\mathcal {X}}}({\\bf g}_n^x(y), x) \\le {\\bf d}_{{\\mathcal {X}}}({\\bf g}_n^x(y), v_1) + {\\bf d}_{{\\mathcal {X}}}(x, v_1) < \\delta _1 ~ , $ so that ${\\mathcal {K}}_1 =\\psi _1({\\mathcal {J}}_1) \\subset [x - \\delta _1, x+ \\delta _1]$ , as was to shown.",
"This completes the proof of Proposition REF ."
],
[
"Hyperbolic sets with positive measure",
"The main result of this section is: THEOREM 6.1 Let ${\\mathcal {F}}$ be a $C^1$ -foliation of codimension-one of a compact manifold $M$ for which ${\\rm E}^+({\\mathcal {F}})$ has positive Lebesgue measure.",
"Then ${\\mathcal {F}}$ has a hyperbolic resilient leaf, and hence the geometric entropy 0.",
"The assumption that the Lebesgue measure $|{\\rm E}^+({\\mathcal {F}})| > 0$ is used in two ways.",
"First, the set ${\\rm E}^+({\\mathcal {F}})$ is an increasing union of the sets ${\\rm E}^+_a({\\mathcal {F}})$ for $a > 0$ , so $|{\\rm E}^+({\\mathcal {F}})| > 0$ implies $|{\\rm E}^+_a({\\mathcal {F}})| > 0$ for some $a> 0$ .",
"For each $x \\in {\\rm E}^+_a({\\mathcal {F}})$ , we obtain from Proposition REF uniform hyperbolic contractions with fixed-points arbitrarily close to the given $x \\in E$ , and with prescribed bounds on their domains.",
"Secondly, almost every point of a measurable set is a point of positive Lebesgue density, hence $|{\\rm E}^+_a({\\mathcal {F}})| > 0$ implies that ${\\rm E}^+_a({\\mathcal {F}})$ has a “pre-perfect” subset of points with expansion greater than $a$ .",
"This observation enables us to construct an infinite sequence of hyperbolic fixed-points arbitrarily close to the support of ${\\rm E}^+_a({\\mathcal {F}})$ , whose domains have to eventually overlap since the closure $\\overline{{\\mathcal {X}}}$ is compact.",
"This yields the existence of a resilient orbit for ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ , hence a ping-pong game dynamics as defined in Section REF , which implies that 0.",
"DEFINITION 6.2 A set ${\\mathcal {E}}$ is said to be pre-perfect if it is non-empty, and its closure $\\overline{{\\mathcal {E}}}$ is a perfect set.",
"Equivalently, ${\\mathcal {E}}$ is pre-perfect if it is not empty, and no point is isolated.",
"The following observation is a standard property of sets with positive Lebesgue measure.",
"LEMMA 6.3 If $X \\subset {\\mathbb {R}}^q$ has positive Lebesgue measure, then there is a pre-perfect subset ${\\mathcal {E}}\\subset X$ .",
"Let ${\\mathcal {E}}\\subset X$ be the set of points with Lebesgue density 1.",
"Recall that this means that for each $x \\in X$ and each $\\delta > 0$ , the Lebesgue measure $|B_X(x,\\delta ) \\cap X| > 0$ , and $\\displaystyle \\lim _{\\delta \\rightarrow 0}\\frac{ |B_X(x,\\delta ) \\cap X|}{|B_X(x,\\delta )|} = 1$ .",
"It is a standard fact of Lebesgue measure theory that $|{\\mathcal {E}}| = |X|$ , so that $|X| > 0$ implies that ${\\mathcal {E}}\\ne \\emptyset $ .",
"Moreover, if $x \\in {\\mathcal {E}}$ is isolated in ${\\mathcal {E}}$ , then $x$ is a point with Lebesgue density 0, thus each $x \\in {\\mathcal {E}}$ cannot be isolated.",
"It follows that ${\\mathcal {E}}$ is pre-perfect.",
"Theorem REF now follows from Lemma REF and the following result: PROPOSITION 6.4 Let $a>0$ , and suppose there exists a pre-perfect subset ${\\mathcal {E}}\\subset {\\rm E}^+_a({\\mathcal {F}})$ , then ${\\mathcal {F}}$ has a resilient leaf contained in the closure $\\overline{{\\rm E}^+_a({\\mathcal {F}})}$ .",
"Let $a > 0$ and let ${\\mathcal {E}}\\subset {\\rm E}^+_a({\\mathcal {F}})$ be a pre-perfect set.",
"The saturation of a pre-perfect set under the action of the holonomy pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is pre-perfect, so we can assume that ${\\mathcal {E}}$ is saturated.",
"We assume that ${\\mathcal {F}}$ does not have a resilient leaf in $\\overline{{\\rm E}^+_a({\\mathcal {F}})}$ , and show this leads to a contradiction.",
"We follow the notation introduced in the proof of Proposition REF , which will be invoked repeatedly, and the resulting maps and constants will be labeled according to the stage of the induction.",
"Choose $0 < {\\epsilon }_1 < \\min \\lbrace {\\epsilon }_0, a/100\\rbrace $ , and let $\\delta _0$ be chosen as in Definition REF .",
"Fix a choice of $0 < \\mu < 1$ , and choose $0 < \\delta _1 < \\delta _0$ and $x_1 \\in {\\mathcal {E}}\\cap {\\mathcal {X}}_{\\alpha }$ .",
"Then by Proposition REF , there exists holonomy maps $\\phi _1, \\psi _1 \\in {{\\mathcal {G}}}_{{\\mathcal {F}}}$ and points $u_1 \\in \\overline{{\\mathcal {X}}}$ and $v_1 = \\psi _1(u_1)$ , such that ${\\bf d}_{{\\mathcal {X}}}(x_1,v_1) < \\delta _1$ and which are fixed-points for the maps $\\Phi _1$ , $\\Psi _1$ respectively.",
"Moreover, we have the sets ${\\mathcal {J}}_1 \\equiv [u_1 - \\delta _0, u_1+ \\delta _0]$ ${\\mathcal {I}}_1 \\equiv \\Phi _1({\\mathcal {J}}_1) \\subset (u_1 - \\delta _0, u_1+ \\delta _0)$ ${\\mathcal {K}}_1 \\equiv \\psi _1({\\mathcal {J}}_1) \\subset (x_1 - \\delta _1, x_1+ \\delta _1)$ whose properties were given in Proposition REF .",
"In particular, $\\Phi _1 \\colon {\\mathcal {J}}_1 \\rightarrow {\\mathcal {I}}_1 \\subset {\\mathcal {J}}_1$ is a hyperbolic contraction with fixed-point $u_1$ .",
"In particular, note that $\\displaystyle \\bigcap _{\\ell > 0} \\, \\Phi _1^{\\ell }({\\mathcal {J}}_1) = \\lbrace u_1 \\rbrace $ .",
"If the orbit of $u_1$ under ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ intersects ${\\mathcal {J}}_1$ in a point other than $u_1$ , then by definition, $u_1$ is a hyperbolic resilient point, which by assumption does not exist.",
"Therefore, the ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ -orbit of $u_1$ intersects the interval ${\\mathcal {J}}_1$ exactly in the interior point $u_1$ , and intersects ${\\mathcal {K}}_1$ exactly in the interior point $v_1$ .",
"Note that $x_1 \\in {\\mathcal {K}}_1 \\cap {\\mathcal {E}}$ so there exists $x_2 \\in ({\\mathcal {K}}_1 - \\lbrace x_1, v_1\\rbrace ) \\cap {\\mathcal {E}}$ as ${\\mathcal {E}}$ is pre-perfect.",
"Choose $0 < \\delta _2 < \\delta _1$ so that $\\displaystyle (x_2 - \\delta _2, x_2 + \\delta _2) \\subset ({\\mathcal {K}}_1 - \\lbrace x_1, v_1\\rbrace )$ .",
"The ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ -orbit of $v_1$ intersects ${\\mathcal {K}}_1$ only in the point $v_1$ , thus the interval $(x_2 - \\delta _2, x_2 + \\delta _2)$ is disjoint from the orbit of $v_1$ .",
"We then repeat the construction in the proof of Proposition REF , to obtain holonomy maps $\\phi _2, \\psi _2 \\in {{\\mathcal {G}}}_{{\\mathcal {F}}}$ and points $u_2 \\in \\overline{{\\mathcal {X}}}$ and $v_2 = \\psi _2(u_2)$ , such that ${\\bf d}_{{\\mathcal {X}}}(x_2,v_2) < \\delta _2$ and which are fixed-points for the maps $\\Phi _2$ , $\\Psi _2$ respectively.",
"Again, define the sets ${\\mathcal {J}}_2 \\equiv [u_2 - \\delta _0, u_2+ \\delta _0]$ ${\\mathcal {I}}_2 \\equiv \\Phi _2({\\mathcal {J}}_2) \\subset (u_2 - \\delta _0, u_2+ \\delta _0)$ ${\\mathcal {K}}_2 \\equiv \\psi _2({\\mathcal {J}}_2) \\subset [x_2 - \\delta _2, x_2+ \\delta _2]$ .",
"We then repeat this construction recursively.",
"Let $\\lbrace u_1, u_2, \\ldots \\rbrace \\subset \\overline{{\\mathcal {X}}}$ be the resulting centers of contraction for the hyperbolic maps $\\lbrace \\Phi _{i} \\mid i > 0\\rbrace $ .",
"As $\\overline{{\\mathcal {X}}}$ is compact, there exists an accumulation point $u_* \\in \\overline{{\\mathcal {X}}}$ .",
"In particular, there exists distinct indices $i_1, i_2 > 0$ such that ${\\bf d}_{{\\mathcal {X}}}(u_*, u_{i_1}) < \\delta _0/10$ and ${\\bf d}_{{\\mathcal {X}}}(u_*, u_{i_2}) < \\delta _0/10$ and thus ${\\bf d}_{{\\mathcal {X}}}(u_{i_1}, u_{i_2}) < \\delta _0/5$ .",
"Recall that the intervals ${\\mathcal {J}}_{i_1} = [u_{i_1} - \\delta _0, u_{i_1}+ \\delta _0]$ and ${\\mathcal {J}}_{i_2} = [u_{i_2} - \\delta _0, u_{i_2}+ \\delta _0]$ have uniform width, and moreover $\\displaystyle \\lbrace u_{i_1}, u_{i_2}\\rbrace \\subset {\\mathcal {J}}_{i_1} \\cap {\\mathcal {J}}_{i_2}$ .",
"As $u_{i_1}$ and $u_{i_2}$ are disjoint fixed-points of hyperbolic attractors, we can choose integers $m_1, m_2 > 0$ so that $\\displaystyle \\Phi _{i_1}^{m_1}({\\mathcal {J}}_{i_1}) \\cap \\Phi _{i_2}^{m_2}({\\mathcal {J}}_{i_2}) = \\emptyset $ and $\\Phi _{i_j}^{m_j}({\\mathcal {J}}_{i_j}) \\subset {\\mathcal {J}}= {\\mathcal {J}}_{i_1} \\cap {\\mathcal {J}}_{i_2}$ for $j=1,2$ .",
"Then the action of the contracting maps ${\\bf H}= \\Phi _{i_1}^{m_1}$ and ${\\bf G}= \\Phi _{i_2}^{m_2}$ on ${\\mathcal {J}}$ define a “ping-pong game” as in Definition REF .",
"Now let $x = u_{i_1}$ , $y = {\\bf G}(x) \\ne x$ , then ${\\bf H}^{\\ell }(y) \\rightarrow x$ as $\\ell \\rightarrow \\infty $ , so that the orbit of $x$ under the action ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is resilient, contrary to assumption.",
"Hence, if there exists a pre-perfect set ${\\mathcal {E}}\\subset {\\rm E}^+_a({\\mathcal {F}})$ for $a > 0$ , then there exists a resilient leaf."
],
[
"Open manifolds",
"In this section, we extend the methods above from compact manifolds to open manifolds, using the techniques of [38].",
"THEOREM 7.1 Let ${\\mathcal {F}}$ be a codimension-one $C^2$ -foliation of an open complete manifold $M$ .",
"If the Godbillon-Vey class $GV({\\mathcal {F}}) \\in H^3(M; {\\mathbb {R}})$ is non-zero, the ${\\mathcal {F}}$ has a hyperbolic resilient leaf.",
"The class $GV({\\mathcal {F}}) \\in H^3(M; {\\mathbb {R}})$ is determined by its pairing with the compactly supported cohomology group $H^{m-3}_c(M; {\\mathbb {R}})$ , so $GV({\\mathcal {F}}) \\ne 0$ implies there exists a closed $m-3$ form $\\xi $ with compact support on $M$ such that $\\langle GV({\\mathcal {F}}), [\\xi ]\\rangle \\ne 0$ .",
"Let $|\\xi | \\subset M$ denote the support of $\\xi $ , which is a compact set.",
"As the support $|\\xi |$ is compact, there is a finite open cover of $|\\xi |$ by a regular foliation atlas $\\lbrace (U_{\\alpha },\\phi _{\\alpha }) \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ for ${\\mathcal {F}}$ on $M$ (as in Section above).",
"Let $M_0$ denote the union of the sets $\\lbrace U_{\\alpha } \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ , then the closure $\\overline{M_0}$ is a compact subset of $M$ and $|\\xi | \\subset M_0$ .",
"Thus we have $GV({\\mathcal {F}}|M_0) \\ne 0$ .",
"If $M_0$ is not connected, we can choose a connected component $M_1 \\subset M_0$ for which $GV({\\mathcal {F}}|M_1) \\ne 0$ .",
"Thus, we may assume that $M_0$ is connected.",
"The proof of Theorem REF used only the properties of the pseudogroup generated by a regular foliation atlas $\\lbrace (U_{\\alpha },\\phi _{\\alpha }) \\mid \\alpha \\in {\\mathcal {A}}\\rbrace $ – the compactness of $M$ was not used except in the construction of this atlas.",
"The definition and properties of the Godbillon measure also apply to open manifolds, as was discussed in [38].",
"Hence, by the same proof we obtain that the set ${\\rm E}({\\mathcal {F}}|M_0)$ has positive measure.",
"The proofs of Propositions REF and REF use only the assumption that the pseudogroup ${{\\mathcal {G}}}_{{\\mathcal {F}}}$ is compactly generated, as defined by Haefliger [32], and do not require the compactness of $M$ , hence apply directly to show that ${{\\mathcal {G}}}{\\mathcal {F}}|M_0$ has a hyperbolic resilient point if ${\\rm E}({\\mathcal {F}}|M_0)$ has positive measure.",
"Thus, ${\\mathcal {F}}|M_0$ must have a resilient leaf, and so also must ${\\mathcal {F}}$ .",
"Here is an application of Theorem REF .",
"Let ${\\bf B}\\Gamma _1^{(2)}$ denote the Haefliger classifying space of codimension–one $C^2$ -foliations [30], [31].",
"There is a universal Godbillon-Vey class $GV \\in H^3({\\bf B}\\Gamma _1^{(2)}; {\\mathbb {R}})$ such that for every codimension–one $C^2$ -foliation ${\\mathcal {F}}$ of a manifold $M$ , there is a classifying map $h_{{\\mathcal {F}}} \\colon M \\rightarrow {\\bf B}\\Gamma _1^{(2)}$ such that $h_{{\\mathcal {F}}}^* GV = GV({\\mathcal {F}})$ (see [6], [53].)",
"The first two integral homotopy groups $\\pi _1({\\bf B}\\Gamma _1^{(2)}) = 0 = \\pi _2({\\bf B}\\Gamma _1^{(2)})$ , while Thurston showed in [72] that the Godbillon-Vey class defines a surjection $GV \\colon \\pi _3({\\bf B}\\Gamma _1^{(2)}) \\rightarrow {\\mathbb {R}}$ .",
"It follows from Thurston's work in [73], that for a closed oriented 3-manifold $M$ and any $a >0$ , there exists a codimension–one foliation ${\\mathcal {F}}_a$ on $M$ such that $\\langle GV({\\mathcal {F}}_a) , [M]\\rangle = a$ .",
"Each such foliation ${\\mathcal {F}}_a$ for $a \\ne 0$ must then have resilient leaves.",
"More generally, given any finite CW complex $X$ , a continuous map $h \\colon X \\rightarrow {\\bf B}\\Gamma _1^{(2)}$ defines a foliated microbundle over $X$ , whose total space $M$ is an open manifold with a codimension–one foliation ${\\mathcal {F}}_h$ such that $h^*GV = GV({\\mathcal {F}}_h)$ .",
"This is discussed in detail by Haefliger [31], who introduced the technique.",
"(See also Lawson [53].)",
"Thus, using homotopy methods to construct the map $h$ so that $h^*GV \\ne 0$ , one can construct many examples of open foliated manifolds with non-trivial Godbillon-Vey classes.",
"Theorem REF implies that all such examples have resilient leaves."
]
] | 1403.0494 |
[
[
"Dark Matter as a Trigger for Periodic Comet Impacts"
],
[
"Abstract Although statistical evidence is not overwhelming, possible support for an approximately 35 million year periodicity in the crater record on Earth could indicate a nonrandom underlying enhancement of meteorite impacts at regular intervals.",
"A proposed explanation in terms of tidal effects on Oort cloud comet perturbations as the Solar System passes through the galactic midplane is hampered by lack of an underlying cause for sufficiently enhanced gravitational effects over a sufficiently short time interval and by the time frame between such possible enhancements.",
"We show that a smooth dark disk in the galactic midplane would address both these issues and create a periodic enhancement of the sort that has potentially been observed.",
"Such a disk is motivated by a novel dark matter component with dissipative cooling that we considered in earlier work.",
"We show how to evaluate the statistical evidence for periodicity by input of appropriate measured priors from the galactic model, justifying or ruling out periodic cratering with more confidence than by evaluating the data without an underlying model.",
"We find that, marginalizing over astrophysical uncertainties, the likelihood ratio for such a model relative to one with a constant cratering rate is 3.0, which moderately favors the dark disk model.",
"Our analysis furthermore yields a posterior distribution that, based on current crater data, singles out a dark matter disk surface density of approximately 10 solar masses per square parsec.",
"The geological record thereby motivates a particular model of dark matter that will be probed in the near future."
],
[
"Dark Matter as a Trigger for Periodic Comet Impacts Lisa Randall and Matthew Reece Department of Physics, Harvard University, Cambridge, MA, 02138 Although statistical evidence is not overwhelming, possible support for an approximately 35 million year periodicity in the crater record on Earth could indicate a nonrandom underlying enhancement of meteorite impacts at regular intervals.",
"A proposed explanation in terms of tidal effects on Oort cloud comet perturbations as the Solar System passes through the galactic midplane is hampered by lack of an underlying cause for sufficiently enhanced gravitational effects over a sufficiently short time interval and by the time frame between such possible enhancements.",
"We show that a smooth dark disk in the galactic midplane would address both these issues and create a periodic enhancement of the sort that has potentially been observed.",
"Such a disk is motivated by a novel dark matter component with dissipative cooling that we considered in earlier work.",
"We show how to evaluate the statistical evidence for periodicity by input of appropriate measured priors from the galactic model, justifying or ruling out periodic cratering with more confidence than by evaluating the data without an underlying model.",
"We find that, marginalizing over astrophysical uncertainties, the likelihood ratio for such a model relative to one with a constant cratering rate is 3.0, which moderately favors the dark disk model.",
"Our analysis furthermore yields a posterior distribution that, based on current crater data, singles out a dark matter disk surface density of approximately 10 $M_\\odot /{\\rm pc}^2$ .",
"The geological record thereby motivates a particular model of dark matter that will be probed in the near future.",
"Large meteorite strikes on Earth cause big impact craters that are very likely responsible for some mass extinctions [1].",
"Possible evidence of $\\approx 35$ million year periodicity in the dates of these events suggest a nonrandom underlying origin [2], [6], [7], [3], [4], [5], [8], [9], [10], [11], [12].",
"Although not yet clearly established, it is of interest to explore possible underlying causes, especially if they have other measurable consequences.",
"Two suggestions were made simultaneously by multiple groups to explain a periodic enhancement of Oort cloud induced comets hitting the Earth.",
"One, known as the “Nemesis hypothesis,” was that the Sun has a so-far undetected companion star [3], [4].",
"No companion has been detected.",
"The other suggestion involves the Sun moving through the plane of the galaxy.",
"The Milky Way, like other spiral galaxies, has a large fraction of its normal (baryonic) matter arranged in the shape of a flattened disk, with the density falling off exponentially over a characteristic distance of 3 kpc in the radial direction but in a much shorter characteristic distance of about 300 parsecs in the vertical direction [13], [14].",
"The flattened shape arises because normal matter cools by emitting photons that carry kinetic energy away from the galaxy.",
"This lowers the velocity of ordinary matter and the less energetic particles move in a smaller volume due to their reduced velocities and their gravitational interactions.",
"Such particles do however retain angular momentum, so the phase space doesn't shrink in the radial direction.",
"Matter therefore forms a flattened disk with small vertical height.",
"The idea for explaining periodic cratering is that the Sun, as it orbits the Galactic Center, oscillates up and down through the plane of the galaxy, leading to periodic perturbations of the Oort cloud from enhanced density near the plane.",
"These perturbations cause comets to enter the inner Solar System resulting in comet showers [6], [8].",
"However, to date no suggested mechanism for the enhanced density is successful in explaining the timing and magnitude of the periodicity.",
"Molecular clouds have been suggested [6], [8] but they have been shown to be spread too far from the plane to justify periodic cratering [15].",
"The period is in any case too short to be accounted for by conventional baryonic matter, which as mentioned above also does not have a large enough vertical density gradient to explain a strong periodic signal.",
"Remarkably, a dark matter disk could address both these issues.",
"Despite the apparent lack of fundamental explanation, studies have searched for periodic phenomena by fitting ad hoc sinusoidal templates without an underlying physical model.",
"These were recently reviewed in Ref. [12].",
"Recent analyses of the crater data usually find that a period of about 35 Myr is most consistent with the data, although the statistical evidence is weak and disappears completely when the look elsewhere effect is taken into account (if there is no prior favoring particular periods).",
"In this letter, we conjecture that thin dark matter disks, which would form if a species of dark matter has dissipative dynamics [16], could affect meteorite impacts and address both of the above issues.",
"The bulk of dark matter, based on observed rotation curves and expected properties of weakly interacting particles, is known to be arranged in a roughly spherical halo, gradually growing less dense over distances of order 20 kpc.",
"However, this has been established only for the majority of dark matter.",
"A small fraction might have interactions similar to those of baryons, emitting “dark photons” and dissipating energy, thereby cooling into an even thinner dark disk embedded in the ordinary baryonic disk [16].",
"The existence or nonexistence of such a disk will be probed most directly over the next decade through extensive measurements of stellar kinematics in the Milky Way [17], [18].",
"Assuming the dominant perturbing mechanism is the tidal force, which is proportional to the density of the disk [19], the Sun's passage through the dark matter disk would cause enhanced periodic Oort cloud perturbations.",
"We find that the observed crater dates agree with such a model better than with a constant cratering rate by a likelihood ratio of 3.0, and single out a dark matter disk surface density of approximately 10 $M_\\odot /{\\rm pc}^2$ .",
"This proposal will be tested by upcoming measurements from the Gaia satellite that will narrow the range of priors, and hence the possible cratering predictions.",
"More precise measurements of the Milky Way's properties will thereby provide a sharper statistical test of the comet shower hypothesis.",
"The results could ultimately reveal a strong dark matter influence on the history of our Solar System and even of life here on Earth.",
"We reframe the problem of testing galactic influences on the terrestrial impact crater record in a form that is more robust than testing the data for periodicity.",
"The observation of possible periodicity was an important impetus for the original hypotheses of astrophysical influences on life on Earth.",
"However, the science will be vastly improved by setting priors with current and future data about our galaxy.",
"We show how to use all available measured data to pin down the shape of the galaxy and derive a detailed trajectory of the Sun as it oscillates through.",
"Figure: The Sun's height above the galactic plane as a function of time, extrapolated backward via Eq. .",
"The corresponding cratering probability is shown in Fig. .",
"Inset: an illustration of how the Sun moves around the galactic center while also oscillating vertically; the vertical oscillation is exaggerated for visibility.Our focus in this letter is the influence of a dark disk in this context.",
"We take the mass distribution of the disk to be an isothermal sheet, with density: $\\rho _{\\rm disk}(R,z) = \\frac{M_{\\rm disk}}{8\\pi R_d^2 z_d} \\exp \\left(-R/R_d\\right) {\\rm sech}^2\\left(z/\\left(2 z_d\\right)\\right).",
"$ This form, with density falling exponentially with radius and height, can be derived from the Poisson equation for a gravitationally interacting set of particles that have a Maxwellian vertical velocity distribution [14].",
"In reality, a thin disk will fragment into smaller clouds or clumps, the final size of which depends on details of cooling and angular momentum transfer.",
"We assume their distribution to be uniform enough to approximate a smooth tidal field.",
"We characterize the matter in a disk via its surface density $\\Sigma $ , which is the integral of $\\rho (R,z)$ over $z$ at fixed radius $R$ .",
"We assume an equal scale radius for baryons and dark disk matter, $R_d \\approx 3$ kpc [13].",
"We use a one-dimensional model of the Sun's motion through the galaxy, assuming small vertical oscillations around a circular orbit, with acceleration determined by the local density: $a(z) \\equiv \\ddot{z} = - \\frac{\\partial \\Phi }{\\partial z} \\approx - \\int dz~4\\pi G\\rho (z) .$ This equation relies on the fact that the Milky Way's rotation curve is flat at the radius of the Sun's orbit, so that $\\frac{\\partial }{\\partial R}\\left(R \\frac{\\partial \\Phi }{\\partial R}\\right) \\approx 0$ .",
"An example of the vertical motion derived in this approximation is shown in Fig.",
"REF .",
"We assume, as a first approximation, that the probability that a comet shower begins at a time $t$ is proportional to the total local matter density near the Sun at that time, $\\rho (t)$ .",
"This assumption is motivated by Refs.",
"[19], [20], [21], which argue that perturbations to the Oort cloud are a result of tidal forces.",
"The initial paper by Heisler and Tremaine [19] demonstrates that tidal effects dominate over stellar perturbations.",
"However, because they assume a uniform disk, comet showers came only from the combined effects of stars and the tide [20] and occur only infrequently.",
"With a dark disk the tidal effect still dominates, and with a thin disk the temporal variation can suffice to explain even a 35 million year period.",
"The tidal forces gradually alter the angular momentum of the comet by modifying its transverse velocity $v_T$ : up to factors depending on time-dependent angles, $dv_T/dt \\sim r \\partial _z^2 \\Phi (z) \\sim 4\\pi G r\\rho $ , with $\\Phi $ the gravitational potential and $r$ the Sun-comet distance.",
"Comets with $v_T$ small enough compared to the circular velocity move on approximately radial orbits, falling into the inner Solar System.",
"Thus, tidal forces gradually strip comets with small transverse velocities out of the Oort cloud at a rate proportional to the local density at any given time.",
"These comets near the edge of the loss cone enter the inner Solar System in a time of order their orbital time of $\\stackrel{<}{{}_\\sim }1$ Myr, which is less than or approximately the time of transit of the dark disk.",
"We model this time delay based on a published result that used Monte Carlo simulation to deduce the longevity of the perturbation's influence [5], illustrated in Fig.",
"REF .",
"The convolution of this time delay with the density $\\rho (t)$ near the Sun defines $r(t)$ , the rate for impact craters at time $t$ .",
"We confront the model with observations of craters listed in the Earth Impact Database [22].",
"We (arbitrarily) choose to focus on craters greater than 20 km in diameter (since smaller craters occur much more frequently and don't necessarily require large comet-induced impacts) within the last 250 million years (as a minimal way to model the fact that older craters are eroded and rarely found).",
"Ultimately we would want to be able to distinguish impacts due to asteroids vs. comets to obtain a better test of the hypothesis.",
"There is also data on $^{3}$ He in dust from comets that can ultimately lend support to (or refute) an assumed periodicity [23].",
"We find this possibility exciting but neglect this data for the time being.",
"Figure: Comet shower profile: dashed line from ; solid line an ansatz we use in the numerics.We focus on the likelihood ratio between two models, $\\frac{P\\left({\\rm data}|{\\rm model}_1\\right)}{P\\left({\\rm data}|{\\rm model}_2\\right)},$ which via Bayes' theorem is also the ratio of posterior probabilities for the models if we begin with equal prior probabilities.",
"We are interested in whether the data provide support to a model in which the rate of impact crater events over time, $r(t)$ , is driven by the Sun's motion through the galaxy.",
"We compute the likelihood $P({\\rm data}|{\\rm model})$ as a product of the probabilities for each event, which are given by the overlap of the Gaussian characterizing the observed crater age with the model's rate function $r(t)$ : $P_{\\rm event}(E_i) = \\int _{t_{\\rm begin}}^{t_{\\rm end}}\\frac{ (t_{\\rm end} - t_{\\rm begin})r(t)}{\\sqrt{2\\pi }\\sigma _i}e^{-\\frac{(t - t_i)^2}{2\\sigma _i^2}} dt.",
"$ The factor $t_{\\rm end} - t_{\\rm begin}$ is present so the result will be dimensionless; we will compare ratios of likelihoods, so this factor will drop out.",
"For any given set of galactic parameters, we normalize $r(t)$ so that the average expected number of craters in 250 Myr matches the number in our sample, which provides an optimal fit.",
"Ideally, in the future a detailed model of the Oort cloud would specify the normalization of $r(t)$ , in which case a factor $P_{\\rm gap}(t_0, t_1) = \\exp \\left(-\\int _{t_0}^{t_1} r(t) dt\\right)$ is required.",
"The likelihood ratio allows us to quantify the evidence for a hypothesis relative to a different hypothesis, which we take to be a constant probability per unit time.",
"The periodic fits in the literature to date with a period of about 35 million years match the data better than an assumed constant rate of meteorite hits, but the statistical significance seems to disappear when the “look elsewhere” effect is taken into account [12].",
"That is, there are so many possible periodic functions that the fact that some do better is not a significant result.",
"That conclusion changes when a model with measured priors is used, rather than a random periodic model.",
"Constraints on the galactic density select a range of reasonable periods.",
"Table: Summary of the factors making up the prior probability distribution.Figure: An example of a model that provides a good fit.",
"The parameters of the dark disk are Σ D =13M ⊙ / pc 2 \\Sigma _D = 13 M_\\odot /{\\rm pc}^2 and z d D =5.4 pc z_d^D = 5.4~{\\rm pc}.",
"The baryonic disk is 350 pc thick with total surface density 58 M ⊙ / pc 2 M_\\odot /{\\rm pc}^2.",
"The local dark halo density is 0.037 GeV/cm 3 ^3.Z ⊙ =20Z_\\odot = 20 pc and W ⊙ W_\\odot = 7.8 km/s.",
"In this case, the period between disk crossings is about 35 Myr.",
"In orange is the rate r(t)r(t) of comet impacts (with arbitrary normalization).",
"This is approximately proportional to the local density, but convolved with the shower profile from Fig. .",
"The various blue curves each correspond to one recorded crater impact.Figure: Preferred parameters.",
"One-dimensional projections of the prior (blue, dashed) and posterior (orange, solid) probability distributions.",
"(a) The surface density of the dark disk, which the posterior distribution prefers to be between about 10 and 15 M ⊙ / pc 2 M_\\odot /{\\rm pc}^2.",
"(b) The dark disk thickness, which fits best at about 10 parsec scale height but extends to thinner disks.",
"The posterior distribution is flat even for very thin disks, because comet showers last for around a million years even if the Solar System passes through the disk in a shorter time.",
"(c) The local density of disk dark matter (relevant for solar capture or direct detection), which has significant weight up to several GeV/cm 3 ^3.",
"(d) The interval between times when the Sun passes through the dark disk, which fits best at values of about 35 Myr.Our assumed parameters are: baryonic disk parameters $\\Sigma _B$ and $z^B_d$ , the dark disk parameters $\\Sigma _D$ and $z^D_d$ , the dark halo parameter $\\rho _{\\rm halo}$ , the Sun's position $Z_\\odot $ and velocity $W_\\odot $ .",
"Collectively, these seven quantities parameterize the model, and to assign a likelihood we marginalize over them, i.e.",
"integrate over the space of parameters weighted by the prior distribution.",
"The seven parameters are straightforwardly related to the first seven constrained quantities in Table 1, with one extra constraint on the total surface density.",
"(By Eq.",
"REF , $\\Sigma _B = \\Sigma _B^{1.1}/\\tanh (1.1~{\\rm kpc}/(2 z_d^B))$ .)",
"We sample random numbers directly from the distributions in each row of Table 1 except for $\\Sigma _{\\rm tot}^{1.0}$ .",
"We then compute this total density for the sample parameters and apply Monte Carlo sampling (keeping the point if a random number is less than the weight assigned to $\\Sigma _{\\rm tot}^{1.0}$ in the last line of the table).",
"Thus, in the end $\\Sigma _D$ does not have a flat distribution, but has been reweighted to penalize choices with too much total density.",
"After marginalizing over all parameters, we find a likelihood ratio of 3.0 for the dark disk model compared to a uniform cratering rate.",
"In other words, if we assigned equal prior probabilities, then in light of the data our model is more likely by a factor of 3.",
"This Bayes factor is not large enough to be decisive, but it is intriguing.",
"It indicates we should find the dark disk moderately more plausible than we did a priori.",
"An example of parameters with larger likelihood is shown in Fig.",
"REF .",
"Although the likelihood ratio favors the model of a dark disk over a uniform rate, it does not tell us if either fits well.",
"Hence, we perform a Cramér–von Mises test to find a $p$ -value for the data (comparing empirical and theoretical cdfs).",
"For constant $r(t)$ , we find $p \\approx 0.09$ .",
"For the model in Fig.",
"REF , this improves to $p \\approx 0.13$ .",
"Thus, these models give reasonable (but not perfect) fits to the data: we cannot reject them at 95% confidence level.",
"As such, it makes sense to compare them, and the likelihood ratio gives a mild preference to the disk model.",
"For a different perspective we consider the Akaike information criterion [29] as modified for small sample sizes .",
"This compares maximum $\\log {\\cal L}$ but penalizes models with more parameters: in our case seven parameters with a dark disk versus five without.",
"Our maximum likelihood difference is $\\Delta \\log {\\cal L} \\approx 6$ when $\\Sigma _D \\approx 13~M_\\odot /{\\rm pc}^2$ , and the modified AIC criterion asks for $\\Delta \\log {\\cal L} > 3.6$ , so again we find a preference for the dark disk model.",
"Furthermore, a Bayesian analysis makes predictions for the values of parameters that can be measured in the future.",
"We show the prior and posterior distributions for a few of our parameters in Figure REF .",
"The posterior distribution strongly favors a dark disk surface density of $\\Sigma _D \\sim 10~M_\\odot /{\\rm pc}^2$ and scale height $z_d^D \\sim 10$ pc.",
"These parameters are not yet tested, but involve a large enough dark matter disk density that we expect measurements of stellar kinematics from the Gaia satellite [17], [18] to be a stringent test of the proposal in the near future.",
"Once such measurements are in hand, we can turn the question around and predict a cratering rate, strengthening the link between galactic and terrestrial data.",
"This dark disk surface density is consistent with current observational constraints once the overall uncertainty in the dynamically determined surface density and the large uncertainty in the ISM is accounted for.",
"The ISM value of 13 $M_\\odot /{\\rm pc}^2$ includes 2 $M_\\odot /{\\rm pc}^2$ of hot gas and furthermore has an uncertainty that is not precisely given but can be reasonably taken as 50% .We thank Jo Bovy and Ruth Murray-Clay for discussions on this point.",
"Furthermore more recent textbooks and give values of 5.5 and 7.6 $M_\\odot /{\\rm pc}^2$ respectively.",
"The argument against a dark disk in did not include this source of uncertainty.Again we thank Jo Bovy for discussions.",
"The posterior distribution for the current volume density of dissipative dark matter near the Sun peaks at low values but is significant and relatively flat between 1 and 5 GeV/cm$^3$ .",
"These densities are significantly larger than those generally assumed in direct detection experiments on the basis of a spherical dark matter halo, leading to interesting model-dependent prospects for detecting low-energy nuclear recoils induced by disk dark matter , , .",
"We will present details elsewhere of a study with disks not necessarily aligned, although we find the new parameters do not lead to a larger likelihood ratio.",
"We conclude that if a dark disk exists, it could play a significant role in explaining the observed pattern of craters, and possibly even mass extinctions.",
"We have also demonstrated how to use measurements of the galaxy and Solar System to weight models with different parameters and ascertain the statistical significance of our hypothesis.",
"With the prospect of better data that will further constrain the model in the future, the statistical tests will become even more stringent, validating or ruling out our proposal.",
"Meanwhile we find this a fascinating possibility worthy of further exploration.",
"Even though crater data is hard to come by, data about the galaxy will be much more abundant in the near future.",
"When we pin this down we will be better able to unambiguously predict the motion of the Solar System and thereby constrain possibilities for nonrandom structure in crater timing.",
"Acknowledgments.",
"We thank Paul Davies for suggesting this intriguing idea, and for subsequent related correspondence.",
"We thank Oded Aharonson, Jo Bovy, Matthew Buckley, Sean Carroll, Ken Farley, Marat Freytsis, Fiona Harrison, Arthur Kosowsky, Eric Kramer, David Krohn, Avi Loeb, Ruth Murray-Clay and Scott Tremaine for useful discussions or correspondence.",
"We thank the referees for useful comments and suggested references.",
"The work of LR was supported in part by NSF grants PHY-0855591 and PHY-1216270.",
"MR thanks the KITP in Santa Barbara for its hospitality while a portion of this work was completed.",
"The KITP is supported in part by the National Science Foundation under Grant No.",
"NSF PHY11-25915."
]
] | 1403.0576 |
[
[
"Heisenberg versus standard scaling in quantum metrology with Markov\n generated states and monitored environment"
],
[
"Abstract Finding optimal and noise robust probe states is a key problem in quantum metrology.",
"In this paper we propose Markov dynamics as a possible mechanism for generating such states, and show how the Heisenberg scaling emerges for systems with multiple `dynamical phases' (stationary states), and noiseless channels.",
"We model noisy channels by coupling the Markov output to `environment' ancillas, and consider the scenario where the environment is monitored to increase the quantum Fisher information of the output.",
"In this setup we find that the survival of the Heisenberg limit depends on whether the environment receives `which phase' information about the memory system."
],
[
"Introduction",
"The accurate estimation of unknown parameters is a fundamental task in quantum technologies, with applications ranging from spectroscopy [1] and interferometry [2], [3], to atomic clocks [4], [5], [6] and gravitational wave detectors [7], [8].",
"In a typical metrological protocol [9], a quantum transformation $T_\\theta $ is applied (in parallel) to each component of a `probe' ensemble of $n$ quantum systems initially prepared in the joint state $|\\Psi ^n\\rangle $ .",
"The ensemble is subsequently measured and an estimator $\\hat{\\theta }_n$ of $\\theta $ is computed.",
"While for uncorrelated states the mean square error scales as $1/n$ (standard scaling), if quantum resources such as entanglement or squeezing are used in the preparation stage, the precision can be enhanced to $1/n^2$ (Heisenberg scaling) if $T_\\theta $ is unitary [2], [9], [10].",
"However, when noise and decoherence are taken into account, they typically lead to a `downgrading' of the Heisenberg scaling to the standard one, but a `quantum enhancement' is nevertheless achievable in the form of a constant factor that increases with decreasing noise level [13], [11], [12], [15], [14].",
"In this setup, new tools for deriving upper bounds on the quantum Fisher information of the final state have been developed in [15], [16], [17], [18].",
"We note also that the Heisenberg limit can be preserved for some noise models [19], [20] or by using quantum error correction techniques [21], [22], [23], [24] in certain modified metrological settings.",
"The aim of this paper is to explore quantum metrology in a novel setup characterised by two key features.",
"Firstly, we model the channel $T_\\theta $ as coupling with an ancilla (environment) and we assume that the latter can be monitored by means of measurements, as illustrated in Figure REF .",
"The outcome $\\underline{c}$ of the measurement provides additional information, which generally improves the estimation efficiency, and even restores the Heisenberg limit in certain models.",
"Secondly, the n-partite probe state is generated as output of a quantum Markov chain, which is similar to the matrix product states (MPS) ansatz proposed in [25].",
"More concretely, the probe systems are initially independent and identically prepared, and interact successively with a `memory' system which imprints correlations into the ensemble.",
"This specific preparation method allows us to apply system identification techniques for quantum Markov dynamics [26], [27], and to identify the mechanism responsible for the Heisenberg scaling and its degrading.",
"Figure: (Color online) Quantum metrology with nn noisy channels acting on a pure state input |Ψ n 〉|\\Psi ^n\\rangle .",
"The channel T θ (i) T^{(i)}_\\theta is monitored by measuring the associated `environment' and the result c i c_i is obtained.Conditional on the measurement record c ̲:=(c 1 ,⋯,c n )\\underline{c}:= (c_1, \\dots , c_n), the final probe state has quantum Fisher information F(Ψ n (θ|c ̲))F(\\Psi ^n(\\theta |\\underline{c})).In this setting, we observe that that if the Markov transition operator used for generating the probe state is primitive (irreducible and aperiodic), then the quantum Fisher information (QFI) scales linearly with $n$ even with full access to the environment, and the standard limit holds.",
"We therefore consider Markov models with multiple `dynamical phases' (invariant spaces), and investigate the evolution of the probe state QFI, conditional on the environment measurement record.",
"We show that one of the following two scenarios can occur.",
"If the environment measurement does not distinguish between the dynamical phases, and the memory is started in a superposition of different phases, then the QFI of the conditional output state scales as $n^2$ , and the Heisenberg scaling holds.",
"In a two-phases Markov dynamics for instance, for large $n$ the conditional memory-probe state becomes a `macroscopic' superposition $|\\Psi ^n(\\theta |\\underline{c}) \\rangle = |\\Psi ^{n,0}(\\theta |\\underline{c})\\rangle +|\\Psi ^{n,1}(\\theta |\\underline{c})\\rangle $ of components whose weights remain constant even when the environment is observed.",
"The quantum Fisher information of this superposition is proportional to the variance of the generator $\\bar{G}$ responsible for the parameter change.",
"For each component $a=0,1$ , the mean value of the generator increases linearly with $n$ as $ng_a$ , so that if $g_0\\ne g_1$ , the variance of $\\bar{G}$ with respect tot the superposition, grows as $n^2(g_0-g_1)^2$ .",
"Alternatively, if the environment receives `which phase' information about the memory, the QFI may have an initial quadratic scaling but becomes linear as the memory system is `collapsed' to one of the phases.",
"In this case, by simulating measurement trajectories we can estimate the average conditional QFI as a function of $n$ and identify the optimal number of iterations of the Markov dynamics.",
"In particular, this provides an upper bound to the QFI of the probe state in the absence of monitoring.",
"In section we discuss the monitored environment setup, the associated notion of Fisher information, and present a toy example where monitoring restores the Heisenberg scaling.",
"Section describes the full setup including the Markov generated probe states.",
"In section we analyse a dephasing channel example, and show how standard or Heisenberg scaling ca be achieved depending on the chosen Markov dynamics.",
"We finish with comments on possible further investigations."
],
[
"Metrology with monitored environment",
"In this section we describe the environment monitoring scheme, and compare the associated Fisher information to that of the standard metrology setup.",
"We refer to the appendix for a brief review of the definition and statistical interpretation of the quantum Fisher information.",
"The noisy channels $T_\\theta $ are modelled by coupling each probe system unitarily to an individual ancilla representing its environment, as illustrated in Figure REF .",
"If the ancilla is initially prepared in state $|\\chi \\rangle $ , and the interaction is described by the unitary $W_\\theta $ , then $T_\\theta $ has Kraus decomposition $T_\\theta (\\rho ) = \\sum _c A_c^\\theta \\rho A_c^{\\theta \\dagger },\\qquad A_c^\\theta := \\langle c | W_\\theta | \\chi \\rangle ,$ where $|c\\rangle $ is chosen to be the basis in which the environment is measured.",
"By monitoring the ensemble of $n$ ancillas we obtain the outcome $\\underline{c}=(c_1,\\dots ,c_n)$ , an the conditional final state of the probe ensemble is $|\\Psi ^n(\\theta |\\underline{c}) \\rangle :=\\frac{A^\\theta _{c_n}\\otimes \\dots \\otimes A^\\theta _{c_1} |\\Psi ^n \\rangle }{\\sqrt{p^n(\\underline{c}|\\theta ) }},$ where $p^n(\\underline{c}|\\theta ) $ is the probability of the outcome $\\underline{c}$ $p^n(\\underline{c}|\\theta ) :=\\Vert A^\\theta _{c_n}\\otimes \\dots \\otimes A^\\theta _{c_1} \\Psi ^n\\Vert ^2 .$ The measurement data $\\underline{c}$ is fed into the design of the final measurement which aims to extract the maximum amount of information about $\\theta $ .",
"Our figure of merit for estimation is the total Fisher information of the available `data' consisting of the classical result $\\underline{c}$ and the conditional quantum state of the probe $|\\Psi ^n(\\theta | \\underline{c})\\rangle $ .",
"This can be written as (see appendix) $F^n_{total}(\\theta ) = F^n_{cl} (\\theta ) + \\sum _{\\underline{c}} p^n(\\underline{c}|\\theta ) F_q ( \\Psi ^n(\\theta |\\underline{c}))$ where the first term on the right side is the classical Fisher information of $\\underline{c}$ , while the second is the average quantum Fisher information of the final conditional state.",
"By comparison, the figure of merit for the standard (no monitoring) quantum metrology setting is the quantum Fisher information $F_q(\\rho ^n_\\theta )$ of the (average) final probe state $\\rho ^n_\\theta =\\sum _{\\underline{c}}p^n(\\underline{c}) | \\Psi ^n(\\theta |\\underline{c})\\rangle \\langle \\Psi ^n(\\theta |\\underline{c})|=T_\\theta ^{\\otimes n}(|\\Psi ^n\\rangle \\langle \\Psi ^n|).$ Since monitoring provides additional information, the following inequality holds $F^n_{total}(\\theta ) \\ge F_q(\\rho ^n_\\theta ).$ We illustrate our setup with the following toy example.",
"The qubit channel $T_\\theta $ is the convex combination of unitary rotations $T_\\theta \\left(\\rho \\right)=\\sum _{j \\in \\lbrace 0,\\dots ,3\\rbrace }\\lambda _j e^{i\\theta \\sigma _j}\\rho e^{-i\\theta \\sigma _j}$ where $\\sigma _j$ are the Pauli matrices.",
"Since $T_\\theta $ is an interior point of the convex space of qubit channels, it can be represented as a mixture of extremal channels with a smooth $\\theta $ -dependent probability distribution over such channels.",
"By applying the `classical simulation' argument of [15], we conclude that the QFI $F(\\rho ^n_\\theta )$ grows at most linearly with $n$ and therefore, the estimation rate in the standard metrology setup is $n^{-1}$ .",
"Consider now that by monitoring the environment, we know which of the unitaries has been applied on each qubit.",
"Recall that for a rotation family of pure states $|\\psi _\\theta \\rangle = \\exp (i\\theta G)|\\psi \\rangle $ the QFI has the expression $F_q(|\\psi _\\theta \\rangle ) = 4{\\rm Var}(G):= 4 (\\langle G^2 \\rangle - \\langle G\\rangle ^2)$ where $\\langle \\cdot \\rangle $ denotes the expectation with respect to $|\\psi \\rangle $ .",
"If the probe is prepared in the state $|\\Psi ^{n}\\rangle =\\left(|0\\rangle ^{\\otimes n} +|1\\rangle ^{\\otimes n} \\right)/\\sqrt{2}$ then $F_{cl}(\\theta )=0$ and using (REF ) we find $F^n_{total} (\\theta )= 4 n^2 \\lambda _3^2 + 4n( \\lambda _3(1-\\lambda _3) +\\lambda _1 + \\lambda _2)$ which scales quadratically in $n$ .",
"Before proceeding to the preparation stage, we would like to briefly comment on the physical realizability of our setting.",
"Although it is not our purpose to construct concrete physical models, we point out that `environment monitoring' and continuous time filtering (or quantum trajectories) are well established tools in quantum optics [28], which been used successfully e.g.",
"for mitigating decoherence [29], speeding up purification [30], or preparing a target state by means of feedback control [31].",
"Therefore we believe that the input-output formalism offers a natural framework for continuous time metrology with open systems."
],
[
"Markov generated probe states",
"We now introduce the second main ingredient of our analysis: a Markovian mechanism for generating the initial probe state $|\\Psi ^n\\rangle $ .",
"This ansatz is partly motivated by the close relationship to finitely correlated states [34] and matrix product states (MPS) [35], which provide efficient and tractable approximations of complex many-body states [36].",
"The preparation stage and subsequent metrology protocol is illustrate in Figure REF .",
"The top row represents a `memory system' $A$ which interacts sequentially (moving from right to left) with a chain of $n$ identically prepared probe systems (row B), by applying the same unitary $U^{AB}$ .",
"After the interaction, the chain B together with the memory are in the state $|\\Psi ^n_{AB}\\rangle = \\sum _{f, \\underline{b}}\\langle f | K_{\\underline{b}}| i\\rangle |f\\rangle \\otimes |\\underline{b}\\rangle ,$ where $|i\\rangle $ is the initial state of $A$ , $\\underline{b}= (b_1, \\dots , b_n)$ is the index of the product basis for the B row, $K_{\\underline{b}}:=K_{b_n}\\dots K_{b_1}$ , and $K_{b}:=\\langle b| U^{AB} |\\xi \\rangle $ are the Kraus operators associated to the unitary $U^{AB}$ and the initial state $|\\xi \\rangle $ of the $B$ systems.",
"After the preparation stage, each system undergoes a separate unitary interaction $W_\\theta ^{BC}$ with an ancilla (environment) in row C, prepared initially in state $|\\chi \\rangle $ , as described in the previous section.",
"In particular the channel $T_\\theta $ and its Kraus operators are given by equation (REF ).",
"By commutativity, the final ABC state is the same irrespective of whether the unitaries $W_\\theta ^{BC}$ are applied at the end of the preparation stage or each of them is applied immediately after the corresponding $U^{AB}$ .",
"With similar notations as above, the joint $ABC$ final state is $&&|\\Psi _{ABC}^{n}(\\theta )\\rangle = \\sum _{f, \\underline{b},\\underline{c}}\\langle f| K^{\\theta }_{\\underline{b},\\underline{c}}| i\\rangle |f\\rangle \\otimes |\\underline{b}\\rangle \\otimes |\\underline{c}\\rangle ,\\\\&=& \\sum _{\\underline{c}} \\left( \\mathbf {I}_A\\otimes A_{\\underline{c}}^{\\theta } |\\Psi _{AB}^{n}\\rangle \\right) \\otimes |\\underline{c}\\rangle = \\sum _{\\underline{c}} |\\tilde{\\Psi }_{AB}^n(\\theta |\\underline{c}) \\rangle \\otimes | \\underline{c}\\rangle \\nonumber $ where $|\\tilde{\\Psi }_{AB}^n(\\theta |\\underline{c}) \\rangle $ is the unnormalised conditional state of $AB$ , for a given outcome $\\underline{c}$ , and $K_{b,c}^{\\theta } $ are `extended' Kraus operators.",
"Figure: (Color online) Model of discrete dynamics with Markov generated probe state.The memory system AA interacts successivly (from right to left) with the probe systems BB via the unitary U AB U^{AB}.The channel T θ T_\\theta on BB is implemented by unitary coupling of the probe systems with ancillas CC.The reduced evolution of $A$ is obtained either by tracing out the $B$ systems in $|\\Psi ^n_{AB}\\rangle $ or the $B$ and $C$ systems in $|\\Psi _{ABC}^{n}\\rangle $ , and its one step transition operator is $Z_\\theta (\\rho ) = \\sum _{b} K_{b} \\rho K_{b}^{\\dagger } =\\sum _{b,c} K_{b,c}^\\theta \\rho K_{b,c}^{\\theta \\dagger }.$ To summarise, the metrological probe is prepared via the Markov dynamics involving the memory A and the row B.",
"The channels $T_\\theta $ are modelled via the subsequent interaction $W^{BC}$ between the B and the corresponding C systems.",
"We distinguish two scenarios: the experimentalist has access only to the $B$ systems (which leads to standard rates for non-unitary channels $T_\\theta $ ), or the environment $C$ can be monitored and the collected data can be used to improve the estimation rates.",
"From a system identification perspective, this set-up has been investigated in [26], [27], which show that if the transition operator $Z_\\theta $ is primitive [32] (memory $A$ converges to a unique stationary state) then the quantum Fisher information of the state $|\\Psi ^n_{ABC}(\\theta )\\rangle $ increases linearly with $n$ , so $\\theta $ can only be estimated with rate $n^{-1}$ .",
"Therefore, from the metrology viewpoint it is interesting to consider models in which the memory $A$ has several invariant subspaces, or `dynamical phases'.",
"In this case, the quantum Fisher information of the full state $|\\psi ^n_{ABC}\\rangle $ may increase as $n^2$ [26], as we will explain below.",
"For two dynamical phases for instance, the memory space decomposes into orthogonal subspaces $\\mathcal {H}_A= \\mathcal {H}_{A}^0\\oplus \\mathcal {H}_{A}^1,$ such that the Kraus operators $K_b$ (and similarly for $K_{b,c}$ ) are block-diagonal with respect to this decomposition $K_{b}=\\langle i | U^{AB} |\\xi \\rangle =\\begin{pmatrix}K_b^0& 0 \\\\0 &K_b^1\\end{pmatrix},\\quad $ and the restricted evolutions are primitive, with unique stationary states $\\rho _{ss}^{0},\\rho _{ss}^{1}$ .",
"Assuming that the initial state of $A$ is a coherent superposition of states from the two phases, e.g.",
"$|i\\rangle = (|i,0 \\rangle +|i,1 \\rangle )/\\sqrt{2} \\in \\mathcal {H}_A ,$ the joint states $|\\Psi _{AB}^n\\rangle $ and $|\\Psi _{ABC}^n(\\theta )\\rangle $ have a similar decomposition $|\\Psi _{AB}^n\\rangle = \\frac{1}{\\sqrt{2}}\\left(|\\Psi _{AB}^{n,0} \\rangle + |\\Psi _{AB}^{n,1}\\rangle \\right)\\in \\mathcal {H}_A^0 \\otimes \\mathcal {H}_B^{\\otimes n} \\oplus \\mathcal {H}_A^1 \\otimes \\mathcal {H}_B^{\\otimes n}.$ For concreteness we consider a unitary $W^{BC}_\\theta $ of the form $W^{BC}_\\theta = (U^B_\\theta \\otimes \\mathbf {I}^C) V^{BC}$ where $U^B_\\theta = \\exp (-i\\theta G) $ is a phase rotation on the probe system $B$ with generator $G$ and $V^{BC}$ is a fixed unitary describing the interaction with the environment.",
"Since $|\\Psi ^n_{ABC}\\rangle $ is a rotation family, $|\\Psi ^n_{ABC} (\\theta )\\rangle = \\exp (- i \\theta \\bar{G} ) |\\Psi ^n_{ABC}(0)\\rangle ,\\quad \\bar{G}= \\sum _{i=1}^n G^{(i)},$ its quantum Fisher information is proportional to the variance of the `total generator' $\\bar{G}$ .",
"For simplicity, in the sequel we will identify the total generator with the random variable obtained by measuring $\\bar{G}$ .",
"Its probability distribution with respect to $|\\Psi ^n_{ABC} (\\theta )\\rangle $ is the mixture $(\\mathbb {P}^{0} +\\mathbb {P}^{1})/2$ of the distributions corresponding to the two phases, computed from the states $|\\Psi ^{n,0}_{ABC}\\rangle $ and $|\\Psi ^{n,1}_{ABC}\\rangle $ .",
"Under each $\\mathbb {P}_0$ and $\\mathbb {P}_1$ separately, the following convergence in law to the normal distribution (Central Limit Theorem) holds [26] $\\frac{1}{\\sqrt{n}} (\\bar{G} - n g_a) \\overset{\\mathcal {L}}{\\longrightarrow } N(0, V^a), \\quad a=0,1.$ for certain means $g_a$ and variances $V_a$ .",
"Therefore, if $g_0\\ne g_1$ the distribution of $\\bar{G}$ with respect to the output state $|\\Psi ^n_{ABC}(\\theta )\\rangle $ has variance of the order $n^2$ , and we are in the Heisenberg scaling regime, cf.",
"Figure REF .",
"We now investigate what happens when the ancillas in row $C$ are measured, as described in our environment monitoring scheme.",
"Note that the measurement data $\\underline{c}$ on its own, carries no information about $\\theta $ , i.e.",
"$F_{cl}(\\theta )=0$ since the unitary rotation is applied at the end, and only on the B row.",
"However, as in the toy example of section , the results do contribute to a larger quantum Fisher information, by identifying the pure components $|\\Psi ^n_{AB}(\\theta |\\underline{c})\\rangle $ of the mixed probe state $\\rho ^n_\\theta $ .",
"These conditional states have a similar phase decomposition $|\\Psi _{AB}^n( \\theta |\\underline{c} )\\rangle =\\sqrt{p^n_{0}( \\underline{c} )} |\\Psi _{AB}^{n,0}(\\theta |\\underline{c}) \\rangle +\\sqrt{p^n_{1}( \\underline{c} )} |\\Psi _{AB}^{n,1}(\\theta |\\underline{c}) \\rangle $ where $p^n_{a}(\\underline{c})$ is the probability that $A$ is in phase $a$ , given the outcome $\\underline{c}$ , and $|\\Psi _{AB}^{n,a}(\\underline{c})\\rangle $ is the posterior states corresponding to the initial state $|i,a\\rangle \\in \\mathcal {H}_A^a$ .",
"Figure: (Color online) The distribution of the `total generator' G ¯\\bar{G} in |Ψ ABC n 〉|\\Psi ^n_{ABC}\\rangle is a mixture of (approximately) Gaussian distributions centred at ng 0 ng_0 and ng 1 ng_1, cf.",
"().",
"When g 0 ≠g 1 g_0\\ne g_1 the distance between the two peaks is n(g 1 -g 0 )n(g_1-g_0) and the quantum Fisher information (F=4 Var (G ¯)F= 4{\\rm Var}(\\bar{G})) scales as n 2 n^2.",
"If phase purification occurs due to `which phase' information leaking to the environment, one of the peaks decays and the variance scales as nn.By the same argument as above, the variance of $\\bar{G}$ with respect to the conditional state $|\\Psi _{AB}^n( \\underline{c})\\rangle $ increases as $n^2$ provided that the `weights' $p^n_{0}(\\underline{c}) $ and $p^n_{1}(\\underline{c})$ of the two components of the mixture stay away from the extreme values $0,1$ .",
"Unfortunately however, this can happen only in special situations as the following argument shows.",
"Let $\\mathbb {Q}^n_0(\\underline{c})$ and $\\mathbb {Q}^n_1(\\underline{c})$ be the probability distributions of measurements on the environment corresponding to the two phases.",
"If these distributions are different in the stationary regime, the observer can distinguish between them at a certain exponential rate, similarly to the case of discrimination between two coins with different bias.",
"This means that the conditional probability $p^n_{0}( \\underline{c} )$ will converge (almost surely along any trajectory) either to zero or to one in the limit of large $n$ .",
"We call this phenomenon phase purification, in analogy with that of state purification for a system monitored through the environment [37].",
"Essentially, phase purification occurs when the environment learns about the phase of $A$ .",
"Therefore, keeping the coherence between the two outputs $ |\\Psi _{AB}^{n,0}(\\underline{c}) \\rangle $ and $|\\Psi _{AB}^{n,1}(\\underline{c}) \\rangle $ requires that the environment measurement does not provide any information to distinguish between the two.",
"In conclusion, total Fisher information $F^n_{total}(\\theta )$ scales as $n^2$ if phase purification does not occur, and scales linearly in $n$ otherwise.",
"In the latter case, the long time behaviour is determined by the average of the Fisher informations corresponding to the two phases.",
"However, in the short term the Fisher information may increase quadratically in $n$ until phase purification destroys the coherence between the two phases."
],
[
"Example of a Heisenberg limited system",
"In this section we present a concrete example exhibiting the two behaviours described above, depending on the choice of Markov dynamics.",
"Inspired by [25] we consider a minimalistic example with a two dimensional memory whose phases are the basis vectors $|0\\rangle $ and $|1\\rangle $ , and a two dimensional probe unit $B$ with initial state $|\\xi \\rangle $ .",
"The Kraus operators are of the form $K_{b}=\\langle b | U^{AB} |\\xi \\rangle =\\begin{pmatrix}\\sqrt{\\alpha _b} & 0 \\\\0 & \\sqrt{\\beta _b}\\end{pmatrix},\\quad b=0,1,$ with $\\alpha _0+\\alpha _1=\\beta _0+\\beta _1=1$ .",
"The initial state of $A$ is the superposition $|i\\rangle =\\frac{1}{\\sqrt{2}}(|0\\rangle _A+|1\\rangle _A)$ , and the unitary rotation on the probe system B is $U^B_\\theta = \\exp (i\\theta |1\\rangle \\langle 1|)$ .",
"In the absence of noise, the output state is $|\\Psi ^n_{AB}(\\theta )\\rangle =\\frac{1}{\\sqrt{2}}\\left( |0 \\rangle \\otimes |\\psi _0(\\theta )\\rangle ^{\\otimes n}+|1 \\rangle \\otimes |\\psi _1(\\theta )\\rangle ^{\\otimes n}\\right)$ where $&&|\\psi _0(\\theta )\\rangle =\\sqrt{\\alpha }_0 |0\\rangle +e^{i\\theta }\\sqrt{\\alpha }_1 |1 \\rangle , \\\\&&|\\psi _1(\\theta )\\rangle =\\sqrt{\\beta }_0 |0\\rangle +e^{i\\theta }\\sqrt{\\beta }_1 |1\\rangle .$ The distribution of the generator $\\bar{G}$ with respect to this state is a mixture of two binomial distributions ${\\rm Bin}(n, g_0= \\alpha _1)$ and ${\\rm Bin}(n, g_1= \\beta _1)$ , and the quantum Fisher information is $F(|{\\Psi }^n_{AB}\\rangle )=2n(\\alpha _0\\alpha _1+\\beta _0\\beta _1)+n^2(\\alpha _1-\\beta _1)^2.$ We add now a noise model given by phase damping in the direction $v$ on the Bloch sphere $\\Lambda ^v[\\rho ]= \\sum _{c\\in \\lbrace 0,\\pm 1\\rbrace } A_c \\rho A_c^{\\dagger }$ with Kraus operators $A_0=\\sqrt{p} \\mathbf {I}$ , $A_{\\pm 1}^v=\\sqrt{1-p}|v_\\pm \\rangle \\langle v_\\pm |$ for $p \\in [0,1]$ .",
"By (REF ), the record of ancilla measurement outcomes indicate which of the Kraus operators defined $A_j$ acted on each of the probe systems B.",
"Therefore, the unnormalised conditional output state is $\\begin{split}|\\tilde{\\Psi }^n_{AB}(\\theta |\\underline{c})\\rangle =&&|0\\rangle _A \\otimes U^B_\\theta A_{c_n} |\\psi _0\\rangle \\otimes \\dots \\otimes U^B_\\theta A_{c_1} |\\psi _0\\rangle \\\\&+&|1\\rangle _A \\otimes U^B_\\theta A_{c_n} |\\psi _1\\rangle \\otimes \\dots \\otimes U^B_\\theta A_{c_1} |\\psi _1\\rangle \\end{split}$ The probability distribution $\\mathbb {Q}^n(\\underline{c})=\\Vert \\tilde{\\Psi }^n_{AB}(\\theta |\\underline{c})\\Vert ^2 $ is the mixture $(\\mathbb {Q}^n_0 + \\mathbb {Q}^n_1)/2$ where both components are product measures (independent samples from $\\lbrace 0,\\pm 1\\rbrace $ ) with probabilities $q^{0,1}_0 = p, \\quad q^{0}_{\\pm } =(1-p) \\alpha _\\pm , \\quad q^{1}_{\\pm } =(1-p) \\beta _\\pm ,$ where $\\alpha _{\\pm }=|\\langle v_{\\pm }| \\psi _0\\rangle |^2, \\beta _{\\pm }=|\\langle v_{\\pm }| \\psi _1\\rangle |^2$ .",
"In particular, the weights $p^n_a(\\underline{c})$ of the two phases and the quantum Fisher information of the conditional state $|\\Psi ^n_{AB}(\\theta |\\underline{c})\\rangle $ depend only on the total number for each outcome $n_{\\pm }$ and $n_0$ , the latter being the number of systems which have not been affected by the noise.",
"The Fisher information has the following expression $F\\left(n_\\pm , n_0\\right) &=& (n_+ + n_-) F_\\pm + 4 n_0 (p^n_0 \\alpha _0\\alpha _1 + p^n_1 \\beta _0\\beta _1)\\nonumber \\\\&+& 4 n_0^2 p^n_0 p^n_1 (\\alpha _1 - \\beta _1)^2$ where $F_\\pm = 4 |\\langle v_+|1\\rangle |^2 |\\langle v_-|1\\rangle |^2 $ , and the phase weights are $p^n_1= p^n_1 (n_\\pm , n_0) = 1-p^n_0$ with $p^n_{0}= p^n_0(n_\\pm , n_0)=\\alpha _+^{n_+}\\alpha _-^{n_-} / (\\alpha _+^{n_+}\\alpha _-^{n_-}+\\beta _+^{n_+}\\beta _-^{n_-}).$ Figure: (Color online) Phase purification and quantum Fisher information for noise in xx direction with parameters α 0 =0.1,β 0 =0.7\\alpha _0=0.1, \\beta _0=0.7 and p=0.6p=0.6.",
"Main plot: the `scaled' Fisher information F(|Ψ AB n (θ|c ̲)〉)/nF(|\\Psi ^n_{AB}(\\theta |\\underline{c})\\rangle ) /n as function of `time' nn, for two trajectories with different limiting phases (continuous blue and dotted red curves); the corresponding averages are plotted in black.",
"Inset plot:-the weights of the limiting phase p 1 n (c ̲)p^n_1(\\underline{c}) and p 0 n (c ̲)p^n_0(\\underline{c}) for the two given trajectories converge to 1 due to phase purification.Since $n_0\\approx pn$ , the state has Heisenberg scaling if and only if the last term in (REF ) does not converge to zero, i.e.",
"the means of $\\bar{G}$ in the two phases are different ($\\alpha _1\\ne \\beta _1)$ , and phase purification does not occur ($q^0_\\pm = q^1_\\pm $ ).",
"Now, it is easy to verify that for any noise direction $v$ different from $z$ , there exist Kraus operators $K_b$ such that these conditions are met.",
"In terms of the Bloch sphere, the requirement is that the Bloch vectors of $|\\psi _0\\rangle $ and $|\\psi _1\\rangle $ lie symmetrically with respect to $v$ such that the (environment) measurement in this direction cannot distinguish between the two states.",
"In the special case when $v$ is along the $z$ axis, the means of $G$ in the two phases are equal and therefore the variance scales linearly with $n$ .",
"On the other hand, when the environment can distinguish the two phases, either the probabilities $p^n_{0}(\\underline{c})$ converges to 1 exponentially with rate equal to the relative entropy $S(q^1 | q^0)$ , or $p^n_{1}(\\underline{c})$ converges to 1 exponentially with rate $S(q^0 | q^1)$ .",
"This implies that for small $n$ the weights of the two phases are comparable, and quantum Fisher information per probe system $f^n(\\underline{c})= \\mathbb {E} F_q(|\\Psi ^n_{AB}(\\theta |\\underline{c})\\rangle )/n$ increases linearly with $n$ ; after that, the exponential decay kicks in and $f^n(\\underline{c})$ converges to the Fisher information of the corresponding limiting phase.",
"In Figure REF we illustrate this behaviour with two trajectories having different limiting phases, with the phases weights shown in the inset plot, and the black lines showing the average Fisher information over trajectories converging to either phase.",
"The average of these two is the scaled total information $F^n_{total}(\\theta )/n$ ."
],
[
"Conclusions and outlook",
"We proposed Markov dynamics as a mechanism for generating probe states for quantum metrology, and showed how the Heisenberg scaling emerges for systems with multiple `dynamical phases', and noiseless channels.",
"Additinally, we modelled noisy channels by coupling the Markov output to `environment' ancillas, and considered the scenario where the environment is monitored to increase the quantum Fisher information of the output.",
"In this setup we found that the survival of the Heisenberg limit depends on whether the environment receives `which phase' information about the memory system.",
"If `phase purification' occurs, the quantum Fisher information of the conditional output state has an initial quadratic scaling, but in the long run the environment wins, and the massive coherent superposition of `output phases' is destroyed leading to the standard scaling.",
"However, in a simple example we showed that the Heisenberg scaling is preserved if the Markov dynamics is chosen such that the environment cannot distinguish between the dynamical phases.",
"These preliminary results open several lines of investigation in the input-output setting with monitored environment, e.g.",
"finding the `optimal' Markov dynamics and `stopping times' which maximise the constant of the standard scaling, analysing the use of feedback control based on the measurement outcomes.",
"An appropriate framework for answering these questions may be that of 'thermodynamics of trajectories' and `dynamical phase transitions' [38], [39].",
"Acknowledgements.",
"We thank Rafal Demkowicz-Dobrzanski, and Katarzyna Macieszczak for useful discussions.",
"This work was supported by the EPSRC grant EP/J009776/1."
],
[
"Appendix",
"For reader's convenience we briefly review here some general notions of quantum parameter estimation and quantum Fisher information used in the paper.",
"In quantum estimation (or quantum tomography) we are given a system prepared in the state $\\rho _\\theta $ , where $\\theta $ is an unknown parameter which for our purposes can be chosen to be one-dimensional, and we would like to estimate $\\theta $ by measuring the system and computing an estimator $\\hat{\\theta }= \\hat{\\theta }(X)$ based on the measurement result $X$ .",
"If the map $\\theta \\mapsto \\rho _\\theta $ is smooth, then the following quantum Cramér-Rao bound [33] holds for any unbiased estimator (i.e.",
"$\\mathbb {E}_\\theta (\\hat{\\theta })=\\theta $ ) $\\mathbb {E}[ (\\hat{\\theta }- \\theta )^2 ] \\ge F_q(\\rho _\\theta )^{-1}.$ The left side is the mean square error (MSE) of $\\hat{\\theta }$ , which is the standard figure of merit for estimation, while the right hand side is the inverse of the quantum Fisher information (QFI), defined below.",
"Note that the left side is an intrinsic property of the quantum statistical model $\\theta \\mapsto \\rho _\\theta $ , and therefore is a lower bound for the MSE of any unbiased estimator.",
"The QFI is given by $F_q(\\rho _\\theta )=\\rm Tr(\\rho _\\theta L_\\theta ^2)$ where $L_\\theta $ is the operator (called symmetric logarithmic derivative) defined by the equation $2\\frac{d\\rho _\\theta }{d\\theta }=\\lbrace L_\\theta ,\\rho _\\theta \\rbrace .$ In particular, if $\\rho _\\theta = |\\psi _\\theta \\rangle \\langle \\psi _\\theta |$ is a rotation family with $|\\psi _\\theta \\rangle = \\exp (i\\theta G)|\\psi \\rangle $ then $F_q(\\rho _\\theta ) = 4{\\rm Var}(G):= 4 (\\langle G^2 \\rangle - \\langle G\\rangle ^2)$ where $\\langle \\cdot \\rangle $ denotes the expectation with respect to $|\\psi \\rangle $ .",
"In general, the quantum Cramér-Rao bound (REF ) may not be achievable, and the restriction to unbiased estimators is not desirable.",
"However, in practice one usually estimates the parameter $\\theta $ by performing repeated measurements on an ensemble of $N$ identically prepared systems, and computing an estimator $\\hat{\\theta }_N$ based on the collected data.",
"In this case, asymptotically optimal estimators (e.g.",
"the maximum likelihood estimator) achieve the Cramér-Rao in the sense that as $N\\rightarrow \\infty $ $\\sqrt{N}(\\hat{\\theta }_N - \\theta ) \\overset{\\mathcal {L}}{\\longrightarrow } N(0, F_q(\\rho _\\theta )^{-1})$ where $\\mathcal {L}$ denotes convergence in distribution and $N(\\mu , V)$ is the normal distribution of mean $\\mu $ and variance $V$ .",
"In particular $\\lim _{n\\rightarrow \\infty } N \\mathbb {E} [ (\\hat{\\theta }_N- \\theta )^2 ] = F(\\rho _\\theta )^{-1}$ and no estimator can improve on this limit for all $\\theta $ .",
"Therefore the inverse quantum Fisher information is the optimal constant in the large samples scenario which most relevant for practical purposes.",
"In the `environment monitoring' setting, we consider the scenario where the parameter dependent state is bipartite $\\rho _\\theta = \\rho _\\theta ^{12}$ .",
"Suppose that a projective measurement with outcome $X$ is performed on the second system, and let $\\rho ^1(\\theta |X)$ be the conditional state of the first system, given the outcome.",
"Then the following inequalities hold $F_q(\\rho ^1_\\theta ) \\le F_{cl}(X;\\theta ) + \\sum _x p_\\theta (x) F_q(\\rho ^1(\\theta |x)) \\le F_q(\\rho _\\theta ^{12})$ where $\\rho ^1_\\theta $ be the reduced state of the first system, $F_{cl}(X;\\theta )$ is the classical Fisher information of the measurement result $X$ , and $p_\\theta (x)$ is the probability of the outcome $X=x$ .",
"Obviously the interpretation is that if we measure system 2, the classical information plus the quantum Fisher information of the remaining state is smaller than the full quantum information of both systems and larger than that of the first system alone.",
"In the case of quantum metrology the situation is somewhat more complicated, due to the fact that the probe systems are in general correlated rather than independent.",
"Indeed, unitary channels can be estimated with MSE scaling as $N^{-2}$ with the size of the probe ensemble [10].",
"With the notations of the introduction, let us consider an $n$ systems probe ensemble with final state $\\rho ^n_\\theta = T^{\\otimes n}_\\theta (|\\Psi ^n\\rangle \\langle \\Psi ^n|)$ and QFI $F_q(\\rho ^n_\\theta )$ .",
"Since the systems may be correlated, the above asymptotic results do not apply automatically.",
"One can however consider a larger ensemble of $N= k\\cdot n$ systems consisting of $k$ independent batches of identically prepared sub-ensembles in state $|\\Psi ^n\\rangle $ .",
"For large $k$ and given $n$ , the quantum Cramér-Rao bound is achievable in the sense that $N \\mathbb {E} [ (\\hat{\\theta }_N- \\theta )^2 ] \\approx \\frac{n}{F_q(\\rho ^n_\\theta )}.$ By optimising over $|\\Psi ^n\\rangle $ one can in principle minimise the right side and obtain the optimal MSE for a fixed ensemble size $n$ .",
"We now distinguish two situations i) Heisenberg scaling: $F_q(\\rho ^n_\\theta )$ scales as $n^2$ and therefore the left side of (REF ) decreases as $n^{-1}$ ; this indicates that the overall MSE does not scale as $N^{-1}$ but rather as $N^{-2}$ .",
"However, the Fisher information theory cannot be used to find the correct asymptotic constant.",
"ii) Standard scaling: for a large class of noisy channels the quantum Fisher information $F_q(\\rho ^n_\\theta )$ scales as $n$ in the sense that the increasing sequence $f_n := \\sup _{|\\Psi ^n\\rangle } \\, \\frac{F_q(\\rho ^n_\\theta )}{n}$ has a finite limit $f$ .",
"By choosing sufficiently large $n$ , one can achieve asymptotic MSEs scaling as $1/(f^\\prime N)$ for any constant $f^\\prime <f$ ; we conjecture that than by increasing both $k$ and $n$ one can achieve the optimal asymptotic MSE $1/(f N)$ .",
"In conclusion, we found that while for Heisenberg scaling the QFI predicts the right scaling of the MSE but not the constant factor, in the standard scaling case the quantum Fisher information predicts the correct asymptotic behaviour of the MSE, including the optimal constant.",
"In Markov generated setting investigated here, the maximum (average conditional) quantum Fisher information is achieved at a finite $n$ and therefore the optimal strategy within this setting is to prepare independent batches Markov correlated states of optimal length."
]
] | 1403.0116 |
[
[
"Temporal Image Fusion"
],
[
"Abstract This paper introduces temporal image fusion.",
"The proposed technique builds upon previous research in exposure fusion and expands it to deal with the limited Temporal Dynamic Range of existing sensors and camera technologies.",
"In particular, temporal image fusion enables the rendering of long-exposure effects on full frame-rate video, as well as the generation of arbitrarily long exposures from a sequence of images of the same scene taken over time.",
"We explore the problem of temporal under-exposure, and show how it can be addressed by selectively enhancing dynamic structure.",
"Finally, we show that the use of temporal image fusion together with content-selective image filters can produce a range of striking visual effects on a given input sequence."
],
[
"Introduction",
"The limitations placed by the sensor on a camera's dynamic range have been widely studied.",
"We know how to estimate the response of a sensor to light, and can predict the minimum and maximum amounts of radiance that will be properly recorded under a given exposure setting.",
"We can combine multiple, differently exposed images to create radiance maps covering a much wider range of values than the sensor would otherwise allow, and we know how to render such radiance maps onto images that better approximate the human visual perception of scenes with widely varying radiance regions.",
"However, relatively little attention has been given to the limitations that the sensor's dynamic range imposes on our ability to capture and render dynamic scene content.",
"The majority of HDR techniques assume that scene content remains static during the capture process.",
"For a scene requiring more than a few exposures, or for a scene requiring long exposure times, this means that any moving objects will be captured at different locations in each of the individual shots and will afterwards cause artifacts during the radiance estimation and tone mapping steps.",
"In turn, this limitation of HDR means that for typical consumer digital cameras, rendering moving content without artifacts, whether in still photographs or video, still relies on standard single exposure capture.",
"This places tight constraints on the range of exposure times that can be achieved in practice during photography or filming, and therefore limits the photographer's creative freedom to choose how dynamic content is to be rendered onto an image.",
"Capturing long exposure effects on full frame rate video, achieving arbitrarily long photographic exposures, and enhancing the rendering of fleeting temporal phenomena are examples of photographic effects not currently achievable using standard image processing techniques.",
"This paper proposes temporal image fusion (TIF) as a means for expanding the camera's temporal dynamic range.",
"While temporal image fusion can not expand the sensor's light-gathering ability or the overall dynamic range of each frame, it can blend information from multiple frames taken over some interval of time to render events of varying duration in a precisely controlled way.",
"Full frame rate video showing long exposure effects, and very long exposure photography are easily produced from a discrete set of input frames as shown in Fig.",
"REF .",
"More interestingly, TIF provides a means for controlling how strongly moving content will contribute to the output frames.",
"Fleeting phenomena can either be enhanced to show structure that would otherwise be lost under regular long-exposure photography, or suppressed to remove transient content from video.",
"With the addition of simple content-dependent filters, TIF can be used to achieve a wide variety of striking visual effects.",
"TIF should provide photographers with a significantly larger amount of freedom in choosing how to present moving content.",
"Figure: Temporal blending of a sequence of 32 HDR photographs taken over a period of 1 hour around sunset.",
"Four of the sourceframes are also shown.",
"The blended image contains detail and structure from all source images, simulating an exposure length that can not beachieved in practice given the illumination conditions and the capture process of HDR photography.In what follows, we will provide a brief review of existing work on image blending and HDR, then introduce the exposure fusion algorithm which is at the heart of TIF.",
"We will then show how to control virtual exposure time, and explore the problem of properly capturing fast-moving objects and other transient structure under long exposures via a simple temporal distinctness filter.",
"Finally, we show how a wide range of visual effects can be easily obtained by applying TIF selectively over particular image regions.",
"This paper expands on preliminary work in [10] by improving the overall temporal fusion framework, discussing conditions during image capture that can lead to artifacts in the final blends -along with proposals for minimizing resulting artifacts-, and more importantly; by introducing the temporal distinctness filter to control the rendering of fleeting phenomena."
],
[
"High-Dynamic-Range Imaging and Exposure Fusion",
"The problem of blending information from differently exposed images to capture and render a larger dynamic range has been studied at length over the past two decades.",
"High-Dynamic-Range imaging addresses the problem of estimating a scene radiance map from a set of differently exposed images, all subject to under- and over-exposure.",
"The radiance map can be used to render a view of the scene in such a way that it better approximates what a human observer would perceive looking at the scene.",
"There is a large body of work dealing with the generation, encoding, and rendering of HDR images.",
"Early work by Mann and Picard [24], Debevec and Malik [7], and Mitsunaga and Nayar [29], [30] among others set the base framework for creating images with extended dynamic range, recovering a camera’s response curve, generating range radiance maps, and rendering these maps for display on a low dynamic range device such as a computer display.",
"Since then, a large volume of research on algorithms and techniques for HDR has been published.",
"Thorough studies of existing techniques with an extensive bibliographic reference can be found in [2], [14].",
"HDR techniques involve two general steps.",
"First, a radiance map for the scene must be computed.",
"This is typically done at the level of RAW image intensities captured by the camera sensor.",
"Secondly, the radiance map is processed by a tone-mapping operator to compress the radiance range in the scene to a range that can be stored in a regular image format and displayed on a standard computer display.",
"The tone-mapping step has a strong influence on the visual quality of the resulting scene.",
"The operator can be chosen to either maximize the similarity between the resulting image and an observer's perception of the original scene, or to achieve an artistic effect at the expense of photo-realism.",
"Tone mapping operators include logarithmic curves, gamma-correction, histogram equalization, and biologically inspired methods based on the human visual system [33], [9], [11], [25], [19], [27], [20].",
"Parallel to the development of HDR techniques, sensor and image fusion techniques have been proposed to enhance or expand the information content of images.",
"Photo stacking and depth from defocus [5], [12], [13], coded aperture photography [21], [36], [32], photo montages [1], and multi-spectral imaging [4], [22], [16], [34] are examples of such image fusion techniques (see [28] for a thorough treatment of image fusion techniques).",
"Among image fusion techniques, Mertens et al.",
"[26] propose a simple exposure fusion technique as an alternative for generating images that closely resemble HDR photographs, but without the need for radiance map estimation or tone-mapping.",
"Like HDR, exposure fusion creates an image from a set of differently exposed images, unlike HDR, the final image is computed directly from the source pictures as a weighted linear combination of pixels from the different frames.",
"Because of its reliance on simple weighted combinations of pixels based on feature maps, their method is well suited for application to the temporal aspect of the imaging process.",
"Given a set of input images $\\lbrace I_1, I_2, \\ldots , I_K\\rbrace $ corresponding to differently exposed shots of the same scene, the exposure fusion method computes weight maps based on the contrast, saturation, and well-exposedness features at each pixel.",
"Contrast computation is based on local edge energy, saturation is defined as the standard deviation of the RGB colour components of each pixel, and well-exposedness gives higher weights to pixels away from brightness extremes.",
"Given these components, pixel weights are computed as $W_{i,j,k}=C_{i,j,k}^{\\alpha _c} \\cdot S_{i,j,k}^{\\alpha _s} \\cdot E_{i,j,k}^{\\alpha _e},$ where the indices $i,j,k$ refer to pixel $(i,j)$ in image $k$ , $C_{i,j,k}$ , $S_{i,j,k}$ , and $E_{i,j,k}$ are the pixel's contrast, saturation, and well-exposedness values respectively, and the $\\alpha $ exponents control the influence of each of these terms.",
"Pixel weights are normalized to that the weight of pixels at $(i,j)$ across all frames sums to 1 $\\hat{W}_{i,j,k}=\\frac{W_{i,j,k}}{\\sum _{k^{\\prime }=1}^K W_{i,j,k^{\\prime }}}.$ Given the pixel weights for all pixels in all source frames, the simplest exposure fusion algorithm would compute the colour of the final blended pixel $R(i,j)$ as $\\hat{R}_{i,j}=\\sum _{k=1}^K \\hat{W}_{i,j,k}I_{i,j,k}.$ However, this can lead to artifacts around image edges and other fine structure due to high frequency variations in the weight maps themselves.",
"To avoid this problem, Eq.",
"REF is actually implemented using a pyramid blending scheme similar to that described in [3].",
"Each input image is processed to obtain a Laplacian $L\\lbrace I_k\\rbrace $ pyramid with $d$ levels.",
"The corresponding weight map is processed to obtain a Gaussian pyramid $G\\lbrace \\hat{W}_k\\rbrace $ with the same number of levels $d$ .",
"From these Laplacian and Gaussian pyramids, each level $l=\\lbrace d,d-1,\\ldots , 1\\rbrace $ of the blended Laplacian pyramid for the result frame is computed as $L_l\\lbrace R_{i,j}\\rbrace = \\sum _{k=1}^K G_l\\lbrace \\hat{W}_{i,j,k}\\rbrace L_l\\lbrace I_{i,j,k}\\rbrace .$ The result frame $R$ is finally obtained by pyramid reconstruction from $L\\lbrace R\\rbrace $ .",
"Because of its relative simplicity, exposure fusion has been adopted in applications such as panoramic imaging through an open source project called enfuse.",
"Of particular interest for us is the fact that since it removes the need for radiance map estimation.",
"Because of this step, and as mentioned above, HDR techniques generally perform poorly in the presence of fast changing image content.",
"They produce visual artifacts (so-called ghosts) wherever moving content is present.",
"Though ghost removal techniques do exist [18], such methods eliminate moving content from the scene.",
"To render moving content in HDR, a more complex setup is required.",
"One option is to use multiple sensors to capture the different exposures simultaneously (for example Tocci et al.",
"[35]), another is to limit the number of exposures to reduce temporal changes between HDR frames, and to incorporate a registration step to align all moving structures as in Kang et al. [17].",
"Contrary to HDR, our goal here is to actually preserve (or even enhance) motion blur.",
"In this regard, the exposure fusion algorithm provides a larger degree of flexibility since moving content is blended-in more naturally than with HDR.",
"By carefully controlling the way the blending is performed, we can produce a wide variety of visual effects other than dynamic range expansion.",
"These effects are interesting for their photographic quality, as well as for their potential for highlighting patterns in the motion within an scene.",
"Already in current practice, photographers use a very limited analog to exposure fusion in order to simulate very long exposures.",
"The technique is called exposure stacking [6] and involves blending a set of input images by taking the brightest pixels from each one.",
"This is typically done manually on photo editing software such as Photoshop or Gimp.",
"However, the technique is very limited in scope, and is most commonly used to create long exposures of star trails and similarly bright structures on dark backgrounds.",
"Figure: An overview of the TIF framework.",
"An input image sequence, be it from video or still photographs, is processed by a sliding window with an associated Gaussian weight profile.",
"Frames are filtered to detect edge structure, high contrast or colorful regions, moving content, and regions matching a user-selected target colour.",
"Weight maps associated with these features are used to blend the input images into a single output frame for each position of the sliding window.In this paper, we extend the original exposure fusion formulation and formalize the process of using the resulting TIF algorithm to create long-exposure effects for video and photographs.",
"Figure REF provides an overview of the TIF workflow.",
"TIF differs from traditional Time Lapse Photography in that while time lapses are also created from a set of images taken over a possibly very long interval of time, time lapses assemble individual frames sequentially into a motion sequence without blending frames.",
"The visual effect is that of fast-forwarding: Hours turn to minutes, minutes turn to seconds.",
"Because of the lack of blending between frames, time lapses do not exhibit the long-exposure effects the TIF algorithm is designed to enhance.",
"Indeed, TIF can be applied to time lapse sequences to further highlight motion and temporal variation where desired.",
"In what follows, we describe the components of the TIF algorithm and demonstrate the kinds of visual effects that can be achieved with this technique."
],
[
"Temporal Image Fusion",
"For exposure fusion, the set of input frames is assumed to correspond to a sequence of differently exposed shots of the same (static) scene.",
"We now turn to the processing of a sequence of frames taken over time.",
"This can be a set of video frames, or a time-lapse photograph sequence.",
"In either case.",
"It is assumed that the images are registered, either having been taken with the camera mounted on a sturdy tripod or via post-processing.",
"For video applications, it is further assumed that the exposure time for each frame is close to $1/{\\rm fps}$ seconds.",
"Under these conditions any static structures in the scene will be rendered without blurring in the output frames.",
"We will discuss further on in the paper what happens when either of these assumptions is not met.",
"The simplest form of temporal image fusion uses a sliding temporal window over the input sequence $I_1,\\ldots ,I_K$ .",
"The result is an output sequence $R_1,\\ldots ,R_k$ in which image $R_t$ is produced by blending source frames $I_{t},I_{t-1},\\ldots ,I_{t-\\tau }$ effectively incorporating visual components from the $\\tau $ previous frames as well as the current one.",
"The value of $\\tau $ is set by the user, and controls the length of the virtual exposure time for each result frame.",
"For video, the effective virtual exposure time (VET) of each frame is given by ${\\rm VET}=(\\tau +1)/({\\rm fps})\\;s.$ .",
"For photographic sequences, the virtual exposure time is dependent on the interval between shots as well as the duration of each exposure.",
"In order to avoid sharp changes between successive blended frames, the sliding window has a Gaussian profile, giving larger weight to pixels in frames closer to the current one.",
"With the addition of the Gaussian sliding window, the blending weights in Eq.",
"REF become $W_{i,j,k}=(C_{i,j,k}^{\\alpha _c}\\cdot S_{i,j,k}^{\\alpha _s}\\cdot E_{i,j,k}^{\\alpha _e})\\cdot T(k,t),$ with $T(k,t)=e^{-\\frac{(t-k)^2}{2*\\sigma ^2}},$ where $t$ the index of the current frame, and $\\sigma =\\tau /3$ .",
"The rest of the exposure fusion framework remains unchanged: Input frames are processed to compute contrast, saturation, and exposure; and the pixels from all $\\tau +1$ frames are blended using pyramid blending to obtain the final image with the desired virtual exposure.",
"Figure REF shows a couple of blended frames resulting from applying temporal image fusion to a video sequence of fireworks.",
"Note the sharp detail on static regions of the scene, and the long-exposure effect that creates smooth trails for the fireworks themselves.",
"Compare this with the results of simply averaging the same set of frames, which yields blurry images with little detail in changing regions.",
"Figure REF shows different virtual exposure times applied to the same input sequence of waves.",
"For full frame-rate video, the virtual exposure can be controlled in increments of $1/({\\it frame rate})\\; s.$ .",
"This is much finer control than is allowed by typical DSLR exposure controls.",
"Figure: Left column: Original input frames from a movie sequence of fireworks.",
"Middle column: Resultsof averaging each frame with the previous 25 in the sequence.",
"Right column: Temporal blendingresults with τ=25\\tau =25 for a virtual exposure time of .83 seconds.Averaging generates blur and blends structure with the background.",
"Temporal image fusion produces a pleasing long-exposure effect and preservessharp detail.Figure: Left column: Original input frame from a movie sequence of waves and a detail crop.",
"Middle column: Temporal blendingresults for τ=15\\tau =15 (virtual exposure time .5 seconds).",
"Right column: Temporal blendingwith τ=55\\tau =55 (virtual exposure time 1.83 seconds).",
"Exposure length can be controlled withan accuracy of 1/(𝑓𝑝𝑠)1/({\\it fps}) seconds.This simple scheme is enough to produce long-exposure effects on video that can not be produced by the camera due to frame-rate restrictions.",
"For time-lapse photography, where the input frames are spaced by intervals that range from seconds to minutes, there is significantly more freedom in generating the input frame set.",
"For example, the blending process can be fed a sequence of tone-mapped HDR images.",
"Figure REF was created in just this way.",
"An input set of 96 LDR images was processed to yield 32 HDR frames, these were in turn processed via TIF to yield the final result.",
"No single HDR exposure could have captured both the afterglow and clouds in the sky, and the city lights which are turned on only after the sky turns dark.",
"Further, the 1 hour exposure time that results in the smoothing effect on the clouds could not have been achieved in practice in any other way, due to the strong illumination provided by the sunset sky.",
"In this way, TIF provides a way to achieve arbitrarily long exposures while allowing the photographer to retain control over the aperture and exposure settings of the camera.",
"Exposure times ranging from tens to hundreds of minutes are easy to achieve even under bright daylight.",
"Examples of long exposure photographs created from time-lapse sequences are shown on Fig.",
"REF .",
"Figure: Very-long exposure photography from time-lapse sequences.",
"Virtual exposure times range in the tens to hundreds of minutes."
],
[
"Fast-moving structure and temporal underexposure",
"The simple algorithm described above is sufficient in most instances to create pleasing visual effects.",
"However, for longer virtual exposure times, it has a tendency to replace fleeting phenomena with background content.",
"For video applications, fast moving objects will appear at a specific location for a limited amount of time, and will likely be replaced or at the very least strongly attenuated by the background structure from the remaining frames in the temporal window.",
"This is a form of temporal under-exposure: Fast-moving objects are imaged for too short a time at any image location to leave a lasting impression in the final blend.",
"Figure REF shows an example of this problem on a sequence of falling leaves.",
"Figure: Temporal under-exposure on a sequence of falling leaves.",
"On the left: a detail crop from one frame of the original video.",
"Notice the bright yellow leaf.",
"On the right: the temporal blending result for the entire sequence (20 seconds long).",
"Due to the short interval for which the leaf is imaged at each location, in the resulting blend the leaf has mostly disappeared into the background.",
"We would like to control, either to enhance or to suppress, the contribution of such dynamic structure to the blended results.In general, moving objects regardless of their speed will will be affected by some amount of blending with the background.",
"It would be desirable for the blending process to provide control over how much blending takes place.",
"One way to do this would be to use optical flow to detect moving image regions, and to increase their weight in the blending process proportional to the magnitude of their motion.",
"However, we find that it is sufficient to use a simple temporal distinctness map based on the difference between the current frame, and the average of all the frames seen thus far.",
"Specifically, for each pixel we compute a temporal distinctness value: $TD_{i,j,k}=\\max _{R,G,B}|I_{i,j,k}-\\mu _{i,j,k}|,$ where $\\mu (i,j,k)$ is the mean image from all frames seen thus far, and we take the maximum difference over the three colour channels.",
"Temporal distinctness values for the entire frame are then normalized to $[0,1]$ .",
"In the resulting map, pixels that correspond to fast moving objects and other transient structures will have distinctness value close to $1.0$ .",
"We use these values to modify the blending weights for pixels in the frame using an exponential envelope $\\hat{TD}_{i,j,k}=e^{\\alpha _{d} \\cdot TD_{i,j,k}}.$ For positive values of $\\alpha _{d}$ , $\\hat{TD}_{i,j,k} \\ge 1.0$ and has much larger magnitudes for pixels that have higher distinctness.",
"For negative $\\alpha _{d}$ , $\\hat{TD}_{i,j,k} \\le 1.0$ and is much smaller for the most temporally-distinct pixels.",
"The temporal distinctness term multiplies the blending weights from Eq.",
"REF .",
"$W_{i,j,k}=\\hat{TD}_{i,j,k}\\cdot C_{i,j,k}^{\\alpha _c}\\cdot S_{i,j,k}^{\\alpha _s}\\cdot E_{i,j,k}^{\\alpha _e}\\cdot T(k,t),$ Figure: Controlling temporal under-exposure.",
"These results show the effect of the temporal distinctness term on the final blending results.",
"The left column shows standard temporal blending (α d =0\\alpha _{d}=0).",
"The middle column shows the effect of increasing the blending weights for fast-moving structures (α d =25\\alpha _{d}=25).",
"The right column shows the effect of significantly reducing the weights for the same fast-moving content (α d =-25\\alpha _{d}=-25).",
"The blends incorporate all frames in the sequence, approximately 20 seconds in length.Figure REF shows results on a sequence of falling leaves illustrating the effect of the temporal distinctness component on the blended results.",
"The original temporal blending weights result in blurring of the falling leaves with the background.",
"The leaf trails are faint and look short.",
"Using a positive $\\alpha _{d}$ the weights of the leaves are greatly increased and the final blend shows much more strongly their trajectories over time.",
"Using a negative $\\alpha _{d}$ the reverse effect is obtained; the leaves are blended out and replaced by the background.",
"With the addition of the temporal distinctness component, we can manage temporal underexposure and selectively enhance or suppress the contributions of dynamic content to the output frames.",
"This provides an extra degree of control over the blending process along the temporal dimension."
],
[
"Content Selective Blending",
"Up to this point, the temporal discount factor $T(k,t)$ has been applied uniformly over entire frames.",
"However, we can easily change the blending process so that each pixel in the frame receives a different temporal decay, thus providing pixel-level control over exposure length.",
"We can use this to highlight specific scene content based on its appearance, much like we did in the previous section based on motion distinctness.",
"The simplest form of this process uses a threshold on RGB similarity between image pixels and a user selected colour to create a binary mask.",
"This mask is then multiplied by the blending weights from Eq.",
"REF for all previous frames in the current temporal window so that the temporal blending applies only to pixels similar enough to the selected colour, leaving the rest of the scene untouched.",
"Pixels in the current frame are unaffected by the threshold, so current scene content is always included in the final blend.",
"Figure REF shows an example of this process.",
"In this case the algorithm was set to a very light shade of red, and the threshold was adjusted so that the binary mask contains both head and tail lights.",
"The resulting blended output produces long trails for the lights while the remaining parts of the scene are left unchanged.",
"Figure: Content-selective blending.",
"Left: Original video frame.",
"Middle: Temporal blending results without content-dependent weights.",
"Every moving object is uniformly blurred in the resulting blend.",
"Right: Content-dependent fusion which selectively smooths car lights while leaving the rest of the scene unchanged.",
"Exposure length for both blended frames is 1 second.Figure: Content-dependent blending.",
"Top-left: Input frame from a slow-motion glass breaking sequence.",
"Top-right: Full-frame temporal blending result.",
"Bottom-left: Content-dependent blending of glass shards only.",
"Note that the coloured liquids are not affected.",
"Bottom-right: Content-dependent blending applied only to the green liquid.",
"The rest of the scene is unaffected and only the green liquid is rendered as if under long-exposure.",
"Original video courtesy of Zach King (http://www.finalcutking.com) used with permission.While a threshold on RGB distance may appear too simple, in practice it suffices to create a wide variety of visual effects.",
"Fig REF shows results of applying different RGB thresholds to the same input scene.",
"It is possible to achieve fine control over what regions of the image are temporally blended with only a minimum effort in terms of selecting a colour of interest and a suitable threshold.",
"More complex content detectors (e.g.",
"texture or object detectors) could be used to select what parts of the image are affected by the temporal blending, it is also possible to have the virtual exposure length be proportional to the strength of the response of the detector, instead of using a simple binary mask."
],
[
"Discussion and Future Work",
"The components of the temporal image fusion framework provide a large degree of control over the blending process.",
"Not only does TIF enable the rendering of long-exposure effects on video, but it also allows the user to select how strongly a transient structure contributes to the final blend, and to apply temporal blending selectively by visual appearance over specific image regions.",
"The technique can produce striking visual results assuming a small amount of care has been taken in setting up the camera during capture.",
"Most importantly, and as discussed above, the algorithm expects the input sequence to be registered.",
"This requirement can be difficult to meet in practice.",
"Fortunately, existing tools for panoramic photography can be used to register images with small misalignments (e.g.",
"due to camera or tripod shake) prior to blending.",
"We have successfully used the align image stack component of the Hugin panorama stitching software to align sequences in which strong wind caused a significant amount of camera shake.",
"In fact, Fig.",
"REF was created in just this way.",
"See the crops on Fig.REF for a comparison of the blended output with and without alignment.",
"In addition to an aligned image sequence, to create seamless blends on video it is necessary for each frame to have an exposure time as close as the camera permits to the inverse of the frame rate.",
"Thus, for a video shot at 30fps the individual frames should have an exposure time close to $1/30$ sec.",
"If the exposure time is much smaller than this, small moving objects will be imaged at separate, non-overlapping positions in successive frames, and the final blend will show a streaky pattern as shown in Fig.",
"REF .",
"Unfortunately, it is not always possible to set the exposure time to the desired value, and for fast moving objects this problem may still happen during the lag incurred by the camera while reading-off the current frame and clearing the sensor array in preparation for capturing the next.",
"Future work will look at applying motion compensation methods (e.g.",
"[15], [8], [31]) between consecutive frames to provide seamless blending.",
"We are also working on extending the content-dependent blending component to incorporate texture and motion information to help reduce artifacts on sequences in which colour is not sufficient to specify what regions should be affected (and how strongly) by the algorithm.",
"Figure: Detail crops from the final blend used in Fig. .",
"The image on the top shows the result of blending the input frames directly.",
"Due to a strong wind causing tripod shake, the blended result shows artifacts from misalignment between frames.",
"The image at the bottom is the result of blending the frames after processing with Hugin's align image stack which successfully removed the initial registration errors.Figure: Detail crop from the full-frame blend used in Fig. .",
"The camera's exposure time was significantly shorter than the interval between frames, leading to streaky pattern for fast moving objects.In terms of practical implementation: The TIF algorithm itself involves only low-level filtering operations and can be readily implemented without the need for specialized numerical or image processing libraries.",
"Currently, the main practical limitation for the TIF framework is the amount of memory required to store the Laplacian and Gaussian pyramids for the frames in the current blending window, as well as their weights.",
"This means that in practice the maximum virtual exposure for video is limited by how many frames can be stored in memory at a given time.",
"While the blending process is demanding enough that real-time performance is not yet feasible, it is realistic to expect that it can be implemented as an image processing App for use with recent Android-based cameras which have sufficient computational power and on-board memory to handle the load.",
"We will work on the development of a practical App for Android devices in the near term.",
"A working implementation of temporal image fusion, providing the basic temporal blending, is available at http://www.cs.utoronto.ca/~strider/TLF.",
"The source code provided should compile and run on Linux systems without modification, and porting it to Windows should be straightforward.",
"A complete implementation including the structure dependent blending and the temporal distinctness component will be made available in short order.",
"Except for the broken glasses sequence, all the input videos shown here were taken by the author and will be made available on request.",
"As a final note.",
"It is worth noting that current photo/video cameras are beginning to support HDR video capture.",
"Indeed, current DSLRs can be used to shoot video in HDR (with some limitations) via a firmware extension called MagicLantern [23].",
"Since temporal image fusion is a superset of exposure fusion, such HDR video can be processed with negligible extra effort to create HDR blended sequences that incorporate the dynamic range extension from HDR and all the visual effects provided by the temporal fusion framework as discussed above."
],
[
"Conclusions",
"This paper presented Temporal Image Fusion.",
"The framework is intended to provide control over the way in which dynamic scene content is captured and rendered onto photographs or video, thereby expanding the temporal dynamic range of current photo and video cameras.",
"TIF can render long-exposure photographic effects onto full frame-rate video, generate arbitrarily long exposures for photography, enhance or suppress dynamic content, and selectively blend image regions by visual similarity.",
"The framework is implemented via low-level processing and filtering, making it a good candidate for on-camera implementation.",
"TIF has applications in video editing and photography, and the results shown here serve to illustrate the wide range of striking visual effects that are possible via this technique."
]
] | 1403.0087 |
[
[
"The joint law of the extrema, final value and signature of a stopped\n random walk"
],
[
"Abstract A complete characterization of the possible joint distributions of the maximum and terminal value of uniformly integrable martingale has been known for some time, and the aim of this paper is to establish a similar characterization for continuous martingales of the joint law of the minimum, final value, and maximum, along with the direction of the final excursion.",
"We solve this problem completely for the discrete analogue, that of a simple symmetric random walk stopped at some almost-surely finite stopping time.",
"This characterization leads to robust hedging strategies for derivatives whose value depends on the maximum, minimum and final values of the underlying asset."
],
[
"Introduction.",
"Suppose given 0, and suppose that $(\\xi _t, {\\mathcal {F}} _t)_{t\\in ^+} $ is a symmetric simple random walk on the grid $$ , started at zero.",
"Define $S_t \\equiv \\sup _{s\\le t} \\xi _s$ , $I_t \\equiv \\inf _{s\\le t} \\xi _s$ , $g^+_t \\equiv \\inf \\lbrace u \\le t: \\xi _u = S_u\\rbrace $ , $g^-_t \\equiv \\inf \\lbrace u \\le t: \\xi _u = I_u\\rbrace $ , and let $\\sigma _t &=& +1 \\qquad \\hbox{\\rm if $g^+_t > g^-_t$}\\nonumber \\\\&=& -1 \\qquad \\hbox{\\rm else}.$ The process $S$ records the running maximum of the martingale, and the process $\\sigma $ records whether the martingale is currently on an excursion down from its running maximum ($\\sigma = +1$ ) or on an excursion up from its running minimum ($\\sigma = -1$ ).",
"We refer to the process $\\sigma $ as the signature of the random walk.",
"Suppose that $T$ is an almost-surely finite $( {\\mathcal {F}} _t)$ -stopping time, and write $X_t \\equiv \\xi _{t\\wedge T}$ for the stopped process.",
"The paper is concerned with the possible joint laws $m$ of the quadruple $(I_T, X_T, S_T, \\sigma _T)$ , which we will abbreviate to $(I,X,S,\\sigma )$ where no confusion may arise.",
"Clearly the law $m$ must be defined on the set $ {\\mathcal {X}} \\equiv (-h\\mathbb {Z} ^+) \\times \\times ^+ \\times \\lbrace -1,+1\\rbrace $ , and evidently we must have $m( I \\le X \\le S) = 1$ ; but beyond this, is it possible to state a set of necessary and sufficient conditions for a probability $m$ on $ {\\mathcal {X}} $ to be the joint distribution of $(I_t,X_T,S_T,\\sigma _T)$ ?",
"The motivation for this attempt is twofold.",
"Firstly, the joint law of $(X,S)$ has been characterized completely (for general local martingales, not assumed to be continuous or uniformly integrable) in [7]; can the methods of that paper be extended to deal with the running minimum also?",
"The second reason to look at this problem is the interesting recent work of Cox & Obloj [3] which finds extremal martingales for various derivatives whose payoffs depend on the maximum, minimum and terminal value of the underlying asset.",
"This builds to some extent on the earlier work of Hobson and others ([6], [1], [2]), which addresses similar questions for derivatives whose payoffs depend only on the maximum and terminal value of the underlying asset.",
"Many of the results of this literature can be derived alternatively using the results of [7], by converting the problem into a linear program.",
"This approach is more general, but leads to less explicit answers in the specific instances analyzed to date.",
"What we shall find here is that it is possible to generalize the results of [7] to cover the joint law of $(I,X,S,\\sigma )$ , but that the statements are more involved.",
"For this reason, we shall restrict our analysis to a symmetric simple random walk taking values in a grid $$ for some 0, stopped at an almost-surely finite stopping time.",
"The main result is presented in Section .",
"The proof of necessity is in Section REF , and requires only the judicious use of the Optional Sampling Theorem.",
"The proof of sufficiency, in Section , is constructive, and requires suitable modification of some of the techniques of [7].",
"We then show in Section how this characterization can lead to robust hedging schemes and extremal prices for derivatives whose payoff depends on the maximum, minimum, terminal value and signature."
],
[
"The main result.",
"We take a symmetric simple random walk $(\\xi _t, {\\mathcal {F}} _t)_{t\\in ^+} $ on $$ for some fixed $h>0$ ; in general, the filtration $( {\\mathcal {F}} _t)$ is larger than the filtration of the random walk, to allow for additional randomization.",
"Stopping $\\xi $ at the almost-surely finite stopping time $T$ creates the martingale $X_t = \\xi _{t \\wedge T}$ .",
"We use the notation of the Introduction, and notice that $g^+_t \\equiv \\sup \\lbrace u \\le t: S_u > S_{u-\\rbrace ,\\qquad g^-_t \\equiv \\sup \\lbrace u \\le t: I_u < I_{u-\\rbrace ,}emphasizing the fact that we are dealing with {\\em strict}ascending/descending ladder epochs, to use the language ofFeller \\cite {Feller}.",
"The process \\sigma is defined asbefore at (\\ref {sigdef}).",
"}$ Definition 2.1 We say that the probability measure $m$ on $ {\\mathcal {X}} \\equiv -^+ \\times \\times ^+ \\times \\lbrace -1,+1\\rbrace $ is consistent if there is some almost-surely finite $( {\\mathcal {F}} _t)$ -stopping time $T$ such that $m$ is the law of $(I_T,X_T,S_T,\\sigma _T)$ ."
],
[
"Necessity.",
"For $x \\in $ we define the hitting time $H_x = \\inf \\lbrace u: \\xi _u = x\\rbrace ,$ with the usual convention that the infimum of the empty set is $+\\infty $ .",
"In what follows, we will let $a$ , $b$ stand for two generic members of $^+$ , and will be studying the exit time $H_b \\wedge H_{-a} \\equiv \\inf \\lbrace u: \\xi _u \\notin (-a,b)\\rbrace $ and related stopping times.",
"The measure $m$ says nothing directly about these stopping times, but by way of the Optional Sampling Theorem we are able to deduce quite a lot of information about them if the law $m$ is consistent.",
"Indeed, assuming that $m$ is consistent, we are able to find the probability that $H_{-a}<H_b$ (for example) in terms of $m$ -expectations of functions defined on $ {\\mathcal {X}} $ .",
"The expressions derived make perfectly good sense even if $m$ is not consistent, but it may be that the expressions do not in general satisfy positivity or other properties which would hold if $m$ were consistent.",
"For this reason, we will denote by $ \\bar{m} (Y)$ the expression for the $m$ -expectation of a random variable $Y$ which would be correct if $m$ were consistent; if $m$ is not consistent, all we have is an algebraic expression without the desired probabilistic meaning, and the use of the symbol $ \\bar{m} $ warns us not to assume properties which need not hold.",
"The first result we need is the following, which illustrates the use of this notational convention.",
"Proposition 1 For any $a, b \\in ^+$ we have $ \\bar{m} (H_b < H_{-a}) &=&\\frac{ a - m(a+X; S<b, I>-a) }{a+b}\\;\\equiv \\varphi (b,-a) ,\\\\ \\bar{m} (H_{-a} < H_b)&=&\\frac{b - m(b-X; S<b, I>-a) }{a+b}\\;\\equiv \\varphi (-a,b) .$ Proof.",
"We use the Optional Sampling Theorem at the time $H_b \\wedge H_{-a}$ to derive the two equations $1 &=& \\bar{m} (H_{-a} < H_b) + \\bar{m} (H_b < H_{-a}) + m(S<b, I>-a)\\\\0 &=& -a\\, \\bar{m} (H_{-a} < H_b) +b\\, \\bar{m} (H_b < H_{-a}) + m(X;S<b, I>-a).$ Solving this pair of linear equations leads to the conclusion that $ \\bar{m} (H_b < H_{-a}) &=& \\bigl \\lbrace \\, a - m(a+X; S<b, I>-a) \\,\\bigr \\rbrace /(a+b) \\; ,\\\\ \\bar{m} (H_{-a} < H_b) &=& \\bigl \\lbrace \\, b - m(b-X; S<b, I>-a) \\, \\bigr \\rbrace /(a+b) \\; ,$ as claimed.",
"$\\square $ If $m$ is consistent, then we would have for any $a, b \\in ^+$ not both zero that $ \\bar{m} (H_{-a} < H_b < H_{-a-) &=& \\bar{m} (H_{-a} \\le H_b < H_{-a-)\\\\&=& \\bar{m} (H_b < H_{-a-) - \\bar{m} (H_b < H_{-a})\\\\&=& \\bar{m} (H_b < \\infty , I(H_b) = -a).",
"}This is because on the event \\lbrace H_{-a} < H_b < H_{-a- \\rbrace the hitting time H_b is finite, and so cannot be equal toH_{-a}; the second equality follows from the inclusion\\lbrace H_b < H_{-a} \\rbrace \\subseteq \\lbrace H_b < H_{-a- \\rbrace .",
"We willtherefore introduce the notation\\begin{eqnarray}\\psi _+(-a,b) &=& \\varphi (b,-a- - \\varphi (b,-a),\\\\\\psi _-(-a,b) &=& \\varphi (-a,b+ - \\varphi (-a,b).\\end{eqnarray}Notice that \\psi _+(-a,b) is {\\em defined}as an algebraic expression in terms of mvia (\\ref {psip}) and (\\ref {m+}); if m is {\\em consistent}, then\\psi _+(-a,b) is equal to \\bar{m} (H_b<\\infty , I(H_b) = -a),but no such interpretation holds in general.",
"}}\\vspace{14.45377pt}}The necessary condition we derive comes from considering whatmay happen if the event B_+ = \\lbrace H_b < \\infty , I(H_b) = -a\\rbrace occurs.",
"When this event occurs, the martingale X does reachb before being stopped, and at that time H_b the minimumvalue is -a.",
"Thereafter, one of three things will happen:\\begin{itemize}\\item [(i)] X reaches b+ before reaching -a- andbefore T;\\item [(ii)] T happens before X reaches either -a-or b+;\\item [(iii)] X reaches -a- before reaching b+ andbefore T.\\end{itemize}The next result derives a necessary condition from the OptionalSampling Theorem applied at H_{-a- \\wedge H_{b+ \\wedge T.}\\begin{proposition}Define the events\\begin{equation}B_+ = \\lbrace H_b < \\infty , I(H_b) = -a)\\rbrace , \\qquad B_- = \\lbrace H_{-a}<\\infty , S(H_{-a}) = b \\rbrace ,\\end{equation}set p_\\pm = \\bar{m} (B_\\pm ) = \\psi _\\pm (-a,b),and set\\begin{equation}p_{+0} = m(S = b, I = -a, \\sigma =+1),\\qquad p_{-0} = m(S = b, I = -a, \\sigma =-1).\\end{equation}If we denote\\begin{equation}v_{\\pm } \\equiv \\frac{m(X; S=b,I=-a,\\sigma = \\pm 1)}{p_{\\pm 0} }\\equiv m(X \\, \\vert \\, S=b,I=-a,\\sigma = \\pm 1),\\end{equation}then the conditions\\footnote {If either of p_\\pm is zero, thenthe inequalities (\\ref {nc5}), (\\ref {nc6}) have to be understood incross-multiplied form, when they state vacuously that 0 \\le 0.",
"}\\begin{eqnarray}\\frac{p_{+0}}{p_+} & \\le &\\frac{{b+v_+}\\\\\\frac{p_{-0}}{p_-}& \\le &\\frac{{a+v_-}}{a}re necessary for m to be consistent.",
"}{}\\end{eqnarray}\\end{proposition}\\medskip {\\sc Proof.",
"}We introduce the notation\\begin{gather}p_{++} = \\bar{m} ( H_{-a}< H_b<H_{b+ < H_{-a- ),\\quad p_{+-} = \\bar{m} ( H_{-a}< H_b<H_{-a- < H_{b+ ),\\nonumber \\\\p_{--} = \\bar{m} ( H_{b}< H_{-a}<H_{-a- < H_{b+ ),\\quad p_{-+} = \\bar{m} ( H_{b}< H_{-a}<H_{b+ < H_{-a- ).\\nonumber }Using the Optional Sampling Theorem,we have similarly to (\\ref {p1}), (\\ref {OST1}) the equations\\begin{eqnarray}p_+ &=& p_{++} + p_{+0} + p_{+-}\\\\bp_+ &=& (b+ p_{++}-(a+ p_{+-}+m(X; S=b,I=-a,\\sigma = +1).\\end{eqnarray}If we write \\tilde{p}_{xy} = p_{xy}/p_x for x\\in \\lbrace -, + \\rbrace , y \\in \\lbrace -, 0 , +\\rbrace the equations (\\ref {p2}), (\\ref {OST2}) are expressed more simplyin conditional form:\\begin{eqnarray}1 &=& \\tilde{p}_{++} + \\tilde{p}_{+-} + \\tilde{p}_{+0}\\\\b &=& (b+\\tilde{p}_{++}-(a+ \\tilde{p}_{+-}+\\tilde{p}_{+0} v_+.\\end{eqnarray}The value of p_{+0} is known from m, as is the valueof v_+, and since we assume that m is consistent thevalues of p_\\pm = \\psi _\\pm (-a,b) are also known from m.Therefore we can solve the linear system(\\ref {p3}), (\\ref {OST3}) to discover\\begin{eqnarray}\\tilde{p}_{++} &=& \\frac{b+a+ (a+v_+)\\, \\tilde{p}_{+0}}{b+a+2\\\\\\tilde{p}_{+-} &=& \\frac{h - (b+v_+)\\, \\tilde{p}_{+0}}{b+a+2.",
"}In order that \\tilde{p}_{+-} as given by (\\ref {p+-}) should benon-negative, we require that\\begin{equation}\\tilde{p}_{+0} \\equiv \\frac{m(S = b, I = -a, \\sigma =+1)}{p_+ } \\le \\frac{{b+v_+},}{w}hich is condition (\\ref {nc5}).",
"Necessity of (\\ref {nc6}) isderived similarly.\\end{equation}}\\hfill \\square \\end{eqnarray}}\\medskip {\\sc Remarks.}",
"(i) The necessary conditions (\\ref {nc5}), (\\ref {nc6})come from the requirement that \\tilde{p}_{+-} and \\tilde{p}_{-+}should be non-negative.",
"Do we know for sure that \\tilde{p}_{++}and \\tilde{p}_{--} are non-negative?",
"The definition (\\ref {vdef})of v_\\pm guarantees that -a \\le v_\\pm \\le b, so if (\\ref {p+0})holds then we know that \\tilde{p}_{+0} \\le 1.",
"From (\\ref {p++})we see then that \\tilde{p}_{++} \\ge 0.",
"Since all the summands on theright-hand side of (\\ref {p3}) are non-negative, we learn thatthey are probabilities summing to 1.",
"}\\medskip (ii) Notice that we have two expressions for \\bar{m} (H_{b+<\\infty ,I(H_{b+) = -a), either as p_{++} + p_{-+}, or as\\psi _+(-a,b+.Confirming that these are the same is an important step in the proofof sufficiency.",
"}\\bigbreak \\subsection {Sufficiency.",
"}We have now identified necessary conditions(\\ref {nc5}) and (\\ref {nc6}) for m to be consistent.The main result of this paper is that these conditions are alsosufficient.\\begin{theorem}The probability measure m on {\\mathcal {X}} \\equiv -^+ \\times \\times ^+ \\times \\lbrace -1,+1\\rbrace isconsistentif and only if m(I \\le X\\le S) = 1 and necessary conditions(\\ref {nc5}) and (\\ref {nc6}) hold.\\end{theorem}}\\bigbreak {\\sc Proof.}",
"Necessity has been proved, sowhat remains is to show that conditions(\\ref {nc5}) and (\\ref {nc6}) aresufficient.",
"Not surprisingly, the proof of this is constructive.",
"}We require a probability space (\\Omega , {\\mathcal {F}} , P) rich enough tocarry an IID sequence U_0, U_1, \\ldots of U[0,1] randomvariables, and an independent standard Brownian motion (B_t).Let {\\mathcal {U}} = \\sigma (U_0, U_1, \\ldots ), and let ( {\\mathcal {G}} _t) bethe usual augmentation of the filtration( {\\mathcal {U}} \\vee \\sigma (B_s: s\\le t)).",
"Define ( {\\mathcal {G}} _t)-stopping times\\begin{equation*}\\alpha _0 \\equiv 0, \\qquad \\alpha _{n+1} \\equiv \\inf \\lbrace t> \\alpha _n : |B_t-B_{\\alpha _n}| > ,\\end{equation*}the process \\xi _{n \\equiv B(\\alpha _n) and the filtration {\\mathcal {F}} _{n \\equiv {\\mathcal {G}} _{\\alpha _n}, so that (\\xi _t, {\\mathcal {F}} _t)_{t \\in ^+} is a symmetric simple random walk.",
"As before,defineS_t \\equiv \\sup _{s\\le t} \\xi _s, I_t \\equiv \\inf _{s\\le t} \\xi _sfor t \\in ^+.",
"}}The construction borrows the technique of \\cite {R1},where we firstly modify the given law m so that theconditional distribution of X_T given \\lbrace S_T=b, I_T = -a, \\, \\sigma _T = s \\rbrace is a unit mass on theexpected value m[ X_T \\,|\\, S_T=b, I_T = -a, \\, \\sigma _T = s\\, ].If we can construct a martingale with this degenerate conditionallaw, then we can build the required distribution of X_Tgiven \\lbrace S_T=b, I_T = -a, \\, \\sigma _T = s \\rbrace by Skorokhod embeddingin a Brownian motion.",
"So we may and shall supposethat\\footnote {There is no reason why v need be a multipleof h, but this does not matter; if s = +, say, we shall usethe Brownian motion living in the original probability space,starting at b and rununtil it first hits either theupper barrier b+ or the lower barrier, which will be{\\em randomized}, taking value v_+ withsuitably-chosen probability \\theta , otherwise taking value -a-.",
"}\\begin{equation}m[X_T = v \\, |\\, S_T=b, I_T = -a, \\, \\sigma _T = s\\, ] = 1,\\end{equation}where v = m[ X_T \\,|\\, S_T=b, I_T = -a, \\, \\sigma _T = s\\, ].",
"}The construction is sequential, and the proof that it succeeds isinductive.",
"Let \\tau _n \\equiv \\inf \\lbrace t: S_t - I_t = n,and set \\sigma _n = \\alpha _{\\tau _n}, the corresponding stoppingtime for the Brownian motion.",
"The construction of T beginsby setting T = 0 if U_0< m(S=I=0), otherwise T \\ge h = \\tau _1.The sequential construction supposes\\footnote {Weprovide details of what happens if S_{\\tau _n} = \\xi _{\\tau _n};the treatment of the case I_{\\tau _n} = \\xi _{\\tau _n} isanalogous.}",
"we have found thatT \\ge \\tau _n, and S_{\\tau _n} = \\xi _{\\tau _n} = b,I_{\\tau _n}=-a.",
"Then we placea lower barrier \\ell \\in [-a-h,b+h] by the recipe\\begin{eqnarray*}\\ell &=& v_+ \\qquad \\textrm {if } U_n < \\theta \\\\&=& -a-h \\qquad \\textrm {else}\\end{eqnarray*}where v_+ is defined in terms of m by (\\ref {vdef}), and\\theta is defined by\\begin{equation}\\tilde{p}_{+0} \\equiv \\frac{m(S=b, I = -a, \\sigma = +1)}{\\psi _+(-a,b)}= \\frac{m(S=b, I = -a, \\sigma = +1)}{ \\bar{m} (H_b<\\infty ,I(S_b) = -a) }= \\theta \\; \\frac{{b+v_+}}{w}ith the notation of Proposition \\ref {prop2}; in view of thefact that we have assumed the necessary conditions (\\ref {nc5})and (\\ref {nc6}), {\\em we can assert\\footnote {We shall establish in the inductive proof that \\psi _\\pm are non-negative.}",
"that \\theta so defined {\\em is}a probability}: 0 \\le \\theta \\le 1.",
"We now run the Brownianmotion B forward from time \\sigma _n until it first hits\\ell or b+h.",
"If \\ell = v_+ and B hits \\ell beforeb+h, then we will stop everything at that time, anddeclare that X_T = v_+; otherwise, we will reach either-a-h or b+h and declare that T \\ge \\tau _{n+1}.",
"If wedetermine that T \\ge \\tau _{n+1}, we take a further step of theconstruction.\\end{equation}For each n \\ge 1, let Q_n be the combined statement\\footnote {The functions \\psi _\\pm are defined in terms of m by(\\ref {m+}), (\\ref {m-}), (\\ref {psip}), (\\ref {psim}).",
"}\\begin{center}\\begin{itemize}\\item [(i)] for all a, b \\in ^+, 0 < a+b \\le n\\begin{eqnarray}P(H_b \\le T,\\; I(H_b) = -a) &=& \\psi _+(-a,b)\\\\P(H_{-a} \\le T,\\; S(H_{-a}) = b)&=& \\psi _-(-a,b)\\end{eqnarray}\\item [(ii)]\\begin{equation}P(S=x, I = -y, X = z, \\sigma = s)= m(S=x, I = -y, X = z, \\sigma = s)\\end{equation}for alls \\in \\lbrace -1,1\\rbrace , x, y, z, \\in ,x, y \\ge 0, x+y < n.\\end{itemize}\\end{center}We shall prove by induction that Q_n is true for all n>0,establishing the statement first for n=1.",
"We prove (\\ref {Qnp}),leaving the analogous proof of (\\ref {Qnm}) to the diligent reader.Taking b = 0, \\; a = ,(\\ref {Qnp}) says thatP(H_0 \\le T, \\; I(H_0) = - = \\psi _+(-0),and both sides are readily seen to be equal to zero; taking b=h,\\;a = 0,(\\ref {Qnp}) says that\\begin{eqnarray*}P(H_T, \\; I(H_ = 0) &=& \\psi _+(0,\\\\&=& \\varphi (- - \\varphi (0)\\\\&=&\\frac{ m(X; S< I>- }{2\\; - 0\\\\&=& \\frac{1}{2} \\bigl [ \\, 1 - m(S=X=I=0) \\, ]}which is clearly true, because if the construction does notstop immediately at time 0 (an event of probability m(I=X=S=0))then with equal probability the process steps at time 1 to\\pm .",
"The second statement (\\ref {Qnii}) holds becausewe have constructed the probability of I=X=S=0 correctly.\\end{eqnarray*}\\vspace{14.45377pt}Now suppose that Q_k has been proved to hold for k \\le n; we haveto prove (\\ref {Qnp}), (\\ref {Qnm}) and (\\ref {Qnii}) for n+1.To prove (\\ref {Qnii}), suppose that x, \\; y \\in ^+ andx+y = n. By construction, the random walk will be stopped beforethe range S-I increases to (n+1) if and only if the barrier\\ell happens to be positioned at v_+ {\\em and} that barrieris hit before the Brownian motion rises to b+h.Conditional on the event B_+ =\\lbrace T \\ge \\tau _n, \\; S_{\\tau _n} = \\xi _{\\tau _n} = b,\\; I_{\\tau _n} = -a \\rbrace ,the probabilityof that joint event is\\begin{equation}\\theta \\times \\frac{h}{b+h-v_+}.\\end{equation}By the inductive hypothesis (\\ref {Qnp}) we have that the probabilityof the conditioning event B_+ is \\psi _+(-a,b); so from thedefinition (\\ref {thetadef}) of \\theta we learn thatP(S_T = b, \\; I_T = -a, \\; \\sigma = +1 )=m(S=b, I=-a, \\; \\sigma = +1).Given that this event happens, the conditional distribution ofX_T is correct, by the Skorohod embedding construction of X_Twith mean v_+.",
"Therefore (\\ref {Qnii}) has been proven for anyx, \\; y \\in with x+y = n, and for any z \\in ,s \\in \\lbrace -1, 1\\rbrace .",
"}\\vspace{10.84006pt}It remains to prove assertion (i) of Q_{n+1}, and for this we recallsome of the notation of the proof of Proposition \\ref {prop2}.",
"Fora, \\; b \\in ^+, a+b = n, wewrite\\begin{eqnarray*}p_+ &=& P(B_+ ) \\equiv P(H_b \\le T, \\; I(H_b) = -a),\\\\p_- &=& P(B_-) \\equiv P(H_{-a} \\le T, \\; S(H_{-a}) = b)\\end{eqnarray*}which in view of the truth of Q_n we know are equal to\\psi _+(-a,b) and \\psi _-(-a,b) respectively.If we now define\\begin{eqnarray*}p_{++} &=& P(B_+, H_{b+ \\le T \\wedge H_{-a- )\\\\p_{+-} &=& P(B_+, H_{-a- \\le T \\wedge H_{b+ )\\\\p_{+0} &=& P(B_+, T <\\tau _{n+1})\\\\p_{-+} &=& P(B_-, H_{b+ \\le T \\wedge H_{-a- )\\\\p_{--} &=& P(B_-, H_{-a- \\le T \\wedge H_{b+ )\\\\p_{-0} &=& P(B_-, T < \\tau _{n+1})}then by exactly the same Optional Sampling argument which led to(\\ref {p++}), (\\ref {p+-}), we conclude that\\begin{eqnarray}p_{++} &=& \\frac{(b+a+p_+ - (a+v_+)\\, p_{+0}}{b+a+2\\\\p_{+-} &=& \\frac{p̉_+ - (b+v_+)\\, p_{+0}}{b+a+2\\\\p_{-+} &=& \\frac{p̉_- - (a+v_-) p_{-0}}{a+b+2\\\\p_{--} &=& \\frac{(a+b+p_- - (b+v_-)p_{-0}}{a+b+2}and now the task is to prove (after cross-multiplyingby a+b+2) that\\begin{equation}(a+b+2\\lbrace \\,p_{++} + p_{-+}\\, \\rbrace = (a+b+2\\psi _+(-a, b+,\\end{equation}and the minus analogue, which is just the same argument{\\it mutatis mutandis.",
"}Firstly we develop the left-hand side using (\\ref {ppp}),(\\ref {ppm}) and their analogues for B_- to obtain\\begin{eqnarray*}LHS &=& (a+b+\\psi _+(-a,b) - (a+v_+)p_{0+}+_-(-a,b) - (a+v_-) p_{-0}\\\\&=& (a+b+\\lbrace \\, \\varphi (b,-a- - \\varphi (b,-a)\\, \\rbrace + \\, \\varphi (-a,b+ - \\varphi (-a,b)\\, \\rbrace \\\\&&\\qquad \\qquad \\qquad -(a+ m(S=b,I=-a) - m(X; S = b, I= -a)\\\\&=& a+m(a+X; S<b, I>-a- - \\lbrace \\, a - m(a+X; S<b, I>-a)\\,\\rbrace \\\\&&\\qquad \\qquad - \\varphi (b-a) + \\varphi (-a,b)) + (-a,b+-m(a+X; S=b, I = -a)\\\\&=& m(a+X; S<b, I>-a- +m(a+X; S<b, I>-a)\\\\&&\\qquad \\qquad -1 - m(S<b,I>-a)\\rbrace + (-a,b+-m(a+X; S=b, I = -a)\\\\&=& -m(a+X; S<b, I>-a- +m(a+X; S<b, I>-a)\\\\&&\\qquad \\qquad -m(a+X; S=b, I = -a) + (-a,b+\\\\&=& -m(a+X: (A_2\\cup A_3) \\backslash A_1) + (-a,b+\\end{eqnarray*}where A_1 = \\lbrace S<b, I > -a\\rbrace , A_2 = \\lbrace S<b, I>-a- and A_3 = \\lbrace S=b, I = -a\\rbrace .",
"Noticing thatA_1 \\subseteq A_2 and A_3 is disjoint from A_1,the region of integration is(A_2\\cup A_3) \\backslash A_1= \\lbrace S<b, I = -a\\rbrace \\cup A_3= \\lbrace S \\le b, I = -a\\rbrace = \\lbrace S < b+ I = -a\\rbrace .Hence the left-hand side is equal to\\begin{equation}LHS = -m(a+X; S<b+ I = -a) + (-a,b+.\\end{equation}}\\medskip Turning now to the right-hand side of (\\ref {toprove}), we have\\begin{eqnarray}RHS &=& (a+b+2\\lbrace \\, \\varphi (b+-a- - \\varphi (b+-a)\\,\\rbrace \\nonumber \\\\&=& a+m(a+X:S<b+I>-a- -(b+-a)\\nonumber \\\\&&\\qquad \\qquad \\qquad - \\lbrace \\, a - m(a+X:S<b+I>-a)\\,\\rbrace \\nonumber \\\\&=& m(a+X:S<b+I>-a-+m(a+X;S<b+I>-a)\\nonumber \\\\&&\\qquad \\qquad -m̉(S<b+I>-a) - (b+-a)\\nonumber \\\\&=&\\, 1-m(S<b+I>-a) - \\varphi (b+-a)\\, \\rbrace \\nonumber \\\\&&\\qquad \\qquad \\qquad - m(a+X;S<b+ I = -a).\\end{eqnarray}Comparing (\\ref {LHS}) and (\\ref {RHS}), we see that we have to prove\\begin{equation} \\varphi (b+-a) + \\varphi (-a,b+ = 1- m(S<b+I>-a),\\end{equation}which is evidently true from the definition (\\ref {m+}),(\\ref {m-}) of \\varphi .",
"}\\hfill \\square }\\end{eqnarray}\\section {Hedging.",
"}Theorem \\ref {thm1} provides us with necessary and sufficientconditions for a measure m on {\\mathcal {X}} to be consistent.",
"Inprinciple, this allows us to construct extremal martingales, androbust hedges for derivatives.",
"}Let us firstly see how this works in the context of the joint lawof (S,X) studied in \\cite {R1}.",
"We begin by recalling some ofthe results of that paper.We let X_t = B_{t \\wedge T} bea Brownian motion stopped as an almost-surely finite stoppingtime T, with S_t = \\sup _{u \\le t} X_u, and withS \\equiv S_\\infty , X \\equiv X_\\infty .",
"With this terminology,Theoren 3.1 of \\cite {R1} says the following.",
"}\\begin{theorem}The probability measure \\mu on \\mathbb {R} ^+ \\times \\mathbb {R} ^+ is thejoint law of (S, S-X) for some almost-surely finitestopping time T if and only if\\begin{equation}\\biggl ( \\iint _{ (t,\\infty ) \\times \\mathbb {R} ^+} \\mu (ds,dy)\\biggr ) dt \\ge \\int _{(0,\\infty )} y \\; \\mu (dt, dy).\\end{equation}If (X_t)_{t\\ge 0} is also uniformly integrable, theninequality (\\ref {R1_3.1}) holds with equality:\\begin{equation}\\biggl ( \\iint _{ (t,\\infty ) \\times \\mathbb {R} ^+} \\mu (ds,dy)\\biggr ) dt = \\int _{(0,\\infty )} y \\; \\mu (dt, dy).\\end{equation}Finally, if (\\ref {R1_3.2}) holds, and if X \\in L^1,\\begin{equation}\\iint |t-y| \\; \\mu (dt,dy) < \\infty ,\\end{equation}then \\mu is the joint law of (S, S-X) for auniformly integrable martingale (X_t)_{t\\ge 0}.\\end{theorem}}\\medskip {\\sc Proof.}",
"See \\cite {R1}.",
"The final assertion is notin \\cite {R1}, but can easily be deduced.",
"In view of the firstassertion, there is some stopping time T<\\infty such that\\mu is the joint law of (S,S-X).",
"By multiplying (\\ref {R1_3.2})by some non-negative test function \\varphi and integratingwith respect to t we discover that\\begin{equation}\\mu (\\Phi ) = \\mu ( \\, (S-X)\\varphi (S) \\,)\\end{equation}where \\Phi (t) = \\int _0^t \\, \\varphi (y) \\; dy.",
"Taking\\varphi (x) = I_{\\lbrace x > b \\rbrace } for some b\\ge 0 we find that\\begin{equation}b \\mu (S>b) = \\mu ( X: S>b).\\end{equation}Using the fact that X \\in L^1, we can let b \\uparrow \\infty in (\\ref {eq35}) to prove that \\lim _{b \\uparrow \\infty }b \\mu (S>b) =0.",
"Lemma 2.3 of \\cite {R1} gives the result.",
"}\\hfill \\square }\\medskip {\\sc Remark.}",
"Standard monotone class arguments show that(\\ref {R1_3.1}) is equivalent to the statement that\\begin{equation}\\mu (\\Phi ) \\ge \\mu ( \\, (S-X)\\varphi (S) \\,)\\end{equation}for all non-negative test functions, which again is equivalentto the statement that\\begin{equation}b \\mu (S>b) \\ge \\mu ( X: S>b)\\end{equation}for all b \\ge 0.",
"Likewise, (\\ref {R1_3.2}) is equivalentto (\\ref {eq34}) for all non-negative test functions \\varphi ,which again is equivalent to the statement (\\ref {eq35}):\\begin{equation}\\mu ( \\, X-b : S>b ) = 0\\qquad \\forall b \\ge 0.\\end{equation}}\\vspace{7.22743pt}An important and typical\\footnote {The papers Hobson\\cite {Hobson1}, ... give examples of thiskind.}",
"use of this would be to try to findan {\\em extremal} martingale, which would in turn lead to amaximum possiblederivative price and a robust hedging strategy.",
"So, forexample, suppose that we observe call option prices C(K) for everystrike K at a common fixed expiry time\\footnote {Let us suppose that the expiry is 1.}",
"for some (discounted)asset, and suppose that the asset has continuous paths(X_t)_{0 \\le t\\le 1}, andis a uniformly-integrable martingale in thepricing measure.",
"}Suppose now that we are given somederivative whose payoff at time 1 is G(S_1, X_1), whereS_1 = \\sup _{0 \\le t \\le 1} X_t; {\\em what is the mostexpensive the time-0 price of this derivative can be?",
"}\\end{eqnarray*}The time-0 price of the derivative is given by\\begin{equation}\\iint G(s, x) \\; q(ds,dx)\\end{equation}where q is the joint law\\footnote {As before, when the time subscript of a process is omitted, weunderstand it to be 1.}",
"of (S, X).",
"Now provided the law q satisfies the conditions\\begin{equation}\\iint (x-K)^+ \\; q(ds,dx) = C(K) \\qquad \\forall K\\end{equation}and (see (\\ref {suff3}))\\begin{equation}\\iint _{s>b} (x-b) \\; q(ds,dx) = 0 \\qquad \\forall b>0\\end{equation}then q is the joint distribution of (S, X) for {\\em some }continuous martingale whose law at time 1 agrees with the datacontained in the call prices.",
"The problem of finding the mostexpensive time-0 price is therefore the problem of maximizing the{\\em linear} objective (\\ref {obj1}) over non-negative probabilitymeasures q subject to the {\\em linear}constraints (\\ref {cons1}) and(\\ref {cons2}).Writing the problem in Lagrangian form\\footnote {This linear programming approach to the problem is also usedin \\cite {DOR}.",
"}, we seek\\begin{eqnarray}L(\\alpha ,\\eta ,\\lambda ) &=& \\sup _{ q \\ge 0} \\biggl [\\;\\iint \\bigl \\lbrace \\, G(s,x) - \\alpha - \\int (x-K)^+ \\;\\eta (dK) + \\int _0^\\infty (x-b)I_{\\lbrace s > b\\rbrace } \\; \\lambda (db)\\bigr \\rbrace \\; q(ds,dx)\\nonumber \\\\&&\\qquad \\qquad \\qquad + \\alpha + \\int C(K) \\; \\eta (dK) \\; \\biggr ].\\end{eqnarray}From standard linear programming results, we would expect thatfor dual feasibility we must have\\begin{equation}G(s,x) \\le \\alpha + \\int (x-K)^+ \\;\\eta (dK) - \\int _0^\\infty (x-b)I_{\\lbrace s > b\\rbrace } \\; \\lambda (db)\\end{equation}everywhere, with equality everywhere that the optimal q placesmass; and that the dual problem will be\\begin{equation}\\inf \\biggl [ \\;\\alpha + \\int C(K) \\; \\eta (dK) \\; \\biggr ]\\end{equation}over (\\alpha ,\\eta ,\\lambda ) satisfying (\\ref {robust_hedge}).These equations have a simple and beautiful interpretation.The dual-feasibility relation (\\ref {robust_hedge}) expressesa {\\em robust hedge}; if we hold \\alpha in cash, \\eta (dK)calls of strike K, and {\\em sell forward \\lambda (db) units ofthe underlying when S reaches the level b}, then we generatea contingent claim at the terminal time which will alwaysdominate the claim G which we have to pay out.",
"The dualform of the linear program (\\ref {dualLP}) says that the cost ofconstructing such a hedge, which is of course\\alpha + \\int C(K) \\; \\eta (dK), must be minimized.",
"}The primal problem seeks to find the most expensive that thederivative G(S,X) can be, given the market prices C(K); and thedual problem seeks the cheapest super-replicating hedge.",
"Thecharacterization (\\ref {suff3}) of the possible joint laws of (S,X){\\em tells us what the form of the hedge (\\ref {robust_hedge}) must be.",
"}}\\vspace{14.45377pt}\\end{gather}}Our goal now is to try to use Theorem \\ref {thm1} to similarlybound the price of, and to super-replicate, contingent claimswhich depend on the maximum, terminal value, {\\em minimum,and direction of the final excursion} for a stopped symmetricsimple random walk.",
"To understand how this is to be done, wefocus on the `plus^{\\prime } versions of the necessary and sufficientconditions (\\ref {nc5}).",
"We shall also suppose that themartingale X is {\\em uniformly integrable}, to avoidhaving to bother about side issues.",
"}The condition (\\ref {nc5}) can be restated in terms of the measurem as\\begin{eqnarray}m(b+h-X: S=b, I=-a, \\sigma =+1) &\\le & h \\psi _+(-a,b)\\\\&=& h \\lbrace \\,\\varphi (b,-a-h) - \\varphi (b,-a) \\, \\rbrace \\nonumber \\end{eqnarray}in the notation of Section \\ref {S1}.",
"From the definition(\\ref {m+}) of \\varphi (b,-a), from the fact that m(X)=0, andthe Optional Sampling Theorem result that m(a+X:I \\le -a)=0,we have\\begin{eqnarray*}(a+b)\\varphi (b,-a) &=& a - m(a+X: S <b, I > -a)\\\\&=& m(a+X: \\textrm { S\\ge b or I \\le -a} )\\\\&=& m(a+X: S \\ge b, I > -a)\\\\&=& (a+b) m(S \\ge b, I > -a) - m(b-X: S \\ge b, I > -a).\\end{eqnarray*}Thus the inequality (\\ref {HE1}) may be re-expressed aftersome simple rearrangement as\\begin{eqnarray}0 &\\le & h m(S\\ge b, I = -a) - \\frac{h}{a+b+h} \\; m(b-X:S\\ge b, I > -a-h) +\\nonumber \\\\&&\\qquad + \\frac{h}{a+b} \\; m(b-X: S \\ge b, I > -a)- m(b+h-X: S=b, I=-a, \\sigma =+1).\\nonumber \\end{eqnarray}This inequality for all a,\\, b \\in h\\mathbb {Z} ^+ not both zero, togetherwith the `minus^{\\prime } analogues, is necessary and sufficient for aprobability measure m to be the joint law of (I,X,S,\\sigma ).Just as we did at (\\ref {lagr1})for derivatives depending only on (X,S),we can construct the Lagrangian for this problem, which wouldgive us terms of the form\\begin{eqnarray}\\lambda ^+_{ab} \\;(Z-w) &\\equiv &\\lambda ^+_{ab}\\biggl [ \\;h I_{\\lbrace S \\ge b, I = -a\\rbrace } - \\frac{h}{a+b+h} \\;(b-X)I_{\\lbrace S \\ge b, I > -a-h\\rbrace } +\\nonumber \\\\&&+ \\frac{h}{a+b} \\; (b-X) I_{\\lbrace S \\ge b, I > -a \\rbrace }- (b+h-X)I_{\\lbrace S=b, I=-a, \\sigma =+1 \\rbrace } -w\\;\\biggr ],\\end{eqnarray}where w\\ge 0 is a non-negative slack variable to handle theinequality constraint.",
"Dual feasibility will therefore requirethat \\lambda ^+_{ab}\\ge 0, and at optimality we will havethe complementary slackness condition \\lambda ^+_{ab}\\, w = 0.$ In the situation of derivatives depending only on $(X,S)$ , we had terms of the form $\\lambda _a (X-a) I_{\\lbrace S>a\\rbrace }$ , which were interpreted as forward purchase of the underlying asset when the supremum process reaches a new level.",
"This forward purchase interpretation determines a hedging strategy which can be implemented in an adapted fashion.",
"However, it is very far from clear that the random variable $Z$ defined at () can be realized by some adapted trading strategy.",
"For example, the term involving $(b-X) I_{\\lbrace S \\ge b, I > -a \\rbrace }$ could be interpreted as a forward sale of the underlying when the price first gets to $b$ ; but this trade should only be put on if $I>-a$ , and it is not known at time $H_b$ whether or not the ultimate infimum $I$ will be greater than $-a$ or not.",
"Nevertheless, we can specify an adapted trading strategy which will subreplicate the random variable $Z$ , as follows.",
"We construct a random variable $Y$ which is the final value of the adapted hedging strategy made up of three component positions: At $H_b$ , buy forward $h/(a+b+h)$ units of the underlying if $I(H_b)>-a-h$ , and come out of the position at time $H_{-a-h}$ ; At $H_b$ , buy forward $-h/(a+b)$ units of the underlying if $I(H_b)>-a$ , and come out of the position at time $H_{-a}$ ; At $H_b$ , buy forward 1 unit of the underlying if $I(H_b)=-a$ , and come out of the position at time $H_{b+h}\\wedge H_{-a-h}$ .",
"Now clearly the random variable $Z &\\equiv & h I_{\\lbrace S \\ge b, I = -a\\rbrace } - \\frac{h}{a+b+h} \\;(b-X)I_{\\lbrace S \\ge b, I > -a-h\\rbrace } +\\nonumber \\\\&&+ \\frac{h}{a+b} \\; (b-X) I_{\\lbrace S \\ge b, I > -a \\rbrace }- (b+h-X)I_{\\lbrace S=b, I=-a, \\sigma =+1 \\rbrace }$ will be zero if $S < b$ or if $I\\le -a-h$ , so to understand $Z$ we may suppose that $H_b<\\infty = H_{-a-h}$ .",
"But before we narrow our attention down to the event $\\lbrace H_b<\\infty = H_{-a-h}\\rbrace $ , we should consider what happens off that event to $Y$ .",
"If $H_b =\\infty $ , then none of the component positions of $Y$ is ever entered, so $Y=0$ in that case.",
"If $H_b<\\infty $ and $H_{-a-h}< \\infty $ , then we have three cases to consider: (i) When $I(H_b)>-a$ , the strategy enters positions 1 and 2 at time $H_b$ , and closes out both when the infimum falls to $-a$ and then to $-a-h$ ; position 1 loses $h$ , position 2 gains $h$ , so altogether $Y=0$ ; (ii) When $I(H_b)=-a$ , the strategy enters positions 1 and 3.",
"If $H_{b+h}< H_{-a-h}$ , then position 3 makes a gain of $h$ when it is closed out, but position 1 makes a loss of $h$ when it is closed out, so overall zero gain.",
"On the other hand, if $H_{-a-h}<H_{b+h}$ , then position 1 makes a loss of $h$ when it is closed out, and position 3 makes a loss of $(a+b+h)$ when it is closed out, so overall $Y = -(a+b+h)-h<0$ , and as we shall subsequently see, this is the only situation in which $Y$ is strictly less than $Z$; (iii) When $I(H_b) \\le -a-h$ , none of the positions is entered, and $Y=0$ .",
"We now have to compare the values of $Z$ and $Y$ on the event $\\lbrace H_b<\\infty = H_{-a-h}\\rbrace $ , breaking the comparison down into seven cases as presented in the following table.",
"In the first two rows, we see what happens if $I>-a$ , and in the remaining rows, we are considering situations where $I=-a$ .",
"The reader is invited to check through each of the entries of the table, and confirm the findings reported there.",
"The only entry that requires comment is the penultimate row, in the column for $Z$ .",
"In this row, we are in the situation where $S=b$ and $I=-a$ , so we get a contribution to $Z$ from the first term in (REF ), and from the second term, none from the third term, and none from the fourth term, because if $H_b<H_{-a}<H_{b+h} = \\infty $ it must be that the signature $\\sigma $ is $-1$ !",
"What we see from the table is that in every case the value of $Z$ is equal to the value of $Y$ .",
"Table: NO_CAPTIONThus we may conclude that $Y \\le Z$ in all instances, and the only situation in which the inequality is strict is when $H_{-a} < H_b< H_{-a-h} < H_{b+h}$ .",
"Now we explain how these observations lead to a super-replicating hedging strategy.",
"For this, let us denote by $Z^+_{ab}$ then random variable we have been calling $Z$ up til now; this is because in the Lagrangian we have to consider such random variables (and their `minus' analogues) for all $a, \\, b \\in h\\mathbb {Z} ^+$ not both zero.",
"Suppose that we have some derivative $G(I,X,S,\\sigma )$ whose price we wish to maximize subject to the distribution of $X$ matching call price data, just as we did for derivatives depending only on $(X,S)$ in the first part of our discussion in this Section.",
"We would find ourselves with a Lagrangian form similar to (): $L(\\alpha ,\\lambda ,\\eta ) &=&\\sup _{m\\ge 0}\\biggl [\\;\\int \\bigl \\lbrace \\;G(I,X,S,\\sigma ) - \\alpha - \\int (X-K)^+ \\eta (dK)+\\nonumber \\\\&& \\qquad + \\sum _{a,b,\\pm } \\lambda ^\\pm _{ab}(Z^{\\pm }_{ab} - w^\\pm _{ab})\\;\\bigr \\rbrace \\; dm(I,X,S,\\sigma )+ \\alpha + \\int C(K) \\; \\eta (dK)\\;\\bigr \\rbrace \\;\\biggr ]$ with obvious notation.",
"Now dual feasibility imposes the condition $G(I,X,S,\\sigma ) & \\le &\\alpha + \\int (X-K)^+ \\eta (dK)-\\sum _{a,b,\\pm } \\lambda ^\\pm _{ab}\\,Z^{\\pm }_{ab}\\\\&\\le &\\alpha + \\int (X-K)^+ \\eta (dK)-\\sum _{a,b,\\pm } \\lambda ^\\pm _{ab}\\,Y^{\\pm }_{ab}$ in another obvious notation.",
"The interpretation of () is that the derivative $G$ is super-replicated by the adaptively-realizable hedge given by a position in calls and a position in the $Y$ -hedges.",
"At optimality, complementary slackness tells us that if $\\lambda ^+_{ab}>0$ then $w^+_{ab}=0$ , and therefore the inequality () must hold with equality.",
"Tracing this back to the condition (), and its derivation from (), we find that equality in () is equivalent to the statement that $\\tilde{p}_{+-}=0$ .",
"What this means is that on the event $\\lbrace H_{-a}<H_b <H_{-a-h}\\rbrace $ we cannot have $H_{-a-h}<H_{b+h}$, and as we saw, this was the only situation where $Y<Z$.",
"We may therefore conclude that for the optimal $m^*$ , not only will (REF ) hold with equality $m^*$ -a.e., but also () will hold with equality $m^*$ -a.e..",
"In other words, if the joint law $m$ is the optimal joint law, the hedging strategy expressed by () is a perfect replication of the contingent claim - there is no slack."
]
] | 1403.0220 |
[
[
"ADHM data for the Hilbert scheme of points of the total space of\n $\\mathcal O_{\\mathbb P^1}(-n)$"
],
[
"Abstract Relying on a monadic description of the moduli space of framed sheaves on Hirzebruch surfaces, we construct ADHM data for the Hilbert scheme of points of the total space of the line bundle $\\mathcal O(-n)$ on $\\mathbb P^1$."
],
[
"Introduction",
"Let $X$ be a smooth quasi-projective irreducible surface over $\\mathbb {C}$ .",
"The Hilbert scheme of points $\\operatorname{Hilb}^c(X)$ , which parameterizes 0-dimensional subschemes of $X$ of length $c$ , is well known to be quasi-projective [7] and smooth of dimension $2c$ [5]; indeed, the so-called Hilbert-Chow morphism $\\operatorname{Hilb}^c(X)\\longrightarrow S^c X$ onto the $c$ -th symmetric product of $X$ is a resolution of singularities.",
"Hilbert schemes of points on surfaces were extensively studied from many perspectives over the past two decades (see e.g.",
"[12], [9], [11]), however there are relatively few cases in which they are susceptible of an explicit description.",
"Arguably, the most significant examples are the spaces $\\operatorname{Hilb^c}(\\mathbb {C}^2)$ , which can be described by means of linear data, the so-called ADHM (Atiyah-Drinfel'd-Hitchin-Manin) data [12].",
"Also the Hilbert schemes of points of multi-blowups of $\\mathbb {C}^2$ admit an ADHM description, as provided by the work of A.A. Henni [8] specialized to the rank one case.",
"The goal of this paper is to provide an ADHM-type construction for the Hilbert schemes of points over the total space $\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))$ of the line bundle $\\mathcal {O}_{\\mathbb {P}^1}(-n)$ on $\\mathbb {P}^1$ .",
"These spaces are the rank 1 case of the moduli spaces of framed sheaves of the Hirzebruch surface $\\Sigma _n$ (by framing to the trivial bundle on a divisor linearly equivalent to the section of $\\Sigma _n\\rightarrow \\mathbb {P}^1$ of positive self-intersection) which were studied in [3], [2].",
"These modules spaces were considered in physics in connection with the so-called D4-D2-D0 brane system in topological string theory (cf.",
"[13], [1] and [3] for a concise discussion).",
"To construct the ADHM data for the Hilbert scheme of points of $\\operatorname{Hilb}^c(\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n)))$ we identify it with the moduli space $\\mathcal {M}^{n}(1,0,c)$ of framed sheaves on the Hirzebruch surface $\\Sigma _n$ that have rank 1, vanishing first Chern class, and second Chern class $c_2 = c$ , and exploit the description of $\\mathcal {M}^{n}(1,0,c)$ in terms of monads given in [2].",
"Theorem REF states that the moduli space $\\mathcal {M}^{n}(1,0,c)$ is isomorphic to the quotient $P^n(c) / \\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ , where $P^n(c)$ is a quasi-affine variety contained in the linear space $\\operatorname{End}(\\mathbb {C}^{c})^{\\oplus n+2}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})$ .",
"This result relies on the fact that the partial quotient $P^n(c) / \\operatorname{GL}(c, \\mathbb {C})$ can be assembled glueing $c+1$ open sets, each one isomorphic to the space of ADHM data for $\\operatorname{Hilb^c}(\\mathbb {C}^2)$ (Theorem REF ).",
"Since the proof of Theorem REF is based on the description of the moduli spaces of framed sheaves on $\\Sigma _n$ worked out in [2], for the reader's convenience we briefly recall here the fundamental ingredients of that construction."
],
[
"Acknowledgments",
"This work was partially supported by PRIN “Geometria delle varietà algebriche\", by the University of Genoa's project “Aspetti matematici della teoria dei campi interagenti e quantizzazione per deformazione\" and by GNSAGA-INDAM.",
"U.B.",
"is a member of the VBAC group.",
"Let $\\Sigma _n$ be the $n$ -th Hirzebruch surface, i.e., the projective closure of the total space of the line bundle $\\mathcal {O}_{\\mathbb {P}^1}(-n)$ ; we restrict ourselves to the case $n > 0$ .",
"We denote by $F$ the class in $\\operatorname{Pic}(\\Sigma _n)$ of the fibre of the natural ruling $\\Sigma _n\\longrightarrow \\mathbb {P}^1$ , by $H$ the class of the section of the ruling squaring to $n$ , and by $E$ the class of the section squaring to $-n$ .",
"We fix a curve $\\ell _{\\infty }\\simeq \\mathbb {P}^1$ in $\\Sigma _n$ linearly equivalent to $H$ and think of it as the “line at infinity”.",
"A framed sheaf on $\\Sigma _n$ is a pair $(\\mathcal {E}, \\theta )$ , where $\\mathcal {E}$ is a torsion-free sheaf that is trivial along $\\ell _{\\infty }$ , and $\\theta \\colon \\mathcal {E}\\vert _{\\ell _{\\infty }}\\stackrel{\\sim }{\\longrightarrow }\\mathcal {O}_{\\ell _\\infty }^{\\oplus r}$ is a fixed isomorphism, $r$ being the rank of $\\mathcal {E}$ .",
"A morphism between the framed sheaves $(\\mathcal {E}, \\theta )$ , $(\\mathcal {E}^{\\prime }, \\theta ^{\\prime })$ is by definition a morphism $\\Lambda \\colon \\mathcal {E}\\longrightarrow \\mathcal {E}^{\\prime }$ such that $\\theta ^{\\prime }\\circ \\Lambda \\vert _{\\ell _{\\infty }} = \\theta $ .",
"The moduli space parameterizing isomorphism classes of framed sheaves $(\\mathcal {E}, \\theta )$ on $\\Sigma _n$ with $\\textrm {ch}(\\mathcal {E}) = (r, aE, -c -\\frac{1}{2} na^2)$ will be denoted by $\\mathcal {M}^{n}(r,a,c)$ .",
"We assume that the framed sheaves are normalized in such a way that $0\\le a\\le r-1$ .",
"A description of the moduli space $\\mathcal {M}^{n}(r,a,c)$ in terms of monads was provided in [2], generalizing work by Buchdahl [4].",
"If $[(\\mathcal {E}, \\theta )]$ lies in $\\mathcal {M}^{n}(r,a,c)$ , the sheaf $\\mathcal {E}$ is isomorphic to the cohomology of a monad ${M(\\alpha ,\\beta ):&0 [r] & \\mathcal {U}_{\\vec{k}}[r]^-{\\alpha } & {\\vec{k}}[r]^-{\\beta } & \\mathcal {W}_{\\vec{k}}[r] & 0}\\,, $ where $\\vec{k} = (n,r,a,c)$ ; in others words, the terms of (REF ) depend only on the Chern character of $\\mathcal {E}$ .",
"More precisely, if we put ${\\left\\lbrace \\begin{array}{ll}\\begin{aligned}k_1&=c+\\dfrac{1}{2}na(a-1)\\\\k_2&=k_1+na\\\\k_3&=k_1+(n-1)a\\\\k_4&=k_1+r-a\\,,\\end{aligned}\\end{array}\\right.",
"}$ we have $\\left\\lbrace \\begin{aligned}\\mathcal {U}_{\\vec{k}}&:=\\mathcal {O}_{\\Sigma _n}(0,-1)^{\\oplus k_1}\\\\{\\vec{k}}&:=\\mathcal {O}_{\\Sigma _n}(1,-1)^{\\oplus k_2} \\oplus \\mathcal {O}_{\\Sigma _n}^{\\oplus k_4}\\\\\\mathcal {W}_{\\vec{k}}&:=\\mathcal {O}_{\\Sigma _n}(1,0)^{\\oplus k_3}\\,.\\end{aligned}\\right.$ This procedure yields a map $(\\mathcal {E},\\theta )\\longmapsto \\operatorname{Hom}(\\mathcal {U}_{\\vec{k}},{\\vec{k}})\\oplus \\operatorname{Hom}({\\vec{k}},\\mathcal {W}_{\\vec{k}})\\,,$ whose image $L_{\\vec{k}}$ is a smooth variety, which can be completely characterized by imposing suitable conditions on the pairs $(\\alpha , \\beta ) \\in \\operatorname{Hom}(\\mathcal {U}_{\\vec{k}},{\\vec{k}})\\oplus \\operatorname{Hom}({\\vec{k}},\\mathcal {W}_{\\vec{k}})$ [2].",
"One can construct a principal $\\operatorname{GL}(r,\\mathbb {C})$ -bundle $P_{\\vec{k}}$ over $L_{\\vec{k}}$ whose fibre over a point $(\\alpha ,\\beta )$ is naturally identified with the space of framings for the cohomology of the complex (REF ).",
"Hence, the map (REF ) can be lifted to a map $(\\mathcal {E},\\theta )\\longmapsto \\theta \\in P_{\\vec{k}}\\,.$ The algebraic group $G_{\\vec{k}}=\\operatorname{Aut}(\\mathcal {U}_{\\vec{k}})\\times \\operatorname{Aut}({\\vec{k}})\\times \\operatorname{Aut}(\\mathcal {W}_{\\vec{k}})$ of isomorphisms of monads of the form (REF ) acts freely on $P_{\\vec{k}}$ , and the moduli space $\\mathcal {M}^{n}(r,a,c)$ can be described as the quotient $P_{\\vec{k}}/G_{\\vec{k}}$ [2].",
"This space is nonempty if and only if $c + \\frac{1}{2} na(a-1) \\ge 0$ , and, in this case, is a smooth algebraic variety of dimension $rc + (r-1) na^2$ .",
"If the sheaf $\\mathcal {E}$ has rank $r=1$ , by normalizing we can assume $a=0$ .",
"Hence, the double dual $\\mathcal {E}^{\\ast \\ast }$ of $\\mathcal {E}$ , being locally free with $\\textrm {c}_1(\\mathcal {E}^{\\ast \\ast }) = \\textrm {c}_1(\\mathcal {E})= 0$ , is isomorphic to structure sheaf $\\mathcal {O}_{\\Sigma _n}$ .",
"As a consequence, since $\\mathcal {E}$ is trivial on $\\ell _{\\infty }$ , the correspondence $\\mathcal {E}\\longmapsto \\mbox{schematic support of}\\ \\mathcal {E}^{\\ast \\ast }/ \\mathcal {E}$ yields an isomorphism $\\mathcal {M}^{n}(1,0,c) \\simeq \\operatorname{Hilb}^c (\\Sigma _n \\setminus \\ell _{\\infty }) = \\operatorname{Hilb}^c (\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n)))\\,.$ In the following, we shall denote the moduli space $\\mathcal {M}^{n}(1,0,c)$ simply by $\\mathcal {M}^{n}(c)$ ."
],
[
"Statement of the Main Theorem",
"We call $P^{n}(c)$ the subset of the vector space $\\operatorname{End}(\\mathbb {C}^{c})^{\\oplus n+2}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})$ whose points $\\left(A_1,A_2;C_1,\\dots ,C_{n};e\\right)$ satisfy the following conditions: ${\\left\\lbrace \\begin{array}{ll}A_1C_1A_{2}=A_2C_{1}A_{1}&\\qquad \\text{when $n=1$}\\\\[15pt]\\begin{aligned}A_1C_q&=A_2C_{q+1}\\\\C_qA_1&=C_{q+1}A_2\\end{aligned}\\qquad \\text{for}\\quad q=1,\\dots ,n-1&\\qquad \\text{when $n>1$}\\end{array}\\right.",
"}\\,;$ there exists $[\\nu _{1},\\nu _{2}]\\in \\mathbb {P}^1$ such that $\\det (\\nu _1A_1+\\nu _2A_2)\\ne 0$ ; for all values of the parameters $\\left([\\lambda _1,\\lambda _2],(\\mu _1,\\mu _{2})\\right)\\in \\mathbb {P}^1\\times \\mathbb {C}^{2}$ such that $\\lambda _{1}^{n}\\mu _{1}+\\lambda _{2}^{n}\\mu _{2}=0$ there is no nonzero vector $v\\in \\mathbb {C}^c$ such that $\\left\\lbrace \\begin{aligned}\\left(\\lambda _2{A_1}+\\lambda _1{A_2}\\right)v&=0\\\\(C_{1}A_{2}+\\mu _{1}\\mathbf {1}_{c})v&=0\\\\(C_{n}A_{1}+(-1)^{n-1}\\mu _{2}\\mathbf {1}_{c})v&=0\\\\ev&=0\\,.\\end{aligned}\\right.$ We define an action of $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ on $P^n(c)$ by the equations $\\left\\lbrace \\begin{array}{ccll}C_j & \\longmapsto & \\phi _1C_j\\phi _2^{-1} &\\qquad j=1,\\dots ,n\\\\A_i & \\longmapsto & \\phi _2A_i\\phi _1^{-1} &\\qquad i=1,2\\\\e & \\longmapsto & e\\phi _1^{-1}&\\end{array}\\right.\\qquad (\\phi _1,\\phi _2)\\in \\operatorname{GL}(c, \\mathbb {C})^{\\times 2}\\,.$ Theorem 2.1 There is an isomorphism of complex varieties $P^n(c)\\left\\bad.\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}\\right.\\simeq \\mathcal {M}^n(c)=\\operatorname{Hilb}^c(\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n)))\\,,$ and $P^{n}(c)$ is a locally trivial principal $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -bundle over $\\mathcal {M}^n(c)$ ."
],
[
"A consistency check",
"Before proving Theorem REF we check its consistency in the simplest case $c=1$ , by verifying that the quotient $P^{n}(1)/(\\mathbb {C}^{*})^{\\times 2}$ is isomorphic to the total space of $\\mathcal {O}_{\\mathbb {P}^1}(-n)$ .",
"Indeed, one has $\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))\\simeq \\widetilde{T}_{n}/\\mathbb {C}^{*}$ , where $\\widetilde{T}_{n}=\\left\\lbrace \\left.",
"((y_{1},y_{2}),(u_{1},u_{2}))\\in \\left(\\mathbb {C}^{2}\\setminus \\lbrace 0\\rbrace \\right)\\times \\mathbb {C}^{2}\\right|u_{1}y_{1}^{n}=u_{2}y_{2}^{n}\\right\\rbrace $ and the $\\mathbb {C}^{*}$ -action is $\\left\\lbrace \\begin{aligned}(y_{1},y_{2})&\\longmapsto \\lambda (y_{1},y_{2})\\\\(u_{1},u_{2})&\\longmapsto (u_{1},u_{2})\\end{aligned}\\right.\\qquad \\lambda \\in \\mathbb {C}^{*}$ (cf. eq.",
"(REF )).",
"Proposition 2.2 $P^{n}(1)/(\\mathbb {C}^{*})^{\\times 2}\\simeq \\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))\\,.$ When $c=1$ , the matrices $(A_{1},A_{2},C_{1},\\dots ,C_{n},e)$ are complex numbers, and condition (P2) is equivalent to requiring that $(A_{1},A_{2})\\ne (0,0)$ .",
"When $n=1$ condition (P1) is identically satisfied, while when $n>1$ it is equivalent to ${\\left\\lbrace \\begin{array}{ll}C_q=\\left(\\frac{A_{2}}{A_{1}}\\right)^{n-q}C_{n}\\qquad \\text{for}\\quad q=1,\\dots ,n-1& \\qquad \\text{if}\\quad A_{1}\\ne 0\\\\[7pt]C_q=\\left(\\frac{A_{1}}{A_{2}}\\right)^{q-1}C_{1}\\qquad \\text{for}\\quad q=2,\\dots ,n&\\qquad \\text{if}\\quad A_{2}\\ne 0\\,.\\end{array}\\right.",
"}$ Using these equations it is possible to show that condition (P3) reduces to $e\\ne 0$ .",
"By acting with $\\left(\\mathbb {C}^{*}\\right)^{\\times 2}$ we can fix $e=1$ , and the maximal subgroup preserving this condition is clearly $\\lbrace 1\\rbrace \\times \\mathbb {C}^{*}$ .",
"We introduce the variety $\\widetilde{Y}_{n}=\\left\\lbrace \\left.",
"((y_{1},y_{2}),(x_{1},x_{2}))\\in \\left(\\mathbb {C}^{2}\\setminus \\lbrace 0\\rbrace \\right)\\times \\mathbb {C}^{2}\\right|x_{1}y_{1}^{n-1}=x_{2}y_{2}^{n-1}\\right\\rbrace \\,,$ with $n\\ge 1$ , and we let $\\mathbb {C}^{*}$ act on $\\widetilde{Y}_{n}$ as follows: $\\left\\lbrace \\begin{aligned}(y_{1},y_{2})&\\longmapsto \\lambda (y_{1},y_{2})\\\\(x_{1},x_{2})&\\longmapsto \\lambda ^{-1}(x_{1},x_{2})\\end{aligned}\\qquad \\lambda \\in \\mathbb {C}^{*}\\right.\\,.$ We cover $\\widetilde{Y}_{n}$ with the two $\\mathbb {C}^{*}$ -invariant subsets $\\widetilde{Y}_{n,i}=\\lbrace y_{i}\\ne 0\\rbrace $ , and analogously we cover $P^{n}(1)$ with the $(\\mathbb {C}^{*})^{\\times 2}$ -invariant subsets $P^{n}(1)_{i}=\\lbrace A_{i}\\ne 0\\rbrace $ , $i=1,2$ .",
"Next, we define the morphisms $\\begin{array}{ccl}\\widetilde{Y}_{n,i}&\\longrightarrow &P^{n}(1)_{i}\\\\[5pt]((y_{1},y_{2}),(x_{1},x_{2}))&\\longmapsto &{\\left\\lbrace \\begin{array}{ll}\\left(y_{1},y_{2},\\left(\\frac{y_{2}}{y_{1}}\\right)^{n-1}x_{2},\\left(\\frac{y_{2}}{y_{1}}\\right)^{n-2}x_{2},\\ldots ,x_{2},1\\right) & i=1\\\\[8pt]\\left(y_{1},y_{2},x_{1},\\left(\\frac{y_{1}}{y_{2}}\\right)x_{1},\\ldots ,\\left(\\frac{y_{1}}{y_{2}}\\right)^{n-1}x_{1},1\\right) & i=2\\,.\\end{array}\\right.",
"}\\end{array}$ These glue together providing a $\\mathbb {C}^{*}$ -equivariant closed immersion $\\widetilde{Y}_{n}\\rightarrow P^{n}(1)$ , which induces an isomorphism $P^{n}(1)/(\\mathbb {C}^{*})^{\\times 2}\\simeq \\widetilde{Y}_{n}/\\mathbb {C}^{*}\\,.$ Finally, the $\\mathbb {C}^{*}$ -equivariant morphism $\\begin{array}{ccl}\\widetilde{Y}_{n}&\\longrightarrow &\\left(\\mathbb {C}^{2}\\setminus \\lbrace 0\\rbrace \\right)\\times \\mathbb {C}^{2}\\\\((y_{1},y_{2}),(x_{1},x_{2}))&\\longmapsto &((y_{1},y_{2}),(u_{1},u_{2}))=((y_{1},y_{2}),(x_{1}y_{2},x_{2}y_{1}))\\,.\\end{array}$ establishes the required isomorphism."
],
[
"Glueing ADHM data",
"In this section we provide an ADHM description for each open set of a suitable open cover of $\\mathcal {M}^{n}(c)$ .",
"If we fix $c+1$ distinct fibres $f_0,\\dots ,f_{c}\\in |F|$ , for any $[(\\mathcal {E},\\theta )]\\in \\mathcal {M}^{n}(c)$ there exists at least one $m\\in \\lbrace 0,\\dots ,c\\rbrace $ such that $\\mathcal {E}|_{f_{m}}\\simeq \\mathcal {O}_{f_{m}}$ .",
"With this in mind, we choose the fibres $f_{m}$ cut in $\\Sigma _n = \\left\\lbrace ([y_1,y_2],[x_1,x_2,x_3])\\in \\mathbb {P}^1\\times \\mathbb {P}^2\\;|\\;x_1y_1^n=x_2y_2^n \\right\\rbrace $ by the equations $f_m=\\lbrace [y_1,y_2]=[c_m,s_m]\\rbrace \\qquad m=0,\\dots ,c$ where $c_m=\\cos \\left(\\pi \\frac{m}{c+1}\\right)\\qquad \\text{,}\\qquad s_m=\\sin \\left(\\pi \\frac{m}{c+1}\\right)\\,.$ Then we get an open cover $\\left\\lbrace \\mathcal {M}^n(c)_m\\right\\rbrace _{m=0}^{c}$ for $\\mathcal {M}^n(c)$ by letting $\\mathcal {M}^n(c)_m:=\\left\\lbrace [(\\mathcal {E},\\theta )]\\in \\mathcal {M}^n(c)\\left|\\begin{array}{l}\\text{the restricted sheaf $\\mathcal {E}|_{f_m}$}\\\\\\text{is isomorphic to $\\mathcal {O}_{f_m}$}\\end{array}\\right.\\right\\rbrace .$ Each of these spaces is isomorphic to the Hilbert scheme of points of $\\mathbb {C}^2$ , so that it admits the ADHM description [12], which we briefly recall.",
"The variety $c)$ of ADHM data is defined as the space of triples $(b_1,b_2,e)\\in \\operatorname{End}(\\mathbb {C}^c)^{\\oplus 2}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})$ such that (T1) $[b_{1},b_{2}]=0\\,;$ (T2) for all $(z,w)\\in \\mathbb {C}^2$ there is no nonzero vector $v\\in \\mathbb {C}^{c}$ such that $\\left\\lbrace \\begin{aligned}(b_{1}+z\\mathbf {1}_c)v&=0\\\\(b_{2}+w\\mathbf {1}_c)v&=0\\\\ev&=0\\,.\\end{aligned}\\right.$ A $\\operatorname{GL}(c, \\mathbb {C})$ -action on $c)$ is naturally defined as follows: $\\left\\lbrace \\begin{array}{rcl}b_{i}&\\longmapsto & \\phi b_{i}\\phi ^{-1}\\qquad i=1,2 \\\\e&\\longmapsto & e\\phi ^{-1}\\\\\\end{array}\\right.\\qquad \\phi \\in \\operatorname{GL}(c, \\mathbb {C})\\,.$ The ADHM data for the open set $\\mathcal {M}^{n}(c)_{m}$ will be denoted by $(b_{1m},b_{2m},e_{m})$ ; the transition functions on the intersections $\\mathcal {M}^n(c)_{ml}=\\mathcal {M}^n(c)_m\\cap \\mathcal {M}^n(c)_l$ are explicitly described in the next Theorem.",
"Theorem 3.1 The intersection $\\mathcal {M}^n(c)_{ml}=\\mathcal {M}^n(c)_m\\cap \\mathcal {M}^n(c)_l$ is characterized by the condition $\\det \\left(c_{m-l}\\mathbf {1}_c+s_{m-l}b_{1l}\\right) \\ne 0 \\quad (\\text{or, equivalently,}\\ \\det \\left(c_{l-m}\\mathbf {1}_c+s_{l-m}b_{1m}\\right) \\ne 0)\\,,$ where $c_{m}$ and $s_{m}$ are the numbers defined in eq.",
"(REF ).",
"On any of these intersections, the ADHM data are related by the transition functions $\\begin{aligned}\\varphi _{lm}\\colon \\mathcal {M}^n(c)_{ml} &\\longrightarrow \\mathcal {M}^n(c)_{ml}\\\\\\left[(b_{1m}, b_{2m}, e_{m})\\right]&\\longmapsto \\left[(b_{1l},b_{2l},e_{l})\\right]\\,,\\end{aligned}$ $\\text{where}\\qquad \\left\\lbrace \\begin{aligned}b_{1l}&=\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1m}\\right)^{-1}\\left(s_{m-l}\\mathbf {1}_c+c_{m-l}b_{1m}\\right)\\\\b_{2l}&=\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1m}\\right)^{n}b_{2m}\\\\e_{l}&=e_{m}\\,.\\end{aligned}\\right.$ To prove Theorem REF we observe that $\\operatorname{GL}(c, \\mathbb {C})$ can be embedded as a closed subgroup of $G_{\\vec{k}}$ by means of the homomorphism $\\begin{array}{rccl}\\iota \\colon &\\operatorname{GL}(c, \\mathbb {C})&\\longrightarrow &G_{\\vec{k}}\\\\[8pt]&\\phi &\\longmapsto &\\left({}^{t}{-3mu}\\phi ^{-1},\\begin{pmatrix}{}^{t}{-3mu}\\phi ^{-1} & 0 & 0\\\\0 & {}^{t}{-3mu}\\phi ^{-1} & 0\\\\0 & 0 & 1\\end{pmatrix},{}^{t}{-3mu}\\phi ^{-1}\\right)\\,.\\end{array}$ Let $\\pi \\colon P_{\\vec{k}}\\longrightarrow \\mathcal {M}^n(c)$ be the canonical projection.",
"The open subsets $P_{\\vec{k},m}=\\pi ^{-1}\\left(\\mathcal {M}^n(c)_m\\right)\\,,\\qquad m=0,\\dots ,c\\,,$ provide a $G_{\\vec{k}}$ -invariant open cover of $P_{\\vec{k}}$ ; $\\operatorname{GL}(c, \\mathbb {C})$ acts on each $P_{\\vec{k},m}$ via the immersion (REF ).",
"Proposition 3.2 There are $\\operatorname{GL}(c, \\mathbb {C})$ -equivariant closed immersions $j_{m}\\colon c)\\rightarrow P_{\\vec{k},m}\\,\\qquad \\text{for $m=0,\\dots ,c$}\\,.$ These induce isomorphisms $\\eta _m\\colon c)/\\operatorname{GL}(c, \\mathbb {C})\\longrightarrow P_{\\vec{k},m}/G_{\\vec{k}}\\simeq \\mathcal {M}^n(c)_m\\qquad \\text{for $m=0,\\dots ,c$}\\,.$ See Section REF .",
"We introduce the open subsets $c)_{m,l}=j_{m}^{-1}\\left(\\operatorname{Im}j_{m}\\cap P_{\\vec{k},l}\\right)\\qquad \\text{for}\\quad m,l=0,\\dots ,c\\,.$ Lemma 3.3 $c)_{m,l}=\\left\\lbrace (b_{1},b_{2},e)\\in c)\\left| \\det \\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1}\\right)\\ne 0\\right.\\right\\rbrace \\,.$ The intersection $\\operatorname{Im}j_{m}\\cap P_{\\vec{k},l}$ is the set of points $(\\alpha ,\\beta ,\\xi )\\in \\operatorname{Im}j_m$ such that $\\det \\left(\\left.\\beta _1\\right|_{f_l}\\right)$ $\\ne 0$ , where $\\beta _1$ is the first component of $\\beta $ .",
"From the fact that $(\\alpha ,\\beta ,\\xi )\\in \\operatorname{Im}j_m$ it follows that $\\beta _1=\\mathbf {1}_cy_{1m}+{}^{t}{-3mu}{2mu}b_{1}y_{2m}=\\begin{pmatrix}\\mathbf {1}_c & {}^{t}{-3mu}{2mu}b_{1}\\end{pmatrix}\\begin{pmatrix}y_{1m}\\\\y_{2m}\\end{pmatrix}=\\begin{pmatrix}\\mathbf {1}_c & {}^{t}{-3mu}{2mu}b_{1}\\end{pmatrix}\\begin{pmatrix}c_m & s_m\\\\-s_m & c_m\\end{pmatrix}\\begin{pmatrix}y_{1}\\\\y_{2}\\end{pmatrix}\\,.$ Since $[y_{1},y_{2}]=[c_l,s_l]$ along $f_{l}$ , the thesis follows.",
"Proposition 3.4 The map $\\begin{array}{rccl}\\tilde{\\varphi }_{lm}\\colon &c)_{m,l}&\\longrightarrow &c)_{l,m}\\\\&\\begin{pmatrix}b_{1}\\\\b_{2}\\\\e\\end{pmatrix}&\\longmapsto &\\begin{pmatrix}\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1}\\right)^{-1}\\left(s_{m-l}\\mathbf {1}_c+c_{m-l}b_{1}\\right)\\\\\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1}\\right)^{n}b_{2}\\\\e\\end{pmatrix}\\end{array}$ is $\\operatorname{GL}(c, \\mathbb {C})$ -equivariant, and induces an isomorphism $\\varphi _{lm}\\colon c)_{m,l}/\\operatorname{GL}(c, \\mathbb {C})\\longrightarrow c)_{l,m}/\\operatorname{GL}(c, \\mathbb {C})\\,,$ such that the triangle ${c)_{m,l}/\\operatorname{GL}(c, \\mathbb {C}) [r]^-{\\varphi _{lm}} [rd]_-{\\eta _{m,l}} & c)_{l,m}/\\operatorname{GL}(c, \\mathbb {C}) [d]^{\\eta _{l,m}}\\\\& \\mathcal {M}^n(c)_{ml}}$ is commutative, where $\\eta _{m,l}$ is the restriction of $\\eta _{m}$ to $c)_{m,l}/\\operatorname{GL}(c, \\mathbb {C})$ (see eq.",
"(REF )).",
"See Section REF .",
"Theorem REF is now a direct consequence of Proposition REF ."
],
[
"Proof of the Main Theorem",
"We introduce the matrices $\\begin{aligned}A_{1m}&=c_mA_1-s_mA_2\\,,\\\\A_{2m}&=s_mA_1+c_mA_2\\,,\\\\E_{m}&=\\left[\\sum _{q=1}^{n}\\binom{n-1}{q-1}c_{m}^{n-q}s_{m}^{q-1}C_q\\right]A_{2m}\\,,\\end{aligned}$ for $m=0,\\dots ,c$ .",
"Since the polynomial $\\det (A_1\\nu _{1}+A_2\\nu _{2})$ has at most $c$ distinct roots in $\\mathbb {P}^1$ , the $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -invariant open subsets $P^n(c)_m=\\left\\lbrace \\left(A_1,A_2,C_{1},\\dots ,C_{n},e\\right)\\in P^n(c)\\left|\\det A_{2m}\\ne 0\\right.\\right\\rbrace \\,,\\qquad m=0,\\dots ,c\\,,$ cover $P^{n}(c)$ .",
"On $P^{n}(c)_{m}$ one can also define the matrix $B_{m}=A_{2m}^{-1}A_{1m}\\,.$ The matrices $(B_{m},E_{m},A_{2m},e)$ provide local affine coordinates for $P^n(c)$ .",
"Proposition 4.1 The morphism $\\begin{array}{rccl}\\zeta _{m}\\colon &P^{n}(c)_{m}&\\longrightarrow &\\left[\\operatorname{End}(\\mathbb {C}^{c})^{\\oplus 2}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})\\right]\\times \\operatorname{GL}(c, \\mathbb {C})\\\\&(A_{1},A_{2};C_{1},\\dots ,C_{n};e)&\\longmapsto &\\left(B_{m},E_{m},e;A_{2m}\\right)\\end{array}$ is an isomorphism onto $c)\\times \\operatorname{GL}(c, \\mathbb {C})$ .",
"The induced $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -action is given by $\\left\\lbrace \\begin{array}{rcl}B_{m} & \\longmapsto & \\phi _{1}B_{m}\\phi _{1}^{-1}\\\\E_{m} & \\longmapsto & \\phi _{1}E_{m}\\phi _{1}^{-1}\\\\A_{2m} & \\longmapsto & \\phi _{2}A_{2m}\\phi _{1}^{-1}\\\\e & \\longmapsto & e\\phi _{1}^{-1}\\,.\\end{array}\\right.$ We divide the proof of this Proposition into several steps.",
"First we define the matrices $\\sigma ^h_{m}=(\\sigma ^{h}_{m;pq})_{0\\le p,q \\le h}$ for all $h\\ge 0$ and $m\\in \\mathbb {Z}$ by means of the equations $(s_{m}\\mu _{1}+c_{m}\\mu _{2})^{p}(c_{m}\\mu _{1}-s_{m}\\mu _{2})^{h-p}=\\sum _{q=0}^{h}\\sigma ^{h}_{m;pq}\\mu _{2}^{q}\\mu _{1}^{h-q}$ for any $(\\mu _{1},\\mu _{2})\\in \\mathbb {C}^{2}$ and $p=0,\\dots ,h$ .",
"Notice that $\\sigma ^h_{m}\\sigma ^h_{l}=\\sigma ^h_{m+l}$ and $\\sigma ^h_{0}=\\mathbf {1}_{h+1}$ .",
"In particular, $\\sigma ^h_{m}$ is invertible for all $h\\ge 0$ and $m\\in \\mathbb {Z}$ .",
"To prove the injectivity of $\\zeta _{m}$ — which is trivial only when $n=1$ — we need the following Lemma.",
"Lemma 4.2 Assume $n>1$ .",
"If the matrices $A_{1},A_{2}\\in \\operatorname{End}(\\mathbb {C}^{c})$ satisfy the condition (P2), the system $\\begin{pmatrix}A_{1} & -A_{2}\\\\& \\ddots & \\ddots \\\\& & A_{1} & -A_{2}\\end{pmatrix}\\begin{pmatrix}C_{1}\\\\\\vdots \\\\\\vdots \\\\C_{n}\\end{pmatrix}=0\\,,$ with $C_{q}\\in \\operatorname{End}(\\mathbb {C}^{c})$ , has maximal rank, namely, $(n-1)c^{2}$ .",
"In particular, if $\\det A_{2m}\\ne 0$ , the general solution is $\\begin{pmatrix}C_{1}\\\\\\vdots \\\\\\vdots \\\\C_{n}\\end{pmatrix}=(\\sigma ^{n-1}_{m}\\otimes \\mathbf {1}_{c})\\begin{pmatrix}\\mathbf {1}_{c}\\\\B_{m}\\\\\\vdots \\\\B_{m}^{n-1}\\end{pmatrix}D_m\\,,$ where we have chosen as free parameter the matrix $D_m=\\sum _{q=1}^{n}\\binom{n-1}{q-1}c_{m}^{n-q}s_{m}^{q-1}C_q\\,.$ First we show by induction that the $(n-1)c\\times nc$ matrices ${A}_{n}=\\begin{pmatrix}A_{1} & -A_{2}\\\\& \\ddots & \\ddots \\\\& & A_{1} & -A_{2}\\end{pmatrix}\\qquad {A}^{\\prime }_{n}=\\begin{pmatrix}-{}^{t}{-2mu}A_{2} & {}^{t}{-2mu}A_{1}\\\\& \\ddots & \\ddots \\\\& & -{}^{t}{-2mu}A_{2} & {}^{t}{-2mu}A_{1}\\end{pmatrix}$ have maximal rank for all $n>1$ .",
"For $n=2$ condition (P2) ensures the existence of a point $[\\nu _{1},\\nu _{2}]\\in \\mathbb {P}^1$ such that $\\det (A_{1}\\nu _{1}+A_{2}\\nu _{2})\\ne 0$ ; it follows that the columns of $A_{1}$ and $A_{2}$ span a vector space of dimension $c$ , so that $\\operatorname{rk}{A}_{2}=c$ .",
"The case of ${A}^{\\prime }_{2}$ is analogous.",
"Assume that the claim holds true for some $k>1$ , and observe that ${A}_{k+1}=\\left(\\begin{array}{c|ccc|c}\\begin{matrix}A_{1}\\\\0\\\\\\vdots \\\\0\\end{matrix} & &{}^{t}{A}^{\\prime }_{k} & &\\begin{matrix}0\\\\\\vdots \\\\0\\\\-A_{2}\\\\\\end{matrix}\\end{array}\\right)\\,.$ Let $v\\in \\mathbb {C}^{(k+1)c}$ , and decompose it as $v=\\begin{pmatrix}v_{1}\\\\\\begin{matrix}\\\\[-10pt]v_{2}\\\\[-10pt]\\\\\\end{matrix}\\\\v_{3}\\end{pmatrix}\\begin{array}{l}\\updownarrow c\\\\\\left\\updownarrow \\begin{matrix}\\\\[-10pt](k-1)c\\\\[-10pt]\\\\\\end{matrix}\\right.\\\\\\updownarrow c\\\\\\end{array}\\,.$ If ${A}_{k+1}v=0$ , we get $\\left\\lbrace \\begin{aligned}A_{1}v_{1}+{}^{t}{A}^{\\prime }_{k}v_{2}&=0\\\\{}^{t}{A}^{\\prime }_{k}v_{2}&=0\\\\{}^{t}{A}^{\\prime }_{k}v_{2}-A_{2}v_{3}&=0\\,.\\end{aligned}\\right.$ Since $\\ker {}^{t}{A}^{\\prime }_{k}=0$ by inductive hypothesis, it follows that ${A}_{k+1}$ has maximal rank.",
"The case of ${A}^{\\prime }_{k+1}$ is analogous.",
"Eq.",
"(REF ) is checked by direct computation and eq.",
"(REF ) is obtained by using the invertibility of $\\sigma ^{n-1}_{m}$ .",
"Since $E_{m}=D_{m}A_{2m}$ , the morphism $\\zeta _{m}$ is injective.",
"Next we prove that $\\operatorname{Im}\\zeta _{m}\\subseteq c)\\times \\operatorname{GL}(c, \\mathbb {C})$ via the following two Lemmas.",
"Lemma 4.3 For all $(B_{m},E_{m},e;A_{2m})\\in \\operatorname{Im}\\zeta _{m}$ one has $[B_{m},E_{m}]=0\\,.$ For all $n\\ge 1$ condition (P1) implies that $A_1C_qA_2-A_2C_qA_1=0\\qquad \\text{for}\\quad q=1,\\dots ,n\\,.$ By recalling eqs.",
"(REF ) and (REF ), the thesis follows from the identity $A_{1}CA_{2}-A_{2}CA_{1}=A_{1m}CA_{2m}-A_{2m}CA_{1m}\\,,$ which holds true for all $C\\in \\operatorname{End}(\\mathbb {C}^{c})$ and for $m=0,\\dots ,c$ .",
"Lemma 4.4 Let $(A_{1},A_{2};C_{1},\\dots ,C_{n};e)\\in \\operatorname{End}(\\mathbb {C}^{c})^{\\oplus (n+2)}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})$ be an $(n+3)$ -tuple which satisfies the condition (P1) and $\\det A_{2m}\\ne 0$ .",
"Then if $[\\lambda _1,\\lambda _2]=[c_m,s_m]$ , the condition (P3) is trivially satisfied; if $[\\lambda _1,\\lambda _2]\\ne [c_m,s_m]$ , the condition (P3) holds true if and only if the condition (T2) holds true for the triple $(B_{m},E_{m},e)$ .",
"One has $\\lambda _{2}A_{1}+\\lambda _{1}A_{2}={\\left\\lbrace \\begin{array}{ll}\\lambda A_{2m} &\\qquad \\text{if}\\quad [\\lambda _1,\\lambda _2]=[c_m,s_m]\\\\\\lambda A_{2m}(B_{m}+z\\mathbf {1}_{c}) &\\qquad \\text{if}\\quad [\\lambda _1,\\lambda _2]\\ne [c_m,s_m]\\end{array}\\right.",
"}$ for some $\\lambda \\ne 0$ , where $z=\\frac{c_{m}\\lambda _{1}+s_{m}\\lambda _{2}}{-s_{m}\\lambda _{1}+c_{m}\\lambda _{2}}\\,.$ This proves the first statement.",
"As for the second statement, eq.",
"(REF ) yields $\\begin{aligned}C_{1}&=(c_{m}\\mathbf {1}_{c}-s_{m}B_{m})^{n-1}E_{m}A_{2m}^{-1}\\\\C_{n}&=(s_{m}\\mathbf {1}_{c}+c_{m}B_{m})^{n-1}E_{m}A_{2m}^{-1}\\,.\\end{aligned}$ Moreover, whenever $[\\lambda _1,\\lambda _2]\\ne [c_m,s_m]$ , the condition $\\lambda _{1}^{n}\\mu _{1}+\\lambda _{2}^{n}\\mu _{2}=0$ is satisfied if and only if $\\left\\lbrace \\begin{aligned}\\mu _{1}&=(s_{m}z+c_{m})^{n}w\\\\\\mu _{2}&=-(c_{m}z+s_{m})^{n}w\\end{aligned}\\right.$ for some $w\\in \\mathbb {C}$ .",
"Eqs.",
"(REF ) and (REF ) show the equivalence of the following systems: $\\left\\lbrace \\begin{aligned}\\left(\\lambda _2{A_1}+\\lambda _1{A_2}\\right)v&=0\\\\(C_{1}A_{2}+\\mu _{1}\\mathbf {1}_{c})v&=0\\\\(C_{n}A_{1}+(-1)^{n-1}\\mu _{2}\\mathbf {1}_{c})v&=0\\end{aligned}\\right.\\qquad \\Longleftrightarrow \\qquad \\left\\lbrace \\begin{aligned}(B_{m}+z\\mathbf {1}_{c})v&=0\\\\(s_{m}z+c_{m})^{n}(E_{m}+w\\mathbf {1}_{c})v&=0\\\\(-c_{m}z+s_{m})^{n}(E_{m}+w\\mathbf {1}_{c})v&=0\\,.\\end{aligned}\\right.$ Since the polynomials $s_{m}z+c_{m}$ and $-c_{m}z+s_{m}$ are coprime in $\\mathbb {C}[z]$ , the right-hand system is equivalent to $\\left\\lbrace \\begin{aligned}(B_{m}+z\\mathbf {1}_{c})v&=0\\\\(E_{m}+w\\mathbf {1}_{c})v&=0\\,.\\end{aligned}\\right.$ Finally we prove that $c)\\times \\operatorname{GL}(c, \\mathbb {C})\\subseteq \\operatorname{Im}\\zeta _{m}$ .",
"Let $(b_{1},b_{2},e;A)\\in c)\\times \\operatorname{GL}(c, \\mathbb {C})$ ; if we set $\\begin{aligned}A_{1}&=A(c_{m}b_{1}+s_{m}\\mathbf {1}_{c})\\,,\\\\A_{2}&=A(-s_{m}b_{1}+c_{m}\\mathbf {1}_{c})\\,,\\end{aligned}$ $\\begin{pmatrix}C_{1}\\\\\\vdots \\\\\\vdots \\\\C_{n}\\end{pmatrix} =(\\sigma ^{n-1}_{m}\\otimes \\mathbf {1}_{c})\\begin{pmatrix}\\mathbf {1}_{c}\\\\b_{1}\\\\\\vdots \\\\b_{1}^{n-1}\\end{pmatrix}b_{2}A^{-1}\\,,$ then $(A_1,A_2;C_1,\\dots ,C_n;e)\\in P^n(c)_m$ and $\\zeta _{m}(A_1,A_2;C_1,\\dots ,C_n;e)=(b_{1},b_{2},e;A)$ .",
"It is an easy matter to verify by substitution that the condition (P1) holds.",
"Notice now that by substituting (REF ) into eq.",
"(REF ) one gets $A_{1m}=Ab_{1}\\qquad \\text{,}\\qquad A_{2m}=A\\qquad \\text{,}\\qquad E_{m}=b_{2}\\,.$ This shows that $A_{2m}$ is invertible, and in particular the condition (P2) holds true.",
"By eq.",
"(REF ) one has that $B_{m}=b_{1}$ , so that the validity of the condition (P3) follows from the condition (T2) by Lemma REF .",
"This concludes the proof of Proposition REF .$\\Box $ We now compute the transition functions on the intersections $P^{n}(c)_{ml}=P^{n}(c)_{m}\\cap P^{n}(c)_{l}$ , for $m,l=0,\\dots ,c$ .",
"First observe that $\\zeta _{m}\\left(P^{n}(c)_{ml}\\right)=c)_{m,l}\\times \\operatorname{GL}(c, \\mathbb {C})\\,.$ This fact is a consequence of the identity $\\begin{split}A_{2l}&=\\begin{pmatrix}s_{l}\\mathbf {1}_{c} & c_{l}\\mathbf {1}_{c}\\end{pmatrix}\\begin{pmatrix}c_{m}\\mathbf {1}_{c} & s_{m}\\mathbf {1}_{c}\\\\-s_{m}\\mathbf {1}_{c} & c_{m}\\mathbf {1}_{c}\\end{pmatrix}\\begin{pmatrix}A_{1m}\\\\A_{2m}\\end{pmatrix}=A_{2m}(c_{m-l}\\mathbf {1}_{c}-s_{m-l}B_{m})\\,.\\end{split}$ Proposition 4.5 One has the commutative triangle ${& P^{n}(c)_{ml} [ld]_{\\zeta _{m,l}} [rd]^{\\zeta _{l,m}}\\\\c)_{m,l}\\times \\operatorname{GL}(c, \\mathbb {C}) [rr]^-{\\omega _{lm}} & & c)_{l,m}\\times \\operatorname{GL}(c, \\mathbb {C})\\,,}$ where $\\zeta _{m,l}$ and $\\zeta _{l,m}$ are the restrictions of $\\zeta _{m}$ and $\\zeta _{l}$ , respectively, and $\\omega _{lm}(B_{m},E_{m},e;A_{2m})=\\left(\\tilde{\\varphi }_{lm}(B_{m},E_{m},e),A_{2m}(c_{m-l}\\mathbf {1}_{c}-s_{m-l}B_{m})\\right)\\,,$ the functions $\\tilde{\\varphi }_{lm}$ being defined as in Proposition REF .",
"The transition functions $\\omega _{lm}$ are $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -equivariant.",
"We want to express $(B_l,E_l,e;A_{2l})$ in terms of $(B_m,E_m,e;A_{2m})$ .",
"We already have eq.",
"(REF ); analogously, one can prove $A_{1l}=A_{2m}(s_{m-l}\\mathbf {1}_{c}+c_{m-l}B_{m})$ .",
"From that, it follows that $B_{l}=(c_{m-l}\\mathbf {1}_{c}-s_{m-l}B_{m})^{-1}(s_{m-l}\\mathbf {1}_{c}+c_{m-l}B_{m})$ .",
"As for $E_{l}$ , one has $\\begin{split}E_{l}&=\\left[\\sum _{p=1}^{n}\\sigma ^{n-1}_{-l;0,p-1} C_{p}\\right]A_{2l}=\\\\&=\\left[\\sum _{p=0}^{n-1} \\sigma ^{n-1}_{m-l;0p}B_{m}^{p}\\right]E_{m}A_{2m}^{-1}A_{2l}=\\\\&=(c_{l-m}\\mathbf {1}_{c}-s_{l-m}B_{m})^{n}E_{m}\\,,\\end{split}$ where we have used eq.",
"(REF ), the relation $\\sigma ^{n-1}_{m-l}=\\sigma ^{n-1}_{-l}\\sigma ^{n-1}_{m}$ and Lemma REF .",
"The equivariance of $\\omega _{lm}$ is straightforward, and this completes the proof.",
"By Proposition REF and Lemma REF the immersion $c)\\rightarrow c)\\times \\lbrace \\mathbf {1}_{c}\\rbrace $ induces an isomorphism $P^{n}(c)_{m}/\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}\\simeq c)/\\Delta \\,,$ where $\\Delta \\subset \\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ is the diagonal subgroup.",
"By comparing eqs.",
"(REF ) and (REF ), it turns out that $c)/\\Delta =c)/\\operatorname{GL}(c, \\mathbb {C})$ .",
"It follows that $P^{n}(c)_{m}/\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}\\simeq \\mathcal {M}^{n}(c)_{m}\\,.$ Recall that $c)$ is a principal $\\operatorname{GL}(c, \\mathbb {C})$ -bundle over $c)/\\operatorname{GL}(c, \\mathbb {C})$ [12].",
"Now, by Proposition REF there is an isomorphism $P^{n}(c)_{m}\\simeq c)\\times \\operatorname{GL}(c, \\mathbb {C})$ which is well-behaved with respect to the group actions; as a consequence, $P^{n}(c)_{m}$ is a locally trivial principal $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -bundle over $\\mathcal {M}^{n}(c)_{m}$ .",
"Finally, Propositions REF and REF ensure that this property globalizes, in the sense that $P^n(c)$ is a locally trivial principal $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ -bundle and this completes the proof of Theorem REF ."
],
[
"Some geometrical constructions",
"The projection $q_n\\colon \\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n)) \\longrightarrow \\mathbb {P}^1$ induces a morphism $p_{n,c}\\colon \\operatorname{Hilb}^c(\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))) \\longrightarrow \\mathbb {P}^c$ defined as the composition $ \\operatorname{Hilb}^c(\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))) \\xrightarrow{} S^c \\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n)) \\xrightarrow{} S^c \\mathbb {P}^1= \\mathbb {P}^c\\,,$ where $\\pi _{n,c}$ is the Hilbert-Chow morphism.",
"This morphism can be described in terms of ADHM data, as the following result essentially shows.",
"Let $N(c)$ be the space of pairs $(A_1,A_2)$ of $c\\times c$ complex matrices satisfying property (P2), see the beginning of Section .",
"The group $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ acts on $N(c)$ as in equation (REF ).",
"Proposition 5.1 There is a commutative diagram of morphisms of schemes ${P^n(c) [r] [d]_{h_{n,c}} & \\operatorname{Hilb}^c(\\operatorname{Tot}(\\mathcal {O}_{\\mathbb {P}^1}(-n))) [d]^{p_{n,c}} \\\\N(c) [r]^{g_c} & \\mathbb {P}^c\\,,}$ where $g_c$ is the categorical quotient (in the sense of geometric invariant theory), and $h_{n,c}$ , with reference to the notation in the beginning of Section , is the morphism $h_{n,c}(A_1,A_2;C_1,\\dots ,C_n;e) = (A_1,A_2)\\,.$ We introduce the open affine cover $\\lbrace U_m\\rbrace _{m=0,\\dots c}$ of $\\mathbb {P}^c$ $ U_m = \\lbrace [x_0,\\dots ,x_c]\\in \\mathbb {P}^c \\, \\vert \\, \\sum _{p=0}^c \\sigma ^c_{m;p0}\\,x_p \\ne 0\\rbrace \\simeq \\mathbb {C}^c\\,,$ where the matrices $\\sigma ^c_{m}$ are defined in (REF ).",
"The inverse images $N_m=g_c^{-1}(U_m)$ yield an affine open cover of $N(c)$ .",
"The open subsets $h_{n,c}^{-1}(N_m)\\subset P^n(c)$ are exactly the sets $P^n(c)_m$ defined in equation (REF ).",
"The composition $g_c\\circ h_{n,c}$ on $P^n(c)_m$ can be identified with the map that to the quadruple $(B_m,E_m,e;A_{2m})$ (cf.",
"Proposition REF ) associates the evaluation of the symmetric elementary functions on the eigenvalues of $B_m$ .",
"Since checking the commutativity of the diagram (REF ) is a local matter, and locally our ADHM data coincide with those for the Hilbert scheme of $\\mathbb {C}^2$ as in [12], we can proceed as in [12]."
],
[
"Proofs of Propositions ",
"In this Appendix, after giving some preliminary results, we provide proofs of Propositions REF and REF ."
],
[
"A lemma about quotients",
"If $X$ is a smooth algebraic variety over $\\mathbb {C}$ with a (left) action $\\gamma \\colon X\\times G \\rightarrow X\\times X$ of a complex algebraic group $G$ , the set-theoretical quotient $X/G$ has a natural structure of ringed space induced by the quotient map $q\\colon X\\longrightarrow X/G$ .",
"If the action is free, and the image of the morphism $\\gamma $ is closed, $X/G$ is a smooth algebraic variety, and the pair $(X/G,p)$ is a geometric quotient of $X$ modulo $G$ .",
"More precisely, $X$ is a (locally isotrivial) principle $G$ -bundle over $X/G$ .",
"A proof of this fact was given in [2].",
"Let $X$ be a smooth algebraic variety over $\\mathbb {C}$ , let $Y$ be a smooth closed subvariety; moreover let $G$ be a complex linear algebraic group, and $H$ a closed subgroup.",
"Assume that $G$ acts on $X$ and $H$ on $Y$ so that the inclusion $j\\colon Y \\hookrightarrow X$ is $H$ -equivariant.",
"We consider the quotients $q\\colon X\\longrightarrow X/G$ and $p\\colon Y\\longrightarrow Y/H$ as ringed spaces with the quotient topology, and structure sheaves given by the sheaves of invariant functions.",
"Lemma 1.1 Assume that the action of $G$ on $X$ is free, and that the image of $\\gamma \\colon X\\times G \\rightarrow X\\times X$ is closed.",
"Moreover, assume that the intersection of $\\operatorname{Im}j$ with every $G$ -orbit in $X$ is nonempty; for all $G$ -orbits $O_{G}$ in $X$ , one has $\\operatorname{Stab}_{G}(O_{G}\\cap \\operatorname{Im}j)=\\operatorname{Im}\\iota $ .",
"Then $j$ induces an isomorphism $\\bar{\\jmath }$ of algebraic varieties between $Y/H$ and $X/G$ .",
"Corollary 1.2 $X \\rightarrow X/G$ and $Y\\rightarrow Y/H$ are both principal bundles, and the second is a reduction of the structure group of the first.",
"If $X \\rightarrow X/G$ is locally trivial (and not only locally isotrivial), the same is true for $Y\\rightarrow Y/H$ .",
"Since $\\gamma $ is a closed immersion, it is proper.",
"Hence by [11, Prop.",
"0.7] the morphism $q$ is affine.",
"Then if $U\\subset X/G$ is an open affine subset, $V=q^{-1}(U)$ is affine, $V=\\operatorname{Spec}A$ , so that $U= \\operatorname{Spec}(A^{G})$ , and the restricted morphism $q|_{V}$ is induced by the canonical injection $q^{\\sharp }\\colon A ^{G}\\hookrightarrow A$ .",
"Since $j$ is an affine morphism [6], the counterimage $W=j^{-1}(V)$ is affine, $W=\\operatorname{Spec}B$ , and by the equivariance of $j$ it is $H$ -invariant.",
"It follows that its image $p(W)=\\operatorname{Spec}(B^{H})$ is affine, and the restricted morphism $p|_{W}$ is induced by the canonical injection $p^{\\sharp }\\colon B^{G}\\hookrightarrow B$ .",
"Let $j^{\\sharp }\\colon A \\rightarrow B $ be the homomorphism associated to $j$ .",
"It is easy to prove that $\\operatorname{Im}\\left(j^{\\sharp }\\circ p^{\\sharp }\\right)\\subseteq A^{G}$ , and that this composition is an isomorphism, which induces $\\bar{\\jmath }$ ."
],
[
"Premilinaries",
"As we recalled in the Introduction, for any isomorphism class $[(\\mathcal {E}, \\theta )]$ in the moduli space $\\mathcal {M}^n(r,a,c)$ of framed sheaves on $\\Sigma _n$ , the underlying sheaf $\\mathcal {E}$ is isomorphic to the cohomology of a monad ${M(\\alpha ,\\beta ):&0 [r] & \\mathcal {U}_{\\vec{k}}[r]^-{\\alpha } & {\\vec{k}}[r]^-{\\beta } & \\mathcal {W}_{\\vec{k}}[r] & 0}\\,,$ where $\\vec{k} = (n,r,a,c)$ .",
"To express the pair of morphisms $(\\alpha ,\\beta )$ as a pair of matrices, we select suitable bases for the space $\\operatorname{Hom}&(\\mathcal {U}_{\\vec{k}},{\\vec{k}}) \\oplus \\operatorname{Hom}({\\vec{k}},\\mathcal {W}_{\\vec{k}})=\\\\=&\\left[\\operatorname{Hom}\\left(\\mathbb {C}^{k_1},\\mathbb {C}^{k_2}\\right)\\otimes H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(1,0)\\right)\\right]\\oplus \\left[\\operatorname{Hom}\\left(\\mathbb {C}^{k_1},\\mathbb {C}^{k_4}\\right)\\otimes H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(0,1)\\right)\\right]\\oplus \\\\&\\left[\\operatorname{Hom}\\left(\\mathbb {C}^{k_2},\\mathbb {C}^{k_3}\\right)\\otimes H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(0,1)\\right)\\right]\\oplus \\left[\\operatorname{Hom}\\left(\\mathbb {C}^{k_4},\\mathbb {C}^{k_3}\\right)\\otimes H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(1,0)\\right)\\right],$ where the integers $k_1$ , $k_2$ , $k_3$ , $k_4$ are specified in eq.",
"(REF ).",
"To this aim, after fixing homogeneous coordinates $[y_1,y_2]$ for $\\mathbb {P}^1$ , we introduce additional $c$ pairs of coordinates $[y_{1m},y_{2m}]=[c_{m}y_{1}+s_{m}y_{2},-s_{m}y_{1}+c_{m}y_{2}]\\qquad m=0,\\dots ,c\\,,$ where $c_m$ and $s_m$ are the real numbers defined in eq.",
"(REF ).",
"The set $\\left\\lbrace y_{2m}^{q}y_{1m}^{h-q}\\right\\rbrace _{q=0}^{h}$ is a basis for $H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(0,h)\\right)=H^{0}\\left(\\pi ^{*}\\mathcal {O}_{\\mathbb {P}^1}(h)\\right)$ for all $h\\ge 1$ , where $\\pi \\colon \\Sigma _{n}\\longrightarrow \\mathbb {P}^1$ is the canonical projection.",
"Furthermore if we call $s_{E}$ the (unique up to homotheties) global section of $\\mathcal {O}_{\\Sigma _n}(E)$ , it induces an injection $\\mathcal {O}_{\\Sigma _n}(0,n)\\rightarrowtail \\mathcal {O}_{\\Sigma _n}(1,0)$ , so that the set $\\left\\lbrace (y_{2m}^{q}y_{1m}^{n-q})s_{E}\\right\\rbrace _{q=0}^{n}\\cup \\lbrace s_{\\infty }\\rbrace $ is a basis for $H^{0}\\left(\\mathcal {O}_{\\Sigma _n}(1,0)\\right)$ , where $s_{\\infty }$ is the section characterized by the condition $\\lbrace s_{\\infty }=0\\rbrace =\\ell _\\infty $ .",
"We get $\\begin{aligned}\\alpha &=\\begin{pmatrix}\\sum _{q=0}^{n}\\alpha _{1q}^{(m)}\\left(y_{2m}^{q}y_{1m}^{n-q}s_{E}\\right)+\\alpha _{1,n+1}s_{\\infty }\\\\[7pt]\\alpha _{20}^{(m)}y_{1m}+\\alpha _{21}^{(m)}y_{2m}\\end{pmatrix}\\\\\\beta &=\\begin{pmatrix}\\beta _{10}^{(m)}y_{1m}+\\beta _{11}^{(m)}y_{2m} &\\sum _{q=0}^{n}\\beta _{2q}^{(m)}\\left(y_{2m}^{q}y_{1m}^{n-q}s_{E}\\right)+\\beta _{2,n+1}s_{\\infty }\\end{pmatrix}\\,.\\end{aligned}$ By restricting the display of the monad $M(\\alpha ,\\beta )$ to $\\ell _\\infty $ , twisting by $\\mathcal {O}_{\\ell _\\infty }(-1)$ and taking cohomology, one finds the diagram ${0 [r] & H^{0}({\\vec{k},\\infty }(-1)) [r] & H^{0}(\\mathcal {A}_{\\infty }(-1)) [d]^-{\\Phi } [r] & H^{1}(\\mathcal {U}_{\\vec{k},\\infty }(-1)) [r] & 0\\\\& & H^{0}(\\mathcal {W}_{\\vec{k},\\infty }(-1))}\\,,$ where $\\mathcal {A}_{\\infty }=(\\operatorname{coker}\\alpha )|_{\\ell _\\infty }$ .",
"One of the conditions that characterize $L_{\\vec{k}}$ is the invertibility of $\\Phi $ (see [2]).",
"By suitably splitting the short exact sequence which appears in (REF ), the morphism $\\Phi $ becomes $\\Phi ={\\left\\lbrace \\begin{array}{ll}\\beta _{11}^{(m)}\\alpha _{10}^{(m)}+\\beta _{21}^{(m)}\\alpha _{20}^{(m)} &\\text{for $n=1$;}\\\\[15pt]\\left(\\begin{array}{cccc|c}\\beta _{10}^{(m)} & & & & \\beta _{11}^{(m)}\\alpha _{10}^{(m)}+\\beta _{21}^{(m)}\\alpha _{20}^{(m)}\\\\\\beta _{11}^{(m)} & \\beta _{10}^{(m)} & \\text{\\raisebox {0pt}[0pt][0pt]{\\raisebox {.25cm}{\\makebox{[}0pt]{\\hspace{19.91684pt}\\huge 0}}}} & & \\beta _{22}^{(m)}\\alpha _{20}^{(m)}\\\\& \\beta _{11}^{(m)} & \\ddots & & \\vdots \\\\& \\text{\\raisebox {0pt}[0pt][0pt]{\\raisebox {-.4cm}{\\makebox{[}0pt]{\\hspace{-25.6073pt}\\huge 0}}}} & \\ddots & \\beta _{10}^{(m)} & \\beta _{2,n-1}^{(m)}\\alpha _{20}^{(m)}\\\\& & & \\beta _{11}^{(m)} & \\beta _{2n}^{(m)}\\alpha _{20}^{(m)}\\end{array}\\right)&\\text{for $n>1$.}\\end{array}\\right.",
"}$ Let us now consider the principal $\\operatorname{GL}(r,\\mathbb {C})$ -bundle $\\tau \\colon P_{\\vec{k}}\\longrightarrow L_{\\vec{k}}$ , whose fibre over a point $(\\alpha ,\\beta )$ is naturally identified with the space of framings for the cohomology of the monad (REF ).",
"By inspecting the display of $M(\\alpha ,\\beta )$ , one sees that fixing a framing in the fibre $\\tau ^{-1}(\\alpha ,\\beta )$ is equivalent to choosing a basis for $H^{0}\\left(\\ker \\beta |_{\\ell _\\infty }\\right)=\\ker H^{0}\\left(\\beta |_{\\ell _\\infty }\\right)$ .",
"So, $P_{\\vec{k}}$ can be described as the quasi-affine variety of the triples $(\\alpha ,\\beta ,\\xi )$ , where $(\\alpha ,\\beta )$ is a point of $L_{\\vec{k}}$ and $\\xi \\colon \\mathbb {C}^{r}\\longrightarrow V_{\\vec{k}}:=H^{0}({\\vec{k},\\infty })$ is an injective vector space morphism such that $H^{0}\\left(\\beta |_{\\ell _\\infty }\\right)\\circ \\xi =0$ ."
],
[
"Proof of Proposition ",
"We now are in the case where $r=1$ (hence, $a=0$ ).",
"We begin by constructing the immersion $j_{m}$ for any fixed $m\\in \\lbrace 0,\\dots ,c\\rbrace $ .",
"We define the morphism $\\begin{aligned}\\tilde{\\jmath }_{m}\\colon \\operatorname{End}(\\mathbb {C}^{c})^{\\oplus 2}\\oplus \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})&\\longrightarrow \\operatorname{Hom}(\\mathcal {U}_{\\vec{k}},{\\vec{k}})\\oplus \\operatorname{Hom}({\\vec{k}},\\mathcal {W}_{\\vec{k}})\\oplus \\operatorname{Hom}(\\mathbb {C}^{r},V_{\\vec{k}})\\\\(b_{1},b_{2},e)&\\longmapsto (\\alpha ,\\beta ,\\xi )\\end{aligned}$ $\\text{where}\\qquad \\left\\lbrace \\begin{aligned}\\alpha &=\\begin{pmatrix}\\mathbf {1}_c(y_{2m}^{n}s_{E})+{}^{t}{-3mu}{2mu}b_{2}s_\\infty \\\\\\mathbf {1}_cy_{1m}+{}^{t}{-3mu}{2mu}b_{1}y_{2m}\\\\0\\end{pmatrix}\\\\\\beta &=\\begin{pmatrix}\\mathbf {1}_cy_{1m}+{}^{t}{-3mu}{2mu}b_1y_{2m}&-\\left(\\mathbf {1}_c(y_{2m}^{n}s_{E})+{}^{t}{-3mu}{2mu}b_2s_\\infty \\right)&{}^{t}{-3mu}es_\\infty \\end{pmatrix}\\\\\\xi &=\\begin{pmatrix}0\\\\ \\vdots \\\\ 0 \\\\1\\end{pmatrix}\\end{aligned}\\right.$ and we let $j_{m}$ be the restriction of $\\tilde{\\jmath }_{m}$ to $c)$ .",
"Lemma 1.3 The morphism $j_{m}$ is a $\\operatorname{GL}(c, \\mathbb {C})$ -equivariant closed immersion of $c)$ into $P_{\\vec{k},m}$ .",
"Since it is clear that $\\tilde{\\jmath }_{m}$ is a closed immersion, it is enough to prove that $\\operatorname{Im}\\tilde{\\jmath }_{m}\\cap P_{\\vec{k},m}=\\operatorname{Im}j_{m}\\,.$ Let $(\\alpha ,\\beta ,\\xi )=\\tilde{\\jmath }_{m}(b_{1},b_{2},e)$ be any point in the intersection $\\operatorname{Im}\\tilde{\\jmath }_{m}\\cap P_{\\vec{k},m}$ ; the equation $\\beta \\circ \\alpha =0$ implies that the triple $(b_{1},b_{2},e)$ satisfies the condition (T1), while the fact that $\\beta \\otimes k(x)$ has maximal rank for all $x\\in \\Sigma _{n}$ implies condition (T2).",
"It follows that $\\operatorname{Im}\\tilde{\\jmath }_{m}\\cap P_{\\vec{k},m}\\subseteq \\operatorname{Im}j_{m}\\,.$ To get the opposite inclusion, note that for all $(\\alpha ,\\beta ,\\xi )\\in \\operatorname{Im}\\tilde{\\jmath }_{m}$ the following conditions are satisfied: (i) the morphism $\\alpha \\otimes k(x)$ fails to have maximal rank at most at a finite number of points $x\\in \\Sigma _{n}$ ; hence, $\\alpha $ is injective; (ii) the morphisms $\\alpha \\otimes k(x)$ and $\\beta \\otimes k(x)$ have maximal rank for all points $x\\in \\ell _\\infty \\cup f_{m}$ ; (iii) the morphism $\\Phi $ is invertible; (iv) one has $\\beta _{1}|_{f_{m}}=\\mathbf {1}_{c}$ ; (v) the morphism $\\xi $ has maximal rank.",
"If $(\\alpha ,\\beta ,\\xi )\\in \\operatorname{Im}j_{m}$ , the condition (T2) implies that $\\beta \\otimes k(x)$ has maximal rank for all $x\\in \\Sigma _{n}\\setminus (\\ell _\\infty \\cup f_{m})$ : by the condition (ii) this is sufficient to ensure that $\\beta $ is surjective.",
"Condition (T1) implies that $\\beta \\circ \\alpha =0$ , so that we can define the quotient sheaf $\\mathcal {E}=\\ker \\beta /\\operatorname{Im}\\alpha $ .",
"By condition (i) $\\mathcal {E}$ is torsion free, by conditions (ii) and (iii) it is trivial at infinity, and by condition (iv) $\\mathcal {E}|_{f_{m}}$ is trivial as well.",
"The $\\operatorname{GL}(c, \\mathbb {C})$ -equivariance of $j_{m}$ is readily checked.",
"Lemma REF will now allow us to prove that $j_{m}$ induces an isomorphism between the quotients of $c)$ and $P_{\\vec{k},m}$ under the actions of $\\operatorname{GL}(c, \\mathbb {C})$ and $\\operatorname{GL}(c, \\mathbb {C})^{\\times 2}$ , respectively.",
"Thus, we have to show that for any $G_{\\vec{k}}$ -orbit $O_{G_{\\vec{k}}}$ in $P_{\\vec{k},m}$ the intersection $O_{G_{\\vec{k}}}\\cap \\operatorname{Im}j_{m}$ is not empty and that its stabilizer in $G_{\\vec{k}}$ coincides with $\\operatorname{Im}\\iota $ .",
"To this aim, we build up a strictly descending chain of closed subvarieties $P_{\\vec{k},m}=:P^0\\supsetneqq P^1 \\supsetneqq \\cdots \\supsetneqq P^h=\\operatorname{Im}j_{m}\\,,$ for a certain $h>0$ , such that there exists a strictly descending chain of subgroups $G_{\\vec{k}}=:G^0\\supsetneqq G^1 \\supsetneqq \\cdots \\supsetneqq G^h=\\operatorname{Im}\\iota $ with the property that $G^{i}$ is the stabilizer inside $G_{\\vec{k}}$ of the intersection $O_{G_{\\vec{k}}}\\cap P^{i}$ for all $G_{\\vec{k}}$ -orbits in $P_{\\vec{k},m}$ .",
"Note that for each point $(\\alpha ,\\beta ,\\xi )\\in P_{\\vec{k}}$ one has an exact sequence ${0 [r] & \\mathcal {E}[r] & \\mathcal {O}_{\\Sigma _n}[r] & \\mathcal {O}_{Z} [r] & 0}$ where $\\mathcal {E}=\\mathcal {E}_{\\alpha ,\\beta }$ and $Z$ is the singular locus of $\\mathcal {E}$ .",
"If we restrict this sequence to $f_{m}$ , twist it by $\\mathcal {O}_{f_{m}}(-1)$ and take cohomology, we find out that $Z\\cap f_{m}=\\emptyset $ if and only if $H^{i}(\\mathcal {E}|_{f_{m}}(-1))=0$ for $i=0,1$ .",
"By using the display of the monad $M(\\alpha ,\\beta )$ one sees that this condition is equivalent to the condition $\\det (\\beta _{10}^{(m)})\\ne 0$ (the coefficient $\\beta _{10}^{(m)}$ is defined in eq.",
"(REF )).",
"By acting with $G_{\\vec{k}}$ on $(\\alpha ,\\beta ,\\xi )$ we can assume that $\\left\\lbrace \\begin{aligned}\\beta _{10}^{(m)}&=\\mathbf {1}_{c}\\\\\\beta _{2q}^{(m)}&=0\\qquad q=0,\\dots ,n-1\\,.\\end{aligned}\\right.$ These equations define the subvariety $P^{1}$ , whose stabilizer $G^{1}$ is the subgroup of $G_{\\vec{k}}$ determined by the conditions $\\psi _{11} =\\chi $ and $\\psi _{12} =0$ .",
"Let ${}^{t}{-3mu}{2mu}b_1:=\\beta _{11}^{(m)}$ .",
"The equation $\\beta \\circ \\alpha =0$ implies that $\\alpha _{1q}^{(m)}&=0\\qquad q=0,\\dots ,n-1\\\\\\alpha _{1n}^{(m)}&=-\\beta _{2n}^{(m)}\\alpha _{20}^{(m)}\\,.$ The invertibility of $\\Phi $ is equivalent to the condition $\\det \\alpha _{1n}^{(m)}\\ne 0$ , and by acting with $G^{1}$ we can assume that $\\alpha _{1n}^{(m)}=\\mathbf {1}_c$ .",
"This equation cuts the subvariety $P^{2}$ inside $P^{1}$ , and the stabilizer $G^{2}$ is the subgroup of $G^{1}$ where $\\chi =\\phi $ .",
"From eq.",
"(REF ) we deduce that $\\mathbf {1}_c&=-\\beta _{2n}^{(m)}\\alpha _{20}^{(m)}\\,,\\qquad \\text{so that}\\qquad \\operatorname{rk}\\beta _{2n}^{(m)}=\\operatorname{rk}\\alpha _{20}^{(m)}=c\\,.$ Therefore, by acting with $G^{2}$ we can assume that $\\alpha _{20}^{(m)}=\\begin{pmatrix}\\mathbf {1}_c\\\\0\\end{pmatrix}\\,.$ This equation cuts the subvariety $P^{3}$ inside $P^{2}$ , and the stabilizer $G^{3}$ is the subgroup of $G^{2}$ , where $\\psi _{22}=\\begin{pmatrix}\\phi & g_{12}\\\\0 & g_{22}\\end{pmatrix}$ for some $g_{12}\\in \\operatorname{Hom}(\\mathbb {C},\\mathbb {C}^{c})$ and $g_{22}\\in \\mathbb {C}^{*}$ .",
"Eq.",
"(REF ) implies that $\\beta _{2n}^{(m)}$ is of the form $\\begin{pmatrix}-\\mathbf {1}_c & *\\end{pmatrix}$ , but by acting with $G^{3}$ we can assume that $\\beta _{2n}^{(m)}=\\begin{pmatrix}-\\mathbf {1}_c & 0\\end{pmatrix}$ .",
"This equation characterizes $P^{4}$ inside $P^{3}$ , and the stabilizer $G^{4}$ is the subgroup of $G^{3}$ where $g_{12}=0$ .",
"The equation $H^{0}\\left(\\beta |_{\\ell _\\infty }\\right)\\circ \\xi =0$ implies that $\\xi ^{(m)}=\\begin{pmatrix}0\\\\\\theta ^{-1}\\end{pmatrix}\\,.$ By acting with $G^{4}$ we can assume that $\\theta =1$ : this cuts $P^{5}$ inside the variety $P^{4}$ , and the stabilizer $G^{5}$ is the subgroups of $G^{4}$ where $g_{22}=1$ .",
"It is not difficult to show that $G^{5}$ concides with $\\operatorname{Im}\\iota $ .",
"To prove that $P^{5} = \\operatorname{Im}j_{m}$ we use once more the constraint $\\beta \\circ \\alpha =0$ and get the system $\\left\\lbrace \\begin{aligned}{}^{t}{-3mu}{2mu}b_1+\\begin{pmatrix}-\\mathbf {1}_c & 0\\end{pmatrix}\\alpha _{21}^{(m)}&=0\\\\\\alpha _{1,n+1}+\\beta _{2,n+1}\\begin{pmatrix}\\mathbf {1}_c\\\\0\\end{pmatrix}&=0\\\\\\beta _{11}^{(m)}\\alpha _{1,n+1}+\\beta _{2,n+1}\\alpha _{21}^{(m)}&=0\\,.\\end{aligned}\\right.$ From the first two equations we deduce that $\\alpha _{21}^{(m)}=\\begin{pmatrix}{}^{t}{-3mu}{2mu}b_1\\\\{}^{t}{-3mu}e_2\\end{pmatrix}\\qquad \\text{and}\\qquad \\beta _{2,n+1}=\\begin{pmatrix}-\\alpha _{1,n+1} & {}^{t}{-3mu}e\\end{pmatrix}$ for some $e\\in \\operatorname{Hom}(\\mathbb {C}^{c},\\mathbb {C})$ and $e_{2}\\in \\operatorname{Hom}(\\mathbb {C},\\mathbb {C}^{c})$ .",
"Only the last equation is not identically satisfied, and is equivalent to ${}^{t}{-3mu}{2mu}b_1{}^{t}{-3mu}{2mu}b_{2}-{}^{t}{-3mu}{2mu}b_{2}{}^{t}{-3mu}{2mu}b_1+{}^{t}{-3mu}e{}^{t}{-3mu}e_2=0\\,,$ where we have put ${}^{t}{-3mu}{2mu}b_{2}=\\alpha _{1,n+1}$ .",
"Since the morphism $\\beta \\otimes k(x)$ has maximal rank for all $x\\in \\Sigma _{n}$ , the quadruple $\\left({}^{t}b_{1},{}^{t}b_{2},{}^{t}e,{}^{t}e_{2}\\right)$ satisfies the hypotheses of [12], which implies $e_{2}=0$ .",
"It follows that $P^{5}=\\operatorname{Im}j_{m}$ ."
],
[
"Proof of Proposition ",
"Lemma 1.4 For any $l,m=0,\\dots ,c$ and for any point $\\vec{b}_{m}=(b_{1m},b_{2m},e_{m})\\in c)_{m}$ , there exists a unique element $\\psi _{l}(\\vec{b}_{m})=(\\phi ,\\psi ,\\chi )\\in G_{\\vec{k}}$ such that $\\chi =\\mathbf {1}_{c}$ ; the point $(\\alpha ^{\\prime },\\beta ^{\\prime },\\xi ^{\\prime })=\\psi _{l}(\\vec{b}_{m})\\cdot j_{m}(\\vec{b}_{m})$ lies in the image of $j_{l}$ .",
"If we set $(b_{1l},b_{2l},e_{l})=j_{l}^{-1}(\\alpha ^{\\prime },\\beta ^{\\prime },\\xi ^{\\prime })$ , we have $\\left\\lbrace \\begin{aligned}b_{1l}&=\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1m}\\right)^{-1}\\left(s_{m-l}\\mathbf {1}_c+c_{m-l}b_{1m}\\right)\\\\b_{2l}&=\\left(c_{m-l}\\mathbf {1}_c-s_{m-l}b_{1m}\\right)^{n}b_{2m}\\\\e_{l}&=e_{m}\\,.\\end{aligned}\\right.$ If we set $(\\alpha ,\\beta ,\\xi )=j_{m}(\\vec{b}_{m})$ , by expressing $[y_{1m},y_{2m}]$ as functions of $[y_{1l},y_{2l}]$ we get $\\begin{aligned}\\alpha &=\\begin{pmatrix}\\sum _{q=0}^{n}(\\sigma _{q}\\mathbf {1}_{c})(y_{2l}^{q}y_{1l}^{n-q}s_{E})+{}^{t}{-3mu}{2mu}b_{2m}s_\\infty \\\\d_{1m}y_{1m}+d_{2m}y_{2m}\\\\0\\end{pmatrix}\\,,\\\\\\beta &=\\begin{pmatrix}d_{1m}y_{1m}+d_{2m}y_{2m}&-\\sum _{q=0}^{n}(\\sigma _{q}\\mathbf {1}_{c})(y_{2l}^{q}y_{1l}^{n-q}s_{E})-{}^{t}{-3mu}{2mu}b_{2m}s_\\infty &{}^{t}{-3mu}e_{m}s_\\infty \\end{pmatrix}\\,,\\end{aligned}$ where $d_{1m}=c_{m-l}\\mathbf {1}_c-s_{m-l}{}^{t}{-3mu}{2mu}b_{1m}\\qquad \\qquad d_{2m}=s_{m-l}\\mathbf {1}+c_{m-l}{}^{t}{-3mu}{2mu}b_{1m}$ and we have put $\\sigma _{q}=\\sigma ^{n}_{l-m;nq}$ for $q=0,\\dots ,n$ (see eq.",
"(REF )).",
"The explicit form of $\\psi _{l}(\\vec{b}_{m})$ is obtained by imposing the equality $(\\phi ,\\psi ,\\mathbf {1}_{c})\\cdot (\\alpha ,\\beta ,\\xi )=j_{l}(b_{1l},b_{2l},e_{l})$ for some $(b_{1l},b_{2l},e_{l})\\in c)_{l}$ .",
"One gets $\\begin{aligned}\\phi &=d_{1m}^{-(n-1)}\\\\\\psi &=\\begin{pmatrix}d_{1m} & \\psi _{12,1} & 0\\\\0 & d_{1m}^{-n} & 0\\\\0 & 0 & \\mathbf {1}_{r}\\end{pmatrix}\\,,\\end{aligned}\\\\[5pt]\\text{where}\\qquad \\psi _{12,1}=-\\sum _{q=0}^{n-1}\\sum _{p=0}^{q}\\sigma _{q-p}\\left(-d_{2m}d_{1m}^{-1}\\right)^{p}y_{1l}^{q}y_{2l}^{n-1-q}\\,.$ Eq.",
"(REF ) follows from eq.",
"(REF ).",
"Since $j_{m}$ and $j_{l}$ are injective, the map $\\vec{b}_{m}\\longmapsto \\psi _{l}(\\vec{b}_{m})\\cdot \\vec{b}_{m}$ induces the morphism $\\tilde{\\varphi }_{lm}$ in eq.",
"(REF ).",
"This completes the proof of Proposition REF ."
]
] | 1403.0460 |
[
[
"First Millimeter-wave Spectroscopy of the Ground-state Positronium"
],
[
"Abstract We report on the first measurement of the Breit-Wigner resonance of the transition from {\\it ortho-}positronium to {\\it para-}positronium.",
"We have developed an optical system to accumulate a power of over 20 kW using a frequency-tunable gyrotron and a Fabry-P\\'{e}rot cavity.",
"This system opens a new era of millimeter-wave spectroscopy, and enables us to directly determine both the hyperfine interval and the decay width of {\\it p-}Ps."
],
[
"Introduction",
"Positronium (Ps) [1] is a bound state of an electron and a positron.",
"Ground-state positronium has two spin eigenstates: ortho-positronium (o-Ps, Spin $=1$ , $3\\gamma $ -decay, lifetime = 142 ns [2]) and para-positronium (p-Ps, Spin $=0$ , $2\\gamma $ -decay, lifetime = 125 ps [3]).",
"The energy level of o-Ps is higher than that of p-Ps by the hyperfine structure ($\\Delta ^{\\rm Ps}_{\\rm HFS}$ ).",
"Compared with the hyperfine structure of hydrogen (about 1.4 GHz), $\\Delta ^{\\rm Ps}_{\\rm HFS}$ is very large, about 203 GHz (wavelength $=1.5$ mm), due to light mass of Ps and an s-channel contribution (87 GHz).",
"Since the transition from o-Ps to p-Ps is forbidden, high-power (over 10 kW) millimeter-wave radiation is required to measure the resonance around the hyperfine structure.",
"Many technological difficulties regarding the use of millimeter waves have prevented a direct measurement of this resonance.",
"Measurements using the Zeeman effect in a static magnetic field ($\\sim $ 1 T) have been intensively studied instead of direct measurements.",
"However, it has been strongly desired to directly examine the Breit-Wigner resonance from free o-Ps to p-Ps because properties of Ps is derived in a fundamental way by Quantum Electrodynamics without any external fields.",
"In this paper, we present the first results of the measurement of the resonance transition in ground-state free Ps.",
"We developed a very challenging system of high-power and frequency tunable millimeter-wave devices for this measurement.",
"As a result of the measurement of the Breit-Wigner resonance, we can directly determine both $\\Delta ^{\\rm Ps}_{\\rm HFS}$ and the decay width of p-Ps ($\\Gamma _{\\rm p\\text{--}Ps}$ ).",
"It should be noted that determination of these two values is the first achievement for free Ps.",
"The present work is the first demonstration of spectroscopy by scanning the frequency of high-power millimeter waves.",
"This new method also paves the way for various measurements in material and life science, such as the DNP-NMR spectroscopy [4].",
"The spectroscopy of the transition from o-Ps to p-Ps requires frequency-tunable (201–205 GHz) and high-power (over 20 kW) millimeter waves.",
"Previously [5], a millimeter-wave source, gyrotron, was totally monochromatic (202.89 GHz) and its output profile is an impure Gaussian (about 30%).",
"A Fabry-Pérot cavity which accumulates millimeter waves from the gyrotron was unable to store over 11 kW.We developed two innovative devices in the millimeter-wave range: A frequency tunable gyrotron with output of a Gaussian profile (purity is over 95%).",
"A high-gain Fabry-Pérot cavity withstanding very high power.",
"Figure REF shows the apparatus of our setup.",
"The Fabry-Pérot cavity is placed inside a gas chamber which will be described later.",
"Figure: Schematic view of the experimental setup.Top and side views of the gas chamber are shown.The gyrotron is a cyclotron-resonance-maser fast wave device, whose output power ($P_g$ ) is highest ($>100$ W) in the millimeter-wave range.",
"An electron beam gyrating in a strong magnetic field ($\\sim 7$ T) bunches to a deceleration phase and excites a resonant mode (millimeter waves) of a cavity in the gyrotron.",
"We have successfully developed a new gyrotron (FU CW GI) operating in the TE$_{52}$ mode with an internal mode converter [6].This gyrotron works in pulsed operation (duty ratio 30%, repetition rate 5 Hz), with which all data are acquired in synchronization (events collected during and outside gyrotron pulses are defined as beam-ON and beam-OFF, respectively).",
"The output millimeter-wave beam has a Gaussian profile.",
"The electron beam current is monitored and fed back to control the voltage of the heater of the gyrotron's electron gun.",
"The power of the output beam was thus stabilized to within $\\sim $ 10 % during each measurement (lasting a few days).",
"The frequency of the gyrotron is tuned between 201 GHz and 205 GHz by using gyrotron cavities of different radius ($R_0$ ).",
"The values of the frequency and cavity radius are summarized in Table REF .",
"A far off-resonance point ($180.56$ GHz) is obtained by using a different operating mode (TE$_{42}$ mode).",
"When the cavity of 2.467 mm was used, we changed the strength of gyrotron's magnetic field so that oscillation frequency moved within Q-value of the cavity.",
"The frequency is precisely measured ($\\pm $ 1 kHz) using a heterodyne detector (Virginia Diodes Inc., WR5.1 Even Harmonic Mixer).",
"Using this method, we have successfully overcome many difficulties in tuning high-power millimeter waves, which are basically monochromatic.",
"Table: Operating points.As shown in Fig.",
"REF , the beam from the gyrotron is guided into the Fabry-Pérot cavity, which consists of a gold mesh plane mirror (diameter $= 50$ mm, line width $= 200$ $\\mu $ m, separation $= 140$ $\\mu $ m, thickness $= 1$ $\\mu $ m) and a copper concave mirror (diameter $= 80$ mm, curvature $= 300$ mm, reflectivity $= 99.85$ %).",
"The cavity length (156 mm) is precisely controlled ($\\sim $ 100 nm) by a piezoelectric stage under the copper mirror (side view of Fig.",
"REF ).",
"The accumulated power in the cavity is measured using the radiation transmitted through a hole (diameter $= 0.6$ mm) at the center of the copper mirror.",
"This transmitted radiation is monitored by a pyroelectric detector.",
"The gold mesh mirror is fabricated on a high-resistivity silicon plate (thickness $= 1.96$ mm).",
"This silicon substrate, blocking optical photons, is also used as the window of the gas chamber.",
"The use of silicon as a base is a technical breakthrough to withstand at most 80 kW effective power with water cooling.",
"Thanks to this high effective power, we can obtain enough signals of the transition from o-Ps to p-Ps to determine the resonance shape.",
"One disadvantage of the silicon is a severe interference of millimeter waves between the mesh mirror and the silicon plate due to its high refractive index ($3.45$ ).",
"In order to reduce this effect, CST Microwave Studio [7] is used to simulate the interference and to optimize the structure of the mesh mirror.",
"A high reflectivity ($\\sim $ 99.1 %) and low loss ($\\sim $ 0.3 %) are obtained at frequencies around 203 GHz.",
"The power stored in the Fabry-Pérot cavity ($P_{\\rm eff}$ ) is designed to be over 20 kW when the power of input radiation is over 100 W."
],
[
"Power estimation",
"Absolute power estimation of high-power millimeter waves is very difficult.",
"Moreover, we should calibrate the power stored inside the Fabry-Pérot cavity.",
"The absolute accumulated power is measured as shown in Fig.",
"REF .",
"The ratio between the accumulated power and the radiation transmitted through the hole on the copper mirror is calibrated using the beam from the gyrotron.",
"A chopper splits the beam in order to simultaneously measure the transmitted signal and the beam power.",
"This reduces systematic uncertainties due to time-dependent (a few minutes) instability of the gyrotron output.",
"The chopper is synchronized to the gyrotron pulses, and switches the propagation direction from one pulse to the next.",
"Half of the pulses are totally absorbed in a Teflon box filled with water (46 ml).",
"Transparency of Teflon was measured in advance (95% $\\pm $ 5%).",
"Heat dissipation from water was corrected by fitting cooling curves by theory, and obtained cooling rates were confirmed by additional measurements.",
"The power $P$ is estimated by a temperature increase of the water.",
"The other half are passed to the copper mirror, where the power transmitted through its hole is measured by the pyroelectric detector (output voltage $=V_{\\rm tr}$ ).",
"The calibration factor $C$ is defined as $C\\equiv 2P/V_{\\rm tr}$ [kW/V].",
"The factor 2 comes from back-and-forth waves in the Fabry-Pérot cavity.",
"Using this method, the accumulated power $P_{\\rm eff}=CV_{\\rm tr}$ is measured (Table REF ).",
"The stored power is always over 20 kW, which is twice as high as previous power (11 kW).",
"The result is consistent with a rough estimated $P_{\\rm eff}$ considering the finesse (400-600) and coupling (about 60%) of the cavity [8].",
"To control $P_{\\rm eff}$ for all frequency points, we placed a wire grid polarizer between the gyrotron and the Fabry-Pérot cavity.",
"We measure $C$ before and after the transition measurement at each frequency; since these two results are consistent, their mean value is used in the analysis.",
"Figure: Schematic view of the setup used to estimate the absolute accumulated power."
],
[
"Formation of positronium",
"Positronium is formed in the gas chamber in which the Fabry-Pérot cavity is placed (Fig.",
"REF ).",
"A positron emitted from a $^{22}$ Na source (1 MBq) is tagged by a thin plastic scintillator (thickness $=0.1$ mm, NE-102 equivalent), and the $\\gamma $ rays produced in its annihilation are detected by four LaBr$_{3}$ (Ce) crystals.",
"Time spectra of Ps is obtained as a time difference between the positron and $\\gamma $ -ray signals.",
"Photomultipliers (HAMAMATSU R5924-70) are used to detect optical photons from the scintillators, and charge-sensitive ADCs are used to measure the energy.",
"The temperature of the chamber is maintained at less than $30 ^{\\circ }$ C using water-cooling.",
"There has been a long-standing problem of increasing ratio of Ps production in gas when irradiated by electromagnetic waves.",
"It was firstly reported by Ref.",
"[9], and was studied in a static electric field using the Boltzmann equation [10].",
"We have also observed the same phenomenon using millimeter waves [5].",
"In order to further investigate this phenomenon, we measured time spectra of o-Ps and slow positrons (positron with energy below threshold for Ps production [11]) in pure N$_2$ .",
"The spectra in N$_2$ shown in Fig.",
"REF (a) clearly demonstrates the increase of o-Ps and decrease of slow positrons.",
"This phenomenon is due to a slow positron accelerated by the strong millimeter-wave fields [12].",
"The accelerated slow positrons collide randomly with gas molecules with a rate comparable to 203 GHz and finally exceed the threshold of Ps production (Ore gap).",
"Figure REF (b) shows that increasing ratio of Ps is almost proportional to the stored power.",
"This phenomenon does not strongly depend on frequency and causes fake signals at off-resonance points; therefore it would distort the Breit-Wigner resonance by a few % level.",
"Figure: (a) Time spectra subtracted beam-OFF events from beam-ON ones in which frequency is 201.8 GHz and stored power is 24 kW.Slow positron (selected by back-to-back γ\\gamma rays of 511 keV) and o-Ps (γ\\gamma rays from 340 keV to 450 keV) in pure N 2 _2 (1 atm) are shown.",
"(b) Power dependence of increasing ratio of o-Ps in N 2 _2 gas and neopentane.A solid line shows a linear fit for o-Ps data and a dashed line does that for neopentane.The origin of neopentane data is defined as zero due to lack of data at low power.The cause of this phenomenon is an elastic scattering of slow positrons with N$_2$ gas molecules.",
"Target gas is required to have many vibrational and rotational modes because its cross-section of inelastic scattering is large and drastically decelerates the accelerated slow positrons as indicated in Ref. [10].",
"We selected pure neopentane (C-(CH$_{3}$ )$_{4}$ ) gas (25$^\\circ $ , 1 atm) which has much more internal degrees of freedom than N$_2$ .",
"No increase of o-Ps in neopentane (Fig.",
"REF (b)) justifies our hypothesis.",
"The use of neopentane also provides high stopping power and efficient Ps production.",
"Furthermore, neopentane does not absorb millimeter waves, differently from isobutane (mixed to N$_2$ in Ref.",
"[5]) which has an absorption line at 202.5 GHz [13].",
"To confirm that the use of neopentane surely eliminates the problem in Ps production, a far off-resonance point ($180.56$ GHz) was measured.",
"To enhance o-Ps events, we require that the time difference between the plastic scintillator signal and the coincidence signal of the LaBr$_3$ (Ce) detectors is between 50 ns and 250 ns.",
"Pileup events in which two different positrons signal the plastic scintillator are reduced by requiring that the charge measured by long (1000 ns) and short (60 ns) gate ADCs are consistent.",
"The number of accidental coincidence is estimated using the time window between 850 ns and 900 ns, and is subtracted from the signal sample.",
"We also apply an energy selection, between 494 keV and 536 keV, to select the two back-to-back $\\gamma $ rays.",
"Figure REF shows the measured time spectra at a frequency of $203.51$ GHz and accumulated power of $67.4$ kW.",
"Data are shown separately for events in beam-ON or beam-OFF of gyrotron pulses.",
"The beam-OFF spectrum consists of pick-off annihilation (quenching by an electron in a gas molecule [14]) and $3\\gamma $ -decays of o-Ps.",
"The lifetime shortened by the transition from o-Ps to p-Ps ($\\tau _{\\rm OFF}=131.3\\pm 2.7\\, {\\rm ns} \\rightarrow \\tau _{\\rm ON}=108.2\\pm 3.1\\, {\\rm ns}$ ) is observed as shown in Fig.",
"REF .",
"This decrease in lifetime is consistent with the theoretical prediction, and results in an enhancement of the event rate during the beam-ON period.",
"The event rates in beam-ON and beam-OFF periods are $R_{\\text{ON}}=548$ mHz and $R_{\\text{OFF}}=455$ mHz, respectively.",
"Figure: (color on-line)Time spectra of the LaBr 3 _3(Ce) scintillator at 203.51203.51 GHz and 67.467.4 kW,after the rejection of accidental events and the energy selection.The solid lines show the results of fits to exponential functions.The chosen time window is shown by the two dashed lines.The reaction cross-section $\\sigma $ of the transition from o-Ps to p-Ps is obtained by comparing the measured $S/N \\equiv (R_{\\text{ON}}-R_{\\text{OFF}})/R_{\\text{OFF}}$ with the value simulated using the stored power.",
"The calibrated effective power in the Fabry-Pérot cavity is continuously monitored by measuring the $V_{\\rm tr}$ waveform using a sampling ADC (sampling rate of 0.5 kHz).",
"We estimate the position of Ps formation and relative detection efficiencies of 2$\\gamma $ - and 3$\\gamma $ -decays using GEANT4 simulation [15].",
"The transition probability is calculated using the Ps positions and the theoretical distribution of the electromagnetic field within the cavity.",
"We then obtain the relation between $S/N$ and $\\sigma $ , and numerically solve the equation $S/N (\\sigma ) = (R_{\\text{ON}}-R_{\\text{OFF}})/R_{\\text{OFF}}$ .",
"The advantage of using $S/N$ is that the least well constrained parameters used in the simulation (absolute source intensity, detector misalignment, and stopping position of positrons) are canceled out.",
"We also measure $S/N$ when the Fabry-Pérot cavity does not accumulate millimeter waves, in which case $S/N$ is consistent with zero."
],
[
"Result",
"Figure REF shows the obtained result of cross-sections versus frequency.",
"Data at far off-resonance ($180.56$ GHz) demonstrate the absence of fake signals.",
"A clear resonance is obtained.",
"The data are fitted by a Breit-Wigner function of the angular frequency $\\omega $ $g(\\omega ) = 3A \\frac{\\pi c^2}{\\hbar ^2\\omega _0^2} \\cdot \\frac{1}{\\pi } \\frac{\\Gamma /2}{(\\omega -\\omega _0)^2 + (\\Gamma /2)^2},$ where $\\omega _0$ is $2\\pi \\Delta ^{\\rm Ps}_{\\rm HFS}$ , $A$ is the Einstein A coefficient of this transition, and $\\Gamma $ is the natural width of the transition.",
"Using the decay width of o-Ps ($\\Gamma _{\\rm o\\text{--}Ps}$ ) and $\\Gamma _{\\rm p\\text{--}Ps}$ , $\\Gamma $ is expressed by $\\Gamma = A + \\Gamma _{\\rm p\\text{--}Ps} + \\Gamma _{\\rm o\\text{--}Ps}.$ Since $A$ and $\\Gamma _{\\rm o\\text{--}Ps}$ are much smaller than $\\Gamma _{\\rm p\\text{--}Ps}$ , $\\Gamma $ is approximated by $\\Gamma _{\\rm p\\text{--}Ps}$ .",
"We therefore treat $\\Delta ^{\\rm Ps}_{\\rm HFS}$ , $\\Gamma _{\\rm p\\text{--}Ps}$ and $A$ as the three parameters to be determined in the fit.",
"Figure: Measured reaction cross-section of the direct transition.The solid line is the best fit (using only statistical errors) to a Breit-Wigner function.Systematic errors are summarized in Table REF .",
"The second largest systematic is about the power calibration factor $C$ .",
"The systematic error on $C$ is from the measurement of the water temperature (10%) and correction of the spatial distribution (10%).",
"This was combined with the variations of $C$ observed under different reflection conditions.",
"The standard deviation of this fluctuation is between 9% and 20% for the different gyrotron cavities.",
"At each frequency, we propagate uncertainty of $C$ to the three fitting parameters.",
"The Stark effect due to the electric field of gas molecules induces a shift in $\\Delta ^{\\rm Ps}_{\\rm HFS}$ .",
"This effect is estimated from the measurements in N$_2$ gas used in Ref.",
"[16], assuming that it depends linearly on the number of density and the scattering cross-section obtained in Doppler-broadening measurements [17].",
"The shift is corrected ($+460$ ppm) and the amount of this correction is conservatively assigned as a systematic error.",
"A linear extrapolation is sufficient at the current experimental precision, however, as has recently been pointed out [18][19], the effect of non-thermalized Ps distorts the linearity by around 10–20 ppm, and may be problematic for more precise measurements.",
"We also estimate an uncertainty due to detection efficiencies obtained using GEANT4 simulation.",
"Since $S/N$ is used to obtain the cross-sections, only the relative efficiency between 2$\\gamma $ - and 3$\\gamma $ -decays takes part in the uncertainty.",
"Energy spectra of beam-OFF events are fitted with the simulated spectra of 2$\\gamma $ - and 3$\\gamma $ -decays, in which their ratio is taken as a free parameter.",
"This ratio is nothing but the pick-off annihilation probability of beam-OFF events, given by fitting the time spectra.",
"The lifetime of o-Ps decreases from 142 ns to approximately 131 ns due to this effect (the pick-off annihilation probability is about 8%).",
"Relative differences of the two pick-off annihilation probabilities determined using these different methods are between 1% and 17% at the different frequencies, and are assigned as a systematic uncertainty to $S/N$ .",
"These errors are propagated to obtained cross-sections and then to the three fitting parameters.",
"Table: Summary of the systematic errors.Table: Summary of results.",
"The first error is statistical and the second is systematic.The systematic errors discussed above are independent, and are therefore summed quadratically to calculate the total systematic error.",
"The obtained fitting parameters are listed in Table REF .",
"This is the first direct measurement of both $\\Delta ^{\\rm Ps}_{\\rm HFS}$ and $\\Gamma _{\\rm p\\text{--}Ps}$ .",
"These all are consistent with the theoretical predictions [20][21][22]."
],
[
"Discussion",
"In this paper, we firstly demonstrate that $\\Delta ^{\\rm Ps}_{\\rm HFS}$ can be directly determined with the millimeter-wave spectroscopy.",
"A conventional method is using the Zeeman shifted levels caused by a static magnetic field.",
"In a static magnetic field ($\\sim $ 1 T), one of the o-Ps states are mixed with p-Ps and the energy level of the mixed o-Ps state rises by about 3 GHz compared to the original state.",
"This Zeeman splitting can be precisely measured by an RF, being scanned by strength of the magnetic field.",
"The value of $\\Delta ^{\\rm Ps}_{\\rm HFS}$ is calculated via the Breit-Rabi formula [23].",
"In the 1970s and 1980s, the measurements with the Zeeman effect reached accuracies of ppm level [16].",
"It should be noted that the obtained $\\Delta ^{\\rm Ps}_{\\rm HFS}$ significantly differs by 13 ppm from theoretical predictions calculated in the 2000s [20].",
"It may be due to underestimated systematic errors in the previous measurement.",
"For example, non-uniformity of the static magnetic field is a candidate of the systematic uncertainty.",
"Some independent experiments (using quantum interference [24], optical lasers [25], and a new method using a precise magnetic field [19]) have been performed.",
"All of them are measurements using the Zeeman intervals.",
"It is of great importance to re-measure $\\Delta ^{\\rm Ps}_{\\rm HFS}$ using a method totally different from the previous experiments.",
"Determination of $\\Delta ^{\\rm Ps}_{\\rm HFS}$ by directly measuring the transition from free o-Ps to p-Ps is a complementary approach to the measurements using the Zeeman effect.",
"We now discuss three improvements to achieve accuracy of 10 ppm level for $\\Delta ^{\\rm Ps}_{\\rm HFS}$ : Using a high-intensity positron beam (intensity of 7$\\times 10^7$ e$^{+}$ /s is available in KEK [26]) would increase the statistics by four orders of magnitude because only a few kHz Ps is formed inside the Fabry-Pérot cavity using the $^{22}$ Na source.",
"The statistical error becomes smaller than 10 ppm.",
"Positronium will be formed in vacuum using an efficient Ps converter (conversion efficiency is around 20–50% [27]).",
"The Stark effect (460 ppm at 1 atm) and non-thermalization effect of Ps (about 10–20 ppm) can be eliminated.",
"Since there is no pick-off annihilation in vacuum, $S/N$ will be also improved significantly by a factor of two.",
"Using a megawatt (MW) class gyrotron [28][29] would enable us to precisely (better than $0.3$ %) monitor the real power with a calorimeter.",
"The present accuracy (20 %) of the power estimation is mainly limited by uncertainty of the effective power in the Fabry-Pérot cavity.",
"The systematic error due to the power can be better than 10 ppm.",
"All these improvements have been technically achieved in the region of positron science and millimeter-wave technology.",
"Therefore, we can further investigate the disagreement of 4.0 standard deviations between measured $\\Delta ^{\\rm Ps}_{\\rm HFS}$ and QED theory with the direct measurement firstly reported in this paper."
],
[
"Conclusion",
"We firstly measured the Breit-Wigner resonance of the transition from o-Ps to p-Ps with the frequency-tunable millimeter-wave system.",
"Both $\\Delta ^{\\rm Ps}_{\\rm HFS}$ and $\\Gamma _{\\rm p\\text{--}Ps}$ of free Ps were directly and firstly determined through this resonance.",
"We pointed out the displacement of $\\Delta ^{\\rm Ps}_{\\rm HFS}$ between the previous experiments using the Zeeman effect and the theoretical calculations can be tested by improving accuracy of this direct experiment.",
"Both direct and indirect measurements would be required to conclusively solve the long-standing problem on the ground-state hyperfine structure of Ps."
],
[
"Acknowledgements",
"We would like to express our sincere gratitude to Dr. Daniel Jeans for useful discussions.",
"This experiment is a joint research between Research Center for Development of Far-Infrared Region in University of Fukui and the University of Tokyo.",
"This research is supported by JSPS KAKENHI Grant Number 20340049, 22340051, 20840010, 21360167, 23740173, 24840011, 25800129, and 11J07131."
]
] | 1403.0312 |
[
[
"Octet negative parity to octet positive parity electromagnetic\n transitions in light cone QCD"
],
[
"Abstract Light cone QCD sum rules for the electromagnetic transition form factors among positive and negative parity octet baryons are derived.",
"The unwanted contributions of the diagonal transitions among positive parity octet baryons are eliminated by combining the sum rules derived from different Lorentz structures.",
"The $Q^2$ dependence for the transversal and longitudinal helicity amplitudes are studied."
],
[
"Introduction",
"According to the quark-gluon picture baryons are represented as their bound states, and for this reason what is measured in experiments is actually indirect manifestation of their realizations.",
"The experimental investigation for obtaining information about the internal structure of baryons is based on measurement of the form factors.",
"One main direction in getting useful information in order to understand the internal structure of baryons is to study the electromagnetic properties of the baryons.",
"At present, except the proton and neutron, the electromagnetic form factors of octet spin-1/2 baryons have not yet been studied experimentally.",
"The experiments conducted at Jefferson Laboratory (JLab) and Mainz Microton facilty play the key role in study of the electromagnetic structure of baryons via the scattering of electrons on nucleons, i.e., $e N \\rightarrow e N^\\ast $ , where $N^\\ast $ is the nucleon excitation.",
"These reactions proceed through the $\\gamma ^\\ast N \\rightarrow N^\\ast $ , where $\\gamma ^\\ast $ is the virtual photon, and these transitions are described by the electromagnetic form factors.",
"The study of the properties of nucleon excitations, in particular the form factors constitute one of the main research programs of the above-mentioned laboratories.",
"In one of our previous works we analyzed the $\\gamma ^\\ast N \\rightarrow N^\\ast (1535)$ transition form factors [1] within the light cone QCD sum rules method (LCSR) [2] (for an application of the LCSR on baryon form factors, see also [3]).",
"The experiments that have been conducted at JLab for this experiment have collected a lot of data.",
"The future planned experiments at JLab will allow the chance to study the structure of $N^\\ast $ at high photon virtualities up to $Q^2=12~GeV^2$ (see [4]).",
"This transition has already been studied in framework of the covariant quark model [5] and lattice gauge theory [6].",
"In the present work we extend our previous work for the $\\gamma ^\\ast N \\rightarrow N^\\ast (1535)$ transition [1] to all members of the positive and negative parity spin-1/2 octet baryons.",
"In this regard we analyze the $\\gamma ^\\ast \\Sigma \\rightarrow \\Sigma ^\\ast $ and $\\gamma ^\\ast \\Xi \\rightarrow \\Xi ^\\ast $ transitions and calculate their transition form factors within framework of the LCSR.",
"All these transitions are customarily denoted as $\\gamma ^\\ast B \\rightarrow B^\\ast $ , where $B$ represents the spin-1/2 positive parity; and $B^\\ast $ represents the spin-1/2 negative parity baryons.",
"The paper is organized in the following way.",
"In section 2, the sum rules for the form factors of the $\\gamma ^\\ast B \\rightarrow B^\\ast $ transitions are derived in LCSR.",
"Section 3 is devoted to the numerical analysis, summary and conclusions."
],
[
"Transition form factors of $\\gamma ^\\ast B \\rightarrow B^\\ast $",
"It is well known that in describing the $\\gamma ^\\ast B \\rightarrow B^\\ast $ transition, the electromagnetic current $J_\\mu ^{el}$ is sandwiched between $B$ with momentum $p$ and $B^\\ast $ with momentum $p^\\prime $ , i.e., $\\left<B^\\ast (p^\\prime ) \\left|J_\\mu ^{el} (q) \\right|B(p) \\right>$ .",
"This matrix element is parametrized in terms of two form factors as: $\\left<B^\\ast (p^\\prime ) \\left|J_\\mu ^{el} (q) \\right|B(p) \\right>\\!\\!\\!",
"&=& \\!\\!\\!\\bar{u}_{B^\\ast } (p^\\prime ) \\Bigg [ \\Bigg ( \\gamma _\\mu - {\\unknown.",
"/{q} q_\\mu \\over q^2} \\Bigg ) F_1^\\ast (Q^2) \\nonumber \\\\&-& \\!\\!\\!",
"{i \\over m_B + m_{B^\\ast } }\\sigma _{\\mu \\nu } q^\\nu F_2^\\ast (Q^2) \\Bigg ] \\gamma _5 u_B(p)~,$ where $F_1^\\ast (Q^2)$ ; and $ F_2^\\ast (Q^2)$ are the Dirac and Pauli form factors, respectively; $q=p-p^\\prime $ ; and $Q^2=-q^2$ .",
"Appearance of the second term in $F_1^\\ast $ is dictated by the conservation of the electromagnetic current.",
"In order to derive the LCSR for the form factors $F_1^\\ast (Q^2)$ and $F_2^\\ast (Q^2)$ we consider the following vacuum-to-ground state positive parity baryon correlation function: $\\Pi _\\alpha (p,q) \\!\\!\\!",
"&=& \\!\\!\\!i \\int d^4x e^{i q x}\\left<0 \\left|{\\cal T} \\lbrace \\eta (0)J_\\alpha ^{el}(x) \\rbrace \\right|B(p) \\right>~.$ Here $\\eta $ is the interpolating current of the octet baryon, and $J_\\alpha ^{el}(x) = e_u \\bar{u} \\gamma _\\alpha u + e_d \\bar{d} \\gamma _\\alpha d+ e_s \\bar{s} \\gamma _\\alpha s $ is the electromagnetic current.",
"The general form of the interpolating currents for light octet baryons are given as: $\\eta _{\\Sigma ^+} \\!\\!\\!",
"&=& \\!\\!\\!2 \\varepsilon ^{abc} \\sum _{\\ell =1}^2 (u^{aT} C A_1^\\ell s^b)A_2^\\ell u^c~, \\nonumber \\\\\\eta _{\\Sigma ^-} \\!\\!\\!",
"&=& \\!\\!\\!\\eta _{\\Sigma ^+} (u \\rightarrow d)~, \\nonumber \\\\\\eta _{\\Sigma ^0} \\!\\!\\!",
"&=& \\!\\!\\!\\sqrt{2} \\varepsilon ^{abc} \\sum _{\\ell =1}^2 \\left[(u^{aT} CA_1^\\ell s^b) A_2^\\ell d^c + (d^{aT} C A_1^\\ell s^b) A_2^\\ell u^c \\right]~, \\nonumber \\\\\\eta _{\\Xi ^0} \\!\\!\\!",
"&=& \\!\\!\\!\\eta _{\\Sigma ^+} (u \\leftrightarrow s)~, \\nonumber \\\\\\eta _{\\Xi ^-} \\!\\!\\!",
"&=& \\!\\!\\!\\eta _{\\Sigma ^-} (d \\leftrightarrow s)~,$ where $C$ is the charge conjugation operator; and $A_1^1=I$ ; $A_1^2=A_2^1=\\gamma _5$ ; and $A_2^2=\\beta $ .",
"We consider now the hadronic transitions involving negative parity baryons.",
"According to the standard procedure for obtaining sum rules for the corresponding physical quantities, we substitute in Eq.",
"(REF ) the total set of negative and positive parity baryon states between the interpolating and electromagnetic currents.",
"The resulting hadronic dispersion relations contain contributions from the lowest positive parity baryon and its negative parity partner.",
"The matrix element of the interpolating current between the vacuum and one-particle positive parity baryon states is determined as: $\\left<0 \\left|\\eta \\right|B(p) \\right>= \\lambda _B u_B(p)~,$ where $\\lambda _B$ is the residue of the corresponding baryon.",
"Similarly, the matrix element of the interpolating current between the vacuum and one-particle negative parity baryon states is defined as: $\\left<0 \\left|\\eta \\right|B^\\ast (p) \\right>= \\lambda _{B^\\ast } \\gamma _5u_{B^\\ast }(p)~.$ The hadronic matrix elements $\\left<B (p-q) \\left|J_\\alpha ^{el} \\right|B(p)\\right>$ and $\\left<B^\\ast (p-q) \\left|J_\\alpha ^{el} \\right|B(p) \\right>$ are defined in terms of the form factors.",
"The second of these matrix elements which is written in terms of the Dirac and Pauli type form factors is given in Eq.",
"(REF ).",
"The first matrix element describing the electromagnetic transition among positive parity baryons can be obtained from Eq.",
"(REF ) by making the following replacements $F_{1,2}^\\ast (Q^2) \\rightarrow F_{1,2}(Q^2)$ , and then omitting the $\\unknown.",
"/{q}q_\\alpha /q^2$ terms and replacing $\\gamma _5$ with the unit matrix.",
"Using the equation of motion $(\\unknown.",
"/{p} - m_B) u_B(p)=0$ , the correlation function can be represented in terms of six independent invariant functions as: $\\Pi _\\alpha ((p-q)^2,q^2) \\!\\!\\!",
"&=& \\!\\!\\!\\Pi _1((p-q)^2,q^2) \\gamma _\\alpha +\\Pi _2((p-q)^2,q^2) q_\\alpha + \\Pi _3((p-q)^2,q^2)q_\\alpha \\unknown.",
"/{q} \\nonumber \\\\&+& \\!\\!\\!\\Pi _4((p-q)^2,q^2) p_\\alpha +\\Pi _5((p-q)^2,q^2) p_\\alpha \\unknown.",
"/{q} +\\Pi _6((p-q)^2,q^2) \\gamma _\\alpha \\unknown.",
"/{q}~,$ where all invariant functions depend on $(p-q)^2$ and $q^2$ .",
"Using the definition of the form factors and residues, and performing summation over the baryon spin we get the following expressions for the invariant functions: $\\Pi _1((p-q)^2, q^2) \\!\\!\\!",
"&=& \\!\\!\\!- {\\lambda _{B^\\ast } (m_{B^\\ast }+m_B) \\over m_{B^\\ast }^2 - (p-q)^2 } F_1^\\ast (q^2) -{\\lambda _{B^\\ast } (m_{B^\\ast }-m_B) \\over m_{B^\\ast }^2 - (p-q)^2 } F_2^\\ast (q^2) + \\cdots \\nonumber \\\\ \\nonumber \\\\\\Pi _2((p-q)^2, q^2) \\!\\!\\!",
"&=& \\!\\!\\!",
"{\\lambda _{B^\\ast } (m_{B^\\ast }^2 - m_B^2) \\over q^2[m_{B^\\ast }^2 -(p-q)^2] } F_1^\\ast (q^2) +{\\lambda _{B^\\ast } (m_{B^\\ast }-m_B) \\over (m_{B^\\ast } + m_B) [m_{B^\\ast }^2 -(p-q)^2] } F_2^\\ast (q^2) \\nonumber \\\\&+& \\!\\!\\!",
"{\\lambda _B \\over m_B^2 - (p-q)^2} F_2(q^2) + \\cdots \\nonumber \\\\ \\nonumber \\\\\\Pi _3((p-q)^2, q^2) \\!\\!\\!",
"&=& \\!\\!\\!",
"{\\lambda _{B^\\ast } (m_{B^\\ast } + m_B) \\over q^2[m_{B^\\ast }^2 -p^{\\prime 2}] } F_1^\\ast (q^2) +{\\lambda _{B^\\ast } \\over (m_{B^\\ast }+m_B)[m_{B^\\ast }^2 - (p-q)^2] }F_2^\\ast (q^2) \\nonumber \\\\&-& \\!\\!\\!",
"{\\lambda _B \\over 2 m_B [m_B^2 - (p-q)^2]} F_2(q^2) + \\cdots $ Here dots correspond to the contributions of the excited and continuum states with quantum numbers of $B$ and $B^\\ast $ .",
"According to the quark hadron duality these contributions are modeled as perturbative ones starting on from some threshold $s_0$ .",
"Employing the nucleon interpolating current, we now calculate the correlation function from the QCD side.",
"In order to justify the expansion of the product of two current near the light cone $x^2 \\simeq 0$ , the external momenta are taken in deep Eucledian domain.",
"The operator product expansion (OPE) is carried out over twist which involves the distribution amplitudes (DAs) of the baryon with growing twist.",
"The matrix element of the three quark operators between the vacuum and the state of the members of the positive parity octet baryons is defined in terms of the DAs of the baryons, i.e., $\\varepsilon ^{abc} \\left<0 \\left|q_{1\\alpha }^a (a_1x) q_{2\\beta }^b (a_2x)q_{3\\gamma }^c (a_3x) \\right|B(p) \\right>~,$ where $a,b,c$ are the color indices; $a_1,a_2$ ; and $a_3$ are positive numbers.",
"Using the Lorentz covariance, and parity and spin of the baryons this matrix element can be written in terms of 27 DAs as: $4 \\varepsilon ^{abc} \\left<0 \\left|q_{1\\alpha }^a (a_1x) q_{2\\beta }^b (a_2x)q_{3\\gamma }^c (a_3x) \\right|B(p) \\right>= \\sum _{i} {\\cal F}_i\\Gamma _{\\alpha \\beta }^{1i} \\left[\\Gamma ^{2i} B(p) \\right]_\\gamma ~,$ where $\\Gamma ^i$ are certain Dirac matrices; and ${\\cal F}_i$ are the DAs which do not posses definite twist.",
"For completeness, the matrix element (REF ) is presented in Appendix A.",
"The matrix element given in Eq.",
"(REF ) is defined in terms of the definite twist DAs as: $4 \\varepsilon ^{abc} \\left<0 \\left|q_{1\\alpha }^a (a_1x) q_{2\\beta }^b (a_2x)q_{3\\gamma }^c (a_3x) \\right|B(p) \\right>= \\sum _{i} F_i\\Gamma _{\\alpha \\beta }^{\\prime 1i} \\left[\\Gamma ^{\\prime 2i} B(p) \\right]_\\gamma ~, \\nonumber $ and the two sets of DAs are connected to each other by the following relations: $\\begin{array}{ll}{\\cal S}_1 = S_1~,& (2 P \\!\\cdot \\!x) \\, {\\cal S}_2 = S_1 - S_2~, \\\\{\\cal P}_1 = P_1~,& (2 P \\!\\cdot \\!x) \\, {\\cal P}_2 = P_2 - P_1~, \\\\{\\cal V}_1 = V_1~,& (2 P \\!\\cdot \\!x) \\, {\\cal V}_2 = V_1 - V_2 - V_3~, \\\\2{\\cal V}_3 = V_3~,& (4 P \\!\\cdot \\!x) \\, {\\cal V}_4 =- 2 V_1 + V_3 + V_4 + 2 V_5~, \\\\(4 P\\!\\cdot \\!x) \\, {\\cal V}_5 = V_4 - V_3~,&(2 P \\!\\cdot \\!x)^2 \\, {\\cal V}_6 = - V_1 + V_2 + V_3 + V_4 + V_5 - V_6~, \\\\{\\cal A}_1 = A_1~,& (2 P \\!\\cdot \\!x) \\, {\\cal A}_2 = - A_1 + A_2 - A_3~, \\\\2 {\\cal A}_3 = A_3~,&(4 P \\!\\cdot \\!x) \\, {\\cal A}_4 = - 2 A_1 - A_3 - A_4 + 2 A_5~, \\\\(4 P \\!\\cdot \\!x) \\, {\\cal A}_5 = A_3 - A_4~,&(2 P \\!\\cdot \\!x)^2 \\, {\\cal A}_6 = A_1 - A_2 + A_3 + A_4 - A_5 + A_6~, \\\\{\\cal T}_1 = T_1~, & (2 P \\!\\cdot \\!x) \\, {\\cal T}_2 = T_1 + T_2 - 2T_3~, \\\\2 {\\cal T}_3 = T_7~,& (2 P \\!\\cdot \\!x) \\, {\\cal T}_4 = T_1 - T_2 - 2 T_7~, \\\\(2 P\\!\\cdot \\!x) \\, {\\cal T}_5 = - T_1 + T_5 + 2 T_8~,&(2 P \\!\\cdot \\!x)^2 \\, {\\cal T}_6 = 2 T_2 - 2 T_3 - 2 T_4 + 2 T_5 + 2 T_7 + 2T_8~, \\\\(4P \\!\\cdot \\!x) \\, {\\cal T}_7 = T_7 - T_8~, &(2 P\\!\\cdot \\!x)^2 \\, {\\cal T}_8 = - T_1 + T_2 + T_5 - T_6 + 2 T_7 + 2 T_8~.\\end{array}$ We also present the explicit expressions of the DAs with definite twist in Appendix B.",
"The calculation of the invariant amplitudes from the QCD side is tedious but straightforward.",
"The invariant amplitudes in terms of the spectral densities $\\rho _{2i}$ , $\\rho _{4i}$ , and $\\rho _{6i}$ can be written as: $\\Pi _i = N \\int _0^1 dx \\Bigg \\lbrace {\\rho _{2i}(x) \\over (q-px)^2} +{\\rho _{4i}(x) \\over (q-px)^4} +{\\rho _{6i}(x) \\over (q-px)^6} \\Bigg \\rbrace ~,$ where $N=2$ for the $\\Sigma ^+$ , $\\Sigma ^-$ , $\\Xi ^0$ , and $\\Xi ^-$ ; and $\\sqrt{2}$ for $\\Sigma ^0$ baryons.",
"Explicit expressions of the spectral densities $\\rho _{2i}$ , $\\rho _{4i}$ , and $\\rho _{6i}$ are presented in Appendix C. It should be noted that the sum rules derived from combinations of different Lorentz structures are suggested in [7].",
"Equating Eqs.",
"(REF ) and (REF ), and performing Borel transformation over $-(p-q)^2$ we get the following sum rules for the $\\gamma ^\\ast B \\rightarrow B^\\ast $ transition form factors: $\\lambda _{B^\\ast } \\left[m_{B^\\ast }+m_B) F_1^\\ast (Q^2) + m_{B^\\ast }-m_B)F_2^\\ast (Q^2)\\right] e^{-m_{B^\\ast }^2/M^2} \\!\\!\\!",
"&=& \\!\\!\\!-I_1(Q^2,M^2,s_0)~, \\nonumber \\\\\\lambda _{B^\\ast } \\left[ - {m_{B^\\ast }+m_B)\\over Q^2} F_1^\\ast (Q^2) +F_2^\\ast (Q^2)\\right] e^{-m_{B^\\ast }^2/M^2} \\!\\!\\!",
"&=& \\!\\!\\!-I_1(Q^2,M^2,s_0) \\nonumber \\\\&+& \\!\\!\\!2 m_B I_3(Q^2,M^2,s_0)~,$ and $I_i(Q^2,M^2,s_0)$ are determined to be: $I_i(Q^2,M^2,s_0) \\!\\!\\!",
"&=& \\!\\!\\!\\int _{x_0}^1 dx \\Bigg [- {\\rho _{2i}(x)\\over x} + {\\rho _{4i}(x) \\over x^2 M^2} -- {\\rho _{6i}(x) \\over 2 x^3 M^4} \\Bigg ] e^{-s(x)/M^2} \\nonumber \\\\&+& \\!\\!\\!\\Bigg [ {\\rho _{4i}(x_0) \\over Q^2 + x_0^2 m_B^2} -{1\\over 2 x_0}{\\rho _{6i}(x_0) \\over (Q^2 + x_0^2 m_B^2) M^2} \\nonumber \\\\&+& \\!\\!\\!",
"{1\\over 2} {x_0^2 \\over (Q^2 + x_0^2 m_B^2)} \\Bigg ({d\\over dx_0} {\\rho _{6i}(x_0) \\over x_0 (Q^2 + x_0^2 m_B^2)M^2} \\Bigg ) \\Bigg ]e^{-s_0/ M^2}\\Bigg \\rbrace ~,$ where $s(x) = {\\bar{x} Q^2 + x \\bar{x} m_B^2 \\over x}~, \\nonumber $ and $x_0$ is the solution of $s(x) = s_0$ .",
"As has already been noted, the analysis of the experimental data for the nucleon system is performed with the Dirac and Pauli type form factors.",
"The data for the nucleon resonances is mostly analyzed with the help of helicity amplitudes (see for example [4], [8], [9]).",
"In other words, in the analysis of data related to the negative parity baryons it is more suitable to study the helicity amplitudes instead of Dirac and Pauli type form factors.",
"The electro production of negative parity spin-1/2 baryon resonance in the $\\gamma ^\\ast B \\rightarrow B^\\ast $ transitions is described with the help of two independent, transverse amplitude $A_{1/2}$ and longitudinal amplitude $S_{1/2}$ .",
"The relations among the form factors $F_1^\\ast $ and $F_2^\\ast $ , and helicity amplitudes $A_{1/2}$ and $S_{1/2}$ is given as: $A_{1/2} \\!\\!\\!",
"&=& \\!\\!\\!- 2 e \\sqrt{ {(m_{B^\\ast } + m_B)^2 +Q^2 \\over 8 m_B (m_{B^\\ast }^2- m_B^2) } } \\Bigg [ F_1^\\ast (Q^2) + {m_{B^\\ast } - m_B \\over m_{B^\\ast } + m_B}F_2^\\ast (Q^2) \\Bigg ]~, \\\\ \\nonumber \\\\S_{1/2} \\!\\!\\!",
"&=& \\!\\!\\!\\sqrt{2} e \\sqrt{ {(m_{B^\\ast } + m_B)^2 +Q^2 \\over 8 m_B(m_{B^\\ast }^2 - m_B^2) } }{\\left|\\vec{q} \\right|\\over Q^2}\\Bigg [ {m_{B^\\ast } - m_B \\over m_{B^\\ast } + m_B} F_1^\\ast (Q^2) \\nonumber \\\\&-& \\!\\!\\!",
"{Q^2 \\over (m_{B^\\ast } + m_B)^2 } F_2^\\ast (Q^2) \\Bigg ]~,$ where $e$ is the electric charge; and $\\vec{q}$ is the photon three-momentum whose absolute value in the rest frame of $B^\\ast $ is given as: $\\left|\\vec{q} \\right|={\\sqrt{ Q^4 + 2 Q^2(m_{B^\\ast }^2+m_B^2) +(m_{B^\\ast }^2 - m_B^2)^2} \\over 2 m_{B^\\ast }}~.",
"\\nonumber $ In determining the form factors $F_1^\\ast $ and $F_2^\\ast $ from the sum rules given in Eq.",
"(REF ) the residues $\\lambda _{B^\\ast }$ of the negative parity baryons are needed, which is obtained from the two-point correlation function $\\Pi (p^2) = i \\int d^4x e^{ipx} \\left<0 \\left|\\mbox{T} \\left\\lbrace \\eta (x)\\bar{\\eta }(0) \\right\\rbrace \\right|0 \\right>~.\\nonumber $ Following the same procedure presented in [1], we get for the mass and residue of the negative parity spin-1/2 octet baryons $m_{B^\\ast }^2 \\!\\!\\!",
"&=& \\!\\!\\!",
"{ \\displaystyle \\int _0^{s_0} ds \\, e^{-s/M^2} s \\Big [ m_B \\mbox{Im}\\Pi _1(s) - \\mbox{Im} \\Pi _2 (s) \\Big ] \\over \\displaystyle \\int _0^{s_0} ds \\, e^{-s/M^2} \\Big [ m_B \\mbox{Im}\\Pi _1(s) - \\mbox{Im} \\Pi _2 (s) \\Big ] }~, \\\\ \\nonumber \\\\\\left|\\lambda _{B^\\ast } \\right|^2 \\!\\!\\!",
"&=& \\!\\!\\!",
"{e^{m_{B^\\ast }^2/M^2} \\over m_{B^\\ast } + m_B}{1\\over \\pi } \\int _0^{s_0} ds \\,e^{-s/M^2} \\Big [ m_B \\mbox{Im}\\Pi _1(s) - \\mbox{Im} \\Pi _2 (s) \\Big ]~,$ $\\mbox{Im}\\Pi _1(s)$ and $\\mbox{Im} \\Pi _2 (s)$ correspond to the spectral densities for the $\\unknown.",
"/{p}$ and unit operator structures, respectively, and they are calculated in [10]."
],
[
"Numerical analysis",
"In the previous section we have calculated the $\\gamma ^\\ast B \\rightarrow B^\\ast $ transition form factors and helicity amplitudes using within the framework of the LCSR method.",
"In this section we will present our numerical results on the helicity amplitudes.",
"The main nonperturbative contributions to LCSR are realized by the DAs, which are presented in the Appendix.",
"In the numerical analysis we will use the DAs of the $\\Sigma $ , $\\Xi $ and $\\lambda $ baryons which are calculated in [11], [12], [13].",
"The parameters appearing in the expressions of the DAs are determined from the two-point QCD sum rules, and their values are given as: $f_\\Xi \\!\\!\\!",
"&=& \\!\\!\\!",
"(9.9 \\pm 0.4)\\times 10^{-3}~GeV^2~, \\nonumber \\\\\\lambda _1 \\!\\!\\!",
"&=& \\!\\!\\!-(2.1 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _2 \\!\\!\\!",
"&=& \\!\\!\\!",
"(5.2 \\pm 0.2)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _3 \\!\\!\\!",
"&=& \\!\\!\\!",
"(1.7 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\ \\nonumber \\\\f_\\Sigma \\!\\!\\!",
"&=& \\!\\!\\!",
"(9.4 \\pm 0.4)\\times 10^{-3}~GeV^2~, \\nonumber \\\\\\lambda _1 \\!\\!\\!",
"&=& \\!\\!\\!-(2.5 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _2 \\!\\!\\!",
"&=& \\!\\!\\!",
"(4.4 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _3 \\!\\!\\!",
"&=& \\!\\!\\!",
"(2.0 \\pm 0.1)\\times 10^{-2}~GeV^2~,$ We have recalculated these parameters once more and obtained that the sum rules of these parameters given in [9], [10], [11] contain errors.",
"Our reanalysis on these parameters predicts that: $f_\\Xi \\!\\!\\!",
"&=& \\!\\!\\!",
"(11.70 \\pm 0.4)\\times 10^{-3}~GeV^2~, \\nonumber \\\\\\lambda _1 \\!\\!\\!",
"&=& \\!\\!\\!-(3.15 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _2 \\!\\!\\!",
"&=& \\!\\!\\!",
"(6.50 \\pm 0.2)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _3 \\!\\!\\!",
"&=& \\!\\!\\!",
"(2.15 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\ \\nonumber \\\\f_\\Sigma \\!\\!\\!",
"&=& \\!\\!\\!",
"(13.00 \\pm 0.4)\\times 10^{-3}~GeV^2~, \\nonumber \\\\\\lambda _1 \\!\\!\\!",
"&=& \\!\\!\\!-(3.15 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _2 \\!\\!\\!",
"&=& \\!\\!\\!",
"(6.75 \\pm 0.1)\\times 10^{-2}~GeV^2~, \\nonumber \\\\\\lambda _3 \\!\\!\\!",
"&=& \\!\\!\\!",
"(1.80 \\pm 0.1)\\times 10^{-2}~GeV^2~,$ which we shall use in our numerical analysis.",
"The masses of the negative parity baryons are taken from the QCD sum rules estimation give in Eq.",
"() having the values: $m_{\\Sigma ^\\ast }=(1.7\\pm 0.1)~GeV$ $m_{\\Xi ^\\ast }=(1.75\\pm 0.1)~GeV$ , and $m_{\\Lambda ^\\ast }=(1.7\\pm 0.1)~GeV$ which are very close to the experimental results.",
"For the quark condensate we use $\\langle \\bar{q} q \\rangle (1~GeV) = -\\left(246_{-19}^{+28}~MeV\\right)^3$ [14].",
"The domain of the Borel mass parameter used in the calculations for the form factors is chosen to be $M^2=(1.8\\pm 0.4)~GeV^2$ , which is decided with the criteria that the power and continuum contributions are sufficiently suppressed.",
"The value of the continuum threshold is determined in such a way that the mass sum rules prediction reproduce the experimentally measured mass to within the limits of 10-15% accuracy, and this condition leads that $s_0=(3.7\\pm 0.3)~GeV^2$ .",
"The working region of the arbitrary parameter $\\beta $ is determined from the condition that $\\lambda _{B^\\ast }^2$ be positive and exhibit good stability with respect to the variation in $\\beta $ .",
"Our analysis shows that the residues of all negative parity baryons satisfy the above-required conditions in the range $0.4 \\le \\beta \\le 0.8$ , which we shall use in further numerical analysis.",
"In Figs.",
"(1) and (2) we depict the photon momentum square $Q^2$ dependence of the helicity amplitudes $A_{1/2}$ and $S_{1/2}$ for the $\\gamma ^\\ast \\Sigma ^+ \\rightarrow \\Sigma ^{+\\ast }$ , respectively, at $M^2=1.6~GeV^2$ , $s_0=3.5~GeV^2$ , and at three fixed values of $\\beta $ picked from its working region.",
"For the parameters $\\lambda _1$ , $\\lambda _2$ and $\\lambda _3$ appearing in DAs, we use our own results given in Eq.",
"(REF ).",
"In order to keep higher twist, continuum and higher states contributions under control $Q^2$ is restricted vary in the domain $1~GeV^2 \\le Q^2 \\le 10~GeV^2$ .",
"It follows from Fig.",
"(1) that, $A_{1/2}$ decreases with increasing $Q^2$ and tends to zero asymptotically.",
"The situation for $S_{1/2}$ is presented in Fig.",
"(2), from which we observe that it also mimics the behavior of $A_{1/2}$ and tends to zero at large $Q^2$ .",
"We see that the transversal helicity amplitude is 3 to 4 times smaller in modulo compared to the longitudinal helicity amplitude $S_{1/2}$ at all values of $Q^2$ .",
"In Figs.",
"(3) and (4) the dependencies of $A_{1/2}$ and $S_{1/2}$ on $Q^2$ at the same values of $M^2$ and $s_0$ are presented for the $\\gamma ^\\ast \\Sigma ^- \\rightarrow \\Sigma ^{-\\ast }$ transition, respectively.",
"The trends in regard to their dependence on $Q^2$ are same, i.e., both amplitudes decrease with increasing $Q^2$ in modulo.",
"We also observe that the values of the modulo of $A_{1/2}$ and $S_{1/2}$ are small compared to the $\\gamma ^\\ast \\Sigma ^+ \\rightarrow \\Sigma ^{+\\ast }$ transition, at least 2 to 3 times.",
"In Figs.",
"(5) and (6) we present the $Q^2$ dependence of the transversal and longitudinal helicity amplitudes for the $\\gamma ^\\ast \\Sigma ^0 \\rightarrow \\Sigma ^{0\\ast }$ transition.",
"We see from these figures that the magnitude of $A_{1/2}$ seems to be slightly smaller compared to the $\\gamma ^\\ast \\Sigma ^-\\rightarrow \\Sigma ^{-\\ast }$ case, while the magnitude of $S_{1/2}$ appears to be approximately 50% larger compared to the same transition.",
"The $Q^2$ dependence of $A_{1/2}$ and $S_{1/2}$ for the $\\gamma ^\\ast \\Xi ^-\\rightarrow \\Xi ^{-\\ast }$ transition are given in Figs.",
"(7) and (8).",
"We observe from these figures that the values of $A_{1/2}$ are quite similar to the ones predicted for the $\\gamma ^\\ast \\Sigma ^- \\rightarrow \\Sigma ^{-\\ast }$ transition.",
"In the case of $S_{1/2}$ however, the difference between the transitions is around 40%.",
"Finally, Figs.",
"(9) and (10) depict the dependence of the helicity amplitudes on $Q^2$ for the $\\gamma ^\\ast \\Xi ^0 \\rightarrow \\Xi ^{0\\ast }$ transition.",
"It follows from these figures that, $A_{1/2}$ change its sign at $Q^2=1.5~GeV^2$ at the fixed value of the arbitrary parameter $\\beta =0.8$ .",
"The maximum value $A_{1/2}$ is equal to $0.04$ at $Q^2=1~GeV^2$ , when $\\beta =0.4$ .",
"We further see that the magnitude of $S_{1/2}$ is quite close to the one predicted for the $\\gamma ^\\ast \\Sigma ^0 \\rightarrow \\Sigma ^{0\\ast }$ transition.",
"We can summarize our results as follows: The transversal helicity amplitude $A_{1/2}$ seems to be practically insensitive to the values of the arbitrary parameter $\\beta $ for the $\\gamma ^\\ast \\Sigma ^- \\rightarrow \\Sigma ^{-\\ast }$ and $\\gamma ^\\ast \\Xi ^- \\rightarrow \\Xi ^{-\\ast }$ transitions.",
"Contrary to the above behavior, the same amplitude $A_{1/2}$ for the $\\gamma ^\\ast \\Sigma ^+ \\rightarrow \\Sigma ^{+\\ast }$ is quite sensitive to the value of $\\beta $ .",
"The value of $A_{1/2}$ at $Q^2=1~GeV^2$ doubles itself when $\\beta $ changes from $0.4$ to $0.8$ .",
"The longitudinal amplitude $S_{1/2}$ does weakly depend on $\\beta $ for all considered transitions.",
"Of course measurement of these electromagnetic form factors is quite difficult due to the short life-time of hyperons.",
"We hope that along with further developments in experimental techniques, measurement of these transition form factors could become possible.",
"It should be noted here that, our results can be improved further with the help of more reliable calculations of DAs and with the inclusion of perturbative ${\\cal O}(\\alpha _s)$ corrections, and the first attempt in this direction has already been made in [15].",
"In conclusion, we investigate the electromagnetic transition among octet positive and negative parity baryons within LCSR method.",
"We calculate the transversal and longitudinal helicity amplitudes described by these transitions.",
"The $Q^2$ dependence of these amplitudes are studied.",
"We show that the longitudinal helicity amplitude seems to be practically insensitive to the variations in the arbitrary parameter $\\beta $ for all considered transitions."
],
[
"Appendix A",
"In this appendix we present the general Lorentz decomposition of the matrix element of the three-quark operators between the vacuum and the octet baryon states in terms of the DAs [16].",
"${ 4 \\left<0 \\left|\\varepsilon ^{ijk} u_\\alpha ^i(a_1 x) u_\\beta ^j(a_2 x) d_\\gamma ^k(a_3 x)\\right|B(p) \\right>= } \\nonumber \\\\&&{\\cal S}_1 m_B C_{\\alpha \\beta } \\left(\\gamma _5 B\\right)_\\gamma +{\\cal S}_2 m_B^2 C_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} \\gamma _5 B\\right)_\\gamma +{\\cal P}_1 m_B \\left(\\gamma _5 C\\right)_{\\alpha \\beta } B_\\gamma +{\\cal P}_2 m_B^2 \\left(\\gamma _5 C \\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!\\left({\\cal V}_1+\\frac{x^2m_B^2}{4}{\\cal V}_1^M \\right)\\left(\\!\\lnot \\!",
"{p}C \\right)_{\\alpha \\beta } \\left(\\gamma _5 B\\right)_\\gamma +{\\cal V}_2 m_B \\left(\\!\\lnot \\!",
"{p} C \\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} \\gamma _5 B\\right)_\\gamma +{\\cal V}_3 m_B \\left(\\gamma _\\mu C \\right)_{\\alpha \\beta }\\left(\\gamma ^{\\mu } \\gamma _5 B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!",
"{\\cal V}_4 m_B^2 \\left(\\!\\lnot \\!",
"{x}C \\right)_{\\alpha \\beta } \\left(\\gamma _5 B\\right)_\\gamma +{\\cal V}_5 m_B^2 \\left(\\gamma _\\mu C \\right)_{\\alpha \\beta } \\left(i \\sigma ^{\\mu \\nu } x_\\nu \\gamma _5B\\right)_\\gamma + {\\cal V}_6 m_B^3 \\left(\\!\\lnot \\!",
"{x} C \\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} \\gamma _5 B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!\\left({\\cal A}_1+\\frac{x^2m_B^2}{4}{\\cal A}_1^M\\right)\\left(\\!\\lnot \\!",
"{p}\\gamma _5 C \\right)_{\\alpha \\beta } B_\\gamma +{\\cal A}_2 m_B \\left(\\!\\lnot \\!",
"{p}\\gamma _5 C \\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} B\\right)_\\gamma +{\\cal A}_3 m_B \\left(\\gamma _\\mu \\gamma _5 C \\right)_{\\alpha \\beta }\\left( \\gamma ^{\\mu } B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!",
"{\\cal A}_4 m_B^2 \\left(\\!\\lnot \\!",
"{x} \\gamma _5 C \\right)_{\\alpha \\beta } B_\\gamma +{\\cal A}_5 m_B^2 \\left(\\gamma _\\mu \\gamma _5 C \\right)_{\\alpha \\beta } \\left(i \\sigma ^{\\mu \\nu } x_\\nu B\\right)_\\gamma + {\\cal A}_6 m_B^3 \\left(\\!\\lnot \\!",
"{x} \\gamma _5 C \\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!\\left({\\cal T}_1+\\frac{x^2m_B^2}{4}{\\cal T}_1^M\\right)\\left(p^\\nu i \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\gamma ^\\mu \\gamma _5 B\\right)_\\gamma +{\\cal T}_2 m_B \\left(x^\\mu p^\\nu i \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\gamma _5 B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!",
"{\\cal T}_3 m_B \\left(\\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\sigma ^{\\mu \\nu }\\gamma _5 B\\right)_\\gamma + {\\cal T}_4 m_B \\left(p^\\nu \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\sigma ^{\\mu \\rho } x_\\rho \\gamma _5 B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!",
"{\\cal T}_5 m_B^2 \\left(x^\\nu i \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\gamma ^\\mu \\gamma _5 B\\right)_\\gamma +{\\cal T}_6 m_B^2 \\left(x^\\mu p^\\nu i \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\!\\lnot \\!",
"{x} \\gamma _5 B\\right)_\\gamma \\nonumber \\\\&+& \\!\\!\\!",
"{\\cal T}_{7} m_B^2 \\left(\\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\sigma ^{\\mu \\nu } \\!\\lnot \\!",
"{x} \\gamma _5 B\\right)_\\gamma + {\\cal T}_{8} m_B^3 \\left(x^\\nu \\sigma _{\\mu \\nu } C\\right)_{\\alpha \\beta } \\left(\\sigma ^{\\mu \\rho } x_\\rho \\gamma _5 B\\right)_\\gamma ~,\\nonumber $ where $C$ is the charge conjugation operator; and $B$ represents the octet baryon with momentum $p$ .",
"In this appendix we present the expressions for the functions $\\rho _{2i}$ , $\\rho _{4i}$ and $\\rho _{6i}$ which appear in the sum rules for $F_i^\\ast (Q^2)$ for the $\\gamma ^\\ast \\Sigma ^+ \\rightarrow \\Sigma ^{\\ast +}$ transition.",
"$\\rho _{61}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!e_u m_B^3 Q^2 {(Q^2+m_B^2 x^2)\\over x}\\Big [4 (m_{B^\\ast }-m_B) (1+\\beta ) (2-x)+m_u (1-\\beta ) x\\Big ] \\;\\check{\\!\\check{B}}_6 (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_u m_B^2 Q^2 \\Big \\lbrace 4 m_B^2 (m_{B^\\ast }-m_B)(1+\\beta )(2-x) \\Big (\\; \\widetilde{\\!\\widetilde{C}}_6 + \\;\\widetilde{\\!\\widetilde{D}}_6 \\Big ) \\nonumber \\\\&+& \\!\\!\\!",
"{(1-\\beta )\\over x} \\Big [ m_B^2 [ 8 m_{B^\\ast } -m_B( 8-x^2) ] x \\; \\widetilde{\\!\\widetilde{B}}_6+ Q^2 [ 4 m_{B^\\ast } -m_B( 4-x) ] \\; \\widetilde{\\!\\widetilde{B}}_6 \\nonumber \\\\&-& \\!\\!\\!8 m_B^2 (m_{B^\\ast }-m_B) x (2-x)\\; \\widetilde{\\!\\widetilde{B}}_8 \\Big ]\\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B^2 {Q^2 \\over x} \\Big \\lbrace 4 m_s m_B^2(m_{B^\\ast }-m_B) (1-\\beta ) (2-x) x \\Big (\\;\\widehat{\\!\\widehat{C}}_6 - \\; \\widehat{\\!\\widehat{D}}_6 \\Big ) \\nonumber \\\\&+& \\!\\!\\!4 (1+\\beta ) m_B (m_{B^\\ast }-m_B) (2-x) \\Big [ 2 m_s m_B x\\; \\widehat{\\!\\widehat{B}}_8 +(Q^2+m_B^2 x^2) \\;\\widehat{\\!\\widehat{B}}_6 \\Big ] \\nonumber \\\\&+& \\!\\!\\!",
"(1+\\beta ) m_s \\Big [ x m_B^2[8 m_{B^\\ast } - m_B (8 - x^2) ] +Q^2 [4 m_{B^\\ast } - m_B (4 - x) ] \\Big ]\\;\\widehat{\\!\\widehat{B}}_6 \\Big \\rbrace (x) \\nonumber \\\\ \\nonumber \\\\\\rho _{41}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!",
"{1\\over 2} e_u m_B^3 Q^2 (1-\\beta ) \\Big \\lbrace 2 \\left[ 2 m_{B^\\ast } -m_B (2-x)\\right] \\Big (\\;\\check{\\!\\check{C}}_6 +\\;\\check{\\!\\check{D}}_6 \\Big ) + m_u \\;\\check{\\!\\check{B}}_6 \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B^3 {Q^2\\over x} (1+\\beta ) \\Big \\lbrace \\Big [ m_B [ 8 -(5-x) x ] - m_{B^\\ast } (8 - 5 x) \\Big ] \\;\\check{\\!\\check{B}}_6 \\nonumber \\\\&-& \\!\\!\\!3 x [ 2 m_{B^\\ast } - m_B (2-x) ] \\;\\check{\\!\\check{B}}_8 \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B^2 {Q^2 \\over 2 x} (1-\\beta ) \\Big \\lbrace 2 m_B x[ 2 m_{B^\\ast } - m_B (2-x)] \\Big (\\;\\widetilde{\\!\\widetilde{C}}_6 - \\; \\widetilde{\\!\\widetilde{D}}_6 \\Big ) \\nonumber \\\\&-& \\!\\!\\!m_u \\Big [ m_B (8+x) \\; \\widetilde{\\!\\widetilde{B}}_6- 8 m_{B^\\ast } \\; \\widetilde{\\!\\widetilde{B}}_6 - 4 m_B x\\; \\widetilde{\\!\\widetilde{B}}_8 \\Big ] \\Big \\rbrace (x)\\nonumber \\\\&+& \\!\\!\\!e_u m_B^2 {Q^2 \\over 2 x} (1+\\beta ) \\Big \\lbrace Q^2 \\; \\widetilde{\\!\\widetilde{B}}_6 - m_B x [m_B x + 4(m_{B^\\ast } - m_B)] \\Big (\\;\\widetilde{\\!\\widetilde{B}}_6 - 2 \\; \\widetilde{\\!\\widetilde{B}}_8 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_u m_B x \\Big (\\; \\widetilde{\\!\\widetilde{C}}_6 +\\; \\widetilde{\\!\\widetilde{D}}_6 \\Big ) \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B^3 Q^2 (1-\\beta ) \\Big [ m_B x\\Big (\\;\\widehat{\\!\\widehat{C}}_6 + \\; \\widehat{\\!\\widehat{D}}_6\\Big ) + m_s \\Big (\\;\\widehat{\\!\\widehat{C}}_6 - \\;\\widehat{\\!\\widehat{D}}_6 \\Big )\\Big ] (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B^2 {Q^2 \\over 2 x} (1+\\beta ) \\Big \\lbrace 2 m_B x [2(m_s + m_{B^\\ast }) - m_B (2+x)] \\; \\widehat{\\!\\widehat{B}}_8 \\nonumber \\\\&+& \\!\\!\\!\\Big [2 m_B ( m_{B^\\ast } - m_B)(8-3 x) - 2 Q^2 + m_s [8 m_{B^\\ast } - m_B (8+x)] \\Big ]\\; \\widehat{\\!\\widehat{B}}_6\\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B {Q^2 \\over 2 x} (1-\\beta ) \\Big \\lbrace 2 m_B (x^3 m_B^2 - 2 Q^2) \\Big (\\check{C}_2 + \\check{D}_2 \\Big ) \\nonumber \\\\&+& \\!\\!\\!2 m_B x \\Big [ 2 m_B (m_{B^\\ast }-m_B)\\Big ( 2 \\check{C}_2 + \\check{C}_4 -3 \\check{C}_5 + 2 \\check{D}_2 - \\check{D}_4 + 3 \\check{D}_5 \\Big )+ Q^2 \\Big (\\check{C}_2 + \\check{D}_2 \\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!Q^2 \\Big [ 4 m_{B^\\ast } \\Big (\\check{C}_2 + \\check{D}_2 \\Big ) +m_u \\Big (\\check{B}_2 + 5 \\check{B}_4 \\Big ) \\Big ] \\nonumber \\\\&-& \\!\\!\\!m_B^2 x^2 \\Big [2 (m_{B^\\ast }-m_B) \\Big (\\check{C}_4 - 3 \\check{C}_5 - \\check{D}_4 + 3 \\check{D}_5 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_u \\Big ( \\check{B}_2 -\\check{B}_4 + 6 \\check{B}_5 + 12 \\check{B}_7 -2 \\check{E}_1 + 2 \\check{H}_1\\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {Q^2 \\over 2 x} (1+\\beta ) \\Big \\lbrace -[2 (m_{B^\\ast }-m_B) Q^2 + m_B^3 x^3 ]\\Big ( \\check{B}_2 + 5 \\check{B}_4 \\Big ) \\nonumber \\\\&+& \\!\\!\\!2 m_B^2 (m_{B^\\ast }-m_B) x^2 \\Big (\\check{B}_2 - \\check{B}_4 + 6 \\check{B}_5 + 12 \\check{B}_7 -2 \\check{E}_1 + 2 \\check{H}_1\\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B x \\Big [-Q^2 \\Big (\\check{B}_2 + 5 \\check{B}_4 \\Big ) -8 m_B (m_{B^\\ast }-m_B) \\Big (\\check{B}_2 + 2 \\check{B}_4 + 3 \\check{B}_5 + 6 \\check{B}_7 -\\check{E}_1 + \\check{H}_1\\Big )\\nonumber \\\\&+& \\!\\!\\!m_u \\Big [ m_B^2 x^2 \\Big ( \\check{C}_4 - 3 \\check{C}_5- \\check{B}_4 + 3 \\check{B}_5 \\Big ) - Q^2 \\Big ( \\check{C}_2 +\\check{D}_2 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B {Q^2 \\over 2 x} (1-\\beta ) \\Big \\lbrace 2 m_B^2 (m_{B^\\ast }-m_B) (2-x) x\\Big (\\widetilde{C}_4 - \\widetilde{C}_5 + \\widetilde{D}_4 -\\widetilde{D}_5 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_u \\Big [ Q^2 \\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 \\Big )+ 4 m_B (m_{B^\\ast }-m_B) x\\Big ( \\widetilde{B}_2 - \\widetilde{B}_4 + 2 \\widetilde{B}_5\\Big ) \\nonumber \\\\&-& \\!\\!\\!m_B^2 x \\Big ( \\widetilde{B}_2 - \\widetilde{B}_4 + 2\\widetilde{B}_5 + 2 \\widetilde{E}_1 + 2 \\widetilde{H}_1 \\Big ) \\Big ]\\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {Q^2 \\over 2 x} (1+\\beta ) \\Big \\lbrace - [4 (m_{B^\\ast }-m_B) Q^2 + m_B^3 x^3]\\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 \\Big ) \\nonumber \\\\&+& \\!\\!\\!8 m_B^2 (m_{B^\\ast }-m_B) x^2\\Big ( \\widetilde{B}_5 + 2 \\widetilde{B}_7 \\Big ) +m_B x \\Big [ -Q^2 \\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!8 m_B (m_{B^\\ast }-m_B)\\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 + 2\\widetilde{B}_5 + 4 \\widetilde{B}_7 \\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!m_u \\Big [ - m_B x [2 m_{B^\\ast } - m_B (2 -x)]\\Big ( \\widetilde{C}_4 - \\widetilde{D}_4 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B x [2 m_{B^\\ast } - m_B (2 +x)]\\Big ( \\widetilde{C}_5 - \\widetilde{D}_5 \\Big )+ 2 Q^2 \\Big ( \\widetilde{C}_2 + \\widetilde{D}_2 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B {Q^2 \\over 2 x} (1-\\beta ) \\Big \\lbrace 4 (m_{{\\cal O}^\\ast }-m_B) (2 m_B^2 x + Q^2) \\Big ( \\widehat{C}_2 +\\widehat{D}_2 \\Big ) \\nonumber \\\\&-& \\!\\!\\!4 m_B^2 (m_{B^\\ast } - m_B)(2-x) x \\Big ( \\widehat{C}_5 - \\widehat{D}_5 \\Big )+ m_s \\Big [ - 2 Q^2 \\Big ( \\widehat{C}_2 -\\widehat{D}_2 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B x [2 m_{B^\\ast } - m_B (2-x)] \\Big ( \\widehat{C}_4 + \\widehat{D}_4 \\Big ) +m_B x [2 m_{B^\\ast } - m_B (2+x)] \\Big ( \\widehat{C}_5 + \\widehat{D}_5 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B {Q^2 \\over 2 x} (1+\\beta ) \\Big \\lbrace 2 (m_{B^\\ast }-m_B) Q^2 \\Big ( \\widehat{B}_2 - 3 \\widehat{B}_4\\Big ) + m_B^3 x^3 \\Big ( \\widehat{B}_2 + \\widehat{B}_4\\Big ) \\nonumber \\\\&+& \\!\\!\\!2 m_B^2 (m_{B^\\ast }-m_B) x^2\\Big ( \\widehat{B}_2 - \\widehat{B}_4 + 2 \\widehat{B}_5 +4 \\widehat{B}_7 - 2 \\widehat{E}_1 + 2 \\widehat{H}_1 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B x \\Big [ Q^2 \\Big ( \\widehat{B}_2 + \\widehat{B}_4\\Big )- 8 m_B (m_{B^\\ast }-m_B) \\Big ( \\widehat{B}_4 + \\widehat{B}_5 +2 \\widehat{B}_7 - 2 \\widehat{E}_1 + \\widehat{H}_1 \\Big ) \\Big ] \\nonumber \\\\&-& \\!\\!\\!m_s \\Big [ Q^2 \\Big ( \\widehat{B}_2 + \\widehat{B}_4\\Big ) -m_B^2 x^2 \\Big ( \\widehat{B}_2 - \\widehat{B}_4 +2 \\widehat{B}_5 - 2 \\widehat{E}_1 - 2 \\widehat{H}_1 \\Big ) \\nonumber \\\\&-& \\!\\!\\!4 m_B (m_{B^\\ast }-m_B) x \\Big ( \\widehat{B}_2 -\\widehat{B}_4 + 2 \\widehat{B}_5 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B^3 [ 2 m_{B^\\ast } - m_B (2-x)] Q^2\\int _0^{\\bar{x}} dx_3 \\, \\Big [(1-\\beta ) (A_1^M - V_1^M) - 3 (1+\\beta ) T_1^M\\Big ](x,1-x-x_3,x_3) \\nonumber \\\\&+& \\!\\!\\!e_u m_B^2 {Q^2\\over x}\\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace (1-\\beta ) Q^2 (A_1^M+V_1^M) \\nonumber \\\\&+& \\!\\!\\!m_B x (1+\\beta ) [4 m_{B^\\ast } - m_B (4-x)]T_1^M \\Big \\rbrace (x_1,x,1-x_1-x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B^2 Q^2\\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace [2 m_B ( m_B - m_{B^\\ast })x - Q^2] (1-\\beta ) (A_1^M - V_1^M) \\nonumber \\\\&+& \\!\\!\\!m_B x [2 m_{B^\\ast } - m_B(2+x)] (1+\\beta ) T_1^M \\Big \\rbrace (x_1,1-x_1-x,x) \\nonumber \\\\ \\nonumber \\\\\\rho _{21}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!e_u m_B^2 {Q^2 \\over 2 x} (1+\\beta ) \\;\\widetilde{\\!\\widetilde{B}}_6 (x)\\nonumber \\\\&-& \\!\\!\\!e_s m_B^2 {Q^2 \\over x} (1+\\beta ) \\;\\widehat{\\!\\widehat{B}}_6 (x)\\nonumber \\\\&-& \\!\\!\\!e_u m_B {Q^2\\over 2 x} \\Big \\lbrace (1+\\beta ) \\Big [(m_{B^\\ast } - m_B) \\Big ( \\check{B}_2 + 5\\check{B}_4 \\Big )+ m_u \\Big ( \\check{C}_2 + \\check{D}_2 \\Big ) \\Big ]\\nonumber \\\\&+& \\!\\!\\!",
"(1-\\beta ) \\Big [ 4 (m_{B^\\ast } - m_B)\\Big ( \\check{B}_2 + \\check{B}_4 \\Big ) + m_u \\Big ( \\check{B}_2 +5\\check{B}_4 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {Q^2\\over 2 x} \\Big \\lbrace 2 (1+\\beta ) \\Big [-2 (m_{B^\\ast } - m_B) \\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B x \\Big ( \\widetilde{B}_4 + \\widetilde{B}_5 + 2 \\widetilde{B}_7+ \\widetilde{E}_1 - \\widetilde{H}_1 \\Big ) +m_u \\Big ( \\widetilde{C}_2 +\\widetilde{D}_2 \\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!",
"(1-\\beta ) \\Big [ 2 m_B x \\Big ( \\widetilde{C}_2 - \\widetilde{C}_5 -\\widetilde{D}_2 - \\widetilde{D}_5 \\Big ) - m_u \\Big ( \\widetilde{B}_2 +\\widetilde{B}_4 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B {Q^2\\over 2 x} \\Big \\lbrace (1-\\beta ) \\Big [4 (m_{B^\\ast } - m_B) \\Big ( \\widehat{C}_2 - \\widehat{D}_2 \\Big )- m_B x \\Big ( \\widehat{C}_4 - \\widehat{C}_5 - \\widehat{D}_4 +\\widehat{D}_5 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_s \\Big ( \\widehat{C}_2 - \\widehat{D}_2 \\Big ) \\Big ]- (1+\\beta ) \\Big [ 2 m_{B^\\ast } \\Big ( \\widehat{B}_2 - 3\\widehat{B}_4 \\Big ) - 2 m_B (1-x) \\widehat{B}_2 +6 m_B \\widehat{B}_4 \\nonumber \\\\&+& \\!\\!\\!2 m_B x \\Big ( \\widehat{B}_4 +2 \\widehat{B}_5 + 4 \\widehat{B}_7 \\Big ) -m_s \\Big ( \\widehat{B}_2 + \\widehat{B}_4 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B Q^2\\lbrace [2 m_{B^\\ast } - m_B (2-x)](1+\\beta ) + m_u (1-\\beta )\\rbrace \\nonumber \\\\&\\times & \\!\\!\\!\\int _0^{\\bar{x}}dx_3 \\, (P_1 + S_1 + 3 T_1 - 6 T_3)(x,1-x-x_3,x_3) \\nonumber \\\\&-& \\!\\!\\!e_u m_B Q^2 \\lbrace [2 m_{B^\\ast } - m_B (2-x)](1-\\beta ) + m_u (1+\\beta )\\rbrace \\nonumber \\\\&\\times & \\!\\!\\!\\int _0^{\\bar{x}} dx_3 \\,( A_1 + 2 A_3 - V_1 + 2 V_3 ) (x,1-x-x_3,x_3) \\nonumber \\\\&+& \\!\\!\\!e_u m_B Q^2 (1+\\beta ) \\int _0^{\\bar{x}} dx_1 \\,\\Big [4 (m_{B^\\ast } - m_B)(T_1-2 T_3) +m_B x (P_1 + S_1 +T_1 - 2 T_3) \\nonumber \\\\&+& \\!\\!\\!m_u (A_1 + A_3 - V_1 + V_3) \\Big ] (x_1,x,1-x_1-x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {Q^2 \\over x} (1-\\beta ) \\int _0^{\\bar{x}} dx_1 \\,\\Big [ m_B^2 A_1^M + 2 m_B (m_{B^\\ast } - m_B) x(A_3-V_3) + Q^2 (A_1+V_1) \\nonumber \\\\&+& \\!\\!\\!m_u m_B x (P_1-S_1+T_1) \\Big ](x_1,x,1-x_1-x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B Q^2 (1+\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace m_B \\Big [ (2-x) (P_1+S_1) + (2+x) (T_1-2 T_3) \\Big ] \\nonumber \\\\&-& \\!\\!\\!2 m_{B^\\ast } (P_1 + S_1 +T_1 - 2 T_3)+ m_s (P_1 - S_1 - T_1) \\Big \\rbrace (x_1,1-x_1-x,x) \\nonumber \\\\&-& \\!\\!\\!e_s {Q^2 \\over x}(1-\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace m_B x \\Big [2 m_{B^\\ast } (A_1 + A_3 - V_1 + V_3) +m_s (A_1 + A_3 + V_1 -V_3) \\Big ] \\nonumber \\\\&-& \\!\\!\\!Q^2 (A_1-V_1) - m_B^2 \\Big [(A_1^M - V_1^M) + 2 x(A_1+A_3-V_1+V_3) \\Big ] \\Big \\rbrace (x_1,1-x_1-x,x) \\nonumber \\\\ \\nonumber \\\\\\rho _{62}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!-e_u m_B^3 {(Q^2+m_B^2 x^2)\\over x}[4 Q^2 (2-x) (1+\\beta ) - m_u (m_{B^\\ast }+m_B) x(1-\\beta )] \\;\\check{\\!\\check{B}}_6 (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_u m_B^2 {1\\over x} \\Big \\lbrace 4 m_B^2 Q^2 (2-x)x (1+\\beta ) \\Big (\\; \\widetilde{\\!\\widetilde{C}}_6 + \\;\\widetilde{\\!\\widetilde{D}}_6 \\Big ) \\nonumber \\\\&-& \\!\\!\\!",
"(1-\\beta ) [ m_B^3(m_{B^\\ast }+m_B) x^3 + m_B (m_{B^\\ast } -7 m_B) Q^2 x - 4 Q^4 ] \\; \\widetilde{\\!\\widetilde{B}}_6 \\nonumber \\\\&+& \\!\\!\\!8 m_B^2 Q^2 (2-x) x (1-\\beta ) \\; \\widetilde{\\!\\widetilde{B}}_8 \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!4 e_s m_s m_B^4 Q^2 (2-x) (1-\\beta ) \\Big (\\;\\widehat{\\!\\widehat{C}}_6 - \\; \\widehat{\\!\\widehat{D}}_6 \\Big ) (x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B^2 {1\\over x} (1+\\beta ) \\Big \\lbrace 4 m_B Q^2 (Q^2+m_B^2 x^2)(2-x) \\;\\widehat{\\!\\widehat{B}}_6 \\nonumber \\\\&-& \\!\\!\\!m_s [ m_B^3(m_{B^\\ast }+m_B) x^3 + m_B (m_{B^\\ast } -7 m_B) Q^2 x - 4 Q^4 ] \\; \\widehat{\\!\\widehat{B}}_6 \\nonumber \\\\&-& \\!\\!\\!8 m_s m_B^2 Q^2 (2-x) x \\; \\widehat{\\!\\widehat{B}}_8\\Big \\rbrace (x) \\nonumber \\\\ \\nonumber \\\\\\rho _{42}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!- e_u m_B^3 {1\\over 2 x} \\Big \\lbrace 2 (1+\\beta ) \\Big [8 Q^2 \\;\\check{\\!\\check{B}}_6 - x Q^2 \\Big (5\\;\\check{\\!\\check{B}}_6 - 6 \\;\\check{\\!\\check{B}}_8\\Big ) +m_B (m_{B^\\ast }+m_B) x^2 \\Big (\\;\\check{\\!\\check{B}}_6 - 3 \\;\\check{\\!\\check{B}}_8\\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!x (1-\\beta ) \\Big [2 [m_B (m_{B^\\ast }+m_B) x - 2 Q^2]\\Big ( \\;\\check{\\!\\check{C}}_6 + \\;\\check{\\!\\check{D}}_6\\Big ) -m_u (m_{B^\\ast }+m_B) \\;\\check{\\!\\check{C}}_6 \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B^2 {1\\over 2 x} \\Big \\lbrace (1+\\beta ) \\Big [(m_{B^\\ast }+m_B) Q^2 \\; \\widetilde{\\!\\widetilde{B}}_6 -m_B [m_B (m_{B^\\ast }+m_B) x - 4 Q^2] x \\Big ( \\;\\widetilde{\\!\\widetilde{B}}_6 - 2 \\; \\widetilde{\\!\\widetilde{B}}_8 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_u m_B (m_{B^\\ast }+m_B) x \\Big ( \\;\\widetilde{\\!\\widetilde{C}}_6 + \\; \\widetilde{\\!\\widetilde{D}}_6 \\Big )\\Big ] \\nonumber \\\\&+& \\!\\!\\!",
"(1-\\beta ) \\Big [ 2 m_B [m_B (m_{B^\\ast }+m_B) x - 2 Q^2] x\\Big ( \\; \\widetilde{\\!\\widetilde{C}}_6 - \\; \\widetilde{\\!\\widetilde{D}}_6 \\Big ) \\nonumber \\\\&-& \\!\\!\\!8 m_u Q^2 \\; \\widetilde{\\!\\widetilde{B}}_6 - m_u m_B(m_{B^\\ast }+m_B) x \\Big ( \\;\\widetilde{\\!\\widetilde{B}}_6 - 4 \\; \\widetilde{\\!\\widetilde{B}}_8\\Big )\\Big ] \\Big \\rbrace (x)\\nonumber \\\\&-& \\!\\!\\!e_s m_B^3 (m_{B^\\ast }+m_B) {1\\over 2 x} \\Big \\lbrace 2 x (1-\\beta ) \\Big [ m_B x \\Big ( \\; \\widehat{\\!\\widehat{C}}_6 +\\; \\widehat{\\!\\widehat{D}}_6 \\Big ) + m_s \\Big ( \\; \\widehat{\\!\\widehat{C}}_6- \\; \\widehat{\\!\\widehat{D}}_6 \\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!",
"(1+\\beta ) \\Big [ 2 m_B [m_B (m_{B^\\ast }+m_B) x+ 2 Q^2 - 2 m_s (m_{B^\\ast }+m_B)] x \\; \\widehat{\\!\\widehat{B}}_8 \\nonumber \\\\&+& \\!\\!\\!2 Q^2 [m_{B^\\ast } + 3 m_B (3-x)] \\;\\widehat{\\!\\widehat{B}}_6 + m_s [m_B (m_{B^\\ast }+m_B)x + 8 Q^2 ] \\; \\widehat{\\!\\widehat{B}}_6 \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B {1\\over 2 x} (1-\\beta ) \\Big \\lbrace 2 m_B^3 x^3 (m_{B^\\ast }+m_B) \\check{C}_2 +2 m_B^2 Q^2 x^2 \\Big (\\check{C}_4 - 3 \\check{C}_5 -\\check{D}_4 + 3 \\check{D}_5 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_B Q^2 x \\Big [m_B \\Big ( 3 \\check{C}_2 + 2\\check{C}_4- 6 \\check{C}_5 - 2 \\check{D}_4 + 6 \\check{D}_5\\Big ) -m_{B^\\ast } \\check{C}_2 \\Big ] \\nonumber \\\\&-& \\!\\!\\!4 Q^4 \\check{C}_2 +2 [m_B^3 x^3 (m_{B^\\ast }+m_B) + m_B x(m_{B^\\ast }- 3 m_B) Q^2 - 2 Q^4] \\check{D}_2 \\nonumber \\\\&+& \\!\\!\\!m_u (m_{B^\\ast }+m_B) \\Big [ Q^2 \\Big (\\check{B}_2 + 5 \\check{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!m_B^2 x^2 \\Big (\\check{B}_2 - \\check{B}_4 + 6 \\check{B}_5 + 12 \\check{B}_7 - 2 \\check{E}_1+ 2 \\check{H}_1 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {1\\over 2 x} (1+\\beta ) \\Big \\lbrace [2 Q^4 - m_B^3 (m_{B^\\ast }+m_B) x^3] \\Big (\\check{B}_2 + 5 \\check{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_B^2 Q^2 x^2 \\Big (\\check{B}_2 - \\check{B}_4 + 6 \\check{B}_5 + 12 \\check{B}_7 - 2 \\check{E}_1+ 2 \\check{H}_1 \\Big ) - m_B Q^2 x \\Big [ m_{B^\\ast } \\Big (\\check{B}_2 + 5 \\check{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!m_B\\Big ( 7 \\check{B}_2 + 11 \\check{B}_4 + 24 \\check{B}_5 + 48 \\check{B}_7 - 8\\check{E}_1 + 8 \\check{H}_1 \\Big ) \\Big ]\\nonumber \\\\&+& \\!\\!\\!m_u (m_{B^\\ast }+m_B) \\Big [m_B^2x^2 \\Big (\\check{C}_4 - 3 \\check{C}_5 - \\check{D}_4 + 3 \\check{D}_5 \\Big ) - 2 Q^2\\Big ( \\check{C}_2 + \\check{D}_2 \\Big ) \\Big ] \\Big \\rbrace (x)\\nonumber \\\\&+& \\!\\!\\!e_u m_B {1\\over 2 x} (1-\\beta ) \\Big \\lbrace 2 m_B^2 Q^2 (2-x) x \\Big (\\widetilde{C}_4 - \\widetilde{C}_5 +\\widetilde{D}_4 - \\widetilde{D}_5 \\Big ) \\nonumber \\\\&-& \\!\\!\\!m_u \\Big [ (m_{B^\\ast }+m_B) Q^2 \\Big (\\widetilde{B}_2 +\\widetilde{B}_4\\Big ) + 4 m_B Q^2 x \\Big (\\widetilde{B}_2 -\\widetilde{B}_4 + 2 \\widetilde{B}_5\\Big ) \\nonumber \\\\&-& \\!\\!\\!m_B^2 x^2 (m_{B^\\ast }+m_B) \\Big (\\widetilde{B}_2 -\\widetilde{B}_4 + 2 \\widetilde{B}_5 +2 \\widetilde{E}_1 + 2\\widetilde{H}_1 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {1\\over 2 x} (1+\\beta ) \\Big \\lbrace [4 Q^4 - m_B^3 (m_{B^\\ast }+m_B) x^3] \\Big (\\widetilde{B}_2 + \\widetilde{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!8 m_B^2 Q^2 x^2 \\Big (\\widetilde{B}_5 + 2 \\widetilde{B}_7 \\Big ) -m_B Q^2 x \\Big [ m_{B^\\ast } \\Big (\\widetilde{B}_2 +\\widetilde{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!m_B \\Big (7 \\widetilde{B}_2 + 7 \\widetilde{B}_4 + 16 \\widetilde{B}_5 + 32\\widetilde{B}_7\\Big ) \\Big ] -m_u \\Big [ m_B^2 (m_{B^\\ast }+m_B) x^2 \\Big (\\widetilde{C}_4 - \\widetilde{C}_5 - \\widetilde{D}_4 + \\widetilde{D}_5 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_B Q^2 x \\Big (\\widetilde{C}_4 + \\widetilde{C}_5 -\\widetilde{D}_4 - \\widetilde{D}_5 \\Big ) - 2 (m_{B^\\ast }+m_B) Q^2\\Big (\\widetilde{C}_2 + \\widetilde{D}_2 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B {1\\over 2 x} (1-\\beta ) \\Big \\lbrace 4 m_B^2 Q^2 x \\Big [ 2 \\Big ( \\widehat{C}_2 + \\widehat{D}_2 \\Big )- (2-x) \\Big ( \\widehat{C}_5 -\\widehat{D}_5 \\Big ) \\Big ] \\nonumber \\\\&+& \\!\\!\\!4 Q^4 \\Big ( \\widehat{C}_2 +\\widehat{D}_2 \\Big ) -m_s \\Big [ m_B^2 x^2 (m_{B^\\ast }+m_B)\\Big ( \\widehat{C}_4 - \\widehat{C}_5 +\\widehat{D}_4 - \\widehat{D}_5\\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_B Q^2 x \\Big (\\widehat{C}_4 +\\widehat{C}_5 +\\widehat{D}_4 + \\widehat{D}_5\\Big ) +(m_{B^\\ast }+m_B) Q^2 \\Big (\\widehat{C}_2 - \\widehat{D}_2 \\Big )\\Big ]\\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B {1\\over 2 x} (1+\\beta ) \\Big \\lbrace m_B^3 (m_{B^\\ast }+m_B) x^3 \\Big ( \\widehat{B}_2 +\\widehat{B}_4 \\Big ) - 2 Q^4\\Big ( \\widehat{B}_2 - 3 \\widehat{B}_4 \\Big ) \\nonumber \\\\&-& \\!\\!\\!2 m_B^2 Q^2 x^2 \\Big (\\widehat{B}_2 - \\widehat{B}_4 + 2 \\widehat{B}_5 + 4 \\widehat{B}_7 - 2 \\widehat{E}_1+ 2 \\widehat{H}_1 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_B Q^2 x \\Big [ m_{B^\\ast } \\Big ( \\widehat{B}_2 + \\widehat{B}_4\\Big ) + m_B \\Big ( \\widehat{B}_2 + 9 \\widehat{B}_4 + 8 \\widehat{B}_5 + 16\\widehat{B}_7 - 8 \\widehat{E}_1 + 8 \\widehat{H}_1 \\Big ) \\Big ] \\nonumber \\\\&-& \\!\\!\\!m_s \\Big [(m_{B^\\ast }+m_B) Q^2 \\Big ( \\widehat{B}_2 +\\widehat{B}_4 \\Big ) - m_B^2 (m_{B^\\ast }+m_B) x^2\\Big ( \\widehat{B}_2 - \\widehat{B}_4 + 2 \\widehat{B}_5 - 2 \\widehat{E}_1- 2 \\widehat{H}_1 \\Big ) \\nonumber \\\\&+& \\!\\!\\!4 m_B Q^2 x \\Big ( \\widehat{B}_2 - \\widehat{B}_4 +2 \\widehat{B}_5 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&-& \\!\\!\\!e_u m_B^3 [ m_B (m_{B^\\ast }+m_B) x - 2 Q^2]\\int _0^{\\bar{x}} dx_3 \\, \\Big [ (1-\\beta ) (A_1^M-V_1^M) \\nonumber \\\\&-& \\!\\!\\!3 (1+\\beta ) T_1^M\\Big ] (x,1-x-x_3,x_3) \\nonumber \\\\&+& \\!\\!\\!e_u m_B^2 {1\\over x}\\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace (1-\\beta ) (m_{B^\\ast }+m_B)Q^2 (A_1^M+V_1^M) \\nonumber \\\\&+& \\!\\!\\!",
"(1+\\beta ) m_B [m_B (m_{B^\\ast } - m_B) x -4 Q^2] x T_1^M \\Big \\rbrace (x_1,x,1-x_1-x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B^2 {1\\over x} \\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace (1-\\beta ) [m_{B^\\ast }+m_B (1+2 x)] Q^2 (A_1^M - V_1^M) \\nonumber \\\\&-& \\!\\!\\!",
"(1+\\beta ) m_B [m_B (m_{B^\\ast }+m_B) x+ 2 Q^2] x T_1^M \\Big \\rbrace (x_1,1-x_1-x,x) \\nonumber \\\\ \\nonumber \\\\\\rho _{22}^{\\Sigma ^{\\ast +}} (x)\\!\\!\\!",
"&=& \\!\\!\\!e_u m_B^2 (m_{B^\\ast }+m_B) {1\\over 2 x} (1+\\beta )\\;\\widetilde{\\!\\widetilde{B}}_6 (x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B^2 (m_{B^\\ast }+m_B) {1\\over 2 x} (1+\\beta )\\;\\widehat{\\!\\widehat{B}}_6 (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {1\\over 2 x} \\Big \\lbrace 2 (1+\\beta ) \\Big [Q^2 \\Big ( \\check{B}_2 + 5 \\check{B}_4 \\Big ) -m_u (m_{B^\\ast }+m_B) \\Big ( \\check{C}_2 + \\check{D}_2 \\Big )\\Big ] \\nonumber \\\\&+& \\!\\!\\!",
"(1-\\beta ) \\Big [ 4 Q^2 \\Big (\\check{C}_2 + \\check{D}_2 \\Big ) -m_u (m_{B^\\ast }+m_B) \\Big ( \\check{B}_2 + 5 \\check{B}_4\\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B {1\\over 2 x} \\Big \\lbrace 2 (1+\\beta )\\Big [ 2 Q^2 \\Big ( \\widetilde{B}_2 + \\widetilde{B}_4 \\Big )+ m_B (m_{B^\\ast }+m_B) x \\Big (\\widetilde{B}_4 + \\widetilde{B}_5 + 2 \\widetilde{B}_7 + \\widetilde{E}_1- \\widetilde{H}_1 \\Big ) \\nonumber \\\\&+& \\!\\!\\!m_u (m_{B^\\ast }+m_B)\\Big (\\widetilde{C}_2 + \\widetilde{D}_2 \\Big ) \\Big ]+ (m_{B^\\ast }+m_B) (1-\\beta ) \\Big [2 m_B x \\Big (\\widetilde{C}_2 - \\widetilde{C}_5 -\\widetilde{D}_2 - \\widetilde{D}_5\\Big ) \\nonumber \\\\&-& \\!\\!\\!m_u \\Big (\\widetilde{B}_2 + \\widetilde{B}_4 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B {1\\over 2 x} \\Big \\lbrace (1-\\beta )\\Big [ m_B (m_{B^\\ast }+m_B) x \\Big (\\widehat{C}_4 - \\widehat{C}_5 - \\widehat{D}_4 + \\widehat{D}_5\\Big ) + 4 Q^2 \\Big ( \\widehat{C}_2 + \\widehat{D}_2 \\Big ) \\nonumber \\\\&+& \\!\\!\\!2 m_s (m_{B^\\ast }+m_B) \\Big ( \\widehat{C}_2 -\\widehat{D}_2 \\Big ) \\Big ] \\nonumber \\\\&-& \\!\\!\\!",
"(1+\\beta ) \\Big [2 Q^2 \\Big ( \\widehat{B}_2 - 3 \\widehat{B}_4 \\Big )- 2 m_B (m_{B^\\ast }+m_B) x \\Big (\\widehat{B}_2 + \\widehat{B}_4 + 2 \\widehat{B}_5 + 4 \\widehat{B}_7\\Big ) \\nonumber \\\\&+& \\!\\!\\!m_s (m_{B^\\ast }+m_B) \\Big ( \\widehat{B}_2 +\\widehat{B}_4 \\Big ) \\Big ] \\Big \\rbrace (x) \\nonumber \\\\&+& \\!\\!\\!e_u m_B (1+\\beta )\\int _0^{\\bar{x}} dx_3 \\,\\Big \\lbrace [m_B (m_{B^\\ast }+m_B) x - 2 Q^2](P_1 + S_1 + 3 T_1 - 6 T_3) \\nonumber \\\\&-& \\!\\!\\!m_u (m_{B^\\ast }+m_B)(A_1+2 A_3 - V_1 + 2 V_3) \\Big \\rbrace (x,1-x-x_3,x_3) \\nonumber \\\\&-& \\!\\!\\!e_u m_B (1-\\beta )\\int _0^{\\bar{x}} dx_3 \\, \\Big \\lbrace [m_B (m_{B^\\ast }+m_B) x - 2 Q^2](A_1 + 2 A_3 - V_1 + 2 V_3) \\nonumber \\\\&+& \\!\\!\\!m_u (m_{B^\\ast }+m_B)(P_1 + S_1 + 3 T_1 - 6 T_3) \\Big \\rbrace (x,1-x-x_3,x_3) \\nonumber \\\\&-& \\!\\!\\!e_u m_B (1+\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big [4 Q^2 (T_1-2 T_3) - m_B (m_{B^\\ast }+m_B) x(P_1+S_1+T_1-2 T_3) \\nonumber \\\\&-& \\!\\!\\!m_u (m_{B^\\ast }+m_B) (A_1+A_3-V_1+V_3) \\Big ](x_1,x,1-x_1-x) \\nonumber \\\\&-& \\!\\!\\!e_u {1\\over x} (1-\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace - m_B^2 (m_{B^\\ast }+m_B) (A_1^M+V_1^M) \\nonumber \\\\&-& \\!\\!\\!Q^2 \\Big [ (m_{B^\\ast }+m_B) (A_1+V_1) -2 m_B x (A_3-V_3) \\Big ] \\nonumber \\\\&-& \\!\\!\\!m_u m_B (m_{B^\\ast }+m_B) x (P_1-S_1+T_1)\\Big \\rbrace (x_1,x,1-x_1-x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B (1+\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big [2 Q^2 (P_1+S_1+T_1-2 T_3) - m_B (m_{B^\\ast }+m_B) x(P_1+S_1-T_1+2 T_3) \\nonumber \\\\&+& \\!\\!\\!m_s (m_{B^\\ast }+m_B) (P_1-S_1-T_1)\\Big ] (x_1,1-x_1-x,x) \\nonumber \\\\&-& \\!\\!\\!e_s m_B (1+\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big [2 Q^2 (P_1+S_1+T_1-2 T_3) - m_B (m_{B^\\ast }+m_B) x(P_1+S_1-T_1+2 T_3) \\nonumber \\\\&+& \\!\\!\\!m_s (m_{B^\\ast }+m_B) (P_1-S_1-T_1)\\Big ] (x_1,1-x_1-x,x) \\nonumber \\\\&+& \\!\\!\\!e_s m_B {1\\over x} (1-\\beta ) \\int _0^{\\bar{x}} dx_1 \\, \\Big \\lbrace m_B^2 (m_{B^\\ast }+m_B) (A_1^M-V_1^M) +Q^2 \\Big [ (m_{B^\\ast }+m_B + 2 x m_B) (A_1-V_1) \\nonumber \\\\&+& \\!\\!\\!2 m_B x (A_3+V_3) \\Big ] -m_s m_B (m_{B^\\ast }+m_B) x (A_1+A_3+V_1-V_3)\\Big \\rbrace (x_1,1-x_1-x,x) \\nonumber $ In the above expressions for $\\rho _{2i}$ , $\\rho _{4i}$ and $\\rho _{6i}$ the functions ${\\cal F}(x_i)$ are defined in the following way: $\\check{\\cal F}(x_1) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_1}\\!\\!dx_1^{^{\\prime }}\\int _0^{1- x^{^{\\prime }}_{1}}\\!\\!dx_3\\,{\\cal F}(x_1^{^{\\prime }},1-x_1^{^{\\prime }}-x_3,x_3)~, \\nonumber \\\\\\check{\\!\\!\\!\\;\\check{\\cal F}}(x_1) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_1}\\!\\!dx_1^{^{\\prime }}\\int _1^{x^{^{\\prime }}_{1}}\\!\\!dx_1^{^{\\prime \\prime }}\\int _0^{1- x^{^{\\prime \\prime }}_{1}}\\!\\!dx_3\\,{\\cal F}(x_1^{^{\\prime \\prime }},1-x_1^{^{\\prime \\prime }}-x_3,x_3)~, \\nonumber \\\\\\widetilde{\\cal F}(x_2) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_2}\\!\\!dx_2^{^{\\prime }}\\int _0^{1- x^{^{\\prime }}_{2}}\\!\\!dx_1\\,{\\cal F}(x_1,x_2^{^{\\prime }},1-x_1-x_2^{^{\\prime }})~, \\nonumber \\\\\\widetilde{\\!\\widetilde{\\cal F}}(x_2) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_2}\\!\\!dx_2^{^{\\prime }}\\int _1^{x^{^{\\prime }}_{2}}\\!\\!dx_2^{^{\\prime \\prime }}\\int _0^{1- x^{^{\\prime \\prime }}_{2}}\\!\\!dx_1\\,{\\cal F}(x_1,x_2^{^{\\prime \\prime }},1-x_1-x_2^{^{\\prime \\prime }})~, \\nonumber \\\\\\widehat{\\cal F}(x_3) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_3}\\!\\!dx_3^{^{\\prime }}\\int _0^{1- x^{^{\\prime }}_{3}}\\!\\!dx_1\\,{\\cal F}(x_1,1-x_1-x_3^{^{\\prime }},x_3^{^{\\prime }})~, \\nonumber \\\\\\widehat{\\!\\widehat{\\cal F}}(x_3) \\!\\!\\!",
"&=& \\!\\!\\!\\int _1^{x_3}\\!\\!dx_3^{^{\\prime }}\\int _1^{x^{^{\\prime }}_{3}}\\!\\!dx_3^{^{\\prime \\prime }}\\int _0^{1- x^{^{\\prime \\prime }}_{3}}\\!\\!dx_1\\,{\\cal F}(x_1,1-x_1-x_3^{^{\\prime \\prime }},x_3^{^{\\prime \\prime }})~.\\nonumber $ Definitions of the functions $B_i$ , $C_i$ , $D_i$ , $E_1$ and $H_1$ that appear in the expressions for $\\rho _i(x)$ are given as follows: $B_2 \\!\\!\\!",
"&=& \\!\\!\\!T_1+T_2-2 T_3~, \\nonumber \\\\B_4 \\!\\!\\!",
"&=& \\!\\!\\!T_1-T_2-2 T_7~, \\nonumber \\\\B_5 \\!\\!\\!",
"&=& \\!\\!\\!- T_1+T_5+2 T_8~, \\nonumber \\\\B_6 \\!\\!\\!",
"&=& \\!\\!\\!2 T_1-2 T_3-2 T_4+2 T_5+2 T_7+2 T_8~, \\nonumber \\\\B_7 \\!\\!\\!",
"&=& \\!\\!\\!T_7-T_8~, \\nonumber \\\\B_8 \\!\\!\\!",
"&=& \\!\\!\\!-T_1+T_2+T_5-T_6+2 T_7+2T_8~, \\nonumber \\\\C_2 \\!\\!\\!",
"&=& \\!\\!\\!V_1-V_2-V_3~, \\nonumber \\\\C_4 \\!\\!\\!",
"&=& \\!\\!\\!-2V_1+V_3+V_4+2V_5~, \\nonumber \\\\C_5 \\!\\!\\!",
"&=& \\!\\!\\!V_4-V_3~, \\nonumber \\\\C_6 \\!\\!\\!",
"&=& \\!\\!\\!-V_1+V_2+V_3+V_4+V_5-V_6~, \\nonumber \\\\D_2 \\!\\!\\!",
"&=& \\!\\!\\!-A_1+A_2-A_3~, \\nonumber \\\\D_4 \\!\\!\\!",
"&=& \\!\\!\\!-2A_1-A_3-A_4+2A_5~, \\nonumber \\\\D_5 \\!\\!\\!",
"&=& \\!\\!\\!A_3-A_4~, \\nonumber \\\\D_6 \\!\\!\\!",
"&=& \\!\\!\\!A_1-A_2+A_3+A_4-A_5+A_6~, \\nonumber \\\\E_1 \\!\\!\\!",
"&=& \\!\\!\\!S_1-S_2~, \\nonumber \\\\H_1 \\!\\!\\!",
"&=& \\!\\!\\!P_2-P_1~.",
"\\nonumber $ Fig.",
"1 The dependence of the helicity amplitude $A_{1/2}$ for the $\\gamma ^\\ast \\Sigma ^+ \\rightarrow \\Sigma ^{+\\ast }$ transition on $Q^2$ at $M^2=1.6~\\mbox{GeV}^2$ , $s_0=3.5~\\mbox{GeV}^2$ , and at several fixed values of the auxiliary parameter $\\beta $ .",
"Fig.",
"2 The same as in Fig.",
"1, but for the helicity amplitude $S_{1/2}$ .",
"Fig.",
"3 The same as in Fig.",
"1, but for the $\\gamma ^\\ast \\Sigma ^- \\rightarrow \\Sigma ^{-\\ast }$ transition.",
"Fig.",
"4 The same as in Fig.",
"3, but for the helicity amplitude $S_{1/2}$ .",
"Fig.",
"5 The same as in Fig.",
"1, but for the $\\gamma ^\\ast \\Sigma ^0 \\rightarrow \\Sigma ^{0\\ast }$ transition.",
"Fig.",
"6 The same as in Fig.",
"5, but for the helicity amplitude $S_{1/2}$ .",
"Fig.",
"7 The same as in Fig.",
"1, but for the $\\gamma ^\\ast \\Xi ^- \\rightarrow \\Xi ^{-\\ast }$ transition, at $M^2=1.8~\\mbox{GeV}^2$ , $s_0=4.0~\\mbox{GeV}^2$ .",
"Fig.",
"8 The same as in Fig.",
"7, but for the helicity amplitude $S_{1/2}$ .",
"Fig.",
"9 The same as in Fig.",
"7, but for the $\\gamma ^\\ast \\Xi ^0 \\rightarrow \\Sigma ^{-\\ast }$ transition.",
"Fig.",
"10 The same as in Fig.",
"9, but for the helicity amplitude $S_{1/2}$ ."
]
] | 1403.0096 |
[
[
"Vector-valued Hilbert transforms along curves"
],
[
"Abstract In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\\mathbb R}^n;X)$ for some $1<p<\\infty$ and some UMD spaces $X$.",
"In particular, we prove that the Hilbert transform along some curves are completely $L^p$-bounded in the terminology from operator space theory.",
"Moreover, we obtain the $L^p(\\mathbb{R}^n;X)$-boundedness of anisotropic singular integrals by using the \"method of rotations\" of Calder\\'{o}n-Zygmund.",
"All these results extend the existing related ones."
],
[
"colorlinks=true,linkcolor=red, anchorcolor=green, citecolor=cyan, urlcolor=red, filecolor=magenta, pdftoolbar=true"
]
] | 1403.0177 |
[
[
"Critical spin-flip scattering at the helimagnetic transition of MnSi"
],
[
"Abstract We report spherical neutron polarimetry (SNP) and discuss the spin-flip scattering cross sections as well as the chiral fraction $\\eta$ close to the helimagnetic transition in MnSi.",
"For our study, we have developed a miniaturised SNP device that allows fast data collection when used in small angle scattering geometry with an area detector.",
"Critical spin-flip scattering is found to be governed by chiral paramagnons that soften on a sphere in momentum space.",
"Carefully accounting for the incoherent spin-flip background, we find that the resulting chiral fraction $\\eta$ decreases gradually above the helimagnetic transition reflecting a strongly renormalised chiral correlation length with a temperature dependence in excellent quantitative agreement with the Brazovskii theory for a fluctuation-induced first order transition."
],
[
"Critical spin-flip scattering at the helimagnetic transition of MnSi J. Kindervater Physik-Department E21, Technische Universität München, D-85748 Garching, Germany W. Häußler Physik-Department E21, Technische Universität München, D-85748 Garching, Germany Heinz Maier-Leibnitz Zentrum, Technische Universität München, D-85748 Garching, Germany M. Janoschek Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA C. Pfleiderer Physik-Department E21, Technische Universität München, D-85748 Garching, Germany P. Böni Physik-Department E21, Technische Universität München, D-85748 Garching, Germany M. Garst Institute for Theoretical Physics, Universiät zu Köln, D-50937 Köln, Germany We report spherical neutron polarimetry (SNP) and discuss the spin-flip scattering cross sections as well as the chiral fraction $\\eta $ close to the helimagnetic transition in MnSi.",
"For our study, we have developed a miniaturised SNP device that allows fast data collection when used in small angle scattering geometry with an area detector.",
"Critical spin-flip scattering is found to be governed by chiral paramagnons that soften on a sphere in momentum space.",
"Carefully accounting for the incoherent spin-flip background, we find that the resulting chiral fraction $\\eta $ decreases gradually above the helimagnetic transition reflecting a strongly renormalised chiral correlation length with a temperature dependence in excellent quantitative agreement with the Brazovskii theory for a fluctuation-induced first order transition.",
"75.25-j, 75.50.-y, 75.10-b It has long been established that B20 transition metal compounds support a well-understood hierarchy of energy scales [1] comprising in decreasing strength ferromagnetic exchange, Dzyaloshinsky-Moriya spin-orbit interactions and, finally, magnetic anisotropies due to higher-order spin-orbit coupling.",
"In turn, zero temperature spontaneous magnetic order in these materials appears essentially ferromagnetic on short distances with a long wave-length homochiral twist on intermediate distances that propagates along directions favoured by the magnetic anisotropies on the largest length scales.",
"In recent years the nature of the associated helimagnetic transition at $T_{\\rm c}$ , which makes contact with areas ranging from nuclear matter over quantum Hall physics to soft matter systems, has been the topic of heated scientific controversy since critical helimagnetic spin fluctuations are able to drive the transition first order.",
"However, considerable differences exists as to the proposed character of these fluctuations.",
"Based on a minimal description taking only into account the three scales mentioned above two scenarios may be distinguished.",
"First, according to Bak and Jensen [2] when the magnetic anisotropies are sufficiently strong, anisotropic critical paramagnons develop along the easy axis already at $T>T_{\\rm c}$ .",
"Second, in the opposite limit when the magnetic anisotropies are weak, critical spin fluctuations soften isotropically on the surface of a sphere in momentum space giving rise to a fluctuation-driven first order transition in the spirit of a proposal by Brazovskii [3].",
"In addition, a third and completely different scenario by Rößler, Bogdanov and Pfleiderer [4] has generated great interest, which requires, however, an additional phenomenological parameter beyond the minimal model.",
"In this model the generic formation of a skyrmion liquid phase between the paramagnetic and helimagnetic state is predicted, which implies an additional phase transition at a temperature $T_{\\rm sk}>T_{\\rm c}$ .",
"Eearly experimental studies of the electrical resistivity [5], specific heat [6], thermal expansion [7], [8], ultrasound attenuation [9] and neutron scattering [10] in MnSi, as the most extensively studied B20 compound, were interpreted in terms of the scenario by Bak and Jensen.",
"Comprehensive elastic neutron scattering, demonstrating critical fluctuations on the surface of a sphere, together with specific heat, susceptibility and magnetisation measurements, recently changed this view, providing quantitatively consistent evidence of a fluctuation-driven first order transition as proposed by Brazovskii [11], [12], [13], [14], see also Ref.",
"[15] for related work on Cu$_2$ OSeO$_3$ .",
"Following the discovery of a skyrmion lattice phase in small applied magnetic fields just below $T_c$ [16], [17], [18], [19], several authors have argued that the specific heat and susceptibility provide evidence for further complex phases including a skyrmion liquid phase at zero field [20], [21], [22], [23], [24].",
"Most importantly, it has been claimed that the observation of a chiral fraction $\\eta \\approx 1$ up to at least 1 K above $T_c$ in a seminal SNP study in MnSi by Pappas et al.",
"[25], [26] provides microscopic evidence supporting a skyrmion liquid phase.",
"However, as explained in our Letter $\\eta $ provides a measure of the asymmetry of magnetic spin-flip scattering, assuming extreme values $\\eta = \\pm 1$ if one of the spin-flip scattering processes, i.e., $\\uparrow $ to $\\downarrow $ or vice versa, is forbidden.",
"Hence $\\eta $ shows to what extent a magnetic system is homochiral.",
"In contrast, by definition $\\eta $ is neither a direct measure of the topological winding as the defining new property of the skyrmion liquid, nor of the phase relationship of the underlying multi-$q$ modulations determined recently in the skyrmion lattice phase in MnSi [27].",
"Moreover, to the best of our knowledge a theoretical link between $\\eta $ and the formation of a skyrmion liquid phase has not been reported either.",
"Motivated by the broad interest in the helimagnetic transition of chiral magnets and the special attention paid to $\\eta $ we have revisited the entire issue from a more general point of view in an experimental and theoretical study of the critical spin-flip scattering in MnSi.",
"We thereby do not find any evidence suggesting an additional phase transition above $T_{\\rm c}$ .",
"As our main conclusion, our SNP results are in excellent quantitative agreement with the minimal model of a fluctuation-induced first order transition as predicted by Brazovskii, establishing also quantitative consistency with previous specific heat, magnetisation, susceptibility and elastic neutron scattering studies [11], [12], [13], [14].",
"Carefully considering the experimental requirements to go beyond previous SNP studies, presented in detail below, revealed as most prominent aspect the need to track incoherent signal contributions.",
"To meet these requirements we have developed a versatile miniaturised SNP device [28], which in its present version offers great flexibility at scattering angles up to $15^{\\circ }$ .",
"In particular, as opposed to the large size of SNP devices such as CryoPAD or MuPAD [29], [30] our entire set-up (diameter 50 mm; height 120 mm) is integrated into a normal sample stick fitting a standard pulse-tube cooler.",
"In turn this reduces the time required for setting up our SNP device to the time needed for a conventional sample change.",
"Moreover, when combined with an area detector fast data collection at various sample orientations and momentum transfers are readily possible.",
"Figure: Schematic depiction of the miniaturised SNP device.",
"(a) Schematic overview of the complete setup with cryostat, polarizer (P), analyzer (A) and detector (D).",
"(b) Close-up view of the SNP device, as composed of the coil bodies (CB) with their neutron window (NW).",
"The coils are surrounded by a mu-metal yoke (MM).",
"(c) Orientation of the precession coils (blue arrows), local magnetic field (red arrow) and sample (S).Shown in Fig.",
"REF (a) is a schematic depiction of the miniaturised SNP set-up developed for our study.",
"Pairs of crossed precession coils (PC) before and after the sample generate Larmor precessions of the neutron polarisation that permit to rotate the polarisation of an incoming and scattered neutron beam in any arbitrary direction.",
"The precession coils are wound on coil bodies (CB) with a large neutron window (NW), see Fig.",
"REF (b).",
"As its main advantage parasitic rotations of the polarisation are minimised by the very compact geometry and the mu-metal yokes (MM) around the precession coils, which short-circuit both external fields and the precession fields.",
"The measurements reported in this Letter were performed on the beam-line RESEDA [31] at the Heinz Maier-Leibnitz Zentrum of the Technische Universität München.",
"Neutrons were polarised with a cavity providing a polarisation of 95 % and analysed with a bender at an efficiency of 98 %.",
"Data were recorded at a neutron wavelength $\\lambda =4.5\\,{\\rm Å}$ with a wavelength spread $\\Delta \\lambda /\\lambda =0.16$ .",
"Using a CASCADE [32] area detector (PSD) with $200\\,{\\rm mm}\\times 200\\,{\\rm mm}$ active area ($128 \\times 128$ pixels) at a distance to the sample of 1596 mm our set-up corresponded effectively to small angle neutron scattering.",
"The FWHM of the resolution and the wavelength band $\\sigma _{\\textrm {fCol}}=3.6\\cdot 10^{-3}\\,\\textrm {Å}^{-1}$ and $\\sigma _{\\textrm {W}}=2.6\\cdot 10^{-3}\\,\\textrm {Å}^{-1}$ , respectively, as considered at the wave vector of the helical modulation, $k=0.039$ Å of MnSi, resulted in a momentum uncertainty $\\Delta Q/Q\\approx 10\\,\\%$ .",
"Figure: Operating principle of MiniMuPAD demonstrating the spin-selective Bragg scattering off helimagnetic order in MnSi.",
"(a) Wave vectors, 𝐤 1 \\mathbf {k}_1 and 𝐤 2 \\mathbf {k}_2, of two different helimagnetic domains are within the plane orthogonal to the neutron beam n ^\\hat{n}.",
"(b) The angle Ω\\Omega defines the relative orientation of the spin polarisation of the incoming neutrons e ^ in \\hat{e}_{\\rm in} orthogonal to n ^\\hat{n}.",
"(c) - (f) SANS patterns obtained for specific polarizations e ^ in \\hat{e}_{\\rm in}.",
"(g) Intensity of magnetic Bragg peaks at ±𝐤 1,2 \\pm \\mathbf {k}_{1,2} as a function of e ^ in \\hat{e}_{\\rm in}, i.e., Ω\\Omega , where Ω 1 =arccos1 3\\Omega _1 = \\arccos \\frac{1}{3}.For our study we used the MnSi single crystal investigated in Ref. [11].",
"The sample was oriented by x-ray Laue diffraction such that two $\\langle 111\\rangle $ axes, in the following denoted by $k_1\\parallel [1\\bar{1}\\bar{1}]$ and $k_2\\parallel [1\\bar{1}1]$ , were orthogonal to the neutron beam $\\hat{n}\\parallel [110]$ , see Fig.",
"REF (a).",
"The combination of the large window of the PCs with a PSD detector and the possibility for rotations of the sample with respect to its vertical $[1 \\bar{1} 2]$ axis by an angle $\\Phi =0^\\circ ,3^\\circ ,25^\\circ $ , see also Fig.",
"REF (c), allowed to track various points in reciprocal space going well beyond previous work.",
"For a physically transparent theoretical account of the spin-flip scattering in chiral magnets it is helpful to begin with the spin-resolved energy-integrated neutron scattering cross section, $\\sigma _{\\hat{e}_{\\rm out}, \\hat{e}_{\\rm in}}(\\mathbf {Q}) \\equiv \\frac{d \\sigma }{d \\Omega }$ with momentum transfer $\\mathbf {Q}$ , which describes the change of the spin eigenstate of the neutron from $|\\hat{e}_{\\rm in} \\rangle $ to $|\\hat{e}_{\\rm out} \\rangle $ with $\\vec{\\sigma }| \\hat{e}_\\alpha \\rangle = \\hat{e}_\\alpha | \\hat{e}_\\alpha \\rangle $ .",
"It comprises, in general, a nuclear and a magnetic contribution, as well as a nuclear–magnetic interference term [33], [34].",
"The spin-polarisation dependence of the cross-section, $\\sigma (\\mathbf {Q},\\hat{e}_{\\rm in}) = \\sum _{\\tau =\\pm 1} \\sigma _{\\tau \\hat{e}_{\\rm out}, \\hat{e}_{\\rm in}}(\\mathbf {Q})$ may then be resolved, when scattering a polarised neutron beam.",
"The magnetic contribution to $\\sigma (\\mathbf {Q},\\hat{e}_{\\rm in})$ consists thereby of a symmetric and an antisymmetric part $\\sigma _{\\rm mag}(\\mathbf {Q},\\hat{e}_{\\rm in}) = \\sigma ^S_{\\rm mag}(\\mathbf {Q}) + (\\hat{Q} \\hat{e}_{\\rm in}) \\sigma ^A_{\\rm mag}(\\mathbf {Q})$ , where the latter is weighted by the scalar product $\\hat{Q} \\hat{e}_{\\rm in}$ with $\\hat{Q} = \\mathbf {Q}/Q$ .",
"The so-called chiral fraction $\\eta $ , is finally defined as $\\eta (\\mathbf {Q}) = \\sigma ^A_{\\rm mag}(\\mathbf {Q})/\\sigma ^S_{\\rm mag}(\\mathbf {Q})$ , measuring the chirality of the magnetic scattering.",
"Finite values of $\\sigma ^A_{\\rm mag}$ (and thus $\\eta $ ) originate in the polarization-dependent scattering characteristics of chiral magnetic systems like MnSi.",
"In the helimagnetically ordered phase of MnSi, the magnetisation may be described as $\\vec{M}(\\mathbf {r}) = M_{\\rm hel} (\\hat{e}_1 \\cos \\mathbf {k}\\mathbf {r} + \\hat{e}_2 \\sin \\mathbf {k}\\mathbf {r})$ , with wave vector $\\mathbf {k}$ and orthonormal unit vectors $\\hat{e}_i \\hat{e}_j = \\delta _{ij}$ , with $i,j = 1,2$ orthogonal to $\\mathbf {k}$ ; for a left-handed helix $\\hat{e}_1 \\times \\hat{e}_2 = -\\hat{k}$ .",
"The magnetic Bragg scattering by a magnetic helix possesses equal magnitudes of symmetric and antisymmetric contributions, $\\sigma ^S_{\\rm mag}(\\mathbf {k})|_{\\rm Bragg} = \\pm \\sigma ^A_{\\rm mag}(\\mathbf {k})|_{\\rm Bragg}$ where the sign indicates the handedness of the magnetic helix with $\\eta _{\\rm Bragg}(\\mathbf {k}) = 1$ being left- and $\\eta _{\\rm Bragg}(\\mathbf {k}) = -1$ being right-handed.",
"For left-handed helimagnetic order we thus expect $\\sigma _{\\rm mag}(\\mathbf {k},\\hat{e}_{\\rm in})|_{\\rm Bragg} \\propto (1+(\\hat{k} \\hat{e}_{\\rm in})) = 2 \\cos ^2(\\delta \\Omega /2)$ , i.e., a sinusoidal dependence as a function of the angle $\\delta \\Omega $ enclosed by $\\hat{k}$ and $\\hat{e}_{\\rm in}$ .",
"In order to confirm the proper functioning of our SNP device this dependence was experimentally verified by rotating the polarisation $\\hat{e}_{\\rm in}$ of the incident neutron in the plane perpendicular to the beam as depicted in Fig.",
"REF (b), where the orientation is denoted by the angle $\\Omega = \\angle (\\hat{e}_{\\rm in}, \\mathbf {k}_1)$ .",
"Well below $T_{\\rm c}$ weak cubic anisotropies align the helimagnetic wave vector $\\mathbf {k}$ along crystallographic $\\langle 111\\rangle $ directions.",
"In turn the intensity displays extrema whenever the polarisation $\\hat{e}_{\\rm in}$ aligns with a wave vector of a helimagnetic domain $\\pm \\mathbf {k}_{1,2}$ for $\\Omega _0=0$ , $\\Omega _1=\\arccos \\frac{1}{3}$ , $\\Omega _2=\\pi $ and $\\Omega _3=\\Omega _1 + \\pi $ .",
"The associated intensity patterns at the PSD are shown in Fig.",
"REF (c) through (f).",
"Apart from a constant background, the variation of the intensity as a function of $\\delta \\Omega = \\Omega - \\Omega _n$ , with $n=0,1,2,3$ , shown in Fig.",
"REF (g), may be fitted in perfect agreement with chiral Bragg scattering, underscoring the precision of the alignment of $\\hat{e}_{\\rm in}$ .",
"This suggests, in particular, a negligible interference between magnetic and nuclear contributions.",
"Differences of the maximum peak intensity are thereby due to a small misalignment of the $\\mathbf {k}_2$ crystallographic axis with respect to the SNP set-up.",
"Table: Polarization matrix in the helimagnetically ordered phase of MnSi at 6.5 Kdetermined experimentally with our miniaturised SNP device.Using in addition an analyser the fully spin-resolved scattering in the helical state may be discussed, where the polarisation matrix is defined as $\\mathbb {P}_{\\alpha ,\\beta }(\\mathbf {Q}) =\\frac{\\sigma _{\\hat{e}_{\\alpha },\\hat{e}_{\\beta }}(\\mathbf {Q}) - \\sigma _{-\\hat{e}_{\\alpha },\\hat{e}_{\\beta }}(\\mathbf {Q}) }{\\sigma _{\\hat{e}_{\\alpha },\\hat{e}_{\\beta }}(\\mathbf {Q}) + \\sigma _{-\\hat{e}_{\\alpha },\\hat{e}_{\\beta }}(\\mathbf {Q}) }$ and $\\hat{e}_{\\alpha }$ with $\\alpha = x,y,z$ forms a right-handed triad.",
"Far below $T_{\\rm c}$ Bragg scattering prevails close to the magnetic wave vector $\\mathbf {k}$ thus allowing to approximate $\\sigma _{\\hat{e}_{\\rm out},\\hat{e}_{\\rm in}}(\\mathbf {k}) \\approx \\sigma ^{\\rm mag}_{\\hat{e}_{\\rm out},\\hat{e}_{\\rm in}}(\\mathbf {k})|_{\\rm Bragg}$ .",
"In this case the polarisation matrix simplifies to $\\mathbb {P}_{x,\\beta }(\\mathbf {k}) \\approx -1$ for all $\\beta = x,y,z$ and zero otherwise, where we assumed $\\hat{k} = \\hat{e}_x$ without loss of generality.",
"The experimental values for $\\mathbb {P}_{\\alpha ,\\beta }(\\mathbf {k}_1)$ recorded at 6.5 K accounting for the analyser efficiency are summarised in table REF .",
"The deviation by $\\sim 4\\%$ from purely magnetic Bragg scattering may be attributed to the flip efficiency of the precession coils and non-magnetic scattering by the coating of the precession coils and aluminium in the beam.",
"This brings us finally to the spin-resolved scattering close and above the helimagnetic transition at $T_c \\approx 29$ K. Here the spin-polarisation of the in- and out-going neutron beam were longitudinal to the transferred momentum, $\\sigma ^\\parallel _{\\tau _{\\rm out}, \\tau _{\\rm in}}(\\mathbf {Q}) = \\sigma _{\\tau _{\\rm out} \\hat{Q},\\tau _{\\rm in}\\hat{Q}}(\\mathbf {Q})$ with $\\tau _{\\rm out}, \\tau _{\\rm in} \\in \\lbrace +1,-1\\rbrace \\equiv \\lbrace \\uparrow ,\\downarrow \\rbrace $ .",
"The spin-flip scattering $\\sigma ^\\parallel _{\\pm ,\\mp }(\\mathbf {Q}) = \\sigma ^S_{\\rm inc}(\\mathbf {Q}) + \\sigma ^{S}_{\\rm mag}(\\mathbf {Q}) \\mp \\sigma ^{A}_{\\rm mag}(\\mathbf {Q})$ is then goverened by critical chiral magnetism apart from an incoherent spin-flip background contribution $\\sigma ^S_{\\rm inc}$ .",
"From the theory of chiral magnets [35] one expects the spin-flip scattering to assume the following simple form for $T>T_c$ $ \\sigma ^{S}_{\\rm mag}(\\mathbf {Q}) \\mp \\sigma ^{A}_{\\rm mag}(\\mathbf {Q}) = \\frac{\\mathcal {A}\\, k_B T}{(|\\mathbf {Q}| \\pm k)^2 + \\kappa ^2(T)},$ where $k_B$ is the Boltzmann constant and $\\mathcal {A}$ is a constant that depends on the form factor of MnSi.",
"This expression applies to chiral systems tending to develop a left-handed helix with a pitch vector, $k>0$ .",
"The inverse correlation length, $\\kappa (T)$ , represents the point of contact with the different theoretical proposals of the helimagnetic transition that motivated our study.",
"In particular, for very weak cubic anisotropies in a Brazovskii scenario chiral paramagnons develop isotropically and become soft on a sphere in momentum space as $\\kappa (T) \\rightarrow 0$ , see Eq.",
"(REF ) [11].",
"These chiral paramagnons effectively display an one-dimensional character resulting in strong renormalizations of $\\kappa (T)$ according to the cubic equation $\\kappa ^2/\\kappa _{\\rm Gi}^2 = \\frac{T-T_{\\rm MF}}{\\tilde{T}_0} + \\kappa _{\\rm Gi}/\\kappa $ .",
"For MnSi $T_{\\rm MF} \\approx 30.5$ K is the expected mean-field transition temperature, where $\\tilde{T}_0 \\approx 0.5$ K and $\\kappa _{\\rm Gi} \\approx 0.018$ Å$^{-1}$ is the inverse Ginzburg length [11].",
"In turn, these strongly interacting paramagnons suppress the transition by $\\Delta T = T_{\\rm MF} - T_c \\approx 1.5$ K driving it first order, with characteristic signatures in physical quantities around $T_{\\rm MF}$ .",
"Figure: (a) Temperature dependence of the spin-flip scattering cross section of MnSi σ ±,∓ ∥ (𝐐)\\sigma ^\\parallel _{\\pm ,\\mp }(\\mathbf {Q}) with |𝐐|=k|\\mathbf {Q}| = k close to the critical temperature, T>T c T>T_c, for different orientations specified by the angle φ=0 ∘ ,3 ∘ \\phi = 0^\\circ , 3^\\circ , and 25 ∘ 25^\\circ .",
"For φ=0\\phi =0, Q ^∥[11 ¯1 ¯]\\hat{Q} \\parallel [1 \\bar{1} \\bar{1}] and other orientation are obtained by an anticlockwise rotation by φ\\phi around the [11 ¯2][1 \\bar{1} 2] axis, see also Fig. (c).",
"Data for φ=25 ∘ \\phi = 25^\\circ was corrected for transmission.",
"Panel (b) shows the same data but on a different intensity scale; the dashed line is the fitted incoherent spin-flip background value, σ inc S \\sigma ^S_{\\rm inc}.",
"Panel (c) shows the chiral fraction η\\eta of Eq. ().",
"The solid lines are fits to Brazovskii theory, see text, predicting a turning point of η(T)\\eta (T) at T * T^*.Fig.",
"REF (a) and (b) display the temperature dependence of the spin-flip scattering $\\sigma ^\\parallel _{\\pm \\mp }(\\mathbf {Q})$ measured on a sphere with radius $|\\mathbf {Q}| = k$ for different orientations $\\hat{Q} = \\mathbf {Q}/Q$ .",
"Approaching $T_c$ , the chiral magnetic ordering indeed develops isotropically resulting in a negligible dependence on $\\hat{Q}$ .",
"Here $\\sigma ^\\parallel _{-+}$ reflects the strong $T$ -dependence of $\\kappa $ close to $T_c$ , while $\\sigma ^\\parallel _{+-}$ is barely temperature dependent as it is suppressed by the additional factor $4k^2$ in the denominator of Eq.",
"(REF ).",
"Using the published results for $\\kappa (T)$ and $k\\approx 0.039$ Å$^{-1}$ obtained in the same sample [11], we are left with a single fitting parameter, namely the magnitude $\\mathcal {A}$ in Eq.",
"(REF ), in addition to a temperature and $\\mathbf {Q}$ -independent incoherent background $\\sigma ^S_{\\rm inc}$ shown as the dotted line in Fig.",
"REF (b).",
"We find a remarkably good fit for both cross sections as shown by the solid lines.",
"Subtracting the incoherent background $\\sigma ^S_{\\rm inc}$ determined experimentally, we obtain the chiral fraction $ \\eta \\equiv \\frac{\\sigma ^{A}_{\\rm mag}(\\mathbf {Q})}{\\sigma ^{S}_{\\rm mag}(\\mathbf {Q})}\\Big |_{|\\mathbf {Q}|=k} = \\frac{1}{1 + \\kappa ^2(T)/(2k^2)}$ shown in Fig.",
"REF (c).",
"It is essential to note that the experimental values depend sensitively on $\\sigma ^S_{\\rm inc}$ (likewise the error bars of $\\eta $ derive mainly from $\\sigma ^S_{\\rm inc}$ ).",
"Within the error bars, however, the chiral fraction is in very good agreement with the Brazovskii theory of $\\kappa (T)$ .",
"In particular, $\\eta (T)$ displays a characteristic point of inflection at a temperature $T^*-T_c \\approx 2$ K. It is finally instructive to note that $\\eta (T)$ reported by Pappas et al.",
"[25], [26] differs substantially from up to $\\sim 2\\,{\\rm K}$ above $T_{\\rm c}$ as shown in Fig.",
"REF (c).",
"Based on the information given in Refs.",
"[25], [26] we strongly suspect that this difference is due to an overestimate of $\\sigma ^S_{\\rm inc}$ .",
"In conclusion, we have investigated the critical spin-flip scattering with an emphasis on the chiral fraction $\\eta $ close to the helimagnetic transition in MnSi.",
"For our study we have developed a miniaturised, low-cost SNP device for very fast experiments at scattering angles up to $15^{\\circ }$ .",
"Considering carefully the importance of incoherent background scattering we find excellent quantitative agreement of the temperature dependence of the chiral fraction $\\eta $ at various sample orientations with the Brazovski scenario of a fluctuation-induced first order transition.",
"Our study thereby provides for the first time a quantitative connection of $\\eta $ with elastic neutron scattering as well as the magnetisation, susceptibility and specific heat [11], completing a remarkably comprehensive account in a minimal model that does not require any additional phenomenological parameters.",
"We wish to thank A. Rosch, K. Pappas and S. Grigoriev for helpful discussions.",
"JK acknowledges financial support through the TUM Graduate School.",
"Financial support through DFG TRR80 and ERC-AdG (291079 TOPFIT) are gratefully acknowledged."
]
] | 1403.0551 |
[
[
"Tomography of a multimode quantum black box"
],
[
"Abstract We report a technique for experimental characterization of an $M$-mode quantum optical process, generalizing the single-mode coherent-state quantum-process tomography method [M. Lobino et al., Science 322, 563 (2008); A. Anis and A.I.",
"Lvovsky, New J. Phys.",
"14, 105021 (2012)].",
"By measuring effect of the process on multi-mode coherent states via balanced homodyne tomography, we obtain the process tensor in the Fock basis.",
"This rank-$4M$ tensor, which predicts the effect of the process on an arbitrary density matrix, is iteratively reconstructed directly from the experimental data via the maximum-likelihood method.",
"We demonstrate the capabilities of our method using the example of a beam splitter, reconstructing its process tensor within the subspace spanned by the first three Fock states.",
"In spite of using purely classical probe states, we recover quantum properties of this optical element, in particular the Hong-Ou-Mandel effect."
],
[
"Introduction",
"Precise understanding of the performance of individual quantum systems is a key requirement for the development of compound devices, e.g.",
"quantum computers or secure communication networks.",
"This requirement gives rise to the problem of experimentally characterizing quantum systems as `black boxes: learning to predict their effect on arbitrary quantum states by measuring their effect on a limited number of “probe\" states.",
"The art of solving this problem is referred to as quantum process tomography (QPT).",
"A straightforward approach to QPT consists of measuring the action of the black box on a set of states whose density operators form a spanning set in the space of all operators over a particular Hilbert space.",
"Because any quantum process is a linear map with respect to density operators, this information is sufficient to fully characterize the process [1].",
"However, such a direct method typically requires a large set of difficult-to-prepare probe states, and is consequently restricted to systems of very low dimension.",
"Another possibility is the ancilla-assisted method [2] utilizing an input state that is a part of a fully entangled state in a larger Hilbert space.",
"Although in this case only a single input is necessary thanks to the Jamiolkowski isomorphism [3], both preparation of this state and tomography of the output state is, again, complicated, which dramatically limits the practicality of the method.",
"In application to optics, the coherent-state quantum process tomography (csQPT) [4], [5], [6] offers a practical solution.",
"While being a member of the direct methods family described above, this technique uses only coherent states $|\\alpha \\rangle $ for probing the process $\\mathcal {E}$ , relying on the fact that these states span the space of operators over the optical Hilbert space (the optical equivalence theorem) [7], [8].",
"The prediction for the output $\\mathcal {E}\\left(\\hat{\\rho }\\right)$ of the black box in response to to an arbitrary input state $\\hat{\\rho }$ then involves integration of the measured output states $\\mathcal {E}\\left(|\\alpha \\rangle \\langle \\alpha | \\right)$ , weighted by corresponding Glauber-Sudarshan function $P_{\\hat{\\rho }}(\\alpha )$ , over the phase space.",
"A similar coherent-state based approach can also be used for the tomography of quantum measurements [9], [10].",
"While being a case of the direct method described above, csQPT is relatively easy to implement in an experiment, since coherent states are readily obtained from lasers, and their amplitudes and phases are easy to control.",
"On the other hand, $P_{\\hat{\\rho }}(\\alpha )$ is a generalized function, typically highly singular.",
"Therefore the process reconstruction involving that function may either suffer from inaccuracies or involve an unreasonably large number of required probe states.",
"Moreover, the procedures proposed in Refs.",
"[4], [6] evaluate each element of the process tensor individually, and can hence lead to unphysical (non-trace preserving or non-positive) process tensors.",
"The above shortcomings are absent in a method known as MaxLik csQPT, which exploits the Jamiolkowski isomorphism to reduce the QPT problem to the well-studied problem of the quantum state estimation, and applies the likelihood maximization technique to estimate the process tensor [11].",
"In this way, one can perform the reconstruction without leaving the physically plausible space.",
"MaxLik csQPT has been proposed in Ref.",
"[12] and successfully realized for nondeterministic singe-mode processes [13], [14].",
"In this work, we expand csQPT beyond the “single input — single output” case, which covers only a few of practically relevant quantum optical black boxes.",
"The need for our study is dictated by the growing fields of quantum optical communication and logic, which are impossible without multimode processing.",
"Examples include multimode quantum memories [15], [16] and logic gates for processing photonic qubits [17], [18], to name a few.",
"Although our experiment is in the optical domain, the theory and methodology of csQPT can be employed on a much broader scale.",
"It applies to any physical system whose Hamiltonian is equivalent to that of the harmonic oscillator — such as superconducting cavities, atomic spin ensembles and nanomechanical systems.",
"In all of these, coherent states are the simplest to prepare and are hence most suitable as probe states in QPT."
],
[
"Multimode MaxLik csQPT",
"Our method generalizes the single-mode MaxLik csQPT approach [12], which we briefly outline below.",
"We work in the Fock basis and represent a general $M$ -mode quantum process $\\mathcal {E}$ by a tensor of rank $4M$ which maps the input density matrix into the output one: $\\rho ^{out}_{\\underline{j},\\underline{k}} = \\langle {\\underline{j}|\\mathcal {E}(\\rho ^{in})|\\underline{k}}\\rangle = \\sum _{\\underline{n},\\underline{m}}{\\mathcal {E}^{\\underline{n},\\underline{m}}_{\\underline{j},\\underline{k}}\\rho ^{in}_{\\underline{n},\\underline{m}}},$ where underlined symbols $|\\underline{i}\\rangle =|i_1,\\dots ,i_M\\rangle $ refer to multimode Fock states.",
"In practice, the infinite dimensions of both input $\\mathcal {H}$ and output $\\mathcal {K}$ optical Hilbert spaces are truncated to the $N+1$ lowest Fock states, so that $i_k \\in 0\\dots N$ .",
"In the experiment, the black box is tested with a set of $M$ -mode coherent probe states $|{\\underline{\\alpha }}\\rangle = |\\alpha _1,\\dots ,\\alpha _K\\rangle $ .",
"For every probe state, the output channels are examined by homodyne measurements, which gives a set of quadrature data $\\lbrace \\underline{X}_i, \\underline{\\theta }_i\\rbrace $ , where ${\\underline{\\theta }}_i=(\\theta _{i1},\\dots ,\\theta _{iM})$ is the set of local oscillator (LO) phases associated with the $i$ th measurement.",
"To provide enough information about the process, the probe states should cover the volume of interest in the multimode phase space corresponding to the energies up to the chosen photon truncation number $N$ .",
"Because the mean quadrature variance of the $N$ -photon state equals $N+1/2$ , this volume corresponds to a $2M$ -dimensional hypersphere of radius $\\sqrt{N+1/2}$ .",
"On the other hand, a single multimode set of coherent states corresponds to a hypersphere of radius $\\sqrt{1/2}$ .",
"Therefore the number of the necessary probe states can be estimated as $(2N+1)^M$ .",
"Our process reconstruction method relies on the Jamiolkowski isomorphism, relates the superoperator $\\mathcal {E}$ to an operator $\\hat{E}$ on the product of $\\mathcal {H}$ and $\\mathcal {K}$ spaces: $\\hat{E} = \\sum _{\\underline{n},\\underline{m},\\underline{j},\\underline{k}}{\\mathcal {E}^{\\underline{n},\\underline{m}}_{\\underline{j},\\underline{k}} \\mathinner {|{\\underline{n}}\\rangle } \\mathinner {\\langle {\\underline{m}}|} \\otimes \\mathinner {|{\\underline{j}}\\rangle } \\mathinner {\\langle {\\underline{k}}|}}.$ In this way, the process reconstruction is reduced to a more familiar problem of state reconstruction.",
"The physicality of the process $\\mathcal {E}$ requires it to be completely positive and trace preserving.",
"These conditions are equivalent to the requirement that the corresponding Jamiolkowski operator be positive semidefinite and that $\\mathrm {Tr}_\\mathcal {K} [\\hat{E}] = \\hat{I}_\\mathcal {H}$ , where $\\hat{I}$ is identity operator.",
"The latter condition is readily extended to conditional (trace-reducing) processes as discussed in Refs.",
"[12], [13].",
"The maximum likelihood reconstruction consists of finding an operator $\\hat{E}$ which maximizes the probability of obtaining the harvested data set $\\lbrace \\underline{X}_i, \\underline{\\theta }_i\\rbrace $ .",
"Mathematically, this is equivalent to maximization of the functional $\\mathcal {L}(\\hat{E}) = \\sum \\limits _{i,j} \\ln p(\\alpha _j, i) - \\mathrm {Tr}[\\hat{\\Lambda }\\hat{E}],$ where $\\hat{\\Lambda }$ is Hermitian matrix of Lagrange multipliers incorporating the trace-preservation condition, and $\\begin{split}p(\\alpha , i)&=\\mathrm {Tr}\\left[\\mathcal {E} (\\mathinner {|{\\alpha }\\rangle }\\mathinner {\\langle {\\alpha }|}) \\hat{\\Pi }_{{\\underline{\\theta }}_i}(\\underline{X}_i)\\right] = \\\\&=\\mathrm {Tr}\\left[\\hat{E} \\mathinner {|{\\alpha }\\rangle }\\mathinner {\\langle {\\alpha }|} \\otimes \\hat{\\Pi }_{{\\underline{\\theta }}_i}(\\underline{X}_i) \\right]\\end{split}$ is probability of registering $i$ th outcome for the probe state $\\mathinner {|{\\alpha }\\rangle }$ and $\\hat{\\Pi }_{\\underline{\\theta }_i}(\\underline{X}_i)=\\mathinner {|{\\underline{X}_i, \\underline{\\theta }_i}\\rangle }\\mathinner {\\langle {\\underline{X}_i, \\underline{\\theta }_i}|}$ is the projector corresponding to the $i$ th measurement outcome.",
"For deterministic processes, operator $\\hat{E}$ maximizing the likelihood (REF ) satisfies the extremal condition [12], [11] $\\hat{E} = \\hat{\\Lambda }^{-1} \\hat{R} \\hat{E} \\hat{R} \\hat{\\Lambda }^{-1},$ where $\\hat{R} = \\sum \\limits _{i,j} \\frac{\\mathinner {|{\\alpha _j^*}\\rangle }\\mathinner {\\langle {\\alpha _j^*}|}\\otimes \\hat{\\Pi }_{\\underline{\\theta }_i}(\\underline{X}_i)}{p(\\alpha _j, i)}, \\\\\\hat{\\Lambda } = \\left(\\mathrm {Tr}_\\mathcal {K} \\left[\\hat{R}\\hat{E}\\hat{R}\\right] \\right)^{1/2} \\otimes \\hat{\\mathcal {I}}_\\mathcal {K}.$ Equations (REF )–() can be solved iteratively, starting from an unbiased $\\hat{E}^{(0)}{=}\\hat{\\mathcal {I}}_{\\mathcal {H}\\otimes \\mathcal {K}}/\\mathrm {dim} \\mathcal {K}$ .",
"Due to the Hermitian nature of operators $\\hat{R}$ and $\\hat{\\Lambda }$ , $\\hat{E}$ remains positive semidefinite at each iteration.",
"Together with the trace preservation constraint, this assures physicality of the reconstructed process.",
"The likelihood functional (REF ) is convex over the space of positive semidefinite operators, which eliminates the possibility of the iteration process stopping at a local maximum."
],
[
"Tomography of beam splitting",
"The process of choice for testing the capability of our method is beam splitting.",
"Its paramount importance in quantum optics needs no proof: all linear optical devices (interferometers, waveguide couplers, loss channels, etc.)",
"are equivalent to single beam splitters (BSs) or sets thereof.",
"Any single BS was recently shown to be generator of universal linear optics [19].",
"Accompanied by single photon sources and photon detectors, BSs enable quantum computation [20].",
"In some form, a BS is present in any imaginable optical setting.",
"In addition, the BS Hamiltonian is paramount in interfacing quantum information between harmonic oscillator systems of different nature, e.g.",
"between an electromagnetic mode and either an atomic ensemble [21], or an electromagnetic mode and a nanomechanical oscillator [22].",
"Although the operation of the BS is consistent with classical physics (coherent state inputs lead to coherent state outputs, and vice versa), its response to nonclassical input gives rise to quantum phenomena.",
"A striking example is the Hong-Ou-Mandel effect: when two photons impinge upon a symmetric BS, they appear only in pairs at one of its outputs [23].",
"Our technique reveals this quantum effect in spite of using only classical states in measurements.",
"Figure: Experimental setup.The BS process (encompassed by a green dashed line) is implemented in the polarization basis by an EOM to which a quarter-wave voltage is applied, and subsequent PBS.The input channels of the process are the horizontal and vertical modes of spatial mode 1; the output channels are the horizontal polarization of spatial mode 3 and the vertical polarization of spatial mode 4.The LOs for homodyne measurements are incident onto the PBS in the two polarization modes of spatial mode 2, thereafter emerging in the vertical polarization of spatial mode 3 and the horizontal polarization of spatial mode 4.The BS has previously been characterized by QPT in the role of a Bell-state filter [24] and an amplitude damping channel [25].",
"In both these studies, tomography of the BS as a process on a multimode Hilbert space has been incomplete: limited to a specific photon number subspace of that space.",
"Our technique is free of this shortcoming.",
"It allows one to predict output of the process for any arbitrary Fock states and their superpositions in the input, up to a certain cut-off photon number.",
"Our technique is different from a recently developed methods for characterizing linear optical networks [26] and Gaussian processes [27] in that it makes no assumptions about the content of the black box, in particular about its Gaussianity or linear-optical character.",
"Although for the demonstration we do use a device which is both linear and Gaussian, our approach can be successfully applied to a multimode process of any nature.",
"The light source in our experiment is a mode-locked Ti:Sapphire laser (Coherent Mira 900), which emits pulses at 780 nm with a repetition rate of 76 MHz and a pulse width of $\\sim 1.8$ ps.",
"In order to stabilize and control the relative phases of the inputs and outputs, we realize symmetric beam splitting with respect to the horizontal and vertical polarization modes in the same spatial channel, marked 1 in Fig.",
"REF .",
"The polarizations are mixed using an electrooptical modulator (EOM) with its optical axis oriented at $45^\\circ $ to horizontal and a $\\lambda /4$ voltage applied to it.",
"A polarizing beam splitter (PBS) subsequently separates the output modes spatially for detection.",
"Our black box is thus implemented by combination EOM + PBS.",
"In the Heisenberg picture, this process has the form $\\left[ {\\begin{array}{c}a_1^{out} \\\\a_2^{out} \\\\\\end{array} } \\right]=\\frac{1}{2}\\left[ {\\begin{array}{cc}1 + i & 1 - i \\\\1 - i & 1 + i \\\\\\end{array} } \\right]\\left[ {\\begin{array}{c}a_1^{in} \\\\a_2^{in} \\\\\\end{array} } \\right],$ where $a_{1,2}^{in,out}$ are photon annihilation operators of the input and output modes.",
"The relative amplitudes and phases of the input coherent states are set using a $\\lambda /2 + \\lambda /4$ waveplate pair.",
"The measurement of the output is performed using balanced homodyne detectors (BHDs) [28] in both output channels.",
"To this end, we introduce two LOs in orthogonal polarizations in spatial mode 2, so the central PBS directs them into the two output spatial channels (Fig.",
"REF ).",
"In each output channel of the PBS, we then find the signal and LO in orthogonal polarizations.",
"For homodyne detection, these polarizations are mixed in each channel using a combination of a $\\lambda /2$ plate oriented at $22.5^\\circ $ to the horizontal and an additional PBS.",
"The relative phases the LOs can be controlled by two wave plates, while their common phase is slowly scanned using a piezo-mounted mirror in the signal channel.",
"Figure: Reconstructed (top) and theoretically expected (bottom) process tensor in the Fock space up to N ' =2N^{\\prime }=2.a) Elements of the tensor corresponding to the diagonal elements of the input and output density matrices.Numbers give amplitudes of the non-zero cells.The element 1,1→1,1\\left| 1,1\\right\\rangle \\rightarrow \\left| 1,1\\right\\rangle corresponds to the coincidence probability in a Hong-Ou-Mandel measurement.b) The real (left) and imaginary (right) parts of the full tensor.",
"Each large cell corresponds to a specific element of the input density matrix, while the content of each large cell gives the output density matrix.Insets show the magnified output for the input state 1,1\\left| 1,1\\right\\rangle ."
],
[
"Evaluating the process tensor",
"The process reconstruction is simplified by its invariance with respect to the global phase shift.",
"That is, if both input phases are shifted by some phase $\\theta $ , so will be the output state.",
"This invariance is a consequence of the symmetric nature of time: a global phase shift by $\\theta $ is equivalent to a shift in time by $\\theta /\\omega $ , where $\\omega $ is the optical frequency.",
"If the “black box\" is not connected to any external clock (such as in our case), it will respond to a signal that is shifted in time by the same amount in the output.",
"The effect of phase invariance on the process tensor can be determined from the fact that a phase shift of both modes will transform density matrix elements according to $\\rho ^{in}_{n_1,n_2,m_1,m_2}&\\rightarrow &\\rho ^{in}_{n_1,n_2,m_1,m_2}e^{i\\theta (n_1 + n_2 - m_1 - m_2)}, \\nonumber \\\\\\rho ^{out}_{j_1,j_2,k_1,k_2}&\\rightarrow &\\rho ^{out}_{j_1,j_2,k_1,k_2}e^{i\\theta (j_1 + j_2 -k_1 - k_2 )}.",
"\\nonumber $ Reconciling this with Eq.",
"(REF ), we find that only elements such that $j_1 + j_2 -k_1 - k_2 = n_1 + n_2 - m_1 - m_2$ can be nonzero in tensor $\\mathcal {E}^{\\underline{n},\\underline{m}}_{\\underline{j},\\underline{k}}$ .",
"The process reconstruction requires knowledge of the LO phase vector $\\underline{\\theta }_{i}$ , $i=1,\\ldots ,M$ at each moment in time both for the input and output of the black box.",
"For a general phase-invariant process, this is equivalent to $2M-1$ unknown phase relations.",
"This requirement makes a marked difference between the reconstruction of single-mode and multi-mode processes.",
"In the single-mode case, many relevant processes exhibit intensity-independent phase behavior, which, in combination with the phase invariance, allows one to disregard phase relations between the input and output modes altogether.",
"In multimode processes, however, this is almost always not the case: even in the relatively simple case of the present work, total phase control is essential for successful reconstruction.",
"We acquire the phase vector $\\underline{\\theta }_{i}$ by periodically setting the EOM voltage to zero, so the black box becomes the identity process and the quadrature measurements correspond to the input states.",
"This allows us to monitor all three required phase relations in real time.",
"The inverse sine of the mean quadrature value for each set yields the differences $\\underline{\\theta }_{i}-\\underline{\\theta }_{in,i}$ of the LO and input state phases for both modes.",
"The switching between the BS and identity processes is performed with a period of 0.1 s, which is much faster than the characteristic time of phase fluctuations caused by air movements in the two interferometer channels.",
"In this way, the evaluated LO phases can be translated to the process output measurements by taking into account the linear motion of the piezo.",
"We acquire a total of 48 sets of $10^6$ quadrature samples for three different relative phases of the LOs: $0.67$ , $2.64$ and $5.29$ rad and, in addition to the vacuum, 16 pairs of input coherent states, obtained by setting each waveplate at $0^\\circ , 15^\\circ , 30^\\circ $ and $45^\\circ $ .",
"Each pair of the input states has the same total energy corresponding to $0.9$ photons.",
"This set of probe states is sufficient to reliably reconstruct the process up to a cut-off photon number of 2.",
"We implement a two-step reconstruction process as prescribed by Ref. [12].",
"In the first step, we artificially inflate the reconstruction Hilbert space by choosing the cut-off point at $N=4$ .",
"This is necessary to ensure that both the input probe states and the output states are well accommodated in that space, which is required for the proper function of the reconstruction algorithm.",
"However, the fraction of 3- and 4-photon terms in the Fock decomposition of the probe coherent states is relatively low, and so is their contribution to the log-likelihood functional.",
"As a result, the corresponding terms of the process tensor are not estimated accurately.",
"To eliminate these inaccuracies, we truncate the reconstructed tensor to a lower maximum photon number $N^{\\prime }=2$ after the iterations have been completed [12].",
"The phase invariance property of the process kills about $90\\%$ of $\\approx 4\\times 10^5$ tensor elements.",
"The resulting dimensionality of the optimization space is close to that in the 8-ion tomography done in work [29] and is computationally intensive.",
"The iterative algorithm runs on an Intel Core i7 processor.",
"Paralleled onto 4 of 8 computing cores, each iteration takes about 2 hours.",
"The maximum-likelihood reconstruction algorithm appears to converge at around 100 iterations."
],
[
"Results",
"Fig.",
"REF shows the result of the process reconstruction with $N^{\\prime }=2$ in comparison with the theoretical expectation according to Eq.",
"(REF ) with an additional common phase delay of 0.8 rad.",
"The elements of the process tensor associated with the diagonal elements of the input and output density matrices [Fig.",
"REF (a)] have transparent physical meaning as probabilities of the corresponding transitions.",
"In particular, the Hong-Ou-Mandel effect is represented by the probability of $\\left| 1,1\\right\\rangle \\rightarrow \\left| 1,1\\right\\rangle $ transition, which is zero for ideally symmetrical BS and amounts to $0.01$ in the reconstructed tensor.",
"The data in Fig.",
"REF (a) are only a small fraction of the full tensor shown in Fig.",
"REF (b), which has $\\sim 10^3$ non-zero, generally complex elements.",
"These elements determine the phase behavior of the black box, and are equally important in the description of the process.",
"The left and right columns of the grid present, respectively, the real and imaginary parts of the tensor, while the top and bottom rows correspond to the reconstruction result and the theoretical expectation.",
"The insets in each panel shows the response of our black box to the Hong-Ou-Mandel query, the $\\left| 1,1\\right\\rangle $ input state.",
"The diagonal of the left (real) panel in Fig.",
"REF (b) corresponds to the full panel in Fig.",
"REF (a).",
"To characterize the quality of the reconstructed tensor shown in Fig.",
"REF , we calculate the fidelity between the ideal and reconstructed processes in the Jamiolkowski state representation: $\\mathcal {F} \\left( \\mathcal {E}, \\mathcal {E}_{\\rm est} \\right) = \\mathrm {Tr} \\left[ \\sqrt{ \\sqrt{\\mathcal {E}} \\mathcal {E}_{\\rm est} \\sqrt{\\mathcal {E}} }\\right] = 0.95.$ We perform a few tests to find the source of this non-ideality.",
"First, we quantify the physical imperfections of our black box by fitting the observed phase-dependent mean quadrature data by the theoretical prediction corresponding to an arbitrary BS.",
"We obtain that the power transmittance corresponding to the best fit is 0.502.",
"The fidelity between the processes associated with that slightly asymmetric BS and a symmetric one is 0.998, which shows that the physical errors (at least those which manifest in change of splitting ratio) are insignificant.",
"Second, we evaluate the statistical and systematic errors of the reconstruction using bootstrapping.",
"Specifically, we simulate the quadrature data expected from a model BS and apply the MaxLik reconstruction algorithm to them to calculate a set of tensors $\\mathcal {E}^{\\prime }_{{\\rm est},i}$ .",
"The numbers of simulated data points, the dimensionality of the reconstruction space and the number of algorithm iterations were taken the same as in the real reconstruction procedure.",
"We find $\\mathcal {F} \\left( \\mathcal {E}, \\mathcal {E}^{\\prime }_{{\\rm est},i} \\right)\\sim 0.95$ for all $i$ .",
"Similar values are observed for the pairwise fidelities $\\mathcal {F} \\left( \\mathcal {E}^{\\prime }_{{\\rm est},i}, \\mathcal {E}^{\\prime }_{{\\rm est},j} \\right)$ as well as for the fidelity $\\mathcal {F} \\left(\\overline{ \\mathcal {E}^{\\prime }_{{\\rm est},i}}, \\mathcal {E} \\right)$ between the mean of the bootstrapping tensors and the theoretical one.",
"These statistics show that the experimental fidelity of 0.95 results from both the inaccuracy of the numerical reconstruction algorithm and the statistical error conditioned by the limited amount of experimental data."
],
[
"Summary",
"We presented experimental csQPT reconstruction of the most common multimode optical process, the beam splitter.",
"Our technique can be readily generalized to other processes, other physical systems and scaled up to a higher number of channels and larger state spaces thanks to the simplicity of the required optical measurements and probe state preparation."
],
[
"Acknowledgment",
"We thank Russian Quantum Center for support as well as A. Masalov, D. Mylnikov, A. Bozhenko, and S. Snigirev for helping with the experiment.",
"AKF is a Fellow of the Dynasty foundation.",
"AIL is a CIFAR Fellow."
]
] | 1403.0432 |
[
[
"Disorder-induced Floquet Topological Insulators"
],
[
"Abstract We investigate the possibility of realizing a disorder-induced topological Floquet spectrum in two-dimensional periodically-driven systems.",
"Such a state would be a dynamical realization of the topological Anderson insulator.",
"We establish that a disorder-induced trivial-to-topological transition indeed occurs, and characterize it by computing the disorder averaged Bott index, suitably defined for the time-dependent system.",
"The presence of edge states in the topological state is confirmed by exact numerical time-evolution of wavepackets on the edge of the system.",
"We consider the optimal driving regime for experimentally observing the Floquet-Anderson topological insulator, and discuss its possible realization in photonic lattices."
],
[
"Floquet-Bloch Theory: Definitions",
"Let us start with the Hamiltonian $H(t)$ that is periodic in time, $H({\\bf k},t)&=&H_0({\\bf k})+V(t), \\\\ H({\\bf k},t)&=&H({\\bf k},t+T), \\ \\ {\\rm with} \\ \\ T=2\\pi /\\Omega , \\nonumber $ as the time-period, $\\Omega $ being the frequency.",
"Here, $H_0$ contains the time-independent terms of the Hamiltonian.",
"The states are given by the solution to the full time-dependent Schrödinger equation, $i\\hbar \\frac{\\partial }{\\partial t}\\psi ({\\bf k},t)=H(t) \\psi ({\\bf k},t)$ The Floquet-Bloch theorem states that, the time-evolution operator can be written as $U(t,0)=\\exp \\left(-iH^Ft\\right)W(t),\\ \\ {\\rm with} \\ \\ W(t+T)=W(t), $ and $H^F$ is a time-independent Hermitian operator.",
"The form of Eq.",
"(REF ) allows us to identify $H^F$ as an effective time-independent Floquet Hamiltonian.",
"In order to define $H^F$ , for the case at hand, the Fourier decomposition of the solution to Eq.",
"(REF ) is used, $\\psi ({\\bf k},t)&=&\\sum _n \\psi _n({\\bf k}) e^{i n \\Omega t},\\\\&=&\\sum _n \\langle n|\\psi ^F\\rangle \\langle t|n\\rangle , \\ \\ {\\rm with}\\ \\ \\langle t|n\\rangle =e^{i n \\Omega t}.",
"$ In Eq.",
"(), we have introduced an additional register particle, $\\lbrace |n\\rangle \\rbrace $ , where $n \\in {Z}$ .",
"The Floquet Hamiltonian, $H^F$ , is defined in a way such that $|\\psi ^F\\rangle $ are eigenstates.",
"Necessarily, it is defined in an extended Hilbert space ${H} \\otimes \\lbrace |n\\rangle \\rbrace $ , where ${H}$ is the original Hilbert space of the Hamiltonian (see Eq.",
"(REF )).",
"The time-dependent Schrödinger equation is rewritten in an effective time-independent form, $H^F|\\psi ^F\\rangle =\\epsilon |\\psi ^F \\rangle ,$ where $H^F$ is infinite dimensional.",
"The eigenvalues ($\\epsilon $ ) are referred to as quasi-energies, and the eigenfunctions of the Floquet Hamiltonian, defined in Eq.",
"(REF ), are the quasi-energy states.",
"The spectrum of $H^F$ is unbounded; however, we note that in Eq.",
"(REF ), the eigenvalues ($\\epsilon $ ) of $H^F$ describe the non-periodic evolution of these states as a function of time.",
"Therefore, they are unique modulo $\\Omega $ , $\\epsilon \\equiv \\epsilon +m\\Omega $ .",
"The explicit form of the Floquet Hamiltonian for $H(t)$ defined in Eq.",
"(REF ) is, $& (H^F({\\bf k}))_{mn}&\\equiv \\langle m|H^F({\\bf k})|n\\rangle , \\nonumber \\\\& &=(H_0({\\bf k})+n\\Omega ) \\delta _{mn}+\\tilde{V}_{mn}, \\\\{\\rm where,} & & \\nonumber \\\\&\\tilde{V}_{mn}&= \\frac{1}{T} \\int _0^T dt V(t)e^{i(m-n)\\Omega t }$ The integers $m$ and $n$ indexes a particular Floquet block in the matrix $H^F$ .",
"In this representation, the time-independent terms, like $H_0$ , are diagonal, but the time-dependent potential, $V(t)$ , acts as a hopping amplitude between various Floquet blocks.",
"These Floquet blocks are like replicas of the original Hamiltonian shifted in quasi-energy by $\\Omega $ , and the indices will also be referred as the replica index.",
"The quasi-energies can be computed by truncating the matrix after a certain number of Floquet blocks and diagonalizing it.",
"The Floquet Green function is defined as $G^F=\\frac{1}{(E{1}-H^F)}.",
"$ All elements of $G_F$ can be rewritten in a closed analytical formula [3] for the special case where the only non-zero components of $\\tilde{V}_{mn}$ are $V_-=\\tilde{V}_{m+1,m}$ and $V_+=\\tilde{V}_{m,m+1}$ with $m \\in {Z}$ .",
"We mostly restrict ourselves to the $(0,0)$ Floquet block, $& &(G_F)_{00}=\\frac{1}{E{1}-H_0-V_{\\rm eff}^+-V_{\\rm eff}^-} , \\\\& &V_{\\rm eff}^\\pm =V_+\\frac{1}{E\\pm \\Omega -H_0-V_+\\frac{1}{E\\pm 2 \\Omega -H_0-V_+\\frac{1}{\\vdots } V_-}V_-}V_-.\\nonumber $ The Green function can be obtained perturbatively to any order in $V$ by truncating the continued fraction at that order."
],
[
"Floquet Topological Insulators: Haldane model and higher Chern insulators",
"The topological behavior in the non-equilibrium situation is obtained by choosing a drive of appropriate frequency.",
"We show the non-trivial topology of the quasi-energy band-structure for the graphene based model in the presence of circularly polarized light.",
"The tight-binding model on a hexagonal lattice with nearest neighbor hopping and without radiation, in the low energy and linearized momentum regime reduces to $H_0=v_F(k_x\\sigma _z\\tau _z+k_y\\sigma _y)+M\\sigma _z $ where $\\sigma _x$ and $\\tau _z$ refer to sub-lattice isospin and valley degree of freedom respectively, $v_F$ is the Fermi velocity at the Dirac points and $M$ is the sub-lattice mass term.",
"In the presence of circularly polarized light, using Pierels substitution, we have $H(t) &=& v_F((k_x-A_x)\\sigma _x\\tau _z+(k_y-A_y)\\sigma _y) +M\\sigma _z \\\\{\\bf A}(t)&=&A_0(sin(\\Omega t),cos(\\Omega t)) \\nonumber $ where ${\\bf A}$ is the vector potential for incident radiation.",
"Consider a general form of the external drive defined in Eq.",
"(REF ), $V(t)=V_+e^{i\\Omega t}+V_-e^{-i\\Omega t}, $ where $V_\\pm $ are time-independent operators.",
"Therefore, in the model considered, we have, $V_+&=&A_0\\left(\\frac{i}{2}\\sigma _x\\tau _z-\\frac{1}{2}\\sigma _y \\right), \\\\V_-&=&V_+^\\dag .",
"$ Note that the analysis discussed here (see Eq.",
"(REF ) to Eq.",
"()) is valid only in the perturbative low energy regime with $|{\\bf A}|\\ll 1$ .",
"This model breaks time-reversal symmetry, and is classified by the Chern number.",
"We explore two cases, (a) zero resonances, and (b) a single resonance due to radiation and their effects on the topology."
],
[
"(A) No resonances",
"This case corresponds to irradiating the system with off-resonant light.",
"The incident frequency of the drive, $\\Omega \\gg W$ , where $W$ is the bandwidth of the time-independent band-structure.",
"The correction to the energies of the non-equilibrium states are obtained by inspecting the poles of the Floquet Green function.",
"In this case, to lowest order in the radiation potential, the off-diagonal terms in $G^F$ can be ignored.",
"The diagonal element, $G^F_{00}$ , to $O(V^2)$ is, $G^F_{00}=\\left(E{1}-H_0-\\frac{[V_+,V_-]}{\\Omega }\\right)^{-1}=(E{1}-H_{\\rm eff})^{-1}$ where we have a new effective Hamiltonian, $H_{\\rm eff}$ .",
"Using equations (REF ), () and (REF ), we note that $H_{\\rm eff}$ is equivalent to the Haldane model for anomalous quantum Hall effect with a topological mass $\\Delta _0=\\frac{v_F^2 A_0^2}{\\Omega }$ , $& & H_{\\rm eff}=v_F(k_x \\sigma _x \\tau _z +k_y \\sigma _y)+M\\sigma _z+\\Delta _0\\sigma _z\\tau _z \\\\&=&\\left(\\begin{array}{cccc}\\Delta _+ & k_x-ik_y & 0 & 0\\\\k_x+ik_y& -\\Delta _+ & 0&0 \\\\0 & 0 & \\Delta _-& -k_x-ik_y\\\\0 & 0 & -k_x+ik_y& -\\Delta _-\\end{array}\\right),$ where $\\tau $ denotes the valley space, and $\\Delta _{\\pm }=M\\pm \\Delta _0$ .",
"The mass gap opens at the Dirac points of the band-structure near $\\epsilon =0$ .",
"The bands will be topological or trivial when $M<\\Delta _0 $ and $M>\\Delta _0$ respectively.",
"Specifically, for $M<\\Delta _0$ , the Chern number, $C_n=1$ , when measured at quasi-energies in the gap, $-(\\Delta _0-M)<\\epsilon <\\Delta _0-M$ , and is zero at all other quasi-energies."
],
[
"(B) Single resonance",
"This scenario corresponds to the driving frequency in the regime $W/2<\\Omega <W$ .",
"The quasi-energy band-structure has two gaps at (i)$\\epsilon =0$ , and (ii) $\\epsilon =\\pm \\Omega /2$ , where the topologically non-trivial features may be measured.",
"The gap at $\\epsilon =0$ is the same as that discussed in case (A) and is equal to $\\Delta _0$ .",
"We incorporate the effect of off-resonant processes on the quasi-energy band-structure by making the replacement $H_0 \\rightarrow H_{\\rm eff}$ in the Floquet Hamiltonian defined in Eq.",
"(REF ).",
"For quasi-energies close to resonance, $\\epsilon \\sim \\Omega /2$ , adjacent diagonal Floquet blocks, $H_{\\rm eff}$ , and $H_{\\rm eff}-\\Omega $ , are nearly degenerate.",
"Therefore, to lowest order, $H_F$ must be diagonalized in this subspace of two adjacent Floquet blocks, to obtain the correct quasi-energies.",
"The effective two band Hamiltonian is given by, $(H^F)_{\\rm eff}= P_\\Omega \\left(\\begin{array}{cc}H_{\\rm eff}-\\Omega & V_+ \\\\V_- & H_{\\rm eff} \\end{array} \\right)P_\\Omega ,$ where $P_\\Omega $ is the projector onto the bands with quasi-energies in the range $0<\\epsilon <\\Omega $ .",
"This is exactly the same as degenerate first order perturbation theory, and therefore, the gap exactly at resonance, $\\epsilon =\\Omega /2$ , is proportional to $|V_\\pm |$ .",
"The quasi-energies of $H^F$ are periodic in $\\Omega $ .",
"To properly define a the Chern number for a band ($C_n$ ), we must specify its upper and lower bound in quasi-energies.",
"An alternative is to measure the Chern number ($C_n^{\\rm trunc}$ ) of all bands below a particular quasi-energy, for a truncated $H^F$ .",
"It has been shown [2] that this number corresponds to the number of edge states that will be observed at that particular quasi-energy irrespective of chirality.",
"For the case of single resonance, the Chern number of the truncated $H^F$ for $M<\\Delta _0$ is $C_n^{\\rm trun}={\\left\\lbrace \\begin{array}{ll}1 &\\mbox{if } \\epsilon =0 \\\\2 & \\mbox{if } \\epsilon =\\pm \\Omega /2,\\end{array}\\right.",
"}$ and for $M>\\Delta _0$ $C_n^{\\rm trun}={\\left\\lbrace \\begin{array}{ll}0 &\\mbox{if } \\epsilon =0 \\\\2 & \\mbox{if } \\epsilon =\\pm \\Omega /2,\\end{array}\\right.",
"}$ The Chern number of the bands are $C_n=\\pm 3$ when $M<\\Delta _0$ , and $C_n=\\pm 2$ when $M>\\Delta _0$ ."
],
[
"Born Approximation: Details",
"The transition from a trivial state to a topological state is due to renormalization of parameters of the Hamiltonian due to disorder.",
"In the lowest order Born approximation, the correction to the density of states are obtained from exact analytical expressions for the self energy.",
"This provides an accurate description for the density of states as a function of disorder at dilute disorder.",
"The disorder averaged Floquet Green function is given by, $\\langle G^F(i\\omega _n,{\\bf k}) \\rangle = \\frac{1}{i\\omega _n-H^F({\\bf k})-\\Sigma (E)},$ and, $\\Sigma (i\\omega _n,{\\bf k})=\\int _{{\\rm FBZ}} d{\\bf k^{\\prime }}\\ \\ \\langle U_{\\rm dis}({\\bf k},{\\bf k^{\\prime }})G^F(i\\omega _n,{\\bf k^{\\prime }})U_{\\rm dis}({\\bf k^{\\prime }},{\\bf k})\\rangle ,$ where, $\\langle \\dots \\rangle $ denotes disorder averaging, $U_{\\rm dis}({\\bf k},{\\bf k^{\\prime }})$ is the disorder potential in Fourier space, and $\\Sigma $ is the self energy.",
"We are interested at the physics of the topological transition near $\\epsilon =0$ as a function of disorder.",
"For the case of zero resonances, this is correctly modeled by the effective Hamiltonian, $H_{\\rm {eff}}$ , defined in Eq.",
"(REF ).",
"Therefore, instead of using the Floquet Green function, $G^F$ , we use the effective Green function given by, $G_0^{\\rm eff}(i\\omega _n, {\\bf k})=\\frac{1}{i\\omega _n-H_{\\rm eff}({\\bf k})}.$ The disorder potential, $U_{\\rm dis}$ , is modeled as $\\delta $ -correlated point scatterers.",
"The short range of scattering implies that both inter- and intra-valley processes must be taken into account.",
"It is assumed that, in the linearized regime, the disorder matrix in real space is [4], $U_{\\rm dis}(\\vec{r})=\\sum _{i}\\left(\\begin{array}{cccc}U_{i}^{A} & 0 & U_{i}^{A}e^{i\\phi _{i}^{A}} & 0\\\\0 & U_{i}^{B} & 0 & U_{i}^{B}e^{i\\phi _{i}^{B}}\\\\U_{i}^{A}e^{-i\\phi _{i}^{A}} & 0 & U_{i}^{A} & 0\\\\0 & U_{i}^{B}e^{-i\\phi _{i}^{B}} & 0 & U_{i}^{B}\\end{array}\\right)$ where, $U_{i}^{A,B}&=&u_{i}^{A,B}\\delta ({\\bf r}-{\\bf r}_{i}^{A,B}), \\\\\\phi _{i}^{A,B}&=&({\\bf K}^{\\prime }-{\\bf K})\\cdot {\\bf r}_{i}^{A,B}.$ $A$ and $B$ refer to the different sub-lattices, ${\\bf K}$ and ${\\bf K^{\\prime }}$ are the two valleys, and $i$ is summed over the unit cells.",
"The disorder potentials $u_i^{A,B}$ are taken from an uniform distribution in the range $[-U_0/2,U_0/2]$ and are $\\delta $ -correlated.",
"Therefore, $\\langle u_{i}^{A}\\rangle &=&\\langle u_{i}^{B}\\rangle =0, \\\\\\langle u_{i}^{\\nu }u_{j}^{\\nu ^{\\prime }}\\rangle &=&\\frac{U_0^2}{12}\\delta _{ij}\\delta _{\\nu \\nu ^{\\prime }}, \\ \\ \\nu ,\\nu ^{\\prime }\\equiv A,B,$ where we have used that the variance of the uniform distribution is $U_0^2/12$ .",
"The diagonal and off-diagonal terms in Eq.",
"(REF ) account for intra- and inter-valley scattering respectively and are assumed to have the same magnitude.",
"The self energy can be calculated by rewriting $U_{\\rm dis}$ (see Eq.",
"(REF )) in Fourier space and using Eq.",
"(REF ) .",
"In the limit of $|{\\bf k^{\\prime }}|\\ll |{\\bf K}-{\\bf K^{\\prime }}|$ , it can be assumed that, the fast oscillating exponents in the off-diagonal terms in the self energy Eq.",
"(REF ) averages to zero [4], i.e., $\\langle \\sum _i e^{i ({\\bf k^{\\prime }}.r_i^\\nu \\pm \\phi _i^\\nu )}\\rangle =0.$ Therefore, the self energy is diagonal in valley space and independent of momentum ${\\bf k}$ .",
"Consequently, after integrating out the momentum, ${\\bf k^{\\prime }}$ , in the first Brillouin zone, the four main contribution to the self energy are, $\\Sigma =\\Sigma _I {I}+\\Sigma _M\\sigma _z+ \\Sigma _\\Delta \\sigma _z\\tau _z+\\Sigma _0 \\tau _z,$ with, $&\\Sigma _I& =-nu^{2}\\frac{i\\omega _{n}}{4\\pi v_{F}^{2}}\\log \\left(\\frac{v_{F}^{4}D^{4}}{f_+f_-}\\right),\\\\&\\Sigma _M& = -\\frac{nu^{2}}{4\\pi v_{F}^{2}}\\left[M\\log \\left(\\frac{v_{F}^{4}D^{4}}{f_+f_-}\\right)+\\Delta \\log \\left(\\frac{f_-}{f_+}\\right)\\right], \\\\&\\Sigma _\\Delta & = \\Sigma _0=0, $ where $f_\\pm =\\omega _{n}^{2}+(M\\pm \\Delta _0)^{2}$ .",
"Therefore, the parameters in $H_{\\rm eff}(t)$ get renormalized as $i\\tilde{\\omega }_{n} &=& i\\omega _{n}-\\Sigma _{0}^{I}, \\\\\\tilde{M} &=& M+\\Sigma _{0}^{z}, \\\\{\\rm and},\\ \\ \\tilde{\\Delta _0} &=& \\Delta _0.$ Figure: The expected quasi-energy gap as a function of disorder given by the Born approximation.",
"This is obtained by plotting the solution to the ω ˜=M ˜-Δ 0 ˜\\tilde{\\omega }=\\tilde{M}-\\tilde{\\Delta _0} as a function of disorder.The parameters for the system are A 0 =1.434A_0=1.434, Δ 0 =0.75\\Delta _0=0.75 and M=0.85M=0.85.The renormalized mass, $\\tilde{M}$ , reduces with increasing disorder.",
"The renormalized quasi-energy is obtained by analytical continuation of $i\\omega _n \\rightarrow \\omega $ and the band gap as a function of disorder is the solution to the equation $\\tilde{\\omega }=\\tilde{M}-\\tilde{\\Delta _0}$ .",
"This is shown in Fig.",
"(REF ) with parameters $v_f=3/2$ , $\\Delta =0.75$ $M=0.85$ and $D=4\\pi /3$ .",
"These parameters correspond to the case (I) of zero resonances.",
"The topological phase transition occurs at the point where the band gap vanishes, which happens when $\\tilde{M}=\\tilde{\\Delta }$ .",
"For stronger disorder, the gap reopens in the topological phase and a non-vanishing Chern number must therefore be measured at the quasi- energies in the gap.",
"Figure: Finite size effect of the disorder averaged Bott index at disorder strength, U 0 =3.5U_0=3.5, for the zero resonance case,as a function of quasi-energy for sizesL x =L y =20L_x=L_y=20, 30, 40 and 50.",
"The index has been averaged over 400 disorder realizations."
],
[
"Bott Index for Floquet Hamiltonian.",
"We outline the method to obtain the Chern number of bands for disordered periodically driven Hamiltonians.",
"The Bott index as a measure to obtain the Chern number, was defined by Hastings and Loring [1] for time-independent Hamiltonian.",
"We generalize this formula for periodically driven systems by using the eigenstates of the truncated Floquet Hamiltonian, $H^F$ .",
"This index will measure the number of edge states at a given quasi-energy[2].",
"Consider a Hamiltonian, $H(t)$ defined on a lattice with periodic boundary conditions (torus geometry).",
"Given two diagonal matrices $X_{ij}=x\\delta _{ij}$ and $Y=y\\delta _{ij}$ for the $x$ and $y$ coordinates of the lattice sites, let us define two unitary matrices, $U_X&=& \\exp (i2\\pi X/L_x),\\\\U_Y &=& \\exp (i2\\pi Y/L_y),$ where $L_{x,y}$ are the dimensions of the system.",
"In the extended Floquet Hilbert space, the analogous definition for the unitary matrices are, $\\left(U^F_X\\right)_{mn}&=& U_X \\delta _{mn},\\\\\\left(U^F_Y\\right)_{mn} &=& U_Y \\delta _{mn},$ where $(m,n)$ refer to a particular Floquet block.",
"For a band of quasi-energies, $\\epsilon _l<\\epsilon <\\epsilon _h$ , the Bott index is an integer, and it is well defined as long as the lower and upper bounds, $\\epsilon _{l,h}$ , are in a mobility gap of the quasi-energy band-structure.",
"The topological invariant is calculated using the unitary matrices projected onto a band.",
"Let $P$ be the projector onto the chosen band of quasi-energies.",
"In our system, we will be calculating the Bott index of all states with quasi-energies, $\\epsilon <0$ , in the truncated Floquet Hamiltonian, $H^F$ .",
"The projected unitary matrices are defined as, $\\tilde{U}^F_{X,Y}= PU^F_{X,Y}P.$ For a given disorder configuration, the Bott index of the band is given as, $C_b=\\frac{1}{2\\pi }{\\rm Im}\\left[{\\rm Tr}\\left(\\log \\left(\\tilde{U}^F_{Y}\\tilde{U}^F_{X}\\tilde{U}^{F\\dag }_{Y} \\tilde{U}^{F\\dag }_{X} \\right)\\right)\\right].$ The Bott index is a measure of commutativity of these projected unitary matrices, and it can be shown to be equivalent to the Kubo formula for the Hall conductivity[1].",
"For a given disorder strength, the Bott index must be averaged over a large number of configurations."
],
[
"Finite Size effect",
"We investigate the finite size effect on the non-quantized region of the Bott index when in the Floquet-Anderson topological insulator (FATI) phase.",
"In Fig.",
"REF , we have plotted the disorder averaged Bott index as a function of quasi-energy for different system sizes.",
"It is clear that with increasing system size, the non-quantized region of the Bott index becomes sharper.",
"This is in agreement with the expectation of a sharp extended state in the bulk quasi-energy band, analogous to a quantum hall state.",
"Therefore, we expect the localization transition to be in the quantum Hall universality class.",
"The current accessible system size is not sufficient to obtain the critical exponents of this transition."
],
[
" Obtaining experimental parameters associated with the FTAI",
"The equation describing the paraxial diffraction of light through an array of waveguides is a Schrödinger equation [5], [6]: $i\\partial _z \\psi = -\\frac{1}{2k}\\nabla _\\perp ^2\\psi - \\frac{k\\Delta n (x,y,z)}{n_0} \\psi ,$ where $z$ is the distance of propagation along the waveguide axis; $k$ is the ambient wavenumber in the medium; $\\nabla ^2_\\perp $ is the Laplacian in the transverse (x,y) plane; $n_0$ is the refractive index of the ambient medium, and $\\Delta n(x,y,z)$ is the refractive index variation as a function of position that describes the waveguides.",
"Each waveguide in the lattice is best described by a hypergaussian function, where the refractive index pattern (for a single straight waveguide) can be written as $\\Delta n^{(1)}(x,y,z) =\\Delta n_1 \\exp \\left( -[(x/\\sigma _x)^2 + (y/\\sigma _y)^2]^3 \\right)$ .",
"The fabrication procedure referred to in the main text [26] allows for a range of values for $\\Delta n_1$ , depending on the speed of laser writing - taking values between $0.5\\times 10^{-3}$ and $1.1\\times 10^{-3}$ .",
"The writing procedure leads to waveguides that are elliptical in shape (because the size of the focus of the writing beam in the transverse direction can be made much smaller than that in the longitudinal direction), with $\\sigma _x \\sim 2\\mu m$ and $\\sigma _y \\sim 5.5\\mu m$ .",
"In order to calculate coupling parameters between waveguides, a plane-wave expansion procedure [6], [7] is used in order to exactly numerically diagonalize Eq.",
"(REF ).",
"This procedure is carried out for both a single waveguide and for two waveguides spaced at a distance $d$ from one another.",
"The former allows the calculation of the on-site energy (or “propagation constant” in optics terminology), which is associated with the chosen refractive index of the waveguide: this is just the eigenvalue associated with the bound mode of a single-mode waveguide.",
"The latter allows for the calculation of the evanescent coupling/hopping constant between two neighboring waveguides: this is just half the splitting between the eigenvalues of the two bound modes associated with the coupled waveguides.",
"In order to fully examine the realizability of the experimental setup, we calculate - using standard numerical techniques [6] - that the hopping parameter can be tuned over an extremely large range ($0.083 cm^{-1}$ through $2.7 cm^{-1}$ ) because the nature of the coupling between adjacent waveguides is evanescent (these values correspond to the waveguides discussed in Ref.",
"[5], at lattice spacings $30\\mu m$ and $12 \\mu m$ , respectively).",
"Furthermore, the on-site energies may be varied significantly by varying the refractive index difference of the waveguides relative to the background (which can be realistically varied in the range of $5.0 \\times 10^{-4}$ through $1.1 \\times 10^{-3}$ - as discussed above).",
"Assuming a waveguide helix pitch of $1cm$ as in Ref.",
"[5], the parameter $U_0/\\Omega $ may vary over a range of $\\sim \\pm 1.8$ - this incorporates the full range of parameters discussed in the main section of this paper.",
"For a typical hopping parameter of $t_1=1.5cm^{-1}$ , the dimensionless parameter $U_0/t$ (the degree of on-site disorder in units of the hopping) can take on values anywhere from $U_0/t = 0$ through $1.6$ - again, this includes the range discussed in the present paper.",
"It is important to demonstrate that the strengths of the gauge field, $A_0$ , as used here are directly realizable under experimental conditions.",
"As shown in Ref.",
"[5], the dimensionless expression for the strength of the gauge field is $A_0=kR\\Omega a$ , where $k$ is the wavenumber in the ambient medium ($k=2\\pi n_0/\\lambda $ , where $n_0=1.45$ is the refractive index of fused silica, and $\\lambda =0.633\\mu m$ is the wavelength of laser light used); $R$ is the radius of the waveguide helices; $\\Omega =2\\pi /1cm$ is the spatial frequency of the helices, and $a=15\\mu m$ is the lattice spacing between nearest neighbor waveguides.",
"In the experimental work [5], the helix radius was tuned (with a constant helix pitch of $1cm$ ) from $0\\mu m$ through $16\\mu m$ .",
"This corresponds to a gauge field strength (in the dimensionless units of the present paper) of $A_0=0$ through $A_0=2.17$ .",
"The experimental work showed that this range was fully accessible experimentally.",
"In the present paper, we perform calculations at a number of different gauge field strengths, including 0.28, 0.48, 0.9 and 1.43 (for the calculations shown in Fig.",
"2); 1.43 (for the calculations in Fig.",
"3); and 0.75 (for the calculations in Fig.",
"4).",
"All of these are clearly experimentally accessible.",
"Taken together, we have shown here that the parameters proposed in this work are directly amenable to experimental realization."
]
] | 1403.0592 |
[
[
"Bayesian search for low-mass planets around nearby M dwarfs. Estimates\n for occurrence rate based on global detectability statistics"
],
[
"Abstract Due to their higher planet-star mass-ratios, M dwarfs are the easiest targets for detection of low-mass planets orbiting nearby stars using Doppler spectroscopy.",
"Furthermore, because of their low masses and luminosities, Doppler measurements enable the detection of low-mass planets in their habitable zones that correspond to closer orbits than for Solar-type stars.",
"We re-analyse literature UVES radial velocities of 41 nearby M dwarfs in a combination with new velocities obtained from publicly available spectra from the HARPS-ESO spectrograph of these stars in an attempt to constrain any low-amplitude Keplerian signals.",
"We apply Bayesian signal detection criteria, together with posterior sampling techniques, in combination with noise models that take into account correlations in the data and obtain estimates for the number of planet candidates in the sample.",
"More generally, we use the estimated detection probability function to calculate the occurrence rate of low-mass planets around nearby M dwarfs.",
"We report eight new planet candidates in the sample (orbiting GJ 27.1, GJ 160.2, GJ 180, GJ 229, GJ 422, and GJ 682), including two new multiplanet systems, and confirm two previously known candidates in the GJ 433 system based on detections of Keplerian signals in the combined UVES and HARPS radial velocity data that cannot be explained by periodic and/or quasiperiodic phenomena related to stellar activities.",
"Finally, we use the estimated detection probability function to calculate the occurrence rate of low-mass planets around nearby M dwarfs.",
"According to our results, M dwarfs are hosts to an abundance of low-mass planets and the occurrence rate of planets less massive than 10 M$_{\\oplus}$ is of the order of one planet per star, possibly even greater.",
"..."
],
[
"Introduction",
"In recent years, planets have been discovered around the least massive stars, M dwarfs, in a diversity of different configurations with widely varying orbital properties and masses [19], [11].",
"For instance, there are several high-multiplicity systems around M dwarfs consisting of only low-mass planets that can be referred to as super-Earths or Neptunes, such as those orbiting GJ 581 [10], [73], [50]We note that the number of planets around GJ 581 is uncertain with different authors reporting different numbers from three to six [75], [76], [30], [67], [7]., GJ 667C [1], [2], [16], and GJ 163 [12], [69].",
"Recent precision velocity surveys have also revealed the existence of more massive planetary companions orbiting nearby M dwarfs [61], [4] showing that such companions do exist, but not in abundance [11], [53], and are less common than for K, G, and F stars [19].",
"However, the most interesting planetary companions around these stars are the low-mass ones that orbit their hosts with such separations that, under certain assumptions regarding atmospheric properties, they can be estimated to enable the existence of water in its liquid form on the planetary surfaces [64], [43].",
"Planets of this type – sometimes called habitable-zone super-Earths – are easier to detect around M dwarfs than around more massive stars because the planet-star mass-ratios give rise to signals with sufficiently high amplitudes, and the shorter orbital periods allow for more orbital phases to be sampled in data covering a fixed length of time, to enable their detections [50], [2], [69].",
"Recently, accurate estimates for the occurrence rate of planets in the Kepler's field have been reported in several studies [34], [18], [55].",
"One of the most interesting features in the Kepler sample is that the occurrence rate of planets around stars appears to increase from roughly 0.05 planets per star around F2 stars to 0.3 per star around M0 dwarfs [34], although the functional form of this relation is far from certain.",
"This increase applies to planets with orbital periods below 50 days because of the available baseline of the Kepler data.",
"While Kepler will be able to provide occurrence rates for longer orbital periods, possibly up to 200-300 days, radial velocity surveys will be needed to probe the occurrence rate of planets on orbits longer than that.",
"Moreover, unlike planets around more massive K, G, and F stars that have been targeted by the Kepler space-telescope in abundance, M dwarfs are not bright enough to be found in comparable numbers in the Kepler's field, which makes it difficult to estimate the occurrence rates and statistical properties of planets around such stars in detail.",
"According to [18], the Kepler's sample contains 3897 stars with estimated effective temperatures below 4000 K, out of which 64 are planet candidate host stars with a total of 95 candidate planets orbiting them.",
"[18] concluded that with periods ($P$ ) less than 50 days, the occurrence rate of planets with radii between 0.5 R$_{\\oplus } < r_{p} < 4 $ R$_{\\oplus }$ is 0.90$^{+0.04}_{-0.03}$ planets per star; with radii between 0.5 R$_{\\oplus } < r_{p} <$ 1.4 R$_{\\oplus }$ is 0.51$^{+0.13}_{-0.06}$ planets per star, although this estimate might be underestimated as much as by a factor of two [55]; and that the occurrence rate of planets with $r_{p} > 1.4$ R$_{\\oplus }$ decreases as a function of decreasing stellar temperature.",
"Furthermore, the occurrence rate of planets appears to decrease heavily between 2 and 4 R$_{\\oplus }$ , which is indicative of overabundance of planets with low radii and therefore low masses [55].",
"These findings challenge the results obtained using radial velocity surveys that should be able to detect planets with similar statistics, although the comparison with Kepler's results is difficult due to the challenges in comparing populations described in terms of planetary radii and minimum masses in the absense of accurate population models for planetary compositions and therefore densities.",
"The estimates based on transits detected by using the Kepler telescope might also be contaminated by a false positive rate of $\\sim $ 10% due to astrophysical effects such as stellar binaries in the background [54], [26].",
"Far fewer planets around M dwarfs are known from radial velocity surveys of such stars [11].",
"However, the ones that are known are among the richest and the most interesting extrasolar planetary systems in terms of numbers of planets, their orbital spacing and dynamical packedness, and their low masses [50], [61], [4], [2], [69].",
"To a certain extent, this lack of known planets around M dwarfs is due to observational biases arising from the fact that early radial velocity surveys did not target low-mass stars because of the difficulties in obtaining sufficiently high signal-to-noise observations due to a lack of photons in the V band to enable high quality radial velocity measurements.",
"Another reason was that – based on a sample size of unity – Solar-type stars were considered more promising hosts to planetary systems.",
"This observational bias is also likely caused by the fact that – in comparison with stars of the spectral classes F, G, and K – massive giant planets are not as abundant around M dwarfs [11], and the planets that exist, if they indeed do exist, are likely so small that they induce radial velocity signals that have amplitudes comparable to the current high-precision measurement noise levels, which makes their detection difficult at best.",
"[11] reported estimates for the occurrence rates of planetary companions orbiting M dwarfs based on radial velocity measurements obtained by using the High Accuracy Radial velocity Planet Searcher (HARPS) spectrograph.",
"According to their results, super-Earths with minimum masses between 1 and 10 M$_{\\oplus }$ are abundant around M dwarfs with an occurrence rate of 0.36$^{+0.25}_{-0.10}$ for periods between 1 and 10 days and 0.52$^{+0.50}_{-0.16}$ for periods between 10 and 100 days, respectively.",
"Furhtermore, they reported an estimate for the occurrence rate of super-Earths in the habitable zones (HZs) of M dwarfs of 0.41$^{+0.54}_{-0.13}$ planets per star.",
"M dwarfs are the most abundant type of stars in the Solar neighbourhood.",
"Therefore, the occurrence rate of planets around these stars will dominate any general estimates of the occurrence rate of planets.",
"For this reason, we re-analyse the radial velocities obtained using the Ultraviolet and Visual Echelle Spectrograph (UVES) at VLT-UT2 of a sample of M dwarfs of [79] using posterior sampling techniques in our Bayesian search for planetary signals.",
"We also extract HARPS radial velocities for these stars from the publicly available spectra in the European Southern Observatory (ESO) archive and analyse the combined UVES and HARPS velocities.",
"The methods are presented in Section in detail and we show the results based on combined HARPS and UVES data in Section .",
"We present the statistics of the new planet candidates we detect and compare the obtained occurrence rates to other planet surveys targeting M dwarfs in Section , describe some of the interesting new planetary systems and the evidence in favour of their existence in greater detail in Section , and discuss the results in Section ."
],
[
"Statistical methods and benchmark model",
"We analyse the radial velocity data sets by using posterior sampling algorithms and estimations of Bayesian evidences for models with $k = 0, 1, ...$ Keplerian signals.",
"Throughout the analyses we apply a fully Bayesian data analysis framework as discussed and applied in several astronomy papers over the recent years [23], [24], [66], [22], [29], [48], [68], [71], [72].",
"In particular, we apply the adaptive Metropolis posterior sampling algorithm of [31] that can be applied readily for analyses of radial velocity data [68], [71], [72].",
"We estimate the Bayesian evidences in favour of a given number of signals (i.e.",
"in favour of a model $\\mathcal {M}_{k}$ that contains $k$ signals) by using the estimate of [57] based on statistical samples drawn from both the prior and the posterior densities, and report parameter estimates using the maximum a posteriori (MAP) estimates and 99% Bayesian credibility intervals (BCS).",
"We note that the acronym BCS stands for Bayesian credibility set, which is represented by an interval in a single dimension when the posterior density does not have multiple significant modes.",
"This set with a probability threshold of $\\delta $ is a set $\\mathcal {D}_{\\delta }$ defined for a posterior density $\\pi (\\theta | m)$ as $\\mathcal {D}_{\\delta } = & \\bigg \\lbrace \\theta \\in C \\subset \\Omega : \\int _{\\theta \\in C} \\pi (\\theta | m) d \\theta = \\delta , \\nonumber \\\\& \\pi (\\theta | m) \\big |_{\\theta \\in \\partial C} = c \\bigg \\rbrace ,$ where $\\Omega $ is the parameter space of the parameter vector $\\theta $ , $\\pi (\\theta | m)$ is a posterior probability density function given measurements $m$ , $\\partial \\mathcal {D}_{\\delta }$ represents the edge of the set $\\mathcal {D}_{\\delta }$ , and $c$ is some positive constant.",
"Formal definition and discussion can be found in textbooks of Bayesian statistics, e.g.",
"[8] and [40].",
"Our benchmark statistical model contains linear acceleration and correlation components.",
"In the context of this model, we describe the radial velocity measurement ($m_{i,l}$ ) made at epoch $t_{i}$ with instrument $l$ as in [71], [72] and write it as $m_{i,l} = f_{k}(t_{i}) + \\gamma _{l} + \\dot{\\gamma }t_{i} + \\epsilon _{i,l} + \\phi _{l} \\exp \\big \\lbrace \\alpha (t_{i-1}-t_{i}) \\big \\rbrace \\epsilon _{i-1,l},$ where $\\gamma _{l}$ and $\\dot{\\gamma }$ are free parameters of the model representing the reference velocity of the $l$ th instrument and the linear acceleration, respectively, and $\\epsilon _{i,j}$ is a Gaussian random variable with a zero mean and variance $\\sigma _{i}^{2} + \\sigma _{l}^{2}$ describing the amount of Gaussian white noise in the $i$ th measurement of the instrument $l$ .",
"Parameter(s) $\\phi _{l}$ represents the correlation between the deviations of the $i$ th and $i-1$ th measurement from the mean, i.e.",
"the dependence of the $i$ th measurement on the deviation of the $i-1$ th measurement from the mean because there can be no causal relationship the other way aroundIt is not necessary to assume causality as the model simply aims at removing correlations and is therefore only a statistical model that describes the data reasonably accurately..",
"The parameter vector $\\theta $ of a “baseline” model without Keplerian signals is then $\\theta = (\\gamma _{l}, \\dot{\\gamma }, \\sigma _{l}, \\phi _{l})$ for one instument.",
"In our notation, function $f_{k}$ represents the superposition of $k$ Keplerian signals.",
"In Eq.",
"(REF ), parameter $\\alpha $ corresponds to the timescale of the exponential smoothing in the moving average (MA) component [72].",
"We chose this parameter such that correlations on the time-scale of few days were taken into account because correlations in this time-scale are known to occur in radial velocities [7], [72] but that measurements sufficiently far in time from one another, i.e.",
"more than a dozen days or so, are unlikely to be correlated.",
"Therefore, we chose $\\alpha = 0.01$ hours$^{-1}$ – a value that was supported by the largest UVES datasets (GJ 551 and GJ 699).",
"This value was also found to describe the data sets well because changing the value resulted in, at most, equally good performance of the statistical model for the data sets with large number of measurements.",
"It was not necessary to treat this parameter as a free parameter of the statistical model because in most of the cases, it could not be constrained at all due to a small number of data points and in such cases the principle of parsimony should be applied to decrease the number of free parameters and e.g.",
"fix the time-scale parameter as we have done in the current work.",
"The value $\\alpha = 0.01$ hours$^{-1}$ corresponds to a correlation time-scale of roughly 4 days.",
"It is possible that this choice for the time-scale affects the analysis results of some data sets that in fact have correlations on a much shorter (longer) time-scale of few hours (dozen days) instead of days.",
"However, we consider this to be rather unlikely because of the low number of measurements in most data sets.",
"We also note that there are thus $3j+5k+1$ free parameters in our model if $j$ is the number of instruments that have been used to obtain the data.",
"While our benchmark model in Eq.",
"(REF ) cannot be expected to be a perfect description of the radial velocities [70], [72], we still estimate that it contains most of the important features of a good radial velocity model.",
"Particularly, it contains the linear acceleration that could be present due to a previously unknown long-period substellar companion.",
"Even without evidence of such a companion (i.e.",
"of such a trend), we include this linear acceleration in the model to construct a standard model that can be applied to all the data sets we analyse.",
"If there is no linear acceleration in a given data set, its inclusion in the model makes the model overparameterised because dropping the corresponding term from the formulation would provide a better model due to the principle of parsimony.",
"However, we are willing to risk such overparameterisation to be able to use the same general benchmark model for all the data sets and to obtain as trustworthy results as possible.",
"The same principle applies to the parameter $\\phi _{l}$ that quantifies the amount of intrinsic correlation in the data obtained using the $l$ th instrument [70], [72].",
"We use this MA component in the model even if we cannot show that it is significantly different from zero, which makes the model more complicated but enables us to see consistently whether the excess noise in the data has a significant red-colour component.",
"We note that even if there is no evidence for neither acceleration nor correlation in the sense that the parameters $\\dot{\\gamma }$ and $\\phi _{l}$ have posterior densities that are consistent with zero, including the corresponding terms in the model makes the results more robust because the uncertainties of these terms can be taken into account directly.",
"Furthermore, this ensures that none of the signals we detect are spuriously caused by the combination of these two factors."
],
[
"Prior choice",
"Because prior densities of the model parameters are an integral part of Bayesian analyses, we discuss our prior choices briefly.",
"Throughout the analyses, we use prior probability densities as described in [68] with one small but possibly significant exception.",
"In comparison to the choice of [68], we follow [69] and use more restrictive eccentricity priors in our analyses by adopting $\\pi (e) \\propto \\mathcal {N}(0, 0.1^2)$ .",
"In addition to being a more approppriate functional form for an eccentricity prior than a uniform one [41], we make this choice because the data sets we analyse have already been analysed by [79] who did not report any planetary signals in the UVES data sets.",
"Therefore, if there are any significant signals in these data sets, they are likely to be deeply embedded in the noise in the sense that their amplitudes are comparable to the noise levels, which results in overestimated orbital eccentricities [78].",
"Yet, because low-mass planets are mostly found on close-circular orbits [69], we prefer a slight underestimation of orbital eccentricities over their overestimation.",
"We note that this prior choice is still much more conservative than the commonly made decision to fix eccentricities to zero a priori that correspond to choosing a delta-function prior such that $\\pi (e) = \\delta (e)$ .",
"For further justification of this prior choice, we refer to Appendix A in [2].",
"There is also the possibility that our eccentricity prior decreases the significance of a signal that is actually caused by e.g.",
"a planet on a moderately eccentric orbit.",
"However, we consider that to be unlikely because in the current sample, any planetary signals we detect have amplitudes that are comparable to the measurement noise.",
"This means that the corresponding eccentricities are ill-constrained from below and that using a prior density as described above, or indeed the choice proposed by [41], is thus unikely to affect the results significantly even if some of the stars in the sample have low-mass planets on eccentric orbits."
],
[
"Posterior samplings",
"While the adaptive Metropolis algorithm is an efficient tool in drawing statistically representative samples in cases of unimodal posteriors with little non-linear correlations between the parameters, it is not necessarily well-suited for such samplings of multimodal posteriors that are typical, especially with respect to the period parameters of the signals, when searching for periodic signals of planets.",
"Therefore, when searching for a $k$ th signal we simply divide the period space of this signal into parts that only contain one significant maximum that would, because of its significance, effectively prevent the chain from “jumping” between the different modes.",
"This enables us to use the adaptive Metropolis algorithm.",
"These different parts can then be sampled independently and treated as different (a priori) models to assess the relative significances of the corresponding maxima.",
"However, in practice, such cases were rare and we only applied such divisions of the period space to the data of two targets.",
"When estimating the significances of the solutions, i.e.",
"that the Markov chains were sufficiently close to convergence, we used the Gelman-Rubin statistics [27] as also described in [24].",
"In particular, we required that the test statistics $R(\\theta )$ that approaches unity from above as the Markov chains approach convergence was below 1.1 based on at least four chains to state that the chains were sufficiently close to convergence for all parameters $\\theta $ .",
"This corresponds to a situation where the variance of the parameters is lower between chains than within them indicating that all the chains have identified the same stationary distribution."
],
[
"Signal detection criteria",
"Throughout the analyses, we used the signal detection criteria of [68].",
"These criteria have recently been applied in e.g.",
"[2], [71], [72], and [69], and they appear to be trustworthy in the sense that they are not particularly prone to false positives [70], and enable the detections of signals that cannot be found by using more traditional detection criteria based on false alarm probabilities (FAPs) in the power spectrum of the model residuals [4], [2], [72].",
"As an example, we refer to [69], who independently discovered the same three planet candidates around GJ 163 as [12] with only $\\sim $ 35% of the data.",
"This indicates that the Bayesian signal detection criteria indeed are very sensitive and robust ones in detecting weak signals of low-mass planets in radial velocity data.",
"The first criterion is that a model with $k+1$ signals, denoted as $\\mathcal {M}_{k+1}$ , has a posterior probability that is at least $s$ times greater than the corresponding probability of a model with only $k$ signals, i.e.",
"$P(\\mathcal {M}_{k+1} | m) \\ge s P(\\mathcal {M}_{k} | m)$ .",
"If this criterion is not satisfied, the $k+1$ th signal has a considerable probability of being produced by noise instead of being a genuine periodicity in the data.",
"When analysing the data with $k=0, 1, ...$ it might happen that $P(\\mathcal {M}_{k} | m) < s P(\\mathcal {M}_{k-1} | m)$ but that $P(\\mathcal {M}_{k+1} | m) \\ge s P(\\mathcal {M}_{k} | m)$ .",
"This means that it cannot be said there are significantly $k$ signals, and if the model $\\mathcal {M}_{k+1}$ was not tested, the conclusion would be that there are only $k-1$ signals in the data.",
"Therefore, if we observe putative probability maxima in the parameter space for model $\\mathcal {M}_{k}$ that do not satisfy the detection criterion, we typically analyse the data with models $\\mathcal {M}_{k+1}$ and $\\mathcal {M}_{k+2}$ as well to see if the superposition of more than $k$ signals makes the detection of only $k$ signals impossible.",
"We note that the probability threshold $s$ has to be chosen subjectively.",
"A typical choice would be $s = 150$ because it has been interpreted as corresponding to strong evidence and recommended by e.g.",
"[39] based on the arguments of [35], although it has been argued that a more conservative threshold of 1000 should be chosen [21].",
"This threshold of $s = 150$ has been applied in analyses of radial velocity data succesfully [22], [30], [68], [71], [72].",
"However, this choice might result in overinterpretation of the significance when the statistical model used to describe the data does not represent the data well.",
"For the purpose of population studies such as the current work, one should be more cautious than usual in accepting new planet candidates to avoid the contamination of occurrence rate estimates by false positives.",
"Therefore, we remain cautious about the interpretation of signals that are detected with low thresholds, and use a more conservative threshold of $s = 10^{4}$ as a requirement for a planet candidate.",
"The second criterion is related to the first one in the sense that it states that the radial velocity amplitude has to be statistically significantly (with e.g.",
"99% level) greater than zero.",
"If this was not the case, there would be a considerable probability that the amplitude of the signal was actually negligible and that the signal did not thus exist in reality.",
"The third criterion states that the period of a signal has to be well-constrained from below and above to consider it a genuine periodicity.",
"Again, the justification of this criterion is simple because if the period parameter was not well constrained, it could not be claimed that the corresponding variations in the data were indeed of periodic nature.",
"We note that in practice, based on the analyses in the present work, a signal whose significance exceeds $s = 10^{4}$ is always well constrained in the parameter space.",
"Conversely, if a given signal is constrained in the period and amplitude space, its significance always exceeds some $s > 1$ , though not necessarily $s = 150$ or $s = 10^{4}$ .",
"This means that the detection criteria are complementary and robust in detecting low-amplitude signals.",
"To assess whether any given signal observed in the combined data set is supported by both data sets, we examine the residuals of each model using common weighted root-mean-square (RMS) statistics.",
"If these statistics are decreased for both data sets when adding a signal to the model, we say that both data sets support the existence of the corresponding signal even when the signal cannot be detected in both data sets independently.",
"This choice also serves as a test of whether a given weak signal could be a spurious one generated by noise and the correspondingly insufficiently accurate model.",
"For a given signal to be a genuine one of stellar origin (as opposed to a spurious signal of instrumental origin) – possibly caused by the Doppler variations induced by an orbiting planet – it should result in a decrease in the RMS estimates of both data sets, although this is also subject to chance and does not necessarily hold when there are only few measurements.",
"Finally, to make the detections as robust as possible, we set the following criteria for signals to be able to call them planet candidates.",
"First, we require that such signals are detected with a choice of $s = 10^{4}$ and can be concluded to be present in the corresponding combined data set(s) very significantly.",
"If counterparts of these signals cannot be found in the activity indices calculated from the UVES and HARPS spectra, they are considered planet candidates and we refer to them according to the standard nomenclature by assigning letters b, c, ... to them.",
"If, however, (1) the corresponding signals are detected only with a significance of $s < 10^{4}$ , (2) if there is data available from only one spectrograph such that the existence of a signal is not supported by an independent data set because such a set is not available, or (3) if a signal is present only in one of the data sets (in the sense that the RMS of the other does not decrease when adding the signal to the model) that dominates the posterior density given the combined data because of the low number or poor quality of the measurements in the other set, we do not call them candidate planets.",
"We note that if the posterior density consists of several other (local) maxima in the parameter space at several different periods with reasonably high probabilities (e.g.",
"with $n$ local maxima at $\\theta ^{i}_{\\rm L}, i = 1, ..., n$ , that have $\\pi (\\theta ^{i}_{\\rm L} | m) \\ge 0.01 \\pi (\\theta _{\\rm MAP} | m)$ for all $i$ , where $\\theta _{\\rm MAP}$ is the MAP estimate), it is possible that there are also several significant solutions in the period space that satisfy the detection criteria.",
"Such a situation is hard to interpret reliably, but can arise for two main reasons.",
"First, all the significant periodicities can correspond to genuine periodic signals in the velocity data – several planetary signals in the data may have similar amplitudes and be detected roughly equally confidently [2].",
"Alternatively, such a situation might be representative of poor noise modelling where periodic/quasiperiodic features and/or correlations in the noise are falsely interpreted as genuine signals.",
"In such cases, we analyse the data by increasing $k$ further but report the solution corresponding to the global maximum if there is no strong evidence in favour of additional signals in the sense of our detection criteria."
],
[
"Search for significant periodicities",
"We performed the searches for significant periodicities in the data sets in several stages.",
"First, we obtained a sample from the posterior density of the model without any Keplerian signals.",
"In the second step, we sampled the posterior density of the model with $k=1$ by using tempered chains such that the posterior density was raised to a power of $\\beta \\in (0,1)$ .",
"In particular, we used $\\beta =$ 0.3 - 0.8 to find the most promising areas in the period space because such a choice of $\\beta $ enabled a rapid search of the whole period space $[T_{0}, T_{\\rm obs}$ ], where $T_{\\rm obs}$ is the data baseline and we selected $T_{0} =$ 1 day.",
"However, if there were clear indications of probability maxima in excess of a period equal to $T_{\\rm obs}$ , we increased the upper limit to $2T_{\\rm obs}$ .",
"We did not use the adaptive Metropolis algorithm [31] but the standard Metropolis-Hastings version [51], [32] in these periodicity searches to prevent the proposal density from adapting to the possibly very narrow global maximum in the posterior density because the purpose of these samplings was to identify the positions of the most significant probability maxima in the parameter (period) space and not to enable the chain to adapt to one of them.",
"For most data sets, we started by a test sampling with a “cold” chain, i.e.",
"a chain with $\\beta = 1$ .",
"When the number of measurements was lower than $\\sim $ 15, these cold samplings were sufficient in exploring the whole period-space because typically there were no significant maxima in the parameter space that would have prevented the chain from exploring the whole period space rapidly.",
"When such cold samplings were not possible due to an abundance of reasonably high maxima in the parameter space and thus poor mixing of the Markov chains in the period space, we used tempered chains by setting $\\beta = 0.3$ to 0.5, depending on the number of measurements.",
"The lowest values of 0.3 were used for the largest data sets of GJ 699 ($N_{\\rm UVES} = 226$ ), GJ 551 ($N_{\\rm UVES} = 229$ ), and GJ 433 ($N_{\\rm UVES} = 166$ ).",
"These samplings resulted in estimates for the positions of the most significant maxima in the period space.",
"However, it must be noted that using tempered samplings corresponds actually to using different prior and likelihood models in the analysis such that $\\pi ^{\\beta }(\\theta )$ and $l^{\\beta }(m | \\theta )$ are used as a prior density and likelihood function instead of $\\pi (\\theta )$ and $l(m | \\theta )$ , respectively.",
"Therefore, the shape of the posterior density, as estimated based on tempered samplings, does not necessarily correspond to the actual posterior density.",
"However, the positions of the maxima are unchanged.",
"Given rough estimates for the positions of the probability maxima in the period space based on tempered samplings, we started several cold samplings with initial values in the vicinity of the observed maxima to enable fast convergence.",
"If one (or some) of these maxima corresponded to a significant periodic signal(s) in the sense of the detection criteria we described above (Section REF ), we continued by increasing $k$ and by performing tempered samplings of the parameter space of a model with one more Keplerian signal.",
"These samplings were performed by using cooler chains because after the strongest signal was accounted for by the model, samplings with $\\beta =$ 0.6 - 0.8 were typically found to visit all areas of the period space of the additional signal with $\\sim $ few 10$^{6}$ chain members.",
"Finally, we obtained samples from the posterior density of a model with $k+1$ Keplerian signals when the corresponding data was found to contain significant evidence in favour of only $k$ signals, i.e.",
"that only $k$ signals satisfied the detection criteria.",
"These samplings were performed to estimate which additional signals could be allowed by the data and which ones could thus be ruled out [68].",
"We note that some data sets have only two UVES measurements and/or less than two HARPS measurements.",
"Therefore, because we used a linear trend in the model, we did not analyse UVES data sets with less than three mesurements.",
"Furthermore, we did not analyse the combined set when the number of measurements in either UVES or HARPS set is less than two.",
"This means that we might analyse an individual data set with three measurements and a combined one with four (two UVES and two HARPS) measurements.",
"Even in such extreme cases, when the number of free parameters in the statistical model exceeds the number of measurements, we expect to be able to obtain meaningful results because we use informative prior densities and because the Bayesian statistical techniques we use take the principle of parsimony, and thus the possible effects of such overparameterisation, into account and do not rely on assumptions regarding the number of parameters or measurements."
],
[
"Example: GJ 229",
"To demonstrate the practicality of our periodicity search technique, we show the analysis results in detail for the UVES velocities of GJ 229 because it is a typical target in the sample with a reasonably large number of measurements ($N_{\\rm UVES} = 73$ ) that show evidence in favour of periodic variations.",
"We started by analysing the data with the benchmark model with $k=0$ .",
"We found that the UVES data of GJ 229 contained a significant amount of correlation with $\\phi _{\\rm UVES} =$ 0.85 [0.45, 1.00]; evicence for a linear trend with $\\dot{\\gamma } =$ 1.26 [0.39, 2.17] ms$^{-1}$ year$^{-1}$ , small part of which is caused by secular accelerationThe secular acceleration was not subtracted from the UVES data of [79] at this stage.",
"However, we subtracted it when analysing the UVES data in combination with the HARPS velocities from which the secular acceleration was subtracted by the TERRA processing.",
"of 0.070 ms$^{-1}$ year$^{-1}$ [79]; and excess Gaussian white noise with $\\sigma _{\\rm UVES} =$ 2.51 [1.29, 3.74] ms$^{-1}$ , which is rather low and implies that the star expresses very low levels of velocity variability and/or the UVES instrument uncertainties in the [79] velocities might be overestimated.",
"Removing the MAP trend and correlations, we plotted the resulting residuals in Fig.",
"REF .",
"These residuals can indeed be described as “flat”, which indicates that there cannot be periodic variations in these velocities with amplitudes greater than roughly 10 ms$^{-1}$ .",
"The RMS of the residuals is 3.61 ms$^{-1}$ , which is considerably lower than that of the original velocities of 5.26 ms$^{-1}$ .",
"Figure: Residuals of the benchmark model without Keplerian signals for the UVES data of GJ 229.We performed a tempered search for periodicities, and obtained a sample from the posterior density corresponding to a choice of $\\beta = 0.3$ .",
"We plotted an example of a corresponding sampling in Fig.",
"REF .",
"As can be seen, the chain identifies a global maximum in the period space of the one-Keplerian model at a period of 2.8 days (red arrow).",
"The chain also visits the whole period space between 1 day and $T_{\\rm obs} = 2325$ days for GJ 229.",
"The chain in Fig.",
"REF also shows that there are additional local maxima in the period space exceeding the 10%, 1% and/or 0.1% probability levels (dotted, dashed, and solid horisontal lines, respectively) with respect to the global maximum at periods of roughly 1.5, 10, 200, and 450 days.",
"The existence of such multiple maxima in the scaled posterior suggests that none of them can be confidently considered a solution and thus a periodic signal that is reliably detected in the data.",
"Our samplings also suggest that there are likely no other considerable probability maxima in the period space, or that if they exist, they are so narrow that the chains are unlikely to visit them.",
"We performed such samplings several times and obtained consistent results with no indications of additional maxima with similar posterior values.",
"Figure: Log-posterior density as a function of the log-period parameter of a Keplerian signal from a tempered sampling (β=0.3\\beta = 0.3) of the GJ 229 UVES data.",
"The horizontal lines indicate the 10% (dotted), 1% (dashed), and 0.1% (solid) probability thresholds with respect to the MAP value that is denoted by using the red arrow.Because of the candidate periodicities, we started cold chains with initial periods in the vicinity of the highest maxima (Fig.",
"REF ) and were able to identify two periodicities that satisfied the detection criteria discussed above.",
"These periodicities were found at periods of 1.5 and 206 days with amplitudes of 5.3 and 4.3 ms$^{-1}$ days, respectively, but it cannot be concluded that we have identified such signals confidently in the data because the posterior density has multiple maxima comparable to the posterior at the MAP estimate.",
"Therefore, we can only conclude that if there are periodic signals in the UVES data, they are likely present at the periods corresponding to the maxima in the posterior density (Fig.",
"REF ).",
"We performed samplings of the UVES data with a model containing two Keplerian signals, but could not find significant two-Keplerian solutions to the data.",
"It should be noted that the samples from the (scaled) posterior densities in Fig.",
"REF can be roughly interpreted in a similar manner as e.g.",
"common Lomb-Scargle [47], [62] periodograms.",
"This is because the plotted density represents the relative significances of the different periods and can thus be broadly interpreted according to the probabilities presented in the vertical axis showing the log-posterior values.",
"However, the relative probabilities of the corresponding periodicities, and indeed whether they are significant enough to satisfy the detection criteria, can only be assessed by additional samplings and by seeing whether the periods can be constrained from above and below.",
"One last test for a significance of a signal is whether it is present in two independent data sets or not.",
"According to our results, the putative signals in the UVES data at periods of 1.5 and 206 days fail this test as the HARPS data effectively rules them out.",
"This can be seen by looking at the corresponding log-posterior density as a function of the period of a signal in the combined data (Fig.",
"REF , top panel).",
"Accordingly, there is a strong and isolated global maximum at a period of roughly 470 days that satisfies all the detection criteria discussed above.",
"Because this signal is clearly present as a local maximum in the UVES data alone, and because there are no comparable local maxima in the period space given the combined data, we conclude that there is a significant periodic signal in the combined HARPS and UVES velocities of GJ 229 at a period of 471 [459, 493] days with an amplitude of 3.83 [2.15, 5.57] ms$^{-1}$ that corresponds to a new and previously unknown planet candidate orbiting the star with a minimum mass of 32 [16, 49] M$_{\\oplus }$ .",
"This signal can be called a candidate planet because its existence is supported by both data sets according to our requirements and because it is detected very confidently in the combined data.",
"Furthermore, as we discuss in Section REF , the signal does not correspond to variations in the activity data of GJ 229.",
"To demonstrate its significance, we have plotted the phase-folded signal in Fig.",
"REF .",
"We note that there is no evidence in favour of a second signal in the combined data, as can also be seen in the bottom panel of Fig.",
"REF that does not have strong and isolated maxima in the period space.",
"Figure: As in Fig.",
"for the first signal in the combined HARPS and UVES data of GJ 229 (top panel), and for a cold sampling plotted as a function of the second signal of a two-Keplerian model indicating that there are no additional periodic signals (bottom panel).Figure: Phase folded signal in the combined HARPS (blue) and UVES (red) data of GJ 229."
],
[
"Detection probability and planet occurrence",
"To obtain estimates of the underlying occurrence rate of planets in our sample, it is necessary to estimate the detection bias of the current data caused by the fact that radial velocity data sets are more sensitive to greater planetary masses and shorter orbital periods.",
"It is also necessary to account for the quantity and quality of each data set, as well as their respective baselines.",
"For this purpose, we calculated detection probabilities for the combined UVES and HARPS data sets of the sample by using samplings of the parameter space as shown in e.g.",
"the bottom panel of Fig.",
"REF .",
"First, we estimate that in any given data set (the $i$ th data set), the observed number of signals in a given period interval $\\Delta P$ and minimum mass interval $\\Delta M$ , that we denote as $\\Delta _{\\rm P,M}$ for short, can be written as $f_{\\rm obs,i}(\\Delta _{\\rm P,M}) = f_{\\rm occ,i} (\\Delta _{\\rm P,M}) p_{i}(\\Delta _{\\rm P,M})$ , where $f_{\\rm occ,i}$ is the number of planets orbiting the $i$ th star in the sample and $p_{i}$ is the detectability function that indicates whether a planet with parameters in the interval $\\Delta _{\\rm P,M}$ can be detected in the data set $m_{i}$ .",
"However, while $f_{\\rm obs,i}$ is easy to obtain as it is directly the number of planets we detect in the respective mass-period interval in data set $m_{i}$ , $p_{i}$ is more difficult to calculate.",
"To estimate $p_{i}$ , we use the posterior samplings of a model with $k+1$ Keplerian signals when there are only $k$ signals in the data set.",
"The sample drawn by using the posterior sampling enables us to reconstruct the areas of the parameter space where the parameters describing the hypothetical $k+1$ th signal visited.",
"Because the signal cannot be detected in the sense that its amplitude and period could be constrained, the chain visits all areas in the period space allowed by the chosen range of the period (from one day to data baseline).",
"Moreover, the chain visits, given sufficiently good mixing properties such that the chain visits all relevant areas in the parameter space frequently, all amplitudes at all periods that are allowed by the data, i.e.",
"amplitudes that are so low that they cannot be ruled out by the data because the likelihood function has reasonably high values.",
"The chain does not visit amplitudes in excess of some limiting amplitude at each period because they would correspond to such low likelihoods that planetary signals with such amplitudes are effectively ruled out.",
"Therefore, the sampling yields the areas of parameter space where there could be signals.",
"The rest of the parameter space is thus the area where there are (very likely) no additional signals based on the data.",
"We demonstrate this by using GJ 229 as an example.",
"It has one candidate and thus we use a model with two of them to estimate the detectability function.",
"In Fig.",
"REF , the black area shows the subset of the mass-period space where the mass and period parameters of the second signal visited during the Markov chain samplings.",
"This area thus corresponds to signals that could exist in the data but that cannot be detected because we did not find a second signal in the GJ 229 data.",
"The white area corresponds to mass-period space where the chains did not visit and thus rule out planetary signals because the likelihood function is so low in these areas that the chances of a planet existing in the white area can be approximated to be negligible.",
"The planet candidate GJ 229 b is clearly above the threshold as it should because it was detected.",
"We note that while this planet candidate has parameters close to the threshold, it is still several M$_{\\oplus }$ heavier than the minimum-mass planet that could be detected at that period.",
"In fact, if it was much more above the threshold, it is likely that its existence would already have been reported based on UVES or HARPS data sets alone.",
"Figure: Detection threshold of the combined data of GJ 229.",
"The white area corresponds to parameter values in the mass-period space where signals could be detected in the data (detection probability of unity) whereas the black area shows a corresponding mass-period space where signals cannot be detected (negligible detection probability).",
"The candidate GJ 229 b is shown as a red circled dot that is in the former area because it was detected.We thus estimate that planet candidates could have been detected in the areas where the Markov chains did not visit (white areas in Fig.",
"REF ).",
"This is the case because they correspond to the complement of the area where planets could not be detected.",
"In reality, however, the threshold is not so strict because the detection probability is a (continuous) function of the parameters.",
"We do not attempt to estimate this function more accurately in this work and use the ”step-function“ for each data set as shown in Fig.",
"REF for GJ 229 as a first-order approximation.",
"Assuming that there are some areas in the parameter space where the chains did not visit due to e.g.",
"poor sampling even though they should have because these areas have sufficiently high likelihoods, we would effectively overestimate the detection probabilities and thus underestimate the occurrence rates.",
"Thich means that our occurrence rate estimates are lower limits, although unlikely to be considerably lower to bias the results significantly given the rather large uncertainties due to the low number of planets in the sample.",
"We approximate the detectability by setting the function $\\hat{p}_{i} = 1- p_{i}$ equal to unity in the areas where the Markov chains did visit, and equal to zero where they did not.",
"We express the detected frequency of planets for the whole sample of stars, with data sets $m_{1}, ..., m_{N}$ , by summing the number of observed signals $f_{\\rm obs,i}$ of all the data sets and by assuming that the occurrence rate is common for all stars in the sample such that $f_{\\rm occ} = f_{\\rm occ,i}$ for all $i$ (in units of planets per star).",
"Thus we obtain $&& f_{\\rm obs}(\\Delta _{\\rm P,M}) = \\sum _{i=1}^{N} f_{\\rm obs,i}(\\Delta _{\\rm P,M}) \\nonumber \\\\&& = f_{\\rm occ} (\\Delta _{\\rm P,M}) \\Bigg [ N - \\sum _{i=1}^{N} \\hat{p}_{i}(\\Delta _{\\rm P,M}) \\Bigg ] ,$ which implies a simple way of calculating the occurrence rate $f_{\\rm occ}$ for the whole sample.",
"In this equation, the term in square brackets on the right hand side is, when divided by $N$ , the detection probability function of our sample that approximates the probability of being able to detect planets in the interval $\\Delta _{\\rm P,M}$ .",
"Thanks to our samplings, we could estimate this function rather accurately - typically by using a 100$\\times $ 100 grid in the log-parameter space ranging from a minimum period of 1 day to a maximum of 10$^{4}$ days and from a minimum mass of 1 M$_{\\oplus }$ to a maximum of 100 M$_{\\oplus }$ .",
"Such a fine grid was not a practical choice for estimating the occurrence rates, because to obtain any meaningful estimates at all, it would be desirable to have at least one planetary signal in most grid points.",
"For this reason, when calculating occurrence rates, we divided the interval into a 4$\\times $ 4 grid.",
"When estimating the uncertainties for the occurrence rates in a given interval $\\Delta _{\\rm P,M}$ , we use rather conservatively the lowest and highest detection probabilities (obtained by using the finer grid) in this interval to calculate lower and upper limits, respectively."
],
[
"The sample and Keplerian signals",
"The sample of M dwarfs for which the UVES velocities were published in [79] contains a collection of nearby stars including the two nearest M dwarfs, namely GJ 551 and GJ 699 that are the nearest and fourth nearest stars to the Sun, respectively.",
"We have listed the stars in this sample in Table REF together with their estimated physical properties.",
"While the parallaxes are obtained from Hipparcos [74] and the mass estimates from [79], we estimated the effective temperatures and luminosities by using the empirical relations of [14] and [13].",
"Based on the estimated luminosities and effective temperatures, we have also calculated the approximate inner and outer edges of the stellar habitable zones according to the equations of [43] and listed them in Table REF .",
"We note that the stars in the sample have also been extensively searched for brown dwarf companions.",
"For instance, [17] reported that $0.0^{+3.5}_{-0.0}$ % occurrence rate of L and T companions in separations between 10–70 AU.",
"Furthermore, the sample stars show no evidence for warm [5] or cold [46] circumstellar material.",
"Table: Target stars, their Hipparcos parallaxes, and physical properties together with the estimated inner and outer edges of the stellar habitable zones based on .We obtained the HARPS-TERRA (Template Enhanced Radial velocity Re-analysis Application) velocities from the publicly available spectraWe also obtained three additional HARPS spectra of GJ 699 to increase the baseline of the HARPS data.",
"by using the data processing algorithms of [3].",
"We chose to use the TERRA velocities instead of the commonly used HARPS-CCF (cross-correlation function) velocities because of their lower scatter and therefore better precision for M dwarfs [3], [4], [2], [69].",
"While an abundance of such velocities could not be obtained for every star because a large fraction of the stars in our sample have not been primary targets of the HARPS-GTO survey [11], there were still several stars for which the available HARPS data could be readily expected to provide better constraints for the possible planet candidates orbiting them due to its high precision, or help disputing the existence of putative signals in the UVES data as false positives.",
"The numbers of HARPS measurements and the corresponding data baselines are listed in Table REF and we have tabulated the corresponding HARPS-TERRA velocities in the Appendix .",
"Table: Properties of the UVES and HARPS data sets in terms of numbers of measurements (N UVES N_{\\rm UVES}, N HARPS N_{\\rm HARPS}) and data baselines (Δ UVES \\Delta _{\\rm UVES}, Δ HARPS \\Delta _{\\rm HARPS}).",
"Log-Bayesian evidence ratios, or Bayes factors (lnB i+1,i =lnP(m|ℳ i+1 )-lnP(m|ℳ i )\\ln B_{i+1,i} = \\ln P(m | \\mathcal {M}_{i+1}) - \\ln P(m | \\mathcal {M}_{i})), are shown when there is evidence in favour of at least one Keplerian signal according to our detection criteria.",
"GJ 190 and GJ 263 are not in the table because they only had two velocity measurements.",
"The four stars with evidence of massive companions in the radial velocity data are also not shown.We present the results of our comparisons of models containing $k=0, 1, 2$ Keplerian signals in Table REF by presenting the numbers of favoured signals in the combined data sets together with the numbers of measurements and the baselines of the UVES and HARPS data sets.",
"According to these results, there are 10 significant signals in the combined data that satisfy our detection criteria and are supported by both data sets according to our requirements.",
"When publishing the analysis results of the same UVES datasets, [79] did not report detections of planet candidates.",
"According to our periodogram analyses with the standard Lomb-Scargle periodogram [47], [62] of the UVES data, this conclusion is indeed justified because, excluding the massive stellar or substellar companionsApparent as variability of the order of few hundred ms$^{-1}$ , although the corresponding orbits cannot be constrained.",
"around GJ 477, GJ 1046, GJ 3020, and GJ 3916; and excluding GJ 551 and GJ 699 for which [79] reported signals caused by data sampling and activity, none of the UVES data sets had significant powers in their Lomb-Scargle periodograms corresponding to low-mass planetary companions.",
"This inability to detect the signals of the planet candidates we report is not surprising because all these signals have very low amplitudes whose detection is difficult without a large number of measurements and high precision.",
"Interestingly, unlike [79], we could not find the 44-day signal they reported in the velocities of the GJ 699.",
"We believe the reason is that this signal is likely caused by activity-related phenomena [79] and is therefore quasiperiodic and/or time-dependent and explained rather well by correlations in the UVES measurements.",
"Indeed, the estimate for the parameter $\\phi _{\\rm UVES}$ was found to be 0.81 [0.62, 0.99], which implies a considerable amount of correlation in the UVES data that could – when coupled with data sampling – give rise to significant but spurious powers in periodogram when not accounted for.",
"In addition to the signals we find, five targets in our sample show linear acceleration that is not consistent with zero suggesting the existence of yet unknown long-period substellar companions (Table REF )."
],
[
"Analysis of activity indicators",
"To determine whether the signals we observe in the UVES radial velocities are caused by Doppler fingerprints of planetary and/or substellar companions or periodic or aperiodic and/or quasiperiodic phenomena related to the stellar activities, we analysed the line bisector (BIS) spans as obtained from the UVES spectra available in the ESO archive.",
"BIS values indicate the line asymmetry and can be used as a signature of variations caused by stellar activity [58], [9].",
"For most stars for which we observed radial velocity signals, the correlation coefficients between the velocities and BIS values were between -0.1 and 0.1, which implies no significant correlation.",
"However, GJ 842 showed positive correlation coefficient of 0.43 between the radial velocities and BIS values that appears significant although the corresponding data set only contained 16 data pointsWe could only obtain 16 spectra for GJ 842 because the ESO archive did not contain calibration frames although there are 17 radial velocities in the [79] data set (see also Table REF ).. We also obtained the BIS values from the HARPS spectra and tested whether there were correlations between these values and the radial velocities.",
"None of the data sets showed significant correlations between the BIS values and velocities, which indicates that the variations in the velocity data are unlikely to have been induced by activity-related phenomena of the stellar surface such as starspots and/or active regions.",
"The typical correlation coeficients were found to be between -0.1 and 0.1, which is consistent with the interpretation that the signals we observe are genuine Doppler signatures of planetary nature."
],
[
"UVES false positives",
"The UVES data contained some signals that were not supported by the HARPS velocities.",
"This means that our benchmark model might not be optimal in the sense that features in the UVES measurement noise, or indeed biases, instrument-related variations, stability problems, etc.",
"can produce variations that are interpreted as Keplerian signals.",
"Furthermore, according to our analyses, even when we do detect a signal in the UVES data, the period space is typically littered by several other almost equally high local maxima that should be interpreted as alternative solutions before they can be ruled out.",
"Such a multimodality occurs e.g.",
"in the UVES data of GJ 160.2, GJ 229 (see Fig.",
"REF ), GJ 377, and GJ 27.1.",
"We identified only two false positives in the UVES data that were essentially ruled out by the HARPS velocities.",
"The first one of these is a 15-day periodicity in the 14 UVES velocity measurements of GJ 377.",
"This signal increases the model probability with respect to the zero-Keplerian model by a factor of 700, which does not exceed our detection threshold of $10^4$ .",
"In this case, the HARPS data did not confirm the existence of the signal but ruled it out rather confidently.",
"This weak evidence for a signal in the UVES data was also suspect because the period parameter of the signal had local probability maxima around 2, 5, 50-200, and 700 days that exceeded the 0.1% or even 1.0% probability thresholds of the global maximum.",
"We interpret this result as hints of a signal in the UVES data whose exact period or amplitude cannot be determined due to the low number of measurements.",
"Instead, in the combined dataset the global maximum of the period parameter was found at a period of roughly 80 days but it did not satisfy the detection criteria.",
"The other example is provided by the solution to the UVES data of GJ 357 that consists of two periodicities at 5 and 26 days, respectively.",
"The significances of these signals decrease below the detection threshold with only five HARPS velocities.",
"This means that – even though there is not enough HARPS data to constrain any signals in the combined data set – some of the local maxima in the UVES data can be ruled outHere ”ruled out“ cannot be interpreted very strictly, as ruling out the existence of a signal is much harder in general than detecting one., but others might actually correspond to signals whose existence could be verified when future high-precision data becomes available.",
"We note that the iodine cell used in the UVES to obtain precise reference lines to the spectra can give rise to differences with respect to the ThAr method used in the HARPS.",
"This means that some of the spurious signals and/or spurious local maxima that are unrelated to sampling in the UVES data can arise from small-scale instability in the UVES instrument.",
"The same argument applies to the HARPS velocities as well, although HARPS has much greater long-term stability and precision.",
"However, at the moment it is not possible to distinguish whether the signals with low significances that are interpreted as false positives in the UVES data because they do not correspond to global maxima in the combined data are caused by such instability, periodic or quasiperiodic features in the noise, periodicities related to the stellar activity and magnetic phenomena, and/or genuine Doppler signatures caused by low-mass planets.",
"Ideally, signals arising from instrumentation can be ruled out by obtaining support for them from two data sets, and those caused by stellar surface phenomena can in principle be ruled out by showing that they or their primary harmonics do not have any couterparts in the activity-indices of the two sets.",
"However, any remaining signals can still be caused by other unknown sources of periodic variation."
],
[
"Planets around M dwarfs",
"We have plotted the known planetsObtained from the Extrasolar Planets Encyclopaedia [63].",
"(light blue dots), known planets around M dwarfs (blue circled dots), the new candidates we report (red circled dots) in Fig.",
"REF .",
"This figure also shows the estimated detection probability of the combined UVES and HARPS data sets as functions of minimum mass and orbital period for the whole sample (Section REF ).",
"The most remarkable feature in this plot appears to be that the new candidates are concentrated around minimum masses of 10 M$_{\\oplus }$ and periods from few to few dozen days.",
"Considering that such planets can be detected in this sample with rather low probabilities of roughly 10-30%, the abundance of such companions appears strikingly high.",
"Yet, the results from analyses of Kepler's data appear to imply that this is a real feature in general [34], and applies to M dwarfs in particular, as shown in [18].",
"They observed an abundance of transiting planets with the same period range and radii of up to 3 R$_{\\oplus }$ , which could correspond to the same population of planets of which we only observe the planets with the highest masses.",
"Figure: Planet detection probability as determined by using Eq.",
"() in the combined UVES and HARPS data set as functions of orbital period and minimum mass.",
"The various dots represent the known planets orbiting all stars (light blue dots), known planets orbiting M dwarfs (circled blue dots), and planet candidates in our sample (circled red dots).",
"The detection probabilities do not exceed 85% even at the high-mass short-period corner of the plot because there are six data sets where planetary signals could not be detected at all due a combination of low number of measurements and evidence of a massive companion that prevented detections of additional companions due to overparameterisation of the benchmark model.The detection probability of a given combination of minimum mass and period in Fig.",
"REF shows the different areas in the mass-period space where planets can be spotted easily and where it is very difficult (or unlikely) given the current sample.",
"The highest gradient in this probability occurs along the line increasing from 4 M$_{\\oplus }$ at a period of one day to 100 M$_{\\oplus }$ at 1000 days.",
"In our sample, only one candidate can be found above this line, even though planets in this region would be the easiest ones to detect.",
"We note that the rather artificial-looking vertical line at roughly 3000 days in the top right corner of the Fig.",
"REF is in fact a threshold arising from the fact that we limited the period search to periods at most the baselines of the data sets.",
"Furthermore, there is a weakly distinguishable feature that shows decreased detection probabilities for periods around 365-day period that demonstrates how the existence of planetary signals at one year period cannot be ruled out generally as easily as slightly shorter and longer ones due to poor phase-coverage caused by data samplings.",
"Based on the analyses of the combined UVES and HARPS radial velocities of the 41 nearby M dwarfs, these data sets contain the signals of eight new exoplanet candidates out of which seven can be classified as super-Earths due to their minimum masses that are higher than that of the Earth but still lower, for most candidates considerably so, than one Neptune-mass.",
"In addition to these super-Earths, we report a detection of a more massive sub-Saturnian companion orbiting GJ 229, and confirm the existence of the long-period planet orbiting GJ 443 detected by [16].",
"Eight of these candidate planets are thus new and previously unknown.",
"We have listed the obtained orbital parameters together with the inferred minimum masses and semi-major axes of the new planet candidates we report in Table REF .",
"The minimum masses and semi-major axes have been estimated by using the stellar masses as presented by [79] and by assuming conservatively that their standard deviations are 10% of the estimated values.",
"The last column of Table REF shows the estimated locations of the planets, i.e.",
"whether they are in the habitable zone [43], in the cool zone outside the outer edge of the habitable zone, or in the hot zone inside the inner edge of the habitable zone.",
"We have also plotted the estimated log-posterior densities as functions of signal periods, indicative of significant periodicities, in the Appendix (Figs.",
"REF - REF ) together with phase-folded signals and probability distributions for the periods and signal amplitudes that demonstrate that the signals satisfy the detection criteria.",
"Table: Orbital solutions (maximum a posteriori estimates and the 99% Bayesian credibility intervals) of the new planet candidates around M dwarfs and the inferred minimum masses and semi-major axes.We also broadly verified the existence of the signals corresponding to the planet candidates in Table REF by using an independent statistical method.",
"We applied the log-likelihood periodograms of [6] and [2] and used the same statistical model as in the Bayesian analyses.",
"The results of these periodogram analyses are shown in Table REF .",
"Six of the signals are clear with FAPs below a threshold of 0.1%.",
"However, GJ 180 c, GJ 422 b, and the two candidates of GJ 682 are only detected with FAPs greater than this threshold.",
"While we suspect that the high FAP of GJ 422 b might be due to priors that were all assumed to be flat when calculating the log-likelihood periodograms and the assumptions behind the significance tests of these periodograms [6], it appears to be clear why GJ 180 c and the candidates GJ 682 b and c were not detected with the periodograms.",
"This is because for GJ 180 c and GJ 682 c, the solution is actually obtained by assuming that there is only one signal when in reality there are two and the $k=1$ model is therefore fitted to the superposition of the two signals that causes an apparent decrease to the significance of the second signal.",
"Instead, the signal of GJ 682 b could not even be detected (it was also below the detection threshold with the Bayesian tools because the log-Bayesian evidence ratio corresponding to the detection threshold of 10$^{4}$ is 9.21, see Table REF ) because the $k=1$ model fitted the data much more poorly than a $k=2$ model.",
"With the posterior samplings, however, detecting these signals was possible (Figs.",
"REF and REF ).",
"Table: Solutions obtained by using the log-likelihood periodograms.",
"Preferred periods (PP), false alarm probabilities (FAP), and the differences between the (natural) logarithms of the likelihoods of the preferred model with kk signals and a model with k-1k-1 signals (Δ L \\Delta _{\\rm L})."
],
[
"Potential additional signals",
"The sample also contained ten additional signals that were well-constrained in period and amplitude but that did not exceed the detection threshold of $s = 10^{4}$ or were supported by data from only one instrument.",
"We call these signals SRCs (signals requiring confirmation).",
"They did, however, all exceed a less conservative threshold of $s = 150$ and it would thus be wrong to simply ignore the existence of these ”emerging” signals.",
"We took these signals into account when calculating the detection thresholds but we do not (yet) consider them to be candidate planets because we wish to avoid overestimation of the occurrence rates.",
"However, we do calculate the occurrence rates under the assumption that these additional signals are planet candidates but remain cautious when interpreting the corresponding results as some of these signals might be false positives caused by noise, insufficient modelling, stellar activity, and/or instrumental artefacts.",
"The ten additional signals are found in the combined data of GJ 433 (at a period of 36.0 days), GJ 551 (332 and 2200 days), GJ 821 (12.6 days), GJ 842 (190 days), GJ 855 (12.7 and 26.2 days), GJ 1009 (24.5 days), GJ 1100 (34.4 days) and in the UVES data of GJ 891 (30.6 days)."
],
[
"Occurrence rates",
"When calculating the planet occurrence rates as described in Section REF , we obtained some interesting estimates and show them in Table REF .",
"We divided the period space into four bins: between 1 and 10 days, 10 and 100 days, 100 and 1000 days, and 1000 and 10$^{4}$ days and the minimum mass space into three bins between 3 and 10 $M_{\\oplus }$ , 10 and 30 $M_{\\oplus }$ , and 30 and 100 $M_{\\oplus }$ (the last bin for masses above 100 M$_{\\oplus }$ is omitted from the table).",
"The resulting occurrence rates show several features that can be considered as representative of the underlying population of planets around M dwarfs.",
"We also show the detection probabilities of each bin based on the whole sample in the bottom of Table REF for comparison.",
"Table: Expected numbers of planets per star based on the sample of stars studied in the current work (top), potential numbers assuming that the ten signals requiring confirmation (SRCs) correspond to planet candidates (middle), and detection probability (DP) of the whole sample in each mass-period bin (bottom).According to the estimated occurrence rates in Table REF (top), the occurrence rate of super-Earths and more massive planets with $m_{p} \\sin i$ of up to 30 M$_{\\oplus }$ increases dramatically for periods between 10 and 100 days.",
"We find that the occurrence rate of planets with masses between 10 and 30 M$_{\\oplus }$ is 0.06$^{+0.11}_{-0.03}$ for the sample in this period range.",
"The most dramatic occurrence rate can be found for super-Earths (3 M$_{\\oplus } < m_{p} \\sin i \\le 10$ M$_{\\oplus }$ ) on orbits between 10 and 100 days of 1.02$^{+1.48}_{-0.69}$ per star, which indicates that such planets are very common around M dwarfs in the Solar neighbourhood.",
"Comparison with the results of [11], which is a similar survey of a larger sample of nearby M dwarfs in the southern sky, is difficult because they did not detect any candidates with masses between 10 and 100 M$_{\\oplus }$ and periods in excess of 10 days.",
"They did, however, detect two candidates with masses between 1 and 10 M$_{\\oplus }$ and periods between 10 and 100 days, which implied an occurrence rate of such companions of 0.54$^{+0.50}_{-0.16}$ .",
"These estimates are broadly consistent when bearing in mind the different sensitivities of the approaches and the fact that we were able to detect more such companions in the sample than [10] could in their larger sample of 104 M dwarfs.",
"Second, the occurrence rate of planets more massive than 10 M$_{\\oplus }$ on orbits with periods less than 10 days is very low, roughly 0.037$^{+0.011}_{-0.006}$ planets per star for planets less massive than 30 M$_{\\oplus }$ .",
"In comparison to the results obtained by the HARPS M dwarf survey, [11] reported the the occurrence rate of planets with periods from 1 to 10 days and masses between 10 and 100 M$_{\\oplus }$ to be 0.03$^{+0.04}_{-0.01}$ , which is consistent with our estimate of 0.037$^{+0.011}_{-0.006}$ .",
"This result is also consistent with the observation of [18] that the occurrence rate of planets with radii in excess of 2.0 R$_{\\oplus }$ on such orbital periods is roughly 0.05 and is therefore very likely a general feature for M dwarfs.",
"However, comparing our results and the results of [11] with Kepler occurrence rates based on estimated planetary radii cannot be performed confidently without bias.",
"We find only a slighly higher occurrence rate of 0.06$^{+0.11}_{-0.03}$ for planets with masses below 10 M$_{\\oplus }$ because there was only one such candidate in our sample.",
"These results are very difficult to compare with the results of [18] and [55] because the mass-radius relation is far from a well-established one for a range of masses and radii in the super-Earth regime.",
"Yet, [18] found the occurrence rate of planets to increase by a factor of ten when moving from radii interval of 2.8 - 5.7 R$_{\\oplus }$ down to 1.4 - 2.8 R$_{\\oplus }$ , which is as dramatic increase as we find when moving from masses between 10 and 30 M$_{\\oplus }$ down to the interval between 3 and 10 M$_{\\oplus }$ .",
"While this does not mean that the results are consistent, it suggests that this large change in abundance may have the same origin.",
"Obviously, this depends on whether the low-temperature Kepler sample and the M dwarf sample analysed in the current work are drawn from the same population.",
"Unlike the Kepler sample which comprises of more massive and brighter M dwarfs further out from the galactic plane, samples analysed in the current work and in [11] are likely to be approximately volume-limited.",
"Third, our results suggest that planets with masses below 100 M$_{\\oplus }$ and periods longer than 100 days might be abundant around nearby M dwarfs.",
"However, conclusions are very hard to draw at the moment because of the low number of such candidates in the combined UVES and HARPS data.",
"[11] did not observe any such planets in their sample even though the candidate orbiting GJ 433 could have been detected had they combined the HARPS data with the UVES velocities that were published in 2009.",
"The same team of researchers reported this candidate in [16], which means it could have been included in the results of [11].",
"Finally, out of the ten planet candidates in the sample, three appear to have orbits located within the respective stellar habitable zones according to the equations of [43] for the inner (moist greenhouse) and outer (maximum greenhouse) edge of the habitable zone.",
"This enables us to state that 30% of M dwarf planets in the current sample are located within the HZs.",
"However, we have to take into account the detection probability of the current data sets of habitable-zone planet candidates to estimate the occurrence rate of habitable-zone planets in a statistically representative, and thus meaningful, manner.",
"We achieve this by calculating the detection probabilities in Fig.",
"REF as a function of semi-major axis instead of orbital period for each star and combine the resulting probabilities in the same way as in Section REF but by limiting the analysis to the stellar habitable zones listed in Table REF .",
"We find that the occurrence rate of planets with minimum masses between 3 and 10 M$_{\\oplus }$ in the habitable zones of the sample stars is 0.21$^{+0.03}_{-0.05}$ planets per star and that of planets with minimum masses between 10 and 30 M$_{\\oplus }$ is 0.035$^{+0.013}_{-0.007}$ planets per star.",
"These estimates can be compared with the results of [11] according to whom M dwarfs would have on average 0.41$^{+0.54}_{-0.13}$ super-Earths (1 M$_{\\oplus } < m_{p} \\sin i < 10$ M$_{\\oplus }$ ) per star in the habitable zones, which appears to be consistent with our estimate considering the slightly different mass range.",
"These estimates also appear to be an order of magnitude greater than the occurrence rate estimates from the low-temperature sample of Kepler stars [18].",
"In particular, [18] estimated that the occurrence rate of Earth-size planets (0.5 - 1.4 R$_{\\oplus }$ ) is 0.06$^{+0.06}_{-0.03}$ and that of larger ones (1.4 - 4 R$_{\\oplus }$ ) is 0.03$^{+0.05}_{0.02}$ in the habitable zones of such stars.",
"However, [44] revised the estimates of [18] by calculating the habitable zones according to the modified equations of [43].",
"As a result, there are 0.48$^{+0.12}_{-0.24}$ planets with 0.5 R$_{\\oplus } < r_{p} <$ 1.4 R$_{\\oplus }$ per star in the habitable zones of the stars in the sample of [18], which increases to 0.51$^{+0.10}_{-0.20}$ when increasing the radius range to 2 R$_{\\oplus }$ .",
"These estimates are broadly consistent with our results.",
"We also estimated the planetary mass function for low-mass companions with $m_{p} \\sin i < 100$ M$_{\\oplus }$ .",
"We used the same minimum mass bins as in Table REF and plotted the resulting occurrence rate as a function of minimum mass in Fig.",
"REF (top panel) together with the mass distribution of the stars in the sample (bottom panel).",
"This figure shows that the mass function increases dramatically with decreasing mass, similar to the increase found for planets orbiting more massive stars [49], which suggests that the high occurrence rate of super-Earths with $m_{p} \\sin i <$ 10 M$_{\\oplus }$ corresponds to the rapid increase observed in the Kepler transit data for planets with radii less than 4 R$_{\\oplus }$ [55].",
"However, the uncertainties are still too high to quantify this increase in a meaningful way and therefore we do not attempt to estimate the mass function quantitatively.",
"Our occurrence rates denoted using the shaded histogram in Fig.",
"REF (top panel) for minimum masses between 3 - 10, 10 - 30, and 30 -100 M$_{\\oplus }$ are 1.08$^{+2.83}_{-0.72}$ , 0.10$^{+0.35}_{-0.04}$ , and 0.24$^{+0.39}_{0.16}$ planets per star, respectively.",
"Figure: Estimated mass-function of low-mass planets (top) based on the planet candidates in the sample and the mass-distribution of the M dwarfs in the sample (bottom).When calculating the occurrence rates under the assumption that the additional ten emerging signals, or SRCs, in the sample are also caused by planets, we obtained even higher occurrence rates for the low-mass planets around M dwarfs (Table REF , middle).",
"These numbers are consistent with but slightly higher than those obtained for the ten candidate planets in the sample.",
"However, we note that some of these signals could be false positives."
],
[
"New planetary systems",
"We have listed the new planet candidates from our analyses of the sample velocities in Table REF .",
"In this section, we discuss them briefly."
],
[
"GJ 433",
"According to our results, we could also identify the two planetary signals that have been reported in the HARPS data of GJ 433 [16].",
"We did not, however, detect any additional candidate planets satisfying our detection criteria in the combined HARPS and UVES data of the star.",
"We note that [11] also reported signals in the HARPS data around 30 days although there are significant problems with their tabulated solutionsIn their Table 7, [11] listed solutions that (i) have velocity amplitude estimates 100 times lower than their uncertainties, e.g.",
"$K = $ 31.6 $\\pm $ 3800.5 ms$^{-1}$ for Gl 54.1, implying that these cannot be real solutions in any meaningful way, and (ii) eccentricity estimates close to or equal to unity – the latter being impossible for periodicities – with occasional uncertainty estimates in excess of unity, e.g.",
"$e = 0.9 \\pm 8.9$ for Gl 54.1..",
"Nevertheless, the log-posterior of this target shows an emerging maximum at a period of 36 days for a three-Keplerian model together with a local maximum at 50 days, which suggests that a two-Keplerian model might not be a sufficiently accurate description of the combined UVES and HARPS velocities.",
"If either one (or both) of these emerging signals is confirmed by future data, GJ 433 would become one of the highly populated planetary systems around M dwarfs together with the famous planet hosting stars GJ 581, GJ 667C, GJ 163, and GJ 676A."
],
[
"GJ 682",
"In our sample, there are also two stars with two new candidate super-Earths orbiting them in the period space between 10 and 100 days (Table REF ).",
"GJ 682 is orbited by candidates with minimum masses of 4.4 [2.0, 8.1] and 8.7 [4.1, 14.5] M$_{\\oplus }$ on orbits with periods of 17.478 [17.438, 17.540] and 57.32 [56.84, 57.77] days, respectively.",
"The former candidate is located in the stellar habitable zone and can be classified as a habitable-zone super-Earth, although its minimum mass is consistent with (sub) Neptunian structure.",
"An interesting detail in the model probabilities of the GJ 682 velocities is that the one-Keplerian model is only 9900 times more probable than the model without any signals, which implies that a one-Keplerian model is not sufficiently good description of the data to enable the detection of a planet candidate according to our criteria.",
"However, a two-Keplerian model has a 1.8$\\times 10^{7}$ times greater probability than the one-Keplerian model, which implies that there is very strong evidence in favour of two candidates orbiting the star.",
"This means that when there are at least two signals of similar amplitude in the velocities, models with only one signal can be difficult to use to interpret the data because they are not good enough in describing the velocity variations.",
"GJ 682 is an inactive dwarf star with no signs of chromospheric activity [77] and has a projected rotation period of 10.7 days [60].",
"This suggests that the signals we have detected indeed are Doppler signatures of planets.",
"We note that possible additional signals in the GJ 682 data, and in particular their superpositions with the two detected ones, might cause biases to the obtained parameters of the two candidates listed in Table REF ."
],
[
"GJ 180",
"Another system with two candidate planets, GJ 180, corresponds to a remarkable configuration of super-Earths with an orbital period ratio of 7:5 – orbital periods of 17.380 [17.360, 17.398] and 24.329 [24.263, 24.381] days, respectively.",
"This suggests (although does not imply) the existence of a stabilising 7:5 mean motion resonance [37].",
"The outer candidate is located in the stellar habitable zone, which makes this candidate interesting because of its reasonably low minimum mass of 6.4 [2.3, 10.1] M$_{\\oplus }$ that enables its classification as a habitable-zone super-Earth.",
"This system, its detection, dynamical stability, and formation history, are discussed in detail in a separate publication [38]."
],
[
"GJ 422",
"The candidate planet GJ 422 b has a minimum mass of 9.9 [5.9, 15.5] M$_{\\oplus }$ and is thus a super-Earth or a sub-Neptunian planet, depending on the composition and atmospheric properties.",
"Its orbit is likely located within the stellar habitable zone of GJ 422."
],
[
"GJ 27.1",
"We also find a similar candidate orbiting GJ 27.1 with a minimum mass of 13.0 [6.4, 17.1] M$_{\\oplus }$ .",
"This candidate can be classified as a sub-Neptunian planet candidate because it is likely too massive to be considered a super-Earth with rocky composition.",
"With an orbital period of 15.819 [15.793, 15.842] days this signal is unlikely to be caused by stellar activity coupled with the rotation period whose projected estimate is 11.9 days [33]."
],
[
"GJ 160.2",
"In addition to GJ 433 b, GJ 160.2 b is the only candidate in our sample with an orbital period shorter than 10 days.",
"Candidates of this kind are the easiest ones to observe in our sample (see Fig.",
"REF ) and the fact that we only found two of them demonstrates that such planets are not very common around M dwarfs as also quantified by the occurrence rates in Table REF .",
"We note that GJ 160.2 might actually be a K dwarf [42], although [79] classified it as a M0 V star.",
"The candidate GJ 160.2 b is a hot sub-Neptunian planet.",
"The star has a projected rotation period of 43.6 days [33], which indicates that the signal in unlikely to be related to stellar rotation and thus activity."
],
[
"GJ 229",
"Finally, we report a discovery of a planet candidate orbiting GJ 229 with an orbital period of 471 [459, 493] days and a minimum mass of 32 [16, 49] M$_{\\oplus }$ .",
"This discovery makes the GJ 229 system one of the most diverse systems around M dwarfs because [56] reported a brown dwarf companion to the star based on direct imaging."
],
[
"Conclusions",
"We have presented our analysis of UVES velocities of a sample of 41 M dwarfs [79] when combining the velocities with HARPS precision data as obtained from the spectra available in the ESO archive.",
"As a result, we report the existence of eight new planet candidates around the sample stars (Tables REF and REF ) and confirm the existence of the two companions around GJ 433 [16] that exceed our conservative probabilistic detection threshold by making the statistical models more than 10$^{4}$ times more probable than models without the corresponding signals.",
"Among the most interesting targets in our sample are GJ 433, GJ 180, and GJ 682, with at least two candidate planets each.",
"We have also presented estimates for the occurrence rate of low-mass planets around M dwarfs (Table REF ) based on the current sample.",
"We find that low-mass planets are very common around M dwarfs in the Solar neighbourhood and that the occurrence rate of planets with masses between 3 and 10 M$_{\\oplus }$ is 1.08$^{+2.83}_{-0.72}$ per star.",
"This estimate is likely consistent with that suggested based on the Kepler results for a sample of stars with $T_{\\rm eff} < 4000$ K [18], [55], although the comparisons are not easily performed because we could assess the occurrence rates of companions with periods up to the span of the radial velocity data of a few thousand days.",
"On the other hand, we confirm the lack of planets with masses above 3 M$_{\\oplus }$ on orbits with periods between 1-10 days.",
"Such companions to low-mass stars have an occurrence rate of only 0.06$^{+0.11}_{-0.03}$ planets per star based on our sample.",
"There are nine targets in the sample that are also found in the sample of M dwarfs presented in [11]: GJ 1, GJ 176, GJ 229, GJ 357, GJ 433, GJ 551, GJ 682, GJ 699, GJ 846, and GJ 849.",
"Out of these nine stars, we found signals in the velocities of GJ 229, GJ 433, and GJ 682.",
"Our results are essentially similar for GJ 433, for which [11] reported a signal at 7.4 days and the same group reported another long-period signal when analysing the HARPS data in combination with the UVES data analysed here [16].",
"The planet candidates GJ 229 b, GJ 682 b and c have orbital periods of 471 [459, 493], 17.478 [17.438, 17.540], and 57.32 [56.84, 57.77] days.",
"[11] did not report any such periodicities for these stars.",
"We believe the reason is that we obtained HARPS-TERRA velocities from the HARPS spectra that are more precise for M dwarfs [3], combined the HARPS velocities with the UVES ones which provides more information on the underlying periodic signals regardless of whether the signals can be detected in the two data sets independently or not, and accounted for correlations in the velocity data that could disable the detections of low-amplitude signals if not accounted for [7], [70], [72].",
"We have compared our results briefly with those obtained by using the Kepler space-telescope [34], [18] in Section .",
"However, such a comparison is not necessarily reliable because the properties of Kepler's transiting planet candidates can only be discussed in terms of planetary radii and the radial velocity method can only be used to obtain minimum masses.",
"Because of this, it is not surprising that there are remarkable differences that are unlikely to arise by chance alone.",
"For instance, [18] estimated that there are roughly 0.15$^{+0.13}_{0.06}$ Earth-sized planets (radii between 0.5 and 1.4 R$_{\\oplus }$ ) in the habitable zones of cool stars (with $T_{\\rm eff} <$ 4000 K) and that the nearest such planet could be expected to be found within 5 pc with 95% confidence.",
"We calculated a similar estimate for candidates with masses between 3 and 10 M$_{\\oplus }$ and obtained an occurrence rate estimate of 0.21$^{+0.03}_{-0.05}$ planets per star that appears to be higher than the estimate of [18] despite the fact that we cannot assess the occurrence rates of planets with masses below 3 M$_{\\oplus }$ because we did not detect any such candidates orbiting the stars in the sample.",
"However, these estimates can only be compared in detail with a range of robust planet composition and evolution models in hand, and is beyond the current work.",
"According to our results, M dwarfs have very high rates of hosting systems of low-mass planets around them and have a high probability of being hosts to super-Earths in their habitable zones.",
"Together with the fact that radial velocity surveys can be used to obtain evidence for Earth-mass planets orbiting such stars, and the fact that M dwarfs are very abundant in the Solar neighbourhood, this makes them primary targets for searches of Earth-like planets, and possibly life, with current and future planet surveys."
],
[
"Acknowledgements",
"The authors acknowledge M. Zechmeister et al.",
"for making the UVES velocity data of the sample stars available and the significant efforts of the HARPS-ESO team in improving the instrument and its data reduction pipelines.",
"We also acknowledge the efforts of all the individuals that have been involved in observing the target stars with HARPS and UVES spectrographs because without such efforts, the current work would not have been possible.",
"JSJ acknowledges funding by Fondecyt through grant 3110004 and partial support from CATA (PB06, Conicyt), the GEMINI-CONICYT FUND and from the Comité Mixto ESO-GOBIERNO DE CHILE."
],
[
"HARPS-TERRA velocities",
"The HARPS-TERRA velocities obtained from the publicly available spectra in the ESO archive are presented in this section for all the targets for which at least two spectra were available.",
"The secular acceleration has been subtracted from every HARPS-TERRA data set.",
"Table: HARPS-TERRA velocity data of GJ 1.Table: HARPS-TERRA velocity data of GJ 27.1.Table: HARPS-TERRA velocity data ofGJ 118.Table: HARPS-TERRA velocity data of GJ 160.2.Table: HARPS-TERRA velocity data of GJ 173.Table: HARPS-TERRA velocity data of GJ 180.Table: HARPS-TERRA velocity data of GJ 218.Table: HARPS-TERRA velocity data of GJ 229.Table: HARPS-TERRA velocity data of GJ 357.Table: HARPS-TERRA velocity data of GJ 377.Table: HARPS-TERRA velocity data of GJ 422.Table: HARPS-TERRA velocity data of GJ 433.Table: HARPS-TERRA velocity data of GJ 510.Table: HARPS-TERRA velocity data of 551.Table: HARPS-TERRA velocity data of GJ 620.Table: HARPS-TERRA velocity data of GJ 637.Table: HARPS-TERRA velocity data of GJ 682.Table: HARPS-TERRA velocity data of GJ 699"
]
] | 1403.0430 |
[
[
"Magicity of the $^{52}$Ca and $^{54}$Ca isotopes and tensor contribution\n within a mean--field approach"
],
[
"Abstract We investigate the magicity of the isotopes $^{52}$Ca and $^{54}$Ca, that was recently confirmed by two experimental measurements, and relate it to like--particle and neutron--proton tensor effects within a mean--field description.",
"By analyzing Ca isotopes, we show that the like--particle tensor contribution induces shell effects that render these nuclei more magic than they would be predicted by neglecting it.",
"In particular, such induced shell effects are stronger in the nucleus $^{52}$Ca and the single--particle gaps are increased in both isotopes due to the tensor force.",
"By studying $N=32$ and $N=34$ isotones, neutron--proton tensor effects may be isolated and their role analyzed.",
"It is shown that neutron--proton tensor effects lead to increasing $N=32$ and $N=34$ gaps, when going along isotonic chains, from $^{58}$Fe to $^{52}$Ca, and from $^{60}$Fe to $^{54}$Ca, respectively.",
"The mean--field calculations are perfomed by employing one Skyrme parameter set, that was introduced in a previous work by fitting the tensor parameters together with the spin--orbit strength.",
"The signs and the values of the tensor strengths are thus checked within this specific application.",
"The obtained results indicate that the employed parameter set, even if generated with a partial adjustment of the parameters of the force, leads to the correct shell behavior and provides, in particular, a description of the magicity of $^{52}$Ca and $^{54}$Ca within a pure mean--field picture with the effective two--body Skyrme interaction."
],
[
"Introduction",
"We recently introduced tensor parametrizations for the phenomenological effective forces Skyrme and Gogny, by taking into account a tensor force of zero and finite range, respectively [1].",
"The tensor parameters were adjusted on top of already existing Skyrme and Gogny parametrizations, by employing a three-step fitting procedure inspired by Refs.",
"[2], [3] and by modifying also, simultaneously, the spin–orbit strength.",
"This work was done as an exploratory study to identify the correct signs and regions for the values of the tensor parameters, and can be viewed as a preparatory study for a global fit of all the parameters, especially in the Gogny case, where much less work has been done including the tensor force.",
"In the present work, we employ one of the Skyrme parametrizations introduced in Ref.",
"[1] and show that (even if it was not found with an adjustment of all the parameters of the force) it properly accounts, within a mean–field picture, for shell effects that were recently confirmed by experimental measurements, namely the magicity of the calcium isotopes $^{52}$ Ca and $^{54}$ Ca.",
"Experimental studies strongly indicate $N=32$ as a new magic number in Ca isotopes due to the high energy of the first 2$^+$ state in this nucleus [4], [5].",
"More recently, high–precision mass measurements were performed for the neutron–rich Ca isotopes $^{53}$ Ca and $^{54}$ Ca by employing the mass spectrometer of ISOLTRAP at CERN [6].",
"The found results and, in particular, the trend obtained for the two–neutron separation energies $S_{2n}$ , definitely confirmed the magicity of the nucleus $^{52}$ Ca.",
"For the nucleus $^{54}$ Ca, the first experimental spectroscopic study on low–lying states was performed very recently with proton knockout reactions at RIKEN [7].",
"This study (in particular, the high energy of the first $2^+$ state) provided a robust experimental signature indicating the magic nature of the nucleus $^{54}$ Ca.",
"Some comparisons with theoretical calculations have been included in the two experimental studies of Refs.",
"[6], [7].",
"In Ref.",
"[6], the experimental $S_{2n}$ energies have been compared with the results of microscopic calculations based on chiral interactions, with coupled–cluster and shell–model results, as well as with Energy Density Functional (EDF) results based on the mean–field approximation.",
"In Ref.",
"[7], shell–model results [8] have been compared with the experimental data and the important role played by the neutron–proton (n–p) tensor contribution has been investigated.",
"It was also mentioned in Ref.",
"[7] that recent calculations including three–body forces [9], [10] provide a very good agreement with the experimental results.",
"It was stressed that, in the shell–model calculations of Ref.",
"[8], the effect of three–body forces is included empirically.",
"This explains why the obtained results are very similar to those of Refs.",
"[9], [10], where the three-body contribution is taken into account.",
"In this work, we perform Hartree-Fock calculations in spherical symmetry and neglect pairing correlations.",
"The calculations are done in coordinate space.",
"We do not need more sophisticated models because our objective is to isolate the genuine tensor contribution: We show that the tensor force, and its induced like–particle and n–p effects, may describe the magicity of $^{52}$ Ca and $^{54}$ Ca within a simple mean–field scheme.",
"By analyzing Ca isotopes, it is shown in particular that an enhancement of the magicity of $^{52}$ Ca and $^{54}$ Ca may be predicted within mean–field calculations by including the like–particle contribution generated by the tensor force.",
"On the other side, by analyzing $N=32$ and $N=34$ isotones, the role played by the n–p tensor contribution is investigated.",
"One of the Skyrme parametrizations introduced in Ref.",
"[1] is employed, that was constructed on top of the SLy5 [11] Skyrme force.",
"In the three–step fitting procedure adopted in Ref.",
"[1], the first adjustment was done to tune the spin–orbit strength (before tuning the tensor parameters) to reproduce the neutron $f$ spin–orbit splitting in the nucleus $^{40}$ Ca.",
"This nucleus is spin saturated and, consequently, the tensor force does not have any effect on its spectroscopic properties.",
"After this first adjustment, the tensor parameters were tuned to reproduce the neutron $f$ spin–orbit splitting first in the nucleus $^{48}$ Ca and then in the nucleus $^{56}$ Ni.",
"Details about this procedure may be found in Ref.",
"[1].",
"The difference of our Skyrme parameter set with respect to the tensor parametrization published by Colò et al.",
"in Ref.",
"[12] (also introduced on top of SLy5) is that the spin–orbit strength is simultaneously modified in our case.",
"For our Skyrme set, the spin–orbit strength was reduced to 101 MeV fm$^5$ , with respect to the value in the original force, and the parameters responsible for the like–particle and the n–p tensor effect were adjusted to the values $\\alpha _{\\rm T} = -170$ and $\\beta _{\\rm T} = 122$ MeV fm$^5$ , respectively.",
"The parameters $\\alpha _T$ and $\\beta _T$ are related to the parameters $U$ and $T$ as follows, $\\nonumber \\alpha _{\\rm T}&=&\\frac{5}{12} \\, U, \\\\\\beta _{\\rm T}&=&\\frac{5}{24} \\left( T+U \\right) ,$ where $T$ and $U$ are the strengths of the Skyrme zero–range tensor force in even and odd states of relative motion, respectively [13].",
"The article is organized as follows.",
"In Sec.",
"we analyze the enhancement of magicity in the isotopes $^{52}$ Ca and $^{54}$ Ca, that is related to the like–particle tensor contribution.",
"In Sec.",
", $N=32$ and $N=34$ isotones are analyzed and the n–p tensor effects are investigated.",
"In Sec.",
"the two–neutron separation energies are compared with the experimental data for the nuclei $^{50}$ Ca, $^{52}$ Ca, and $^{54}$ Ca.",
"In Sec.",
"conclusions are drawn."
],
[
"Magicity of the isotopes $^{52}Ca$ and {{formula:0aef31ca-7f03-4e81-9ab7-80d9e8e2bb6e}} . Like–particle tensor effects",
"We evaluate the single–particle neutron gap for four Ca isotopes, the closed–shell nuclei $^{40}$ Ca and $^{48}$ Ca and the two systems that we wish to analyze here, $^{52}$ Ca and $^{54}$ Ca.",
"Ca isotopes are spin saturated in protons.",
"This means that tensor effects on neutron single–particle states may be induced in practice only by the like–particle neutron–neutron (n–n) contribution along the isotopic chain.",
"We show in Fig.",
"1 the single–particle gaps obtained for the four isotopes with our effective Skyrme force by quenching the like–particle tensor strength (dashed line) and by switching it on (solid line).",
"The reported gaps refer to $N=20$ , $N=28$ , $N=32$ , and $N=34$ for $^{40}$ Ca, $^{48}$ Ca, $^{52}$ Ca, and $^{54}$ Ca, respectively.",
"Two effects may be observed.",
"First, a global enhancement of the single–particle gaps is visible due to the tensor contribution.",
"This enhancement is found also for the closed–shell nucleus $^{48}$ Ca, where the gap is increased by 1.3 MeV.",
"The corresponding experimental value is 5.4 MeV [14].",
"This means that the inclusion of the n–n tensor contribution leads to the correct shift, towards the experimental result.",
"For the nucleus $^{40}$ Ca, the results obtained with and without the n–n tensor effect are obviously almost the same, since this nucleus is fully spin saturated and the tensor force is not active there.",
"We notice that the gap variation due to the inclusion of the n–n tensor contribution is more important for $^{52}$ Ca than for $^{48}$ Ca and $^{54}$ Ca, as will be explained below.",
"In the figure, also the results obtained with the original SLy5 force are reported (blue dotted line).",
"We may observe that our new set of parameters provides a global improvement of the results: it can be seen that both single–particle gaps obtained for $^{40}$ Ca and $^{48}$ Ca are closer to the experimental values (red triangles).",
"It is interesting to notice that, in the case of $^{40}$ Ca, where the tensor force does not play any role, the better agreeement with the experimental gap is entirely due to the reduced strength of the spin–orbit contribution.",
"Figure: (Color online) Single–particle neutron gaps for 40 ^{40}Ca, 48 ^{48}Ca, 52 ^{52}Ca, and 54 ^{54}Ca.",
"The black solid linecorresponds to the results obtained with the new Skyrme force; the black dashed line corresponds to the results obtained with the new Skyrme force by neglecting the n–n tensor contribution.The results obtained by using the original SLy5 force are also reported (blue dotted line).",
"The experimental values for 40 ^{40}Ca and 48 ^{48}Ca are represented by red triangles.Let us investigate in detail the tensor effect for each of the three isotopes where the tensor force contributes.",
"For $^{48}$ Ca, the neutron $1f_{7/2}$ state is filled.",
"Both high-$j$ (by using the terminology of Ref.",
"[15]) single–particle states $1f_{7/2}$ and $2p_{3/2}$ are pushed downwards, more strongly for the first state than for the second [Fig.",
"2(a)].",
"The net effect is that the gap between the two levels is increased.",
"Figure: (Color online) Neutron single–particle states for 48 ^{48}Ca (a), 52 ^{52}Ca (b), and 54 ^{54}Ca (c)with and without the like–particle tensor contribution.For $^{52}$ Ca, the neutron $2p_{3/2}$ state is filled.",
"The high-$j$ $2p_{3/2}$ state is pushed downwards whereas the low-$j$ $2p_{1/2}$ state is pushed upwards [Fig.",
"2(b)].",
"The single–particle gap is increased.",
"For $^{54}$ Ca, the neutron $2p_{1/2}$ state is filled.",
"Both low-$j$ single–particle states $1f_{5/2}$ and $2p_{1/2}$ are pushed upwards, more strongly for the first state than for the second [Fig.",
"2(c)].",
"The gap increases also this time.",
"In particular, the stronger enhancement of the single–particle gap obtained in $^{52}$ Ca is due to the fact that each single–particle state is pushed in the opposite direction by the tensor contribution.",
"Figs.",
"1 and 2 clearly show that the single–particle gaps are increased in $^{52}$ Ca and $^{54}$ Ca due to the like–particle tensor contribution.",
"This provides an enhancement of magicity for the two isotopes, with respect to what found by neglecting the tensor term.",
"In particular, the gap is increased for $^{52}$ Ca from 1.24 to 3.06 MeV and for $^{54}$ Ca from 0.45 to 1.86 MeV, providing in this way nuclei that have a stronger closed–shell nature.",
"With the opposite sign of the tensor strength, the effect would be the opposite, that is, the gap would be decreased by including the tensor contribution and would be in particular more strongly shrinked for the nucleus $^{52}$ Ca, where the $2p_{3/2}$ state would be shifted upwards and the $2p_{1/2}$ state would be pushed downwards.",
"The presently used signs of the tensor parameters are thus the correct ones that allow us to enhance the magic character of the nuclei under study."
],
[
"Magicity of the isotopes $^{52}Ca$ and {{formula:a2ee6e79-651f-4659-ab5a-0d56e09544b8}} . Neutron–proton tensor effects",
"To analyze the effects of the n–p tensor contribution, isotonic chains containing $^{52}$ Ca and $^{54}$ Ca have to be analyzed.",
"Let us start with the $N=32$ isotones $^{52}$ Ca, $^{54}$ Ti, $^{56}$ Cr, and $^{58}$ Fe.",
"In these nuclei, the last occupied neutron state is the state $2p_{3/2}$ , the upper level is the state $2p_{1/2}$ , and the neutron gap $N=32$ thus coincides with the spin–orbit splitting of the $2p$ neutron states.",
"Going from $^{58}$ Fe to $^{52}$ Ca, the $Z$ number is reduced from 26 to 20.",
"The occupation of the proton state $1f_{7/2}$ is thus decreased from 6 to 0.",
"The n–p tensor effect is expected to provide an attractive interaction between the proton state $1f_{7/2}$ and the neutron state $2p_{1/2}$ and a repulsive interaction between the proton state $1f_{7/2}$ and the neutron state $2p_{3/2}$ in a given nucleus.",
"Such tensor contribution is thus expected to induce a reduction of the neutron $p$ spin–orbit splitting, that is, a reduction of the $N=32$ gap.",
"This effect is however expected to be weakened along the isotonic chain, as far as the occupation of the proton $1f_{7/2}$ state is reduced, that is, going from $^{58}$ Fe to $^{52}$ Ca.",
"With the reduction of the n–p effect, the gap is expected to increase going from $^{58}$ Fe to $^{52}$ Ca.",
"This is shown in Fig.",
"3, where the gap is plotted (black solid line) and in Fig.",
"4, where the involved proton and neutron single–particle energies are displayed.",
"In Fig.",
"4 one can see that, moving from the left to the right (that is, towards more neutron–rich nuclei along the isotonic chain), the energy of the proton state moves towards lower values and the energies of the neutron states are pushed upwards, as expected in mean–field calculations.",
"On top of the global mean–field evolution, we should isolate the tensor contribution.",
"From Fig.",
"3, one observes that the expected effect of enhancement of the gap going from $^{58}$ Fe to $^{52}$ Ca looks very weak.",
"To better understand how the tensor force acts in this particular case, we compare the found results with those obtained by switching off the n–p tensor strength.",
"The corresponding values are shown in Fig.",
"3 (red dashed line).",
"One can observe that, without the n–p tensor contribution, the values are similar but the trend is the opposite, that is, the gap is reduced when passing from $^{58}$ Fe to $^{52}$ Ca.",
"The tensor force changes this trend.",
"To isolate all the tensor contributions, we have repeated the same calculations by quenching also the parameter responsible for the like–particle tensor effect.",
"The corresponding results are shown in the figure by a green dot-dashed line.",
"It is clear that the like–particle tensor contribution provides an enhancement of the gap but the change of trend is due only to the n–p contribution.",
"It is also interesting to compare the present results with the values obtained with the original SLy5 force (blue dotted line in Fig.",
"3).",
"We can see that the trend is very similar to that obtained in our case by switching off the n–p tensor strength or both tensor strengths.",
"There is however a shift of the values: the results obtained with the original parametrization are located between the green dot–dashed and the red dashed curves.",
"The effect of the reduction of the spin–orbit induces the shift from the dotted to the dot–dashed curve and the inclusion of the like–particle tensor strength strongly pushes the values upwards.",
"Finally, the inclusion of the n–p tensor contribution determines the change of slope and provides increasing (instead of decreasing) gaps going from $^{58}$ Fe to $^{52}$ Ca.",
"This is what expected following the arguments based on shell–model calculations and presented in Ref.",
"[7], where this analysis is done for $N=34$ isotones.",
"Figure: (Color online) Neutron gap N=32N=32 calculated for the isotones 52 ^{52}Ca, 54 ^{54}Ti, 56 ^{56}Cr, and 58 ^{58}Fe.Figure: (Color online) Single–particle energies of the proton state 1f 7/2 1f_{7/2} and of the neutron states2p 3/2 2p_{3/2} and 2p 1/2 2p_{1/2} for the isotones 52 ^{52}Ca, 54 ^{54}Ti, 56 ^{56}Cr, and 58 ^{58}Fe.We repeat the same analysis for $N=34$ isotones for the nuclei $^{54}$ Ca, $^{56}$ Ti, $^{58}$ Cr, and $^{60}$ Fe.",
"The results are plotted in Figs.",
"5 and 6.",
"The difference with respect to the previous case is that the last occupied neutron state is now the state $2p_{1/2}$ and the neutron gap $N=34$ is calculated between the states $1f_{5/2}$ and $2p_{1/2}$ .",
"Now, the n–p tensor effect associated to the proton state $1f_{7/2}$ induces for both neutron states an attractive interaction, stronger for the $f$ state than for the $p$ state.",
"This means that the gap is expected also in this case to be reduced in a given nucleus owing to the n–p tensor contribution.",
"Again, when this contribution is weakened going from $^{60}$ Fe to $^{54}$ Ca, the gap is eventually expected to increase.",
"This effect may be seen in Fig.",
"5 (black solid line).",
"In Fig.",
"6 the involved neutron and proton single–particle energies are represented.",
"Again, the global mean–field evolution provides a reduction of the proton energy and an enhancement of the neutron energies, when going towards more neutron–rich nuclei along the isotonic chain.",
"On top of this general effect, the tensor effect has to be disentangled.",
"This is done by plotting in Fig.",
"5 the results obtained by switching off the n–p tensor strength (red dashed line).",
"The gap still increases, but much less strongly than in the full calculations.",
"By switching off also the like–particle tensor strength, the obtained values are reported by a green dot-dashed line.",
"The value for the nucleus $^{60}$ Fe is not reported.",
"The Hartree–Fock calculation does not converge in this case.",
"This is probably due to the fact that the neutron states $2p_{1/2}$ and $1f_{5/2}$ become almost degenerate and cross each other at each iteration.",
"The values obtained with the original SLy5 force are also presented (blue dotted line).",
"Also this time one observes that the slope in the results obtained with the original SLy5 force is very similar to that obtained by switching off, in our case, the n–p tensor strength or both tensor strenghts.",
"Also in this case there is a shift between the dotted curve and the dashed and dot–dashed curves.",
"The spin–orbit reduction in the new parametrization determines the shift to lower values from the dotted to the dot–dashed curve.",
"The inclusion of the like–particle tensor contribution induces a strong shift to higher energies (dashed curve).",
"Finally, the inclusion of the n–p tensor contribution leads to a change of slope (solid curve).",
"We notice that this time, even without the inclusion of the n–p tensor contribution, the mean–field calculations provide an enhancement of the gap when going from $^{60}$ Fe to $^{54}$ Ca.",
"The tensor force leads however to a much stronger effect, affecting significantly the slope.",
"As was mentioned in Sec.",
"I, the calculations are done in spherical symmetry and the effect of possible deformations is thus neglected.",
"The inclusion of possible small deformations for some $N=32$ and $N=34$ isotones would probably modify quantitatively our predictions but we do not expect that the qualitative trends and the general conclusions of this work would be changed: the tensor force would always shift in the same direction the single–particle energies; the qualitative evolution of the corresponding gaps would not be expected to be stronlgy modified.",
"Figure: (Color online) Neutron gap N=34N=34 calculated for the isotones 54 ^{54}Ca, 56 ^{56}Ti, 58 ^{58}Cr, and 60 ^{60}Fe.Figure: (Color online) Single–particle energies of the proton state 1f 7/2 1f_{7/2} and of the neutron states2p 1/2 2p_{1/2} and 1f 5/2 1f_{5/2} for the istones 54 ^{54}Ca, 56 ^{56}Ti, 58 ^{58}Cr, and 60 ^{60}Fe."
],
[
"Masses and separation energies",
"It was stressed in Ref.",
"[1] that the new sets may induce non negligeable effects on the masses of some nuclei, because the spin–orbit strength was modified without refitting also the other parameters of the force.",
"We have thus checked whether the values of the masses predicted for the Ca isotopes under study are reasonable with the new parametrization.",
"We have found the following binding energies: 343.83, 415.91, 438.62, and 445.21 MeV for the nuclei $^{40}$ Ca, $^{48}$ Ca, $^{52}$ Ca, and $^{54}$ C, respectively.",
"These values have to be compared with the binding energies obtained with the original SLy5 force, that are 344.07, 415.92, 437.25, and 444.94 MeV for $^{40}$ Ca, $^{48}$ Ca, $^{52}$ Ca, and $^{54}$ C, respectively.",
"We observe that the deviations are not very important.",
"The largest deviation between the new Skyrme values and those obtained with SLy5 is 0.3%.",
"In Ref.",
"[6], the two–neutron separation energies are compared with several theoretical models.",
"Experimentally, a clear signature of shell closure for the nucleus $^{52}$ Ca is the important reduction of the $S_{2n}$ value going from $^{52}$ Ca to $^{54}$ Ca.",
"This new experimental value is now available owing to the high–precision mass measurement of $^{54}$ Ca reported in Ref.",
"[6]: going from $^{50}$ Ca to $^{52}$ Ca, the value of $S_{2n}$ remains almost the same, and it has a sudden decrease from $A=52$ to $A=54$ .",
"Several EDF results are reported in Fig.",
"3(b) of Ref.",
"[6].",
"It is commented there that, in general, these models do not provide the correct trend for the $S_{2n}$ energies going from $A=50$ to $A=54$ : they predict a very smooth (almost linear) change in the $S_{2n}$ values.",
"We have thus computed the two–neutron separation energies $S_{2n}= E(N,Z)-E(N-2,Z)$ , where $E(N,Z)$ is the binding energy of the nucleus $(N,Z)$ , and compared them with the experimental values for the isotopes $^{50}$ Ca, $^{52}$ Ca, and $^{54}$ Ca.",
"This is shown in Fig.",
"7 where also the experimental values (including those of Ref.",
"[16] and the new data of Ref.",
"[6]) are presented.",
"Figure: (Color online) Two–neutron separation energies for the isotopes 50 ^{50}Ca, 52 ^{52}Ca, and 54 ^{54}Ca calculated with the new Skyrme set.",
"The experimental values are reported , .",
"The theoretical value for the nucleus 56 ^{56}Ca is also shown.",
"In the inset the difference between the theoretical and the experimental values is displayed.We observe that the agreement with the experimental values is quite good and that the predicted change with respect to $A$ is not linear.",
"The good agreement can also be seen in the inset of the figure, where the differences between the theoretical and the experimental values of $S_{2n}$ are reported.",
"We have to mention that the correct trend for the $S_{2n}$ values may be found also by using the original SLy5 force, without including the tensor term.",
"However, the signatures of magicity are slightly stronger with the present set of parameters.",
"With the present modified SLy5, the so–called two–neutron shell gap, calculated as $S_{2n}(N,Z)-S_{2n}(N+2,Z)$ , is equal to 5.1 MeV for $^{52}$ Ca, to be compared with an experimental value of almost 4 MeV [6].",
"The reduction of the $S_{2n}$ value, found going from $A=54$ to $A=56$ , cannot be compared with any experimental result because the mass of the nucleus $^{56}$ Ca was not yet measured experimentally.",
"The experimental $S_{2n}$ value is thus still unknown at $A=56$ .",
"However, our theoretical prediction, that leads to a quite significant reduction from $A=54$ to $A=56$ , is coherent with the new spectroscopic data of Ref.",
"[7], that indicate a magic nature for the nucleus $^{54}$ Ca.",
"One should mention however that the inclusion of possible pairing correlations in the nucleus $^{56}$ Ca would render this system more bound and would thus provide a higher value of $S_{2n}$ at $A=56$ .",
"We can conclude that the present EDF results display the correct trend for the values of $S_{2n}$ in the region from $^{50}$ Ca to $^{54}$ Ca (weak change from $A=50$ to $A=52$ and significant change from $A=52$ to $A=54$ )."
],
[
"Conclusions",
"In this work, we have examined some shell effects that may be clearly related to the contributions induced by a tensor force within the mean–field framework with the Skyrme interaction.",
"The employed tensor force is of zero range.",
"We use one parametrization that was introduced in a previous work [1] on top of the Skyrme SLy5 force [11].",
"In this parameter set, the spin–orbit strength was adjusted together with the tensor parameters.",
"With this parameter set, we have performed Hartree–Fock calculations in spherical symmetry.",
"We are aware that many effects are disregarded by using this simple theoretical model, but our objective is to isolate the genuine effects coming from the inclusion of the tensor force in a mean–field model.",
"The magicity of the two Ca isotopes $^{52}$ Ca and $^{54}$ Ca has been confirmed by two recent experimental measurements [6], [7].",
"We have shown here that the introduction of the tensor force leads to an enhancement of magicity in the two isotopes.",
"By analyzing Ca isotopes, it is shown that the like–particle tensor contribution renders these nuclei more magic than they would be predicted by neglecting it.",
"By studying $N=32$ and $N=34$ isotones, the neutron–proton tensor effects are identified and isolated with respect to the effects coming from the like–particle tensor contribution.",
"It is shown that the $N=32$ and $N=34$ neutron gaps are predicted to increase along isotonic chains, going from $^{58}$ Fe to $^{52}$ Ca, and from $^{60}$ Fe to $^{54}$ Ca, respectively.",
"In the case $N=32$ , the n–p tensor contribution is responsible for the increasing trend (without this contribution, the gap decreases).",
"In the case $N=34$ , the gap increases even without the n–p tensor contribution, but the n–p tensor contribution determines an important change of the slope.",
"A check on the masses of the Ca isotopes under study is performed and the two–neutron separation energies are compared with the experimental data for $^{50}$ Ca, $^{52}$ Ca, and $^{54}$ Ca.",
"A good agreement is found: from $A=50$ to $A=52$ , the $S_{2n}$ value does not change strongly, whereas it is significantly reduced from $A=52$ to $A=54$ .",
"This provides an evidence for the shell closure $N=32$ in Ca isotopes.",
"The application described in this work confirms the robustness of the findings of Ref.",
"[1] about the signs and the values for the tensor parameters.",
"The employed parameter set, even if introduced with a partial adjustment of the parameters (the other parameters of the forces are not modified, with the exception of the spin–orbit strength), provides the correct expected shell effects.",
"Different signs of the tensor parameters would generate un uncorrect behavior if compared with the experimental results.",
"The present tensor parametrization (and its induced like–particle and neutron–proton effects) allows us in particular to describe, within a simple mean–field picture, the magicity of the Ca isotopes $^{52}$ Ca and $^{54}$ Ca."
]
] | 1403.0366 |
[
[
"Baryon femtoscopy in heavy-ion collisions at ALICE"
],
[
"Abstract In this report, femtoscopic measurements with proton-proton, antiproton-antiproton, proton-antiproton, proton-antilambda, antiproton-lambda and lambda-antilambda pairs in Pb-Pb collisions at sqrt(s_NN)=2.76 TeV registered by ALICE at the LHC are presented.",
"Emission source sizes extracted from the correlation analysis with (anti)protons grow with the event multiplicity, as expected.",
"A method to extract the interaction potentials (e.g.",
"for proton-antilambda and antiproton-lambda pairs) based on femtoscopy analysis is discussed.",
"The importance of taking into account the so-called residual correlations induced by pairs contaminated by secondary particles is emphasized for all analyses mentioned above."
],
[
"Introduction",
"ALICE, A Large Ion Collider Experiment at the Large Hadron Collider (LHC) at CERN is dedicated to the study of the properties of the quark-gluon plasma (QGP), the state of matter described by partonic degrees of freedom [1].",
"Such a state may be created in ultra-relativistic collisions of heavy ions.",
"The information about the spatio-temporal characteristics of the final hadronic system of the collision can be deduced from analysis techniques [2] based on the measured two-particle correlations at low relative momenta and the knowledge of their sources; namely, Final State Interactions (Coulomb and Strong) and Quantum Statistics (in the case of identical particles).",
"The two-particle correlation can be expressed by the following equation [3]: $C({k^*}) = \\int S({k^*},{r^*}) \\Psi ({k^*},{r^*}) \\mathrm {d}^4{r^*},$ where $C$ is the measured correlation, ${k^*}$ is half of the pair relative momentum, ${r^*}$ is the pair relative separation, $S$ is the source function and $\\Psi $ is the pair wave function.",
"Generally, the two-particle interaction is known for the most commonly analysed types of pairs, e.g.",
"for pions and kaons.",
"Hence, the measured correlation function may be used to extract the information about the source function (e.g.",
"a Gaussian shape of the source can be assumed to determine the width of the distribution).",
"For instance, $\\Psi $ for pp pairs contains components from Fermi-Dirac statistics, Coulomb interactions (both leading to anticorrelation) as well as strong Final State Interactions (resulting in a characteristic peak at ${k^* \\approx 20}$ MeV/$c$ , whose height is sensitive to the femtoscopic radius).",
"The goal of the baryon-(anti)baryon femtoscopic analysis is to complement the measurements of the transverse mass ($m_{\\mathrm {T}}$ ) dependence of the ”homogeneity lengths” [12] (the sizes of the phase space of emitted particles with specific velocities) extracted so far from correlations of pions and kaons.",
"Based on such measurements, one is able to verify the $m_{\\mathrm {T}}$ scaling of the source size [13], commonly explained in the framework of hydrodynamic models as a signature of collective behaviour of matter created in heavy-ion collisions.",
"However, the strong FSI parameters for many systems such as p$\\bar{\\mathrm {\\Lambda }}$ , $\\bar{\\mathrm {p}} \\Lambda $ and $\\Lambda \\bar{\\mathrm {\\Lambda }}$ are not known precisely.",
"Experimental data of cross-sections for these pairs are very limited.",
"Only phenomenological models exist, predicting the values of the parameters of the strong interaction [4], [5], [6], [7].",
"Therefore, femtoscopic techniques may be helpful in constraining the strong FSI parameters for the aforementioned systems.",
"This can be achieved by inverting the standard femtoscopic procedure by fixing $S$ in Eq.",
"(REF ) to extract $\\Psi $ .",
"For instance, one can measure the $\\bar{\\mathrm {p}} \\Lambda $ correlations and assume that the source size for this pair is the same as for the pp system (it should be similar indeed, due to the comparable masses of the p and $\\Lambda $ baryons).",
"Then, the Lednicky-Lyuboshitz analytical model [8] might be used to infer the information about the FSI parameters.",
"In this model the correlation function is calculated as the square of the wave function averaged over the total spin and the relative separation of the points of particles emision in the pair rest frame.",
"One should emphasize that the results of such an analysis can be applied in modelling the phase of hadronic rescatterings.",
"In particular, one can use the extracted values of the parameters which account for annihilation in the baryon-antibaryon systems.",
"This may explain the observed decrease of baryon yields which are below expectations from thermal models at the LHC energies [9], [10].",
"Also, more precise knowledge of the parameters of strong interactions for baryon pairs is of great importance for astrophysics; in particular in understanding the properties of neutron stars [11].",
"The aforementioned analyses require taking into account the so-called residual correlations [14], [15], [16].",
"The reason for that is the following: in heavy-ion collisions in a collider setup, a significant number of secondary baryons (products of weak decays) cannot be distinguished from the primary ones (prompt particles produced in the collision).",
"Therefore, a sizeable number of analysed pairs are composed of at least one secondary particle.",
"Hence, the correlations of such pairs arise due to the interactions between the parent particles.",
"The effect becomes more and more important when the momentum of the decay products, in the rest frame of the parent particle, is relatively small.",
"Then, the daughter particle will move in a direction similar to the direction of the parent particle and the correlation, which diluted somewhat, will still be observed for such a pair."
],
[
"Data analysis",
"The analysis was performed over a sample of about 30 million Pb–Pb collisions at $\\sqrt{s_{\\mathrm {NN}}}$$=2{.",
"}76$ TeV registered by ALICE [1].",
"The VZERO detector (the forward scintillator arrays) was used for centrality determination and triggering.",
"Events with an interaction vertex within 8 cm from the nominal interaction point in the beam axis were selected.",
"The Time Projection Chamber (TPC) was used for track reconstruction.",
"The identification of (anti)protons was based on the measurements of the specific ionisation energy loss by the TPC and the time-of-flight by the TOF and T0 detectors, in conjunction with the momentum of the particle inferred from the curvature of its trajectory in the magnetic field.",
"$\\Lambda $ and $\\bar{\\Lambda }$ baryons were selected using their decay topology by identifying the daughter tracks measured in the TPC and TOF.",
"Only particles within the pseudorapidity range $|\\eta | < 0.8$ were accepted in the analysis which corresponds to the region of uniform TPC acceptance.",
"To reduce the contribution from secondary particles, a cut on the distance of closest approach of the particle trajectory to the primary vertex was applied.",
"The correlation function was obtained by dividing the signal and the uncorrelated distributions.",
"The former was formed by calculating the relative momentum (in one dimension, in the Pair Rest Frame) $q_{\\mathrm {inv}}=2 \\cdot k^{*} = |q-P(m_1^2-m_2^2)/P^2|$ ($q$ is the pair relative momentum, $P$ is the pair total momentum and $m_1$ , $m_2$ denote the masses) for particles from the same event.",
"The uncorrelated distributions were created by pairing particles from different events.",
"Pair selections were applied to account for fake pairs with low relative momentum due to the so-called splitting (i.e., one particle reconstructed as two tracks) and two-track inefficiency due to merging, (i.e., two tracks reconstructed as one).",
"These selection criteria made use of the ratio of the detector signals (clusters) shared by two tracks to all clusters and the angular distance between two tracks inside the TPC."
],
[
"Results",
"In fig.",
"REF , $\\bar{\\mathrm {p}} \\Lambda $ correlations obtained using the Lednicky-Lyuboshitz analytical model [8] are shown.",
"They are calculated using the formula given by Eq.",
"(REF ).",
"$C({k^*}) = 1+ \\sum _{S} \\rho _S \\left[{1 \\over 2} \\left| {f^S(k^{*})}\\over {R}\\right|^2 \\left( 1-{{d_0^S}\\over {2 \\sqrt{\\pi } R} } \\right) + {{2 \\Re f^S(k^{*})}\\over {\\sqrt{\\pi } R}} F_1 (2k^{*}R) - {{\\Im f^S(k^{*})}\\over {R}}F_2(2k^{*}R) \\right],$ where $F_1(z) = {\\int _0^z \\mathrm {d}x {{\\mathrm {e}^{x^2-z^2}}\\over {z}}}$ , $F_2(z)=1-{{1-\\mathrm {e}^{-z^2}}\\over {z}}$ , $f^S(k^{*}) = (1/{f_0^S+{1 \\over 2} d_0^S k^{*} -ik^{*}})^{-1}$ is the spin-dependent scattering amplitude, $\\rho _s$ is the fraction of pairs in each total spin state S. In this analysis only spin-averaged values are considered.",
"The effective radius of the interaction $d_0^S$ is set to zero, following the procedure applied in [17].",
"Therefore, the scattering amplitude depends on the scattering length $f_0^S$ .",
"The contribution from the real part of the scattering amplitude can lead to either positive or negative correlation but it is always narrow in $k^{*}$ in comparison to the imaginary part, which accounts for annihilation in the Final State Interactions.",
"Therefore, only the non-zero $\\Im f^S(k^{*})$ may lead to an anticorrelation wide in $k^{*}$ .",
"Figure: p ¯Λ\\bar{\\mathrm {p}} \\Lambda correlation functions and their components calculated from the Lednicky-Lyuboshitz analytical model for different FSI parameters (see text for details).",
"The left plot shows the results of calculations with the imaginary part of f S (k * )f^S(k^{*}) set to zero.",
"The right plot is the example of the calculations with the non-zero imaginary part of f S (k * )f^S(k^{*}).The p$\\bar{\\mathrm {p}}$ correlation functions, for different centrality classes are presented in fig.",
"REF .",
"As extensively studied and discussed in the literature [3], they show a peak at low relative momenta, due to the Coulomb attraction.",
"For larger values of $k^{*}$ a wide anticorrelation is observed.",
"It is caused by the annihilation component of the strong FSI, similarly to the $\\bar{\\mathrm {p}} \\Lambda $ case described above with the Lednicky-Lyuboshitz analytical model.",
"Both effects are necessary to describe the data which is compatible with the baryon annihilation in the final state.",
"Moreover, one can notice that the anticorrelation is stronger for more peripheral events which reflects the smaller size of the source.",
"Figure: pp ¯\\bar{\\mathrm {p}} correlation functions from the Pb–Pb collisions at s NN \\sqrt{s_{\\mathrm {NN}}}=2.76=2.76 TeV.The $\\bar{\\mathrm {p}} \\Lambda $ and p$\\bar{\\mathrm {\\Lambda }}$ correlation functions are shown in fig.",
"REF .",
"For these systems, the strong FSI is the only source of femtoscopic correlations.",
"No significant difference between $\\bar{\\mathrm {p}} \\Lambda $ and p$\\bar{\\mathrm {\\Lambda }}$ is observed, as expected.",
"A wide negative correlation can be noticed.",
"Based on the explanation by the Lednicky-Lyuboshitz model, it is qualitatively consistent with the annihilation from the FSI.",
"Figure: pΛ ¯\\bar{\\mathrm {\\Lambda }} and p ¯Λ\\bar{\\mathrm {p}} \\Lambda correlation functions from the Pb–Pb collisions at s NN \\sqrt{s_{\\mathrm {NN}}}=2.76=2.76 TeV.In fig.",
"REF , the $\\Lambda \\bar{\\mathrm {\\Lambda }}$ correlation functions are presented.",
"The suppression at low relative momenta for three centrality ranges is clearly visible.",
"Again, it is compatible with baryon-antibaryon annihilation (see above).",
"The strength of the correlation decreases for more central events which indicates that the emitting source size becomes larger for such events.",
"Figure: ΛΛ ¯\\Lambda \\bar{\\mathrm {\\Lambda }} correlation functions from the Pb–Pb collisions at s NN \\sqrt{s_{\\mathrm {NN}}}=2.76=2.76 TeV.The femtoscopic correlations for pp and $\\bar{\\mathrm {p}}\\bar{\\mathrm {p}}$ (shown in the left panel of fig.",
"REF ) are due to the interplay of Fermi-Dirac statistics, Coulomb and Strong FSI resulting in a distinctive maximum for ${q_{\\mathrm {inv}}=2 k^{*}~\\approx ~40}$ MeV/$c$ [8].",
"Since the feed-down from weak decays cannot be neglected in ultra-relativistic heavy-ion collisions, the residual correlations from the p$\\Lambda $ system in the pp correlations ought to be taken into consideration.",
"As a $\\Lambda $ baryon decays into a proton and a $\\pi ^{-}$ with a small decay momentum with respect to the proton mass, femtoscopic correlations between a primary p and a $\\Lambda $ may be detected for a pair composed of the primary p and the secondary p from the $\\Lambda $ decay.",
"The method of simultaneously fitting the pp ($\\bar{\\mathrm {p}}\\bar{\\mathrm {p}}$ ) and p$\\Lambda $ ($\\bar{\\mathrm {p}} \\bar{\\Lambda }$ ) correlations was applied to extract the femtoscopic radii from proton femtoscopy.",
"As one can observe in fig.",
"REF the contribution from genuine $\\bar{\\mathrm {p}}\\bar{\\mathrm {p}}$ correlations in the measured correlation function describes the maximum at $k^{*}~\\approx ~20$ MeV/$c$ .",
"Also the wide correlation excess is fairly well reproduced by the residual correlations from the $\\bar{\\mathrm {p}} \\bar{\\Lambda }$ system.",
"The same combination of effects was needed to describe the p$\\bar{\\mathrm {p}}$ correlations (see the right panel of fig.",
"REF ).",
"Figure: Example of the fit to the p ¯p ¯\\bar{\\mathrm {p}}\\bar{\\mathrm {p}} and pp ¯\\bar{\\mathrm {p}} correlation function, taking into account contributions from residual correlations.The shape and strength of these residual correlations might be sensitive to the FSI parameters, though.",
"One can incorporate these parameters from existing models and particles' momenta generated by a model of heavy-ion collision (e.g.",
"THERMINATOR 2 [18]) to simulate the impact of residual correlations on the measured correlations for baryon pairs.",
"Then, the same analysis procedure (as the one depicted in fig.",
"REF ) may be performed for the obtained correlation functions to fit not only the radii but also the FSI parameters.",
"Therefore, the described method may be used to constrain their values.",
"This is especially important for baryon-antibaryon pairs where the FSI parameters are known with large uncertainty or even unknown.",
"The pair transverse momentum ($k_{\\mathrm {T}}$ =${1 \\over 2} {|\\vec{p}_{\\mathrm {T,1}}+\\vec{p}_{\\mathrm {T,2}}}|$ ) dependence of the invariant radii deduced from proton femtoscopy is presented in fig.",
"REF .",
"The extracted radii increase with event multiplicity and slightly decrease with $k_{\\mathrm {T}}$ .",
"Figure: Pair transverse momentum dependence of the radius parameter extracted from correlations of protons."
],
[
"Summary",
"Femtoscopic correlation functions for baryon-(anti)baryon pairs are presented.",
"The evident anticorrelation observed in p$\\bar{\\mathrm {p}}$ , p$\\bar{\\mathrm {\\Lambda }}$ , $\\bar{\\mathrm {p}} \\Lambda $ and $\\Lambda \\bar{\\mathrm {\\Lambda }}$ correlations is consistent with the hypothesis that annihilation from the strong FSI may cause that lower baryon yields are observed compared to thermal models at LHC.",
"The importance of the residual correlations in baryon femtoscopy is stressed.",
"The radii deduced from proton femtoscopy analysis show an increase with event multiplicity and decrease with pair transverse momentum."
],
[
"Acknowledgements",
"This work has been financed by the Polish National Science Centre under decision no.",
"2011/01/B/ST2/03483."
]
] | 1403.0462 |
[
[
"Top Cross-Sections and Single Top"
],
[
"Abstract This paper summarizes top quark cross-section measurements at the Tevatron and the LHC.",
"Top quark pair production cross-sections have been measured in all decay modes by the ATLAS and CMS collaborations at the LHC and by the CDF and D0 collaborations at the Tevatron.",
"Single top quark production has been observed at both the Tevatron and the LHC.",
"The t-channel and associated Wt production modes have been observed at the LHC and evidence for s-channel production has been reported by the Tevatron collaborations."
],
[
"Introduction",
"The top quark is central to understanding physics in the Standard Model (SM) and beyond.",
"This paper summarizes top-quark related measurements from the Tevatron proton-antiproton collider at Fermilab and from the Large Hadron Collider (LHC), the proton-proton collider at CERN.",
"The top quark couplings to the gluon, to the $W$ boson, and now also to the photon and $Z$ boson are all probed in these measurements.",
"Searches for new physics in the top quark final state look for new particles and new interactions.",
"The Tevatron operation ended in 2011, with CDF and D0 each collecting 10 fb$^{-1}$ of proton-antiproton data [1] at a center-of-mass (CM) energy of 1.96 TeV.",
"The ATLAS [2] and CMS [3] collaborations at the LHC have reported measurements at CM energies of 7 TeV and 8 TeV, with up to 4.9 fb$^{-1}$ and up to 20 fb$^{-1}$ , respectively.",
"This paper reports recent measurements of top pair production, of single top production, as well as recent searches for new physics in top quark final states.",
"Giving a complete overview of all activities is not possible here, but I will highlight relevant measurements and new developments.",
"Section summarizes top quark pair production measurements, Section summarizes single top quark production, Section presents new physics searches in the top quark sector, and Section gives a summary."
],
[
"Top quark pair production",
"Top quark pair production proceeds mainly via gluon initial states at the LHC, shown in Fig.",
"REF (a), and mainly via quark-antiquark annihilation at the Tevatron, shown in Fig.",
"REF (b).",
"Figure: Feynman diagrams for top pair production (a) via gluon fusionand (b) via quark-antiquark annihilation, and (c) for top quark decay.The production cross-section has been calculated at next-to-next-to leading order (NNLO), including next-to-next-to leading log (NNLL) soft gluon resummation [4].",
"The top quark decays to a $W$ boson and a $b$ quark, and the final state topology in top quark pair events is determined by the subsequent decay of the two $W$ bosons, as shown in Fig.",
"REF (c).",
"About a third of top pairs decay to the lepton+jets final state where one $W$ boson decays to an electron or muon and the other to a quark pair.",
"The background to this final state is mainly from $W$ +jets production and QCD multi-jet events where one quark jet is mis-identified as a lepton.",
"This final state topology has reasonable statistics and a manageable background while also allowing for the reconstruction of the two top quarks.",
"A small fraction of about 6% of top pair events decay to the dilepton ($ee$ , $e\\mu $ and $\\mu \\mu $ ) final state, which has small backgrounds from $Z$ +jets and diboson production.",
"This topology is attractive for its clean signature, though the individual top quarks can not be reconstructed directly due to the presence of two neutrinos.",
"About 46% of top pair events decay to an all-hadronic final state which is overwhelmed by a large QCD multi-jet background.",
"Other top pair decays involve $\\tau $ leptons, and in particular hadronic $\\tau $ decays are of interest because they provide sensitivity to non-SM top decays.",
"Leptonic $\\tau $ decays are included in the lepton+jets and dilepton final states, though the lower lepton $p_T$ and the presence of additional neutrinos modifies the event kinematics."
],
[
"Lepton+jets final state",
"The lepton+jets final state (where the lepton is an electron or a muon) has backgrounds that can be controlled and higher event statistics than the dilepton final state.",
"Cross-section measurements both at the Tevatron and the LHC by ATLAS [5] and CMS [6] rely on $b$ -quark identification ($b$ -tagging) as well as multivariate analysis techniques to separate the top pair signal from the background sources, mainly $W$ +jets and QCD multi-jet production.",
"The CMS analysis at 7 TeV utilizes the secondary vertex mass to discriminate the top quark pair signal from the backgrounds [6].",
"This distribution is shown in Fig.",
"REF for different jet- and $b$ -tag multiplicities.",
"The measured cross-section is $158.1\\pm 11.0$ pb for an uncertainty of only 7%.",
"Figure: Secondary vertex mass distribution in electron+jet events in the measurementof the topDifferential cross-sections have also been measured in top pair production by ATLAS at 7 TeV [7] and by CMS at 7 TeV [8] and 8 TeV [9], [10].",
"The differential cross-section measured by ATLAS at 7 TeV as a function of the top pair transverse momentum is shown in Fig.",
"REF (left).",
"The differential cross-section is normalized to the total cross-section, thus canceling many systematic uncertainties."
],
[
"Dilepton final state",
"The dilepton final state (di-electron, di-muon and electron-muon) is clean with small backgrounds and small uncertainties, hence it provides high-precision measurements of the production cross-section.",
"The CMS measurement at 7 TeV has an uncertainty of 4.2%, currently the single most precise measurement [11].",
"The differential cross-section has also been measured in the dilepton final state.",
"Figure REF (right) shows the relative differential cross-section of the transverse momentum of the top quark pair.",
"Figure: (Left) ATLAS differential cross-section normalized to the total cross-sectionversus the"
],
[
"Pair production summary",
"The Tevatron measurements are summarized in Fig.",
"REF .",
"The combined Tevatron top pair cross-section is measured to be $7.60 \\pm 0.41$ pb, an uncertainty of only 5.4% [12].",
"Figure: Tevatron top quark pair production cross-section measurements and theirThe measurements by ATLAS and CMS are shown as a function of the collider energy in Figs.",
"REF and REF , respectively.",
"Note that the CMS summary figures do not yet include the latest CMS dilepton result [11].",
"The measured cross-sections are consistent with each other and with the theory predictions [13], [14].",
"The LHC top pair cross-section combination from Fall 2012 [15] is also shown in Fig.",
"REF .",
"Figure: ATLAS top quark pair production cross-section measurementsas a function of colliderFigure: CMS top quark pair production cross-section measurements as a function ofcolliderTop quark pair production in association with one or more quarks or with a $W$ or $Z$ boson provides a measurement of the top quark strong and weak interactions and is an important background in new physics searches and Higgs boson measurements in $t\\overline{t}H$ .",
"The jet multiplicity in top pair events has been measured by both ATLAS [18] and CMS [19], [20].",
"Figure REF shows the jet multiplicity in lepton+jets top pair events.",
"At low jet multiplicities, the measurement agrees within the large uncertainty band with the theoretical predictions, while at high jet multiplicities the agreement is poor.",
"Figure: Event yields in the tri-lepton channels inThe production of top quark pairs in association with $Z$ bosons has a small cross-section and is difficult to measure.",
"ATLAS had a first search for $t\\overline{t}Z$ production at 7 TeV [22].",
"CMS found evidence for top pair production in association with a boson in two analyses [21] using 7 TeV data: A search for $t\\overline{t}Z$ and a search for $t\\overline{t}V$ , where $V$ can be a $W$ boson or a $Z$ boson.",
"Figure REF shows the two cross-section measurements and their uncertainties."
],
[
"Single top production",
"Single top quark production proceeds via the $t$ -channel exchange of a $W$ boson between a heavy quark line and a light quark line, shown in Fig.",
"REF (a) or via the $s$ -channel production and decay of a virtual $W$ boson, shown in Fig.",
"REF (b) or as the production of a top quark in association with a $W$ boson, shown in Fig.",
"REF (c).",
"At the Tevatron, the $t$ -channel cross-section is largest, followed by the $s$ -channel, while the $Wt$ cross-section is too small to be observed.",
"At the LHC, $t$ -channel production benefits from the $qb$ initial state, with a large cross-section.",
"The $s$ -channel has a smaller cross-section and has not been seen yet.",
"The associated production has a $gb$ initial state and can be observed.",
"Figure: Feynman diagrams for single top quark production in the (a) tt-channel,(b) ss-channel, (c) in association with a WW boson."
],
[
"s-channel production",
"Evidence for single top quark production in the $s$ -channel was reported recently by the D0 [23] and CDF [24] collaborations at the Tevatron.",
"Both collaborations measure a cross-section that is consistent with the SM expectation, and both report an observed significance of 3.8 standard deviations.",
"This is a challenging analysis that relies on multivariate analysis techniques in order to separate the signal from the large backgrounds.",
"The $s$ -channel signal region is shown in Fig.",
"REF .",
"A comparison of the CDF and D0 measurements is shown in Fig.",
"REF .",
"CDF also has a measurement using missing transverse energy plus jets events [25] and a combination of the two results [26].",
"A Tevatron combination of the CDF and D0 results is in progress.",
"Figure: Summary ofTevatron ss-channel single top cross-section measurements."
],
[
"Wt associated production",
"The production cross-section of a single top quark in association with a $W$ boson has been measured both at 7 TeV and at 8 TeV by both ATLAS [27] and CMS [28], [29].",
"The final state in $Wt$ events is categorized by lepton multiplicity, similar to top pair production.",
"The difference to top pair production is that $Wt$ production results in exactly one high-$p_T$ $b$ -quark jet.",
"Hence top pair events comprise the largest background in $Wt$ dilepton events, with smaller contributions for $Z$ +jets, dibosons, and events with fake leptons.",
"Multivariate techniques are required to separate the signal from these backgrounds, and systematic uncertainties are large.",
"Nevertheless, the cross-section has now been measured with a relative uncertainty of 25%.",
"The 7 TeV ATLAS analysis measures a cross-section of $16.8 \\pm 2.9$ (stat) $\\pm 4.9$ (syst) pb [27].",
"The multivariate discriminant for the CMS 8 TeV analysis is shown in Fig.",
"REF , CMS cross-section measurement is $23.4 \\pm 5.5$ pb [29].",
"Both are consistent with the SM expectation.",
"Figure: Ratio of top to anti-top quark production"
],
[
"t-channel production",
"Single top quark production through the $t$ -channel is sensitive to the parton distribution function (PDF) of the light quarks in the proton.",
"The $t$ -channel final state is comprised of a lepton, neutrino and $b$ quark from the top quark decay, a high-$p_T$ forward jet, and possibly a third jet.",
"The main backgrounds to this signature are from $W$ +jets and top pair production.",
"The ratio of top to antitop quark production in the $t$ -channel is a particularly sensitive variable because many of the experimental uncertainties cancel.",
"ATLAS has measured this ratio in $t$ -channel events at 7 TeV using a neural network to separate the $t$ -channel events from the background [30].",
"The measured ratio is $1.81^{ +0.23}_{ -0.22}$ and is compared to several different PDFs in Fig.",
"REF .",
"CMS has measured the same ratio using 8 TeV data to be $1.76\\pm 0.27$ , again consistent with expectations [31].",
"The precision of these measurements is not yet sufficient to constrain PDFs, but future measurements should be able to improve on this situation.",
"Along with the ratio, ATLAS and CMS have also measured the total $t$ -channel cross section, with an uncertainty of 14% from ATLAS at 8 TeV [30] and an uncertainty of 9% from CMS at 7 TeV [32]."
],
[
"New physics searches",
"Searches for new physics in the top quark sector are particularly sensitive to models that have an enhanced coupling to the third generation of fermions.",
"New heavy bosons $Z^{\\prime }$ and $W^{\\prime }$ appear in many models of new physics, and searches for these heavy resonances are a priority at hadron colliders.",
"The reconstructed mass of the top quark pair system in the ATLAS 7 TeV analysis is shown in Fig.",
"REF [33].",
"The transverse momentum of the top quark produced in the decay of a heavy $Z^{\\prime }$ boson is sufficient to collimate quark jets from the top quark decay such that they can no longer be resolved individually.",
"Recent searches for $Z^{\\prime }$ bosons employ algorithms to identify such boosted top quark jets [33], [34].",
"The $Z^{\\prime }$ mass at which such algorithms perform better than resolved algorithms that reconstruct each of the top quark decay jets individually is around 1 TeV as can be seen by the vertical dashed line in Fig.",
"REF .",
"The mass range probed by these searches extends up to masses of 2 TeV.",
"The mass range below 0.75 TeV is also probed at the Tevatron where a CDF search currently provides the best sensitivity [35].",
"Figure: Upper limit on Z ' →tt ¯Z^{\\prime }\\rightarrow t\\overline{t} from CMS atA new charged heavy boson $W^{\\prime }$ can decay to a top quark together with a $b$ quark, leading to a single top final state.",
"The $W^{\\prime }$ boson may have SM-like left-handed couplings or it may have right-handed couplings to the top quark and the $b$ quark.",
"The ATLAS [36] and CMS [37] analyses probe these couplings separately, with CMS also providing two-dimensional limits as a function of the two couplings."
],
[
"Summary",
"The top quark pair production cross-section has been measured in many final states and with high precision by the CDF and D0 collaborations at the Tevatron proton-antiproton collider and by the ATLAS and CMS collaborations at the 7 TeV and 8 TeV LHC proton-proton collider.",
"The single top quark cross-sections have been measured in the $t$ -channel and now also in the $s$ -channel at the Tevatron, and in the $t$ -channel and the $Wt$ associated production at the LHC.",
"Many of these measurements are now at the level of precision of the theory predictions, and higher-precision results are yet to come with 8 TeV data.",
"Searches for new physics in top quark final states have reached a sensitivity to high-mass resonances of over 2 TeV.",
"This reach will be extended significantly at the 14 TeV LHC."
]
] | 1403.0513 |
[
[
"High-frequency asymptotics for path-dependent functionals of Ito\n semimartingales"
],
[
"Abstract The estimation of local characteristics of Ito semimartingales has received a great deal of attention in both academia and industry over the past decades.",
"In various papers limit theorems were derived for functionals of increments and ranges in the infill asymptotics setting.",
"In this paper we establish the asymptotic theory for a wide class of statistics that are built from the incremental process of an Ito semimartingale.",
"More specifically, we will show the law of large numbers and the associated stable central limit theorem for the path dependent functionals in the continuous and discontinuous framework.",
"Some examples from economics and physics demonstrate the potential applicability of our theoretical results in practice."
],
[
"Introduction",
"In the last decade limit theory for high frequency observations of Itô semimartingales has received a lot of attention in the scientific literature.",
"Such observation scheme of semimartingales, also called infill asymptotics, naturally appears in financial, biological and physical applications among many others.",
"For instance, a seminal work of Delbaen and Schachermayer [1] states that price processes must follow a semimartingale model under nor arbitrage conditions.",
"A general Itô semimartingale exhibits a representation of the form $X_t = X_0 + \\int _0^t\\!\\mu _s\\,\\text{d}s + \\int _0^t\\!\\sigma _{s}\\,\\text{d}W_s + \\delta 1_{\\lbrace |\\delta | \\le 1\\rbrace } \\star (m - n)_t + \\delta 1_{\\lbrace |\\delta | > 1\\rbrace } \\star m_t,$ where $\\mu $ represents the drift, $\\sigma $ is the volatility, $W$ is a Brownian motion, $m$ denotes the jump measure associated with $X$ and $n$ is its compensator.",
"Furthermore, for any measure $\\pi $ , we use the short hand notation $f \\star \\pi :=\\int f d\\pi $ whenever the latter is well defined.",
"Irrespective of the application field, researchers are interested in understanding the fine structure of the underlying Itô semimartingale model based on high frequency observations $X_{0},X_{\\Delta _n}, X_{2\\Delta _n}, \\ldots , X_{\\Delta _n\\lfloor t/\\Delta _n\\rfloor },$ where $\\Delta _n\\rightarrow 0$ , which refers to infill asymptotics.",
"For various testing and estimation problems, the class of generalised multipower variations turned out to be a very important probabilistic tool.",
"In their most general form, generalised multipower variations are defined as $\\sum _{i=1}^{\\lfloor t/\\Delta _n\\rfloor -d+1} f\\Big (a_n (X_{i\\Delta _n} - X_{(i-1)\\Delta _n}), \\ldots ,a_n (X_{(i+d-1)\\Delta _n} - X_{(i+d-2)\\Delta _n}) \\Big ),$ where $f:\\mathbb {R}^d \\rightarrow \\mathbb {R}$ is a smooth function and the scaling $a_n$ depends on whether the process $X$ has jumps or not.",
"In the continuous case the proper scaling is $a_n=\\Delta _n^{-1/2}$ .",
"Probabilistic properties of generalised multipower variations in continuous and discontinuous settings have been studies in [2], [3], [4] among many others.",
"We refer to a recent book [5] for a comprehensive study of high frequency asymptotics for Itô semimartingales.",
"Such probabilistic results found manifold applications in the statistical analysis of semimartingale models.",
"Estimation of the quadratic variation (see e.g.",
"[3]), volatility forecasting (see e.g.",
"[6], [7]), and tests for the presence of the jump component (see e.g.",
"[8], [9]) are the most prominent applications among many others.",
"The aim of this paper is to study the asymptotic behaviour of path dependent high frequency functionals of Itô semimartingales.",
"This framework is motivated by the fact that in some situations we can not directly observe the semimartingale $X$ , but only its path dependent functional over short time windows.",
"Let us give two examples.",
"In various applied sciences integrated diffusions (i.e.",
"integrated Itô semimartingales) appear as a natural class of models for a given random phenomena.",
"For example in physics, when a medium's surface (such as the arctic's sea ice) is modelled as a stochastic process, a sonar's measurement of the reflection of this surface is given by the local time of the surface's slope process (see e.g.",
"[10], [11]).",
"Since this local time process is typically an Itô process again (see e.g.",
"[12], [13]), limit theorems for local averages are required in order to make inference on the structure of the original surface process (see e.g.",
"[14] for a detailed discussion).",
"Because only discrete (high frequency) observations of such integrated diffusions are available, one can not recover the original path of the underlying Itô semimartingales from it.",
"Another example of path dependent functionals are ranges whose statistical properties have been studied in [15], [16] in the case of law frequency observations of a scaled Brownian motion.",
"We also refer to an early result by William Feller [17], which characterises the distribution of the range of the Brownian motion.",
"In this paper we will consider functionals of the incremental process built from $X$ , i.e.",
"$V(X,g)_t^n=\\Delta _n \\sum _{i=1}^{\\lfloor t/\\Delta _n\\rfloor } g\\Big (\\big \\lbrace a_n \\big (X_{(i-1\\text{+}s)\\Delta _n}-X_{(i-1)\\Delta _n}\\big );\\, s\\in [0,1]\\big \\rbrace \\Big ),$ where $g$ is now operating on $C([0,1])$ and the scaling $a_n$ equals $\\Delta _n^{-1/2}$ when $X$ is a continuous Itô semimartingale.",
"Obviously, this class of statistics extends the classical concept of power variations to path dependent functionals.",
"The function $g(x)=\\sup _{t\\in [0,1]} x(t) - \\inf _{t\\in [0,1]} x(t)$ recovers the case of realised ranges as the have been considered in [18], [19] in the context of quadratic variation estimation.",
"In this work we will prove the law of large numbers for the functional $V(X,g)_t^n$ and show the associated stable central limit theorem in continuous and discontinuous framework.",
"We remark that extending the analysis to general path dependent functionals increases the complexity of the proofs, which is due to the topological structure of the space $C([0,1])$ .",
"Furthermore, a general asymptotic statement in the discontinuous case seems to be out of reach (in contrast to very general results for classical power variations studied in [3]).",
"For this reason, we restrict our attention to range statistics of discontinuous Itô semimartingales, as they seem to be useful in financial applications (see [19]).",
"Finally, we present some applications of the probabilistic results, in particular in the context of integrated diffusions and realised ranges.",
"The paper is organised as follows.",
"In Section 2 we state the two main theorems for general functionals of continuous Itô semimartingales, establishing the limits in probability as well as the associated stable central limit theorem.",
"In Section 3 we apply the limit theory to three most prominent practical examples including general range statistics and integrated diffusions.",
"Section 4 is devoted to the limit theorems for realised ranges of discontinuous Itô semimartingales.",
"The proofs of the main results are collected in Section 5."
],
[
"Limit Theorems for Continuous Itô Semimartingales",
" Before we present the main results we start by introducing some notation.",
"We denote by $C([0,1])$ the space of continuous real valued functions on the interval $[0,1]$ , and by $\\Vert \\cdot \\Vert _{\\infty }$ the supremum norm on $C([0,1])$ .",
"A function $f:C([0,1])\\rightarrow \\mathbb {R}$ is said to have polynomial growth if $|f(x)|\\le C(1+\\Vert x\\Vert _{\\infty }^p)$ for some $C,p>0$ .",
"For any $x,y\\in C([0,1])$ and $f:C([0,1])\\rightarrow \\mathbb {R}$ , the expression $f^{\\prime }_{y}(x)$ denotes the Gâteaux derivative of $f$ at point $x$ in the direction of $y$ , i.e.",
"$f^{\\prime }_y(x):= \\lim _{h\\rightarrow 0}(f(x+hy) -f(x))/h$ .",
"For any processes $Y^n, Y$ we denote by $Y^n \\stackrel{ucp}{\\rightarrow }Y$ the uniform convergence in probability, i.e.",
"$\\sup _{t\\in [0,T]}|Y^n_t - Y_t| \\stackrel{\\mathbb {P}}{\\rightarrow }0$ for all $T>0$ .",
"Throughout this paper we frequently use the notion of stable convergence, which is due to Renyi [20].",
"A sequence of random variables $(Y_n)_{n\\ge 1}$ on $(\\Omega , \\mathcal {F}, \\mathbb {P})$ with values in a Polish space $(E, \\mathcal {E})$ is said to converge stably in law to $Y$ ($Y_n \\stackrel{d_{st}}{\\rightarrow }Y$ ), where $Y$ is defined on an extension $(\\Omega ^{\\prime }, \\mathcal {F}^{\\prime }, \\mathbb {P}^{\\prime })$ of the original probability space, if and only if for any bounded, continuous function $f$ and any bounded $\\mathcal {F}$ -measurable random variable $Z$ it holds that $ \\mathbb {E}[ f(Y_n) Z] \\rightarrow \\mathbb {E}^{\\prime }[ f(Y) Z], \\quad n \\rightarrow \\infty .$ Typically, we will deal with spaces $E=\\mathbb {D}([0,T] , \\mathbb {R})$ equipped with the uniform topology when the process $Y$ is continuous.",
"Notice that stable convergence is a stronger mode of convergence than weak convergence.",
"In fact, the statement $Y_n \\stackrel{d_{st}}{\\rightarrow }Y$ is equivalent to the joint weak convergence $(Y_n, Z) \\stackrel{d}{\\rightarrow }(Y,Z)$ for any $\\mathcal {F}$ -measurable random variable $Z$ ."
],
[
"Law of Large Numbers",
"Throughout this section we are considering a stochastic process $X$ defined on a filtered probability space $(\\Omega , \\mathcal {F}, \\mathbb {F}\\!=\\!",
"(\\mathcal {F}_t)_{t\\ge 0}, \\mathbb {P})$ satisfying the usual conditions that follows the distribution of a diffusion $X_t=X_0 + \\int _0^t \\mu _s\\, \\text{d}s + \\int _0^t \\sigma _{s}\\, \\text{d}W_s$ for $t\\ge 0$ , where $X_0$ is a constant, $W$ is a Brownian motion, $\\mu $ is a predictable, locally bounded process and $\\sigma $ is an adapted, cádlág process.",
"Given a function $g:C([0, 1]) \\rightarrow \\mathbb {R}$ and a vanishing sequence $(\\Delta _n)_{n\\in \\mathbb {N}}$ we define the sequence of processes $ V(X, g)^n_t &:= \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } g\\big (\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\big ), \\\\[1.5 ex]d_i^n (X)&:= \\big \\lbrace X_{(i-1+s)\\Delta _n} - X_{(i-1)\\Delta _n}\\big \\rbrace _{s\\in [0,1]}.$ For any $z\\in \\mathbb {R}$ and $g\\in C([0,1])$ we introduce the quantity $ \\rho _z(g):=\\mathbb {E}\\left[g(\\left\\lbrace z\\, W_s;\\ s\\in [0, 1]\\right\\rbrace )\\right],$ whenever the latter expectation is finite.",
"Our first result is the law of large numbers for the functional $V(X, g)^n_t$ .",
"Theorem 2.1 (Law of Large Numbers) Let $g$ be a locally uniformly continuous functional, i.e.",
"for $x,y \\in C([0,1])$ , given $K,\\epsilon >0$ there exists $ \\delta >0$ such that for $\\Vert x\\Vert _\\infty , \\Vert y\\Vert _\\infty \\le K$ , $\\Vert x-y\\Vert _\\infty \\le \\delta $ it follows that $|g(x)-g(y)| \\le \\epsilon $ , and have polynomial growth.",
"Then it holds that $V(X, g)^n_t \\ \\stackrel{ucp}{\\rightarrow } \\ V(X,g)_t := \\int _0^t \\rho _{\\sigma _s}(g) \\,\\text{d}s,$ where the quantity $\\rho _z(g)$ is defined at (REF ).",
"Remark 2.1 Our notion of locally uniform continuity is slightly unusual.",
"Instead of requiring uniform continuity on neighbourhoods or compact sets we demand it on balls $B_{\\le K}(0)=\\lbrace x\\in C([0,1]);\\ \\Vert x\\Vert _\\infty \\le K\\rbrace $ for $K>0$ , which are not compact with respect to the uniform topology.",
"This type of locally uniform continuity is not required in the classical limit theory for functionals of increments of $X$ (see e.g.",
"[2]) since on finite dimensional spaces continuity on closed balls implies uniform continuity.",
"We remark that our locally uniform continuity assumption is satisfied whenever $|g(x)-g(y)|\\le C\\Vert x-y\\Vert _{\\infty }^\\delta $ for all $x,y\\in C[0,1]$ and some $C,\\delta >0$ .",
"This condition is satisfied for all practical examples."
],
[
"Central Limit Theorem",
"Having determined the limit in probability we now turn to the associated stable central limit theorem.",
"Theorem 2.2 (Central Limit Theorem) Let $g$ satisfy the conditions of Theorem REF .",
"Moreover, we assume that given $K, \\epsilon >0$ there exists $\\delta >0$ such that for $\\Vert x\\Vert _\\infty , \\Vert y\\Vert _\\infty \\le K$ , $\\Vert x-y\\Vert _\\infty \\le \\delta , \\Vert v\\Vert _\\infty \\le 1$ it follows that $|g^{\\prime }_v(x)-g^{\\prime }_v(y)| \\le \\epsilon $ , there exist $C, p>0$ such that $|g^{\\prime }_v(x)| \\le C(1+ \\Vert x\\Vert _\\infty ^p)$ for $\\Vert v\\Vert _\\infty \\le 1$ .",
"Let $\\sigma $ be a continuous Itô semimartingale of the form $\\sigma _t = \\sigma _0 + \\int _0^t \\tilde{\\mu }_s \\,\\emph {d}s + \\int _0^t \\tilde{\\sigma }_{s} \\,\\emph {d}W_s + \\int _0^t \\tilde{v}_{s}\\, \\emph {d}V_s,$ where $\\tilde{\\mu },\\ \\tilde{\\sigma }$ and $\\tilde{v}$ are adapted, cádlág processes and $V$ is another Brownian motion independent of $W$ .",
"Then it follows that $\\Delta _n^{-\\frac{1}{2}} \\left(V(X, g)^n - V(X, g) \\right) \\ \\stackrel{d_{st}}{\\rightarrow }\\ U(X, g)$ where $U(X, g)_t := \\int _0^t u^1_s \\,\\emph {d}s + \\int _0^t u^2_s\\,\\emph {d}W_s + \\int _0^t u^3_s \\,\\emph {d}W^{\\prime }_s$ with $u^1_s &:= \\mu _s \\rho _{\\sigma _s}^{(2)}(g^{\\prime }) + \\frac{1}{2} \\tilde{\\sigma }_s \\rho _{\\sigma _s}^{(3)}(g^{\\prime }) - \\frac{1}{2} \\tilde{\\sigma }_s \\rho _{\\sigma _s}^{(2)}(g^{\\prime })\\\\u^2_s &:= \\rho _{\\sigma _s}^{(1)}(g), \\\\u^3_s &:= \\sqrt{\\rho _{\\sigma _s}(g^2)-\\rho ^2_{\\sigma _s}(g)-(\\rho _{\\sigma _s}^{(1)}(g))^2},$ and, for $z\\in \\mathbb {R}$ and $f(x,y):=g^{\\prime }_{y}(x)$ , $\\rho _z^{(1)}(g)&:=\\mathbb {E}\\Big [g\\big (\\left\\lbrace z\\, W_s;\\ s\\in [0, 1]\\right\\rbrace \\big )\\, W_1\\Big ], \\\\\\rho _z^{(2)}(f) &:=\\mathbb {E}\\Big [f\\big (\\left\\lbrace z\\, W_s;\\ s\\in [0, 1]\\right\\rbrace ,\\, \\lbrace s;\\ s\\in [0,1]\\rbrace \\big )\\Big ], \\\\\\rho _z^{(3)}(f) &:=\\mathbb {E}\\Big [f\\big (\\left\\lbrace z\\, W_s;\\ s\\in [0, 1]\\right\\rbrace ,\\, \\lbrace W_s^2;\\ s\\in [0,1]\\rbrace \\big )\\Big ].$ Furthermore, $W^{\\prime }$ is a Brownian motion defined on an extension of $(\\Omega , \\mathcal {F}, \\mathbb {F}, \\mathbb {P})$ , which is independent of $\\mathcal {F}$ .",
"Some remarks on the application of this probabilistic result are in order.",
"Remark 2.2 When $g(x)\\equiv f(x(1))$ for some function $f:\\mathbb {R}\\rightarrow \\mathbb {R}$ such that $f,f^{\\prime }$ have polynomial growth, we recover the stable central limit theorem for functionals of increments of $X$ .",
"More precisely, it holds that $\\rho _z^{(1)}(g) = \\mathbb {E}[f(zW_1) W_1], \\quad \\rho _z^{(2)}(g^{\\prime }) = \\mathbb {E}[f^{\\prime }(zW_1)], \\qquad \\rho _z^{(3)}(g^{\\prime }) = \\mathbb {E}[f^{\\prime }(zW_1) W_1^2],$ and we obtain the one-dimensional analogue of the asymptotic theory presented in [4].",
"Remark 2.3 In general, Theorem REF can not be applied for statistical inference, since the distribution of the limit $U(X, g)_t$ is unknown.",
"However, when $g$ is an even functional, i.e.",
"$g(x)=-g(x)$ for all $x\\in C([0,1])$ , things become different.",
"In this case it holds that $\\rho _z^{(1)}(g) = \\rho _z^{(2)}(g^{\\prime }) =\\rho _z^{(3)}(g^{\\prime }) =0$ for all $z\\in \\mathbb {R}$ , since $W\\stackrel{d}{=} -W$ and expectations of odd functionals of $W$ are 0.",
"Hence, the limiting process $U(X, g)$ has the form $U(X, g)_t = \\int _0^t \\sqrt{\\rho _{\\sigma _s}(g^2)-\\rho ^2_{\\sigma _s}(g)} \\,\\emph {d}W^{\\prime }_s,$ which is, conditionally on $\\mathcal {F}$ , a Gaussian martingale with mean 0.",
"For a fixed $t>0$ , the result of Theorem REF can be transformed into a standard central limit theorem when $g$ is even.",
"A slight modification of Theorem REF shows that $V^n_t &:= \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor -1} \\Big \\lbrace g^2\\big (\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\big )- g\\big (\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\big ) g\\big (\\Delta _n^{-\\frac{1}{2}}\\ d_{i+1}^n (X)\\big ) \\Big \\rbrace \\\\&\\stackrel{ucp}{\\rightarrow }\\int _0^t \\rho _{\\sigma _s}(g^2)-\\rho ^2_{\\sigma _s}(g) \\,\\emph {d}s.$ (This should be compared with the asymptotic theory for bipower variation established in [2].)",
"For any fixed $t>0$ , we then deduce a standard central limit theorem $\\frac{\\Delta _n^{-\\frac{1}{2}} \\left(V(X, g)^n_t - V(X, g)_t \\right)}{\\sqrt{V^n_t}} \\stackrel{d}{\\rightarrow }\\mathcal {N}(0,1)$ by properties of stable convergence.",
"The latter can be used to obtain confidence regions for the quantity $V(X, g)_t$ ."
],
[
"Examples and Applications",
" In this section we present some examples that demonstrate the applicability of the limit theory for path dependent functionals of continuous Itô semimartingales.",
"For comparison reasons we start with the classical results on power variations."
],
[
"Example 1",
"Here we consider the power variation case which corresponds to $g(x)\\equiv f(x(1))$ with $f(x)=|x|^p$ , $p>0$ .",
"Recalling the asymptotic theory from [2] we conclude that $\\Delta _n^{1-\\frac{p}{2}}\\sum _{i=1}^{\\lfloor t/\\Delta _n \\rfloor } \\big |X_{i\\Delta _n}-X_{(i-1)\\Delta _n}\\big |^p\\stackrel{ucp}{\\rightarrow }\\lambda ^{1, p} \\int _0^t |\\sigma _s|^p \\,\\text{d}s$ where $\\lambda ^{1, p}=\\mathbb {E}[|W_1|^p]$ .",
"Moreover, the following stable central limit theorem holds $\\Delta _n^{-\\frac{1}{2}} \\Bigg ( \\frac{\\Delta _n^{1-\\frac{p}{2}}}{\\lambda ^{1, p}} \\sum _{i=0}^{\\lfloor t/\\Delta _n \\rfloor } \\big |X_{i\\Delta _n}-X_{(i-1)\\Delta _n}\\big |^p - \\int _0^t\\!",
"|\\sigma _s|^p\\, \\text{d}s\\Bigg )\\stackrel{d_{st}}{\\rightarrow }\\sqrt{\\Lambda ^{1, p}} \\int _0^t |\\sigma _s|^p\\, \\text{d}W^{\\prime }_s,$ where $\\Lambda ^{1, p}:= \\frac{\\lambda ^{1, 2p}-(\\lambda ^{1, p})^2}{(\\lambda ^{1, p})^2}$ .",
"Later on we will compare the efficiency of power variation with other estimators presented in the following examples."
],
[
"Example 2",
"Let $g:C([0,1])\\rightarrow \\mathbb {R}$ be defined as $g(x):=f(\\int _0^1 x(s)\\,\\text{d}s)$ for a continuously differentiable function $f:\\mathbb {R} \\rightarrow \\mathbb {R}$ such that $f,f^{\\prime }$ have polynomial growth.",
"Then condition (i) of Theorem REF is obviously satisfied.",
"Furthermore, it holds that $g^{\\prime }_y(x)= f^{\\prime }\\left(\\int _0^1\\!x(s)\\,\\text{d}s \\right) \\int _0^1\\!",
"y(s)\\,\\text{d}s, \\qquad \\forall x,y\\in C([0,1]),$ and conditions (ii) and (iii) of Theorem REF are fulfilled since $f^{\\prime }$ is continuous and has polynomial growth.",
"In particular, for $f(x)=|x|^p$ with $p>0$ we obtain that $\\Delta _n^{1-\\frac{p}{2}}\\sum _{i=1}^{\\lfloor t/\\Delta _n \\rfloor } \\Big |\\Delta _n^{-1}\\int _{(i-1)\\Delta _n}^{i\\Delta _n}\\!",
"X_s \\,\\text{d}s-X_{(i-1)\\Delta _n}\\Big |^p\\quad \\stackrel{ucp}{\\rightarrow }\\quad \\lambda ^{2, p} \\int _0^t |\\sigma _s|^p \\,\\text{d}s$ where $\\lambda ^{2, p}=\\mathbb {E}[|\\int _0^1 W_s\\, \\text{d}s|^p]$ .",
"Furthermore, for $p>1$ we deduce the corresponding stable central limit theorem (cf.",
"Remark REF ) $\\Delta _n^{-\\frac{1}{2}} \\Bigg ( \\frac{\\Delta _n^{1-\\frac{p}{2}}}{\\lambda ^{2, p}} &\\sum _{i=0}^{\\lfloor t/\\Delta _n \\rfloor } \\Big |\\Delta _n^{-1}\\int _{(i-1)\\Delta _n}^{i \\Delta _n}\\!X_s\\, \\text{d}s -X_{(i-1)\\Delta _n}\\Big |^p - \\int _0^t\\!",
"|\\sigma _s|^p\\, \\text{d}s\\Bigg ) \\\\&\\stackrel{d_{st}}{\\rightarrow }\\quad \\sqrt{\\Lambda ^{2, p}} \\int _0^t |\\sigma _s|^p\\, \\text{d}W^{\\prime }_s,$ with $\\Lambda ^{2, p}:= \\frac{\\lambda ^{2, 2p}-(\\lambda ^{2, p})^2}{(\\lambda ^{2, p})^2}$ ."
],
[
"Example 3",
"Let us now consider the range-based functionals which has been originally studied in [18].",
"Here the functional $g:C([0,1])\\rightarrow \\mathbb {R}$ is a function of the range, i.e.",
"$g(x)=f(\\sup _{t\\in [0,\\,1]}x(t) - \\inf _{t\\in [0,\\,1]} x(t))$ for a continuously differentiable function $f:\\mathbb {R} \\rightarrow \\mathbb {R}$ , such that $f,f^{\\prime }$ have polynomial growth.",
"Then the law of large numbers in Theorem REF readily applies, but the central limit theorem cannot be directly deduced from Theorem REF , because the range is not Gâteaux differentiable in general.",
"However, we may apply the following result: Let $x,y\\in C([0,1])$ be functions such that the set $M:=\\lbrace t\\in [0,1]:~ t=\\text{argmax}_{s\\in [0,1]} x(s)\\rbrace $ is finite, then it holds that (cf.",
"[18]) $\\frac{1}{h} \\Big (\\sup _{0\\le s \\le 1} \\big (x(s)+ h y(s)\\big ) - \\sup _{0\\le s \\le 1} x(s)\\Big )= \\max _{t\\in M} y(t).$ In the proofs (see again [18]) the function $x$ plays the role of the Brownian motion, which attains its maximum (resp.",
"minimum) at a unique point almost surely.",
"Let $t_{max}:= \\arg \\max _{s\\in [0,1]} W_s$ and $t_{min}:= \\arg \\min _{s\\in [0,1]} W_s$ .",
"Then the assertion of Theorem REF remains valid in the range case when $\\sigma $ is everywhere invertible (cf.",
"[19]) with $\\rho _x^{(1)}(g) &=\\mathbb {E}\\Big [f\\Big (x\\, \\Big (\\sup _{0\\le t \\le 1}W_s - \\inf _{0\\le s \\le 1} W_s\\Big )\\Big )\\, W_1\\Big ], \\\\\\rho _x^{(2)}(g^{\\prime }) &=\\mathbb {E}\\Big [f^{\\prime }\\Big (x\\, \\Big (\\sup _{0\\le t \\le 1}W_s - \\inf _{0\\le s \\le 1} W_s\\Big )\\Big )\\big (t_{max}-t_{min}\\big )\\Big ], \\\\\\rho _x^{(3)}(g^{\\prime }) &=\\mathbb {E}\\Big [f^{\\prime }\\Big (x\\, \\Big (\\sup _{0\\le t \\le 1}W_s - \\inf _{0\\le s \\le 1} W_s\\Big )\\Big )\\big (W_{t_{max}}^2-W_{t_{min}}^2 \\big )\\Big ],$ which extends the asymptotic theory presented in [19] to general functions of the range.",
"In particular, for $f(x)=|x|^p$ with $p>0$ we obtain that $\\Delta _n^{1-\\frac{p}{2}}\\sum _{i=1}^{\\lfloor t/\\Delta _n \\rfloor } \\sup _{s,u\\in [(i-1)\\Delta _n,\\, i \\Delta _n]} (X_s -X_u)^p \\quad \\stackrel{ucp}{\\rightarrow }\\quad \\lambda ^{3, p}\\ \\int _0^t |\\sigma _s|^p \\,\\text{d}s$ where $\\lambda ^{3, p}=\\mathbb {E}[\\sup _{s,u\\in [0,1]} (W_s -W_u)^p]$ .",
"Furthermore, since the function $f$ is even, we deduce the following central limit theorem $\\Delta _n^{-\\frac{1}{2}} \\Bigg ( \\frac{\\Delta _n^{1-\\frac{p}{2}}}{\\lambda ^{3, p}} &\\sum _{i=0}^{\\lfloor t/\\Delta _n \\rfloor }\\sup _{s,u\\in [(i-1)\\Delta _n,\\, i \\Delta _n]} (X_s -X_u)^p - \\int _0^t\\!",
"|\\sigma _s|^p\\, \\text{d}s\\Bigg ) \\\\&\\stackrel{d_{st}}{\\rightarrow }\\quad \\sqrt{\\Lambda ^{3, p}} \\int _0^t |\\sigma _s|^p\\, dW^{\\prime }_s$ where $\\Lambda ^{3, p}:= \\frac{\\lambda ^{3, 2p}-(\\lambda ^{3, p})^2}{(\\lambda ^{3, p})^2}$ .",
"This recovers the analysis presented in [19]."
],
[
"Comparison of Examples 1-3",
"When comparing different estimators of integrated powers of volatility presented in the previous examples, we see that $\\Lambda ^{i, p},\\ i=1,2,3$ serve as a convenient measure of their efficiency.",
"We remark however that this comparison is not fair as the sampling schemes of Example 1 and Examples 2-3 are not comparable.",
"Since $\\int _0^1 W_s \\, \\text{d} s \\sim \\mathcal {N}(0, 1/3)$ , it follows that $\\lambda ^{1, p} = 3^{p/2}\\, \\lambda ^{2,p}$ and $\\Lambda ^{1, p}$ coincides with $\\Lambda ^{2, p}$ .",
"However, $\\Lambda ^{3, p}$ is considerably smaller so as expected, range based estimation is asymptotically superior.",
"For example in the case $p=2$ we find $\\Lambda ^{1, p},\\Lambda ^{2, p}= 2$ , whereas $\\Lambda ^{3, p}\\approx 0.4$ .",
"The smaller $p$ is, the more pronounced this relative difference becomes.",
"Figure 1 illustrates these relationships.",
"Figure: The parameters Λ 1,p \\Lambda ^{1, p}=Λ 2,p \\Lambda ^{2,p}, Λ 3,p \\Lambda ^{3,p} and their ratio."
],
[
"Example 4",
"In various applied sciences integrated diffusions appear as a natural model of a random phenomena.",
"For example in physics, when a medium's surface (such as the arctic's sea ice) is modelled as a stochastic process, a sonar's measurement of the reflection of this surface is given by the local time of the surface's slope process (see e.g.",
"[10], [11]).",
"Since this local time process is typically an Itô process again (see e.g.",
"[12], [13]) and since the observations are given as local averages, limit theorems for local averages are required in order to make inference on the structure of the original surface process (see e.g.",
"[14]).",
"So let's define the local averages of an Itô process $X$ as $\\overline{X}_i^n := \\frac{1}{\\Delta _n}\\int _{(i-1)\\Delta _n}^{i\\Delta _n} X_s ds.$ A natural candidate estimator for the quadratic variation of $X$ is given by $\\sum _{i=2}^{\\lfloor t/\\Delta _n \\rfloor } \\left( \\overline{X}_i^n - \\overline{X}_{i-1}^n\\right)^2.$ We note that this estimator does not directly exhibit a representation as in Example 2.",
"However, when we use the decomposition $\\overline{X}_i^n - \\overline{X}_{i-1}^n &= \\frac{1}{\\Delta _n} \\Big (\\int _{(i-1)\\Delta _n}^{i\\Delta _n} X_s - X_{(i-1)\\Delta _n} ds - \\int _{(i-2)\\Delta _n}^{(i-1)\\Delta _n} X_s - X_{(i-2)\\Delta _n} ds \\\\&+(X_{(i-1)\\Delta _n} - X_{(i-2)\\Delta _n})\\Big ),$ Theorem REF , and the bipower concept of Remark REF , we deduce the ucp convergence $\\sum _{i=2}^{\\lfloor t/\\Delta _n \\rfloor } \\left( \\overline{X}_i^n - \\overline{X}_{i-1}^n\\right)^2\\stackrel{ucp}{\\rightarrow }\\frac{2}{3} \\int _0^t \\sigma _s^2 ds.$ This clearly provides a way of estimating the quadratic variation of $X$ from observations of an integrated diffusion."
],
[
"Limit Theorems for Itô Semimartingales with Jumps",
"In this section we study the behavior of certain path-dependent functionals of discontinuous Itô semimartingales.",
"As the general theory is much more difficult to establish compared to the work of [3], we restrict our attention to ranges of Itô semimartingales with jumps.",
"For simplicity of exposition, we will further restrict ourselves to finite activity jump processes."
],
[
"Law of Large Numbers",
"Consider now a stochastic process $X$ defined on a filtered probability space $(\\Omega , \\mathcal {F}, \\mathbb {F}\\!=\\!",
"(\\mathcal {F}_t)_{t\\ge 0}, \\mathbb {P})$ satisfying the usual conditions that follows the distribution of a diffusion with a jump component in the form of a compound Poisson process $Z_t=\\sum _{i=1}^{N_t} J_i$ where $N$ is a Poisson process with intensity $\\lambda $ and i.i.d.",
"jump sizes $J_i $ , i.e.",
"$X_t=X_0 + \\int _0^t \\mu _s\\, \\text{d}s + \\int _0^t \\sigma _{s}\\, \\text{d}W_s + Z_t,$ where $W$ is a Brownian motion independent of $N$ , $\\mu $ is a predictable, locally bounded process and $\\sigma $ is an adapted, cádlág process.",
"For a positive exponent $p>0$ we define $R(X, p)^n_t := \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\sup _{s,u\\in [(i\\!-\\!1)\\Delta _n, i\\Delta _n]} |X_s - X_u|^p$ for $t\\ge 0,\\ n\\in \\mathbb {N}$ .",
"Our first result is the following law of large numbers.",
"Theorem 4.1 We have that $R(X, p)^n_t \\stackrel{\\mathbb {P}}{\\rightarrow }R(X, p)_t:={\\left\\lbrace \\begin{array}{ll} \\lambda ^{3,2} \\int _0^t \\sigma _s^2 \\,\\text{d}s + \\sum _{i=1}^{N_t} J_i^2 & p = 2 \\\\\\sum _{i=1}^{N_t} |J_i|^p & p>2\\end{array}\\right.",
"}$ where $\\lambda ^{3, 2}=\\mathbb {E}[\\sup _{s,u\\in [0,\\,1]} |W_s -W_u|^2]$ .",
"For $p<2$ we obtain infinity in the limit whenever $\\int _0^t \\sigma _s^2\\, \\text{d}s>0$ .",
"We remark that the first convergence of Theorem REF has been already proved in [19] in the context of range based estimation of quadratic variation.",
"Very similar results has been established for the classical power variations in [3]."
],
[
"Central Limit Theorem",
"Having determined the limit in probability we now turn our attention to the associated stable central limit theorems.",
"In order to introduce the weak limit theory we require some further notation.",
"We denote by $(T_i)_{i\\ge 1}$ the successive jump times of the Poisson process $N$ .",
"Furthermore, we introduce two Brownian motions $(W^{\\prime }_t)_{t\\ge 0}$ , $(\\widetilde{W}_t)_{t\\ge 0}$ and a sequence $(\\kappa _i)_{i\\ge 1}$ of i.i.d.",
"$\\mathcal {U}([0,1])$ -distributed random variables, which are mutually independent, and independent of $\\mathcal {F}$ .",
"Finally, we introduce the process $U(X, p)_t &=\\!",
"p \\sum _{i=1}^{N_t} |J_i|^{p-1} \\left\\lbrace \\sup _{\\begin{array}{c}0\\le s \\le \\kappa _i \\\\ \\kappa _i \\le u \\le 1\\end{array}}\\!\\left( (\\widetilde{W}_{i\\text{+}\\kappa _i}\\!-\\!\\widetilde{W}_{i\\text{+}s})\\sigma _{T_i-}\\!+\\!",
"(\\widetilde{W}_{i\\text{+}u}\\!-\\!\\widetilde{W}_{i\\text{+}\\kappa _i})\\sigma _{T_i}\\right) 1_{\\lbrace J_i>0\\rbrace } \\right.\\nonumber \\\\[1.5 ex]&\\left.",
"+ \\sup _{\\begin{array}{c}0\\le s \\le \\kappa _i \\\\ \\kappa _i \\le u \\le 1\\end{array}}\\!\\left( -(\\widetilde{W}_{i\\text{+}\\kappa _i}\\!-\\!\\widetilde{W}_{i\\text{+}s})\\sigma _{T_i-}\\!-\\!",
"(\\widetilde{W}_{i\\text{+}u}\\!-\\!\\widetilde{W}_{i\\text{+}\\kappa _i})\\sigma _{T_i}\\right) 1_{\\lbrace J_i<0\\rbrace } \\right\\rbrace $ that is defined on the extension of the original space $(\\Omega , \\mathcal {F}, \\mathbb {F}, \\mathbb {P})$ .",
"The central limit theorem is as follows.",
"Theorem 4.2 (Central Limit Theorem) (i) For $p>3$ and fixed $t>0$ we obtain the stable convergence $\\Delta _n^{-\\frac{1}{2}} \\left(R(X, p)^n_t - R(X, p)_t\\right) \\ \\stackrel{d_{st}}{\\rightarrow }\\ U(X,p)_t.$ (ii) Let $p=2$ .",
"Assume that the invertible volatility process $\\sigma $ follows the distribution of a discontinuous Itô semimartingale $\\sigma _t = \\sigma _0 + \\int _0^t \\tilde{\\mu }_s \\,\\emph {d}s + \\int _0^t \\tilde{\\sigma }_{s} \\,\\emph {d}W_s + \\int _0^t \\tilde{v}_{s}\\, \\emph {d}V_s + \\tilde{Z}_t,$ where $\\tilde{\\mu },\\ \\tilde{\\sigma }$ and $\\tilde{v}$ are adapted, cádlág processes, $V$ is another Brownian motion independent of $W$ and $\\tilde{Z}_t= \\sum _{i=0}^{\\tilde{N}_t} \\tilde{J}_i$ is a compound Poisson processes with $\\tilde{N}$ being independent of $W$ ($\\tilde{N}$ and $N$ are possibly correlated).",
"Then, for any fixed $t>0$ , we obtain the stable convergence $\\Delta _n^{-\\frac{1}{2}} \\left(R(X, 2)^n_t - R(X, 2)_t\\right) \\stackrel{d_{st}}{\\rightarrow }U(X,2)_t +\\sqrt{\\lambda ^{3, 4}-(\\lambda ^{3, 2})^2} \\int _0^t \\sigma _s^2\\, dW^{\\prime }_s.$ We remark that Theorem REF is similar in fashion to central limit theorems for classical power variations; see [3].",
"We do believe that REF remains valid for a rather general Itô semimartingale model (i.e.",
"not only in the finite activity case), but the proofs become considerably longer.",
"After local estimation of $\\sigma $ and jump sizes $J_i$ , the conditional law of $U(X, p)_t$ given $\\mathcal {F}$ can be simulated.",
"However, unlike for the mixed normal case in the classical power variation framework, the knowledge of the conditional law of $U(X, p)_t$ is not sufficient for statistical inference (e.g.",
"construction of confidence regions)."
],
[
"Proofs",
" First of all, note that without loss of generality we may assume that the processes $\\mu ,\\sigma , \\tilde{\\mu }, \\tilde{\\sigma },\\tilde{v}$ are bounded.",
"This follows from a standard localization procedure (see e.g.",
"[2]).",
"Below, all positive constants are denoted by $C$ or $C_p$ if they depend on an external parameter $p$ , although they may change from line to line."
],
[
"Proof of Theorem ",
"We begin with some preliminary observations.",
"Denoting $A_t:= \\int _{0}^t\\!",
"\\mu _s \\,\\text{d}s, \\qquad M_t:= \\int _{0}^t\\!",
"\\sigma _s \\,\\text{d}W_s,$ we find that for $p>0$ , $ \\nonumber \\mathbb {E}\\left[\\left(\\Delta _n^{-\\frac{1}{2}}\\ \\Vert d_i^n (X)\\Vert _\\infty \\right)^p\\right] &\\le C_p\\Delta _n^{-\\frac{p}{2}}\\left(\\mathbb {E}\\Big [\\Vert d_i^n (A)\\Vert _\\infty ^p\\Big ] + \\mathbb {E}\\Big [\\Vert d_i^n (M)\\Vert _\\infty ^p\\Big ]\\right) \\\\ &\\le C_p\\left(\\Delta _n^{\\frac{p}{2}} \\Vert \\mu \\Vert _\\infty ^p + \\Delta _n^{-\\frac{p}{2}}\\mathbb {E}\\bigg [\\Big (\\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\sigma _s^2 \\,\\text{d}s\\Big )^{\\frac{p}{2}}\\bigg ]\\right) \\nonumber \\\\ &\\le C_p\\left(\\Delta _n^{\\frac{p}{2}}\\Vert \\mu \\Vert _\\infty ^p + \\Vert \\sigma \\Vert _\\infty ^p\\right) < \\infty $ where we used the Burkholder-Davis-Gundy inequality and the boundedness of $\\mu $ and $\\sigma $ .",
"Now, by the assumption of polynomial growth, $\\left|g(x)\\right| \\le C (1+ \\Vert x\\Vert _{\\infty }^p)$ for $p>0$ so $ \\mathbb {E}\\left[ g(\\Delta _n^{-\\frac{1}{2}} \\ d_i^n (X))\\right] &\\le C(1+ \\Delta _n^{-\\frac{p}{2}}\\mathbb {E}\\Big [ \\Vert d_i^n (X)\\Vert _\\infty ^p\\Big ]) < \\infty .$ Define $\\beta _i^n:= \\Delta _n^{-\\frac{1}{2}} \\sigma _{(i-1)\\Delta _n}\\ d_i^n (W)$ , an approximation of $\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)$ .",
"As in (REF ), (REF ) we find that $\\mathbb {E}\\Big [ \\Vert \\beta _i^n\\Vert _\\infty ^p\\Big ] &\\le C_p ,\\quad p>0, \\\\\\mathbb {E}\\Big [\\big |g(\\beta _i^n)\\big |\\Big ] &\\le C. $ $\\beta _i^n$ will serve as a convenient approximation because of its simple form and $\\nonumber &\\mathbb {E}\\bigg [ \\Vert \\beta _i^n -\\Delta _n^{-\\frac{1}{2}} d_i^n(X)\\Vert _\\infty ^p\\bigg ] \\\\ \\nonumber &=\\Delta _n^{-\\frac{p}{2}}\\, \\mathbb {E}\\bigg [\\sup _{[(i-1)\\Delta _n, i\\Delta _n]} \\Big | \\int _{(i-1)\\Delta _n}^t\\!",
"\\mu _s \\,\\text{d}s + \\int _{(i-1)\\Delta _n}^t\\!",
"\\left(\\sigma _s-\\sigma _{(i-1)\\Delta _n}\\right) \\,\\text{d}W_s\\Big |^p\\bigg ] \\\\ \\nonumber &\\le C \\left(\\Vert \\mu \\Vert _\\infty ^p \\Delta _n^{\\frac{p}{2}}+ \\Delta _n^{- \\frac{p}{2}}\\, \\mathbb {E}\\bigg [\\Big (\\int _{(i-1)\\Delta _n}^{i\\Delta _n}\\!",
"\\left(\\sigma _s-\\sigma _{(i-1)\\Delta _n}\\right)^2 \\,\\text{d}s\\Big )^{\\frac{p}{2}}\\bigg ]\\right) \\\\ &\\rightarrow 0,$ where we used again the Burkholder-Davis-Gundy inequality and for the last step that $\\sigma $ is cádlág.",
"Returning to the claimed convergence in ucp, let $U^n_t &:= \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[g(\\beta _i^n) \\left| \\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\right], \\\\R^{1,n}_t &:= \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\big (g(\\beta _i^n)- \\mathbb {E}\\left[g(\\beta _i^n) \\left| \\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\right]\\big ), \\\\R^{2,n}_t &:= \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\big (g(\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X))-g(\\beta _i^n)\\big ),$ for all $t\\ge 0,\\ n\\in \\mathbb {N}$ .",
"Clearly, $V(X, g)^n_t= U^n_t + R^{1,n}_t+R^{2,n}_t$ .",
"In order to prove (REF ), we will first show that the approximation $U^n$ converges to $V(X, g)$ and afterwards that the error terms $R^1,\\ R^2$ vanish.",
"By definition, $\\mathbb {E}\\left[g(\\beta _i^n) \\left|\\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\right]=\\rho _{\\sigma _{(i-1)\\Delta _n}}\\left(g\\right)$ , and therefore $U^n_t = \\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\rho _{\\sigma _{(i-1)\\Delta _n}}\\left(g\\right)\\stackrel{ucp}{\\rightarrow }V(X, g)_t = \\int _0^{t} \\rho _{\\sigma _s}(g) \\,\\text{d}s$ due to continuity of the function $\\rho (g)$ .",
"Turning to the claimed disappearance of $R^{1, n}$ we exploit its martingale property and apply Doob's maximal inequality to get that $\\mathbb {P}\\left[\\sup _{0\\le t \\le T} \\left| R^{1, n}_t\\right| > \\epsilon \\right] \\le C \\frac{\\Delta _n^2}{\\epsilon ^2} \\sum _{i=1}^{\\left\\lfloor T/\\Delta _n\\right\\rfloor } \\mathbb {E}\\left[g(\\beta _i^n)^2\\right]\\le C_T \\Delta _n \\epsilon ^{-2} \\rightarrow 0$ for each $\\epsilon >0$ .",
"Regarding $R^{2, n}$ , Chebyshev's inequality gives that for $\\epsilon >0$ , $\\mathbb {P}\\bigg [\\sup _{0 \\le t \\le T}\\left|R^{2, n}_t\\right|> \\epsilon \\bigg ] \\le \\frac{\\Delta _n}{\\epsilon } \\sum _{i=1}^{\\left\\lfloor T/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[\\big | g(\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X))-g(\\beta _i^n)\\big |\\right].$ Now we make use of the locally uniform continuity of $g$ .",
"For $K, \\hat{\\epsilon }>0$ choose $\\delta >0$ as in (i).",
"Defining $A^{i, n, K}\\!",
":=\\!\\lbrace \\Vert \\beta _i^n\\Vert _\\infty + \\Vert \\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\Vert _\\infty \\le K\\rbrace $ as well as $A^{i, n, K, \\delta }\\!",
":=A^{i, n, K}\\cap \\lbrace \\Vert \\beta _i^n - \\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\Vert _\\infty \\le \\delta \\rbrace $ and denoting $\\Delta _i^n g:=g(\\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X))-g(\\beta _i^n)$ , we find that $\\mathbb {E}\\left[|\\Delta _i^n g|\\right] &=\\mathbb {E}\\left[|\\Delta _i^n g |\\left( 1_{A^{i,n,K,\\delta }} + 1_{A^{i,n,K}\\setminus A^{i,n,K,\\delta }} + 1_{\\Omega \\setminus A^{i,n, K}}\\right)\\right] \\\\ &\\le \\hat{\\epsilon }+ C \\Big ( \\mathbb {E}\\left[ \\Vert \\beta _i^n - \\Delta _n^{-\\frac{1}{2}}\\ d_i^n (X)\\Vert _\\infty \\right] / \\delta + 1 / K \\Big )\\nonumber $ Hence, choosing $K$ and $n$ large, and then $\\hat{\\epsilon }$ small, we see that $\\mathbb {P}\\left[\\sup _{0 \\le t \\le T}|R^{2, n}_t|> \\epsilon \\right]$ vanishes as $n\\rightarrow \\infty $ and we are done.",
"$\\Box $"
],
[
"Proof of Theorem ",
"Thanks to $\\sigma $ following a diffusion process the approximation $\\beta _i^n$ is now sharper than in (REF ): $\\nonumber \\Delta _n^{-\\frac{p}{2}}\\, &\\mathbb {E}\\bigg [ \\Vert \\beta _i^n-\\Delta _n^{-\\frac{1}{2}} d_i^n(X)\\Vert _\\infty ^p\\bigg ] \\\\ \\nonumber &=\\Delta _n^{- p}\\, \\mathbb {E}\\bigg [\\sup _{t\\in [(i-1)\\Delta _n, i\\Delta _n]} \\Big | \\int _{(i-1)\\Delta _n}^t\\!\\mu _s \\,\\text{d}s + \\int _{(i-1)\\Delta _n}^t \\!\\left(\\sigma _s-\\sigma _{(i-1)\\Delta _n}\\right) \\,\\text{d}W_s\\Big |^p\\bigg ] \\\\ \\nonumber &\\le \\Delta _n^{-p}\\, C \\bigg (\\Vert \\mu \\Vert _\\infty ^p \\Delta _n^p+ C\\, \\mathbb {E}\\bigg [\\Big (\\int _{(i-1)\\Delta _n}^{i\\Delta _n}\\!",
"\\left(\\sigma _s-\\sigma _{(i-1)\\Delta _n}\\right)^2 \\,\\text{d}s\\Big )^{\\frac{p}{2}}\\bigg ]\\bigg ) \\\\ &\\le C \\Big (\\Vert \\mu \\Vert _\\infty ^p + \\Delta _n^{-\\frac{p}{2}} \\mathbb {E}\\Big [\\sup _{s\\in [(i-1)\\Delta _n,i\\Delta _n]}|\\sigma _s-\\sigma _{(i-1)\\Delta _n}|^p\\Big ]\\Big ) \\\\ \\nonumber &\\le C \\bigg (\\Vert \\mu \\Vert _\\infty ^p + \\Delta _n^{\\frac{p}{2}} \\Vert \\tilde{\\mu }\\Vert _\\infty + \\Delta _n^{-\\frac{p}{2}} \\mathbb {E}\\bigg [ \\Big (\\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\!",
"(\\tilde{\\sigma }_s-\\tilde{\\sigma }_{(i-1)\\Delta _n})^2\\,\\text{d}s\\Big )^{\\frac{p}{2}}\\bigg ]\\bigg ) \\\\ \\nonumber & \\le C.$ Again, we used the Burkholder-Davis-Gundy inequality and the cádlág property of $\\tilde{\\sigma }$ .",
"In order to prove (REF ) we split up the original term $\\Delta _n^{-\\frac{1}{2}}\\left(V(X,g)^n_t-V(X,g)_t\\right)$ into an approximation and several error terms: $\\Delta _n^{-\\frac{1}{2}}\\left(V(X,g)^n_t-V(X,g)_t\\right) &=\\Delta _n^{-\\frac{1}{2}} \\left(\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor } g(\\Delta _n^{-\\frac{1}{2}} d_i^n (X)) - \\int _0^t \\rho _{\\sigma _s}(g)\\,\\text{d}s\\right) \\\\ &=:U^n_t + R^{1, n}_t + R^{2, n}_t + R^{3, n}_t + R^{4, n}_t,$ where $U^n_t &= \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\left( g(\\beta _i^n) - \\mathbb {E}\\left[g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right]\\right), \\\\R^{1, n}_t &= \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\left(g(\\Delta _n^{-\\frac{1}{2}} d_i^n(X))-g(\\beta _i^n) - \\mathbb {E}\\left[g(\\Delta _n^{-\\frac{1}{2}} d_i^n(X)) -g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right]\\right), \\\\R^{2, n}_t &= \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[g(\\Delta _n^{-\\frac{1}{2}} d_i^n(X)) -g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right], \\\\R^{3, n}_t &= \\Delta _n^{-\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor } \\left(\\Delta _n \\mathbb {E}\\left[g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right] - \\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\rho _{\\sigma _s}(g)\\,\\text{d}s\\right),\\\\R^{4, n}_t &= \\Delta _n^{-\\frac{1}{2}} \\int _{\\left\\lfloor t/\\Delta _n\\right\\rfloor \\Delta _n}^t \\rho _{\\sigma _s}(g)\\,\\text{d}s.$ Obviously, $R^{4,n}_t \\stackrel{ucp}{\\rightarrow } 0$ due to the boundedness of $\\sigma $ and the continuity of $\\rho $ .",
"Furthermore, we also have Lemma 5.1 Under conditions of Theorem REF we obtain (i) $U^n_t\\ \\stackrel{d_{st}}{\\rightarrow }\\ \\int _0^t \\rho _{\\sigma _s}^{(1)}(g)\\,\\emph {d}W_s + \\int _0^t \\sqrt{\\rho _{\\sigma _s}(g^2)-\\rho _{\\sigma _s}(g)^2-(\\rho _{\\sigma _s}^{(1)}(g))^2}\\, \\emph {d}W^{\\prime }_s $ , (ii) $R^{1,n}_t\\ \\stackrel{ucp}{\\rightarrow }\\ 0$ , (iii) $R^{2,n}_t\\ \\stackrel{ucp}{\\rightarrow }\\ \\int _0^t \\mu _s \\rho _{\\sigma _s}^{(2)}(g^{\\prime }) \\,\\emph {d}s + \\frac{1}{2} \\int _0^t \\tilde{\\sigma }_s \\rho _{\\sigma _s}^{(3)}(g^{\\prime }) \\,\\emph {d}s - \\frac{1}{2} \\int _0^t \\tilde{\\sigma }_s \\rho _{\\sigma _s}^{(2)}(g^{\\prime })\\,\\emph {d}s$ , (iv) $R^{3,n}_t\\ \\stackrel{ucp}{\\rightarrow }\\ 0$ .",
"(i) Defining $\\xi _i^n :=\\Delta _n^{\\frac{1}{2}}(g(\\beta _i^n)-\\mathbb {E}[g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.",
"])$ we have $U^n_t= \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\xi _i^n$ .",
"Now we will verify the conditions of Jacod's theorem of stable convergence for semimartingales (see [21]).",
"Introducing the notation $\\Delta _i^n W := W_{i\\Delta _n}-W_{(i-1)\\Delta _n}$ we find that $\\mathbb {E}[\\xi _i^n\\left| \\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"&= 0,$ $ \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}[(\\xi _i^n)^2\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"&=\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor }\\left( \\mathbb {E}[g(\\beta _i^n)^2\\left| \\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"- \\mathbb {E}[g(\\beta _i^n)\\left| \\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.",
"]^2\\right) \\\\ &=\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\left(\\rho _{\\sigma _{(i-1)\\Delta _n}}(g^2)-\\rho _{\\sigma _{(i-1)\\Delta _n}}(g)^2\\right) \\\\ &\\stackrel{ucp}{\\rightarrow }\\int _0^t \\left(\\rho _{\\sigma _s}(g^2)-\\rho _{\\sigma _s}(g)^2\\right) \\,\\text{d}s ,$ $ \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}[\\xi _i^n\\, \\Delta _i^n W \\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"&=\\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}[g(\\beta _i^n)\\, \\Delta _i^n W \\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"\\\\ &=\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\left.\\mathbb {E}[g(\\left\\lbrace x\\,W_s;\\ s\\in [0, 1]\\right\\rbrace ) W_1]\\right|_{x=\\sigma _{(i-1)\\Delta _n}} \\\\ &\\stackrel{ucp}{\\rightarrow }\\int _0^t\\rho _{\\sigma _s}^{(1)}(g) \\,\\text{d}s ,$ $ \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}[(\\xi _i^n)^21_{\\left\\lbrace \\left|\\xi _i^n\\right|> \\epsilon \\right\\rbrace } \\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"&\\le \\frac{\\Delta _n^2}{ \\epsilon ^2} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}[(g(\\beta _i^n)-\\mathbb {E}[g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.",
"])^4\\left| \\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.]",
"\\\\ &\\le \\Delta _n C / \\epsilon ^2 \\rightarrow 0.$ Finally, let $N\\in \\mathcal {M}_b(W)^\\bot $ , the space of all bounded $(\\mathbb {P}, \\mathbb {F})$ -martingales that have zero quadratic covariation with $W$ .",
"Define $M_u:= \\mathbb {E}[g(\\beta _i^n) | \\mathcal {F}_{u}]$ for $u\\ge (j-1)\\Delta _n$ .",
"By the martingale representation theorem we deduce the identity $M_u = M_{(i-1)\\Delta _n} + \\int _{(i-1)\\Delta _n}^u \\eta _s \\, \\text{d}W_s$ for a suitable predictable process $\\eta $ .",
"By the Itô isometry we conclude that $\\mathbb {E}[g(\\beta _i^n) \\Delta _i^n N | \\mathcal {F}_{(i-1) \\Delta _n}] &=\\mathbb {E}[M_{i\\Delta _n} \\Delta _i^n N | \\mathcal {F}_{(i-1) \\Delta _n}] \\\\&= \\mathbb {E}[ \\Delta _i^n M \\Delta _i^n N | \\mathcal {F}_{(i-1) \\Delta _n}] =0.$ Hence, Jacod's convergence theorem (see [22]) gives $U^n_t\\ \\stackrel{d_{st}}{\\rightarrow }\\ \\int _0^t \\rho _{\\sigma _s}^{(1)}(g)\\,\\text{d}W_s + \\int _0^t\\sqrt{\\rho _{\\sigma _s}(g^2)-\\rho _{\\sigma _s}(g)^2-(\\rho _{\\sigma _s}^{(1)}(g))^2}\\, \\text{d}W^{\\prime }_s.$ $\\Box $ (ii) Let $\\eta _i^n := \\Delta _n^{\\frac{1}{2}}(g(\\Delta _n^{-\\frac{1}{2}}\\,d_i^n (X))-g(\\beta _i^n))$ so $R^{1, n}_t = \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } (\\eta _i^n - \\mathbb {E}[\\eta _i^n |\\ \\mathcal {F}_{(i-1) \\Delta _n}])$ .",
"Since $R^{1, n}$ is a martingale we may apply Doob's inequality to obtain $\\mathbb {P}\\left[\\sup _{t \\le T}\\left|R^{1, n}_t\\right| > \\epsilon \\right] \\le C\\frac{\\Delta _n}{\\epsilon ^2} \\sum _{i=1}^{\\left\\lfloor T/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big (g(\\Delta _n^{-\\frac{1}{2}}\\,d_i^n(X))-g(\\beta _i^n)\\big )^2\\Big ].$ By the same argument as in (REF ), making use of the locally uniform continuity of $g$ , we find that the last term converges to 0.",
"$\\Box $ (iii) By the assumed Gâteaux differentiability of $g$ the mean-value theorem gives $g(y)-g(x) =g^{\\prime }_{y-x}(x+\\hat{t}(y-x))$ for some $\\hat{t}\\in [0,1]$ .",
"Let us again use the notation $f(x;y):=g^{\\prime }_y(x)$ .",
"We expand $R^{2, n}= R^{2.1, n}+R^{2.2, n}$ where $R^{2.1, n}_t&:= \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[f^{\\prime }(\\beta _i^n;\\ \\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n) \\left|\\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\right], \\\\R^{2.2, n}_t&:= \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[\\big (f^{\\prime }(\\chi _i^n;\\ \\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n)-f^{\\prime }(\\beta _i^n;\\ \\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n)\\big ) \\left|\\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\right],$ with $\\chi _i^n= \\beta _i^n + \\hat{t}_i^n (\\Delta _n^{-\\frac{1}{2}}\\, d_i^n(X) -\\beta _i^n)$ and $\\hat{t}_i^n \\in [0, 1]$ .",
"Decompose also $\\Delta _n^{-\\frac{1}{2}}\\, d_i^n(X) -\\beta _i^n= V_i^n(1) + V_i^n(2)$ where $V_i^n(1)_t &:= \\Delta _n^{-\\frac{1}{2}}\\bigg (t\\, \\Delta _n\\, \\mu _{(i-1)\\Delta _n} \\\\&+\\int _{(i-1)\\Delta _n}^{(i-1\\text{+}t)\\Delta _n}\\!\\Big ({\\tilde{\\sigma }}_{(i-1)\\Delta _n}(W_s - W_{(i-1)\\Delta _n}) + {\\tilde{v}}_{(i-1)\\Delta _n}(V_s - V_{(i-1)\\Delta _n})\\Big )\\,\\text{d}W_s\\bigg ) \\\\ &=\\Delta _n^{-\\frac{1}{2}}\\bigg (t\\, \\Delta _n\\, \\mu _{(i-1)\\Delta _n}+ \\frac{1}{2}\\, {\\tilde{\\sigma }}_{(i-1)\\Delta _n}((W_{(i-1\\text{+}t)\\Delta _n}-W_{(i-1)\\Delta _n})^2-t\\, \\Delta _n) \\\\ &+{\\tilde{v}}_{(i-1)\\Delta _n}\\int _{(i-1)\\Delta _n}^{(i-1\\text{+}t)\\Delta _n} (V_s - V_{(i-1)\\Delta _n})\\,\\text{d}W_s\\bigg ), \\\\ V_i^n(2)_t :&=\\Delta _n^{-\\frac{1}{2}}\\Bigg (\\int _{(i-1)\\Delta _n}^{(i-1\\text{+}t)\\Delta _n} \\big (\\mu _s-\\mu _{(i-1)\\Delta _n}\\big )\\,\\text{d}s\\ + \\Big (\\int _{(i-1)\\Delta _n}^s\\big ({\\tilde{v}}_u-{\\tilde{v}}_{(i-1)\\Delta _n}\\big )\\text{d}V_u\\Big ) \\,\\text{d}W_s \\\\ &+\\Big (\\int _{(i-1)\\Delta _n}^s{\\tilde{\\mu }}_u\\, \\text{du }+ \\int _{(i-1)\\Delta _n}^s({\\tilde{\\sigma }}_u-{\\tilde{\\sigma }}_{(i-1)\\Delta _n})\\,\\text{d}W_u\\Big )\\text{d}W_s\\Bigg )$ for $t\\in [0,1]$ .",
"Now, by the linearity of $f$ in the second argument $\\Delta _n^{\\frac{1}{2}}& \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [f(\\beta _i^n;\\ V_i^n(1)) \\left|\\ \\mathcal {F}_{(i-1)\\Delta _n}\\right.\\Big ] \\\\ &=\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor }\\left(\\mu _{(i-1)\\Delta _n} \\left.\\mathbb {E}\\Big [f(\\lbrace x\\, W_s;\\ s\\in [0,1]\\rbrace ;\\ \\lbrace s;\\ s\\in [0, 1]\\rbrace )\\Big ]\\right|_{x=\\sigma _{(i-1)\\Delta _n}}\\right.",
"\\\\ &\\quad \\quad \\quad \\left.+ \\ \\frac{1}{2}\\, {\\tilde{\\sigma }}_{(i-1)\\Delta _n} \\left.\\mathbb {E}\\Big [f(\\lbrace x\\, W_s;\\ s\\in [0,1]\\rbrace ;\\ \\lbrace W_s^2-s;\\ s\\in [0, 1]\\rbrace )\\Big ]\\right|_{x=\\sigma _{(i-1)\\Delta _n}} \\right) \\\\ &=\\Delta _n \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor }\\left(\\mu _{(i-1)\\Delta _n}\\, \\rho _{\\sigma _{(i-1)\\Delta _n}}^{(2)}(f) + \\frac{1}{2}\\, {\\tilde{\\sigma }}_{(i-1)\\Delta _n} \\left(\\rho _{\\sigma _{(i-1)\\Delta _n}}^{(3)}(f) - \\rho _{\\sigma _{(i-1)\\Delta _n}}^{(2)}(f)\\right)\\right) \\\\ &\\stackrel{ucp}{\\rightarrow }\\int _0^t \\mu _s\\, \\rho _{\\sigma _s}^{(2)}(f) \\,\\text{d}s + \\frac{1}{2} \\int _0^t \\tilde{\\sigma }_s\\, \\rho _{\\sigma _s}^{(3)}(f) \\,\\text{d}s - \\frac{1}{2} \\int _0^t \\tilde{\\sigma }_s\\, \\rho _{\\sigma _s}^{(2)}(f)\\,\\text{d}s,$ where we used the independence of $W$ and $V$ .",
"Due to linearity we observe the identity $f(\\beta _i^n;\\ V_i^n(2)) = f(\\beta _i^n;\\ V_i^n(2)/\\Vert V_i^n(2)\\Vert _\\infty )\\Vert V_i^n(2)\\Vert _\\infty ,$ whenever $\\Vert V_i^n(2)\\Vert _\\infty >0$ and 0 otherwise.",
"Hence, we deduce that $\\Delta _n^{\\frac{1}{2}}& \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big |f(\\beta _i^n;\\ V_i^n(2))\\big | \\Big ] \\\\ &=\\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big |f(\\beta _i^n;\\ V_i^n(2)/\\Vert V_i^n(2)\\Vert _\\infty )\\big |\\Vert V_i^n(2)\\Vert _\\infty \\Big ] \\\\ &\\le \\Delta _n^{\\frac{1}{2}} C \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big \\Vert V_i^n(2)\\big \\Vert _\\infty ^2 \\Big ]^{\\frac{1}{2}} \\\\ & \\le C \\bigg (\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big \\Vert V_i^n(2)\\big \\Vert _\\infty ^2 \\Big ]\\bigg )^{\\frac{1}{2}} \\rightarrow 0$ by the Cauchy-Schwarz inequality and the polynomial growth of $f$ .",
"So we are only left to prove that $R^{2.2, n} \\stackrel{ucp}{\\rightarrow } 0$ .",
"Defining $\\xi _i^n:=\\frac{\\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n}{\\Vert \\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n\\Vert _\\infty }$ for $\\Vert \\Delta _n^{-\\frac{1}{2}} d_i^n(X) -\\beta _i^n\\Vert _\\infty >0$ and 0 otherwise, we get $\\big |R^{2.2, n}_t\\big | \\le \\Delta _n^{\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mathbb {E}\\left[\\big |f(\\chi _i^n;\\ \\xi _i^n)-f(\\beta _i^n;\\ \\xi _i^n)\\big |\\big \\Vert \\Delta _n^{-\\frac{1}{2}}\\, d_i^n(X) -\\beta _i^n\\big \\Vert _\\infty \\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right].$ Therefore, $\\mathbb {P}\\Big [\\sup _{t\\le T} &\\big |R^{2.2, n}_t\\big | > \\epsilon \\Big ] \\\\ &\\le \\frac{\\Delta _n^{\\frac{1}{2}}}{\\epsilon } \\sum _{i=1}^{\\left\\lfloor T/\\Delta _n \\right\\rfloor } \\mathbb {E}\\Big [\\big |f(\\chi _i^n;\\ \\xi _i^n)-f(\\beta _i^n;\\ \\xi _i^n)\\big |\\big \\Vert \\Delta _n^{-\\frac{1}{2}}\\, d_i^n(X) -\\beta _i^n\\big \\Vert _\\infty \\Big ] \\\\ &\\le C\\frac{\\Delta _n }{\\epsilon } \\sum _{i=1}^{\\left\\lfloor T/\\Delta _n \\right\\rfloor } \\sqrt{\\mathbb {E}\\Big [\\big (f(\\chi _i^n;\\ \\xi _i^n)-f(\\beta _i^n;\\ \\xi _i^n)\\big )^2\\Big ]} \\rightarrow 0,$ where again we used Jensen's and Cauchy-Schwarz inequality and the local Hölder continuity of $f$ (uniformly on the unit circle in the second argument).",
"Putting everything together, we have thus proven (iii).",
"$\\Box $ (iv) We want to show that $R^{3, n}_t = \\Delta _n^{-\\frac{1}{2}} \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor } \\left(\\Delta _n \\mathbb {E}\\left[g(\\beta _i^n)\\left|\\ \\mathcal {F}_{(i-1) \\Delta _n}\\right.\\right] - \\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\rho _{\\sigma _s}(g)\\,\\text{d}s\\right)\\stackrel{ucp}{\\rightarrow }0.$ Define $\\mu _i^n:=\\Delta _n^{-\\frac{1}{2}}\\int _{(i-1)\\Delta _n}^{i\\Delta _n}(\\rho _{\\sigma _{(i-1)\\Delta _n}}(g)-\\rho _{\\sigma _s}(g))\\, \\text{d}s$ so $R^{3, n}_t=\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor } \\mu _i^n$ .",
"Thanks to the differentiability of $g$ and the polynomial growth of $g$ and $g^{\\prime }$ , we find that $\\lim _{h\\rightarrow 0} \\frac{\\rho _{x+h}(g)-\\rho _x(g)}{h} =\\mathbb {E}[g^{\\prime }_{W}(xW)], $ so the derivative of $\\rho _x(g)=:\\psi (x)$ exists.",
"Similarly, using that $g^{\\prime }$ is continuous, linear in the second argument and of polynomial growth in the first argument, we see that $\\psi ^{\\prime }$ is continuous.",
"This allows us to expand $\\mu _i^n=:\\mu _i^n(1) + \\mu _i^n(2)$ where $\\mu _i^n(1)&= \\Delta _n^{-\\frac{1}{2}}\\, \\psi ^{\\prime }(\\sigma _{(i-1)\\Delta _n})\\int _{(i-1)\\Delta _n}^{i\\Delta _n}(\\sigma _{(i-1)\\Delta _n}-\\sigma _s) \\,\\text{d}s, \\\\\\mu _i^n(2)&= \\Delta _n^{-\\frac{1}{2}} \\int _{(i-1)\\Delta _n}^{i\\Delta _n}(\\psi ^{\\prime }(\\chi _s^n)-\\psi ^{\\prime }(\\sigma _{(i-1)\\Delta _n}))(\\sigma _{(i-1)\\Delta _n}-\\sigma _s) \\,\\text{d}s$ with $\\left|\\chi _s^n-\\sigma _{(i-1)\\Delta _n}\\right| \\le \\left|\\sigma _{i\\Delta _n}-\\sigma _{(i-1)\\Delta _n}\\right|$ .",
"Now, decompose $-\\mu _i^n(1)$ into a martingale increment $\\mu _i^n(1.2)$ and a remainder term $\\mu _i^n(1.1)$ , i.e.",
"$-\\mu _i^n(1)=\\mu _i^n(1.1) + \\mu _i^n(1.2)$ with $\\mu _i^n(1.1)&= \\Delta _n^{-\\frac{1}{2}} \\psi ^{\\prime }(\\sigma _{(i-1)\\Delta _n})\\int _{(i-1)\\Delta _n}^{i\\Delta _n}\\left(\\int _{(i-1)\\Delta _n}^s \\tilde{\\mu }_u\\, \\text{du}\\right)\\,\\text{d}s, \\\\\\mu _i^n(1.2)&= \\Delta _n^{-\\frac{1}{2}} \\psi ^{\\prime }(\\sigma _{(i-1)\\Delta _n})\\int _{(i-1)\\Delta _n}^{i\\Delta _n}\\left(\\int _{(i-1)\\Delta _n}^s \\tilde{\\sigma }_u \\,\\text{d}W_u+\\int _{(i-1)\\Delta _n}^s \\tilde{v}_u\\, \\text{d}V_u\\right)\\,\\text{d}s.$ Observing that $\\mu _i^n(1.1)\\le \\Delta _n^{\\frac{3}{2}}\\, \\sup _{\\left|x\\right|\\le \\left|\\sigma \\right|}\\psi ^{\\prime }(x)\\, \\Vert \\tilde{\\mu }\\Vert _\\infty $ , its convergence to 0 follows immediately.",
"With the help of Doob's inequality, $\\mathbb {P}\\Big [\\sup _{t\\le T}\\Big |\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor }\\mu _i^n(1.2)\\Big |> \\epsilon \\Big ] & \\le C/\\epsilon ^2\\, \\mathbb {E}\\Big [\\Big (\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor }\\mu _i^n(1.2)\\Big )^2\\Big ] \\\\ &=C / \\epsilon ^2 \\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor } \\mathbb {E}\\left[\\left(\\mu _i^n(1.2)\\right)^2\\right] \\\\ &\\le C\\frac{\\Delta _n^{\\frac{1}{2}}}{\\epsilon ^2} \\sup _{\\left|x\\right|\\le \\Vert \\sigma \\Vert _{\\infty }}\\psi ^{\\prime }(x)\\big (\\Vert \\tilde{\\sigma }\\Vert ^2 + \\Vert \\tilde{v}\\Vert ^2\\big ) \\rightarrow 0$ so $\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n\\right\\rfloor }\\mu _i^n(1)\\ \\stackrel{ucp}{\\rightarrow } 0$ .",
"Regarding $\\mu _i^n(2)$ , since $\\psi ^{\\prime }$ is uniformly continuous on $[-\\Vert \\sigma \\Vert _\\infty , \\Vert \\sigma \\Vert _\\infty ]$ choose $\\delta >0$ for a given $\\epsilon >0$ such that for all $s,\\ t\\le T$ we have $\\left|\\sigma _s-\\sigma _t\\right|\\le \\delta \\Rightarrow \\left| \\psi ^{\\prime }(\\sigma _s)- \\psi ^{\\prime }(\\sigma _t)\\right|\\le \\epsilon $ .",
"Now, $\\left|\\mu _i^n(2)\\right| &\\le \\Delta _n^{-\\frac{1}{2}}\\, \\epsilon \\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\left|\\sigma _{(i-1)\\Delta _n}-\\sigma _s\\right|\\,\\text{d}s \\\\ &+ 2\\, \\Delta _n^{-\\frac{1}{2}}/\\delta \\sup _{\\left|x\\right|\\le \\Vert \\sigma \\Vert _{\\infty }}\\left|\\psi ^{\\prime }(x)\\right| \\int _{(i-1)\\Delta _n}^{i\\Delta _n} \\left|\\sigma _{(i-1)\\Delta _n}-\\sigma _s\\right|^2\\,\\text{d}s,\\multicolumn{2}{l}{\\text{leading to}}\\\\\\mathbb {P}\\Big [\\sup _{t\\le T}\\big |\\sum _{i=1}^{\\left\\lfloor t/\\Delta _n \\right\\rfloor } \\mu _i^n(2)\\big | > \\hat{\\epsilon }\\Big ] &\\le \\Delta _n^{-\\frac{1}{2}}\\, \\epsilon /\\, \\hat{\\epsilon }\\ \\mathbb {E}\\Big [\\int _0^T \\left|\\sigma _{(i-1)\\Delta _n}-\\sigma _s\\right|\\,\\text{d}s\\Big ] \\\\ &+ 2\\, \\Delta _n^{-\\frac{1}{2}} \\sup _{\\left|x\\right|\\le \\Vert \\sigma \\Vert _{\\infty }}\\left|\\psi ^{\\prime }(x)\\right| \\mathbb {E}\\Big [\\int _0^T \\left|\\sigma _{(i-1)\\Delta _n}-\\sigma _s\\right|^2\\,\\text{d}s \\Big ]/(\\hat{\\epsilon }\\delta ) \\\\ &\\le C_T\\Big (\\epsilon \\, /\\, \\hat{\\epsilon }+ \\Delta _n^{\\frac{1}{2}}\\, \\sup _{\\left|x\\right|\\le \\Vert \\sigma \\Vert _{\\infty }}\\left|\\psi ^{\\prime }(x)\\right|\\, /\\, (\\hat{\\epsilon }\\delta )\\Big ), $ where we used Fubini's theorem.",
"So choosing first $\\epsilon $ small and then $n$ large finishes the proof of (iv), the last step in the proof of (REF )."
],
[
"Proof of Theorems ",
"As in the previous proof we may assume without loss of generality that the processes $\\mu ,\\sigma , \\tilde{\\mu },\\tilde{\\sigma },\\tilde{v}$ as well as the jump sizes $J, \\tilde{J}$ are uniformly bounded in $(\\omega ,t)$ .",
"This is again justified by a standard localisation procedure (see e.g.",
"[2]).",
"Moreover, by the same localisation procedure, we may assume without loss of generality that the jump sizes $J$ are bounded from below, i.e.",
"$|J_i|>\\epsilon , \\qquad 1\\le i\\le N_t,$ for some $\\epsilon >0$ .",
"Now, let $I_i^n=[(i-1)\\Delta _n, i \\Delta _n]$ and $\\Omega _n:=\\lbrace \\omega \\in \\Omega : \\#\\lbrace j\\in \\mathbb {N}: T_j \\in J_i^n\\rbrace \\le 1\\rbrace $ , where $T_j$ denotes the the arrival time of the $j$ 'th jump of the Poisson process $N$ .",
"We clearly have $\\lim _{n \\rightarrow \\infty } \\mathbb {P}[\\Omega _{n}]&= \\mathbb {P}[\\Omega ] = 1.$ Note that each interval $I_i^n$ contains at most one jump of $X$ on $\\Omega _n$ and each jump is at least of size $\\epsilon $ ."
],
[
"Proof of Theorem ",
"The assertion $R(X,p)_t^n \\stackrel{\\mathbb {P}}{\\rightarrow }R(X,p)_t$ for $p=2$ has been already proved in [19], so we show the result for $p>2$ .",
"We write $X_t=X^c + Z_t$ , where $X^c$ denotes the continuous part $X$ and $Z$ stands for the jump part.",
"We define $K_n=\\lbrace i \\le \\left\\lfloor t/\\Delta _n\\right\\rfloor : \\exists T_j \\in I_i^n\\rbrace .$ Note that the cardinality of $K_n$ is finite almost surely as it is bounded by $N_t$ .",
"We decompose the statistic $R(X,p)_t^n$ as $ R(X,p)_t^n = \\sum _{i\\in K_n^c} \\sup _{s,u\\in I_i^n} |X_s^c - X_u^c|^p+ \\sum _{i\\in K_n} \\sup _{s,u\\in I_i^n} |X_s - X_u|^p.$ By Burkholder-Davis-Gundy inequality we conclude that $\\mathbb {E}[\\sup _{s,u\\in I_i^n} |X_s^c - X_u^c|^p]\\le C_p \\Delta _n^{p/2}$ , and since $p>2$ , we obtain the convergence $\\sum _{i\\in K_n^c} \\sup _{s,u\\in I_i^n} |X_s^c - X_u^c|^p \\stackrel{\\mathbb {P}}{\\rightarrow }0.$ Moreover, on $\\Omega _n$ , we have that $\\sum _{i\\in K_n} \\sup _{s,u\\in I_i^n} |Z_s - Z_u|^p - R(X,p)_t = \\sum _{i=N_{\\Delta _n \\lfloor t/\\Delta _n \\rfloor }}^{N_t} |J_i|^p \\stackrel{\\mathbb {P}}{\\rightarrow }0.$ Finally, we obtain by mean value theorem that $& \\mathbb {E}\\left| \\sum _{i\\in K_n} \\sup _{s,u\\in I_i^n} |Z_s - Z_u|^p - \\sup _{s,u\\in I_i^n} |X_s - X_u|^p \\right| \\\\[1.5 ex]&\\le p \\mathbb {E}\\sum _{i\\in K_n} \\sup _{s,u\\in I_i^n} \\max (|Z_s - Z_u|,|X_s - X_u|)^{p-1} \\sup _{s,u\\in I_i^n} |X_s^c - X_u^c| \\\\[1.5 ex]& \\stackrel{\\mathbb {P}}{\\rightarrow }0,$ since the set $K_n$ is finite.",
"Due to $\\Omega _n\\rightarrow \\Omega $ , we thus conclude the assertion of Theorem REF .",
"$\\Box $"
],
[
"Proof of Theorem ",
"(i) We again use the decomposition (REF ) of $R(X,p)_t^n$ .",
"It holds that $\\Delta _n^{-1/2} \\mathbb {E}\\Big [\\sum _{i\\in K_n^c} \\sup _{s,u\\in I_i^n} |X_s^c - X_u^c|^p \\Big ]\\le \\Delta _n^{(p-1)/2 -1}\\rightarrow 0,$ since $p>3$ .",
"On the other hand we also have $\\Delta _n^{-1/2} \\left(\\sum _{i\\in K_n} \\sup _{s,u\\in I_i^n} |Z_s - Z_u|^p - R(X,p)_t \\right) \\stackrel{\\mathbb {P}}{\\rightarrow }0.$ Now, since all jump sizes $|J_i|$ are bounded by $\\epsilon $ from below and $X^c$ is continuous, we obtain by mean value theorem for all $i\\in K_n$ on $\\Omega _n$ : $&\\sup _{s,u\\in I_i^n} (X_s - X_u)^p - \\sup _{s,u\\in I_i^n} (Z_s - Z_u)^p \\\\[1.5 ex]&= p |\\Delta Z_{T_i}|^{p-1} \\left(\\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ s<T_i^n\\le u\\end{array}} (X^c_u-X^c_s) 1_{\\lbrace \\Delta Z_{T_i}>0\\rbrace }+ \\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ u<T_i^n\\le s\\end{array}} (X^c_u-X^c_s) 1_{\\lbrace \\Delta Z_{T_i}<0\\rbrace } \\right) \\\\[1.5 ex]&+ o_{\\mathbb {P}} (\\Delta _n^{1/2}),$ where $T_i^n$ denotes the jump time of $N_t$ in the interval $I_i^n$ .",
"Hence, we deduce the decomposition $\\Delta _n^{-1/2} \\sum _{i\\in K_n} \\Big ( \\sup _{s,u\\in I_i^n} (X_s - X_u)^p - \\sup _{s,u\\in I_i^n} (Z_s - Z_u)^p \\Big )= \\sum _{i\\in K_n} (\\zeta _i^n + \\overline{\\zeta }_i^n) + o_{\\mathbb {P}} (1),$ where $\\zeta _i^n &= p \\Delta _n^{-1/2} |\\Delta Z_{T_i^n}|^{p-1} \\Big (\\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ s<T_i^n\\le u\\end{array}} (\\sigma _{T_i^n -}(W_{T_i^n} - W_s) +\\sigma _{T_i^n }(W_u - W_{T_i^n})) 1_{\\lbrace \\Delta Z_{T_i^n}>0\\rbrace } \\\\[1.5 ex]&+ \\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ u<T_i^n\\le s\\end{array}} (-\\sigma _{T_i^n -}(W_{T_i^n} - W_u) -\\sigma _{T_i^n }(W_s - W_{T_i^n})) 1_{\\lbrace \\Delta Z_{T_i^n}<0\\rbrace } \\Big )$ and the quantity $\\overline{\\zeta }_i^n$ is defined via the identity $\\zeta _i^n + \\overline{\\zeta }_i^n = p \\Delta _n^{-1/2}|\\Delta Z_{T_i^n}|^{p-1} \\Big (\\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ s<T_i^n\\le u\\end{array}} (X^c_u-X^c_s) 1_{\\lbrace \\Delta Z_{T_i^n}>0\\rbrace }+ \\sup _{\\begin{array}{c}s,u \\in I_i^n\\\\ u<T_i^n\\le s\\end{array}} (X^c_u-X^c_s) 1_{\\lbrace \\Delta Z_{T_i^n}<0\\rbrace } \\Big ).$ Obviously the term $\\zeta _i^n$ serves as the first order approximation while $\\overline{\\zeta }_i^n$ is the error term.",
"Since $N$ and $W$ are independent, we obtain the stable convergence $(\\kappa _i^n, \\widetilde{W}_i^n)_{i\\ge 1}&:=\\Big ( \\Delta _n^{-1} \\lbrace T_i - \\Delta _n \\lfloor T_i/\\Delta _n \\rfloor \\rbrace ,\\Delta _n^{-1/2} \\lbrace W_{(i-1+s)\\Delta _n} - W_{(i-1)\\Delta _n}\\rbrace _{s\\in [0,1]} \\Big )_{i\\ge 1} \\nonumber \\\\ & \\stackrel{d_{st}}{\\rightarrow }\\Big (\\kappa _i, \\lbrace \\widetilde{W}_{i-1+s} - \\widetilde{W}_{i-1}\\rbrace _{s\\in [0,1]} \\Big )_{i\\ge 1},$ where $\\kappa _i$ and $\\widetilde{W}$ were defined in Section 4.2.",
"This result is an immediate consequence of [5], but it can be easily shown in a straightforward manner.",
"Now, by properties of stable convergence and continuous mapping theorem, we conclude that $\\sum _{i\\in K_n} \\zeta _i^n \\stackrel{d_{st}}{\\rightarrow }U(X,p)_t$ for any fixed $t>0$ .",
"Indeed this can be deduced from the stable convergence in (REF ), by defining the function $f_{i},:\\mathbb {R}^3 \\times [0,1] \\times C([0,1])$ via $f_{i}(x,y,z) = p x_1 \\sup _{0\\le s<y\\le u\\le 1} \\Big ( x_2(z(y)-z(s)) + x_3 (z(u)-z(y)) \\Big )$ and observing that $\\zeta _i^n &= f_{i}\\Big ((|\\Delta Z_{T_i^n}|^{p-1} 1_{\\lbrace \\Delta Z_{T_i^n}>0\\rbrace },\\sigma _{T_i^n -}, \\sigma _{T_i^n }),\\kappa _i^n,\\widetilde{W}_i^n \\Big ) \\\\&+ f_{i}\\Big ((|\\Delta Z_{T_i^n}|^{p-1} 1_{\\lbrace \\Delta Z_{T_i^n}<0\\rbrace },\\sigma _{T_i^n -}, \\sigma _{T_i^n }),\\kappa _i^n,-\\widetilde{W}_i^n \\Big ).$ Hence, to complete the proof we need to show that $\\sum _{i\\in K_n} \\overline{\\zeta }_i^n \\stackrel{\\mathbb {P}}{\\rightarrow }0$ .",
"Since the processes $\\mu ,\\sigma $ and the jump sizes $J_i$ are uniformly bounded, we deduce by Burkholder-Davis-Gundy inequality that $\\mathbb {E}[|\\overline{\\zeta }_i^n|^2] \\le C \\Delta _n^{-1} \\left( \\Delta _n^2 + \\int _{T_i^n}^{i\\Delta _n} (\\sigma _u-\\sigma _{T_i^n})^2 du+ \\int _{(i-1)\\Delta _n}^{T_i^n} (\\sigma _u-\\sigma _{T_i^n-})^2 du \\right),$ where the right side converges to 0, because $\\sigma $ is cádlág.",
"This completes the proof since $K_n$ is finite.",
"$\\Box $ (ii) Now let us consider the case $p=2$ .",
"According to the previous proof and the limiting results of [18] for the continuous case, we obtain the following asymptotic decomposition $\\Delta _n^{-1/2} (R(X,2)_t^n - R(X,2)_t) = \\sum _{i\\in K_n} \\zeta _i^n + \\sum _{i\\in K_n^c} \\tilde{\\zeta }_i^n + o_{\\mathbb {P}}(1),$ where $\\zeta _i^n$ has been defined in the previous step (now with $p=2$ ) and $\\tilde{\\zeta }_i^n$ serves as the first order approximation in the continuous case, i.e.",
"$\\tilde{\\zeta }_i^n = \\Delta _n^{-1/2} \\sigma _{(i-1)\\Delta _n}^2 \\Big ( \\sup _{s,u \\in I_i^n} (W_u-W_s)^2 - \\Delta _n^{-1}\\lambda _{3,2} \\Big ).$ Now, we need to prove joint stable convergence of the vector $\\Big (\\sum _{i\\in K_n} \\zeta _i^n,\\sum _{i\\in K_n^c} \\tilde{\\zeta }_i^n\\Big )$ .",
"This problem is closely related to [3].",
"Indeed, following exactly the same proof steps, which are based on certain conditioning arguments, it is sufficient to prove the stable central limit theorem for each component of the vector (indeed, the two stable limits are independent conditionally on $\\mathcal {F}$ ).",
"But the stable convergence for the first component follows from the previous step and the stable convergence for the second component has been shown in [18] under the conditions of Theorem REF .",
"This completes the proof.",
"$\\Box $"
]
] | 1403.0217 |
[
[
"Dynamical Mass Reduction in the Massive Yang-Mills Spectrum in 1+1\n dimensions"
],
[
"Abstract The (1+1)-dimensional SU}(N) Yang-Mills Lagrangian, with bare mass M, and gauge coupling e, naively describes gluons of mass M. In fact, renormalization forces M to infinity.",
"The system is in a confined phase, instead of a Higgs phase.",
"The spectrum of this diverging-bare-mass theory contains particles of finite mass.",
"There are an infinite number of physical particles, which are confined hadron-like bound states of fundamental colored excitations.",
"These particles transform under irreducible representations of the global subgroup of the explicitly-broken gauge symmetry.",
"The fundamental excitations are those of the SU(N) X SU(N) principal chiral sigma model, with coupling e/M.",
"We find the masses of meson-like bound states of two elementary excitations.",
"This is done using the exact S matrix of the sigma model.",
"We point out that the color-singlet spectrum coincides with that of the weakly-coupled anisotropic SU(N) gauge theory in 2+1 dimensions.",
"We also briefly comment on how the spectrum behaves in the 't Hooft limit, with N approaching infinity."
],
[
"Introduction",
"Yang-Mills theory in $1+1$ dimensions has no local degrees of freedom.",
"Introducing an explicit mass $\\mathcal {M}$ gives a theory of longitudinally-polarized gluons at tree level.",
"It may seem intuitively obvious, for small gauge coupling, that a particle is either a vector Boson, with a mass roughly equal to $\\mathcal {M}$ , or a bound state of such vector Bosons.",
"This intuition, however, is wrong.",
"We show in this paper that the massive Yang-Mills theory describes an infinite number of particles, with masses that are much less than $\\mathcal {M}$ .",
"This can be called dynamical mass reduction.",
"Alternatively, the massive Yang-Mills model can be thought of as a gauge field, coupled to an ${\\rm SU}(N)\\times {\\rm SU}(N)$ principal chiral nonlinear sigma model.",
"The equivalence is seen by choosing the unitary gauge condition.",
"In a perturbative treatment, the spin waves of the sigma model are Goldstone bosons, giving the vector particles a mass through the Higgs mechanism.",
"Bardeen and Shizuya used this formulation in their proof of renormalizability [1].",
"The tree-level description fails because the excitations of the sigma model (without the gauge field) are not Goldstone Bosons.",
"These excitations are massive.",
"Introducing a gauge field produces a confining force between these excitations.",
"There is no Higgs or Coulomb phase.",
"There is only a confined phase.",
"We briefly describe some important earlier investigations of $(1+1)$ -dimensional Yang-Mills theory.",
"Non-Abelian gauge theories coupled to adjoint matter were studied with light-cone methods by Dalley and Klebanov [2].",
"This led to further investigations of gauged massive adjoint fermions [3].",
"Some detailed results for the spectrum of the model with of adjoint scalars were found later [4].",
"Conformal-field-theory methods have recently been applied to the model with adjoint Fermions [5].",
"Much has also been learned about pure Yang-Mills theory in $1+1$ dimensions [6], and its connections with representation theory.",
"Our model differs from the Bosonic matter theory of Refs.",
"[3], [4], in that the matter field has a non-trivial self-interaction.",
"This means that there are two scales in our problem; the mass gap of the sigma model and the gauge coupling.",
"This is why a nonrelativistic analysis, in which the former is assumed much larger than the latter, can work.",
"A full-fledged relativistic analysis is harder, though we discuss this problem in the last section of this paper.",
"We wish to stress that we are not studying a massive deformation of pure Yang-Mills theory [6] at all.",
"In fact, the situation is exactly the opposite.",
"The deformation is the Yang-Mills action, not the mass term.",
"A quantum field theory of an SU($N$ ) gauge field, coupled minimally to an adjoint matter field, can have distinct Higgs and confinement phases [7], separated by a phase boundary, for space-time dimension greater than two.",
"If this dimension is two, however, there is only the confined phase.",
"In the confined phase, the excitations are bound states of the massive particles of the sigma model.",
"These massive particles are color multiplets of degeneracy $N^{2}$ [8].",
"The action of the massive SU($N$ ) Yang-Mills field in $1+1$ dimensions is $S=\\int d^2x \\left(-\\frac{1}{4}{\\rm Tr}\\,F_{\\mu \\nu }F^{\\mu \\nu } +\\frac{e^2}{2g_0^2}{\\rm Tr} \\,A_\\mu A^\\mu \\right),$ where $A_{\\mu }$ is Hermitian and $F_{\\mu \\nu }=\\partial _\\mu A_\\nu -\\partial _\\nu A_\\mu -{\\rm i}e[A_\\mu ,A_\\nu ]$ with $\\mu ,\\nu =0,1$ and indices are raised by $\\eta ^{\\mu \\nu }$ , where $\\eta ^{00}=-\\eta ^{11}=1,\\,\\eta ^{01}=\\eta ^{10}=0$ .",
"If we drop the cubic and quartic terms from (REF ), the particles are gluons with mass ${\\mathcal {M}}=e/g_{0}$ .",
"Let's now consider a closely-related field theory, namely the ungauged principal chiral sigma model, with action $S_{\\rm PCSM}=\\int d^2x\\, \\frac{1}{2g_0^2} \\,{\\rm Tr}\\,\\partial _\\mu U^\\dag (x) \\partial ^\\mu U(x),$ where the field $U(x)$ is in the fundamental representation of ${\\rm SU}(N)$ .",
"The action (REF ) has a global ${\\rm SU}(N)\\times {\\rm SU}(N)$ symmetry, given by the transformation $U(x)\\rightarrow V_L U(x) V_R$ , where $V_{L,R}\\in {\\rm SU}(N)$ .",
"This model is asymptotically free, and has a mass gap, which we call $m$ .",
"It is possible that this mass gap is generated by non-real saddle points of the functional integral [10].",
"The running bare coupling $g_{0}$ is driven to zero, as the ultraviolet cut-off is removed.",
"We promote the left-handed ${\\rm SU}(N)$ global symmetry of the sigma model to a local symmetry, by introducing the covariant derivative $D_\\mu =\\partial _\\mu -{\\rm i} e A_{\\mu }$ , where $A_{\\mu }$ is a new Hermitian vector field that transforms as $A_{\\mu }\\rightarrow V_{L}^{\\dag }(x) A_{\\mu } V_{L}(x)-\\frac{i}{e}V_L^{\\dag }(x) \\partial _{\\mu } V_{L}(x)$ .",
"We do not gauge the right-handed symmetry.",
"The action is now $S=\\int d^2x \\left[-\\frac{1}{4}{\\rm Tr} \\, F_{\\mu \\nu } F^{\\mu \\nu } +\\frac{1}{2g_0^2} {\\rm Tr}\\,(D_\\mu U)^{\\dag } D^\\mu U\\right].$ In the unitary gauge, with $U(x)=1$ everywhere, this action (REF ) reduces to (REF ).",
"In the remainder of this paper, however, we will study (REF ) in the axial gauge.",
"In our opinion, it is best to think of the left-handed symmetry as (confined) color-SU($N$ ) and the right-handed symmetry as flavor-SU($N$ ).",
"Confinement of left-handed color means that only singlets of the left-handed color group exist in the spectrum.",
"There are “mesonic\" bound states, as well as “baryonic\" bound states.",
"The mesonic bound states have one elementary particle of the sigma model and one elementary antiparticle.",
"The simplest baryonic bound states consist of $N$ of these elementary particles, with no antiparticles.",
"There are also more complicated bound states, which exist because there are excitations in the sigma model (with no gauge field) transforming as higher representations of the color group [8].",
"In this paper, we only discuss the mesonic states in detail.",
"Recently Gongyo and Zwanziger have studied the nearest-neighbor lattice version of the action (REF ) using Monte-Carlo simulations [9].",
"They computed the static potential (through the Wilson loop) at different values of the coupling.",
"They find clear evidence of confinement and string breaking at small values of $g_{0}^{-2}$ (this is proportional to the parameter $\\gamma $ , in their notation), but a nearly-flat potential at large values, closer to the continuum limit.",
"They suggest their results may indicate a phase transition to a Higgs phase (although they do not assert that this is the case).",
"We believe the explanation is the essential singularity of the mass gap as a function of the bare coupling.",
"This mass, in an asymptotically-free theory, vanishes faster than any power of of $g_{0}$ as $g_{0}\\rightarrow 0$ .",
"Thus, string breaking occurs so readily, that it may be difficult to distinguish the two phases.",
"In this paper, the distinction is clear, because we take very small gauge coupling, suppressing (though not eliminating) string breaking.",
"The continuum gauge coupling $e$ (with dimensions of mass) is assumed to be much smaller than the mass gap of the sigma model.",
"There should be no phase transition as the gauge coupling is increased.",
"We therefore expect that, for any gauge coupling and any value of $g_{0}$ , there is only the confined phase.",
"Gongyo and Zwanziger also computed the vector-Boson propagator (the two-point function of a composite field), and the order parameter $U$ (in a particular gauge) and the susceptibility of the latter.",
"The lightest bound-state masses could be found in the behavior of the vector-Boson propagator.",
"This would make for an interesting comparison with our results.",
"A mesonic bound state, in the axial gauge, is a sigma-model particle-antiparticle pair, confined by a linear potential.",
"The string tension is $\\sigma ={e^2}C_N,$ where $C_N$ is the smallest eigenvalue of the Casimir operator of ${\\rm SU}(N)$ .",
"The mass gap is $M=2m+E_{0}\\ll {\\mathcal {M}},\\nonumber $ where $E_{0}$ is the smallest (positive) binding energy, and $m$ is the mass of a sigma-model elementary excitation.",
"This mass $M$ is finite, for fixed $m$ , as the ultraviolet cut-off is removed.",
"In contrast, the bare Yang-Mills mass $\\mathcal {M}$ , which is proportional to $1/g_{0}$ , diverges.",
"Our approach is similar to that of Ref.",
"[11].",
"We find the wave function of an unbound particle-antiparticle pair, taking into account scattering at the origin.",
"Next, we generalize this to the wave function of the pair, confined by a linear potential.",
"The method is inspired by the determination of the spectrum of the two-dimensional Ising model in an external magnetic field [13].",
"More sophisticated approaches to this and other two-dimensional models of confinement [14], [15], [16], including fine structure (form factors) of the fundamental excitations, have been developed.",
"We do not take into account decays or corrections to the spectrum from matrix elements with more fundamental excitations [17] in this paper.",
"For a general review, see Ref.",
"[18].",
"We briefly introduce the axial gauge formulation in the next section.",
"In Section III we discuss the S-matrix of the principal chiral nonlinear sigma model, and find the free particle-antiparticle wave function, for color group SU($N$ ), for $N>2$ .",
"In Section IV, we find the wave functions and bound-state spectrum of a confined pair, for $N>2$ (including $N\\rightarrow \\infty $ [19]).",
"We note that the results generalize the result of Ref.",
"[11], on the spectrum of $2+1$ -dimensional anisotropic SU(2) gauge theories, to SU($N$ ).",
"We treat the $N=2$ case separately in Section V. We present some conclusions and proposals for further work in the last section."
],
[
"The axial gauge formulation and the confined phase",
"Care is necessary to understand why the bare mass is not the physical mass.",
"If the axial gauge $A_{1}=0$ , is chosen, the action (REF ) is $S=\\int d^{2}x \\left[\\frac{1}{2}{\\rm Tr}\\,(\\partial _{1}A_{0})^{2}+\\frac{1}{2g_{0}^{2}}{\\rm Tr}\\,(\\partial _{0}U^{\\dagger }+{\\rm i}eU^{\\dagger }A_{0})(\\partial _{0}U-{\\rm i}eA_{0}U)-\\frac{1}{2g_{0}^{2}}{\\rm Tr}\\,\\partial _{1}U^{\\dagger }\\partial _{1}U\\right]\\,.",
"\\nonumber $ Let us introduce the traceless Hermitian generators $t_{a}$ of SU($N$ ), $a=1,\\dots , N^{2}-1$ , with normalization ${\\rm Tr}\\,t_{a}t_{b}=\\delta _{ab}$ and structure coefficients $f_{abc}$ , defined by $[t_{b},t_{c}]={\\rm i}f_{abc}t_{a}$ .",
"If we naively eliminate $A_{0}$ , by its equation of motion (or integrate $A_{0}$ from the functional integral), we obtain the effective action $S=\\int d^{2}x \\left(\\frac{1}{2g_0^2} {\\rm Tr}\\,\\partial _\\mu U^{\\dag } \\partial ^\\mu U +\\frac{1}{2} \\,{j_{0}^{L}}_{a}\\,\\frac{1}{-\\partial _{1}^{2}+e^{2}/g_{0}^{2}}\\, {j_{0}^{L}}_{a}\\right), $ where $j_\\mu ^{L}(x)_{b}=-{\\rm i} {\\rm Tr}\\,t_{b} \\partial _\\mu U(x) U^\\dag (x)$ is the Noether current of the left-handed ${\\rm SU}(N)$ symmetry.",
"The potential induced on the color-charge density, in the second term of (REF ), indicates that charges are screened, instead of confined.",
"This conclusion, however, is based on the fact that $U^{\\dagger }U=1$ .",
"In the renormalized theory, $U$ is not a physical field.",
"The physical scaling field of the principal chiral nonlinear sigma model is not a unitary matrix.",
"This fact is discussed more explicitly in Refs.",
"[20], in the limit $N\\rightarrow \\infty $ , with $g_{0}^{2}N$ fixed.",
"The actual excitations of the principal chiral model are massive, with a left and right color charge [8], so that no screening takes place.",
"A more careful approach is to first find the Hamiltonian in the temporal gauge $A_{0}=0$ .",
"Gauge invariance, or Gauss' law, must be imposed on physical states.",
"The Hamiltonian is $H=\\int dx^{1} \\,\\left\\lbrace \\frac{g_{0}^{2}}{2} [j^{L}_{0}(x^{1})_{b}]^{2}+ \\frac{1}{2g_{0}^{2}}[j^{L}_{1}(x^{1})_{b}]^{2}+ \\frac{1}{2}[E(x^{1})_{b}]^{2}+\\frac{e}{g_{0}^{2}} j^{L}_{1}(x^{1})_{b} A_{1}(x^{1})_{b}\\right\\rbrace , $ where $A_{1}(x^{1})_{b}={\\rm Tr}\\,t_{b}A$ and $E_{a}$ is the electric field, obeying $[E(x^{1})_{a}, A_{1}(y^{1})_{b}]=-{\\rm i}\\delta _{ab}\\delta (x^{1}-y^{1})$ .",
"The Hamiltonian (REF ) must be supplemented by Gauss' law $G(x^{1})_{a}\\Psi =0$ , for any physical state $\\Psi $ , where $G(x^{1})_{a}$ is the generator of spatial gauge transformations: $G(x^{1})_{a}=\\partial _{1}E(x^{1})_{a}+e f_{abc}A_{1}(x^{1})^{b}E(x^{1})_{c}-\\frac{e}{g_{0}^{2}} j^{L}_{0}(x^{1})_{a}\\,.$ If we require that the electric field vanishes at the boundaries $x^{1}=\\pm l/2$ , Gauss' law may be explicitly solved [12], to yield the expression for the electric field: $E(x^{1})_{a}=\\int _{-l/2}^{x^{1}}dy^{1} \\,\\left\\lbrace {\\mathcal {P}}\\exp \\left[ ie\\int _{-l/2}^{y^{1}} dz^{1}{\\mathcal {A}}_{1}(z^{1})\\right]\\right\\rbrace _{a}^{\\;\\;\\;\\;\\;b}\\;\\;\\frac{e}{g_{0}^{2}} j^{L}_{0}(y^{1})_{b}, $ where ${\\mathcal {A}}_{1}(x^{1})_{a}^{\\;\\;\\;b}={\\rm i}f_{abc}A_{1}(x^{1})_{c}$ is the gauge field in the adjoint representation.",
"There remains a global gauge invariance, which must be satisfied by physical states, i.e., $\\Gamma _{a}\\Psi =0$ , where $\\Gamma _{a}= \\int _{-l/2}^{l/2}dy^{1} \\,\\left\\lbrace {\\mathcal {P}}\\exp \\left[ ie\\int _{-l/2}^{y^{1}} dz^{1}{\\mathcal {A}}_{1}(z^{1})\\right]\\right\\rbrace _{a}^{\\;\\;\\;\\;\\;b}\\;\\;\\frac{e}{g_{0}^{2}} j^{L}_{0}(y^{1})_{b} .$ Now we are free to chose $A_{1}(x^{1})_{b}=0$ , which simplifies (REF ) and (REF ).",
"The solution for the electric field yields the Hamiltonian $H=\\int dx^{1} \\,\\left\\lbrace \\frac{g_{0}^{2}}{2} [j^{L}_{0}(x^{1})_{b}]^{2}+ \\frac{1}{2g_{0}^{2}}[j^{L}_{1}(x^{1})_{b}]^{2}\\right\\rbrace -\\frac{e^{2}}{2g_{0}^{4}} \\int dx^{1} \\!\\!\\int dy^{1}\\; \\vert x^{1}-y^{1} \\vert \\; j^{L}_{0}(x^{1})_{b} \\; j^{L}_{0}(y^{1})_{b}, $ where in the last step, we have taken the size $l$ of the system to infinity.",
"The last term is a linear potential which confines left-handed color.",
"Notice that (REF ) is not bounded from below on the full Hilbert space.",
"This is because of the last, nonlocal term; the energy can be lowered by adding pairs of colored particles (or antiparticles) and by separating them.",
"The residual Gauss-law condition $\\Gamma _{a}\\Psi =0$ , forces the global left-handed color to be a singlet, thereby removing the instability,"
],
[
"The Free Particle-Antiparticle Wave Function: $N>2$",
"The quantized principal chiral nonlinear sigma model is integrable.",
"This property, together with physical considerations, has been used to find the exact S-matrix [8].",
"An excitation has rapidity $\\theta $ , related to that excitation's energy and momentum, by $E=m\\sinh \\theta $ and $p=m\\cosh \\theta $ , respectively.",
"Let us consider a state with two excitations.",
"One excitation is an antiparticle of rapidity $\\theta _1$ and left and right ${\\rm SU}(N)$ color indices $a_1,b_1=1,\\dots , N$ , respectively.",
"The second excitation is a particle of rapidity $\\theta _2$ , and left and right color indices $a_2,b_2$ , respectively.",
"Explicitly the state is $\\vert A, \\theta _1, b_1,a_1; P, \\theta _2,a_2,b_2\\rangle _{\\rm in}.\\nonumber $ The S-matrix element, $S(\\theta )_{a_1b_1;b_2a_2}^{d_2c_2;c_1d_1}$ , is defined by $\\,_{\\rm out}\\langle A, \\theta ^\\prime _1,d_1,c_1; P,\\theta ^\\prime _2,c_2,d_2\\vert A, \\theta _1, b_1,a_1; P, \\theta _2,a_2,b_2\\rangle _{\\rm in}=S(\\theta )_{a_1b_1;b_2a_2}^{d_2c_2;c_1d_1}\\,4\\pi \\delta (\\theta _1-\\theta ^\\prime _1)\\,4\\pi \\delta (\\theta _2-\\theta ^\\prime _2),\\nonumber $ where $\\theta =\\theta _1-\\theta _2$ .",
"This S-matrix element is [8] $S(\\theta )_{a_1b_1;b_2a_2}^{d_2c_2;c_1d_1}=S(\\theta )\\left[\\delta _{a_1}^{c_1}\\delta _{a_2}^{c_2}-\\frac{2\\pi {\\rm i}}{N(\\pi {\\rm i}-\\theta )}\\delta _{a_1a_2}\\delta ^{c_1c_2}\\right]\\left[\\delta _{b_1}^{d_1}\\delta _{b_2}^{d_2}-\\frac{2\\pi {\\rm i}}{N(\\pi {\\rm i}-\\theta )}\\delta _{b_1b_2}b^{d_1d_2}\\right],\\nonumber $ where $S(\\theta )=\\frac{\\sinh \\left[\\frac{(\\pi {\\rm i}-\\theta )}{2}-\\frac{\\pi {\\rm i}}{N}\\right]}{\\sinh \\left[\\frac{(\\pi {\\rm i}-\\theta )}{2}+\\frac{\\pi {\\rm i}}{N}\\right]}\\,\\left\\lbrace \\frac{\\Gamma [i(\\pi {\\rm i}-\\theta )/2\\pi +1]\\Gamma [-{\\rm i}(\\pi {\\rm i}-\\theta )/2\\pi -{1}/{N}]}{\\Gamma [{\\rm i}(\\pi {\\rm i}-\\theta )/2\\pi +1-1/N]\\Gamma [-{\\rm i}(\\pi {\\rm i}-\\theta )/2\\pi ]}\\right\\rbrace ^2.$ For $N>2$ , the expression (REF ) may be written in the exponential form [23] : $S(\\theta )=\\exp 2 \\int _0^\\infty \\,\\frac{d\\xi }{\\xi \\sinh \\xi }\\left[ 2(e^{2\\xi /N}-1)-\\sinh (2\\xi /N)\\right]\\sinh \\frac{\\xi \\theta }{\\pi {\\rm i}} \\;.$ We will discuss the $N=2$ case separately in Section V. The wave function of a free antiparticle at $x^1$ and a free particle at $x^{2}$ , with momenta $p_1$ and $p_2$ , respectively, is $\\Psi _{p_1,\\,p_2}(x^1,y^1)_{a_1a_2;b_1b_2}=\\left\\lbrace \\begin{array}{cc}e^{{\\rm i}p_1x^1+{\\rm i}p_2y^1}A_{a_1a_2;b_1b_2},\\;&{\\rm for} \\;x^1<y^1,\\\\ \\\\e^{{\\rm i}p_2x^1+{\\rm i}p_1y^1}S(\\theta )_{a_1b_1;b_2a_2}^{d_2c_2;c_1d_1}A_{c_1c_2;d_1d_2},\\; &{\\rm for}\\; x^1>y^1.\\end{array}\\right.",
"$ where $A_{a_1a_2;b_1b_2}$ is set of arbitrary complex numbers.",
"The residual Gauss' law in the axial gauge, $\\Gamma _{a}\\Psi =0$ , restricts physical states to those which are invariant under global left-handed ${\\rm SU}(N)$ color transformations.",
"This means that the particle-antiparticle state of the form (REF ) must be projected to a global left-color singlet.",
"A left-color-singlet wave function is $\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\Psi _{p_1p_2} (x^1,y^1)_{b_1b_2}=\\delta ^{a_1a_2}\\Psi _{p_1,\\,p_2}(x^1,y^1)_{a_1a_2b_1b_2}.$ There are states of degeneracy $N^{2}-1$ , which resemble massive gluons.",
"These transform as the adjoint representation of the right-handed color symmetry.",
"The wave function of such a state is traceless in the right-handed color indices: $\\delta ^{b_1b_2}\\Psi _{p_1p_2}\\!\\!\\!&(&\\!\\!\\!x^1,y^1)_{b_1b_2}=0.$ We use a non-relativistic approximation $p_{1,2}\\ll m$ .",
"The wave function in this limit becomes $\\Psi _{p_1p_2}(x^1,y^1)_{b_1b_2}=\\left\\lbrace \\begin{array}{c}e^{{\\rm i}p_1x^1+{\\rm i}p_2y^1}A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x^1<y^1,\\\\\\,\\\\e^{{\\rm i}p_2x^1+{\\rm i}p_1y^1}\\exp ({\\rm i}\\pi -\\frac{i h_{N}}{\\pi m}\\vert p_1-p_2\\vert )A_{b_1b_2},\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x^1>y^1.\\end{array}\\right.$ where ${\\rm Tr} A=0$ , and $h_{N}&=&2\\int _0^\\infty \\frac{d\\xi }{\\sinh \\xi }\\left[2(e^{2\\xi /N}-1)-\\sinh (2\\xi /N)\\right]\\nonumber \\\\&=&-4\\gamma -\\psi \\left(\\frac{1}{2}+\\frac{1}{N}\\right)-3\\psi \\left(\\frac{1}{2}-\\frac{1}{N}\\right)-4\\ln 4,$ where $\\gamma $ is the Euler-Mascheroni constant, and $\\psi (x)={d}\\ln \\Gamma (x)/dx$ is the digamma function.",
"The expression in (REF ) must be equal to the wave function of two confined particles for sufficiently small $\\vert x^1-y^1\\vert $ .",
"To compare the two expressions, it is convenient to use center-of-mass coordinates, $X,\\,x$ , and their respective momenta $P,\\,p$ .",
"Explicitly, $X=x^{1}+y^{1}$ , $x=y^{1}-x^{1}$ , $P=p_{1}+p_{2}$ and $p=p_{2}-p_{1}$ .",
"In these coordinates, the wave function is $\\Psi _p(x)_{b_1b_2}=\\left\\lbrace \\begin{array}{c}\\cos (px+\\omega )A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0,\\\\\\,\\\\\\cos [-px+\\omega -\\phi (p)]A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0,\\end{array}\\right.$ for some constant $\\omega $ , with the phase shift $\\phi (p)=\\pi -\\frac{h_N}{\\pi m}\\vert p\\vert $ .",
"Another type of mesonic state is the right-handed color singlet, with $A_{b_1b_2}=\\delta _{b_1b_2}$ .",
"The non-relativistic limit of the wave function in this case is $\\Psi _p(x)_{\\rm singlet}=\\left\\lbrace \\begin{array}{c}\\cos (px+\\omega ),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0,\\\\\\,\\\\\\cos [-px+\\omega -\\chi (p)],\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0,\\end{array}\\right.$ where $\\chi (p)=-\\frac{h_N}{\\pi m}\\vert p\\vert .$"
],
[
"Mesonic States of Massive Yang-Mills Theory: $N>2$",
"The wave function of a particle-antiparticle pair, confined by string tension $\\sigma $ , satisfies the Schroedinger equation $-\\frac{1}{m}\\frac{d^2}{dx^2}\\Psi (x)_{b_1b_2}+\\sigma \\left|x \\right|\\,\\Psi (x)_{b_1b_2}=E\\Psi (x)_{b_1b_2},$ where $E$ is the binding energy [13].",
"The solution to Equation (REF ) is $\\Psi (x)_{b_1b_2}=\\left\\lbrace \\begin{array}{c}C {\\rm Ai}\\left[(m\\sigma )^{\\frac{1}{3}}\\left(x+\\frac{E}{\\sigma }\\right)\\right]A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0\\\\\\,\\\\C^\\prime {\\rm Ai}\\left[(m\\sigma )^{\\frac{1}{3}}\\left(-x+\\frac{E}{\\sigma }\\right)\\right]A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0,\\end{array}\\right.$ where ${\\rm Ai}(x)$ is the Airy function of the first kind, and $C,\\,C^\\prime $ are constants.",
"For $\\vert x\\vert \\ll (m\\sigma )^{-1/3}$ , the potential energy in (REF ) is sufficiently small that the wave function is (REF ), with $\\vert p\\vert =(mE)^{\\frac{1}{2}}$ .",
"The wave function (REF ) is approximated in this region by $\\Psi (x)_{b_1b_2}=\\left\\lbrace \\begin{array}{c}C\\frac{1}{\\left(x+\\frac{E}{\\sigma }\\right)^{\\frac{1}{4}}}\\cos \\left[\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(x+\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}-\\frac{\\pi }{4}\\right] A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0,\\\\\\,\\\\C^\\prime \\frac{1}{\\left(-x+\\frac{E}{\\sigma }\\right)^{\\frac{1}{4}}}\\cos \\left[-\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(-x+\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}+\\frac{\\pi }{4}\\right]A_{b_1b_2},\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0.\\end{array}\\right.\\nonumber $ Let us now consider the $(N^{2}-1)$ -plet of mesonic states.",
"The wave functions (REF ) and (REF ) should be the same for $x\\downarrow 0$ , yielding $\\frac{C}{(\\frac{E}{\\sigma })^{\\frac{1}{4}}}\\cos \\left[\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}-\\frac{\\pi }{4}\\right]=\\cos (\\omega ).$ Equation (REF ) implies $C=\\left(\\frac{E}{\\sigma }\\right)^{\\frac{1}{4}},\\,\\,\\,\\,\\,\\,\\,\\omega =\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}-\\frac{\\pi }{4}.\\nonumber $ The wave functions (REF ) and (REF ) should also be the same for $x\\uparrow 0$ , yielding $\\frac{C^\\prime }{\\left(\\frac{E}{\\sigma }\\right)^{\\frac{1}{4}}}\\cos \\left[-\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}+\\frac{\\pi }{4}\\right]=\\cos \\left[\\omega -\\pi +\\frac{h_N}{\\pi m}(mE)^{\\frac{1}{2}}\\right],$ hence $C^\\prime =C=\\left(\\frac{E}{\\sigma }\\right)^{\\frac{1}{4}}$ .",
"The arguments of the cosine on each side of (REF ) must be the same, modulo $2\\pi $ : $-\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}+\\frac{\\pi }{4}+2\\pi n=\\frac{2}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}-\\frac{5\\pi }{4}+\\frac{h_N}{\\pi m}(m E)^{\\frac{1}{2}},\\nonumber $ for $n=0,1,2,\\dots $ .",
"We simplify this to $\\frac{4}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}+\\frac{h_N}{\\pi m}(m E)^{\\frac{1}{2}}-\\left(n+\\frac{3}{4}\\right)2\\pi =0.$ An analysis which is similar to that of the previous paragraph yields the quantization condition for the right-handed singlet state (REF ).",
"This is $\\frac{4}{3}(m\\sigma )^{\\frac{1}{2}}\\left(\\frac{E}{\\sigma }\\right)^{\\frac{3}{2}}+\\frac{h_N}{\\pi m}(m E)^{\\frac{1}{2}}-\\left(n+\\frac{1}{4}\\right)2\\pi =0.$ Equations (REF ) and (REF ) are depressed cubic equations of the variable $Z_n=E_n^{\\frac{1}{2}}$ .",
"These cubic equations have only one real solution for each value of $n$ , because ${h_{N}}/({\\pi m^{\\frac{1}{2}}})>0$ .",
"The solution of Equations (REF ) and (REF ) is $E_n=\\left\\lbrace \\left[\\epsilon _n+\\left(\\epsilon _n^2+\\beta _N^3\\right)^{\\frac{1}{2}}\\right]^{\\frac{1}{3}}+\\left[\\epsilon _n-\\left(\\epsilon _n^2+\\beta _N^3\\right)^{\\frac{1}{2}}\\right]^{\\frac{1}{3}}\\right\\rbrace ^{\\frac{1}{2}}, $ where $\\epsilon _n=\\frac{3\\pi }{4}\\left(\\frac{\\sigma }{m}\\right)^{\\frac{1}{2}}\\left(n+\\frac{1}{2}\\pm \\frac{1}{4}\\right),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\beta _N=\\frac{h_N\\sigma ^{\\frac{1}{2}}}{4\\pi m}, $ where $\\pm =+$ for the $(N^2-1)$ -plet, and $\\pm =-$ for the singlet.",
"We show in the next section that the expressions (REF ) and (REF ) remain valid for the SU(2) case, with $h_{2}=-4\\ln 2+2$ and, significantly, with a reversal of the sign in (REF ).",
"For $N=2$ only we must take $\\pm =-$ for the $(N^{2}-1)$ -plet (the triplet) and $\\pm =+$ for the singlet.",
"As it happens, the results we have just obtained for the singlet spectrum generalize the result of Ref.",
"[11], on the spectrum of $2+1$ -dimensional anisotropic SU(2) gauge theories, to SU($N$ ) (where $\\sigma $ is replaced by $2\\sigma $ ).",
"Another interesting special case is the 't Hooft limit $N\\rightarrow \\infty $ [20], [24].",
"The mass gap of the sigma model should be fixed in this limit.",
"The string tension $\\sigma $ will be fixed as well [19], provided $e^{2}N$ is fixed.",
"In this limit $h_{N}\\rightarrow 0$ , and we find $E_n=\\left[\\frac{3\\pi }{2}\\left(\\frac{\\sigma }{m}\\right)^{\\frac{1}{2}}\\left(n+\\frac{1}{2}\\pm \\frac{1}{4}\\right) \\right]^{1/3}.$"
],
[
"The $N=2$ case",
"The exponential expression for the S-matrix (REF ) is only correct for $N>2$ .",
"The principal chiral model with $SU(2)\\times SU(2)$ symmetry is equivalent to the $O(4)$ -symmetric nonlinear sigma model.",
"We will express the S matrix, first found in Ref.",
"[21], by an exponential expression [22].",
"A state with one excitation has a left-handed color index $a=1,2$ and a right-handed color index $b=1,2$ .",
"In the $O(4)$ formulation, excitations have a single species index $j=1,2,3,4$ .",
"The $SU(2)\\times SU(2)$ -symmetric states are related to the $O(4)$ -symmetric states by $\\vert P, \\theta , a, b\\rangle _{\\rm in}&=&\\sum _j \\frac{1}{\\sqrt{2}} \\left(\\delta ^{j 4}\\delta _{ab}-i\\sigma _{ab}^j\\right) \\vert \\theta , j\\rangle _{\\rm in},\\nonumber \\\\\\vert A, \\theta , a, b\\rangle _{\\rm in}&=&\\sum _j \\frac{1}{\\sqrt{2}} \\left(\\delta ^{j 4}\\delta _{ab}-i\\sigma _{ab}^j\\right)^* \\vert \\theta , j\\rangle _{\\rm in},\\nonumber $ where $\\sigma ^j$ with $j=1,2,3$ are the Pauli matrices.",
"The $O(4)$ two-excitation S-matrix, $S(\\theta )^{j_1 j_2}_{j^\\prime _1 j^\\prime _2} $ is given by $\\,_{\\rm out}\\langle \\theta ^\\prime _1, j^\\prime _1;\\theta ^\\prime _2, j^\\prime _2\\vert \\theta _1, j_1;\\theta _2,j_2\\rangle _{\\rm in}\\nonumber =S(\\theta )^{j_1 j_2}_{j^\\prime _1 j^\\prime _2}\\, 4\\pi \\delta (\\theta _1-\\theta ^\\prime _1)\\,4\\pi \\delta (\\theta _2-\\theta ^\\prime _2),\\nonumber $ where [22] $S(\\theta )^{j_1 j_2}_{j^\\prime _1 j^\\prime _2} = \\left[\\frac{\\theta +\\pi i}{\\theta -\\pi i}(P^0)^{j_1 j_2}_{j_1^\\prime j_2^\\prime }+\\frac{\\theta -\\pi i}{\\theta +\\pi i} (P^{+})^{j_1 j_2}_{j_1^\\prime j_2^\\prime }+(P^{-})^{j_1 j_2}_{j_1^\\prime j_2^\\prime }\\right] Q(\\theta ),\\nonumber $ $Q(\\theta )=\\exp 2\\int _0^\\infty \\frac{d\\xi }{\\xi }\\frac{e^{-\\xi }-1}{e^\\xi +1}\\sinh \\left(\\frac{\\xi \\theta }{\\pi {\\rm i}}\\right),\\nonumber $ and $P^0,\\,P^{+},$ and $P^{-}$ are the singlet, symmetric-traceless, and antisymmetric projectors, which are $(P^0)^{j_1 j_2}_{j_1^\\prime j_2^\\prime }=\\frac{1}{4}\\delta ^{j_1j_2}\\delta _{j_1^\\prime j_2^\\prime }&,&\\,\\,\\,(P^+)^{j_1 j_2}_{j_1^\\prime j_2^\\prime }=\\frac{1}{2}(\\delta ^{j_1}_{j_1^\\prime }\\delta ^{j_2}_{j_2^\\prime }+\\delta ^{j_1}_{j_2^\\prime }\\delta ^{j_2}_{j_1^\\prime })-\\frac{1}{4}\\delta ^{j_1j_2}\\delta _{j_1^\\prime j_2^\\prime },\\nonumber \\\\(P^-)^{j_1 j_2}_{j_1^\\prime j_2^\\prime }&=&\\frac{1}{2}(\\delta ^{j_1}_{j_1^\\prime }\\delta ^{j_2}_{j_2^\\prime }-\\delta ^{j_1}_{j_2^\\prime }\\delta ^{j_2}_{j_1^\\prime }),\\nonumber $ respectively.",
"We write the left-color-singlet wave function for a free particle and antiparticle: $\\Psi _{p_1,p_2}(x^1,y^1)_{b_1b_2}&=&D_{b_1b_2}^{j_1j_2}\\left\\lbrace \\begin{array}{c}e^{ip_1 x^1+ip_2y^1}A_{j_1 j_2},\\,\\,\\,\\,{\\rm for}\\,\\,x^1>y^1\\\\\\,\\\\e^{ip_2x^1+ip_1y^1}S(\\theta )^{j^\\prime _1 j^\\prime _2}_{j_1 j_2} A_{j^\\prime _1 j^\\prime _2},\\,\\,\\,\\,{\\rm for}\\,\\,x^1<y^1,\\end{array}\\right.$ where $D_{b_1b_2}^{j_1j_2}=\\frac{1}{2}\\delta ^{a_1a_2}\\left(\\delta ^{j_1 4}\\delta _{a_1b_1}-i\\sigma _{a_1b_1}^{j_1}\\right)^*\\left(\\delta ^{j_2 4}\\delta _{a_2b_2}-i\\sigma _{a_2b_2}^{j_2}\\right)\\,.\\nonumber $ There is a triplet of degenerate states and one singlet state.",
"The triplet satisfies $\\delta ^{b_1b_2}\\Psi _{p_1,p_2}(x^1,y^1)_{b_1b_2}=0.$ Substituting (REF ) into (REF ) gives the condition $\\delta ^{b_1b_2}\\,D_{b_1b_2}^{j_1j_2}\\,A_{j_1j_2}=\\delta ^{j_1 j_2}A_{j_1j_2}=0\\,.\\nonumber $ The traceless matrix $A_{j_1j_2}$ can be split into a symmetric and an antisymmetric part, $A^{+}_{j_1j_2}=(A_{j_1j_2}+A_{j_2j_1})/2$ and $A^{-}_{j_1j_2}=(A_{j_1j_2}-A_{j_2j_1})/2$ , respectively.",
"The matrix $A^{+}_{j_1j_2}$ , however, does not contribute to the wave function (REF ), because $D^{j_1j_2}_{b_1b_2}A^{+}_{j_1j_2}=\\frac{1}{2}\\delta _{b_1b_2}{\\rm Tr}\\,A^{+}=0.\\nonumber $ The matrix $A^{-}_{j_1 j_2}$ satisfies [21], [22]: $S(\\theta )^{j_{1} j_{2}}_{j_{1}^{\\prime } j_{2}^{\\prime }}A^{-}_{j_{1}j_{2}}=Q(\\theta )A^{-}_{j_{1}^{\\prime }j_{2}^{\\prime }}.$ Substituting (REF ) into (REF ), in center-of-mass coordinates and the non-relativistic limit, we find $\\Psi _{p}(x)_{b_1b_2}=D_{b_1b_2}^{j_1j_2}\\left\\lbrace \\begin{array}{c}\\cos (px+\\omega )A_{j_1j_2},\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0,\\\\\\,\\\\\\cos [-px+\\omega -\\phi (p)]A_{j_1j_2},\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0,\\end{array}\\right.$ where $\\phi (p)=-\\frac{i h_2}{\\pi m}\\vert p\\vert $ , where $h_2=2\\int _0^\\infty d\\xi \\,\\frac{e^{-\\xi }-1}{e^\\xi +1}=-4\\ln 2+2.$ The wave function of the right-color-singlet bound state is $\\Psi _{p_1,p_2}^{\\rm singlet}(x^1,y^1)=\\left\\lbrace \\begin{array}{c}e^{ip_1 x^1+ip_2y^1},\\,\\,\\,\\,{\\rm for}\\,\\,x^1>y^1,\\\\\\,\\\\e^{ip_2x^1+ip_1y^1}\\frac{\\theta +\\pi {\\rm i}}{\\theta -\\pi {\\rm i}}Q(\\theta ),\\,\\,\\,\\,{\\rm for}\\,\\,x^1<y^1.\\end{array}\\right.$ In center-of-mass coordinates, in the non-relativistic approximation, this becomes $\\Psi _{p}^{\\rm singlet}(x)=\\left\\lbrace \\begin{array}{c}\\cos (px+\\omega ),\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x>0,\\\\\\,\\\\\\cos [-px+\\omega -\\chi (p)],\\,\\,\\,\\,\\,\\,\\,{\\rm for}\\,\\,x<0,\\end{array}\\right.$ where $\\chi (p)=\\pi -\\frac{i h_2}{\\pi m}\\vert p\\vert $ .",
"From this point onward, the analysis is similar to what we've presented in the last two sections.",
"We obtain (REF ), (REF ), except that $h_{N}$ (defined in (REF )) is replaced with $h_{2}$ (defined in (REF )), with one important difference; we have $\\pm =+$ for the singlet and $\\pm =-$ for the triplet in Eq.",
"(REF ).",
"As mentioned at the end of the last section, the singlet spectrum coincides with that of Ref.",
"[11], in which $\\sigma $ must be replaced by $2\\sigma $ ."
],
[
"Conclusions and Outlook",
"We have found the spectrum of massive $(1+1)$ -dimensional SU($N$ ) Yang-Mills theory, for small gauge coupling.",
"To do this, we formulated the model as a principal chiral sigma model coupled to a massless Yang-Mills field.",
"In the axial gauge, there are sigma-model particles and antiparticles which bind to make left-color singlets.",
"We obtained the mesonic spectrum by determining the particle-antiparticle wave function in the non-relativistic limit, taking into account the phase shift at the origin.",
"In the future, we would like to find relativistic corrections to the mass spectrum.",
"This was done in Ref.",
"[16] for the Ising model in an external magnetic field.",
"The goal would be to find mesonic eigenstates of the Hamiltonian (REF ) of the form: $\\vert \\Psi _B\\rangle _{b_1b_2}=\\vert \\Psi _B^{(2)}\\rangle _{b_1b_2}+\\vert \\Psi _B^{(4)}\\rangle _{b_1b_2}+\\vert \\Psi _B^{(6)}\\rangle _{b_1b_2}+\\dots ,\\nonumber $ where the state $\\vert \\Psi _B^{(2M)}\\rangle _{b_1b_2}$ contains $M$ particles and $M$ antiparticles.",
"The multi-particle contributions are included because an electric string may break [17], producing pairs of sigma-model excitations.",
"Nonetheless, for small gauge coupling, the “two-quark\" approximation is valid.",
"In the this approximation, the bound state is treated as $\\vert \\Psi _B\\rangle _{b_1b_2}\\approx \\vert \\Psi _B^{(2)}\\rangle _{b_1b_2}&=&\\frac{1}{2}\\int \\frac{d\\theta _{1}}{4\\pi }\\frac{d\\theta _{2}}{4\\pi }\\Psi (p_1,p_2)_{a_2a_2}\\vert A,\\theta _1,b_1,a_1;P,\\theta _2,a_2,b_2\\rangle ,\\;{\\rm where},\\nonumber \\\\\\Psi (p_1,p_2)_{a_1a_2}&=&S(\\theta )\\left[\\delta _{a_1}^{c_1}\\delta _{a_2}^{c_2}-\\frac{2\\pi {\\rm i}}{N(\\pi {\\rm i}-\\theta )}\\delta _{a_1a_2}\\delta ^{c_1c_2}\\right]\\Psi (p_2,p_1)_{c_1c_2}.\\ $ The spectrum of masses $\\Delta $ , of the states (REF ) is found from the Bethe-Salpeter equation $(H-\\Delta )\\vert \\Psi ^{(2)}_B\\rangle _{b_1b_2}=0$ .",
"Acting on this state with the Hamiltonian (REF ) yields $\\left(m\\cosh \\theta _1+m\\cosh \\theta _2-\\Delta \\right)\\!\\!\\!\\!\\!&&\\!\\!\\!\\!\\!\\Psi (p_1^\\prime ,p_2^\\prime )_{c_1c_2}\\delta _{b_1d_1}\\delta _{b_2d_2}\\nonumber \\\\&=&\\frac{e^2}{4g_0^4}\\int \\frac{d\\theta _{1}}{4\\pi }\\frac{d\\theta _{2}}{4\\pi }\\Psi (p_1,p_2)_{a_1a_2}\\int dx^1 dy^1\\vert x^1-y^1\\vert \\nonumber \\\\&&\\times \\langle A, \\theta _1^\\prime , d_1,c_1;P,\\theta _2^\\prime ,c_2,d_2\\vert {\\rm Tr}\\,\\left[j_0^L(x^1)j_0^L(y^1)\\right]\\vert A,\\theta _1, b_1, a_1;P, \\theta _2,a_2,b_2\\rangle ,$ where the operator ${\\rm Tr}\\,\\left[j_0^L(x^1)j_0^L(y^1)\\right]$ is not time-ordered.",
"The matrix element $\\langle A, \\theta _1^\\prime , d_1,c_1;P,\\theta _2^\\prime ,c_2,d_2\\vert {\\rm Tr}\\,\\left[j_0^L(x^1)j_0^L(y^1)\\right] \\vert A,\\theta _1, b_1, a_1;P, \\theta _2,a_2,b_2\\rangle \\nonumber $ is obtained by inserting a complete set of states between the current operators and using the exact form factors of the currents of the principal chiral sigma model.",
"For finite $N$ , only the leading two-particle form factors of currents are known [23] and only a vacuum insertion can be made.",
"The complete matrix element is known at large $N$ [24], which should help in finding the relativistic corrections to the eigenvalues of Eq.",
"(REF ).",
"A.C.C.",
"would like to thank Davide Gaiotto and Jaume Gomis for interesting discussions, and the Perimeter Institute for their hospitality.",
"P.O.",
"'s work was supported by a grant from the PSC-CUNY."
]
] | 1403.0276 |
[
[
"Robust Nonlinear L2 Filtering of Uncertain Lipschitz Systems via Pareto\n Optimization"
],
[
"Abstract A new approach for robust Hinfty filtering for a class of Lipschitz nonlinear systems with time-varying uncertainties both in the linear and nonlinear parts of the system is proposed in an LMI framework.",
"The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization.",
"The resulting Hinfty filter guarantees asymptotic stability of the estimation error dynamics with exponential convergence and is robust against nonlinear additive uncertainty and time-varying parametric uncertainties.",
"Explicit bounds on the nonlinear uncertainty are derived based on norm-wise and element-wise robustness analysis."
],
[
"Introduction",
"The problem of observer design for nonlinear continuous-time uncertain systems has been tackled in various approaches.",
"Early studies in this area go back to the works of de Souza et.",
"al.",
"where they considered a class of continuous-time Lipschitz nonlinear systems with time-varying parametric uncertainties and obtained Riccati-based sufficient conditions for the stability of the proposed $H_{\\infty }$ observer with guaranteed disturbance attenuation level, when the Lipschitz constant is assumed to be known and fixed, [9], [19].",
"In an $H_{\\infty }$ observer, the $\\mathcal {L}_{2}$ -induced gain from the norm-bounded exogenous disturbance signals to the observer error is guaranteed to be below a prescribed level.",
"They also derived matrix inequalities helpful in solving this type of problems.",
"Since then, various methods have been reported in the literature to design robust observers for nonlinear systems [17], [16], [8], [23], [1], [3], [2], [4], [5], [18], [22], [14].",
"On the other hand, the restrictive regularity assumptions in the Riccati approach can be relaxed using linear matrix inequalities (LMIs).",
"An LMI solution for nonlinear $H_{\\infty }$ filtering is proposed for Lipschitz nonlinear systems with a given and fixed Lipschitz constant [22], [14].",
"The resulting observer is robust against time-varying parametric uncertainties with guaranteed disturbance attenuation level.",
"In a recent paper the authors considered the nonlinear observer design problem and presented a solution that has the following features [1]: (Stability) In the absence of external disturbances the observer error converges to zero exponentially with a guaranteed convergence rate.",
"Moreover, our design is such that it can maximize the size of the Lipschitz constant that can be tolerated in the system.",
"(Robustness) The design is robust with respect to uncertainties in the nonlinear plant model.",
"(Filtering) The effect of exogenous disturbances on the observer error can be minimized.",
"In this article we consider a similar problem but consider the important extension to the case where there exist parametric uncertainties in the state space model of the plant.",
"The extension is significant because uncertainty in the state space model of the plant is always encountered in a any actual application.",
"Ignoring this form of uncertainty requires lumping all model uncertainty on the nonlinear (Lipschitz) term, thus resulting in excessively conservative results.",
"This extension, is though obtained through a completely different solution from that given in [1].",
"The price of robustness against parametric uncertainties is an stability requirement of the plant model which makes the solution, different and yet a non-trivial extension to that of [1].",
"We will see this in detail in Section 3.",
"Our solution is based on the use of linear matrix inequalities and has the property that the Lipschitz constant is one the LMI variables.",
"This property allows us to obtain a solution in which the maximum admissible Lipschitz constant is maximized through convex optimization.",
"As we will see, this maximization adds an extra important feature to the observer, making it robust against nonlinear uncertainties.",
"The result is an $H_{\\infty }$ observer with a prespecified disturbance attenuation level which guarantees asymptotic stability of the estimation error dynamics with guaranteed speed of convergence and is robust against Lipschitz nonlinear uncertainties as well as time-varying parametric uncertainties, simultaneously.",
"Explicit bound on the nonlinear uncertainty are derived through a norm-wise analysis.",
"Some related results were recently presented by the authors in references [1] and [3] for continues-time and for discrete-time systems, respectively.",
"The rest of the paper is organized as follows.",
"In Section 2, the problem statement and some preliminaries are mentioned.",
"In Section 3, we propose a new method for robust $H_{\\infty }$ observer design for nonlinear uncertain systems.",
"Section 4, is devoted to robustness analysis in which explicit bounds on the tolerable nonlinear uncertainty are derived.",
"In Section 5, a combined observer performance is optimized using multiobjective optimization followed by a design example."
],
[
"Problem Statement",
"Consider the following class of continuous-time uncertain nonlinear systems: $\\left(\\sum \\right): \\dot{x}(t)&=(A+\\Delta A(t))x(t)+\\Phi (x,u)+Bw(t)\\\\ y(t)&=(C+\\Delta C(t))x(t)+Dw(t)$ where $x\\in {\\mathbb {R}} ^{n} ,u\\in {\\mathbb {R}} ^{m} ,y\\in {\\mathbb {R}} ^{p} $ and $\\Phi (x,u)$ contains nonlinearities of second order or higher.",
"We assume that the system (REF )-() is locally Lipschitz with respect to $x$ in a region $\\mathcal {D}$ containing the origin, uniformly in $u$ , i.e.",
": $&\\Phi (0,u^{*})=0\\\\&\\Vert \\Phi (x_{1},u^{*})-\\Phi (x_{2},u^{*})\\Vert \\leqslant \\gamma \\Vert x_{1}-x_{2}\\Vert \\hspace{5.69054pt}\\forall \\, x_{1},x_{2}\\in \\mathcal {D}$ where $\\Vert .\\Vert $ is the induced 2-norm, $u^{*}$ is any admissible control signal and $\\gamma >0$ is called the Lipschitz constant.",
"If the nonlinear function $\\Phi $ satisfies the Lipschitz continuity condition globally in $\\mathbb {R}^{n}$ , then the results will be valid globally.",
"$w(t)\\in \\mathfrak {L}_{2}[0,\\infty )$ is an unknown exogenous disturbance, and $\\Delta A(t)$ and $\\Delta C(t)$ are unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the form $\\Delta A(t)= M_{1}F(t)N_{1} \\\\\\Delta C(t)=M_{2}F(t)N_{2}$ where $M_{1}$ , $M_{2}$ , $N_{1}$ are $N_{2}$ are known real constant matrices and $F(t)$ is an unknown real-valued time-varying matrix satisfying $F^{T}(t)F(t)\\le I \\hspace{28.45274pt} \\forall t\\in [0,\\infty ).$ The parameter uncertainty in the linear terms can be regarded as the variation of the operating point of the nonlinear system.",
"It is also worth noting that the structure of parameter uncertainties in (REF )-() has been widely used in the problems of robust control and robust filtering for both continuous-time and discrete-time systems and can capture the uncertainty in a number of practical situations [13], [9], [21]."
],
[
"Disturbance Attenuation Level",
"Considering observer of the following form $\\dot{\\hat{x}}(t)&=& A\\hat{x}(t)+\\Phi (\\hat{x},u)+L(y-C\\hat{x})$ the observer error dynamics is given by $e(t)\\triangleq & \\ x(t)-\\hat{x}(t)\\\\\\begin{split}\\dot{e}(t)=& \\ (A-LC)e+\\Phi (x,u)-\\Phi (\\hat{x},u)\\\\&+(B-LD)w+(\\Delta A-L \\Delta C)x.\\end{split}$ Suppose that $z(t)=He(t)$ stands for the controlled output for error state where $H$ is a known matrix.",
"Our purpose is to design the observer parameter $L$ such that the observer error dynamics is asymptotically stable with maximum admissible Lipschitz constant and the following specified $H_{\\infty }$ norm upper bound is simultaneously guaranteed.",
"$\\Vert z\\Vert \\le \\mu \\Vert w\\Vert .$ Furthermore we want the observer to a have a guaranteed decay rate."
],
[
"Guaranteed Decay Rate",
"Consider the nominal system $\\left(\\sum \\right)$ with $\\Delta A,\\Delta C=0$ and $w(t)=0$ .",
"Then, the decay rate of the system (REF ) is defined to be the largest $\\beta >0$ such that $\\lim _{t\\rightarrow \\infty } \\exp (\\beta t)\\Vert e(t)\\Vert =0$ holds for all trajectories $e$ .",
"We can use the quadratic Lyapunov function $V (e)=e^{T}Pe$ to establish a lower bound on the decay rate of the (REF ).",
"If $\\frac{dV(e(t))}{dt}\\leqslant -2\\beta V (e(t))$ for all trajectories, then $V(e(t)) \\leqslant \\exp (-2\\beta t)V(e(0))$ , so that $\\Vert e(t)\\Vert \\leqslant \\exp (-\\beta t)\\kappa (P)^{\\frac{1}{2}}\\Vert e(0)\\Vert $ for all trajectories, where $\\kappa (P)$ is the condition number of P and therefore the decay rate of the (REF ) is at least $\\beta $ , [6].",
"In fact, decay rate is a measure of observer speed of convergence."
],
[
"$H_{\\infty }$ Observer Synthesis",
"In this section, an $H_{\\infty }$ observer with guaranteed decay rate $\\beta $ and disturbance attenuation level $\\mu $ is proposed.",
"The admissible Lipschitz constant is maximized through LMI optimization.",
"Theorem 1, introduces a design method for such an observer but first we mention a lemma used in the proof of our result.",
"It worths mentioning that unlike the Riccati approach of [9], in the LMI approach no regularity assumption is needed.",
"Lemma 1.",
"[19] Let $\\mathcal {D}$ , $\\mathcal {S}$ and $F$ be real matrices of appropriate dimensions and $F$ satisfying $F^{T}F\\le I$ .",
"Then for any scalar $\\epsilon >0$ and vectors $x,y\\in \\mathbb {R}^{n}$ , we have $2x^{T} \\mathcal {D}F\\mathcal {S}y \\le \\epsilon ^{-1}x^{T}\\mathcal {D}\\mathcal {D}^{T}x+\\epsilon y^{T}\\mathcal {S}^{T}\\mathcal {S}y$ Note.",
"As an standard notation in LMI context, the symbol “$\\star $ ” represents the element which makes the corresponding matrix symmetric.",
"Theorem 1.",
"Consider the Lipschitz nonlinear system $\\left(\\sum \\right)$ along with the observer (REF ).",
"The observer error dynamics is (globally) asymptotically stable with maximum admissible Lipschitz constant, $\\gamma ^{*}$ , decay rate $\\beta $ and $\\mathfrak {L}_{2}(w \\rightarrow z)$ gain, $\\mu $ , if there exists a fixed scalar $\\beta >0$ , scalars $\\gamma >0$ and $\\mu >0$ , and matrices $P_{1}>0$ , $P_{2}>0$ and $G$ , such that the following LMI optimization problem has a solution.",
"$\\hspace{-113.81102pt} \\max (\\gamma ) $ s.t.",
"$&\\left[\\begin{array}{ccc}\\Psi _{1} & 0 & \\Omega _{1} \\\\\\star & \\Psi _{2} & \\Omega _{2} \\\\\\star & \\star & -\\mu ^{2} I \\\\\\end{array}\\right]<0 $ where $Q&=&-\\left(A^{T}P_{1}+P_{1}A+2\\beta P_{1}-C^{T}G^{T}-GC\\right)\\\\R&=&A^{T}P_{2}+P_{2}A+2N^{T}_{1}N_{1}+N^{T}_{2}N_{2}\\\\S&=&(I+M_{1}M_{1}^{T})^\\frac{1}{2}\\\\\\Psi _{1}&=& \\left[\\begin{array}{cccc}H^{T}H-Q & \\gamma I & P_{1}S & GM_{2} \\\\\\star & -I & 0 & 0 \\\\\\star & \\star & -I & 0 \\\\\\star & \\star & \\star & -I \\\\\\end{array}\\right] \\\\\\Psi _{2}&=& \\left[\\begin{array}{ccc}R & \\gamma I & P_{2}S \\\\\\star & -I & 0 \\\\\\star & \\star & -I\\end{array}\\right] \\\\\\Omega _{1}&=&\\left[\\begin{array}{cccc}P_{1}B-GD & 0 & 0 & 0 \\\\\\end{array}\\right]^{T}\\\\\\Omega _{2}&=&\\left[\\begin{array}{ccc}P_{2}B & 0 & 0 \\\\\\end{array}\\right]^{T}$ Once the problem is solved $L&=&P_{1}^{-1}G \\\\\\gamma ^{*} &\\triangleq & \\max (\\gamma )$ Proof: From (REF ), the observer error dynamics is $\\begin{split}\\dot{e}=& \\ (A-LC)e+\\Phi (x,u)-\\Phi (\\hat{x},u)+(B-LD)w\\\\&+(\\Delta A-L\\Delta C)x.\\end{split}$ Let for simplicity $\\Phi (x,u)\\triangleq \\Phi ,\\ \\ \\Phi (\\hat{x},u)\\triangleq \\hat{\\Phi }.$ Consider the Lyapunov function candidate $V=V_{1}+V_{2}$ where $V_{1}=e^{T}P_{1}e,\\ \\ V_{2}=x^{T}P_{2}x$ .",
"For the nominal system, we have then $\\begin{split}\\dot{V}_{1}(t)&=\\dot{e}^{T}(t)P_{1}e(t)+e^{T}(t)P_{1}\\dot{e}(t)\\\\&=-e^{T}Qe+2e^{T}P_{1}(\\Phi (x,u)-\\Phi (\\hat{x},u))^{T}.\\end{split}$ To have $\\dot{V}_{1}(t)\\leqslant -2\\beta V_{1}(t)$ it suffices (REF ) to be less than zero, where: $(A-LC)^{T}P_{1}+P_{1}^{T}(A-LC)+2\\beta P_{1}=-Q .$ The above can be written as $\\\\A^{T}P_{1}+P_{1}A-C^{T}L^{T}P_{1}-P_{1}LC+2\\beta P_{1}=-Q.$ Defining the new variable $\\\\G\\triangleq P_{1}L\\Rightarrow L^{T}P_{1}^{T}=L^{T}P_{1}=G^{T},$ it becomes $\\\\A^{T}P_{1}+P_{1}A-C^{T}G^{T}-GC+2\\beta P_{1}=-Q.$ Now, consider the systems $\\left(\\sum \\right)$ with uncertainties and disturbance.",
"The derivative of $V$ along the trajectories of $\\left(\\sum \\right)$ is $\\begin{split}\\dot{V}_{1}&=\\dot{e}^{T}P_{1}e+e^{T}P_{1}\\dot{e}\\\\&=-e^{T}Qe+2e^{T}P_{1}(\\Phi -\\hat{\\Phi })+2e^{T}P_{1}(B-LD)w\\\\& \\ \\ \\ +2e^{T}P_{1}M_{1}FN_{1}x-2e^{T}GM_{2}FN_{2}x.\\end{split}$ Using Lemma 1, it can be written $&2e^{T}P_{1}M_{1}FN_{1}x\\le e^{T}P_{1}M_{1}M^{T}_{1}P_{1}e+x^{T}N^{T}_{1}N_{1}x \\\\&2e^{T}GM_{2}FN_{2}x\\le e^{T}GM_{2}M^{T}_{2}G^{T}e+x^{T}N^{T}_{2}N_{2}x \\\\&2x^{T}P_{2}M_{1}FN_{1}x\\le x^{T}P_{2}M_{1}M^{T}_{1}P_{2}x+x^{T}N^{T}_{1}N_{1}x \\\\&2e^{T}P_{1}(\\Phi -\\hat{\\Phi })\\le e^{T}P^{2}_{1}e+(\\Phi -\\hat{\\Phi })^{T}(\\Phi -\\hat{\\Phi })\\\\&\\hspace{60.3197pt}\\le e^{T}P^{2}_{1}e+\\gamma ^{2}e^{T}e \\\\&2x^{T}P_{2}\\Phi \\le x^{T}P^{2}_{2}x+\\Phi ^{T}\\Phi \\le x^{T}P^{2}_{2}x+{\\gamma ^{2}x^{T}x}$ substituting from (REF ), () and () $\\begin{split}\\dot{V}_{1}&\\le -e^{T}Q e+e^{T}P^{2}_{1}e+\\gamma ^{2}e^{T}e+e^{T}P_{1}M_{1}M^{T}_{1}P_{1}e\\\\& \\ \\ \\ +x^{T}(N^{T}_{1}N_{1}+N^{T}_{2}N_{2})x+e^{T}GM_{2}M^{T}_{2}G^{T}e\\\\ & \\ \\ \\ +2e^{T}P_{1}(B-LD)w.\\end{split}$ $\\begin{split}\\dot{V}_{2}&=x^{T}(A^{T}P_{2}+P_{2}A)x\\\\ & \\ \\ \\ +2x^{T}P_{2}\\Phi +2x^{T}P_{2}M_{1}FN_{1}x+2x^{T}P_{2}B w\\end{split}$ substituting from (), () $\\begin{split}\\dot{V}_{2}&\\le x^{T}(A^{T}P_{2}+P_{2}A)x+x^{T}P^{2}_{2}x+\\gamma ^{2}x^{T}x\\\\& \\ +x^{T}P_{2}M_{1}M^{T}_{1}P_{2}x+x^{T}N^{T}_{1}N_{1}x+2x^{T}P_{2}B w.\\end{split}$ Thus, $\\begin{split}\\dot{V}&\\le e^{T}\\left[-Q+P_{1}(I+M_{1}M^{T}_{1})P_{1}+GM_{2}M^{T}_{2}G^{T}+\\gamma ^{2} I\\right]e \\\\& \\ \\ +x^{T}\\left[A^{T}P_{2}+P_{2}A+P_{2}(I+M_{1}M^{T}_{1})P_{2}+\\gamma ^{2}I\\right]x\\\\& \\ \\ +x^{T}(2N^{T}_{1}N_{1}+N^{T}_{2}N_{2})x+2e^{T}P_{1}(B-LD)w\\\\& \\ \\ +2x^{T}P_{2}B w.\\end{split}$ So, when $w=0$ , a sufficient condition for the stability with guaranteed decay rate $\\beta $ is that $&-Q+P_{1}SS^{T}P_{1}+GM_{2}M^{T}_{2}G^{T}+\\gamma ^{2}I<0 \\\\&R+P_{2}SS^{T}P_{2}+\\gamma ^{2} I<0 $ $R$ and $S$ are as in (REF ) and ().",
"Note that $I+M_{1}M^{T}_{1}$ is positive definite and so has always a square root.",
"Now, we define $J\\triangleq \\int ^{\\infty }_{0}(z^{T}z-\\zeta w^{T}w) dt$ where $\\zeta =\\mu ^{2}$ .",
"Therefore $J<\\int ^{\\infty }_{0}(z^{T}z-\\zeta w^{T}w+\\dot{V}) dt$ so a sufficient condition for $J\\le 0$ is that $\\forall t\\in [0,\\infty ),\\hspace{14.22636pt} z^{T}z-\\zeta w^{T}w+\\dot{V}\\le 0.$ We have $z^{T}z-\\zeta w^{T}w+\\dot{V}\\le & e^{T}(H^{T}H-Q+P_{1}SS^{T}P_{1}+GM_{2}M^{T}_{2}G^{T}+\\gamma ^{2} I)e\\\\&+x^{T}(R+P_{2}SS^{T}P_{2}+\\gamma ^{2}I)x+2e^{T}P_{1}(B-LD)w+2x^{T}P_{2}B w-\\zeta w^{T}w $ So a sufficient condition for $J\\le 0$ is that the right hand side of the above inequity be less than zero which by means of Schur complements is equivalent to (REF ).",
"Note that (REF ) and () are already included in (REF ).",
"Then, $z^{T}z-\\zeta w^{T}w\\le 0\\rightarrow \\Vert z\\Vert \\le \\sqrt{\\zeta }\\Vert w\\Vert .",
"\\ \\ \\blacksquare $ Remark 1.",
"The proposed LMIs are linear in both $\\gamma $ and $\\zeta (=\\mu ^{2})$ .",
"Thus, either can be a fixed constant or an optimization variable.",
"If one wants to design an observer for a given system with known Lipschitz constant, then the LMI optimization problem can be reduced to an LMI feasibility problem (just satisfying the constraints) which is easier Remark 2.",
"This observer is robust against two type of uncertainties.",
"Lipschitz nonlinear uncertainty in $\\Phi (x,u)$ and time-varying parametric uncertainty in the pair $(A,C)$ while the disturbance attenuation level is guaranteed, simultaneously."
],
[
"Robustness Against Nonlinear Uncertainty",
"As mentioned earlier, the maximization of Lipschitz constant makes the proposed observer robust against some Lipschitz nonlinear uncertainty.",
"In this section this robustness feature is studied and both norm-wise and element-wise bounds on the nonlinear uncertainty are derived.",
"The norm-wise analysis provides an upper bound on the Lipschitz constant of the nonlinear uncertainty and the norm of the Jacobian matrix of the corresponding nonlinear function.",
"Furthermore, we will find upper and lower bounds on the elements of the Jacobian matrix through and element-wise analysis."
],
[
"Norm-Wise Analysis",
"Assume a nonlinear uncertainty as follows $\\Phi _{\\Delta }(x,u)&=&\\Phi (x,u)+\\Delta \\Phi (x,u)\\\\\\dot{x}(t)&=& (A+ \\Delta A)x(t) + \\Phi _{\\Delta }(x,u)$ where $\\Vert \\Delta \\Phi (x_{1},u)-\\Delta \\Phi (x_{2},u)\\Vert \\leqslant \\Delta \\gamma \\Vert x_{1}-x_{2}\\Vert .\\\\$ Proposition 1.",
"Suppose that the actual Lipschitz constant of the system is $\\gamma $ and the maximum admissible Lipschitz constant achieved by Theorem 1, is $\\gamma ^{*}$ .",
"Then, the observer designed based on Theorem 1, can tolerate any additive Lipschitz nonlinear uncertainty with Lipschitz constant less than or equal $\\gamma ^{*}-\\gamma $.",
"Proof: Based on Schwartz inequality, we have $\\begin{split}\\Vert \\Phi _{\\Delta }(x_{1},u)&-\\Phi _{\\Delta }(x_{2},u)\\Vert \\le \\Vert \\Phi (x_{1},u)-\\Phi (x_{2},u)\\Vert \\\\&\\ \\ \\ +\\Vert \\Delta \\Phi (x_{1},u)-\\Delta \\Phi (x_{2},u)\\Vert \\\\&\\le \\gamma \\Vert x_{1}-x_{2}\\Vert +\\Delta \\gamma \\Vert x_{1}-x_{2}\\Vert .\\end{split}$ According to the Theorem 1, $\\Phi _{\\Delta }(x,u)$ can be any Lipschitz nonlinear function with Lipschitz constant less than or equal to $\\gamma ^{*}$ , $\\Vert \\Phi _{\\Delta }(x_{1},u)-\\Phi _{\\Delta }(x_{2},u)\\Vert \\le \\gamma ^{*}\\Vert x_{1}-x_{2}\\Vert $ so, there must be $\\gamma +\\Delta \\gamma \\le \\gamma ^{*}\\rightarrow \\Delta \\gamma \\le \\gamma ^{*}-\\gamma .\\ \\ \\ \\blacksquare $ In addition, we know that for any continuously differentiable function $\\Delta \\Phi $ , $\\Vert \\Delta \\Phi (x_{1},u)-\\Delta \\Phi (x_{2},u)\\Vert \\leqslant \\Vert \\frac{\\partial \\Delta \\Phi }{\\partial x}(x_{1}-x_{2})\\Vert $ where $\\frac{\\partial \\Delta \\Phi }{\\partial x}$ is the Jacobian matrix [15].",
"So $\\Delta \\Phi (x,u)$ can be any additive uncertainty with $\\Vert \\frac{\\partial \\Delta \\Phi }{\\partial x}\\Vert \\le \\gamma ^{*}-\\gamma $ ."
],
[
"Element-Wise Analysis",
"Assume that there exists a matrix $\\Gamma \\in \\mathbb {R}^{n\\times n}$ such that $\\Vert \\Phi (x_{1},u)-\\Phi (x_{2},u)\\Vert \\leqslant \\Vert \\Gamma (x_{1}-x_{2})\\Vert .$ $\\Gamma $ can be considered as a matrix-type Lipschitz constant.",
"Suppose that the nonlinear uncertainty is as in (REF ) and $\\Vert \\Phi _{\\Delta }(x_{1},u)-\\Phi _{\\Delta }(x_{2},u)\\Vert \\leqslant \\Vert \\Gamma _{\\Delta }(x_{1}-x_{2})\\Vert .$ Assuming $\\Vert \\Delta \\Phi (x_{1},u)-\\Delta \\Phi (x_{2},u)\\Vert \\leqslant \\Vert \\Delta \\Gamma (x_{1}-x_{2})\\Vert ,$ based the proposition 1, $\\Delta \\Gamma $ can be any matrix with $\\Vert \\Delta \\Gamma \\Vert \\le \\gamma ^{*}-\\Vert \\Gamma \\Vert $ .",
"In the following, we will look at the problem from a different angle.",
"It is clear that $\\Gamma _{\\Delta }=[{\\gamma _{\\Delta }}_{i,j}]_{n}$ is a perturbed version of $\\Gamma $ due to $\\Delta \\Phi (x,u)$ .",
"The question is that how much perturbation can be tolerated on the element of $\\Gamma $ without loosing the observer features stated in Theorem 1.",
"This is important in the sense that in gives us an insight about the amount of uncertainty that can be tolerated in different directions of the nonlinear function.",
"Here, we propose a novel approach to optimize the elements $\\Gamma $ and provide specific upper and lower bounds on tolerable perturbations.",
"Before stating the result of this section, we need to recall some matrix notations.",
"For matrices $A=[a_{i,j}]_{m\\times n}$ , $B=[b_{i,j}]_{m\\times n}$ , $A\\preceq B$ means $a_{i,j}\\le b_{i,j} \\ \\forall \\ 1\\le i \\le m,1\\le j\\le n$ .",
"For square A, $diag (A)$ is a vector containing the elements on the main diagonal and $diag(x)$ where $x$ is a vector is a diagonal matrix with the elements of $x$ on the main diagonal.",
"$|A|$ is the element-wise absolute value of $A$ , i.e.",
"$[|a_{i,j}|]_{n}$ .",
"$A\\circ B$ stands for the element-wise product (Hadamard product) of $A$ and $B$ .",
"Corollary 1.",
"Consider Lipschitz nonlinear system $\\left(\\sum \\right)$ satisfying (REF ), along with the observer (REF ).",
"The observer error dynamics is (globally) asymptotically stable with the matrix-type Lipschitz constant $\\Gamma ^{*}=[\\gamma ^{*}_{i,j}]_{n}$ with maximized admissible elements, decay rate $\\beta $ and $\\mathfrak {L}_{2}(w\\rightarrow z)$ gain, $\\mu $ , if there exist fixed scalars $\\beta >0$ and $c_{i,j}>0 \\ \\forall \\ 1\\le i,j\\le n$ , scalars $\\omega >0$ and $\\mu >0$ , and matrices $\\Gamma =[\\gamma _{i,j}]_{n}\\succ 0$ , $P_{1}>0$ , $P_{2}>0$ and $G$ , such that the following LMI optimization problem has a solution.",
"$\\hspace{-113.81102pt} \\max \\ \\omega $ s.t.",
"$&c_{i,j}\\gamma _{i,j}>\\omega \\ \\ \\ \\ \\ \\ \\forall \\ 1\\le i,j\\le n\\\\&\\left[\\begin{array}{ccc}\\Psi _{1} & 0 & \\Omega _{1} \\\\\\star & \\Psi _{2} & \\Omega _{2} \\\\\\star & \\star & -\\mu ^{2} I \\\\\\end{array}\\right]<0$ where $\\Psi _{1}$ , $\\Psi _{2}$ , $\\Omega _{1}$ and $\\Omega _{2}$ are as in Theorem 1 replacing $\\gamma I$ by $\\Gamma $ .",
"Once the problem is solved $L&=&P_{1}^{-1}G\\\\\\gamma ^{*}_{i,j} &\\triangleq & \\max (\\gamma _{i,j})$ Proof: The proof is similar to the proof of Theorem 1 with replacing $\\gamma I$ by $\\Gamma $ .",
"$\\blacksquare $ Remark 3.",
"By appropriate selection of the weights $c_{i,j}$ , it is possible to put more emphasis on the directions in which the tolerance against nonlinear uncertainty is more important.",
"To this goal, one can take advantage of the knowledge about the structure of the nonlinear function $\\Phi (x,u)$ .",
"According to the norm-wise analysis, it is clear that $\\Delta \\Gamma $ in (REF ) can be any matrix with $\\Vert \\Delta \\Gamma \\Vert \\le \\Vert \\Gamma ^{*}\\Vert -\\Vert \\Gamma \\Vert $ .",
"We will now proceed by deriving bounds on the elements of $\\Gamma _{\\Delta }$ .",
"Lemma 2.",
"For any $T=[t_{i,j}]_{n}$ and $U=[u_{i,j}]_{n}$ , if $|T|\\preceq U$ , then $TT^{T}\\le UU^{T}\\circ nI$.",
"Proof: Assume any $x=[x_{i}]_{n\\times 1}$ , then, it is easy to show that $T^{T}x=[(\\sum _{i=1}^{n}t_{i,j}x_{i})_{j}]_{n\\times 1}$ .",
"Therefore, $\\begin{split}x^{T}TT^{T}x&=\\langle T^{T}x,T^{T}x\\rangle =\\sum _{j=1}^{n}(\\sum _{i=1}^{n}t_{i,j}x_{i})^{2}\\\\&\\le \\sum _{j=1}^{n}\\sum _{i=1}^{n}t_{i,j}^{2}x_{i}^{2}+\\sum _{j=1}^{n}\\sum _{i=1}^{n-1}\\sum _{k=i+1}^{n}(t_{i,j}^{2}x_{i}^{2}+t_{k,j}^{2}x_{k}^{2})\\\\&\\le \\sum _{j=1}^{n}\\sum _{i=1}^{n}u_{i,j}^{2}x_{i}^{2}+\\sum _{j=1}^{n}\\sum _{i=1}^{n-1}\\sum _{k=i+1}^{n}(u_{i,j}^{2}x_{i}^{2}+u_{k,j}^{2}x_{k}^{2})\\\\&=\\sum _{j=1}^{n}[\\sum _{i=1}^{n}u_{i,j}^{2}x_{i}^{2}+\\sum _{i=1}^{n-1}\\sum _{k=i+1}^{n}(u_{i,j}^{2}x_{i}^{2}+u_{k,j}^{2}x_{k}^{2})]\\\\&=n\\sum _{i=1}^{n}\\sum _{j=1}^{n}u_{i,j}^{2}x_{i}^{2}=n \\ x^{T}diag(diag(UU^{T}))x\\\\&\\Rightarrow \\ TT^{T}\\le n \\ diag(diag(UU^{T}))= UU^{T}\\circ nI.\\ \\blacksquare \\end{split}$ Now we are ready to state the element-wise robustness result.",
"Assume additive uncertainty in the form of (REF ), where $\\Vert \\Phi _{\\Delta }(x_{1},u)-\\Phi _{\\Delta }(x_{2},u)\\Vert \\leqslant \\Vert \\Gamma _{\\Delta }(x_{1}-x_{2})\\Vert .$ It is clear that $\\Gamma _{\\Delta }=[{\\gamma _{\\Delta }}_{i,j}]_{n}$ is a perturbed version of $\\Gamma $ .",
"Proposition 2.",
"Suppose that the actual matrix-type Lipschitz constant of the system is $\\Gamma $ and the maximized admissible matrix-type Lipschitz constant achieved by Corollary 1, is $\\Gamma ^{*}$ .",
"Then, $\\Delta \\Phi $ can be any additive nonlinear uncertainty such that $|\\Gamma _{\\Delta }|\\preceq n^{-\\frac{3}{4}} \\Gamma ^{*}$ .",
"Proof: According to the Proposition 1, it suffices to show that $\\sigma _{max}(\\Gamma _{\\Delta })\\le \\sigma _{max}(\\Gamma ^{*})$ .",
"Using Lemma 2, we have $\\begin{split}\\sigma _{max}^{2}(\\Gamma _{\\Delta })&=\\lambda _{max}(\\Gamma _{\\Delta }\\Gamma _{\\Delta }^{T})\\\\&\\le \\lambda _{max}(n\\ diag(diag(n^{-\\frac{3}{2}}\\Gamma ^{*}{\\Gamma ^{*}}^{T})))\\\\&\\le \\sigma _{max}(n\\ diag(diag(n^{-\\frac{3}{2}} \\Gamma ^{*}{\\Gamma ^{*}}^{T})))\\\\&=\\max _{i}(n^{-\\frac{1}{2}}\\sum _{j=1}^{n}{\\gamma ^{*}_{i,j}}^{2})=\\frac{1}{\\sqrt{n}}\\Vert \\Gamma ^{*}\\circ \\Gamma ^{*}\\Vert _{\\infty }\\\\&\\le |\\Gamma ^{*}\\circ \\Gamma ^{*}\\Vert _{2}\\le |\\Gamma ^{*}\\Vert ^{2}_{2}=\\sigma _{max}^{2}(\\Gamma ^{*}).\\end{split}$ The first inequality follows from Lemma 2 and the symmetry of $\\Gamma _{\\Delta }\\Gamma _{\\Delta }^{T}$ and diag(diag($\\Gamma ^{*}{\\Gamma ^{*}}^{T}))$ , [10].",
"The last two inequalities are due to the relation between the induced infinity and 2 norms [10] and the fact that the spectral norm is submultiplicative with respect to the Hadamard product [11], respectively.",
"Since the singular values are nonnegative, we can conclude that $\\sigma _{max}(\\Gamma _{\\Delta })\\le \\sigma _{max}(\\Gamma ^{*})$ .",
"$\\blacksquare $ Therefore, denoting the elements of $\\Gamma _{\\Delta }$ as ${\\gamma _{\\Delta }}_{i.j}=\\gamma _{i,j}+\\delta _{i,j}$ , the following bound on the element-wise perturbations is obtained $-n^{-\\frac{3}{4}}\\gamma ^{*}_{i,j}-\\gamma _{i,j} \\le \\delta _{i,j}\\le n^{-\\frac{3}{4}}\\gamma ^{*}_{i,j}-\\gamma _{i,j}.$ In addition, $\\Delta \\Phi (x,u)$ can be any continuously differentiable additive uncertainty which makes $|\\frac{\\partial \\Phi _{\\Delta }}{\\partial x}|\\preceq n^{-\\frac{3}{4}}\\Gamma ^{*}$ .",
"It is worth mentioning that the results of Lemma 2 and Proposition 2 have intrinsic importance from the matrix analysis point of view regardless of our specific application in the robustness analysis."
],
[
"Combined Performance using Multiobjective Optimization",
"The LMIs proposed in Theorem 1 are linear in both admissible Lipschitz constant and disturbance attenuation level.",
"So, as mentioned earlier, each can be optimized.",
"A more realistic problem is to choose the observer gain matrix by combining these two performance measures.",
"This leads to a Pareto multiobjective optimization in which the optimal point is a trade-off between two or more linearly combined optimality criterions.",
"Having a fixed decay rate, the optimization is over $\\gamma $ (maximization) and $\\mu $ (minimization), simultaneously.",
"The following theorem is in fact a generalization of the results of [22] and [20] (for the systems in class of $\\sum $ ) in which the Lipschitz constant is known and fixed, in one point of view; and the results of [12] in which a special class of sector nonlinearities is considered and there is no uncertainty in pair (A,C), in another.",
"Theorem 2.",
"Consider Lipschitz nonlinear system $\\left(\\sum \\right)$ along with the observer (REF ).",
"The observer error dynamics is (globally) asymptotically stable with decay rate $\\beta $ and simultaneously maximized admissible Lipschitz constant, $\\gamma ^{*}$ and minimized $\\mathfrak {L}_{2}(w \\rightarrow z)$ gain, $\\mu ^{*}$ , if there exists fixed scalars $\\beta >0$ and $0\\le \\lambda \\le 1$ , scalars $\\gamma >0$ and $\\zeta >0$ , and matrices $P_{1}>0$ , $P_{2}>0$ and $G$ , such that the following LMI optimization problem has a solution.",
"$\\hspace{-113.81102pt} \\min \\left[\\lambda (-\\gamma )+(1-\\lambda )\\zeta \\right]$ s.t.",
"$&\\left[\\begin{array}{ccc}\\Psi _{1} & 0 & \\Omega _{1} \\\\\\star & \\Psi _{2} & \\Omega _{2} \\\\\\star & \\star & -\\zeta I \\\\\\end{array}\\right]<0$ where $\\Psi _{1}$ , $\\Psi _{2}$ , $\\Omega _{1}$ and $\\Omega _{2}$ are as in Theorem 1.",
"Once the problem is solved $L&=&P_{1}^{-1}G\\\\\\gamma ^{*} &\\triangleq & \\max (\\gamma )=\\min (-\\gamma )\\\\\\mu ^{*} &\\triangleq & \\min (\\mu )=\\sqrt{\\zeta }$ Proof: The above is a scalarization of a multiobjective optimization with two optimality criterions.",
"Since each of these optimization problems is convex, the scalarized problem is also convex [7].",
"The rest of the proof is the same as the proof of Theorem 1.",
"$\\blacksquare $ Remark 4.",
"The matrix-type Lipschitz constant $\\Gamma $ may also be considered in place of $\\gamma $ in Theorem 2.",
"Since the observer gain directly amplifies the measurement noise, sometimes, it is better to have an observer gain with smaller elements.",
"There might also be practical difficulties in implementing high gains.",
"We can control the Frobenius norm of $L$ either by changing the feasibility radius of the LMI solver or by decreasing $\\lambda _{min}^{-1}(P_{1})$ which is $\\lambda _{max}(P_{1}^{-1})$ , to decrease $\\bar{\\sigma }(L)$ as in (REF ).",
"The latter can be done by replacing $P_{1}>0$ with $P_{1}>\\theta I$ in which $\\theta >0$ can be either a fixed scalar or an LMI variable.",
"Considering $\\bar{\\sigma }(L)$ as another performance index, note that it is even possible to have a triply combined cost function in the LMI optimization problem of Theorem 2.",
"Now, we show the usefulness of this Theorem through a design example.",
"Example: Consider a system of the form of $\\left(\\sum \\right)$ where $A&=&\\left[\\begin{array}{cc}0 & 1 \\\\-1 & -1 \\\\\\end{array}\\right], \\ \\ \\Phi (x)=\\left[\\begin{array}{c}0 \\\\0.2sin(x_{1}) \\\\\\end{array}\\right]\\\\M_{1}&=&\\left[\\begin{array}{cc}0.1 & 0.05 \\\\-2 & 0.1 \\\\\\end{array}\\right], \\ \\ M_{2}=\\left[\\begin{array}{cc}-0.2 & 0.8 \\\\\\end{array}\\right]\\\\C&=&\\left[\\begin{array}{cc}1 & 0 \\\\\\end{array}\\right], \\ \\ N_{1}=N_{2}=\\left[\\begin{array}{cc}0.1 & 0 \\\\0 & 0.1 \\\\\\end{array}\\right].$ Assuming $\\beta &=&0.35, \\lambda =0.95\\\\B&=&\\left[\\begin{array}{cc}1 & 1 \\\\\\end{array}\\right]^{T}\\\\D&=&0.2\\\\H&=&0.5 I_{2}$ we get $\\gamma ^{*}&=&0.3016, \\ \\mu ^{*}=3.5\\\\L&=&\\left[\\begin{array}{cc}5.0498 & 4.9486 \\\\\\end{array}\\right]^{T}$ Figure REF , shows the true and estimated values of states.",
"Figure: The true and estimated states of the exampleThe values of $\\gamma ^{*}$ , $\\mu ^{*}$ and $\\bar{\\sigma }(L)$ , and the optimal trade-off curve between $\\gamma ^{*}$ and $\\mu ^{*}$ over the range of $\\lambda $ when the decay rate is fixed ($\\beta =0.35$ ) are shown in figure REF .",
"Figure: γ * \\gamma ^{*}, μ * \\mu ^{*} andσ ¯(L)\\bar{\\sigma }(L), and the optimal trade-off curveThe optimal surfaces of $\\gamma ^{*}$ , $\\mu ^{*}$ and $\\bar{\\sigma }(L)$ over the range of $\\lambda $ when the decay rate is variable are shown in figures REF , REF and REF , respectively.",
"The maximum value of $\\gamma ^{*}$ is 0.34 obtained when $\\lambda =1$ .",
"In the range of $0\\le \\lambda \\le 1$ and $0\\le \\beta \\le 0.8$ , the norm of $L$ is almost constant.",
"As $\\beta $ increases over 0.8, $\\bar{\\sigma }(L)$ rapidly increases and for $\\beta =1.2$ , the LMIs are infeasible.",
"Figure: The optimal surface of γ * \\gamma ^{*}Figure: The optimal surface of μ * \\mu ^{*}Figure: The optimal surface of σ ¯(L)\\bar{\\sigma }(L)"
],
[
"Conclusion",
"A new nonlinear $H_{\\infty }$ observer design method for a class of Lipschitz nonlinear uncertain systems is proposed through LMI optimization.",
"The developed LMIs are linear both in the admissible Lipschitz constant and the disturbance attenuation level allowing both two be an LMI optimization variable.",
"The combined performance of the two optimality criterions is optimized using Pareto optimization.",
"The achieved $H_{\\infty }$ observer guarantees asymptotic stability of the error dynamics with a prespecified decay rate (exponential convergence) and is robust against Lipschitz additive nonlinear uncertainty as well as time-varying parametric uncertainty.",
"Explicit bounds on the nonlinear uncertainty are derived through norm-wise and element-wise analysis."
]
] | 1403.0093 |
[
[
"Intrinsic optical conductivity of modified-Dirac fermion systems"
],
[
"Abstract We analytically calculate the intrinsic longitudinal and transverse optical conductivities of electronic systems which govern by a modified-Dirac fermion model Hamiltonian for materials beyond graphene such as monolayer MoS$_2$ and ultrathin film of the topological insulator.",
"We analyze the effect of a topological term in the Hamiltonian on the optical conductivity and transmittance.",
"We show that the optical response enhances in the non-trivial phase of the ultrathin film of the topological insulator and the optical Hall conductivity changes sign at transition from trivial to non-trivial phases which has significant consequences on a circular polarization and optical absorption of the system."
],
[
"introduction",
"Two-dimensional (2D) materials have been one of the most interesting subjects in condensed matter physics for potential applications due to the wealth of unusual physical phenomena that occur when charge, spin and heat transport are confined to a 2D plane [1].",
"These materials can be mainly classified in different classes which can be prepared as a single atom thick layer namely, layered van der Waals materials, layered ionic solids, surface growth of monolayer materials, 2D topological insulator solids and finally 2D artificial systems and they exhibit novel correlated electronic phenomena ranging from high-temperature superconductivity, quantum valley or spin Hall effect to other enormously rich physics phenomena.",
"two-dimensional materials can be mostly exfoliated into individual thin layers from stacks of strongly bonded layers with weak interlayer interaction and a famous example is graphene and hexagonal boron nitride [2].",
"The 2D exfoliates versions of transition metal dichalcogenides exhibit properties that are complementary to and distinct from those in graphene [3].",
"Optical spectroscopy is a broad field and useful to explore the electronic properties of solids.",
"Optical properties can be tuned by varying the Fermi energy or the electronic band structure of 2D systems.",
"Recently, developed 2D systems such as gapped graphene [4], thin film of the topological insulator [5], [6], and monolayer of transition metal dichalcogenides [3] provide the electronic structures with direct band gap signatures.",
"The optical response of semiconductors with direct band gap is strong and easy to explore experimentally since photons with energy greater than the energy gap can be absorbed or omitted.",
"The thin film of the topological insulator, on the other hand, has been fabricated experimentally by using Sb$_2$ Te$_3$ slab [7] and has been shown that a direct band gap can be formed owing to the hybridization of top and bottom surface states.",
"Furthermore, a non-trivial quantum spin Hall phase has been realized experimentally which was predicted previously in this system [8], [9], [10].",
"Although pristine graphene and surface states of the topological insulator reveal massless Dirac fermion physics , by opening an energy gap they become formed as massive Dirac fermions.",
"The thin film of the topological insulator and monolayer transition metal dichacogenides can be described by a modified-Dirac Hamiltonian.",
"A monolayer of the molybdenum disulfide (ML-MoS$_2$ ) is a direct band gap semiconductor [11], however its multilayer and bulk show indirect band gap [3].",
"This feature causes the optical response in ML-MoS$_2$ to increases in comparison with its bulk and multilayer structures [15], [14], [12], [13], [16].",
"One of the main properties of ML-MoS$_2$ is a circular dichroism aspect responding to a circular polarized light where the left or right handed polarization of the light couples only to the $K$ or $K^{\\prime }$ valley and it provides an opportunity to induce a valley polarized excitation which can profoundly be of interest in the application for valleytronics [17], [18], [19].",
"Another peculiarity of ML-MoS$_2$ is the coupled spin-valley in the electronic structure which is owing to the strong spin-orbit coupling originating from the existence of a heavy transition metal in the lattice structure and the broken inversion symmetry too.",
"[20] These two aspects are captured in a minima massive Dirac-like Hamiltonian introduced by Xiao et al.",
"[20] However it has been shown , based on the tight-binding [21], [22] and $k.p$ method [23], that other terms like an effective mass asymmetry, a trigonal warping, and a diagonal quadratic term might be included in the massive Dirac-like Hamiltonian.",
"The effect of the diagonal quadratic term is very important, for instance, if the system is exposed by a perpendicular magnetic field, it will induce a valley degeneracy breaking term [21].",
"The optical properties of ML-MoS$_2$ have been evaluated by ab-initio calculations [24] and studied theoretically based on the simplified massive Dirac-like model Hamiltonian [25], which is by itself valid only near the main absorbtion edge.",
"A part of the model Hamiltonian which describes the dynamic of massive Dirac fermions are known in graphene committee to have an optical response quite different from that of a standard 2D electron gas.",
"Thus it would be worthwhile to generalize the optical properties of such systems by using the modified-Dirac fermion model Hamiltonian.",
"The modified Hamiltonian for ML-MoS$_2$ without trigonal warping effect at $K$ point is very similar to the modified-Dirac equation which has been studied for an ultrathin film of the topological insulator (UTF-TI) around $\\Gamma $ point.",
"[8], [26] The modified-Dirac Hamiltonian reveals non-trivial quantum spin hall (QSH) and trivial phases corresponding to the existence and absence of the edge states, respectively.",
"Those phases have been predicted theoretically [8], [9], [10], [27] and recently observed by experiment [7].",
"An enhancement of the optical response of UTF-TI has been obtained in the non-trivial phase [28] and a band crossing is observed in the presence of the structure inversion asymmetry induced by substrate [29].",
"Since the modified-Dirac Hamiltonian incorporates an energy gap and a quadratic term in momentum which both have topological meaning, it is natural to expect that the topological term of the Hamiltonian plays an important role in the optical conductivity.",
"In this paper, we analytically calculate the intrinsic longitudinal and transverse optical conductivities of the modified-Dirac Hamiltonian as a function of photon energy.",
"This model Hamiltonian covers the main physical properties of ML-MoS$_2$ and UTF-TI systems in the regime where interband transition plays a main role.",
"We analyze the effect of the topological term in the Hamiltonian on the optical conductivity and transmittance.",
"Furthermore, we show that the UTF-TI system has a non-trivial phase and its optical response enhances in addition, the optical Hall conductivity changes sign at a phase boundary, when the energy gap is zero.",
"This changing of the sign has a significant consequence on the circular polarization and the optical absorbtion of the system.",
"The paper is organized as follows.",
"We introduce the low-energy model Hamiltonian of ML-MoS$_2$ and UTF-TI systems and then the dynamical conductivity is calculated analytically by using Kubo formula in Sec.",
".",
"The numerical results for the optical Hall and longitudinal conductivities and optical transmittance are reported and we also provide discussions with circular dichroism in both systems in Sec. .",
"A brief summary of results is given in Sec.",
"."
],
[
"theory and method",
"The low-energy properties of the ML-MoS$_2$ and other transition metal dicalcogenide materials can be described by a modified-Dirac equation [21], [23], [22] and the Hamiltonian around the $K$ and $K^{\\prime }$ points is given by ${\\cal H}_{\\tau s}=\\frac{\\lambda }{2}\\tau s+\\frac{\\Delta -\\lambda \\tau s}{2}\\sigma _z+t_0a_0{\\bf q}\\cdot {\\sigma }_\\tau +\\frac{\\hbar ^2|{\\bf q}|^2}{4m_0}(\\alpha +\\beta \\sigma _z)\\nonumber \\\\$ where the Pauli matrices stand for a pseudospin which indicates the conduction and valence band degrees of freedom, $\\tau =\\pm $ denotes the two independent valleys in the first Brillouin zone, ${\\bf q}=(q_x,q_y)$ and ${\\sigma }_{\\tau }=(\\tau \\sigma _x,\\sigma _y)$ .",
"The numerical values of the parameters will be given in the Sec. .",
"The UTF-TI system, on the other hand, can be described by a modified Dirac Hamiltonian around the $\\Gamma $ point with two independent hyperbola (isospin) degree of freedoms [8], [26] and thus the Hamiltonian reads ${\\cal H}_{\\tau }=\\epsilon _0+\\tau \\frac{\\Delta }{2}\\sigma _z+t_0a_0{\\bf q}\\cdot {\\sigma }+\\frac{\\hbar ^2|{\\bf q}|^2}{4m_0}(\\alpha +\\tau \\beta \\sigma _z)$ Note that the Pauli matrices in this Hamiltonian stand for the real spin where spin is rotated by operator $U={\\rm diag}[1,i]$ which results in $U^\\dagger \\sigma _xU=-\\sigma _y$ and $U^\\dagger \\sigma _yU=\\sigma _x$ and the isospin index of $\\tau =\\pm $ indicates two independent solutions of UTF-TI which are degenerated in the absence of the structure inversion asymmetry and can be assumed as an internal isospin (spin, valley, or sublattice) degree of freedom.",
"Two mentioned models, Eqs.",
"(REF ) and (REF ), are similar to some extent and describe similar physical properties.",
"Generally, the Hamiltonian around the $\\Gamma (\\tau =+)$ and $K(\\tau =+$ ) points for UTF-TI and monolayer MoS$_2$ systems, respectively can be re-written as $H=\\begin{pmatrix}a_1+b(\\alpha +\\beta )q^2&&c q^\\ast \\\\c q && a_2+b(\\alpha -\\beta )q^2\\end{pmatrix}$ where $a_1=\\Delta /2+\\epsilon _0,~a_2=-\\Delta /2+\\epsilon _0$ for UTF-TI and $a_1=\\Delta /2,~a_2=-\\Delta /2+\\lambda s$ for ML-MoS$_2$ .",
"Note that $b=\\hbar ^2/4m_0a_0^2,~c=t_0$ , and we set $a_0q\\rightarrow q$ .",
"The eigenvalue and eigenvector of the Hamiltonian, Eq.",
"(REF ) can be obtained as $&&|\\psi _{c,v}\\rangle =\\frac{1}{D_{c,v}}\\begin{pmatrix}-c q^\\ast \\\\h_{c,v}\\end{pmatrix}\\nonumber \\\\&&h_{c,v}=d\\mp \\sqrt{d^2+c^2 q^2}~~~,~~~d=\\frac{a_1-a_2}{2}+b\\beta q^2\\nonumber \\\\&&D_{c,v}=\\sqrt{c^2 q^2+h^2_{c,v}}\\nonumber \\\\&&\\varepsilon _{c,v}=a_1+b(\\alpha +\\beta )q^2-h_{c,v}$ and velocity operators along the $x$ and $y$ directions are $\\hbar v_x=\\frac{\\partial H}{\\partial q_x}=c\\sigma _x+2b\\alpha q_x+2b\\beta q_x\\sigma _z\\nonumber \\\\\\hbar v_y=\\frac{\\partial H}{\\partial q_y}=c\\sigma _y+2b\\alpha q_y+2b\\beta q_y\\sigma _z$ The intrinsic optical conductivity can be calculated by using the Kubo formula [30], [31], [32] in a clean sample and it is given by $&&\\sigma _{xy}(\\omega )=-i\\frac{e^2}{2\\pi h}\\int {d^2q\\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\lbrace \\frac{\\langle \\psi _c|\\hbar v_x|\\psi _v\\rangle \\langle \\psi _v|\\hbar v_y |\\psi _c\\rangle }{\\hbar \\omega +\\varepsilon _c-\\varepsilon _v+i0^+}+\\frac{\\langle \\psi _v|\\hbar v_x |\\psi _c\\rangle \\langle \\psi _c|\\hbar v_y |\\psi _v\\rangle }{\\hbar \\omega +\\varepsilon _v-\\varepsilon _c+i0^+}\\rbrace }\\nonumber \\\\&&\\sigma _{xx}(\\omega )=-i\\frac{e^2}{2\\pi h}\\int {d^2q\\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\lbrace \\frac{\\langle \\psi _c|\\hbar v_x|\\psi _v\\rangle \\langle \\psi _v|\\hbar v_x |\\psi _c\\rangle }{\\hbar \\omega +\\varepsilon _c-\\varepsilon _v+i0^+}+\\frac{\\langle \\psi _v|\\hbar v_x |\\psi _c\\rangle \\langle \\psi _c|\\hbar v_x |\\psi _v\\rangle }{\\hbar \\omega +\\varepsilon _v-\\varepsilon _c+i0^+}\\rbrace }$ where $f(\\omega )$ is the Fermi distribution function.",
"We include only the interband transitions and the contribution of the intraband transitions, which leads to the fact that the Drude-like term, is no longer relevant in this study since the momentum relaxation time is assumed to be infinite.",
"This approximation is valid at low-temperature and a clean sample where defect, impurity, and phonon scattering mechanisms are ignorable.",
"We also do not consider the bound state of exciton in the systems.",
"After straightforward calculations (details can be found in Appendix A), the real and imaginary parts of diagonal and off-diagonal components of the conductivity tensor at $\\tau =+$ are given by $&&\\sigma ^{\\Re }_{xy}(\\omega )=\\frac{2e^2}{h}\\int {qdq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{c^2}{\\sqrt{d^2+c^2q^2}}(d-2b\\beta q^2)\\rbrace \\lbrace \\mathbb {P}\\frac{-1}{(\\hbar \\omega )^2-(\\varepsilon _c-\\varepsilon _v)^2}\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Im }_{xy}(\\omega )=\\frac{\\pi e^2}{h}\\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace \\frac{c^2}{\\sqrt{d^2+c^2q^2}}(d-2b\\beta q^2)\\rbrace \\lbrace \\delta (\\hbar \\omega +\\varepsilon _v-\\varepsilon _c)-\\delta (\\hbar \\omega +\\varepsilon _c-\\varepsilon _v)\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Im }_{xx}(\\omega )=-\\frac{2e^2}{h}\\hbar \\omega \\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace c^2-\\frac{c^2q^2}{d^2+c^2q^2}[\\frac{c^2}{2}+b\\beta (a_1-a_2)]\\rbrace \\lbrace \\mathbb {P}\\frac{-1}{(\\hbar \\omega )^2-(\\varepsilon _c-\\varepsilon _v)^2}\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Re }_{xx}(\\omega )=-\\frac{\\pi e^2}{h}\\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace c^2-\\frac{c^2q^2}{d^2+c^2q^2}[\\frac{c^2}{2}+b\\beta (a_1-a_2)]\\rbrace \\lbrace \\delta (\\hbar \\omega +\\varepsilon _v-\\varepsilon _c)+\\delta (\\hbar \\omega +\\varepsilon _c-\\varepsilon _v)\\rbrace }$ where $\\Re $ and $\\Im $ refer to the real and imaginary parts of $\\sigma $ and $\\mathbb {P}$ denotes the principle value.",
"It is worthwhile mentioning that the conductivity for ML-MoS$_2$ for $\\tau =-$ can be found by implementing $p_x\\rightarrow -p_x$ and $\\lambda \\rightarrow -\\lambda $ .",
"Using these transformations, the velocity matrix elements around the $K^{\\prime }$ point can be calculated by taking the complex conjugation of the corresponding results for the $\\tau =+$ case.",
"Furthermore, for the UTF-TI case system, we must replace $\\Delta $ and $\\beta $ by their opposite signs which lead to the same results in comparison with the ML-MoS$_2$ case around $K^{\\prime }$ point.",
"More details in this regard are given in Appendix A."
],
[
"Optical conductivity of ML-MoS$_2$",
"Having obtained the general expressions of the conductivity for the modified-Dirac fermion systems, the conductivity of two examples namely the ML-MoS$_2$ and UTF-TI could be obtained.",
"Here, we would like to focus on the ML-MoS$_2$ case and explore its optical properties, although all results can be generalized to the UTF-TI system as well.",
"Therefore, the optical conductivity for each spin and valley components of ML-MoS$_2$ can be obtained by using appropriate substitution in Eq.",
"(REF ) and results are written as $&&\\sigma ^{\\Re ,\\tau s}_{xy}(\\omega )=\\frac{2e^2}{h}\\mathbb {P}\\int {dq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{\\tau (\\Delta ^{\\prime }_{\\tau s}q-\\beta ^{\\prime }q^3)}{\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}[4((\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2)-(\\hbar \\omega /t_0)^2]}\\rbrace }\\nonumber \\\\ \\nonumber \\\\&&\\sigma ^{\\Im ,\\tau s}_{xy}(\\omega )=\\frac{\\pi e^2}{2h}\\int {dq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{\\tau (\\Delta ^{\\prime }_{\\tau s} q-\\beta ^{\\prime }q^3)}{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}\\rbrace \\delta (\\hbar \\omega /t_0-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2})}\\nonumber \\\\ \\nonumber \\\\&&\\sigma ^{\\Im ,\\tau s}_{xx}(\\omega )=-\\frac{2e^2}{h}\\hbar \\omega \\mathbb {P}\\int {dq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{q}{\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}[4((\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2)-(\\hbar \\omega /t_0)^2]}}\\nonumber \\\\&&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-{\\frac{q^3[\\frac{1}{2}+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}]}{((\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2)^{3/2}[4((\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2)-(\\hbar \\omega /t_0)^2]}\\rbrace }\\nonumber \\\\ \\nonumber \\\\&&\\sigma ^{\\Re ,\\tau s}_{xx}(\\omega )=-\\frac{\\pi e^2}{2h}\\int {dq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{q}{\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}}-{\\frac{q^3[\\frac{1}{2}+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}]}{((\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2)^{3/2}}\\rbrace \\delta (\\hbar \\omega /t_0-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2})}\\nonumber \\\\$ where $\\Delta ^{\\prime }_{\\tau s}=(\\Delta -\\lambda \\tau s)/2t_0$ , $\\alpha ^{\\prime }=b\\alpha /t_0$ , $\\beta ^{\\prime }=b\\beta /t_0$ , $\\sigma ^{\\tau ,s}_{xy}=\\sigma ^{\\Re ,\\tau s}_{xy}+i\\sigma ^{\\Im ,\\tau s}_{xy}$ , and $\\sigma ^{\\tau ,s}_{xx}=\\sigma ^{\\Re ,\\tau s}_{xx}+i\\sigma ^{\\Im ,\\tau s}_{xx}$ .",
"Note that in the case of UTF-TI, there is no extra spin index of $s$ as a degree of freedom and $\\Delta ^{\\prime }_{\\tau s}$ might be replaced by $\\Delta ^{\\prime }=\\Delta /2t_0$ , consequently we have $\\sigma ^{\\tau }_{xy}$ and $\\sigma ^{\\tau }_{xx}$ rather than $\\sigma ^{\\tau s}_{xy}(\\omega )$ and $\\sigma ^{\\tau s}_{xx}(\\omega )$ .",
"To be more precise, $\\lambda \\tau s$ , which is located out of the radical in Eq.",
"(REF ), might be replaced by $\\epsilon _0$ in the $\\epsilon _{c,v}$ to achieve desirable results corresponding to the UTF-TI.",
"It is clear that the dynamical charge Hall conductivity vanishes in both the UTF-TI and ML-MoS$_2$ systems due to the presence of the time reversal symmetry.",
"For the MoS$_2$ case, the spin and valley transverse ac-conductivity are given by $&&\\sigma ^s_{xy}=\\frac{\\hbar }{2e}\\sum _{\\tau }[\\sigma ^{\\tau ,\\uparrow }_{xy}-\\sigma ^{\\tau ,\\downarrow }_{xy}]\\nonumber \\\\&&\\sigma ^v_{xy}=\\frac{1}{e}\\sum _{s}[\\sigma ^{K,s}_{xy}-\\sigma ^{K^{\\prime },s}_{xy}]$ and for the longitudinal ac-conductivity case, an electric field can only induce a charge current and corresponding conductivity is given as $\\sigma _{xx}=\\sum _{\\tau }[\\sigma ^{\\tau ,\\uparrow }_{xx}+\\sigma ^{\\tau ,\\downarrow }_{xx}]$ Moreover, the longitudinal conductivity is the same as expression given by Eq.",
"(REF ) for the UTF-TI case however, the Hall conductivity is slightly changed.",
"Owing to the coupling between the isospin and the spin indexes, the hyperbola Hall conductivity is a spin Hall conductivity [8], [7] and it is thus given by $\\sigma ^{hyp}_{xy}=\\frac{1}{e}[\\sigma ^{\\Gamma ^+}_{xy}-\\sigma ^{\\Gamma ^-}_{xy}]$"
],
[
"Intrinsic dc-conductivity",
"To find the static conductivity in a clean sample, we set $\\omega =0$ and thus the interband longitudinal conductivity vanishes.",
"Consequently, we calculate only the transverse conductivity in this case.",
"At zero temperature, the Fermi distribution function is given by a step function, i. e. $f(\\varepsilon _{c,v})=\\Theta (\\varepsilon _{\\rm F}-\\varepsilon _{c,v})$ .",
"We derive the optical conductivities for the case of ML-MoS$_2$ and results corresponding to the UTF-TI can be deduced from those after appropriate substitutions.",
"Most of the interesting transport properties of ML-MoS$_2$ originates from its spin splitting band structure for the hole doped case.",
"Therefore, for the later case, when the upper spin-split band contributes to the Fermi level state, the dc-conductivity is given by $&&\\sigma ^{K \\uparrow }_{xy}=-\\sigma ^{K^{\\prime }\\downarrow }_{xy}=-\\frac{e^2}{2h}\\int ^{q_{c}}_{q_{\\rm F}}{\\frac{(\\Delta ^{\\prime }_{K\\uparrow }q-\\beta ^{\\prime }q^3) dq}{((\\Delta ^{\\prime }_{K\\uparrow }+\\beta ^{\\prime }q^2)^2+q^2)^{\\frac{3}{2}}}}\\nonumber \\\\&&=-\\frac{e^2}{2h}{\\cal C}^{K\\uparrow }+\\frac{e^2}{2h}\\frac{2\\mu +2b(\\alpha -\\beta )q^2_{\\rm F}}{\\Delta -\\lambda +2\\mu +2b\\alpha q^2_{\\rm F}}$ and for the spin-down component we thus have $\\sigma ^{K \\downarrow }_{xy}&=&-\\sigma ^{K^{\\prime }\\uparrow }_{xy}=-\\frac{e^2}{2h}\\int ^{q_{c}}_{0}{\\frac{(\\Delta ^{\\prime }_{K\\downarrow }q-\\beta ^{\\prime }q^3) dq}{((\\Delta ^{\\prime }_{K\\downarrow }+\\beta ^{\\prime }q^2)^2+q^2)^{\\frac{3}{2}}}}\\nonumber \\\\&=&-\\frac{e^2}{2h}{\\cal C}^{K\\downarrow }$ where $q_c$ is the ultra violate cutoff and $\\mu /t_0=\\sqrt{(\\Delta ^{\\prime }_{K\\uparrow }+\\beta ^{\\prime }q^2_{\\rm F})^2+q^2_{\\rm F}}-\\Delta ^{\\prime }_{K\\uparrow }-\\alpha ^{\\prime }q^2_{\\rm F}$ stands for the chemical potential and it is easy to show that ${\\cal C}^{Ks}={\\rm sgn}(\\Delta -\\lambda s)-{\\rm sgn}(\\beta )$ at large cutoff values.",
"In a precise definition, ${\\cal C}^{Ks}$ terms are the Chern numbers for each spin and valley degrees of freedom and the total Chern number is zero owing to the time reversal symmetry.",
"Intriguingly, the quadratic term in Eq.",
"(3), $\\beta $ , leads to a new topological characteristic.",
"When $\\beta \\Delta >0$ , with $\\Delta >\\lambda $ , system has a trivial phase with no edge mode closing the energy gap however for the case that $\\beta \\Delta <0$ , the topological phase of the system is a non-trivial with edge modes closing the energy gap.",
"In the case of the ML-MoS$_2$ , the tight binding model [21], [33] predicts the trivial phase ($\\beta >0)$ with ${\\cal C}^{Ks}=0$ .",
"However, a non-trivial phase is expected by Refs.",
"[Kormanyos13, Liu13] (where $\\beta <0$ ) which leads to ${\\cal C}^{Ks}=2$ .",
"In other words, the term proportional to $\\beta $ has a topological meaning in Z$_2$ symmetry invariant like the UTF-TI system [8] and the sign of $\\beta $ plays important role.",
"The transverse intrinsic dc-conductivity for the hole doped ML-MoS$_2$ case, is given by $\\sigma ^s_{xy}&=&\\frac{\\hbar }{e}[\\sigma ^{K\\uparrow }_{xy}-\\sigma ^{K\\downarrow }_{xy}]=\\frac{e}{2\\pi }\\frac{\\mu +b(\\alpha -\\beta )q^2_{\\rm F}}{\\Delta -\\lambda +2\\mu +2b\\alpha q^2_{\\rm F}}\\nonumber \\\\\\sigma ^v_{xy}&=&\\frac{2}{e}[\\sigma ^{K\\uparrow }_{xy}+\\sigma ^{K\\downarrow }_{xy}]=-\\frac{e}{h}{\\cal C}^{K}+\\frac{2}{\\hbar }\\sigma ^s_{xy}$ where, at large cutoff, ${\\cal C}^{K}=[{\\rm sign}(\\Delta -\\lambda )+{\\rm sign}(\\Delta +\\lambda )]/2-{\\rm sign}(\\beta )$ stands for the valley Chern number and it equals to zero or 2 corresponding to the non-trivial or trivial band structure, respectively.",
"In the case of the UTF-TI, the isospin Hall conductivity is $\\sigma ^{hyp}_{xy}=-\\frac{e}{h}{\\cal C}^{\\Gamma }+\\frac{2e}{h}\\frac{\\mu +b(\\alpha -\\beta )q^2_{\\rm F}}{\\Delta +2\\mu +2b\\alpha q^2_{\\rm F}}$ where $\\mu /t_0=\\sqrt{(\\Delta ^{\\prime }+\\beta ^{\\prime }q^2_{\\rm F})^2+q^2_{\\rm F}}-\\Delta ^{\\prime }-\\epsilon _0-\\alpha ^{\\prime }q^2_{\\rm F}$ and ${\\cal C}^{\\Gamma }={\\rm sgn}(\\Delta )-{\\rm sgn}(\\beta )$ at large cutoff.",
"This result is consistent with that result obtained by Lu et al. [8].",
"It should be noted that in the absence of the diagonal quadratic term, the non-zero valley Chern number at zero doping predicts a valley Hall conductivity, which is proportional to ${\\rm sign}(\\Delta )$ .",
"Therefore, the exitance of edge states , which can carry the valley current, is anticipated.",
"However, Z$_2$ symmetry prevents the edge modes from existing.",
"Since the Z$_2$ topological invariant is zero when the gap is caused only the inversion symmetry breaking [34], thus the topology of the band structure is trivial and there are no edge states to carry the valley current when the chemical potential is located inside the energy gap.",
"Therefore, we can ignore the valley Chern number in $\\sigma ^v_{xy}$ and thus the results are consistent with those results reported by Xiao el al.",
"[20] at a low doping rate where $\\mu \\ll \\Delta -\\lambda $ ."
],
[
"Intrinsic dynamical conductivity",
"In this section, we analytically calculate the dynamical conductivity of the modified-Dirac Hamiltonian which results in the trivial and non-trivial phases.",
"Using the two-band Hamiltonian, including the quadratic term in momentum, the optical Hall conductivity for each spin and valley components are given by $\\sigma ^{\\Re ,\\tau s}_{xy}(\\omega )&=&\\tau \\frac{e^2}{h}[G_{\\tau s}(\\omega ,q_{\\rm F})-G_{\\tau s}(\\omega ,q_c)]\\nonumber \\\\\\sigma ^{\\Im ,\\tau s}_{xy}(\\omega )&=&\\tau \\frac{\\pi e^2}{2h}\\frac{\\Delta ^{\\prime }_{\\tau s}-\\beta ^{\\prime } q_{0,\\tau s}^2}{\\hbar \\omega ^{\\prime }n(\\omega ^{\\prime })}\\nonumber \\\\ &\\times & [\\Theta (2\\varepsilon ^{\\prime }_{\\rm F}-\\lambda ^{\\prime }\\tau s-2\\alpha ^{\\prime } q_{0,\\tau s}^2-\\hbar \\omega ^{\\prime })-(\\omega ^{\\prime }\\rightarrow -\\omega ^{\\prime })]\\nonumber \\\\ &\\times &\\Theta (n(\\omega ^{\\prime })-(1+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}))$ where $\\Re $ and $\\Im $ indicate to the real and imaginary parts, respectively and $G_{\\tau s}(\\omega ,q)$ reads as below (details are given in Appendix B) $&&G_{\\tau s}(\\omega ,q)=\\frac{\\Delta ^{\\prime }_{\\tau s}}{\\hbar \\omega ^{\\prime } n(\\omega ^{\\prime })}\\ln |\\frac{\\hbar \\omega ^{\\prime } \\frac{m(q)}{n(\\omega ^{\\prime })}-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime } q^2)^2+q^2}}{\\hbar \\omega ^{\\prime } \\frac{m(q)}{n(\\omega ^{\\prime })}+2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}|\\nonumber \\\\&&+\\frac{1}{4\\beta ^{\\prime }\\hbar \\omega ^{\\prime } n(\\omega ^{\\prime })}\\ln |\\frac{\\hbar \\omega ^{\\prime } \\frac{m(q)}{n(\\omega ^{\\prime })}-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}{\\hbar \\omega ^{\\prime }\\frac{m(q)}{n(\\omega ^{\\prime })}+2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime } q^2)^2+q^2}}|\\nonumber \\\\&&-\\frac{1}{4\\beta ^{\\prime }\\hbar \\omega ^{\\prime }}\\ln |\\frac{\\hbar \\omega ^{\\prime }-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime } q^2)^2+q^2}}{\\hbar \\omega ^{\\prime }+2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime } q^2)^2+q^2}}|$ where $m(q)=1+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}+2\\beta ^{\\prime 2}q^2$ , $n(\\omega ^{\\prime })=\\sqrt{1+4\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime 2}(\\hbar \\omega ^{\\prime })^2}$ , $\\hbar \\omega ^{\\prime }=\\hbar \\omega /t_0$ , $\\varepsilon ^{\\prime }_{\\rm F}=\\varepsilon _{\\rm F}/t_0$ and $\\lambda ^{\\prime }=\\lambda /t_0$ .",
"The value of $q_{0,\\tau s}$ can be evaluated from $m(q_{0,\\tau s})=n(\\omega ^{\\prime })$ .",
"Note that $q_c$ , the ultra violate cutoff, is assumed to be equal to $1/a_0$ .",
"Some special attentions might be taken for the situation in which there is no intersection between the Fermi energy and the band energy, for instance in a low doping hole case of the ML-MoS$_2$ in which the Fermi energy lies in the spin-orbit splitting interval.",
"In this case, the Fermi wave vector ($q_{\\rm F}$ , which has no contribution to the Fermi level) vanishes.",
"The quadratic terms can also affect profoundly on the longitudinal dynamical conductivity which plays main role in the optical response when the time reversal symmetry is preserved.",
"In this case, one can find $\\sigma ^{\\Re ,\\tau s}_{xx}(\\omega )&=&-\\frac{\\pi e^2}{4h}\\frac{1}{n(\\omega ^{\\prime })}(1-\\frac{1+4\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}}{2}(\\frac{2q_{0,\\tau s}}{\\hbar \\omega ^{\\prime }})^2) \\nonumber \\\\&\\times & [\\Theta (2\\varepsilon ^{\\prime }_{\\rm F}-\\lambda ^{\\prime }\\tau s-2\\alpha ^{\\prime } q_{0,\\tau s}^2-\\hbar \\omega ^{\\prime })-(\\omega ^{\\prime }\\rightarrow -\\omega ^{\\prime })]\\nonumber \\\\ &\\times &\\Theta (n(\\omega ^{\\prime })-(1+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}))\\nonumber \\\\\\sigma ^{\\Im ,\\tau s}_{xx}(\\omega )&=&-\\frac{e^2}{h}[H_{\\tau s}(\\omega ,q_{\\rm F})-H_{\\tau s}(\\omega ,q_c)]$ where $H_{\\tau s}(\\omega ,q)$ is given by (details are given in Appendix B) $H_{\\tau s}(\\omega ,q)&=&\\frac{(1+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s})m(q)-(1+4\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s})}{2\\beta ^{\\prime 2}\\hbar \\omega ^{\\prime }\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}\\nonumber \\\\&+&\\frac{1+4\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}}{2\\beta ^{\\prime 2}(\\hbar \\omega ^{\\prime })^2}\\ln |\\frac{\\frac{\\hbar \\omega ^{\\prime }}{2}-\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}{\\frac{\\hbar \\omega ^{\\prime }}{2}+\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}|\\nonumber \\\\&+&\\frac{(1+2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s})(1+4\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s})+\\beta ^{\\prime 2}(\\hbar \\omega ^{\\prime })^2}{2\\beta ^{\\prime 2}(\\hbar \\omega ^{\\prime })^2n(\\omega ^{\\prime })}\\nonumber \\\\&\\times &\\ln |\\frac{\\frac{\\hbar \\omega ^{\\prime }}{2}\\frac{m(q)}{n(\\omega ^{\\prime })}-\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}{\\frac{\\hbar \\omega ^{\\prime }}{2}\\frac{m(q)}{n(\\omega ^{\\prime })}+\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2}}|$ It is worthwhile mentioning that the $G$ and $H$ functions do not depend on the $\\alpha $ term given in Eq. (3).",
"For $\\beta =0$ in Eq.",
"(3), we have $m(q)/n(\\omega ^{\\prime })\\rightarrow 1$ , $1/n(\\omega ^{\\prime })\\rightarrow 1-2\\beta ^{\\prime }\\Delta ^{\\prime }_{\\tau s}$ , therefore $G_{\\tau s}(\\omega ,q)$ reduces to $g_{\\tau s}(\\omega ,q)$ and in a similar way, $H_{\\tau s}$ reduces to $h_{\\tau s}$ .",
"Here $g_{\\tau s}$ and $h_{\\tau s}$ read as below $&&g_{\\tau s}(\\omega ,q)=\\frac{\\Delta -\\lambda \\tau s}{4\\hbar \\omega }\\ln |\\frac{\\hbar \\omega -\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0q^2}}{\\hbar \\omega +\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0q^2}}|\\nonumber \\\\&&h_{\\tau s}(\\omega ,q)=\\frac{\\Delta -\\lambda \\tau s}{2\\hbar \\omega }\\frac{\\Delta -\\lambda \\tau s}{\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0q^2}}\\nonumber \\\\&&+\\frac{1}{4}[1+(\\frac{\\Delta -\\lambda \\tau s}{\\hbar \\omega })^2]\\ln |\\frac{\\hbar \\omega -\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0q^2}}{\\hbar \\omega +\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0q^2}}|\\nonumber \\\\$ Using Eqs.",
"(REF ) and (REF ), the conductivity simplifies when $\\beta =0$ and the results are $\\sigma ^{\\Re ,\\tau s}_{xy}(\\omega )&=&\\tau \\frac{e^2}{h}[g_{\\tau s}(\\omega ,q_{\\rm F})-g_{\\tau s}(\\omega ,q_c)]\\nonumber \\\\\\sigma ^{\\Im ,\\tau s}_{xy}(\\omega )&=&\\tau \\frac{\\pi e^2}{4h}\\frac{\\Delta -\\lambda \\tau s}{\\hbar \\omega }[\\Theta (2\\varepsilon _{\\rm F}-\\lambda \\tau s-\\hbar \\omega )-(\\omega \\rightarrow -\\omega )]\\nonumber \\\\ &\\times & \\Theta (\\hbar \\omega -(\\Delta -\\lambda \\tau s))$ The longitudinal conductivity for the case of $\\beta =0$ is given by the following relations for the electron doped case $\\sigma ^{\\Re ,\\tau s}_{xx}(\\omega )&=&-\\frac{\\pi e^2}{8h}(1+(\\frac{\\Delta -\\lambda \\tau s}{\\hbar \\omega })^2)\\Theta (\\hbar \\omega -(\\Delta -\\lambda \\tau s))\\nonumber \\\\ &\\times &[\\Theta (2\\varepsilon _{\\rm F}-\\lambda \\tau s-\\hbar \\omega )-(\\omega \\rightarrow -\\omega )]\\nonumber \\\\\\sigma ^{\\Im ,\\tau s}_{xx}(\\omega )&=&-\\frac{e^2}{h}[h_{\\tau s}(\\omega ,q_{\\rm F})-h_{\\tau s}(\\omega ,q_c)]$ These relations are consistent with those results reported in Ref. [Li12].",
"Furthermore, dropping the $\\lambda $ term gives rise to the optical conductivity of gapped graphene and the result is in good agreement with the universal conductivity of graphene [35] for $\\Delta =\\lambda =\\alpha =\\beta =0$ .",
"In most numerical results, we use $set_0: \\lambda =0.08eV,~\\Delta =1.9eV, t_0=1.68eV,~\\alpha =m_0/m_{+}=0.43,~\\beta =m_0/m_{-}-4m_0v^2/(\\Delta -\\lambda )=2.21$ where $m_{\\pm }=m_e m_h/(m_h \\pm m_e)$ and $v=t_0 a_0/\\hbar $ .",
"These values have been obtained in Ref. [Rostami13].",
"Moreover, for the sake of completeness, we introduce two other sets of the parameters as $ t_0=1.51eV,~\\beta =1.77$ and another set $t_0=2.02eV,~\\beta =0$ corresponding to the same effective masses ($\\alpha =0$ for $m_e=-m_h=0.5 m_0$ ) for electron and hole bands.",
"These parameters are calculated by using the procedure reported in Ref. [Rostami13].",
"The later comparison helps us to perceive the validity of the effective mass approximation for the ML-MoS$_2$ system and for this purpose, we assume the same effective masses for electron and hole bands to compare the spin Hall conductivity resulted from the Dirac-like and modified-Dirac Hamiltonians.",
"Notice that all energies are measured from the center of the energy gap.",
"The real part of the optical Hall and longitudinal conductivities for the two set of parameters, with and without quadratic terms, are illustrated in Fig.",
"REF and Fig.",
"REF where top and bottom panels indicate electron and hole doped systems, respectively.",
"The effect of the mass asymmetry between the effective masses of the electron and hole ($\\alpha $ ) bands is neglected and it will be discussed later.",
"It is clear that the quadratic term, $\\beta $ , causes a reduction of the intensity of the optical Hall conductivity with no changing of the position of peaks for both electron and hole doped cases.",
"The position of peaks in the real part of Hall conductivity is given by $\\hbar \\omega =\\sqrt{(\\Delta -\\lambda \\tau s)^2+4t^2_0{q_{\\rm Fs}}^2}$ for $\\beta =0$ case and ${\\hbar \\omega ^{\\prime }}m(q_{\\rm Fs}){n(\\omega ^{\\prime })}^{-1}-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }{q_{\\rm Fs}}^2)^2+{q_{\\rm F s}}^2}=0~~\\text{and}~~{\\hbar \\omega ^{\\prime }}-2\\sqrt{(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }{q_{\\rm Fs}}^2)^2+{q_{\\rm F s}}^2}=0$ for each spin component with corresponding Fermi wave vector $q_{\\rm F s}$ and for the case that $\\beta \\ne 0$ .",
"Surprisingly, the last two equations for the later case are simultaneously fulfilled the equation $m(q_{\\rm F s})=n(\\omega ^{\\prime })$ in frequency.",
"In the energy range shown in the figures, the numerical value of the peak position for both cases are approximately equal and it indicates that the position of peaks and steplike configuration don't change due to the $\\beta $ term in a certain Fermi energy.",
"It should be noticed that the intensity of the real part of $\\sigma _{xx}$ decreases with the quadratic term.",
"Consequently, it indicates that the effective mass approximation of the Hamiltonian for the ML-MoS$_2$ is not completely valid because two sets of parameters with the same effective masses are showing distinct results.",
"Figure: (Color online).",
"Real part of the Hall conductivity (in units of e 2 /ℏe^2/\\hbar )for (a) electron with ε F =1eV\\varepsilon _{\\rm F}=1eV and (b) hole with ε F =-1eV+λ\\varepsilon _{\\rm F}=-1eV+\\lambda doped cases as a function of photon energy (in units of eV) around the KK point.",
"Electron and hole masses are set to be 0.5m 0 0.5m_0 and for two set of parameters, β=0,t 0 =2.02eV\\beta =0, t_0=2.02eV and β=1.77,t 0 =1.51\\beta =1.77, t_0=1.51.Figure: (Color online).",
"Real part of the longitudinal conductivity (in units of e 2 /ℏe^2/\\hbar ) for (a) electron with ε F =1eV\\varepsilon _{\\rm F}=1eV and (b) holewith ε F =-1eV+λ\\varepsilon _{\\rm F}=-1eV+\\lambda doped cases as a function of photon energy (in units of eV) around the KK point.",
"Electron and hole masses are set to be 0.5m 0 0.5m_0 and for two set of parameters, β=0,t 0 =2.02eV\\beta =0, t_0=2.02eV and β=1.77,t 0 =1.51\\beta =1.77, t_0=1.51."
],
[
"Mass asymmetry between electron and hole",
"In this subsection, we consider the mass asymmetry between electron and hole bands and then the conductivity of the ML-MoS$_2$ is calculated for the Hamiltonian given in Eq.",
"(REF ).",
"The results are illustrated in Figs.",
"REF and REF around the $K$ point.",
"Due to the mass asymmetry, a small splitting between electron and hole doped cases takes place in the spin-up component.",
"On the other hand, there is considerable splitting between electron and hole doped cases due to both spin-orbit coupling and mass asymmetry for the spin-down case.",
"We also note a sharp onset in the imaginary part of the conductivity, minimum energy associated with the possible interband optical transition.",
"Moreover, corresponding to the onset in $\\sigma ^{\\Im }_{xy}~(\\sigma ^{\\Re }_{xx})$ where there is a peak in its real (imaginary) part at the same energy as they are related by the Kramers-Kroning relations.",
"Figure: (Color online).",
"(a) Real and (b) imaginary parts of the optical Hall conductivity (in units of e 2 /ℏe^2/\\hbar ) as a function of photon energy (in units of eV) around the KK point.Red (blue) color stands for electron (hole) doped case with ε F =1eV\\varepsilon _{\\rm F}=1eV (ε F =-1eV+λ\\varepsilon _{\\rm F}=-1eV+\\lambda ) and solid (dashed) line indicates the spin up (down).Figure: (Color online).",
"(a) Real and (b) imaginary parts of the optical longitudinal conductivity (in units of e 2 /ℏe^2/\\hbar ) as a function of photon energy (in units of eV) around the KK point.Red(blue) color stands for the electron (hole) doped case with ε F =1eV\\varepsilon _{\\rm F}=1eV (ε F =-1eV+λ\\varepsilon _{\\rm F}=-1eV+\\lambda ) and the solid (dashed) line indicates the spin up (down).The position of peaks or steplike configuration of the dynamical conductivity, can be controlled by the doping rate.",
"Figure.",
"REF shows the difference between the position of those peaks, $\\delta \\omega =\\omega _\\uparrow -\\omega _\\downarrow $ , around the $K$ point for electron and hole doped cases corresponding to the real part of the Hall conductivity for each spin component.",
"As it is clearly shown in this figure, $\\delta \\omega $ increases linearly from a negative value to a positive one up to a saturation value ($2\\lambda )$ for the hole doped case.",
"The linear part of the result originates from the spin splitting in the valence band and the fact that there is two fermi wave vectors in which one component spin has zero Fermi wave vector and does not change by increasing the doping rate.",
"Finally, by increasing the Fermi energy, two Fermi wave vectors contribute to the calculations and the position of both peaks move in the same way and lead to a saturation value for $\\delta \\omega $ .",
"Figure: (Color online) Difference between the position of the peak in the real part of the Hall conductivity, δω=ω ↑ -ω ↓ \\delta \\omega =\\omega _{\\uparrow }-\\omega _{\\downarrow } for the both spin componentsfor the electron doped case including mass asymmetry as a function of the chemical potential.Note that μ 0 \\mu _0, which is the band edge in the conduction and valence bands, is 0.950.95eV and -0.87-0.87eV for the electron and holedoped, respectively."
],
[
"Circular dichroism and Optical transmittance",
"One of the main optical properties of the monolayer transition metal dichalcogenide system is the circular dichroism when it is exposed by a circularly polarized light in which left- or right-handed light can be absorbed only by $K$ or $K^{\\prime }$ valley and it makes the material promising for the valleytronic field.",
"This effect originates from the broken inversion symmetry and it can be understood by calculating the interband optical selection rule ${\\cal P_{\\pm }}=m_0\\langle \\psi _c|v_x\\pm i v_y|\\psi _v\\rangle $ for incident right-(+) and left-(-)handed light.",
"The photoluminescence probability for the modified Dirac fermion Hamiltonian is $|{\\cal P_\\pm }|=\\frac{m_0 t_0 a_0}{\\hbar }(1\\pm \\tau \\frac{d-2b\\beta q^2}{\\sqrt{d^2+c^2 q^2}})$ where $q^2=q_x^2+q_y^2$ .",
"Notice that the mass asymmetry term, $\\alpha $ , has no effect on the optical selection rule.",
"The selection rule can simply prove the circular dichroism in the ML-MoS$_2$ .",
"Another approach which helps us to understand this effect is to calculate the optical conductivity around the $K$ point of two kinds of light polarizations as $\\sigma _{\\pm }=\\sum _{s}\\lbrace \\sigma ^{K s}_{xx}\\pm \\sigma ^{K s}_{xy}\\rbrace $ which has been calculated by using the Dirac-like model [19], [25] and now, we modify that by using the modified-Dirac Hamiltonian.",
"Figure.",
"REF shows the coupling of the light and valleys.",
"Note that $\\Re e [\\sigma _{-}]$ is large and comparable in size for either spin up or down while $\\Re e [\\sigma _{+}]$ is small in comparison.",
"The valley around the $K$ point can couple only to the left-handed light and this effect is washed up by increasing the frequency of the light and the result is in good agreement with recent experimental measurements [12].",
"Figure: (Color online) Real part of the optical conductivity around KK point, for left (solid) hand right (dashed) handed light.It indicates the appearance of the circular dichroism effect for the modified-Dirac equation.The electron (ε F =1eV\\varepsilon _{\\rm F}=1eV) doped case including mass asymmetry.Furthermore, the optical transmittance is an important physical quantity and it can be evaluated stemming from the conductivity.",
"The optical transmittance of a free standing thin film exposed by a linear polarized light is given by [36] $T(\\omega )=\\frac{1}{2}\\lbrace |\\frac{2}{2+Z_0\\sigma _{+}(\\omega )}|^2+|\\frac{2}{2+Z_0\\sigma _{-}(\\omega )}|^2\\rbrace $ where $Z_0=376.73\\Omega $ and $\\sigma _{\\pm }(\\omega )=\\sigma _{xx}(\\omega )\\pm i\\sigma _{xy}$ are the vacuum impedance and the optical conductivity of the thin film, respectively.",
"For the ML-MoS$_2$ case, the total Hall conductivity in the presence of the time reversal symmetry is zero and the total longitudinal conductivity is given by $\\sigma _{xx}=2(\\sigma _{xx}^{K\\uparrow }+\\sigma _{xx}^{K\\downarrow })$ .",
"The optical transmittance of the multilayer of MoS$_2$ systems has been recently measured [33] and it is about $94.5\\%$ for each layer in the optical frequency range.",
"The optical transmittance of the ML-MoS$_2$ is displayed in Fig.",
"REF for both electron and hole doped cases using the numerical value defined as $set_0$ .",
"The result shows that the optical transmittance is about $98\\%$ for the frequency range in which both spin components are active for giving response to the incident light.",
"Importantly, for the electron dope case, there are two minimums with distance about $0.16\\text{eV}/\\hbar $ in frequency which mostly indicates the spin-orbit splitting ($2\\lambda $ ) in the valence band and it is consistent with the results illustrated in Fig.",
"REF .",
"The optical transmittance for electron doped case is about $98\\%$ in all frequency range.",
"Moreover, for the hole dope case as it is shown in Fig.",
"REF , the optical transmittance changes by tuning doping rate.",
"Interestingly, at $\\mu =-0.942$ eV the difference between the position of peaks of two spin components, $\\delta \\omega $ is approximately zero.",
"Consequently, the total optical conductivity enhances in this resonating doping rate which has significant effect on the optical transmittance of the system where the transmittance decreases and particularly reaches to a value less than $90\\%$ at the resonance frequency when $\\delta \\omega \\simeq 0$ .",
"Our numerical calculations show that the hole doped ML-MoS$_2$ is darker than the electron doped one specially close to the resonance frequency.",
"Furthermore, this feature provides an opportunity with measuring the spin-orbit coupling by an optical transmittance measurement.",
"Figure: (Color online) Optical transmittance in a finite frequency for the electron (ε F =1eV\\varepsilon _{\\rm F}=1eV) and hole(ε F =-1eV+λ\\varepsilon _{\\rm F}=-1eV+\\lambda ) doped cases including mass asymmetry."
],
[
"Optical response in the non-trivial phase",
"The modified-Dirac Hamiltonian shows a non-trivial phase when $\\beta \\Delta <0$ and it has been numerically shown that in this phase a light matter interaction enhances due to the change of the parabolic band dispersion into the shape of a Mexican-hat with two extrema [28].",
"To fulfill such a band dispersion, a negative value $\\beta \\Delta $ with a large absolute value is required and it is accessible for an ultrathin film of the topological insulator.",
"The sign and the absolute values of the parameters can be manipulated by the thickness of the thin film, while in the case of the ML-MoS$_2$ , to the best of our knowledge, it is barely possible to create a Mexican hat like dispersion relation even for the model Hamiltonian with a non-trivial topology phase [23], [22].",
"In this case, we plot the optical Hall and longitudinal conductivities of the UTF-TI in its trivial and non-trivial phases.",
"In the UTF-TI [37] system, in which only in-plan components of momentum are relevant, one can find the Hamiltonian given by Eq.",
"(REF ) where the numerical value of the model parameters depends on the thickness of the thin films [8], [26].",
"We consider three different thicknesses for which three sets of parameters [8] are listed in Table I.",
"We also neglect the value of $\\epsilon _0$ which is just a constant shift in the energy.",
"Table: Numerical parameter for the ultra thin film of atopological insulator.As it can be seen from table I, a sample with $L=20Å$ or $L=32Å$ indicates the trivial or non-trivial phases, respectively.",
"However for a sample with $L=25Å$ the energy gap vanishes and thus at critical thickness, $L=25Å$ , the trivial to non-trivial phase transition takes place.",
"Hereafter, we call that a phase boundary.",
"Now, we calculate the real part of the Hall and longitudinal conductivities for $\\tau =+$ and the results are illustrated in Fig.",
"REF .",
"It shows that the conductivity enhances in the non-trivial phase which is consistent with previous numerical work.",
"[28] More interestingly, we are now showing that the Hall conductivity changes sign through changing the thickness and it is very important in the circular dichroism effect.",
"This changing of the sign means a different helicity of the light can be coupled to the system.",
"It is worth mentioning that the circular dichroism effect on the electronic system governing modified-Dirac Hamiltonian is also possible when energy gap is zero [19], [25].",
"The selection rule equation reads as $|{\\cal P_\\pm }|=\\frac{m_0 t_0 a_0}{\\hbar }(1\\mp \\tau {b\\beta q}/{\\sqrt{(b\\beta q)^2+c^2}})$ for the case of zero gap.",
"This expression indicates that the circular polarization is achievable away from the $\\Gamma $ point even in the absence of the energy gap.",
"It might be emphasized that the peak in the optical conductivity at zero energy gap originates from a non-zero Fermi energy in which the low energy part of phase space is no longer available for a photon absorbtion process [38] based on the Pauli exclusion principle.",
"More precisely, there is a peak at energy point $\\hbar \\omega \\approx 2\\varepsilon _{\\rm F}$ in the topological insulator case and it can be seen from Eqs.",
"(REF ) and (REF ).",
"Therefore, the peak disappears at zero Fermi energy for a gapless system.",
"In Fig.",
"REF , we show the optical conductivity for the two helicities of light for $\\tau =+$ .",
"The results show that the circular polarization changes sign for negative value of the gap and it gets more strength in the non-trivial phase rather than the trivial phase.",
"Figure: (Color online) Real part of the Hall (a) and longitudinal (b) conductivity for τ=1\\tau =1 and different values of film thickness.It is clear that in the non-trivial phase the optical response of the system is stronger than that of its trivial one.",
"The Fermi energy is ε F =|Δ|/2+0.03\\varepsilon _{\\rm F}=|\\Delta |/2+0.03eV.Figure: (Color online) Circular dichroism effect for different values of the thickness.",
"The real part of the optical conductivityaround the KK point is shown for (a) L=20ÅL=20Å and (b) L=25ÅL=25Å and 31Å31Å.",
"The Fermi energy is ε F =|Δ|/2+0.03\\varepsilon _{\\rm F}=|\\Delta |/2+0.03eV."
],
[
"summary",
"We have analytically calculated the intrinsic conductivity of the electronic systems which govern a modified-Dirac Hamiltonian by using the Kubo formula.",
"We have studied the effect of the quadratic term in momentum, $\\beta $ , which has been recently predicted, and found the different optical responses.",
"This discrepancy originates from the different topological structures of the systems.",
"Our calculations show that the $\\beta $ -term has no effect on the position of the peak of the optical conductivity but it has considerable effect on its magnitude.",
"Therefore, it shows that the same effective mass approximation for electron and hole bands for monolayer MoS$_2$ can not fully describe the optical properties.",
"The effect of the strong spin-orbit interaction can be traced by the difference of the energy interval between the position of the peak in the optical conductivity for the two spin components in electron and hole doped cases.",
"We have shown that this interval for the electron doped case is approximately constant while for the hole doped case, it increases from a negative value to a positive one, and then it increases linearly up to a saturation value.",
"The effect of the mass asymmetry in monolayer MoS$_2$ induces a small splitting between the conductivity spectrum for the electron and hole doped cases.",
"The circular dichroism effect is investigated for the modified-Dirac Hamiltonian of the monolayer MoS$_2$ by calculating the selection rule and the optical conductivity.",
"We have also obtained the optical transmittance of the monolayer MoS$_2$ for the hole and electron doped cases and the results show that the valence band spin splitting has considerable effect on the intensity of the transmittance.",
"We have also studied the effect of the quantum phase transition, which occurs owing to the reducing of the thickness, on the optical conductivity of the thin film of the topological insulator.",
"We have shown that at the phase boundary, when the energy gap is zero, the diagonal quadratic term plays a significant role on the optical conductivity and selection rule.",
"Moreover, we have illustrated that the optical response enhances and the optical Hall conductivity changes sign in the non-trivial phase (QSH) and the phase boundary.",
"R. A. would like to thank the Institute for Material Research in Tohoku University for its hospitality during the period when the last part of this work was carried out."
],
[
"In this appendix, the details of the calculations deriving Eq.",
"(REF ) are presented.",
"Since $\\langle \\psi _c|\\psi _v\\rangle =0$ , we get $\\langle \\psi _c|\\hbar v_x|\\psi _v\\rangle &=&c\\langle \\psi _c|\\sigma _x|\\psi _v\\rangle +2b\\beta q_x\\langle \\psi _c|\\sigma _z|\\psi _v\\rangle \\nonumber \\\\\\langle \\psi _v|\\hbar v_y|\\psi _c\\rangle &=&c\\langle \\psi _v|\\sigma _y|\\psi _c\\rangle +2b\\beta q_y\\langle \\psi _v|\\sigma _z|\\psi _c\\rangle $ owing to the fact that the mass asymmetry parameter $\\alpha $ plays no role in the velocity matrix elements.",
"Using $h_ch_v=-c^2q^2$ we have $\\langle \\psi _c|\\sigma _x|\\psi _v\\rangle &=&\\frac{-c}{D_cD_v}[qh_v+q^\\ast h_c]\\nonumber \\\\\\langle \\psi _v|\\sigma _y|\\psi _c\\rangle &=&\\frac{ic}{D_cD_v}[qh_c-q^\\ast h_v]\\nonumber \\\\\\langle \\psi _c|\\sigma _z|\\psi _v\\rangle &=&\\langle \\psi _v|\\sigma _z|\\psi _c\\rangle =\\frac{2c^2q^2}{D_cD_v}$ In this case $\\langle \\psi _c|\\hbar v_x|\\psi _v\\rangle &=&\\frac{c^2}{D_cD_v}\\lbrace -[qh_v+q^\\ast h_c]+4b\\beta q_x q^2\\rbrace \\nonumber \\\\\\langle \\psi _v|\\hbar v_y|\\psi _c\\rangle &=&\\frac{c^2}{D_cD_v}\\lbrace i[qh_c-q^\\ast h_v]+4b\\beta q_y q^2\\rbrace $ Consequently, the product of the velocity matrix elements are $&&\\langle \\psi _c|\\hbar v_x |\\psi _v\\rangle \\langle \\psi _v|\\hbar v_y |\\psi _c\\rangle =\\frac{c^4}{(D_cD_v)^2}\\lbrace -i(qh_v+q^\\ast h_c)(q h_c-q^\\ast h_v)\\nonumber \\\\&&+(4b\\beta q^2)^2q_xq_y+4b\\beta q^2(-q_y(qh_v+q^\\ast h_c)+iq_x(q h_c-q^\\ast h_v))\\rbrace \\nonumber \\\\&&\\langle \\psi _c|\\hbar v_x |\\psi _v\\rangle \\langle \\psi _v|\\hbar v_x |\\psi _c\\rangle =\\frac{c^4}{(D_cD_v)^2}\\lbrace |qh_v+q^\\ast h_c|^2\\nonumber \\\\&&+(4bq_x\\beta q^2)^2-4bq_x\\beta q^2(qh_v+q^\\ast h_c+q^\\ast h_v+q h_c)\\rbrace $ Using $\\tan \\phi =q_y/q_x$ , one can find $(q h_v+q^\\ast h_c)(q h_c-q^\\ast h_v)&=&-2ic^2q^4\\sin 2\\phi \\nonumber \\\\&-&4q^2d\\sqrt{d^2+c^2q^2}\\nonumber \\\\-q_y(q h_v+q^\\ast h_c)+iq_x(q h_c-q^\\ast h_v)&=&2q^2[-i\\sqrt{d^2+c^2q^2}\\nonumber \\\\&+&d\\sin 2\\phi ]\\nonumber \\\\qh_v+q^\\ast h_c+q^\\ast h_v+q h_c&=&4qd\\cos \\phi \\nonumber \\\\|qh_v+q^\\ast h_c|^2&=&4q^2(d^2+c^2q^2{\\sin \\phi }^2)\\nonumber \\\\(D_cD_v)^2&=&4c^2q^2[d^2+c^2q^2]$ After substituting Eq.",
"(REF ) into Eq.",
"(REF ), we get $\\langle \\psi _c|\\hbar v_x |\\psi _v\\rangle \\langle \\psi _v|\\hbar v_y |\\psi _c\\rangle &=&\\frac{c^2q^2\\sin 2\\phi }{d^2+c^2q^2}\\lbrace -\\frac{c^2}{2}+2b\\beta (b\\beta q^2+d)\\rbrace \\nonumber \\\\&+&i\\frac{c^2}{\\sqrt{d^2+c^2q^2}}\\lbrace d-2b\\beta q^2\\rbrace \\nonumber \\\\\\langle \\psi _c|\\hbar v_x |\\psi _v\\rangle \\langle \\psi _v|\\hbar v_x |\\psi _c\\rangle &=&c^2-\\frac{c^2q^2{\\cos \\phi }^2}{d^2+c^2q^2}\\lbrace c^2+2b\\beta (a_1-a_2)\\rbrace \\nonumber \\\\$ Using $\\int {d\\phi \\sin 2\\phi }=0,\\int {d\\phi {\\cos \\phi }^2}=\\pi $ , one can find $&&\\sigma _{xy}=\\frac{e^2}{h}\\int {qdq\\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace \\frac{c^2}{\\sqrt{d^2+c^2q^2}}(d-2b\\beta q^2)\\rbrace \\lbrace \\frac{1}{\\hbar \\omega +\\varepsilon _c-\\varepsilon _v+i0^+}-\\frac{1}{\\hbar \\omega +\\varepsilon _v-\\varepsilon _c+i0^+}\\rbrace }\\nonumber \\\\&&\\sigma _{xx}=-i\\frac{e^2}{h}\\int {qdq\\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace c^2-\\frac{c^2q^2}{d^2+c^2q^2}[\\frac{c^2}{2}+b\\beta (a_1-a_2)]\\rbrace \\lbrace \\frac{1}{\\hbar \\omega +\\varepsilon _c-\\varepsilon _v+i0^+}+\\frac{1}{\\hbar \\omega +\\varepsilon _v-\\varepsilon _c+i0^+}\\rbrace }\\nonumber \\\\$ Using $(x+i0^+)^{-1}=\\mathbb {P} x^{-1}-i\\pi \\delta (x)$ where $\\mathbb {P}$ stands for principal value, it is easy to show that the real and imaginary parts of diagonal and off-diagonal components of the conductivity tensor read as below $&&\\sigma ^{\\Re }_{xy}=\\frac{2e^2}{h}\\int {qdq (f(\\varepsilon _c)-f(\\varepsilon _v))\\times \\lbrace \\frac{c^2}{\\sqrt{d^2+c^2q^2}}(d-2b\\beta q^2)\\rbrace \\lbrace \\mathbb {P}\\frac{-1}{(\\hbar \\omega )^2-(\\varepsilon _c-\\varepsilon _v)^2}\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Im }_{xy}=\\frac{\\pi e^2}{h}\\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace \\frac{c^2}{\\sqrt{d^2+c^2q^2}}(d-2b\\beta q^2)\\rbrace \\lbrace \\delta (\\hbar \\omega +\\varepsilon _v-\\varepsilon _c)-\\delta (\\hbar \\omega +\\varepsilon _c-\\varepsilon _v)\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Im }_{xx}=-\\frac{2e^2}{h}\\hbar \\omega \\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace c^2-\\frac{c^2q^2}{d^2+c^2q^2}[\\frac{c^2}{2}+b\\beta (a_1-a_2)]\\rbrace \\lbrace \\mathbb {P}\\frac{-1}{(\\hbar \\omega )^2-(\\varepsilon _c-\\varepsilon _v)^2}\\rbrace }\\nonumber \\\\&&\\sigma ^{\\Re }_{xx}=-\\frac{\\pi e^2}{h}\\int {qdq \\frac{f(\\varepsilon _c)-f(\\varepsilon _v)}{\\varepsilon _c-\\varepsilon _v}\\times \\lbrace c^2-\\frac{c^2q^2}{d^2+c^2q^2}[\\frac{c^2}{2}+b\\beta (a_1-a_2)]\\rbrace \\lbrace \\delta (\\hbar \\omega +\\varepsilon _v-\\varepsilon _c)+\\delta (\\hbar \\omega +\\varepsilon _c-\\varepsilon _v)\\rbrace }$ To find the conductivity around $K^{\\prime }$ point we must implement the following changes: $p_x\\rightarrow -p_x$ and $\\lambda \\rightarrow -\\lambda $ .",
"Using these transformations, the velocity matrix elements around the $K^{\\prime }$ point can be calculated by taking complex conjugation of the corresponding results around the $K$ point.",
"Moreover, according to the following dimensionless parameters, $\\varepsilon _c-\\varepsilon _v=2\\sqrt{d^2+c^2q^2}$ , and thus $\\delta (\\hbar \\omega +\\varepsilon _c-\\varepsilon _v)\\rightarrow 0$ for positive frequency in absorbtion process.",
"Thus Eq.",
"(REF ) for the dynamical transverse and longitudinal conductivity is obtained."
],
[
"In this appendix, the details of calculations for some integrals which appear in our model are presented.",
"Using new variables $y=\\beta ^{\\prime }q^2+\\Delta ^{\\prime }_{\\tau s}+(2\\beta ^{\\prime })^{-1}$ and $a^2=\\Delta ^{\\prime }_{\\tau s}/\\beta ^{\\prime }+(4\\beta ^{\\prime 2})^{-1}$ , it is easy to show that $(\\Delta ^{\\prime }_{\\tau s}+\\beta ^{\\prime }q^2)^2+q^2=y^2-a^2$ and we have $G_{\\tau s}(\\omega ,q)&=&\\frac{1}{\\beta ^{\\prime }}\\lbrace (2\\Delta ^{\\prime }_{\\tau s}+\\frac{1}{2\\beta ^{\\prime }})I_1-I_2\\rbrace \\nonumber \\\\H_{\\tau s}(\\omega ,q)&=&\\frac{\\hbar \\omega ^{\\prime }}{\\beta ^{\\prime }}\\lbrace I_1-(2\\Delta ^{\\prime }_{\\tau s}+\\frac{1}{2\\beta ^{\\prime }})I_3\\nonumber \\\\&+&(2\\Delta ^{\\prime }_{\\tau s}+\\frac{1}{2\\beta ^{\\prime }})(\\Delta ^{\\prime }_{\\tau s}+\\frac{1}{2\\beta ^{\\prime }})I_4\\rbrace $ where $I_1, I_2, I_3$ , and $I_4$ are given by $I_1&=&\\int {\\mathbb {P}\\frac{dy}{\\sqrt{y^2-a^2}[4(y^2-a^2)-(\\hbar \\omega ^{\\prime })^2]}}\\nonumber \\\\I_2&=&\\int {\\mathbb {P}\\frac{y dy}{\\sqrt{y^2-a^2}[4(y^2-a^2)-(\\hbar \\omega ^{\\prime })^2]}}\\nonumber \\\\I_3&=&\\int {\\mathbb {P}\\frac{y dy}{(y^2-a^2)^{\\frac{3}{2}}[4(y^2-a^2)-(\\hbar \\omega ^{\\prime })^2]}}\\nonumber \\\\I_4&=&\\int {\\mathbb {P}\\frac{dy}{(y^2-a^2)^{\\frac{3}{2}}[4(y^2-a^2)-(\\hbar \\omega ^{\\prime })^2]}}$ $I_1$ and $I_4$ can be calculated by defining $u$ as a new variable where $y=\\frac{a}{\\sqrt{1-u^2}}$ and it leads to $I_1&=&\\frac{1}{2\\hbar \\omega ^{\\prime }\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}\\ln |\\frac{u-\\frac{\\hbar \\omega ^{\\prime }}{\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}}{u+\\frac{\\hbar \\omega ^{\\prime }}{\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}}|\\nonumber \\\\I_4&=&\\frac{1}{a^2}\\lbrace -I_1+\\frac{1}{(\\hbar \\omega ^{\\prime })^2u}+\\frac{\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}{(\\hbar \\omega ^{\\prime })^3}\\nonumber \\\\&\\times &\\ln |\\frac{u-\\frac{\\hbar \\omega ^{\\prime }}{\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}}{u+\\frac{\\hbar \\omega ^{\\prime }}{\\sqrt{4a^2+(\\hbar \\omega ^{\\prime })^2}}}|\\rbrace $ By defining $y^2=u^2+a^2$ , $I_2$ and $I_3$ are obtained as $I_2&=&\\frac{1}{4\\hbar \\omega ^{\\prime }}\\ln |\\frac{u-\\frac{\\hbar \\omega ^{\\prime }}{2}}{u+\\frac{\\hbar \\omega ^{\\prime }}{2}}|\\nonumber \\\\I_3&=&\\frac{1}{(\\hbar \\omega ^{\\prime })^2u}+\\frac{1}{(\\hbar \\omega ^{\\prime })^3}\\ln |\\frac{u-\\frac{\\hbar \\omega ^{\\prime }}{2}}{u+\\frac{\\hbar \\omega ^{\\prime }}{2}}|$ Using the above expressions for $I_1, I_2, I_3$ , and $I_4$ , it is easy to prove Eqs.",
"(REF ) and (REF )."
]
] | 1403.0080 |
[
[
"The Herschel Fornax Cluster Survey II: FIR properties of\n optically-selected Fornax cluster galaxies"
],
[
"Abstract The $Herschel$ Fornax Cluster Survey (HeFoCS) is a deep, far-infrared (FIR) survey of the Fornax cluster.",
"The survey is in 5 $Herschel$ bands (100 - 500 $\\mu$m) and covers an area of 16 deg$^2$ centred on NGC1399.",
"This paper presents photometry, detection rates, dust masses and temperatures using an optically selected sample from the Fornax Cluster Catalogue (FCC).",
"Our results are compared with those previously obtained using data from the $Herschel$ Virgo Cluster Survey (HeViCS).",
"In Fornax, we detect 30 of the 237 (13%) optically selected galaxies in at least one $Herschel$ band.",
"The global detection rates are significantly lower than Virgo, reflecting the morphological make up of each cluster - Fornax has a lower fraction of late-type galaxies.",
"For galaxies detected in at least 3 bands we fit a modified blackbody with a $\\beta = 2$ emissivity.",
"Detected early-type galaxies (E/S0) have a mean dust mass, temperature, and dust-to-stars ratio of $\\log_{10}(<M_{dust}>/\\mathrm{M_{\\odot}}) = 5.82 \\pm 0.20$, $<T_{dust}> = 20.82 \\pm 1.77$K, and $\\log_{10}(M_{dust}/M_{stars}) = -3.87 \\pm 0.28$, respectively.",
"Late-type galaxies (Sa to Sd) have a mean dust mass, temperature, and dust-to-stars ratio of $\\log_{10}(<M_{dust}>/\\mathrm{M_{\\odot}}) = 6.54 \\pm 0.19$, $<T_{dust}> = 17.47 \\pm 0.97$K, and $\\log_{10}(M_{dust}/M_{stars}) = -2.93 \\pm 0.09$, respectively.",
"The different cluster environments seem to have had little effect on the FIR properties of the galaxies and so we conclude that any environment dependent evolution, has taken place before the cluster was assembled."
],
[
"Introduction",
"The Fornax cluster is a nearby example of a poor but relatively relaxed cluster.",
"It has a recession velocity of 1379 km s$^{-1}$ and a distance of 17.2 Mpc, a mass of 7$ \\times $ 10$^{13}$ M$_{\\odot }$ and virial radius of 0.7 Mpc [33].",
"It is located away from the Galactic plane with a Galactic latitude of $-53.6^{\\circ }$ in an area of relatively low Galactic cirrus.",
"This makes it ideal for study at all wavelengths.",
"[33] showed that despite Fornax's apparent state of relaxation, it still contains substructure, e.g.",
"a small, in-falling group centred on NGC 1316, 3$^{\\circ }$ to the southwest.",
"However, compared to the Virgo cluster, Fornax is very centrally concentrated and probably at a much later epoch of formation.",
"This is also suggested by the strong morphological segregation that has taken place, leaving the cluster almost entirely composed of early-type galaxies.",
"[33] also noted that there exist two different populations, suggesting that while the giant galaxies are virialised, the dwarf population is still in-falling.",
"Morphological segregation is not the only indicator of evolution in the cluster, the interstellar medium (ISM) of the galaxies also seems to have been affected by the cluster environment.",
"[68] found that 35 Fornax cluster galaxies were extremely H i-deficient in comparison to a field sample.",
"H i is generally loosely bound to galaxies and as such is a good indicator of the effects of environmental processes.",
"Dust, another constituent of the ISM, is also affected by the environment.",
"[20], [18] showed that within a cluster like Virgo, dust can be stripped from the outskirts of a galaxy, truncating the dust disk.",
"Dust is crucial for the lifecycle of a galaxy, as it allows atomic hydrogen to transform on its surface into molecular hydrogen and is thus essential for star formation.",
"Around half the energy emitted from a galaxy is first emitted by stars, then reprocessed by dust, and re-emitted from 1 $\\mu m$ to 1 mm [34].",
"Thus, to better understand the physical processes affecting galaxies it is crucial that we observe and understand the complete `stellar' spectral energy distribution (SED).",
"In contrast to Virgo, Fornax has only very weak X-ray emission [37], [69], which traces the hot intra-cluster gas.",
"Compared to Virgo, this lack of an intra-cluster medium (ICM) along with a lower velocity dispersion ($\\sim $ 300 km s$^{-1}$ ) reduces the efficiency of mechanisms such as ram pressure stripping.",
"We can estimate the efficiency of ram pressure stripping using $E \\propto t_{cross} \\delta v^{2} \\rho _{gas}$ [43], where ($E$ ) is the stripping efficiency of a cluster, with a velocity dispersion ($\\delta v$ ), central gas density ($\\rho _{gas}$ ) and a crossing time $t_{cross}$ .",
"Both Virgo and Fornax have a similar crossing time, $t_{cross}\\sim 10^{9}$ yr which is much less than their relaxation time t$_{relax} \\sim 10^{10}$ yr [9].",
"Virgo has a velocity dispersion which is $\\sim $ 4$\\times $ greater and an ICM $\\sim $ 2$\\times $ as dense as Fornax [11], [9], indicating that Fornax may be $\\sim $ 32$\\times $ less efficient than Virgo in removing a galaxy's ISM via ram pressure stripping.",
"Fornax's higher galaxy density, lower ICM density and lower velocity dispersion suggest that galaxy-galaxy tidal interactions will play a more important role than in a more massive cluster like Virgo [14], [49].",
"Whilst the near-infrared (NIR, 1 - 5 $$ ) and mid-infrared (MIR, 5 - 20 $$ ) emission from a galaxy is dominated by the old stellar population and complex molecular line emission, respectively, the far-infrared (FIR, 20 - 500 $$ ) and sub-mm regime (500 - 1000 $$ ) is dominated by dust emitting as a modified blackbody.",
"Although there are a few small windows in the earth's atmosphere, most of the infrared spectrum is absorbed and is either impractical or not possible to observe from the ground, so the infrared wavelength regime is best studied from space-based observatories.",
"In 1983, the $IRAS$ [59] (10 - 100 $\\mu $ m) all-sky survey opened up the extragalactic infrared sky for the first time.",
"Of particular interest to us is the first detection of FIR sources associated with the Fornax cluster.",
"[76] found 5 $IRAS$ sources matching known Fornax galaxies inside the bounds of our survey and located preferentially towards the outskirts of the cluster.",
"Since $IRAS$ , very little further study has been undertaken of the Fornax cluster in the MIR or FIR.",
"However, other cluster observations with $ISO$ [50](10 - 70 $\\mu $ m) indicated that MIR emission, originating from hot dust ($\\sim $ 60 K) correlates well with H ii regions, implying that it is heated primarily by star formation (SF) [65].",
"In contrast, FIR emission from cold dust ($\\approx $ 20 K) had a nonlinear correlation with H$_{\\alpha }$ luminous regions, indicating a link to the older, more diffuse stellar population.",
"Most significantly, they found a cold dust component that, in some cases, was less than 10 K, though $ISO$ lacked the longer wavelength photometric coverage to constrain the Raleigh-Jeans blackbody tail of this `cold dust emission'.",
"The $Spitzer$ Space Telescope [77] (3 - 160 $\\mu $ m) was launched in 2003.",
"Using the MIPS instrument [39] showed that in the Coma cluster SF is suppressed in the cluster and this suppression decreases with distance from the cluster core.",
"All the instruments described above lacked photometric coverage at wavelengths needed to constrain the temperature and mass of cold dust ($T <20$ K).",
"The $Herschel$ Space Observatory [62] rectified this problem as it was able to survey large areas of sky at longer FIR wavelengths and with superior resolution and sensitivity.",
"The $Herschel$ Fornax Cluster Survey [23] observations discussed in this paper make use of the superior observational characteristics of the $Herschel$ Space observatory to address the problems highlighted above.",
"This paper is one in a series of papers in which we compare the properties of galaxies in both the Virgo and Fornax clusters.",
"Other papers in this series are: Paper I [21] examined the FIR properties of galaxies in Virgo cluster core; Paper II [20] studied the truncation of dust disks in Virgo cluster galaxies; Paper III [13] constrained the lifetime of dust in early-type galaxies; Paper IV [71] investigated the distribution of dust mass and temperature in Virgo's spirals; Paper V [42] examined the FIR properties of Virgo's metal-poor, dwarf galaxies; Paper VI [2] a FIR view of M87; Paper VII [28] detected dust in dwarf elliptical galaxies in the Virgo cluster; Paper VIII [24] presented an analysis of the brightest FIR galaxies in the Virgo cluster; Paper IX [54] examined the metallicity dependence of the molecular gas conversion factor; Paper X [16] investigated the effect of interactions on the dust in late-type Virgo galaxies; Paper XI [60] studied the effect of environment on dust and molecular gas in Virgo's spiral galaxies; Paper XII [1] examined the FIR properties of an optically selected sample of Virgo cluster galaxies; Paper XIII [30] investigated the FIR properties of early-type galaxies in the Virgo cluster; Paper XIV [27] studied Virgo's transition-type dwarfs and Paper XVI [22] presented an analysis of metals, stars, and gas in the Virgo cluster.",
"Six further papers [8], [19], [7], [12], [70], [36] discuss the HeViCS galaxies along with other galaxies observed as part of the Herschel Reference Survey (HRS)."
],
[
"Data and Flux Measurement",
"This paper is based on the methodology of [1].",
"As in [1] we initially use an optical catalogue of cluster galaxies to select our targets and then measure the FIR flux density at those locations."
],
[
"Fornax Cluster Catalogue ",
"The Fornax Cluster Catalogue [40], was created from visual inspection of photographic plates taken with the Du Pont 2.5m reflector at the Las Campanas Observatory.",
"It is complete to m$_{BT}\\sim 18$ , and contains members down to m$_{BT}\\sim 20$ .",
"Although this catalogue is 20 years old it is still the best optical catalogue available.",
"It is equivalent to the Virgo Cluster Catalogue [6] used by [1] and so enables a good comparison between the two clusters.",
"[40] assigned cluster membership mainly based on morphology and the detail that could be observed in the images.",
"There are now 104 radial velocity measurements of FCC galaxies which indicate that 6 of them are outside the cluster, (FCC 97, 141, 189, 233, 257, and 287).",
"These were removed from our sample.",
"We use the FCC's positions, optical sizes and shapes as a starting point from which to fit an aperture and measure the FIR emission for each galaxy (see Figure REF )."
],
[
"SuperCOSMOS Sky Survey",
"Due to the angular resolution of $Herschel$ ($\\sim $ 18\" at 250 $$ ) many sources are either confused or blended with nearby or background galaxies.",
"To help overcome this problem we have used the SuperCOSMOS Sky Survey (SSS) [44].",
"The SSS's comparatively high angular resolution allows us to discern if a FIR source is likely to be associated with a Fornax cluster galaxy.",
"We overlay FIR contours on SSS r-band images to determine whether a FIR source can be clearly associated with a single FCC galaxy."
],
[
"HeFoCS data",
"The HeFoCS observations cover a $4\\,^{\\circ }\\,\\times \\,4\\,^{\\circ }$ tile centred on NGC 1399 [$\\alpha = 03^{h}38^{m}29.8^{s}, \\delta = -35^{o}27^{\\prime }2.7\"$ ], the central elliptical galaxy.",
"This is an area of apparently low galactic cirrus when compared to the Virgo cluster field [23].",
"The region contains $\\sim $ 70 % of the area covered by the FCC catalogue, Figure REF .",
"It should be noted that there is an unavoidable misalignment between SPIRE and PACS due to their respective locations on the $Herschel$ focal plane.",
"This misalignment leads to a loss of 30 galaxies that are not in the PACS maps, and are only observable by SPIRE.",
"In total 60 % of the FCC cluster galaxies are observed in all 5 bands (100, 160, 250, 350 & 500 $$ ).",
"The FIR maps in this paper are identical to those used in [23] and a full description of the data reduction for the HeFoCS data is available in that paper.",
"Briefly, the HeFoCS observations are taken using PACS ($100\\,\\&\\,160$ ) [63] and SPIRE ($250\\, , 350\\,, \\& \\,500$ ) [41] in parallel mode with a fast scan rate (60 arcsec s$^{-1}$ ), and our final maps consist of 4 scans ($2\\times 2$ orthogonal cross-linking scans).",
"PACS data were taken from level 0 to level 1 using the standard pipeline, then the 4 scans were combined with the Scanamorphous map maker [67].",
"SPIRE data were processed with a customised pipeline from level 0 to level 1, which is very similar to the official pipeline.",
"The difference being the use of a method called BriGAdE (Smith et al., in preparation), in place of the standard temperatureDriftCorrection.",
"BriGAdE effectively corrects all the bolometers for thermal drift without removing large extended structures like Galactic cirrus.",
"These scans are then combined using the naïve mapper in the standard pipeline.",
"The final maps have pixel sizes of 2, 3, 6, 8 and 12 arc seconds and $1\\sigma $ noise over the entire image of 0.5, 0.7, 0.7, 0.8 and 0.9 mJy pixel$^{-1}$ (or 9.9, 9.2, 8.9, 9.4 and 10.2 mJy beam$^{-1}$ ) for 100, 160, 250, 350 and 500 $$ , respectivelyIn this paper we present updated values for the global noise.",
"These are lower than those presented by [23].",
"Our method for calculating the gobal noise is described in Section REF ..",
"The approximate Full Width Half Maximum (FWHM) of the $Herschel$ beam is 11, 14, 18, 25 & 36 arc seconds, at 100, 160, 250, 350 & 500 $$ , respectively.",
"At the distance of Fornax 10 arc seconds $\\simeq $ 1 kpc giving us the potential to resolve many Fornax galaxies.",
"For example, the three biggest galaxies in the cluster are NGC 1365, 1399, and 1380 with optical diameters of 5.5, 3.8, and 2.7 arc minutes, respectively."
],
[
"Comparsion with HeViCS",
"As much of this paper is based on a comparison between Fornax and Virgo, it is therefore worthwhile to examine the difference between the HeFoCS and HeViCS data.",
"The FIR maps of both surveys are created using identical data reduction techniques.",
"However, they differ with respect to depth and spatial coverage of the clusters.",
"First we consider the depth of the surveys.",
"The HeViCS maps consist of 8 scans (4 x 4 orthogonal cross-linking scans), twice as many as the HeFoCS maps, leading to a $\\sim $ $\\sqrt{2}$ reduction of instrumental noise.",
"[1] calculated instrumental and confusion noise, showing that the HeViCS SPIRE bands were effectively confusion noise limited (70 % of the overall noise is from the confusion noise at 250 $$ ).",
"Consequently, when planning the HeFoCS we requested 4 scans as this offered almost confusion limited maps with half the time required for a single HeViCS tile.",
"In order to asses the ratio of the global noise in the HeViCS and HeFoCS maps, we measure the pixel-pixel fluctuations and apply an iterative 3$\\sigma $ clip to remove bright sources.",
"The global noise in the HeViCS and HeFoCS at 250 $$ is thus, 7.5 and 8.9 mJy beam$^{-1}$ , respectively, yielding a ratio between the two of 1.19.",
"This ratio is half as much as one would expect from a simple $\\sqrt{2}$ increase in depth if the maps were purely instrumental noise limited, thus showing that the surveys are reasonably well suited for comparison.",
"Second we consider the coverage of the HeViCS and HeFoCS FIR maps of their respective clusters.",
"This is not a straightforward task, the clusters have very different physical sizes and states of relaxation - Virgo is far more `clumpy' than Fornax.",
"The irregular shape of Virgo lead to the HeViCS FIR maps comprising of 4 tiles ($4\\,^{\\circ }\\,\\times \\,4\\,^{\\circ }$ ) running North to South, whereas the HeFoCS is only a single tile ($4\\,^{\\circ }\\,\\times \\,4\\,^{\\circ }$ ).",
"A possible solution is to use the fraction of the VCC and FCC galaxies that lay inside the boundary of each FIR survey, this is incidentally $\\frac{2}{3}$ for both, showing again that the HeViCS and HeFoCS are well suited for a FIR comparison of the two clusters."
],
[
"Missing FIR sources",
"By using an optical catalogue we run the risk of missing a population of FIR sources not detected in the optical.",
"Conversely we could make our selection in the FIR, but then there is no way of determining which sources are in the cluster.",
"Below, we show that a large population of cluster FIR sources without optical counterparts does not exist.",
"We do this by comparing the number counts of sources in the HeFoCS 250 $$ map with data extracted from the H-ATLAS [35] North Galactic Pole (NGP) field, which has no foreground cluster.",
"By comparing histograms of number counts in these two fields, one `looking' through the Fornax cluster and the other a purely background reference, we can look for evidence of a FIR excess of sources in the cluster.",
"We only use the 250 $$ band to do this, as we impose a selection in our FIR catalogue such that each galaxy must be detected at 250 $$ (see below).",
"The NGP data and data reduction are fully described in Valiante et al.",
"(in preparation).",
"Here we provide a brief description.",
"The data consists of a scan and cross scan, which have been reduced using the same pipeline as the HeFoCS data.",
"The only difference is that the NGP 250 $$ map is gridded onto 5\" pixels, whereas the HeFoCS data are gridded onto 6\" pixels.",
"Using the same iterative 3$\\sigma $ clipping method as described above, the global noise in the NGP map at 250 $$ is 13.0 mJy beam$^{-1}$ , a factor of 1.5 greater than the HeFoCS data in the same band.",
"The NGP covers a $\\sim $ 180 square degree area of sky, from which we have extracted a 20 square degree area in the north east of the map.",
"This selection is used to avoid the Coma cluster (D = 100 Mpc) which is located in the south west.",
"We use the software, SExtractor to measure the flux density of all the sources in each map, taking care to use an identical method.",
"SExtractor `grids up' each map and calculates the noise in each sub-grid.",
"The parameter that controls this is `meshsize', we fix at 100 arcmin$^2$ as this is much greater than the size of any of our foreground galaxies.",
"The detection threshold was set at 1$\\sigma $ and 1.6$\\sigma $ above the local background for the NGP and HeFoCS maps, respectively.",
"These different threshold values are used to ensure comparable sensitivity in each field given the smaller pixel sizes and higher noise per pixel in the NGP field.",
"Another requirement was that the detection size of a source was greater than the SPIRE beam area at 250 $$ (450 arcsec$^2$ ).",
"Figure REF shows the number counts generated from using the above approach.",
"The black and red lines represent the NGP and HeFoCS fields, respectively.",
"The vertical dashed line marks the minimum flux density detectable ($\\sim $ 15 mJy) given the minimum detection area and 1.6 $\\sigma $ noise level in the HeFoCS data.",
"The cyan dashed line is for our Fornax FIR catalogue, as presented in this paper (see below).",
"The black and red lines trace each other very well within the $\\sqrt{N}$ errors below about 1 Jy, brighter than this there is a small excess due to the presence of Fornax cluster galaxies.",
"In conclusion we find no evidence for a significant excess population of FIR sources that are not associated with the optical sources in the FCC.",
"There are too few 250$$ detections in Fornax to create a statistically meaningful luminosity function (about 3 galaxies per bin in Figure REF ), however, interestingly it has a similar flat luminosity function as found by [22] for the Virgo cluster."
],
[
"General approach",
"We have used a semi-automated source measurement program written in IDL, to measure the FIR flux density of each galaxy.",
"This program is fully described and extensively tested by [1].",
"The method is briefly described below.",
"Of the FCC galaxies, 237 fall into the SPIRE maps and 201 fall into both the PACS and SPIRE maps.",
"The optical parameters (position, eccentricity, optical diameter $D_{25}$ and position anglePosition angle and eccentricity are not listed in the FCC but were obtained using the online database Hyperleda, [61].)",
"from the FCC were used to make an initial estimate of the shape and size of the FIR emission.",
"Previous studies [20], [64] as well as the equivalent to this study in the Coma cluster (Fuller et al., in preparation), show that FIR emission is well traced by the optical parameters of late-type galaxies.",
"Whereas early-type galaxies typically show more compact dust emission [70].",
"The optical parameters are only used to make an initial estimate for creating masks.",
"The program then iterates, to create masks and apertures that best match the diameter and ellipticity of the FIR emission.",
"For the following explanation, it may serve the reader to consult Figure REF .",
"The flux measurement process starts by extracting a 200 $\\times $ 200 pixel sub-image from the raw map as shown in Figure REF b.",
"To measure the background of the sub-image, all nearby galaxies including the galaxy being measured are initially masked at 1.5 $\\times $ D$_{25}$ .",
"If the optical extent of the galaxy is such that this sub-image is not large enough to give an accurate background estimation, then the program will increase the size of the sub-image, up to 600$\\times $ 600 pixels for SPIRE and 1200$\\times $ 1200 for PACS.",
"The background estimation has to deal with the near confusion limited SPIRE maps and instrumental noise in the PACS maps.",
"This program was originally written for use in the HeViCS maps where galactic cirrus was also a major problem.",
"In order to remove bright background galaxies and galactic cirrus [1] used a 98% flux clip and then fitted the remaining pixels with a 2D polynomial.",
"The flux clip, removes bright background galaxies by masking out the brightest 2% of pixels, this ensures that the 2D polynomial is only fitting the galactic cirrus.",
"Cirrus is not obviously present in the HeFoCS maps and as such the 98% clip has been retained and then the median pixel value of the masked sub-image taken as the background value.",
"Figure: The SPIRE 250 contour map, plotted over the superCOSMOS r-band image of FCC 117, 135 and 136 for sub-figures (a), (b) and (c) respectively.",
"The beam size is shown in the lower left hand corner.",
"The white ellipses indicate the optical (D 25 D_{25}) extent of each galaxy.",
"(a) a galaxy that by-eye we flagged as a good detection as it is coincident with the FIR contours.",
"(b) this galaxy was removed as it is clearly a bright background source that does not appear in the optical image.",
"(c) the FIR source cannot be uniquely identified, it looks to be comprised of more than one source and as such was removed.We measured total flux, surface brightness, aperture noise (fully described in Section REF ) and signal to noise (S/N) along annuli of increasing radius centred on the galaxy optical centre.",
"The shape of the annuli is based on the galaxy optical parameters convolved with the appropriate point spread function (PSF).",
"We plot the corresponding radial profiles in Figure REF c, e & f, respectively.",
"The FIR diameterAs some galaxies are not resolved D$_{FIR}$ in some cases be defined by the PSF of the $Herschel$ beam and will not be representative of the extend of dust in the galaxy.",
"D$_{FIR}$ is defined where the S/N profile drops below 2.",
"This D$_{FIR}$ is used to replace the 1.5$\\times D_{25}$ used to make the initial mask.",
"The process iterates until the mask and the D$_{FIR}$ value converge.",
"Only then were aperture corrections applied according to [47] and Griffin & North (In preparation).",
"The aperture corrections take into account the encircled energy fraction within the chosen aperture size.",
"The median aperture corrections are 1.00, 1.00, 0.83, 0.90 and 0.92 at 100, 160, 250, 350 & 500 $$ , respectively.",
"If the total S/N value was less than 3, the sub-image was then searched optimally for a point source.",
"After convolving with the relevant PSF, the maximum value within the FWHM of the PSF centred on the optical position was taken as the flux.",
"The noise was calculated according to [55] and [10], which involved plotting a histogram of all the pixels in the PSF-convolved sub-image and fitting a Gaussian function to the negative tail.",
"The FWHM of this Gaussian is then used to estimate the combined instrumental and confusion noise.",
"This has been summed in quadrature with the calibration uncertainty (see below) to obtain a value for the total noise.",
"If the S/N was still less than 3 we consider the object undetected and set an upper limit on the flux equal to 3 times the noise in the PSF-convolved sub-image.",
"This marked the end of the automatic source measurement process.",
"The output is in the form of postscript files for each galaxy, as shown in Figure REF ."
],
[
"Dealing with blending and contamination",
"$Herschel^{\\prime }s$ comparatively large FWHM can lead to unavoidable contamination by FIR background sources, which could be falsely identified as Fornax galaxies.",
"The level of this contamination is estimated in Section REF .",
"We have plotted the 250 $$ map as contours over a superCOSMOS image of each galaxy and its immediate environment.",
"As shown in Figure REF , if a galaxy could not be clearly separated from a nearby or background galaxy we removed it from our catalogue.",
"Figure REF a shows a FIR source that is clearly coincident with a Fornax galaxy.",
"Figure REF b shows a background source that is brighter than the 3$\\sigma $ noise limit and has been registered as a detection by our program.",
"Figure REF c may be a detection, however, we cannot separate it from another apparent detection, so it was also removed.",
"For galaxies that have been eliminated from our final catalogue through this process we set an upper limit on their flux density equal to the 3$\\sigma $ noise from the PSF-convolved map.",
"As in [1] we impose a strict criterion that a galaxy must be detected at 250 $$ as this provides the best combination of sensitivity and resolution (see below).",
"Figure: The HeFoCS fluxes plotted against the Davies et al.",
"(2013) values for the bright galaxy sample.",
"The residual plot below shows the percentage deviation from the fitted line."
],
[
"Total uncertainty estimate",
"The total uncertainty is estimated from the calibration uncertainty, $\\sigma _{cal}$ and aperture uncertainty, $\\sigma _{aper}$ , summed in quadrature.",
"For SPIRE, $\\sigma _{cal}$ is based on single scans of Neptune and on an assumed model of its emission.",
"The final error for each band is estimated to include 4% correlated and 1.5% from random variation in repeated measurements, as well as 4 % due to uncertainty in the beam area.",
"The SPIRE observer's manualhttp://herschel.esac.esa.int/Docs/SPIRE/html/spire_om.html suggests that these should be added together, leading to a SPIRE $\\sigma _{cal}$ of 9.5%.",
"For PACS, $\\sigma _{cal}$ is based on multiple sources with different models of emission.",
"The PACS observer's manualhttp://herschel.esac.esa.int/Docs/PACS/html/pacs_om.html lists the uncorrelated uncertainties as 3 % & 4 % for 100 and 160 $$ , respectively, and the correlated uncertainty is given for point sources as 2.2 %.",
"However, the data used for calculating these uncertainties were reduced and analysed in a different way than the HeViCS and HeFoCS PACS data.",
"Here we use the same value for total error as in [1], i.e.",
"12 %.",
"To calculate the aperture uncertainty ($\\sigma _{aper}$ ) a large number of apertures of a fixed size were placed randomly on each sub-image.",
"We measure the total flux in each aperture, then by applying an iterative 3$\\sigma $ clipping procedure use $\\sigma $ as the uncertainty for that size of aperture.",
"Repeating this for a range of aperture sizes allows us to estimate the aperture uncertainty as a function of size [47].",
"This method takes into account, both confusion noise and instrumental noise.",
"Figure REF d shows such a plot of aperture uncertainty against radial distance for FCC312.",
"[1] tested this method over an entire 4$^{\\circ }$ x 4$^{\\circ }$ tile in the southern region of Virgo and compared it to the results obtained on the sub-images.",
"They found very good agreement between the two, within the typical radii of FIR emission for Virgo galaxies.",
"At larger radii this relationship broke down, this was attributed to large scale structure in the HeViCS maps."
],
[
"Flux verification",
"As a verification of our automated process we have compared our fluxes with the Fornax Bright Galaxy Sample (BGS) [23], as shown in Figure REF , and tabulated the gradients and intercepts in Table REF .",
"[23] matched 10 galaxies with IRAS [15] and 5 with PLANCK [53] sources finding good agreement in both cases.",
"Table REF shows overall that the results are consistent with a gradient of 1 and an intercept of 0."
],
[
"SED fitting",
"We have fitted a modified blackbody to every galaxy detected in at least 3 $Herschel$ bands (22 galaxies) in order to estimate dust mass and temperature.",
"The fit is based on the equation: $S_{\\lambda } = \\frac{\\kappa _{abs} M_{dust} B(\\lambda , T_{dust})}{D^{2}}, \\nonumber $ where $S_\\lambda $ is the flux density , $M_{dust}$ is the dust mass, $T_{dust}$ is the dust temperature, $B(\\lambda , T_{dust})$ is the Planck function, D is the distance($D_{Fornax}$ = 17.2 Mpc) and $\\kappa _{abs}$ is the dust absorption coefficient.",
"The latter follows a power law modified by an emissivity ($\\beta $ ), such that: $\\kappa _{abs} = \\kappa _{abs}(\\lambda _{0}) \\times \\left( \\frac{\\lambda _{0}}{\\lambda }\\right) ^{\\beta } \\nonumber $ We assume that emission at these wavelengths is purely thermal and from dust at a single temperature with a fixed $\\beta = 2$ emissivity.",
"We use $\\kappa _{abs}(350 )$ = 0.192 m$^{2}$ kg$^{-1}$ according to [31].",
"Although this is most likely an overly simplistic analysis, this approach has been used in previous works [21], [23], [70], [1], [74] and shown to fit the data very well in the FIR/sub-mm regime.",
"[5] showed that using a single component modified blackbody returns equivalent results to more complex models such as [32].",
"It is intended that in future papers we will explore two component fits as well as a variable beta emissivity.",
"Derived dust masses and temperatures are given in Table REF and the SED of each galaxy is shown in Figure REF ."
],
[
"Dust mass estimation",
"Fornax has far less FIR detections than Virgo the dust mass can be calculated for only 22 of them through the SED fitting technique described above.",
"We have performed much of the analysis in this paper with this sample of 22 Fornax galaxies.",
"However, to best exploit the FIR data we have also used the 250$$ flux density ($S_{250}$ ) as a proxy for dust mass, this added 9 galaxies to our Fornax FIR sample.",
"Furthermore, this allows us to estimate an upper limit on dust mass for other galaxies not detected at 250$$ .",
"In order to derive a relation between $S_{250}$ and $M_{Dust}$ , we use galaxies from both Virgo and Fornax.",
"Fornax galaxies are all assumed to lie at 17 Mpc.",
"For galaxies in Virgo we take $S_{250}$ and $M_{Dust}$ from [1] – who used the same SED fitting method described above – and scale the flux values to the distance of Fornax.",
"In Figure REF we plot $S_{250}$ against $M_{Dust}$ and fitted a single relation to all galaxies irrespective of which cluster they originate from: $\\log _{10}\\left(\\frac{M_{Dust}}{\\mathrm {M_{\\odot }}}\\right) = 0.789 \\times \\log _{10}\\left(\\frac{S_{250}}{\\mathrm {Jy}} \\right) + 6.486 \\nonumber $ We also fit relations to Virgo and Fornax individually and show them with a blue and a red dashed line, respectively.",
"These have the same slope and incept within $1 \\sigma $ .",
"Furthermore, we find a small range of dust temperatures.",
"Consequently, we use a single relation for both clusters."
],
[
"Stellar masses",
"Only 35 galaxies in the FCC have both a (B-V) colour and K-band flux listed in Hyperleda and we have used this to calculate stellar masses using the prescription of [4]: $\\log _{10}\\left(\\frac{M_{Star}}{\\mathrm {M_{\\odot }}}\\right) = - 0.206 + 0.135(B-V) + \\log _{10} \\left( \\frac{L_{K}}{\\mathrm {L_{\\odot }}} \\right) \\nonumber $ Based on these 35 galaxies we find the following best-fitting linear relation between $m_{BT}$ and stellar mass: $\\log _{10}\\left(\\frac{M_{Star}}{\\mathrm {M_{\\odot }}}\\right) = -0.51\\,m_{BT} + 16.6.",
"\\nonumber $ We use this relation and the $m_{BT}$ value listed in Hyperleda to estimate the stellar mass of all remaining galaxies in the sample."
],
[
"Possible background contamination",
"In this section we try and assess whether our source rejection process has been reasonable given the background source counts.",
"We assume that if extended FIR emission is found coincident with a Fornax galaxy it is reliable, and thus only concern ourselves with the point source population.",
"We assume that the background sources are distributed randomly and uniformly across the sky with no cosmic variance.",
"The number of contaminating sources is estimated using the number counts from the HeFoCS data (as described in Section 2.1) and then calculating the probability of a chance alignment with the 250 $$ SPIRE beam.",
"We limit this analysis to the 250 $$ SPIRE band, as this was the band in which we made our by-eye inspection.",
"It should also be noted that while the SPIRE bands are near confusion noise limited, the PACS bands are limited by instrumental noise.",
"Consequently, PACS fluxes are far less likely contaminated by a background source.",
"The contamination has been calculated within various flux intervals, as shown in Table REF .",
"If done correctly we would expect the number of rejected galaxies to be roughly equivalent to the number of expected contaminating sources within the sum of the total area of apertures used.",
"Table REF clearly shows that we have been over zealous in our rejection of sources in the 20 - 45 mJy bin, however, in the 45 - 100 mJy bin we have not rejected as many contaminating sources as the number counts predict.",
"Overall we accept 17, reject 19 and estimate there are 15 contaminating galaxies at 250 $$ .",
"If we assume Poisson root N errors, then these small numbers are within $3 \\sigma $ .",
"In this section we describe the HeFoCS detection rate in each $Herschel$ band and compare our results to those obtained for galaxies in Virgo by [1].",
"We then investigate the location of FIR detected and undetected galaxies within the cluster.",
"Every HeFoCS galaxy detected at 250 $$ is shown in Figure REF , where the grey ellipse shows the extent and location of the optical counterpart."
],
[
"Detection rates",
"Figure REF shows the distribution of optical magnitudes $m_{bt}$ of all (black) and detected (blue) FCC galaxies.",
"Except for one faint galaxy (discussed separately in Section 4.1.1), no galaxies are detected in the FIR below $m_{bt}$ = 18.2.",
"Therefore, we do not expect that a deeper optical catalogue would increase the number of FIR detections in our current data.",
"Table REF indicates how many galaxies were recovered in each band above a 3$\\sigma $ noise level in the FIR maps.",
"The SPIRE bands have higher detection rates than the PACS bands, and 250 $$ has the highest detection rate of all.",
"This is due to a combination of its sensitivity and the typical shape of the FIR SED.",
"Consequently, we use the 250 $$ band to compare Fornax and Virgo.",
"At 250 $$ we detect 30 of 237 (13 %) FCC galaxies.",
"This is significantly less than in Virgo, where 254 of 750 (34 %) VCC galaxies are detected [1].",
"In order to investigate the source of the lower global detection rates in Fornax in comparison to Virgo, we examine the morphological make up of each cluster and the detection rates therein.",
"We separate the galaxies into 1 of 4 morphological groups; dwarf (dE /dS0), early (E / S0), late (Sa / Sb/ Sc / Sd), and irregular (BCD / Sm / Im / dS).",
"The upper panel in Figure REF shows the fraction of galaxies detected in each morphological group, while the lower panel shows the overall morphological make up of each cluster (tabulated in Table REF ).",
"Dwarf galaxies are the most numerous in both clusters, however, only 4 % and 6 % are recovered at 250 $$ for Virgo and Fornax, respectively.",
"Early, lateFCC 176 was originally classified as Sa by [40], however, we did not detect this galaxy in any $Herschel$ bands.",
"Upon further inspection it has a very red colour, B-V = 0.88 [66], placing it well within the red sequence, it has also been reclassified S0, more latterly by [29], we have adopted this reclassification., and irregular-type galaxies are detected at 21 %, 90 %, and 31 % in Fornax and 34 %, 91 %, and 47 % in Virgo, respectively.",
"The lower panel of Figure REF shows that Fornax has a far higher fraction of dwarf galaxies, with the lowest detection rate, and far less late and irregular-type type galaxies with the highest detection rate.",
"Furthermore Figure REF shows the fraction of early-type galaxies is the same in both clusters, having no effect on the global detection rate.",
"What is remarkable, is that within the errors the two clusters match each other very closely with respect to the fraction of detected galaxies in each morphological group.",
"The above implies that the lower global detection rates in Fornax are tracing the morphological make up of the cluster.",
"Figure: Histograms of stellar mass for 4 morphological types; dwarf (dE /dS0), early (E / S0), late (Sa / Sb/ Sc / Sd), and irregular (BCD / Sm / Im /dS).",
"The black and coloured histograms are for undetected and detected galaxies at 250 , respectively.",
"The vertical dashed lines represent our estimated stellar mass detection limits for the indicated dust-to-stars mass ratio.",
"Note the change in the Y-scale for the dwarf galaxies panel.",
"The adjacent plots show the locations within the cluster of the undetected and detected galaxies, with empty and filled markers respectively.In order to better understand the limits of our data we estimate the limiting dust mass required for a detection at 250 $$ .",
"The lowest detected 250 $$ flux in our FIR catalogue is $\\sim $ 15 mJy.",
"Using the relation calibrated in Section REF , the corresponding limiting dust mass is log($M_{Dust}$ /M$_{\\odot }$ ) = 5.1.",
"If we assume that early and late-type galaxies typically have dust-to-stellar mass ratios of approximately, $\\log (M_{Dust}/M_{Stars}) = -5$ and -3, respectively [17], [70], then we should detect galaxies with stellar masses of log($M_{Stars}$ /M$_{\\odot }) \\ge $ 10.1 and 8.1, respectively.",
"We can see this more clearly in Figure REF .",
"The 4 left-hand panels display the distribution of stellar mass in the 4 morphological groups described above, with black and coloured histograms showing galaxies undetected and detected at 250 $$ , respectively.",
"The dashed lines indicate the stellar mass above which we expect to detect galaxies with a dust-to-stellar mass ratio of $\\log (M_{Dust}/M_{Stars}) = -5$ and $-3$ , respectively.",
"The dwarf galaxies are the most challenging morphological group to detect in the FIR, due to their low stellar masses, thus, requiring a substantially higher dust-to-stellar mass ratio for their detection.",
"6%(11) of dwarf galaxies are detected, whereas we would expect to detect 18%(33) of the dwarf galaxies if they had dust-to-stellar mass ratios of $\\log (M_{Dust}/M_{Stars}) = -3$ similar to a typical late-type galaxy.",
"The righthand panel of Figure REF , shows where these galaxies are projected spatially in the cluster.",
"The FIR detected dwarf galaxies generally appear on the outskirts of the cluster.",
"To quantify this, Table REF lists the average projected cluster centric radius as a fraction of the virial radius ($R_{virial}$ =0.7 Mpc) for FIR detected and undetected galaxies.",
"On average detected dwarf galaxies are found at a 0.84 R$_{virial}$ , whereas undetected are at 0.50 R$_{virial}$ .",
"These detected dwarf galaxies are found on the outskirts of the cluster in a similar position to the transition dwarfs identified in the Virgo cluster by [27].",
"Only 21 % of all early-type galaxies are detected by $Herschel$ at 250 $$ , with some of the extremely dust deficient early types having dust-to-stars ratios of below $\\log (M_{Dust}/M_{Stars}) = -6.6$ .",
"Early-type galaxies appear very centrally concentrated when compared to the dwarf and irregular-type galaxies.",
"However, both detected and undetected galaxies are found at an average projected cluster centric radius of $\\sim 0.5$ R$_{virial}$ .",
"It would appear that cluster centric radius has no perceivable effect on whether or not an early-type galaxy is detected by $Herschel$ .",
"There are nine late-type galaxies in the Fornax cluster and they are all detected except for FCC 299.",
"In order to be detected at 250 $$ , the latter would require a dust-to-stars ratio greater than $\\log (M_{Dust}/M_{Stars}) = -3$ due to its low stellar mass of $\\log (M_{Stars}$ /M$_{\\odot }$ ) = 7.8.",
"The detected galaxies have a mean projected cluster centric radius of $\\sim 0.56$ R$_{virial}$ and no late-type galaxy has a radius less than 0.3 R$_{virial}$ .",
"The majority of irregular-type galaxies would be detected at 250 $$ if they had dust-to-stars ratios of $\\log (M_{Dust}/M_{Stars}) = -3$ .",
"Instead, approximately 31 % of the irregular-type galaxies are detected, preferentially with higher stellar masses.",
"There is no obvious trend to where they are located in the cluster, both detected and undetected galaxies having a mean projected radius of $\\sim 0.6$ R$_{virial}$ .",
"Table: A comparison of galaxies detected and undetected in the SPIRE 250 band.",
"Projected radii are given as a fraction of the Fornax cluster virial radius of 0.7 Mpc .From 185 dwarf galaxies identified in the FCC, only 11 were detected in the 250 $$ band, and only FCC 215 was detected in 3 or more $Herschel$ bands.",
"FCC 215 has a very high dust-to-stars ratio (approximately $\\log (M_{Dust}/M_{Stars}) = -1$ ) and a very faint optical magnitude ($m_{bt} \\sim 19$ ), making it an interesting object worthy of further inspection.",
"FCC 215 has a dust mass of $\\log _{10}(M_{dust}/M_{\\odot }) = 5.2$ and a stellar mass of $\\log _{10}(M_{stars}/M_{\\odot } )= 6.5$ .",
"It is just detected in the 3 SPIRE bands at S/N $\\le $ 5.",
"The SED fit is quite poor with $\\chi ^{2}_{dof=3} = 9.94$ .",
"The SED appears very flat, which may indicate it is a background galaxy with a synchrotron component.",
"However, it is listed in NED as having a velocity of 1964 km s$^{-1}$ , which places it inside the cluster.",
"Its optical colour is very blue, B-R = 0.36, suggesting that the galaxy is undergoing/has undergone an episode of recent star formation.",
"Assuming that this is a bonafide detection, how could it have such a high dust-to-stars ratio?",
"Is it possible for a galaxy to produce this much dust?",
"Using a closed box model of a galaxy, i.e.",
"no inflow or outflow of material, [38] derive; $\\Delta _{max, f} = \\eta p f \\log (1/f)$ , where $\\Delta _{max, f}$ is the maximum mass of dust a galaxy could possess with a gas fraction $f$ , a fraction of metals in the dust $\\eta $ and a stellar yield $p$ .",
"The stellar yield is the fraction of metals produced per unit mass of gas freshly formed in nucleosynthesis.",
"Its value has been estimated to lie between 0.004 and 0.0012 [75].",
"The fraction of metals in the dust $\\eta $ has been estimated by [56] and more latterly by [22] as 0.5.",
"The gas fraction is $f = M_{gas} / (M_{stars} + M_{dust} + M_{gas})$ , so using the equation above we can estimate the gas mass required, for FCC 215 to have a dust-to-stars ratio of $\\log (M_{Dust}/M_{Stars}) = -1.5$ .",
"The gas-to-stars ratio would have to be 1 and thus a gas mass of $\\log _{10}(M_{gas}$ /M$_{\\odot }) = 6.5$ , making it also very gas rich.",
"Currently the only 21cm survey that covers this region of sky is the H i Parks All Sky Survey (HIPASS) [3].",
"HIPASS does not detect FCC 215, yet their estimated rms noise of $\\sim $ 15 mJy beam$^{-1}$ approximately corresponds to an H i gas mass $\\log _{10}(M_{HI}$ /M$_{\\odot }) = 8$ at the distance of Fornax, meaning that HIPASS would be unable to detect FCC 215 even if all the gas content was locked up in H i.",
"The HeFoCS has secured time to map the Fornax cluster, using the Australia Telescope Compact Array.",
"The estimated survey detection limit is M$_{HI} \\simeq 10^{7}$ M$_{\\odot }$ at the distance of Fornax, very close to our predicted upper estimate of the gas mass of FCC 215.",
"For the following analysis we split the sample into early and late-type galaxies and initially consider only the 22 galaxies detected in at least 3 $Herschel$ bands.",
"`Early' was classified as anything earlier than Sa and `late' as anything later than (and including) Sa.",
"The SED of all galaxies was fitted with a single temperature modified blackbody with $\\beta =2$ .",
"Only two galaxies, FCC 215 (discussed above) and FCC 306, were poorly fitted using this emissivity, with $\\chi ^{2}_{dof = 3} =$ 9.94 and 18.65.",
"The average for the entire sample was, $<\\chi ^{2}_{dof = 3}> = 2.92$ .",
"If FCC 215 and 306 are removed, then the average for the sample falls to $<\\chi ^{2}_{dof = 3}> = 1.78$ .",
"In Table REF we include all galaxies with measured dust mass and temperature.",
"Figure REF shows the SED fits for each galaxy.",
"Detected late-type galaxies have dust masses ranging from $\\log _{10}(M_{dust}$ /M$_{\\odot }) = 5.5$ to $8.2$ and temperatures of 11.2 to 23.7 K, with mean values $\\log _{10}(M_{dust}$ /M$_{\\odot }) = 6.5$ and 17.5 K. By contrast, detected early types have a narrower range of dust masses of $\\log _{10}(M_{dust}$ /M$_{\\odot }) = 5.4$ to $6.6$ and temperatures of 14.9 to 25.8 K, with mean values $\\log _{10}(M_{dust}$ /M$_{\\odot }) = 5.8$ and 19.3 K. Detected Fornax galaxies have mean dust-to-stellar mass ratios of $\\log _{10}(M_{dust}/M_{stars})$ = -3.87 and -2.93, for early and late-types, respectively.",
"As expected from our previous results for Virgo, late-types have a richer and cooler dust reservoir, and early-types have a relatively depleted and warmer ISM.",
"In Table REF we use the Kolmogorov-Smirnov two sample test (KS) to make a more quantitative comparison between Virgo and Fornax's early and late-type galaxy populations with respect to dust mass, dust-to-stars ratio, and dust temperature.",
"Here we use 140 of the [1] galaxies that had SEDs modelled identically to our sample using a single temperature component with a fixed $\\beta = 2$ emissivity.",
"For Virgo we use stellar masses calculated using H band magnitudes and SDSS g-r colours from [22].",
"Virgo, like Fornax, has early types that have lower dust masses and higher temperatures than its late types.",
"However, a KS test shows that for a given morphological type the FIR properties of galaxies in Fornax and Virgo are statistically identical (with the caveat that we are only sampling the massive galaxies, $\\log _{10}(M_{star}$ /M$_{\\odot }) \\ge 8.2$ ).",
"The above results suggest that the different cluster environments have had very little effect on the dust properties of early or late-type galaxies.",
"[1] compared the Virgo cluster to the Herschel Reference Survey [8], [19], [70].",
"The HRS is a volume limited (15 $\\le $ D $\\le $ 25), K band (K $\\ge 8.7$ ) selected sample.",
"It covers a range of environments from the field to the core of the Virgo cluster, making it an ideal comparison sample.",
"[1] showed that early-type galaxies in the Virgo cluster and HRS field show very similar dust properties.",
"However, late-type galaxies typically have larger dust masses in the field.",
"[1] concluded that the difference in dust mass between field and cluster late-type galaxies was due to dust removal in the cluster environment.",
"The implication of this result as well as the results presented in this paper, is that early-type galaxies appear identical in their FIR properties irrespective of what environment they originated from.",
"Furthermore, the larger dust reservoirs of late-type galaxies in the field and the lack of difference in FIR properties between Fornax and Virgo, suggests that this change in dust mass likely occurred before they entered the cluster environment.",
"It is worth noting that `global' environment on its own may not be the best tracer of the action of physical processes.",
"A quantity more sensitive to direct interaction with the cluster environment is the H i-deficiency.",
"[19] compare the FIR properties of galaxies, separated in both H i-deficiency and global environment.",
"They found an $\\sim $ $8 \\sigma $ difference in $\\log (M_{Dust}/M_{Stars}$ ) when comparing H i-normal and H i-deficient galaxies, whereas only a $\\sim $ $3 \\sigma $ difference is found between samples separated based on the environment (i.e.",
"field and cluster members)"
],
[
"Orgin of dust in galaxies",
"In order to extend our analysis of dust and stellar mass to lower limits, and to study how the dust-to-stars ratio changes with lower dust and stellar masses (Figure REF ), an additional 9 galaxies were included in the analysis.",
"These galaxies had insufficient SED data to be fitted by a modified blackbody and so the 250$$ flux density was used as a proxy for dust mass (described in Section REF ).",
"The diagonal dash line in Figure REF indicates the minimum dust-to-stellar mass detected, given our previous estimate of a minimum detectable dust mass of $\\log _{10}(M_{dust}$ /M$_{\\odot })$ = 5.1 (Section REF ).",
"The same morphological categories are used - `early' was classified as anything earlier than Sa and `late' as anything later than, and including, Sa.",
"Figure REF shows early and late-type galaxies designated by red and blue markers, respectively, from both clusters.",
"Fornax galaxies are indicated by a marker set onto a black square.",
"We have measured the correlation between $M_{dust}$ and $M_{star}$ using the Pearson correlation coefficient (PCC).",
"Late-type galaxies have a PCC of 0.84, early-type galaxies have a PCC of 0.43.",
"The correlation between dust and stellar mass in late-type galaxies most likely finds its origins in the mass-metallicity relation [52], [72], [51], [46].",
"These authors have shown that gas phase metallicity correlates with stellar mass and so we might also expect this to be true for the metals in the dust.",
"Early-type galaxies in Fornax and Virgo have a very large range of dust-to-stellar mass ratios, $-1.3 \\ge \\log _{10}(M_{star}/M_{dust}) \\ge -6.2$ , and the weak correlation of dust to stellar mass could be due to the imposed limiting dust mass, artificially creating a correlation, as shown in Figure REF .",
"However, the PCC for early-type galaxies is far lower than we found for late-type galaxies, implying, that stellar mass is far less if at all correlated with dust mass in early-type galaxies.",
"A clue about origin of these two different correlations may lie in the distribution of the dust within early and late-type galaxies.",
"We have calculated the ratio of the FIR to optical size for Fornax cluster galaxies, where this ratio is defined as the FIR diameter of emission $D_{FIR}$ as defined in Section REF divided by the optical diameter $D_{25}$ .",
"We will use the FIR diameter as measured at 250 $$ , and thus we will for the rest of this section refer to $D_{FIR}$ as $D_{250}$ .",
"Only 1 of 7 early and 2 of 13 late types have FIR emission that is smaller than the FWHM of the $Herschel$ 250 $$ beam, and are thus measured as point sources with a $D_{250}$ equal to the 250 $$ beam size.",
"As the majority have $D_{250}$ greater than the PSF FWHM, we can use them to measure the distribution of the dust in comparison to the stars.",
"The FIR/optical size ratio is 0.464 and 0.903, for early and late-type galaxies, respectively.",
"In order to further test the effect of the $Herschel$ beam we restricted the sample to galaxies with an optical diameter greater than 3 times the 250 $$ beam FWHM.",
"This results in a FIR/optical size ratio of 0.305 and 0.917 for early and late-type galaxies, respectively, thus showing that the beam size has a limited effect on the overall result.",
"This shows that dust in early-type galaxies is very centrally concentrated in comparison to late-type galaxies.",
"This has been demonstrated previously for early-type galaxies by [70] and [30].",
"[20] showed that H i-deficient galaxies in the cluster environment also had smaller FIR/optical size ratios, suggesting that dust had been stripped from these galaxies, this would affect the late-type galaxies far more than early-type galaxies, as the above shows that dust in late-type galaxies is held less deeply in their potential wells.",
"[25] and [45] measure the extent of molecular hydrogen in early-type and late-type galaxies.",
"They find the size ratio of molecular gas to optical radius as $\\sim 0.25$ in both cases.",
"In the case of late-types, the spatial extent of molecular gas is much less than of the dust, whereas early-type galaxies have molecular gas and dust that appear spatially coincident.",
"This suggests that the origin of dust in early-types may be the same as that of molecular gas (see below).",
"Whereas dust in late-types is coincident with the stellar population, and as shown above, dust mass is regulated by stellar mass (i.e.",
"mass-metallicity relation), suggesting an internal origin.",
"Dust in early-type galaxies has two possible origins, either internally produced in the atmospheres of evolved stars [78] and supernovae remnants [58], or externally obtained from mergers with other galaxies.",
"The strongest prediction for dust created internally is that the mass of dust and stars should be spatially correlated.",
"Figure REF shows that this is clearly not the case for early-type galaxies.",
"The stellar population must produce dust, but [13] show that it is destroyed on a short timescale of <50 Myrs.",
"They argue that this is far shorter than the dust-transfer timescale, and thus dust created in outer regions of a galaxy is effectively destroyed “on-the-spot”.",
"However, the dust destruction timescale can be greatly extended if the dust is embedded in a cloud of molecular hydrogen, leading to lifetimes of a few 100 Myrs [48].",
"This indicates that dust created internally cannot be the main source of dust in early-type galaxies.",
"If the dominant source of dust is not internal, [70] argue that it may have an external origin such as mergers with dust rich galaxies.",
"Mergers of different dust masses at different times would explain the large range of dust-to-stars ratios seen in early-type galaxies as well as the $\\sim 75 \\%$ of systems which we do not detect with $Herschel$ .",
"However, as shown above, the FIR properties of early-type galaxies do not change between Virgo and Fornax (Table REF ) or the HRS field [1], suggesting that the flow of dust into and out of these systems must be invariant with environment.",
"Since [13] show that the destruction time in early-type galaxies is determined by thermal sputtering, and thus is largely independent of the environment, our findings would imply that the merger rate is roughly the same in all three environments.",
"This is at odds with the idea that the merger rate depends on environment.",
"For example, [57] shows that mergers are far less common in clusters than in groups or in the field - thus there is a dilemma.",
"The mystery deepens if we compare our FIR results for early-types to the molecular gas component of the ISM.",
"[26] show that the detection rate of the molecular ISM and the molecular gas-to-stars ratios for early-type galaxies are invariant to environment, mirroring the FIR results presented in this paper.",
"However, they discovered that the gas kinematics inside and outside of clusters is different.",
"They found one third of galaxies outside of clusters had gas kinematically misaligned to their stars, supporting an external origin.",
"Interestingly, this was not see in early-types inside the cluster."
],
[
"Summary",
"We have undertaken the deepest FIR survey of the Fornax cluster using the $Herschel$ Space Observatory.",
"Our survey covers over 16 deg$^2$ in 5 bands and extends to the virial radius of the cluster, including 237 of the 340 FCC galaxies.",
"We have used the optical positions and parameters of these FCC galaxies to fit appropriate apertures to measure FIR emission.",
"We have detected 30 of 237 (13 %) cluster galaxies in the SPIRE 250 $$ band, a significantly lower detection rate than in the Virgo cluster ; seeauld12.",
"In order to better understand the global detection rate we separated Fornax and Virgo galaxies into 4 morphological categories: dwarf (dE /dS0), early (E / S0), late (Sa / Sb/ Sc / Sd), and irregular (BCD / Sm / Im / dS).",
"We examined the detection rate for each morphological group in the 250 $$ band as it has the highest detection rate of all the $Herschel$ bands.",
"In Fornax we detect 6%, 21%, 90%, and 31% of dwarf, early, late, and irregular, respectively.",
"These results agrees with the fraction of detected galaxies in each morphological category in the Virgo cluster, indicating that the lower global detection rate in Fornax is due to its lower fraction of late-type galaxies.",
"For galaxies detected in at least 3 bands we fit a modified blackbody with a fixed beta emissivity index of 2, giving dust masses and temperatures for 22 Fornax galaxies.",
"Fornax's early-type galaxies show lower dust masses and hotter temperatures than late-type galaxies.",
"When comparing early-type galaxies from the Fornax cluster to their counter-parts in the Virgo cluster, their FIR properties are statistically identical.",
"The same is true for the late-type galaxies.",
"This may suggest that the effect of the cluster is more subtle than previously thought and that the evolution of the ISM components has mostly taken place before the cluster was assembled.",
"We observe dust mass to be well correlated to stellar mass for late-type galaxies.",
"We suggest that this correlation has its origins in the mass-metallicity relation [52], [72], [51], [46], as the ratio between the mass of metals in the dust and the gas has been found to be 0.5 [56], [22].",
"It therefore follows that any correlation with gas phase metallicity should also be observed between stellar and dust mass.",
"We find early-type galaxies to have a very large range of dust-to-stars ratios, $-1.3 \\ge \\log _{10}(M_{star}/M_{dust}) \\ge -6.2$ .",
"We argue that this supports a scenario where the dust in early-type galaxies is from an external origin, as has been previously suggested by other authors [70].",
"As FIR properties are statistically identical between environments, therefore so must the balance between dust input/creation and removal/destruction.",
"However, this conclusion is perplexing as mergers are thought to be far less common in clusters when compared to groups or the field [57], and dust destruction is largely regulated internally [13], thus invariant with respect to environment."
],
[
"Acknowledgements",
"SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including University of Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, University of Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, University of Sussex (UK); and Caltech, JPL, NHSC, University of Colorado (USA).",
"This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA).",
"HIPE is a joint development (are joint developments) by the Herschel Science Ground Segment Consortium, consisting of ESA, the NASA Herschel Science Center, and the HIFI, PACS and SPIRE consortia.",
"The research leading to these results has received funding from the European Community’s Seventh Framework Programme (/FP7/2007–2013/) under grant agreement No.",
"229517.",
"This research has made use of data obtained from the SuperCOSMOS Science Archive, prepared and hosted by the Wide Field Astronomy Unit, Institute for Astronomy, University of Edinburgh, which is funded by the UK Science and Technology Facilities Council.",
"We acknowledge the usage of the HyperLeda database (http://leda.univ-lyon1.fr).",
"IDL is a postdoctoral researcher of the FWO-Vlaanderen (Belgium).",
"We gratefully acknowledge the contribution of L. Cortese to this work.",
"The Parkes telescope is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO."
]
] | 1403.0589 |
[
[
"Exciton fine structure and spin decoherence in monolayers of transition\n metal dichalcogenides"
],
[
"Abstract We study the neutral exciton energy spectrum fine structure and its spin dephasing in transition metal dichalcogenides such as MoS$_2$.",
"The interaction of the mechanical exciton with its macroscopic longitudinal electric field is taken into account.",
"The splitting between the longitudinal and transverse excitons is calculated by means of the both electrodynamical approach and $\\mathbf k \\cdot \\mathbf p$ perturbation theory.",
"This long-range exciton exchange interaction can induce valley polarization decay.",
"The estimated exciton spin dephasing time is in the picosecond range, in agreement with available experimental data."
],
[
"Exciton fine structure and spin decoherence in monolayers of transition metal dichalcogenides M.M.",
"Glazov []glazov@coherent.ioffe.ru Ioffe Physical-Technical Institute of the RAS, 194021 St. Petersburg, Russia T. Amand X. Marie D. Lagarde L. Bouet B. Urbaszek Université de Toulouse, INSA-CNRS-UPS, LPCNO, 135 Av.",
"de Rangueil, 31077 Toulouse, France We study the neutral exciton energy spectrum fine structure and its spin dephasing in transition metal dichalcogenides such as MoS$_2$ .",
"The interaction of the mechanical exciton with its macroscopic longitudinal electric field is taken into account.",
"The splitting between the longitudinal and transverse excitons is calculated by means of the both electrodynamical approach and $\\mathbf {k} \\cdot \\mathbf {p}$ perturbation theory.",
"This long-range exciton exchange interaction can induce valley polarization decay.",
"The estimated exciton spin dephasing time is in the picosecond range, in agreement with available experimental data.",
"71.35.-y,71.70.Gm,72.25.Rb,78.66.Li Introduction.",
"Monolayers (MLs) of transition metal dichalcogenides, in particular, MoS$_2$ form a class of novel two-dimensional materials with interesting electronic and optical properties.",
"The direct band gap in these systems is realized at the edges of the Brillouin zone at points $\\mathbf {K}_+$ and $\\mathbf {K}_-$ .",
"[1] Strikingly, each of the valleys can be excited by the radiation of given helicity only.",
"[2], [3], [4] Recent experiments have indeed revealed substantial optical orientation in ML MoS$_2$ related to selective excitation of the valleys by circularly polarized light.",
"[5], [6], [7], [8] Strong spin-orbit coupling in this material lifts the spin degeneracy of electron and hole states even at $\\mathbf {K}_+$ and $\\mathbf {K}_-$ points of the Brillouin zone resulting in relatively slow spin relaxation of individual charge carriers, which requires their intervalley transfer.",
"[9], [10], [11], [12], [13] However recent time-resolved measurements revealed surprisingly short, in the picosecond range, transfer times between valleys.",
"[14], [15] This could be due to excitonic effects which are strong in transition metal dichalcogenides[16], [17] Although individual carrier spin flips are energetically forbidden, spin relaxation of electron-hole pairs can be fast enough owing to the exchange interaction between an electron and a hole forming an exciton,[18], [19] in close analogy to exciton dephasing in quantum wells.",
"[20] Here we study the energy spectrum fine structure of bright excitons in MoS$_2$ and similar compounds.",
"We use both the electrodynamical approach where the interaction of the mechanical exciton (electron-hole pair bound by the Coulomb interaction) with the macroscopic longitudinal electric field is taken into account[21], [22] and $\\mathbf {k} \\cdot \\mathbf {p}$ perturbation theory.",
"We obtain the radiative lifetime of excitons as well as the splitting between the longitudinal and transverse modes.",
"We compare the developed theory based on long-range exchange interaction (in contrast to calculations in Ref.",
"Yu:2014uq based on short range interaction) with experimental results on spin decoherence in MoS$_2$ MLs[7], [14], [15] and demonstrate an order of magnitude agreement between the experiment and the theory.",
"Model.",
"The point symmetry group of MoS$_2$ -like dichalcogenide MLs is $D_{3h}$ .",
"Since the direct band gap is realized at the edges of hexagonal Brillouin zone, as shown in the inset to Fig.",
"REF , the symmetry of individual valley $\\mathbf {K}_\\pm $ is lower and is described by the $C_{3h}$ point group.",
"The schematic band structure of ML MoS$_2$ is presented in Fig.",
"REF (a), where the bands are labelled according to the irreducible representations of $C_{3h}$ group in notations of Ref. koster63.",
"The analysis shows that all spinor representations of $C_{3h}$ are one-dimensional, therefore, in each valley all the states are non-degenerate.",
"The Kramers degeneracy is recovered taking into account that the time reversal couples $\\mathbf {K}_+$ and $\\mathbf {K}_-$ valleys.",
"The spin-orbit splitting between the valence subbands in a given valley ($\\Gamma _{10}$ and $\\Gamma _{12}$ at $\\mathbf {K}_+$ point and $\\Gamma _9$ and $\\Gamma _{11}$ at $\\mathbf {K}_-$ point) is on the order of 100 meV.",
"[1] The conduction band in each valley is also split into two subbands $\\Gamma _7$ and $\\Gamma _8$ with the splitting being about several meV.",
"[24], [25], [26] Within the $\\mathbf {k} \\cdot \\mathbf {p}$ model we introduce basic Bloch functions $U_i^\\tau (\\mathbf {r})$ , which describe electron states at $\\mathbf {K}_+$ ($\\tau = 1$ ) or $\\mathbf {K}_-$ ($\\tau =-1$ ) point of the Brillouin zone transforming according to the irreducible representation $\\Gamma _i$ of $C_{3h}$ point group.",
"In the minimal approximation where contributions from distant bands are ignored, the Bloch amplitudes $U^\\tau _{7}$ and $U^\\tau _{8}$ of the conduction band states can be recast as products of spinors $\\chi _{s_z}$ corresponding to the spin $z$ component, $s_z = \\pm 1/2$ , and orbital Bloch amplitudes $U^\\tau _{1}$ .",
"Similarly, the valence band states in $\\mathbf {K}_+$ valley, $U^{+1}_{{10}}$ and $U^{+1}_{{12}}$ are the products of $\\chi _{s_z}$ and orbital function $U^{+1}_{3}$ , while the valence band states in $\\mathbf {K}_-$ valley, $U^{-1}_{11}$ and $U^{+1}_{{9}}$ originate from the orbital Bloch amplitude $U^{-1}_{2}$ .",
"Such a four-band $\\mathbf {k} \\cdot \\mathbf {p}$ model is described by 4 parameters: three energy gaps, namely, the band gap $E_g$ , the spin splittings in the conduction and valence bands $\\lambda _c$ , $\\lambda _v$ , respectively, and interband momentum matrix element $p_{\\rm cv} = \\langle U^{+1}_1 | (p_x + \\mathrm {i} p_y)/\\sqrt{2}| U^{+1}_3\\rangle = \\langle U^{-1}_1 | (p_x - \\mathrm {i} p_y)/\\sqrt{2}| U^{-1}_2\\rangle $ .",
"The latter definition is in agreement with remarkable optical selection rules:[2], [3], [4] At a normal incidence of radiation the optical transitions in $\\sigma ^+$ polarization take place in $\\mathbf {K}_+$ valley only between $\\Gamma _{10}$ and $\\Gamma _7$ or $\\Gamma _{12}$ and $\\Gamma _8$ states, while the transitions in $\\sigma ^-$ polarization take place in $\\mathbf {K}_-$ valley and involve either $\\Gamma _{9}$ and $\\Gamma _8$ or $\\Gamma _{11}$ and $\\Gamma _7$ states.",
"Each conduction subband is mixed by $\\mathbf {k} \\cdot \\mathbf {p}$ interaction with the only valence subband having the same spin component.",
"As an example we present below the wavefunctions obtained in the first order of $\\mathbf {k} \\cdot \\mathbf {p}$ interaction for the bottom conduction subband [cf. Ref.",
"birpikuseng]: $\\psi _{c, \\mathbf {k}}^{+1}(\\mathbf {r}) & = \\mathrm {e}^{\\mathrm {i} \\mathbf {k} \\mathbf {r}} \\left[U_{7,c}^{+1}(\\mathbf {r}) + \\frac{\\hbar ^2 p_{\\rm cv}^* k_{+} }{m_0E_g} U_{10,v}^{+1}(\\mathbf {r}) \\right], \\\\\\psi _{c, \\mathbf {k}}^{-1}(\\mathbf {r}) & = \\mathrm {e}^{\\mathrm {i} \\mathbf {k} \\mathbf {r}} \\left[U_{8,c}^{-1}(\\mathbf {r}) + \\frac{\\hbar ^2 p_{\\rm cv}^* k_{-} }{m_0E_g} U_{9,v}^{-1}(\\mathbf {r}) \\right],$ and for the topmost valence subband $\\psi _{v,\\mathbf {k}}^{+1}(\\mathbf {r})& = \\mathrm {e}^{\\mathrm {i} \\mathbf {k} \\mathbf {r}} \\left[U_{10,v}^{+1}(\\mathbf {r}) - \\frac{\\hbar ^2 p_{\\rm cv} k_{-}}{m_0E_g} U_{7,c}^{+1}(\\mathbf {r}) \\right],\\\\\\psi _{v,\\mathbf {k}}^{-1}(\\mathbf {r}) & = \\mathrm {e}^{\\mathrm {i} \\mathbf {k} \\mathbf {r}} \\left[U_{9,v}^{-1}(\\mathbf {r}) - \\frac{\\hbar ^2 p_{\\rm cv} k_{+}}{m_0E_g} U_{8,c}^{-1}(\\mathbf {r}) \\right].$ Here $\\mathbf {k}$ is the wavevector reckoned from the $\\mathbf {K}_+$ or $\\mathbf {K}_-$ point, $k_{\\pm } = (k_x \\pm \\mathrm {i} k_y)/\\sqrt{2}$ , asterisk denotes complex conjugate.",
"The wavefunctions are taken in the electron representation and the factors $\\exp {(\\mathrm {i} \\mathbf {K}_\\pm \\mathbf {r})}$ are included in the definitions of Bloch amplitudes.",
"Figure: (a) Schematic illustration of MoS 2 _2 band structure.",
"The bands are labelled by the corresponding irreducible representations with arrows in parentheses demonstrating electron spin orientation.",
"Solid and dashed arrows show the transitions active at the normal incidence in σ + \\sigma ^+ and σ - \\sigma ^- polarization, respectively.",
"An inset sketches the Brillouin zone.",
"The order of conduction band states is shown in accordance with Ref. Liu:2013if.",
"(b) Optical selection rules of the two bright A-exciton states with the small center of mass wavevector 𝐊\\mathbf {K} and their long-range Coulomb exchange coupling, Eq. ().",
"The ⇑,⇓\\Uparrow ,\\Downarrow symbols represent the exciton pseudospin in the reducible representation Γ 2 +Γ 3 \\Gamma _2+\\Gamma _3.Optical excitation gives rise to the electron-hole pairs bound into excitons by the Coulomb interaction.",
"Excitonic states transform according to the representations $\\Gamma _c \\times \\Gamma _v^*$ , where $\\Gamma _c$ is the representation of the conduction band state and $\\Gamma _v$ is the representation of the valence band state, and in $C_{3h}$ group the time reversed representation $\\mathcal {K}\\Gamma _v=\\Gamma _v^*$ .",
"As a result, at normal incidence the optically active states are given by [see Fig.",
"REF (b)] $\\Gamma _7 \\times \\Gamma _{10}^* = \\Gamma _8 \\times \\Gamma _{12}^* = \\Gamma _2\\quad \\mbox{active in }\\sigma ^+,\\\\\\Gamma _8 \\times \\Gamma _{9}^* = \\Gamma _7 \\times \\Gamma _{11}^* = \\Gamma _3\\quad \\mbox{active in }\\sigma ^-.$ The aim of the paper is to study the fine structure of bright excitonic states and its consequences on the valley dynamics.",
"Therefore, we do not address here the calculation of the “mechanical” exciton (the direct Coulomb problem),[16], [27] we assume that the relative electron-hole motion can be described by an envelope function $\\varphi (\\mathbf {\\rho })$ .The electrodynamical treatment holds also for the case of strongly bound excitons with the replacement of $e^2 |p_{\\rm cv}|^2|\\varphi (0)|^2/(m_0 \\hbar \\omega _0)$ in Eq.",
"(REF ) by exciton oscillator strength per unit area $f$ .",
"The bright exciton fine structure can be calculated either within the $\\mathbf {k} \\cdot \\mathbf {p}$ perturbation theory taking into account the long-range exchange interaction or by electrodynamical approach where the interaction of the mechanical exciton with generated electromagnetic field is taken into account.",
"We start with the latter approach and then demonstrate its equivalence with the $\\mathbf {k} \\cdot \\mathbf {p}$ calculation.",
"In the electrodynamic treatment the exciton frequencies can be found from the poles of the reflection coefficient of the two-dimensional structure.",
"[28] For simplicity we consider a ML of MoS$_2$ situated in $(xy)$ plane surrounded by dielectric media with high-frequency dielectric constant $\\varkappa _b$ , the contrast of background dielectric constants is disregarded.In fact, $\\varkappa _b$ should also include the contributions of transitions spectrally higher than the A-exciton in the MoS$_2$ layer.",
"The allowance for the dielectric constant as well as for the substrate is straightforward following Ref. ivchenko05a.",
"It does not lead to substantial modifications of the results.",
"The geometry is illustrated in Fig.",
"REF .",
"We solve Maxwell equations for electromagnetic field taking into account the the excitonic contribution to the dielectric polarization, which in the vicinity of the A-exciton resonance reads $\\mathbf {P}(z) = \\frac{ \\delta (z) |\\varphi (0)|^2 \\mathbf {E}(z)}{\\omega _0 - \\omega + \\mathrm {i} \\Gamma } \\frac{e^{2}|p_{\\rm cv}|^2}{\\hbar \\omega _0^2m_0^2}.$ Here $\\omega $ is the incident radiation frequency, $\\omega _0$ is the exciton resonance frequency determined by the band gap and binding energy (its evaluation is beyond the scope of present paper), $\\Gamma $ is its nonradiative damping.",
"Factor $\\delta (z)$ ensures that the dipole moment is induced only in the ML of MoS$_2$ whose width is negligible compared with the radiation wavelength.",
"At a normal incidence of radiation the amplitude reflection coefficient of a ML has a standard form $r(\\omega ) = {\\mathrm {i} \\Gamma _0}/[{\\omega _0 - \\omega - \\mathrm {i} (\\Gamma _0+ \\Gamma )}]$ ,[29] where $\\Gamma _0 = \\frac{2\\pi q e^2 |p_{\\rm cv}|^2}{\\hbar \\varkappa _b \\omega _0^2m_0^2} {|\\varphi (0)|^2},$ is the radiative decay rate of an exciton in the MoS$_2$ ML, $q= \\sqrt{\\varkappa _b} \\omega /c$ is the wavevector of radiation.",
"The parameters of the pole in the reflectivity describe the eigenenergy and decay rate of the exciton with allowance for the light-matter interaction.",
"In agreement with symmetry, the reflection coefficient at a normal incidence is polarization independent.",
"Figure: Schematic illustration the system geometry with ss (TE mode) and PP (TM mode) polarized incident light.Under oblique incidence in the $(xz)$ -plane the solution of Maxwell equation yields two eigenmodes of electromagnetic field: $s$ -polarized (TE-polarized) wave with $\\mathbf {E}\\parallel y$ (perpendicular to the light incidence plane) and $p$ -polarized (TM-polarized) wave with $\\mathbf {E}$ in the incidence plane, see Fig.",
"REF .",
"The reflection coefficients in a given polarization $\\alpha =s$ or $p$ are given by the pole contributions with the modified parameters $r_\\alpha (\\omega ) = \\frac{\\mathrm {i} \\Gamma _{0\\alpha }}{\\omega _{0\\alpha } - \\omega - \\mathrm {i} (\\Gamma _{0\\alpha }+ \\Gamma )},$ where $\\Gamma _{0s} = \\frac{q}{q_z} \\Gamma _0, \\quad \\Gamma _{0p} = \\frac{q_z}{q} \\Gamma _0,$ $q_z = (q^2 - q_\\parallel ^2)^{1/2}$ is the $z$ component of the light wavevector, $q_\\parallel $ is its in-plane component, and $\\omega _{0\\alpha } \\equiv \\omega _0(q_\\parallel )= \\omega _0 + \\hbar q_\\parallel ^2/(2M)$ is the mechanical exciton frequency, $M$ is the exciton effective mass.Note, that at $q_\\parallel \\ne 0$ the transitions $\\Gamma _{10}\\rightarrow \\Gamma _{8}$ and $\\Gamma _{9} \\rightarrow \\Gamma _{7}$ active in $z$ polarization become symmetry-allowed (but requiring a spin flip).",
"It gives rise to an additional pole in $r_p(\\omega )$ at the frequency $\\omega _0^{\\prime } \\approx \\omega _0 + \\lambda _c/\\hbar $ and can be evaluated following Ref. ivchenko05a.",
"The light-matter interaction results in the radiative decay of the excitons with the wavevectors inside the light cone, $q_\\parallel \\leqslant \\sqrt{\\varkappa _b} \\omega /c$ .",
"For the excitons outside the light cone, $q_z$ becomes imaginary and corresponding exciton-induced electromagnetic field decays exponentially with the distance from the ML.",
"Therefore, exciton interaction with the field results in renormalization of its frequency rather than its decay rate.",
"[30], [29], [31] Formally, it corresponds to imaginary $\\Gamma _{0\\alpha }$ in Eqs.",
"(REF ).",
"Introducing the notation $\\mathbf {K} = \\mathbf {q}_{\\parallel }$ for the center of mass wavevector of an exciton, we obtain from the poles of reflection coefficients the splitting between the longitudinal ($\\mathbf {P} \\parallel \\mathbf {K}$ ) and transverse ($\\mathbf {P} \\perp \\mathbf {K}$ ) exciton states: $\\Delta E = \\hbar \\Gamma _0 \\frac{K^2}{q \\sqrt{K^2 - q^2}} \\approx \\hbar \\Gamma _0 \\frac{K}{q},$ where the approximate equation holds for $K\\gg q$ and one can replace $\\omega $ by $\\omega _0(K)$ or even by $\\omega _0(0)$ in the definition of $q$ .",
"For excitons outside the light cone and arbitrary in-plane direction of $\\mathbf {K}$ one can present an effective Hamiltonian describing their fine structure in the basis of states $\\Gamma _2$ and $\\Gamma _3$ [see Eqs.",
"(REF )] as $\\mathcal {H}_{X} (\\mathbf {K})=\\begin{pmatrix}0 & \\alpha (K_x - \\mathrm {i} K_y)^2 \\\\\\alpha (K_x+ \\mathrm {i} K_y)^2 & 0\\end{pmatrix} = \\frac{\\hbar }{2} \\left( \\mathbf {\\Omega }_{\\mathbf {K}} \\cdot \\mathbf {\\sigma }\\right).$ Here $\\alpha = \\hbar \\Gamma _0/(2 K q)$ , $\\mathbf {\\sigma }= (\\sigma _x,\\sigma _y)$ are two pseudospin Pauli matrices and the effective spin precession frequency vector has the following components $\\hbar \\Omega _{x} = \\Delta E \\cos {2\\vartheta }$ and $\\hbar \\Omega _{y} = \\Delta E \\sin {2\\vartheta }$ , where $\\vartheta $ is the angle between $\\mathbf {K}$ and the in-plane axis $x$ .",
"Now we demonstrate that the treatment based on electrodynamics presented above is equivalent to the $\\mathbf {k}\\cdot \\mathbf {p}$ calculation of the long-range exchange interaction.",
"Specifically, we consider the exchange interaction between two electrons $\\psi _m$ and $\\psi _n$ occupying $\\Gamma _7$ band in $\\mathbf {K}_+$ valley and $\\Gamma _9$ electron in $\\mathbf {K}_-$ valley.",
"The wavefunctions of these states are given by Eqs.",
"(REF ) and () with the wavevectors $\\mathbf {k}_1$ and $\\mathbf {k}_2$ , respectively.",
"The final states for the pair are the conduction band $\\Gamma _8$ state in $\\mathbf {K}_-$ valley, $\\psi _{m^{\\prime }}$ , and valence band $\\Gamma _{10}$ state in $\\mathbf {K}_+$ valley, $\\psi _{n^{\\prime }}$ characterized by the wavevectors $\\mathbf {k}_1^{\\prime }$ and $\\mathbf {k}_2^{\\prime }$ and described by the wavefunctions Eqs.",
"(), (REF ), respectively.",
"According to the general theory[21] the exchange matrix element of the Coulomb interaction $V(\\mathbf {r}_1 - \\mathbf {r}_2)$ can be written as $\\langle m^{\\prime } n^{\\prime }| V(\\mathbf {r}_1 - \\mathbf {r}_2)| m n\\rangle = -V_{\\mathbf {k}_1^{\\prime } - \\mathbf {k}_2} \\delta _{\\mathbf {k}_1 + \\mathbf {k}_2, \\mathbf {k}_1^{\\prime }+\\mathbf {k}_2^{\\prime }} \\times \\\\ {\\frac{\\hbar ^2 |p_{\\rm cv}|^2}{m_0^2 E_g^2}}(k_{2,+} - k_{2,+}^{\\prime })(k_{1,+}- k_{1,+}^{\\prime }).$ Here $V_{\\mathbf {q}}$ is the two-dimensional Fourier transform of the Coulomb potential.",
"Standard transformations from the electron-electron representation to the electron-hole representation in Eq.",
"(REF ),[21] averaging over the relative motion wavefunction as well as inclusion of high-frequency screening (see Refs.",
"goupalov98,zhilich72:eng,zhilich74:eng) yields off-diagonal element $\\langle \\Gamma _3 |\\mathcal {H}_X (\\mathbf {K})|\\Gamma _2 \\rangle $ in Eq.",
"(REF ).",
"We stress that the Coulomb interaction is long-range, it does not provide intervalley transfer of electrons, however, the exchange process involves one electron from $\\mathbf {K}_+$ and another one from $\\mathbf {K}_-$ valley.",
"As a result, states active in $\\sigma ^+$ and $\\sigma ^-$ polarizations can be mixed (this fact was ignored in Ref. Yu:2014uq).",
"In order to evaluate theoretically the exciton decoherence rate it is convenient to describe the dynamics of bright exciton doublet within the pseudospin formalism, where the $2\\times 2$ density matrix in the basis of $\\Gamma _2$ and $\\Gamma _3$ excitonic states is decomposed $N_{\\mathbf {K}}/2 + \\mathbf {S}_{\\mathbf {K}} \\cdot \\mathbf {\\sigma }$ with $N_{\\mathbf {K}}$ being the occupation of a given $\\mathbf {K}$ state and $\\mathbf {S}_{\\mathbf {K}}$ being the pseudospin.",
"It satisfies the kinetic equation $\\frac{\\partial \\mathbf {S}_{\\mathbf {K}}}{\\partial t} + \\mathbf {S}_{\\mathbf {K}} \\times \\mathbf {\\Omega }_{\\mathbf {K}} = \\mathbf {Q}\\lbrace \\mathbf {S}_{\\mathbf {K}} \\rbrace ,$ where $\\mathbf {\\Omega }_{\\mathbf {K}}$ is defined by Eq.",
"(REF ) and $\\mathbf {Q}\\lbrace \\mathbf {S}_{\\mathbf {K}} \\rbrace $ is the collision integral.",
"Similarly to the case of free excitons in quantum wells[20], [29] different regimes of spin decoherence can be identified depending on the relation between the characteristic spin precession frequency and scattering rates.",
"Here we assume the strong scattering regime, where $\\Omega \\tau \\ll 1$ , where $\\Omega $ is the characteristic precession frequency and $\\tau $ is the characteristic scattering time, the exciton spin is lost by Dyakonov-Perel' type mechanism.",
"[34] Hence, the spin decay law is exponential and spin relaxation rates are given by[20], [29] $\\frac{1}{\\tau _{zz}} = \\frac{2}{\\tau _{xx}} = \\frac{2}{\\tau _{yy}} = \\langle \\Omega _{\\mathbf {K}}^2 \\tau _2\\rangle ,$ where the angular brackets denote averaging over the energy distribution and $\\tau _2 = \\tau _2(\\varepsilon _{\\mathbf {K}})$ is the relaxation time of second angular harmonics of the distribution function.",
"Experiment and discussion.",
"In this section we discuss exciton spin or valley decoherence due to the long-range exchange interaction and compare theoretical estimates with our experimental data.",
"Figure REF presents the results of photoluminescence (PL) experiments carried out on MoS$_2$ ML deposited on the SiO$_2$ /Si substrate, see Ref.",
"PhysRevLett.112.047401 for details on sample preparation and experimental methods.",
"The sample was excited by short circularly polarized laser pulse with the energy $E_{exc} = 1.965$ eV.",
"Both PL intensity and circular polarization degree were recorded as a function of emission energy.",
"As a simplest possible model, we assume that the stationary (time integrated) polarization is determined by the initially created polarization $P_0$ , the lifetime of the electron-hole pair $\\tau $ and the polarization decay time $\\tau _{zz}$ through $P_c=P_0/(1+\\tau /\\tau _{zz})$ .",
"[35] In Fig.",
"REF (a) we measure an average, time-integrated PL polarization of $P_c\\approx 60\\%$ in the emission energy range $1.82\\ldots 1.91$ eV.",
"The emission time measured in time-resolved PL is $\\tau \\simeq 4.5$ ps, extracted from Fig.",
"REF (b), see also Ref. KornMoS2.",
"Note, that it is not clear at this stage if the measured emission time is an intrinsic, radiative lifetime or limited by non-radiative processes.",
"For $P_0=100\\%$ we find an estimate of $\\tau _{zz}\\simeq 7$ ps.",
"Figure: Photoluminescence experiments on A-exciton in ML MoS 2 _2.",
"(a) Left axis: Time integrated PL intensity as a function of emission energy.",
"Right axis: Polarization of PL emission (b) PL emission intensity (black line) at T = 4 K detected at maximum of A-exciton PL E Det =1.867_{\\text{Det}}=1.867 eV as a function of time.",
"Laser reference pulse (dotted blue line).A theoretical estimate of $\\tau _{zz}$ can be obtained taking into account that, in our experimental conditions, owing to the fast energy relaxation in the system, the spread of excitons in the energy space is limited by the collisional broadening, $\\sim \\hbar /\\tau _2$ , rather than by the kinetic energy distribution.",
"Under this assumption, the trend for the spin decoherence rate can be obtained from Eqs.",
"(REF ) taking into account that [cf.",
"Ref.",
"maialle93] $\\langle \\Omega _{\\mathbf {K}}^2 \\tau _2 \\rangle \\simeq \\Omega _{\\mathbf {K}_\\Gamma }^2 \\tau _2,$ where $K_\\Gamma = \\sqrt{{2M\\Gamma _h}/{\\hbar ^2}}$ and $\\Gamma _h=1/(2\\tau _2)$ describes the $\\mathbf {k}$ -space extension of an excitonic “packet”.",
"It follows then that the scattering time cancels in the right hand side of Eq.",
"(REF ) and $\\tau _{zz} = \\frac{4\\hbar \\left(q \\tau _{rad}\\right)^2}{M}$ For an order of magnitude estimate, we set exciton mass $M$ equal to the free electron mass.",
"[16] We also set $\\tau \\approx \\tau _{rad} = 4.5$ ps.",
"Assuming the PL decay time is governed by radiative processes, realistically $\\tau $ should be regarded as a lower bound of $\\tau _{rad}$ .",
"We estimate the radiation wavevector $q= \\sqrt{\\varkappa _b}\\omega _0/c$ assuming $\\hbar \\omega _0=1.867$ eV and $\\varkappa _b=5$ (being half of the substrate high-frequency dielectric constant), and obtain $\\tau _{zz}\\approx 4$ ps.",
"This value is in reasonable agreement with the value of $\\tau _s\\simeq 7$ ps estimated from PL experiments and with recent pump-probe measurements.",
"[14], [15] Similarly to the circular polarization degree whose decay is governed by $\\tau _{zz}$ , the linear polarization decay for the neutral A-exciton is governed by the in-plane pseudospin relaxation times $\\tau _{xx}$ and $\\tau _{yy}$ .",
"As follows from Eq.",
"(REF ) they are of the same order of magnitude as $\\tau _{zz}$ .",
"Interestingly, the observation of linearly polarized emission under the linearly polarized excitation was reported for the neutral A-exciton transition in ML WSe$_2$ .",
"[37] Since the band structure and parameters of WSe$_2$ and MoS$_2$ are similar, it is reasonable to assume that the decay of linear polarization, i.e.",
"intervalley coherence of excitons, is also strongly influenced by the long-range exchange interaction between an electron and a hole.",
"Conclusions.",
"To conclude we have presented the theory of the bright exciton fine structure splitting and exciton spin decoherence in MLs of transition metal dichalcogenides.",
"Using the electrodynamical approach we have calculated eigenfrequencies of excitons taking into account their interaction with longitudinal electromagnetic field, which gives rise to the LT splitting of the bright excitonic doublet.",
"The magnitude of the splitting is expressed via the exciton center of mass wavevector and its radiative decay rate $\\Gamma _0$ .",
"This splitting acts as an effective magnetic field and provides spin relaxation/decoherence of both free and localized excitons.",
"We provided estimation of spin decoherence rate of A-excitons in MoS$_2$ MLs both from the developed theory and from experimental data on optical orientation.",
"The developed theory is in agreement with experiments probing the exciton valley dynamics.",
"Acknowledgements.",
"We thank E.L. Ivchenko and B.L.",
"Liu for stimulating discussions.",
"Partial financial support from RFBR and RF President grant NSh-1085.2014.2, INSA invited Professorship grant (MMG), ERC Starting Grant No.",
"306719 and Programme Investissements d'Avenir ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT is gratefully acknowledged."
]
] | 1403.0108 |
[
[
"A Wilcoxon-Mann-Whitney type test for infinite dimensional data"
],
[
"Abstract The Wilcoxon-Mann-Whitney test is a robust competitor of the t-test in the univariate setting.",
"For finite dimensional multivariate data, several extensions of the Wilcoxon-Mann-Whitney test have been shown to have better performance than Hotelling's $T^{2}$ test for many non-Gaussian distributions of the data.",
"In this paper, we study a Wilcoxon-Mann-Whitney type test based on spatial ranks for data in infinite dimensional spaces.",
"We demonstrate the performance of this test using some real and simulated datasets.",
"We also investigate the asymptotic properties of the proposed test and compare the test with a wide range of competing tests."
],
[
"Introduction",
"For univariate data, the Wilcoxon–Mann–Whitney test is known to have better power than the t-test for several non-Gaussian distributions (see, e.g., [14]).",
"Various extensions of the Wilcoxon–Mann–Whitney test have been studied for multivariate data in finite dimensional spaces (see, e.g., [23], [24], [20], [8], [5] and [21]), and these extensions too outperform Hotelling's $T^{2}$ test for a number of non-Gaussian multivariate distributions.",
"Nowadays, we often have to analyze data, which are curves or functions, e.g., the ECG curves of patients, the temperature curves of different regions, the spectrometry readings over a range of wavelengths etc.",
"Such data, popularly known as functional data, can be conveniently handled by viewing them as random observations from probability distributions in infinite dimensional spaces, e.g., the space of functions on an interval.",
"For testing the equality of means of two functional datasets, [18] proposed two test statistics based on orthogonal projections of the difference between the sample mean functions.",
"One of those statistics is same as Hotelling's $T^{2}$ statistic based on a finite number of such projections.",
"[11] and [32] studied two $L_{2}$ -norm based tests for functional analysis of variance and functional linear models, respectively.",
"For the problem of testing the equality of two mean functions, these two statistics reduce to a constant multiple of the $L_{2}$ -norm of the difference between the sample mean functions.",
"A two sample test for the equality of the means based on this latter statistic was studied by [31].",
"In a different direction, [2], [12], [7] and [28] studied some tests for comparing the means of two finite dimensional datasets for which the data dimension is larger than the sample size, and it grows with the sample size.",
"These authors worked in a setup, which is different from the infinite dimensional setup considered in this paper.",
"Consequently, the tests and the results obtained by these authors are quite different from ours.",
"All of the above-mentioned tests for functional and high dimensional data perform poorly when the observations have non-Gaussian distributions with heavy-tails.",
"Some of the Wilcoxon–Mann–Whitney type tests for finite dimensional data in $\\mathbb {R}^{d}$ , e.g., those defined using simplices (see, e.g., [20] and [21]) or those based on interdirections (see, e.g., [24]), cannot be extended into infinite dimensional spaces due to their dependence on the finite dimensional coordinate system in $\\mathbb {R}^{d}$ .",
"Further, a test that involves standardization by some covariance matrix computed from the sample (see, e.g., [23] and [21]) cannot be used due to the singularity of such a sample covariance matrix when the data dimension exceeds the sample size.",
"Many of the function spaces, where functional data lie, are infinite dimensional Banach spaces.",
"In Section 2, we develop a Wilcoxon–Mann–Whitney type test based on spatial ranks for data lying in those spaces.",
"We show that the proposed test statistic has an asymptotic Gaussian distribution.",
"We implement the test using this asymptotic distribution and demonstrate its performance using some real benchmark data.",
"In Section 3, we derive the asymptotic distribution of our Wilcoxon–Mann–Whitney type test statistic under some sequences of shrinking location shift models.",
"We carry out an asymptotic power comparison between our test and several other tests for functional data.",
"It is observed that our Wilcoxon–Mann–Whitney type test has superior performance than these competing tests in most of the heavy-tailed models as well as in some of the Gaussian models considered.",
"In Section 4, we report the results from a detailed simulation study comparing the finite sample performance of our test with that of a wide range of two sample tests available in the literature for infinite dimensional data."
],
[
"The construction and the implementation of the test",
"For two random samples $X_{1},\\ldots ,X_{m}$ and $Y_{1},\\ldots ,Y_{n}$ in $\\mathbb {R}$ , the Wilcoxon–Mann–Whitney statistic is defined as $\\sum _{i=1}^{m}\\sum _{j=1}^{n} sign(Y_{j} - X_{i})$ (see, e.g., [14]).",
"Since for any $x \\ne 0$ , $sign(x)$ is the derivative of $|x|$ , we define a notion of spatial rank for probability distributions in Banach spaces as follows.",
"Let ${\\bf X}$ be a random element in a Banach space ${\\cal X}$ , and ${\\cal X}^{*}$ be the dual of ${\\cal X}$ , which is the Banach space of real-valued continuous linear functions on ${\\cal X}$ .",
"Suppose that ${\\cal X}$ is smooth, i.e., the norm $||.||$ in ${\\cal X}$ is Gâteaux differentiable (see, e.g., Section 2, Chapter 4 in [4]) at each ${\\bf x} \\ne {\\bf 0}$ with Gâteaux derivative, say, $SGN_{{\\bf x}} \\in {\\cal X}^{*}$ .",
"In other words, we assume that $\\lim _{t \\rightarrow 0} t^{-1}(||{\\bf x} + t{\\bf h}|| - ||{\\bf x}||) = SGN_{{\\bf x}}({\\bf h})$ for all ${\\bf x} \\ne {\\bf 0}$ and ${\\bf h} \\in {\\cal X}$ .",
"In a Hilbert space ${\\cal X}$ , $SGN_{{\\bf x}} = {\\bf x}/||{\\bf x}||$ .",
"If ${\\cal X} = L_{p}[a,b]$ for some $p \\in (1,\\infty )$ , which is the Banach space of all functions ${\\bf x} : [a,b] \\rightarrow \\mathbb {R}$ satisfying $\\int _{a}^{b} |{\\bf x}(s)|^{p}ds < \\infty $ , then $SGN_{{\\bf x}}({\\bf h}) = \\int _{a}^{b} sign\\lbrace {\\bf x}(s)\\rbrace |{\\bf x}(s)|^{p-1}{\\bf h}(s)ds/||{\\bf x}||^{p-1}$ for all ${\\bf h} \\in L_{p}[a,b]$ .",
"We define $SGN_{{\\bf x}} = {\\bf 0}$ if ${\\bf x} = {\\bf 0}$ .",
"The spatial rank of ${\\bf x} \\in {\\cal X}$ with respect to the distribution of a random element ${\\bf X} \\in {\\cal X}$ is defined as $S_{{\\bf x}} = E(SGN_{{\\bf x} - {\\bf X}})$ , where the expectation is in the Bochner sense (see, e.g., Section 2, Chapter 3 in [1]).",
"In a Hilbert space ${\\cal X}$ , $S_{{\\bf x}} = E\\lbrace ({\\bf x}-{\\bf X})/||{\\bf x} - {\\bf X}||\\rbrace $ , and the spatial rank defined in this way has been studied in $\\mathbb {R}^{d}$ by [6], [8], [22] and [16].",
"Let ${\\bf X}_{1},\\ldots ,{\\bf X}_{m}$ and ${\\bf Y}_{1},\\ldots ,{\\bf Y}_{n}$ be independent observations from two probability measures $P$ and $Q$ on a smooth Banach space ${\\cal X}$ .",
"If we assume that $P$ and $Q$ differ by a shift $\\Delta \\in {\\cal X}$ in the location, our Wilcoxon–Mann–Whitney type statistic for testing the hypothesis $H_{0} : \\Delta = {\\bf 0}$ against $H_{1} : \\Delta \\ne {\\bf 0}$ is defined as $T_{WMW} = (mn)^{-1} \\sum _{i=1}^{m} \\sum _{j=1}^{n} SGN_{{\\bf Y}_{j} - {\\bf X}_{i}}$ .",
"Note that $T_{WMW}$ is a Banach space valued U-statistic (see, e.g., [3]) and is an unbiased estimator of $E(SGN_{{\\bf Y} - {\\bf X}})$ .",
"If $H_{0}$ holds, we have $E(SGN_{{\\bf Y} - {\\bf X}}) = {\\bf 0}$ .",
"So, we reject the null hypothesis for large values of $||T_{WMW}||$ .",
"It is straightforward to verify that for any $c \\in \\mathbb {R}$ , ${\\bf a} \\in {\\cal X}$ and a bijective linear isometry $B$ on ${\\cal X}$ , the hypotheses $H_{0}$ , $H_{1}$ , and the test statistic remain invariant under the transformation ${\\bf X} \\mapsto cB({\\bf X}) + {\\bf a}$ and ${\\bf Y} \\mapsto cB({\\bf Y}) + {\\bf a}$ .",
"We shall now study the asymptotic distribution of the statistic $T_{WMW}$ .",
"A Banach space ${\\cal X}$ is said to be of type 2 if there exists a constant $b > 0$ such that for any $n\\ge 1$ and independent zero mean random elements ${\\bf U}_{1},{\\bf U}_{2},\\ldots ,{\\bf U}_{n}$ in ${\\cal X}$ satisfying $E(||{\\bf U}_{i}||^{2}) < \\infty $ for all $i=1,2,\\ldots ,n$ , we have $E(||\\sum _{i=1}^{n} {\\bf U}_{i}||^{2}) \\le b\\sum _{i=1}^{n} E(||{\\bf U}_{i}||^{2})$ (see, e.g., Section 7, Chapter 3 in [1]).",
"Type 2 Banach spaces are the only Banach spaces, where the central limit theorem holds for every sequence of independent and identically distributed random elements, whose squared norms have finite expectations.",
"It is known that Hilbert spaces and the $L_{p}$ spaces with $p \\in [2,\\infty )$ are type 2 Banach spaces.",
"We denote by $G({\\bf m},{\\bf C})$ the distribution of a Gaussian random element (say, ${\\bf W}$ ) in a separable Banach space ${\\cal X}$ with mean ${\\bf m} \\in {\\cal X}$ and covariance ${\\bf C}$ , where ${\\bf C} : {\\cal X}^{*} \\times {\\cal X}^{*} \\rightarrow \\mathbb {R}$ is a symmetric nonnegative definite continuous bilinear functional.",
"Note that for any ${\\bf l} \\in {\\cal X}^{*}$ , ${\\bf l}({\\bf W})$ has a Gaussian distribution on $\\mathbb {R}$ with mean ${\\bf l}({\\bf m})$ and variance ${\\bf C}({\\bf l},{\\bf l})$ .",
"Define $\\mu = E(SGN_{{\\bf Y} - {\\bf X}})$ .",
"We denote by $\\Gamma _{1}, \\Gamma _{2} : {\\cal X}^{**} \\times {\\cal X}^{**} \\rightarrow \\mathbb {R}$ the symmetric nonnegative definite continous bilinear functionals given by $\\Gamma _{1}({\\bf f},{\\bf g}) = E[{\\bf f}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}}\\mid {\\bf X})\\rbrace {\\bf g}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}}\\mid {\\bf X})\\rbrace ] - {\\bf f}(\\mu ){\\bf g}(\\mu )$ , and $\\Gamma _{2}({\\bf f},{\\bf g}) = E[{\\bf f}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}}\\mid {\\bf Y})\\rbrace {\\bf g}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}}\\mid {\\bf Y})\\rbrace ] - {\\bf f}(\\mu ){\\bf g}(\\mu )$ , where ${\\bf f}, {\\bf g} \\in {\\cal X}^{**}$ .",
"Note that for a random element ${\\bf Z}$ in a Banach space with $E(||{\\bf Z}||) < \\infty $ , the conditional expectation of ${\\bf Z}$ given ${\\bf X}$ exists and can be properly defined (see, e.g., Section 4, Chapter II in [29] for the relevant details).",
"Theorem 2.1 Let $N = m + n$ and $m/N \\rightarrow \\gamma \\in (0,1)$ as $m, n \\rightarrow \\infty $ .",
"Also, assume that the dual space ${\\cal X}^{*}$ is a separable and type 2 Banach space.",
"Then, for any two probability measures $P$ and $Q$ on ${\\cal X}$ , $(mn/N)^{1/2} (T_{WMW} - \\mu )$ converges weakly to $G\\lbrace {\\bf 0},(1-\\gamma )\\Gamma _{1} + \\gamma \\Gamma _{2}\\rbrace $ as $m,n \\rightarrow \\infty $ .",
"The implementation of the test can be done using the asymptotic distribution of $T_{WMW}$ under the null hypothesis.",
"Under $H_{0}$ , we have $\\Gamma _{1} = \\Gamma _{2}$ .",
"Let $c_{\\alpha }$ denote the $100(1-\\alpha )$ th percentile of the distribution of $||G({\\bf 0},\\Gamma _{1})||$ .",
"Thus, our test, which rejects $H_{0}$ if $||(mn/N)^{1/2}T_{WMW}|| > c_{\\alpha }$ has asymptotic size $\\alpha $ .",
"When ${\\cal X}$ is a separable Hilbert space, $\\Gamma _{1}$ has a spectral decomposition (see Theorem IV.$2.4$ in page 213 and Proposition $1.9$ in page 161 in [29]), which implies that $||G({\\bf 0},\\Gamma _{1})||^{2}$ is distributed as a weighted sum of independent chi-square random variables each with 1 degree of freedom, and the weights are the eigenvalues of $\\Gamma _{1}$ .",
"Further details about the implementation of our test are given below when we analyze some real data.",
"It follows from Theorem REF that the asymptotic power of our test will be 1 whenever $\\mu \\ne {\\bf 0}$ .",
"This holds in particular if $Q$ differs from $P$ by a non-zero shift $\\Delta $ in the location, ${\\cal X}$ is a reflexive and strictly convex Banach space, and the distribution of ${\\bf Y} - {\\bf X}$ is nonatomic and not concentrated on a line in ${\\cal X}$ (see, e.g., Theorem $4.14$ in [19]).",
"In other words, the test is consistent for location shift alternatives.",
"We have applied our test based on $T_{WMW}$ to three real datasets, namely, the Coffee data, the Berkeley growth data and the Spectrometry data.",
"The Coffee data is obtained from http://www.cs.ucr.edu/$\\sim $ eamonn/time_series_data/ and contains spectroscopy readings taken at 286 wavelength values for 14 samples of each of the two varieties of coffee, namely, Arabica and Robusta.",
"The Berkeley growth data is available in the R package “fda” (see http://rss.acs.unt.edu/Rdoc/library/fda/html/growth.html) and contains the heights of 39 boys and 54 girls measured at 31 time points between the ages 1 and 18 years.",
"The curves have been pre-smoothed using a monotone spline smoothing technique available in the R package “fda”.",
"The curves are recorded at 101 equispaced ages in the interval $[1,18]$ .",
"The Spectrometry data is available at http://www.math.univ-toulouse.fr/staph/npfda and contains the spectrometric curves for 215 meat units measured at 100 wavelengths between 850 nm and 1050 nm.",
"The data also contains the fat content of each meat unit, which is categorized into two classes, namely, “$\\le 20\\%$ ” and “$> 20\\%$ ”.",
"In all these three datasets, each observation can be viewed as an element in the separable Hilbert space $L_{2}[a,b]$ .",
"For instance, the spectrometric curves in the third dataset can be viewed as elements in the space $L_{2}[850,1050]$ .",
"In view of Theorem REF and the discussion following it, for all three real datasets, the asymptotic null distribution of $||(mn/N)^{1/2}T_{WMW}||$ can be expressed in terms of a weighted sum of independent chi-square random variables.",
"Since only a few eigenvalues of the sample analog of $\\Gamma _{1}$ are positive, we get a finite sum, and use its distribution, which can be simulated, to estimate the critical value of our test statistic.",
"For each dataset, the norm in the definition of $SGN_{{\\bf x}}$ used in $T_{WMW}$ is computed as the norm of the Euclidean space whose dimension is the number of time points over which the sample curves in that dataset are observed.",
"We have also applied the two sample version of the test studied by [11] and the two tests of [18] to these datasets.",
"We have used the usual empirical pooled covariance for the two tests of [18], and the numbers of projection directions used in these two tests are chosen using the cumulative variance method described in their paper.",
"For the Coffee data, the p-value of our test based on $T_{WMW}$ is $0.072$ , that of the test in [11] is $0.169$ , and both the tests in [18] have the same p-value $0.273$ .",
"None of the tests yield a very strong evidence against the null hypothesis, and all of them fail to reject it at the $5\\%$ level.",
"However, among the four p-values, the one obtained using our test bears the strongest evidence in favour of the alternative hypothesis.",
"The p-values of all four tests for both of the Berkeley growth data and the Spectrometry data are 0 upto two decimal places.",
"We have also applied these tests to randomly chosen $20\\%$ subsamples of the two datasets instead of the full datasets in order to investigate whether there is any difference in the results obtained using these tests when the sample sizes are smaller.",
"The random subsampling was repeated 1000 times for each dataset to compute the proportion of times each test rejects the null hypothesis when the level is fixed at $5\\%$ for each test.",
"For the subsamples of the Berkeley growth data and the Spectrometry data, the proportions of rejections of the null hypothesis by our test based on $T_{WMW}$ are $0.829$ and $0.832$ , respectively, while those proportions are $0.476$ and $0.712$ , respectively, for the test in [11].",
"The proportions of rejections of the null hypothesis by one of the two tests in [18] are $0.271$ and $0.744$ for the subsamples of the Berkeley growth data and the Spectrometry data, respectively, while those proportions are $0.292$ and $0.778$ , respectively, for the other test in their paper.",
"Thus, for the Berkeley growth data, our test has the highest rate of rejection of the null hypothesis, and those rates for the other three tests are much lower.",
"For the Spectrometry data, all four tests have fairly high rates of rejection of the null hypothesis, and the rate is highest for our test using $T_{WMW}$ ."
],
[
"Asymptotic powers of different tests under shrinking location shifts",
"In the previous section, we have established the consistency of our test for models with fixed location shifts.",
"We shall now derive the asymptotic distribution of our test statistic under appropriate sequences of shrinking location shifts.",
"Suppose that ${\\bf Y}$ is distributed as ${\\bf X} + \\Delta _{N}$ , where $\\Delta _{N} = \\delta (mn/N)^{-1/2}$ for some fixed non-zero $\\delta \\in {\\cal X}$ and $N \\ge 1$ .",
"Recall that $N = m + n$ is the total size of the two samples.",
"For some of the Wilcoxon–Mann–Whitney type tests studied in the finite dimensional setting, such alternative hypotheses have been shown to be contiguous to the null, and this leads to nondegenerate limiting distributions of the test statistics under those alternatives (see, e.g., [8], [5] and [21]).",
"For our next theorem, we assume that the norm in ${\\cal X}$ is twice Gâteaux differentiable at every ${\\bf x} \\ne {\\bf 0}$ (see, e.g., Chapter 4, Section 6 in [4]).",
"Let us denote the Hessian of the function ${\\bf x} \\mapsto E(||{\\bf Y} - {\\bf X} + {\\bf x}||)$ at ${\\bf x}$ by $J_{{\\bf x}} : {\\cal X} \\rightarrow {\\cal X}^{*}$ when it exists.",
"In other words, for every ${\\bf h} \\in {\\cal X}$ , $E(SGN_{{\\bf Y} - {\\bf X} + {\\bf x} + t{\\bf h}}) = E(SGN_{{\\bf Y} - {\\bf X} + {\\bf x}}) + tJ_{{\\bf x}}({\\bf h}) + {\\bf R}(t), $ where $||{\\bf R}(t)||/t \\rightarrow 0$ as $t \\rightarrow 0$ .",
"It is known that the norms in Hilbert spaces and the $L_{p}$ spaces with $p \\in [2,\\infty )$ are twice Gâteaux differentiable.",
"Let ${\\cal X} = L_{p}[a,b]$ for some $2 \\le p < \\infty $ and $-\\infty < a < b < \\infty $ .",
"If $E(||{\\bf Y} - {\\bf X} + {\\bf x}||^{-1}) < \\infty $ , it can be shown that $J_{{\\bf x}}$ exists and is given by $\\lbrace J_{{\\bf x}}({\\bf z})\\rbrace ({\\bf w}) &=& (p-1)E\\left[\\frac{\\int _{a}^{b} |{\\bf Y}(s)-{\\bf X}(s)+{\\bf x}(s)|^{p-2}{\\bf z}(s){\\bf w}(s)ds}{||{\\bf Y} - {\\bf X} + {\\bf x}||^{p-1}} \\right.",
"\\\\&& - \\ \\left.\\frac{\\left\\lbrace \\int _{a}^{b} |{\\bf Y}(s)-{\\bf X}(s)+{\\bf x}(s)|^{p-1}{\\bf z}(s)ds\\right\\rbrace \\left\\lbrace \\int _{a}^{b} |{\\bf Y}(s)-{\\bf X}(s)+{\\bf x}(s)|^{p-1}{\\bf w}(s)ds\\right\\rbrace }{||{\\bf Y} - {\\bf X} + {\\bf x}||^{2p-1}} \\right],$ where ${\\bf z}, {\\bf w}$ and ${\\bf x} \\in L_{p}[a,b]$ .",
"Theorem 3.1 As before, let $N = m + n$ , $m/N \\rightarrow \\gamma \\in (0,1)$ as $m, n \\rightarrow \\infty $ , and ${\\cal X}^{*}$ is a separable and type 2 Banach space.",
"Also, assume that the distribution of ${\\bf X}$ is nonatomic and $J_{{\\bf 0}}$ exists.",
"Then, under the sequence of shrinking location shifts described at the beginning of this section, $(mn/N)^{1/2} T_{WMW}$ converges weakly to $G\\lbrace J_{{\\bf 0}}(\\delta ),\\Gamma _{1}\\rbrace $ as $m, n \\rightarrow \\infty $ .",
"In order to compare the asymptotic power of our test with those of the tests available in [11] and [18], we shall now study the asymptotic distributions of those test statistics under the sequences of shrinking shifts described at the beginning of this section.",
"For the two sample problem in $L_{2}[a,b]$ , the test statistic studied by [11] reduces to $T_{CFF} = m||\\bar{{\\bf X}} - \\bar{{\\bf Y}}||^{2}$ .",
"[18] studied the test statistics $T_{HKR1} = \\sum _{k=1}^{L} (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{\\psi }_{k}\\rangle )^{2}$ and $T_{HKR2} = \\sum _{k=1}^{L} \\widehat{\\lambda }_{k}^{-1} (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{\\psi }_{k}\\rangle )^{2}$ .",
"Here, $\\langle .,.\\rangle $ denotes the inner product in $L_{2}[0,1]$ , the $\\widehat{\\lambda }_{k}$ 's denote the eigenvalues of the empirical pooled covariance of the ${\\bf X}_{i}$ 's and the ${\\bf Y}_{j}$ 's in descending order of magnitudes, and the $\\widehat{\\psi }_{k}$ 's are the corresponding empirical eigenvectors.",
"If ${\\cal X} = \\mathbb {R}^{d}$ and $L = d$ , $T_{HKR2}$ reduces to Hotelling's $T^{2}$ statistic, and $T_{HKR1} = m^{-1}T_{CFF}$ .",
"We derive the asymptotic distributions of $T_{HKR1}$ and $T_{HKR2}$ in a separable Hilbert space.",
"Since the statistic $T_{CFF}$ can be defined in any Banach space, we derive its asymptotic distribution in a separable and type 2 Banach space.",
"Theorem 3.2 Once again, let $N = m + n$ and $m/N \\rightarrow \\gamma \\in (0,1)$ as $m, n \\rightarrow \\infty $ .",
"Then, under the sequence of shrinking location shifts mentioned at the beginning of this section, we have the following.",
"(a) If $E(||{\\bf X}||^{2}) < \\infty $ , $nN^{-1}T_{CFF}$ converges weakly to $||G(\\delta ,\\Sigma )||^{2}$ as $m,n \\rightarrow \\infty $ , where $\\Sigma $ denotes the covariance of ${\\bf X}$ .",
"(b) Assume that for some $L \\ge 1$ , $\\lambda _{1} > \\ldots > \\lambda _{L} > \\lambda _{L+1} > 0$ , where the $\\lambda _{k}$ 's are the eigenvalues of $\\Sigma $ in decreasing order of magnitudes.",
"If $E(||{\\bf X}||^{4}) < \\infty $ , $mnN^{-1}T_{HKR1}$ converges weakly to $\\sum _{k=1}^{L} \\lambda _{k}\\chi ^{2}_{(1)}(\\beta _{k}^{2}/\\lambda _{k})$ , and $mnN^{-1}T_{HKR2}$ converges weakly to $\\sum _{k=1}^{L} \\chi ^{2}_{(1)}(\\beta _{k}^{2}/\\lambda _{k})$ as $m, n \\rightarrow \\infty $ .",
"Here, $\\beta _{k} = \\langle \\delta ,\\psi _{k}\\rangle $ , $\\chi ^{2}_{(1)}(\\beta _{k}^{2}/\\lambda _{k})$ denotes the noncentral chi-square variable with 1 degree of freedom and noncentrality parameter $\\beta _{k}^{2}/\\lambda _{k}$ , and $\\psi _{k}$ is the eigenvector corresponding to $\\lambda _{k}$ for $k = 1, 2, \\ldots , L$ .",
"For evaluating the asymptotic powers of different tests under shrinking location shifts, we have considered some probability distributions in $L_{2}[0,1]$ .",
"Let ${\\bf X} = \\sum _{k=1}^{\\infty } Z_{k}\\phi _{k}$ , where the $Z_{k}$ 's are independent random variables, and $\\phi _{k}(t) = \\sqrt{2}sin\\lbrace (k-0.5){\\pi }t\\rbrace $ for $k \\ge 1$ , which form an orthonormal basis of $L_{2}[0,1]$ .",
"We have considered two cases, namely, $Z_{k}/\\sigma _{k}$ having a $N(0,1)$ distribution and a t distribution with 5 degrees of freedom, where $\\sigma _{k} = \\lbrace (k-0.5)\\pi \\rbrace ^{-1}$ for each $k \\ge 1$ .",
"Both of these distributions satisfy the assumptions made in Theorems REF and REF .",
"These two cases correspond to the Karhunen-Loève expansions of the standard Brownian motion (the sBm distribution) and the centered t process on $[0,1]$ with 5 degrees of freedom (the t(5) distribution) and covariance kernel $K(t,s) = \\min (t,s)$ (see, e.g., [30]), respectively.",
"Recall that ${\\bf Y}$ is distributed as ${\\bf X} + \\delta (mn/N)^{-1/2}$ , and we have considered three choices of $\\delta $ , namely, $\\delta _{1}(t) = c$ , $\\delta _{2}(t) = ct$ and $\\delta _{3}(t) = ct(1-t)$ , where $t \\in [0,1]$ and $c > 0$ .",
"For evaluating the asymptotic powers of different tests using Theorems REF and REF , the expectations appearing in $J_{{\\bf 0}}(\\delta )$ and $\\Gamma _{1}$ can be evaluated numerically using averages over Monte-Carlo replications of relevant random objects.",
"For an appropriately large $d$ , the eigenvalues and the eigenvectors of $\\Gamma _{1}$ and $\\Sigma $ can be approximated by those of the $d \\times d$ covariance matrices associated with the values of the sample functions at $d$ equispaced points in $[0,1]$ .",
"Figure REF shows the plots of the ratios of the asymptotic powers of the tests based on $T_{CFF}$ , $T_{HKR1}$ and $T_{HKR2}$ to those of our test using $T_{WMW}$ .",
"It is seen that all the tests attain the $5\\%$ nominal level asymptotically for both of the distributions, and the curves in each plot in Figure REF meet at $c = 0$ .",
"The asymptotic powers of the tests based on $T_{CFF}$ and $T_{HKR1}$ are close for all the situations considered.",
"The test using $T_{HKR2}$ outperforms both of them for $\\delta _{1}(t)$ and $\\delta _{3}(t)$ , but is asymptotically less powerful for $\\delta _{2}(t)$ for both the distributions considered.",
"The asymptotic powers of the tests based on $T_{CFF}$ and $T_{HKR1}$ are close to that of our test using $T_{WMW}$ for $\\delta _{2}(t)$ under the sBm distribution, and in all other cases, our test outperforms them.",
"The test based on $T_{HKR2}$ outperforms our test for $\\delta _{3}(t)$ for both the distributions, while it is asymptotically more powerful for small $c$ values for $\\delta _{1}(t)$ under the sBm distribution.",
"In other cases, our test outperforms this test.",
"Figure: Plots of the ratios of the asymptotic powers of the tests based on T CFF T_{CFF} (solid line), T HKR1 T_{HKR1} (dashed line) and T HKR2 T_{HKR2} (dot-dashed line) to those of our test using T WMW T_{WMW} for the sBm and the t(5) distributions under shrinking location shifts."
],
[
"Finite sample powers of different tests",
"In this section, we shall carry out a comparative study of the finite sample empirical powers of the tests considered in Sections and and a few other tests.",
"Once again, let ${\\bf X} = \\sum _{k=1}^{\\infty } Z_{k}\\phi _{k}$ , where the $Z_{k}$ 's and the $\\phi _{k}$ 's are as in Section , and the distributions of ${\\bf Y}$ and ${\\bf X}$ differ by a shift $\\Delta $ .",
"Here, we have considered three cases, namely, $Z_{k}/\\sigma _{k}$ having a $N(0,1)$ distribution (the sBm distribution) and $Z_{k}/\\sigma _{k} = U_{k}(V/r)^{-1/2}$ , where $U_{k}$ 's are independent $N(0,1)$ variables and $V$ has a chi-square distribution with $r$ degrees of freedom for $r = 1$ and 5 independent of the $U_{k}$ 's for each $k \\ge 1$ (the t(1) and the t(5) distributions, respectively), where the $\\sigma _{k}$ 's are as in Section .",
"The t(1) distribution is included to investigate the performance of our test and its competitors when the moment conditions on the probability distribution required by the tests based on $T_{CFF}$ , $T_{HKR1}$ and $T_{HKR2}$ (see [11] and [18]) fail to hold.",
"Note that the conditions assumed for our test (see Theorem REF ) hold for all the distributions considered here.",
"We have chosen $m = n = 15$ , and each sample curve is observed at 250 equispaced points in $[0,1]$ .",
"Three types of shifts are considered, namely, $\\Delta _{1}(t) = c$ , $\\Delta _{2}(t) = ct$ and $\\Delta _{3}(t) = ct(1-t)$ , where $t \\in [0,1]$ and $c > 0$ (cf.",
"Section ).",
"For each simulated dataset, all the test statistics and their critical values are computed in the same way as in Section , where we analyzed some real datasets.",
"All the sizes and the powers are evaluated by averaging the results of 1000 Monte-Carlo simulations.",
"Figure REF shows the plots of the ratios of the finite sample powers of the competing tests to those of our test at the nominal level of $5\\%$ .",
"The sizes of all the tests considered in Sections and are close to the nominal $5\\%$ level for the sBm and the t(5) distributions.",
"For the t(1) distribution, all those tests have sizes around $1.5\\%$ , while our test using $T_{WMW}$ has size $4.4\\%$ .",
"The test using $T_{HKR2}$ outperforms the tests based on $T_{CFF}$ and $T_{HKR1}$ in all the situations considered except for $\\Delta _{2}(t)$ under the sBm and the t(5) distributions, where it is less powerful for larger $c$ values.",
"The tests based on $T_{CFF}$ and $T_{HKR1}$ have similar powers for all the models considered.",
"Their powers coincide for all the shifts under the t(1) distribution, where our test outperforms all three competitors.",
"For all the shifts under the t(5) distribution, our test outperforms both the tests using $T_{CFF}$ and $T_{HKR1}$ .",
"For $\\Delta _{1}(t)$ and $\\Delta _{2}(t)$ under the t(5) distribution, our test is more powerful than the test using $T_{HKR2}$ except for small values of $c$ , and this latter test outperforms our test for $\\Delta _{3}(t)$ .",
"The behaviour of all the tests under the sBm distribution is similar to that under the t(5) distribution, except for $\\Delta _{2}(t)$ .",
"For $\\Delta _{2}(t)$ under the sBm distribution, the three competing tests have an edge over our test.",
"These finite sample results are broadly in conformity with the asymptotic results in Section .",
"Figure: Plots of the ratios of the finite sample powers of the tests based on T CFF T_{CFF} (solid line), T HKR1 T_{HKR1} (dashed line) and T HKR2 T_{HKR2} (dot-dashed line) to those of our test using T WMW T_{WMW} for the sBm, the t(1) and the t(5) distributions under location shifts.We have compared the finite sample powers of our test and some more tests available in the literature.",
"A pointwise t-test with an appropriate p-value correction for multiple testing was studied by [9] for testing the equality of means of two Gaussian functional datasets.",
"[27] studied some $F$ test for linear models involving Gaussian functional data, and we consider the two sample version of this test.",
"[10] studied an analysis of variance test for functional data based on multiple testing using random univariate linear projections of the data.",
"The two sample version of their test reduces to the Wilcoxon–Mann–Whitney test based on such projections of the data.",
"[13] studied a test for comparing two probability distributions on metric spaces, which may not necessarily differ by a shift in the location.",
"We have used their test based on the asymptotic distribution of the unbiased statistic MMD$_{u}^{2}$ (see Section 5 in [13]).",
"For comparing two finite dimensional probability distributions, [15] studied a permutation test based on the ranks of the distances between the sample observations, while [25] studied a test based on a notion of adjacency.",
"The authors of both papers pointed out that these tests can be used for infinite dimensional functional data as well.",
"The asymptotic behaviours of none of the above-mentioned tests under the type of shrinking shifts considered in Section are known in the literature, and such an analysis is beyond the scope of this paper.",
"We only carry out a finite sample empirical power comparison of our test based on $T_{WMW}$ with these tests.",
"We have used the $L_{2}$ -distance between the pointwise ranks as the distance function for implementing the Rosenbaum test.",
"The discrete distribution of the test statistic made the size of this test much less than the nominal significance level for the small sample sizes that we have considered.",
"To rectify this, we considered a randomized version of the test, and this improved the size as well as the power of the test.",
"We have chosen 30 random projections for implementing the Cuesta-Albertos and Febrero-Bande test, as recommended by these authors.",
"We have chosen the radial basis function as the kernel for the Gretton et al.",
"test and used the codes provided by these authors.",
"All the other tests are implemented using our own codes, and all the sizes and the powers are evaluated by averaging the results of 1000 Monte-Carlo simulations.",
"Figure REF shows the plots of the ratios of the powers of these tests to those of our test using $T_{WMW}$ .",
"For all the distributions considered, the sizes of the Rosenbaum test, the Hall–Tajvidi test, the Gretton et al.",
"test and the Cox–Lee test were close to the nominal $5\\%$ level.",
"The sizes of the Cuesta-Albertos and Febrero-Bande test were around $2.6\\%$ in all our simulations.",
"The sizes of the Shen–Faraway test were much smaller than the nominal level for all the distributions considered, and it was zero for the t(1) distribution.",
"Figure REF shows that our test based on $T_{WMW}$ is uniformly more powerful than the Cuesta-Albertos and Febrero-Bande test and the Shen–Faraway test in all the situations considered.",
"Our test outperforms the Rosenbaum test, the Hall–Tajvidi test and the Gretton et al.",
"test in all but the following situations.",
"The Rosenbaum test and the Hall–Tajvidi test are more powerful than our test for small values of $c$ for $\\Delta _{2}(t)$ under all the distributions.",
"The Hall–Tajvidi test is also more powerful than our test for small $c$ values for $\\Delta _{1}(t)$ and $\\ Delta_{3}(t)$ under the t(1) distribution.",
"The Gretton et al.",
"test has a slight edge over our test for all the shifts under the t(1) distribution.",
"Except for small $c$ values, our test using $T_{WMW}$ is more powerful than the Cox–Lee test for $\\Delta _{2}(t)$ under all the distributions and for the shift $\\Delta _{3}(t)$ under the t(1) distribution.",
"For $\\Delta _{3}(t)$ under the sBm and the t(5) distributions, the Cox–Lee test has an edge over our test.",
"For $\\Delta _{1}(t)$ , the Cox–Lee test is far more superior to our test for all of the three distributions considered, and we have not plotted the ratios of its power to those of our test, since the values lie beyond the plotting ranges used in Figure REF .",
"The reason for such a behaviour of this test is that the coordinate random variable at $t = 0.0001$ (which is closest to zero in our computations) has scale parameter equal to $0.0001$ for all the distributions considered.",
"Consequently, for this coordinate and $\\Delta _{1}(t)$ , the adjusted p-values of the t-test used in the Cox–Lee procedure are $\\le 0.05$ for many of the simulations.",
"The Cox–Lee test rejects $H_{0}$ for such simulations resulting in the high power of this test for this shift.",
"Figure: Plots of the ratios of the finite sample powers of the Hall–Tajvidi test (solid line), the Shen–Faraway test (dashed line), the Rosenbaum test (dot-dashed line), the Cox–Lee test (––), the Cuesta-Albertos and Febrero-Bande test (–∘\\circ –) and the Gretton et al.",
"test (–×\\times –) to those of our test using T WMW T_{WMW} for the sBm, the t(1) and the t(5) distributions under location shifts."
],
[
"Acknowledgement",
"Research of the first author is partially supported by the SPM Fellowship of the Council of Scientific and Industrial Research, Government of India.",
"[Proof of Theorem REF ] Observe that $T_{WMW} - \\mu $ is a two-sample Banach space valued U-statistic with kernel ${\\bf h}({\\bf x},{\\bf y}) = SGN_{{\\bf y} - {\\bf x}} - \\mu $ satisfying $E\\lbrace {\\bf h}({\\bf X},{\\bf Y})\\rbrace = {\\bf 0}$ .",
"By the Hoeffding decomposition for Banach space valued U-statistics (see, e.g., Section $1.2$ in [3]), we have $T_{WMW} - \\mu = \\frac{1}{m} \\sum _{i=1}^{m} E\\lbrace {\\bf h}({\\bf X}_{i},{\\bf Y})\\mid {\\bf X}_{i}\\rbrace + \\frac{1}{n} \\sum _{j=1}^{n} E\\lbrace {\\bf h}({\\bf X},{\\bf Y}_{j})\\mid {\\bf Y}_{j}\\rbrace + {\\bf R}_{m,n}.$ So, ${\\bf R}_{m,n} = (mn)^{-1} \\sum _{i=1}^{m}\\sum _{j=1}^{n} \\widetilde{{\\bf h}}({\\bf X}_{i},{\\bf Y}_{j})$ , where $\\widetilde{{\\bf h}}({\\bf x},{\\bf y}) = {\\bf h}({\\bf x},{\\bf y}) - E\\lbrace {\\bf h}({\\bf X},{\\bf Y})\\mid {\\bf X}={\\bf x}\\rbrace - E\\lbrace {\\bf h}({\\bf X},{\\bf Y})\\mid {\\bf Y}={\\bf y}\\rbrace $ .",
"Let $\\Phi ({\\bf X}_{i}) = \\sum _{j=1}^{n} \\widetilde{{\\bf h}}({\\bf X}_{i},{\\bf Y}_{j})$ .",
"Since $E\\lbrace \\widetilde{{\\bf h}}({\\bf X},{\\bf Y})\\mid {\\bf Y}={\\bf y}\\rbrace = {\\bf 0}$ for all ${\\bf y} \\in {\\cal X}$ , using the definition of type 2 Banach spaces mentioned in Section , we get $E(||{\\bf R}_{m,n}||^{2}\\mid {\\bf Y}_{j};j=1,2,\\ldots ,n) &=& \\frac{1}{m^{2}n^{2}} E\\left\\lbrace \\left\\Vert \\sum _{i=1}^{m} \\Phi ({\\bf X}_{i})\\right\\Vert ^{2}\\mid {\\bf Y}_{j};j=1,\\ldots ,n\\right\\rbrace \\nonumber \\\\&\\le & \\frac{b}{m^{2}n^{2}} \\sum _{i=1}^{m} E\\left\\lbrace ||\\Phi ({\\bf X}_{i})||^{2}\\mid {\\bf Y}_{j};j=1,\\ldots ,n\\right\\rbrace .",
"$ Taking expectations of both sides of (REF ) with respect to ${\\bf Y}_{j}$ for $1 \\le j \\le n$ , and using the fact that the ${\\bf X}_{i}$ 's are identically distributed, we get $E(||{\\bf R}_{m,n}||^{2}) &\\le & \\frac{b}{mn^{2}} E\\left\\lbrace \\left\\Vert \\sum _{j=1}^{n} \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{j})\\right\\Vert ^{2}\\right\\rbrace .",
"$ Since $E\\lbrace \\widetilde{{\\bf h}}({\\bf X},{\\bf Y})\\mid {\\bf X}={\\bf x}\\rbrace = {\\bf 0}$ for all ${\\bf x} \\in {\\cal X}$ , once again from the definition of type 2 Banach spaces and the fact that the ${\\bf Y}_{j}$ 's are identically distributed, we get $E\\left\\lbrace \\left\\Vert \\sum _{j=1}^{n} \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{j})\\right\\Vert ^{2}\\right\\rbrace &=& E\\left[E\\left\\lbrace \\left\\Vert \\sum _{j=1}^{n} \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{j})\\right\\Vert ^{2}\\mid {\\bf X}_{1}\\right\\rbrace \\right] \\nonumber \\\\&\\le & bE\\left[\\sum _{j=1}^{n} E\\left\\lbrace \\left\\Vert \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{j})\\right\\Vert ^{2}\\mid {\\bf X}_{1}\\right\\rbrace \\right] \\nonumber \\\\&=& bnE\\left\\lbrace \\left\\Vert \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{1})\\right\\Vert ^{2}\\right\\rbrace .",
"$ Since $||SGN_{{\\bf x}}|| \\le 1$ for all ${\\bf x} \\in {\\cal X}$ , we have $||\\widetilde{{\\bf h}}({\\bf x},{\\bf y})|| \\le 4$ for all ${\\bf x}, {\\bf y} \\in {\\cal X}$ .",
"Combining this fact with (REF ) and (REF ), we have $E(||{\\bf R}_{m,n}||^{2}) &\\le & \\frac{b^{2}}{mn} E\\left\\lbrace \\left\\Vert \\widetilde{{\\bf h}}({\\bf X}_{1},{\\bf Y}_{1})\\right\\Vert ^{2}\\right\\rbrace \\ \\le \\ \\frac{16b^{2}}{mn}.$ Since $m/N \\rightarrow \\gamma \\in (0,1)$ , we get $E\\lbrace ||(mn/N)^{1/2}{\\bf R}_{m,n}||^{2}\\rbrace \\rightarrow 0$ as $m,n \\rightarrow \\infty $ .",
"Hence, $(mn/N)^{1/2}{\\bf R}_{m,n}$ converges to ${\\bf 0}$ in probability as $m,n \\rightarrow \\infty $ .",
"We note here that a similar result has been proved in [3] for Banach space valued U-statistics, but the proof given above is simpler and uses the fact that ${\\bf h}$ is a bounded kernel.",
"Such a result for real-valued U-statistics is discussed in Chapter 5 in [26] under the assumption that the kernel has a finite second moment.",
"Now, $m^{-1/2} \\sum _{i=1}^{m} E\\lbrace {\\bf h}({\\bf X}_{i},{\\bf Y})\\mid {\\bf X}_{i}\\rbrace $ and $n^{-1/2} \\sum _{j=1}^{n} E\\lbrace {\\bf h}({\\bf X},{\\bf Y}_{j})\\mid {\\bf Y}_{j}\\rbrace $ converge weakly to $G({\\bf 0},\\Gamma _{1})$ and $G({\\bf 0},\\Gamma _{2})$ , respectively, as $m, n \\rightarrow \\infty $ by the central limit theorem for independent and identically distributed random variables in a separable and type 2 Banach space (see, e.g, Theorem $7.5(i)$ in [1]).",
"So, the independence of these two sums, the assumption that $m/N \\rightarrow \\gamma $ , and the fact that $(mn/N)^{1/2}{\\bf R}_{m,n}$ converges to ${\\bf 0}$ in probability complete the proof.",
"[Proof of Theorem REF ] Define $\\rho (\\Delta _{N}) = E(SGN_{{\\bf Y} - {\\bf X}})$ .",
"Applying the Hoeffding decomposition for Banach space valued U-statistics as in the proof of Theorem REF , it follows that $T_{WMW} - \\rho (\\Delta _{N}) &=& \\frac{1}{m} \\sum _{i=1}^{m} \\lbrace E(SGN_{{\\bf Y} - {\\bf X}_{i}}\\mid {\\bf X}_{i}) - \\rho (\\Delta _{N})\\rbrace \\nonumber \\\\&& + \\ \\frac{1}{n} \\sum _{j=1}^{n} \\lbrace E(SGN_{{\\bf Y}_{j} - {\\bf X}}\\mid {\\bf Y}_{j}) - \\rho (\\Delta _{N})\\rbrace + {\\bf S}_{m,n}.",
"$ Arguing as in the proof of Theorem REF , it can be shown that $E(||{\\bf S}_{m,n}||^{2}) \\le 16b^{2}/mn$ for each $m, n \\ge 1$ .",
"Thus, $(mn/N)^{1/2}{\\bf S}_{m,n} \\rightarrow {\\bf 0}$ in probability as $m,n \\rightarrow \\infty $ under the sequence of shrinking shifts.",
"Note that $\\rho (\\Delta _{N}) = E(SGN_{{\\bf Z} - {\\bf X} + \\Delta _{N}})$ , where ${\\bf Z}$ is an independent copy of ${\\bf X}$ .",
"So, it follows from (REF ) in Section that $(mn/N)^{1/2}\\rho (\\Delta _{N}) \\longrightarrow J_{{\\bf 0}}(\\delta ) $ as $m, n \\rightarrow \\infty $ .",
"We next show the asymptotic Gaussianity of the first term on the right hand side of (REF ) after it is multiplied by $m^{1/2}$ .",
"Let us write $\\Psi _{N}({\\bf X}_{i}) = m^{-1/2}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}_{i}}\\mid {\\bf X}_{i}) - \\rho (\\Delta _{N})\\rbrace $ .",
"Note that $E\\lbrace \\Psi _{N}({\\bf X}_{i})\\rbrace = {\\bf 0}$ .",
"In order to show the asymptotic Gaussianity of $\\sum _{i=1}^{m} \\Psi _{N}({\\bf X}_{i})$ , it is enough to show that the triangular array $\\lbrace \\Psi _{N}({\\bf X}_{1}),\\ldots ,\\Psi _{N}({\\bf X}_{m})\\rbrace _{m=1}^{\\infty }$ of rowwise independent and identically distributed random elements satisfy the conditions of Corollary $7.8$ in [1].",
"Observe that for any $\\epsilon > 0$ , $\\sum _{i=1}^{m} P(||\\Psi _{N}({\\bf X}_{i})|| > \\epsilon ) \\le \\sum _{i=1}^{m} E\\lbrace ||E(SGN_{{\\bf Y} - {\\bf X}_{i}}\\mid {\\bf X}_{i}) - \\rho (\\Delta _{N})||^{3}\\rbrace /m^{3/2} \\le 8m^{-1/2}.$ Thus, $\\lim _{m \\rightarrow \\infty } \\sum _{i=1}^{m} P(||\\Psi _{N}({\\bf X}_{i})|| > \\epsilon ) = 0$ for every $\\epsilon > 0$ , which ensures that condition (1) of Corollary $7.8$ in [1] holds.",
"We next verify condition (2) of Corollary $7.8$ in [1].",
"Let us fix ${\\bf f} \\in {\\cal X}^{**}$ .",
"Since $||SGN_{{\\bf x}}|| = 1$ for all ${\\bf x} \\ne {\\bf 0}$ , we can choose $\\delta = 1$ in that condition (2).",
"Then, using the linearity of ${\\bf f}$ , we have $\\sum _{i=1}^{m} E[{\\bf f}^{2}\\lbrace \\Psi _{N}({\\bf X}_{i})\\rbrace ] = m^{-1} \\sum _{i=1}^{m} E[\\lbrace W_{N,i} - E(W_{N,i})\\rbrace ^{2}], $ where $W_{N,i} = {\\bf f}\\lbrace E(SGN_{{\\bf Y} - {\\bf X}_{i}}\\mid {\\bf X}_{i})\\rbrace $ .",
"Since the ${\\bf X}_{i}$ 's are identically distributed, the right hand side in (REF ) simplifies to $E[\\lbrace W_{N,1} - E(W_{N,1})\\rbrace ^{2}]$ .",
"Note that $W_{N,1} = {\\bf f}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1} + \\Delta _{N}}\\mid {\\bf X}_{1})\\rbrace $ , where ${\\bf Z}$ is an independent copy of ${\\bf X}_{1}$ .",
"Since the norm in ${\\cal X}$ is assumed to be twice Gâteaux differentiable, it follows from Theorem $4.6.15$ (a) and Proposition $4.6.16$ in [4] that the norm in ${\\cal X}$ is Fréchet differentiable.",
"This in turn implies that the map ${\\bf x} \\mapsto SGN_{{\\bf x}}$ is continuous on ${\\cal X}\\backslash \\lbrace {\\bf 0}\\rbrace $ (see, e.g., Corollary $4.2.12$ in [4]).",
"Using this fact, it can be shown that $E(SGN_{{\\bf Z} - {\\bf X}_{1} + \\Delta _{N}}\\mid {\\bf X}_{1}) \\longrightarrow E(SGN_{{\\bf Z} - {\\bf X}_{1}}\\mid {\\bf X}_{1}) $ as $m, n \\rightarrow \\infty $ for almost all values of ${\\bf X}_{1}$ .",
"Thus, we get the convergence of $E(W_{N,1})$ to $E[{\\bf f}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1}}\\mid {\\bf X}_{1})\\rbrace ]$ as $m,n \\rightarrow \\infty $ .",
"Similarly, it follows that $E(W_{N,1}^{2})$ converges to $E[{\\bf f}^{2}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1}}\\mid {\\bf X}_{1})\\rbrace ]$ as $m,n \\rightarrow \\infty $ .",
"So, $\\sum _{i=1}^{m} E[{\\bf f}^{2}\\lbrace \\Psi _{N}({\\bf X}_{i})\\rbrace ] \\rightarrow \\Gamma _{1}({\\bf f},{\\bf f})$ as $m, n \\rightarrow \\infty $ , where $\\Gamma _{1}$ is as defined before Theorem REF in Section .",
"This completes the verification of condition (2) of Corollary $7.8$ in [1].",
"Finally, for the verification of condition (3) of Corollary $7.8$ in [1], suppose that $\\lbrace {\\cal F}_{k}\\rbrace _{k \\ge 1}$ is a sequence of finite dimensional subspaces of ${\\cal X}^{*}$ such that ${\\cal F}_{k} \\subseteq {\\cal F}_{k+1}$ for all $k \\ge 1$ , and the closure of $\\bigcup _{k=1}^{\\infty } {\\cal F}_{k}$ is ${\\cal X}^{*}$ .",
"Such a sequence of subspaces exists because of the separability of ${\\cal X}^{*}$ .",
"For any ${\\bf x} \\in {\\cal X}^{*}$ and any $k \\ge 1$ , we define $d({\\bf x},{\\cal F}_{k}) = \\inf \\lbrace ||{\\bf x} - {\\bf y}|| : {\\bf y} \\in {\\cal F}_{k}\\rbrace $ .",
"It is straightforward to verify that for every $k \\ge 1$ , the map ${\\bf x} \\mapsto d({\\bf x}, {\\cal F}_{k})$ is continuous and bounded on any closed ball in ${\\cal X}^{*}$ .",
"Thus, using (REF ), it follows that $\\rho (\\Delta _{N}) \\rightarrow 0$ as $m, n \\rightarrow \\infty $ , and we have $\\sum _{i=1}^{m} E[d^{2}\\lbrace \\Psi _{N}({\\bf X}_{i}), {\\cal F}_{k}\\rbrace ] &=& m^{-1} \\sum _{i=1}^{m} E[d^{2}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{i} + \\Delta _{N}}\\mid {\\bf X}_{i}) - \\rho (\\Delta _{N}), {\\cal F}_{k}\\rbrace ] \\\\&=& E[d^{2}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1} + \\Delta _{N}}\\mid {\\bf X}_{1}) - \\rho (\\Delta _{N}), {\\cal F}_{k}\\rbrace ] \\\\&\\longrightarrow & E[d^{2}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1}}\\mid {\\bf X}_{1}), {\\cal F}_{k}\\rbrace ]$ as $m, n \\rightarrow \\infty $ .",
"From the choice of the ${\\cal F}_{k}$ 's, it can be shown that $d({\\bf x}, {\\cal F}_{k}) \\rightarrow 0$ as $k \\rightarrow \\infty $ for all ${\\bf x} \\in {\\cal X}^{*}$ .",
"So, we have $\\lim _{k \\rightarrow \\infty } E[d^{2}\\lbrace E(SGN_{{\\bf Z} - {\\bf X}_{1}}\\mid {\\bf X}_{1}), {\\cal F}_{k}\\rbrace ] = 0,$ and this completes the verification of condition (3) of Corollary $7.8$ in [1].",
"Thus, $\\sum _{i=1}^{m} \\Psi _{N}({\\bf X}_{i})$ converges weakly to a centered Gaussian random element in ${\\cal X}^{*}$ as $m, n \\rightarrow \\infty $ .",
"Further, its asymptotic covariance is $\\Gamma _{1}$ , which was obtained while checking condition (2) of Corollary $7.8$ in [1].",
"It follows from similar arguments that when the second term on the right hand side of (REF ) is multiplied by $n^{1/2}$ , it also converges weakly to a Gaussian random element in ${\\cal X}^{*}$ with the same distribution as $m, n \\rightarrow \\infty $ .",
"Hence, using the independence of the first two terms on the right hand side of (REF ), we have $(mn/N)^{1/2}\\lbrace T_{WMW} - \\rho (\\Delta _{N})\\rbrace \\longrightarrow G({\\bf 0},\\Gamma _{1})$ weakly as $m,n \\rightarrow \\infty $ under the sequence of shrinking shifts.",
"This, together with (REF ), completes the proof of the theorem.",
"[Proof of Theorem REF ] (a) Let us observe that $nN^{-1}T_{CFF} = mnN^{-1}||\\bar{{\\bf X}} - \\bar{{\\bf Y}}||^{2}$ .",
"For each $N \\ge 1$ , ${\\bf Y}$ has the same distribution as that of ${\\bf Z} + \\Delta _{N}$ , where ${\\bf Z}$ is an independent copy of ${\\bf X}$ .",
"Now, by the central limit theorem for independent and identically distributed random elements in a separable and type 2 Banach space (see, e.g.",
"Theorem $7.5$ (i) in [1]), it follows that $(mn/N)^{1/2}(\\bar{{\\bf Z}} - \\bar{{\\bf X}})$ converges weakly to $G({\\bf 0},\\Sigma )$ as $m, n \\rightarrow \\infty $ .",
"Thus, $(mn/N)^{1/2}\\left(\\bar{{\\bf Y}} - \\bar{{\\bf X}}\\right)$ , which has the same distribution as that of $(mn/N)^{1/2}\\left(\\bar{{\\bf Z}} - \\bar{{\\bf X}} + \\Delta _{N}\\right)$ , converges weakly to $G(\\delta ,\\Sigma )$ as $m, n \\rightarrow \\infty $ .",
"This proves part (a) of the proposition.",
"(b) Let ${\\bf v} = (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\psi _{1}\\rangle ,\\ldots ,\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\psi _{L}\\rangle )^{T}$ and $\\beta = (\\beta _{1},\\ldots ,\\beta _{L})^{T}$ .",
"It follows from the central limit theorem in $\\mathbb {R}^{L}$ that $(mn/N)^{1/2}\\lbrace {\\bf v} - (mn/N)^{-1/2}\\beta \\rbrace $ converges weakly to $N_{L}({\\bf 0},\\Lambda _{L})$ as $m, n \\rightarrow \\infty $ under the given sequence of shrinking shifts, where $\\Lambda _{L}$ is the diagonal matrix $Diag(\\lambda _{1},\\ldots ,\\lambda _{L})$ .",
"Thus, under the given sequence of shifts, $(mn/N)^{1/2}{\\bf v}$ converges weakly to a $N_{L}(\\beta ,\\Lambda _{L})$ distribution as $m, n \\rightarrow \\infty $ .",
"From arguments similar to those in the proof of Theorem 5.3 in [17], and using the assumptions in the present theorem, we get $&& \\max _{1 \\le k \\le L} (mn/N)^{1/2} \\left|\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{\\psi }_{k} - \\widehat{c}_{k}\\psi _{k}\\rangle \\right| = o_{P}(1) $ as $m, n \\rightarrow \\infty $ under this sequence of shifts.",
"Here $\\widehat{\\psi }_{k}$ is the empirical version of $\\psi _{k}$ and $\\widehat{c}_{k} = sign(\\langle \\widehat{\\psi }_{k},\\psi _{k}\\rangle )$ .",
"The limiting distribution of $mnN^{-1} \\sum _{k=1}^{L} (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{\\psi }_{k}\\rangle )^{2}$ is the same as that of $mnN^{-1} \\sum _{k=1}^{L} (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{c}_{k}\\psi _{k}\\rangle )^{2}$ in view of ((REF )).",
"Since the $\\widehat{c}_{k}$ 's take values $\\pm 1$ only, $mnN^{-1} \\sum _{k=1}^{L} (\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{c}_{k}\\psi _{k}\\rangle )^{2} = mnN^{-1}||{\\bf v}||^{2}$ , and the latter converges weakly to $||N_{L}(\\beta ,\\Lambda _{L})||^{2}$ as $m, n \\rightarrow \\infty $ .",
"Thus, $mnN^{-1}T_{HKR1}$ converges weakly to $\\sum _{k=1}^{L} \\lambda _{k}\\chi ^{2}_{(1)}(\\beta _{k}^{2}/\\lambda _{k})$ under the given sequence of shrinking shifts as $m, n \\rightarrow \\infty $ .",
"It also follows using similar arguments as in the proof of Theorem 5.3 in [17] that under the assumptions of the present theorem, and for the given sequence of shrinking shifts, we have $&& \\max _{1 \\le k \\le L} (mn/N)^{1/2} \\widehat{\\lambda }_{k}^{-1/2}\\left|\\langle \\bar{{\\bf X}} - \\bar{{\\bf Y}},\\widehat{\\psi }_{k} - \\widehat{c}_{k}\\psi _{k}\\rangle \\right| = o_{P}(1)$ as $m, n \\rightarrow \\infty $ .",
"Similar arguments as in the case of $T_{HKR1}$ now yield the asymptotic distribution of $mnN^{-1}T_{HKR2}$ , and this completes the proof."
]
] | 1403.0201 |
[
[
"Singularity of the varieties of representations of lattices in solvable\n Lie groups"
],
[
"Abstract For a lattice $\\Gamma$ of a simply connected solvable Lie group $G$, we describe the analytic germ in the variety of representations of $\\Gamma$ at the trivial representation as an analytic germ which is linearly embedded in the analytic germ associated with the nilpotent Lie algebra determined by $G$.",
"By this description, under certain assumption, we study the singularity of the analytic germ in the variety of representations of $\\Gamma$ at the trivial representation by using the Kuranishi space construction.",
"By a similar technique, we also study deformations of holomorphic structures of trivial vector bundles over complex parallelizable solvmanifolds."
],
[
"introduction",
"Let $X$ be an analytic germ in ${n}$ at the origin defined by analytic equations $f_{1}(z)=0,\\dots ,f_{k}(z)=0.$ We say that $X$ is cut out by polynomial equations of degree at most $\\nu $ if $f_{1}(z),\\dots ,f_{k}(z)$ are polynomial functions of degree at most $\\nu $ with trivial linear terms.",
"We say that an analytic germ $Y$ is linearly embedded in $X$ if for a subspace $V\\subset {n}$ , the germ $Y$ is equivalent to an analytic germ in $V$ at the origin defined by analytic equations $f_{1}(z)=0,\\dots ,f_{k}(z)=0, \\,\\,\\, z\\in V.$ If $X$ is cut out by polynomial equations of degree at most $\\nu $ and $Y$ is linearly embedded in $X$ , then $Y$ is also cut out by polynomial equations of degree at most $\\nu $ .",
"Let $\\Gamma $ be a finitely generated group, $A$ a linear algebraic group with Lie algebra $a$ and $R(\\Gamma , A)$ the set of homomorphisms $\\Gamma \\rightarrow A$ .",
"Then $R(\\Gamma , A)$ can be considered as an affine algebraic variety.",
"For a representation $\\rho \\in R(\\Gamma , A)$ we are interested in the analytic germ $(R(\\Gamma , A),\\rho )$ .",
"The singularity of the analytic germ $(R(\\Gamma , A),\\rho )$ can be considered as an obstruction of deformations of $\\rho $ .",
"If $\\Gamma $ is the fundamental group of a manifold $M$ , we can geometrically describe the analytic germ $(R(\\Gamma , A),\\rho )$ by using the deformation theory of differential graded Lie algebras (for short, DGLAs) developed by Goldman and Millson [6], [7].",
"By such technique and the Hodge theory of local systems over Kähler manifolds studied by Simpson [14], if $\\Gamma $ is a Kähler group (i.e.",
"a group which can be the fundamental group of a compact Kähler manifold), then for a semisimple representation $\\rho \\rightarrow {\\rm GL}_{m}($ , the analytic germ $(R(\\Gamma , {\\rm GL}_{m}(),\\rho )$ is cut out by polynomial equations of degree at most 2.",
"However in general the analytic germ $(R(\\Gamma , A),\\rho )$ is not cut out by polynomial equations of degree at most 2.",
"In [6], Goldman and Millson observed that for a lattice $\\Gamma $ in the three dimensional real Heisenberg group, the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is equivalent to a cubic cone.",
"In this paper we consider a certain class of groups which contains this example.",
"Let $\\Gamma $ be a lattice in a simply connected solvable Lie group $G$ .",
"Then the solvmanifold $G/\\Gamma $ is an aspherical manifold with the fundamental group $\\Gamma $ .",
"The purpose of this paper is to study the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ .",
"For a manifold $M$ the analytic germ at the trivial representation of the fundamental group of $M$ can be studied by the differential graded algebra (for short, DGA) $A^{\\ast }(M)$ of differential forms on $M$ .",
"The following result is known.",
"Theorem 1.1 ([6], [7],[4]) Let $M$ be a compact manifold with the fundamental group $\\Gamma $ .",
"Suppose that we have a finite dimensional sub-DGA $C^{\\ast }\\subset A^{\\ast }(M)\\otimes such that the inclusion induces a cohomology isomorphism and $ C0=.",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is equivalent to the analytic germ $(F(C^{\\ast }, a), 0)$ at the origin 0 for the affine variety $F(C^{\\ast }, a)=\\left\\lbrace \\omega \\in C^{\\ast }\\otimes a: d\\omega +\\frac{1}{2}[\\omega ,\\omega ]=0\\right\\rbrace .$ Consider a solvmanifold $G/\\Gamma $ , Lie algebra ${g}$ of $G$ and the cochain complex $\\bigwedge {g}^{\\ast }$ which is regarded as a differential graded algebra of left-$G$ -invariant forms on $G/\\Gamma $ .",
"Suppose that $G$ is completely solvable.",
"In [8] Hattori proved that the inclusion $\\bigwedge {g}^{\\ast }\\subset A^{\\ast }(G/\\Gamma )$ induces a cohomology isomorphism.",
"By Theorem REF and Hattori's theorem, in [4], Dimca and Papadima remarked that the analytic germ $(R(\\Gamma , A),{\\bf 1})$ is equivalent to the analytic germ $(F(\\bigwedge {g}^{\\ast }, a), 0)$ at the origin 0.",
"However, for a general solvmanifold $G/\\Gamma $ , the inclusion $\\bigwedge {g}^{\\ast }\\subset A^{\\ast }(G/\\Gamma )$ does not induce a cohomology isomorphism.",
"In this paper, we consider general solvmanifolds.",
"Let ${g}$ be a solvable Lie algebra.",
"Then we can define the nilpotent Lie algebra $u$ called nilshadow of ${g}$ which is uniquely determined by ${g}$ , as shown in [5].",
"Theorem 1.2 ([9]) Let $G$ be a simply connected solvable Lie group with a lattice $\\Gamma $ and ${g}$ the Lie algebra of $G$ .",
"We consider the nilshadow $u$ of ${g}$ .",
"Then we have a sub-DGA $A^{\\ast }_{\\Gamma }\\subset A^{\\ast }(G/\\Gamma )\\otimes such that:\\begin{itemize}\\item The inclusion A^{\\ast }_{\\Gamma }\\subset A^{\\ast }(G/\\Gamma )\\otimes induces an isomorphism in cohomology.\\item A^{\\ast }_{\\Gamma } can be regarded as a sub-DGA of \\bigwedge u^{\\ast }\\otimes .\\end{itemize}$ See Section for the constructions of the nilshadow and DGA $A_{\\Gamma }^{\\ast }$ .",
"By this theorem and Theorem REF , the analytic germ $(R(\\Gamma , A),{\\bf 1})$ is equivalent to the analytic germ $(F(A^{\\ast }_{\\Gamma }, a), 0)$ .",
"By the second assertion of the theorem we have the following theorem.",
"Theorem 1.3 Let $\\Gamma $ be a lattice in a simply connected solvable Lie group $G$ and ${g}$ the Lie algebra of $G$ .",
"Let $A$ be a linear algebraic group with the Lie algebra $a$ .",
"Consider the nilshadow $u$ of ${g}$ .",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is linearly embedded in the analytic germ $(F(\\bigwedge u^{\\ast }, a), 0)$ at the origin 0 for the affine variety $F(\\bigwedge u^{\\ast }, a)=\\left\\lbrace \\omega \\in \\bigwedge u^{\\ast }\\otimes a: d\\omega +\\frac{1}{2}[\\omega ,\\omega ]=0\\right\\rbrace $ This theorem is useful for estimating the singularity of the analytic germ $(R(\\Gamma , A),{\\bf 1})$ .",
"Let ${n}$ be a $\\nu $ -step nilpotent Lie algebra.",
"Consider the lower central series ${n}={n}^{(1)}\\supset {n}^{(2)}\\supset \\dots \\supset {n}^{(\\nu )}(\\ne \\lbrace 0\\rbrace )\\supset {n}^{(\\nu +1)}=\\lbrace 0\\rbrace $ where ${n}^{(i+1)}=[{n},{n}^{(i)}]$ .",
"Take a subspace $a^{(i)}$ such that ${n}^{(i)}={n}^{(i+1)}\\oplus a^{(i)}$ .",
"We have ${n}=a^{(1)}\\oplus a^{(2)}\\oplus \\dots \\oplus a^{(\\nu )}.$ It is known that $[{n}^{(i)},{n}^{(j)}]\\subset {n}^{(i+j)}$ (see [2]).",
"A nilpotent Lie algebra $n$ is called naturally graded if we can choose subspaces $a^{(i)}$ such that $[a^{(i)},a^{(j)}]\\subset a^{(i+j)}$ .",
"We prove the following proposition by using the construction of Kuranishi spaces of DGLAs.",
"Proposition 1.4 Let ${n}$ be a $\\nu $ -step naturally graded nilpotent Lie algebra and $g$ a Lie algebra.",
"Then the analytic germ $(F(\\bigwedge n^{\\ast }, g), 0)$ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"By this proposition and Theorem REF , we have the following theorem.",
"Theorem 1.5 Let $\\Gamma $ be a lattice in a simply connected solvable Lie group $G$ and ${g}$ the Lie algebra of $G$ .",
"Let $A$ be a linear algebraic group with a Lie algebra $a$ .",
"Consider the nilshadow $u$ of ${g}$ .",
"We suppose that the Lie algebra $u$ is $\\nu $ -step naturally graded.",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"Let $n$ be a two-step nilpotent Lie algebra.",
"For any complement $a^{(1)}$ of ${n}^{(2)}$ in $n$ , we have ${n}=a^{(1)}\\oplus {n}^{(2)}$ and $[a^{(1)},a^{(1)}]\\subset {n}^{(2)}$ and so a two-step nilpotent Lie algebra $n$ is naturally graded.",
"Hence as an application of Theorem REF , we have the following Corollary Corollary 1.6 Let $\\Gamma $ be a lattice in a simply connected solvable Lie group $G$ and ${g}$ the Lie algebra of $G$ .",
"Let $A$ be a linear algebraic group with a Lie algebra $a$ .",
"Consider the nilshadow $u$ of ${g}$ .",
"We suppose that the Lie algebra $u$ is two-step nilpotent.",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most 3."
],
[
"Nilshadows and cohomology of solvmanifolds",
"Let ${g}$ be a solvable $K$ -Lie algebra for $K=\\mathbb {R}$ or $.Let $ n$ be the nilradical of $ g$.There exists a subvector space (not necessarily Lie algebra) $ V$ of $ g$ so that$ g=Vn$ as the direct sum of vector spaces and for any $ A,BV$ $ (adA)s(B)=0$ where $ (adA)s$ is the semi-simple part of $ adA$ (see \\cite [Proposition I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.1.1] {DER}).We define the map $ ads:gD(g)$ as$ adsA+X=(adA)s$ for $ AV$ and $ Xn$.Then we have $ [ads(g), ads(g)]=0$ and $ ads$ is linear (see \\cite [Proposition I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.1.1] {DER}).Since we have $ [g,g]n$, the map $ ads:gD(g)$ is a representation and the image $ ads(g)$ is abelian and consists of semi-simple elements.Let $ g =Im adsg$and$$u=\\lbrace X-{\\rm ad}_{sX}\\in \\bar{{g}} \\vert X\\in {g}\\rbrace .$$Then we have $ [g,g]nu$ and $ u$ is the nilradical of $ g$ (see \\cite {DER}).Hence we have $ g= Im adsu$.It is known that the structure of the Lie algebra $ u$ is independent of a choice of a subvector space $ V$ (see \\cite [Corollary I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.3.6]{DER} ).$ Lemma 2.1 ([9]) Suppose ${g}=\\mathbb {R}^{k}\\ltimes _{\\phi } {n}$ such that $\\phi $ is a semi-simple action and ${n}$ is nilpotent.",
"Then the nilshadow $u$ of ${g}$ is the direct sum $\\mathbb {R}^{k}\\oplus {n}$ .",
"Let $G$ be a simply connected solvable Lie group with the $\\mathbb {R}$ -Lie algebra $g$ .",
"We denote by ${\\rm Ad}_{s}:G\\rightarrow {\\rm Aut}({g})$ the extension of ${\\rm ad}_{s}$ .",
"Then ${\\rm Ad}_{s}(G)$ is diagonalizable.",
"Let $X_{1},\\cdots ,X_{n}$ be a basis of ${g}\\otimes { such that {\\rm Ad}_{s} is represented by diagonal matrices.Then we have {\\rm Ad}_{sg}X_{i}=\\alpha _{i}(g)X_{i} for characters \\alpha _{i} of G.Let x_{1},\\dots ,x_{n} be the dual basis of X_{1},\\dots ,X_{n}.",
"}We suppose $ G$ has a lattice $$.Then we consider the sub-DGA $ A$ of the de Rham complex $ A(G/) which is given by $A^{p}_{\\Gamma }=\\left\\langle \\alpha _{I} x_{I} {\\Big \\vert } \\begin{array}{cc}I\\subset \\lbrace 1,\\dots ,n\\rbrace ,\\\\ (\\alpha _{I})_{\\vert _{\\Gamma }}=1 \\end{array}\\right\\rangle .$ where for a multi-index $I=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ we write $x_{I}=x_{i_{1}}\\wedge \\dots \\wedge x_{i_{p}}$ , and $\\alpha _{I}=\\alpha _{i_{1}}\\cdots \\alpha _{i_{p}}$ .",
"Theorem 2.2 ([9]) Let $G$ be a simply connected solvable Lie group with a lattice $\\Gamma $ .",
"Then we have : The inclusion $A^{\\ast }_{\\Gamma }\\subset A^{\\ast }(G/\\Gamma )\\otimes induces an isomorphism in cohomology.\\item $ A$ can be regarded as a sub-DGA of $ u.",
"We explain the second assertion more precisely.",
"We consider the subspace $\\tilde{u}=\\langle \\alpha _{1}^{-1}X_{1},\\dots , \\alpha _{n}^{-1}X_{n}\\rangle $ of the space of complex valued vector fields on $G$ .",
"Then $\\tilde{u}=\\langle \\alpha _{1}^{-1}X_{1},\\dots , \\alpha _{n}^{-1}X_{n}\\rangle $ is a Lie sub-algebra of the Lie algebra of vector fields and the map $\\tilde{u}\\ni \\alpha _{i}^{-1}X_{i}\\mapsto X_{i}-{\\rm ad}_{sX_{i}}\\in u\\otimes is a Lie algebra isomorphismwhere $ $ is the nilshadow of $ g$ (see \\cite [Proof of Lemma 5.2]{K2}).$ Example 1 Let ${g}$ be a 4-dimensional Lie algebra such that ${g}=\\langle T,X,Y,Z\\rangle $ $[T,X]=X$ , $[T,Y]=-Y$ , $[X,Y]=Z$ .",
"Then we have the splitting ${g}=\\langle T\\rangle \\ltimes \\langle X,Y,Z\\rangle $ such that $\\langle X,Y,Z\\rangle $ is the three dimensional real Heisenberg Lie algebra $h(3)$ and the action of $\\langle T\\rangle $ is semi-simple.",
"Hence by Lemma REF , the nilshadow $u$ of ${g}$ is given by $u=\\mathbb {R}\\oplus h(3)$ .",
"Hence as similar to [6], the analytic germ $(F(\\bigwedge u^{\\ast }, a), 0)$ is equivalent to a cubic cone.",
"Consider the simply connected solvable Lie group $G$ whose Lie algebra is ${g}$ .",
"Then $G$ has a lattice $\\Gamma $ [13].",
"We can easily show that the DGA $A^{\\ast }(G/\\Gamma )$ is formal and hence the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most 2.",
"Hence $(R(\\Gamma , A),{\\bf 1})$ is linearly embedded in the analytic germ $(F(\\bigwedge u^{\\ast }, a), 0)$ but its singularity is different from $(F(\\bigwedge u^{\\ast }, a), 0)$ .",
"By Lemma REF , we give one more corollary of Theorem REF .",
"Corollary 2.3 Let ${g}=\\mathbb {R}^{k}\\ltimes _{\\phi } {n}$ such that $\\phi $ is a semi-simple action and ${n}$ is a $\\nu $ -step naturally graded nilpotent Lie algebra.",
"Consider the simply connected solvable Lie group $G$ whose Lie algebra is ${g}$ .",
"Suppose $G$ has a lattice $\\Gamma $ .",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most $\\nu +1$ ."
],
[
"Finite-dimensional DGAs of Poincaré duality type",
"Let $A^{\\ast }$ be a finite-dimensional graded commutative $-algebra.\\begin{definition}[\\cite {KSP}]A^{\\ast } is of Poincaré duality type (PD-type) if the following conditions hold:\\begin{itemize}\\item A^{\\ast <0}=0 and A^{0}= where 1 is the identity element of A^{\\ast }.\\item For some positive integer n, A^{\\ast >n}=0 and A^{n}=v̏ for v\\ne 0.\\item For any 0<i<n the bi-linear map A^{i}\\times A^{n-i}\\ni (\\alpha ,\\beta )\\mapsto \\alpha \\cdot \\beta \\in A^{n} is non-degenerate.\\end{itemize}\\end{definition}$ Suppose $A^{\\ast }$ is of PD-type.",
"Let $h$ be a Hermitian metric on $A^{\\ast }$ which is compatible with the grading.",
"Take $v\\in A^{n}$ such that $h(v,v)=1$ .",
"Define the $-anti-linear map $ : AiAn-i$ as $ =h(,)v$.$ Definition 3.1 ([10]) A finite-dimensional DGA $(A^{\\ast },d)$ is of PD-type if the following conditions hold: $A^{\\ast }$ is a finite-dimensional graded $-algebra of PD-type.\\item $ dAn-1=0$ and $ dA0=0$.$ Let $(A^{\\ast },d)$ be a finite-dimensional DGA of PD-type.",
"Denote $d^{\\ast }=-\\bar{\\ast }d\\bar{\\ast }$ .",
"Lemma 3.2 ([10]) We have $h(d\\alpha , \\beta )=h(\\alpha ,d^{\\ast }\\beta )$ for $\\alpha \\in A^{i-1}$ and $\\beta \\in A^{i}$ .",
"Define $\\Delta =dd^{\\ast }+d^{\\ast }d$ .",
"and ${\\mathcal {H}}^{\\ast }(A)=\\ker \\Delta $ .",
"By Lemma REF and finiteness of the dimension of $A^{\\ast }$ , we can easily show the following lemma.",
"Lemma 3.3 ([10]) We have the Hodge decomposition $A^{r}={\\mathcal {H}}^{r}(A)\\oplus \\Delta (A^{r})={\\mathcal {H}}^{r}(A)\\oplus d(A^{r-1})\\oplus d^{\\ast }(A^{r+1}).$ By this decomposition, the inclusion ${\\mathcal {H}}^{\\ast }(A)\\subset A^{\\ast }$ induces a isomorphism ${\\mathcal {H}}^{p}(A)\\cong H^{p}(A)$ of vector spaces.",
"We denote by $H$ the projection $H: A^{p} \\rightarrow {\\mathcal {H}}^{p}(A)$ and define the operator $G$ as the composition $ \\Delta ^{-1}_{\\vert \\Delta (A^{p})}\\circ ({\\rm id }-H)$ .",
"Let $\\beta :A^{\\ast }\\rightarrow dA^{\\ast -1}$ be the projection for the decomposition $A^{r}={\\mathcal {H}}^{r}(A)\\oplus d(A^{r-1})\\oplus d^{\\ast }(A^{r+1}).$ The restriction map $d: d^{\\ast }(A^{\\ast })\\rightarrow d(A^{\\ast -1})$ is an isomorphism.",
"Take the inverse $d^{-1}:d(A^{\\ast -1})\\rightarrow d^{\\ast }(A^{\\ast })$ .",
"Consider the map $d^{\\ast }G:A^{\\ast }\\rightarrow A^{\\ast -1}$ .",
"Then for $\\omega \\in {\\mathcal {H}}^{r}(A)$ , $d^{\\ast }x\\in d^{\\ast }(A^{r})$ and $d^{\\ast }y \\in d^{\\ast }(A^{r+1})$ , we have $d^{\\ast }G(\\omega +dd^{\\ast }x+d^{\\ast }y)=d^{\\ast }(dd^{\\ast })^{-1}dd^{\\ast }x=d^{\\ast }x.$ Hence we have $d^{\\ast }G=d^{-1}\\circ \\beta $ .",
"Kuranishi spaces of finite-dimensional DGLAs Let $L^{\\ast }$ be a finite-dimensional DGLA with a differential $d$ .",
"Consider the splitting $d(L^{p})\\rightarrow L^{p}$ for the short exact sequence ${0[r]& {\\rm ker} \\,d_{\\vert _{L^{p}}} [r]& L^{p}[r]^d&d(L^{p})[r]&0}$ and the splitting $H^{p}(L^{\\ast })\\rightarrow {\\rm ker} \\,d_{\\vert _{L^{p}}}$ for the short exact sequence ${0[r]& d(L^{p-1})[r]& {\\rm ker} \\,d_{\\vert _{L^{p}}}[r]&H^{p}(L^{\\ast })[r]&0.",
"}$ Denote by $\\mathcal {A}^{p}$ and $\\mathcal {H}^{p}$ the images of the splittings $d(L^{p})\\rightarrow L^{p}$ and $H^{p}(L^{\\ast })\\rightarrow {\\rm ker} \\,d_{\\vert _{L^{p}}}$ respectively.",
"Then we have $L^{p}= {\\mathcal {H}}^{p}\\oplus d(L^{p-1})\\oplus \\mathcal {A}^{p}.$ Consider the projections $\\beta ^{\\ast }: L^{\\ast }\\rightarrow d(L^{\\ast -1})$ , $H: L^{\\ast }\\rightarrow \\mathcal {H}^{\\ast }$ and $\\alpha ^{\\beta }: L^{\\ast }\\rightarrow \\mathcal {A}^{\\ast }$ .",
"Since the restriction $ d : \\mathcal {A}^{p} \\rightarrow d(L^{p})$ is an isomorphism, we have the inverse $d^{-1}: d(L^{p})\\rightarrow \\mathcal {A}^{p}$ of $ d : \\mathcal {A}^{p} \\rightarrow d(L^{p})$ .",
"We define $\\delta =d^{-1}\\circ \\beta :L^{p+1}\\rightarrow L^{p}$ .",
"Define the map $F: L^{1}\\rightarrow L^{1}$ as $F(\\zeta )=\\zeta +\\frac{1}{2}\\delta [\\zeta ,\\zeta ].$ Then by the inverse function theorem, on a small ball $B$ in $L^{1}$ , the map $F$ is an analytic diffeomorphism.",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ is defined by ${\\mathcal {K}}(L^{\\ast })=\\lbrace \\eta \\in F(B)\\cap \\mathcal {H}^{1}: H([F^{-1}(\\eta ),F^{-1}(\\eta )])=0\\rbrace .$ It is known that the analytic germ $({\\mathcal {K}}(L^{\\ast }),0)$ is equivalent to the germ at the origin for the variety $\\left\\lbrace \\zeta \\in L^{1}: d\\zeta +\\frac{1}{2}[\\zeta ,\\zeta ]=0, \\,\\, \\delta \\zeta =0\\right\\rbrace $ (see [7]).",
"In particular, if $d(L^{0})=0$ , then ${\\mathcal {K}}(L^{\\ast })$ is equivalent to the germ at the origin for the variety $\\left\\lbrace \\zeta \\in L^{1}: d\\zeta +\\frac{1}{2}[\\zeta ,\\zeta ]=0 \\right\\rbrace .$ Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $L^{1}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}\\delta [\\phi _{s}(t), \\phi _{r-s}(t)].$ Then $F^{-1}$ is given by $\\phi _{1}(t)\\mapsto \\phi (t)$ and the Kuranishi space $ {\\mathcal {K}}(L^{\\ast })$ is an analytic germ in ${m}$ at the origin defined by equations $H\\left([\\phi (t),\\phi (t)]\\right)=0.$ Let $A$ be a finite-dimensional DGA of PD-type and ${g}$ a Lie algebra, and consider the DGLA $A^{\\ast }\\otimes {g}$ .",
"Then we have the Hodge decomposition $A^{\\ast }\\otimes {g}={\\mathcal {H}}^{p}(A)\\otimes {g}\\oplus d(A^{p-1})\\otimes {g}\\oplus d^{\\ast }(A^{p+1})\\otimes {g}$ as above with $\\delta =d^{\\ast } G\\otimes {\\rm id}$ .",
"Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}(A^{\\ast })\\otimes {g}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $A^{1}\\otimes {g}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}d^{\\ast } G\\otimes {\\rm id}[\\phi _{s}(t), \\phi _{r-s}(t)].$ By the above argument we have the following lemma.",
"Lemma 3.4 The analytic germ $(F(A^{\\ast }, g), 0)$ is equivalent to the analytic germ in ${m}$ at the origin defined by equations $H\\left([\\phi (t),\\phi (t)]\\right)=0.$ Nilpotent Lie algebras Let ${n}$ be a $\\nu $ -step nilpotent $K$ -Lie algebra for $K=\\mathbb {R}$ or $.Consider the lower central series$${n}={n}^{(1)}\\supset {n}^{(2)}\\supset \\dots \\supset {n}^{(\\nu )}\\supset {n}^{(\\nu +1)}=\\lbrace 0\\rbrace $$where $ n(i+1)=[n,n(i)]$.Take a subspace $ a(i)$ such that $ n(i)=n(i+1)a(i)$.We have$${n}=a^{(1)}\\oplus a^{(2)}\\oplus \\dots \\oplus a^{(\\nu )}.$$Consider the dual spaces $ n$ and $ a(i)$ of $ n$ and $ a(i)$ respectively.We consider the cochain complex $ n$ of the Lie algebra with the differential $ d$.Then $ n$ is a a finite-dimensional DGA of PD-type.We have$$\\bigwedge {n}^{\\ast } =\\left(\\bigwedge a^{(1)\\ast }\\right)\\wedge \\dots \\wedge \\left(\\bigwedge a^{(\\nu )\\ast }\\right).$$We have$$H^{1}({n})={\\rm ker}\\, d_{\\bigwedge ^{1} {n}^{\\ast }}= a^{(1)\\ast }.$$$ Lemma 3.5 ${\\rm ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}\\subset \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Let $\\sigma \\in {\\rm ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}$ .",
"For a positive integer $k<\\nu $ , we say that $\\sigma $ is $k$ -decomposable if we have a decomposition $\\sigma =\\sigma _{1}+\\sigma _{2}+\\sigma _{3}$ such that: $\\sigma _{1}\\in \\bigoplus _{i+j\\le \\nu +1, i\\le j, k<j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ $\\sigma _{2}\\in \\bigoplus _{ i\\le k} a^{(i)\\ast }\\wedge a^{(k)\\ast }.$ $\\sigma _{3}\\in \\bigoplus _{ i\\le j, j<k} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ If $k\\le \\frac{\\nu +1}{2}$ , then we have $\\sigma \\in \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Consider the case $\\frac{\\nu +1}{2}<k$ .",
"For $X,Y\\in {n}$ and $Z\\in {n}^{(k)}$ , we have $\\sigma _{1}([X,Y],Z)=0$ , $\\sigma _{2}(X,[Y,Z])=0$ , $\\sigma _{2}(Y,[X,Z])=0$ , $\\sigma _{3}([X,Y],Z)=0$ , $\\sigma _{3}(X,[Y,Z])=0$ and $\\sigma _{3}(Y,[X,Z])=0$ .",
"By $d\\sigma =0$ , we have $\\sigma _{2}([X,Y],Z)=\\sigma _{1}(X,[Y,Z])-\\sigma _{1}(Y,[X,Z]).$ Taking $X\\in {n}$ and $Y\\in {n}^{(l-1)}$ such that $\\nu +1<k+l$ , we have $\\sigma _{2}([X,Y],Z)=0.$ Hence for $W\\in {n}^{(l)}$ and $Z\\in {n}^{(k)}$ such that $\\nu +1<k+l$ , we have $\\sigma _{2}(W,Z)=0.$ Thus we have $\\sigma _{2}\\in \\bigoplus _{i+k\\le \\nu +1, i\\le k} a^{(i)\\ast }\\wedge a^{(k)\\ast }.$ Hence taking $\\sigma _{1}^{\\prime }=\\sigma _{1}+\\sigma _{2}$ and $\\sigma _{3}=\\sigma _{2}^{\\prime }+\\sigma _{3}^{\\prime }$ such that $\\sigma _{2}^{\\prime }\\in \\bigoplus _{ i\\le k-1} a^{(i)\\ast }\\wedge a^{(k-1)\\ast }$ and $\\sigma _{3}^{\\prime }\\in \\bigoplus _{ i\\le j, j<k-1} a^{(i)\\ast }\\wedge a^{(j)\\ast },$ by the decomposition $\\sigma =\\sigma _{1}^{\\prime }+\\sigma ^{\\prime }_{2}+\\sigma ^{\\prime }_{3}$ , $\\sigma $ is $(k-1)$ -decomposable.",
"Thus we can say that if $\\sigma $ is $k$ -decomposable and $\\frac{\\nu +1}{2}<k-l-1$ for an integer $l$ , then $\\sigma $ is also $(k-l)$ -decomposable.",
"Take $l$ such that $k-l\\le \\frac{\\nu +1}{2}$ .",
"Then we can say $\\sigma \\in \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Hence it is sufficient to show the above decomposition of $\\sigma $ for $k=\\nu -1$ .",
"This was shown in [1].",
"Hence the Lemma follows.",
"It is known that $[{n}^{(i)},{n}^{(j)}]\\subset {n}^{(i+j)}$ (see [2]) and hence we have $d \\left(a^{(k)\\ast }\\right)\\subset \\bigoplus _{i+j\\le k, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Definition 3.6 A nilpotent Lie algebra $n$ is called naturally graded if we can take subspaces $a^{(i)}\\subset n$ such that ${n}^{(i)}={n}^{(i+1)}\\oplus a^{(i)}$ and $[a^{(i)},a^{(j)}]\\subset a^{(i+j)}$ for each $i,j$ where ${n}={n}^{(1)}\\supset {n}^{(2)}\\supset \\dots \\supset {n}^{(\\nu )}\\supset {n}^{(\\nu +1)}=\\lbrace 0\\rbrace $ is the lower central series of $n$ .",
"If $n$ is naturally graded, then we have $d \\left(a^{(k)\\ast }\\right)\\subset W_{k}$ where $ W_{k}=\\bigoplus _{i+j= k, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }$ .",
"Let $g$ be a Hermitian metric on ${n}$ such that the sum ${n}=a^{(1)}\\oplus a^{(2)}\\oplus \\dots \\oplus a^{(\\nu )}$ is an orthogonal direct sum.",
"Then $g$ give a Hermitian metric on the finite-dimensional DGA $\\bigwedge {n}^{\\ast }$ of PD-type.",
"Consider the decomposition $\\bigwedge ^{r}{n}^{\\ast }={\\mathcal {H}}^{r}(\\bigwedge {n}^{\\ast })\\oplus d(\\bigwedge ^{r-1}{n}^{\\ast })\\oplus d^{\\ast }(\\bigwedge ^{r+1}{n}^{\\ast }).$ Then $\\bigwedge ^{2}{n}^{\\ast }=W_{1}\\oplus W_{2}\\oplus \\dots \\oplus W_{2\\nu }$ is an orthogonal direct sum and we have $d^{-1}\\circ \\beta (W_{k})\\subset a^{(k)\\ast }$ by $d(a^{(k)\\ast })\\subset W_{k}$ .",
"Proposition 3.7 Let ${n}$ be a $\\nu $ -step naturally graded nilpotent Lie algebra and $g$ a Lie algebra.",
"Then the analytic germ $(F(\\bigwedge u^{\\ast }, g), 0)$ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}(\\bigwedge u^{\\ast })\\otimes {g}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $L^{1}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}\\delta [\\phi _{s}(t), \\phi _{r-s}(t)].$ By Lemma REF , the analytic germ $(F(\\bigwedge u^{\\ast }, g), 0)$ is equivalent to the analytic germ in ${m}$ at the origine defined by equations $ H\\left([\\phi (t),\\phi (t)]\\right)=0$ where $H:\\bigwedge ^{\\ast }{n}^{\\ast }\\otimes {g}\\rightarrow {\\mathcal {H}}^{\\ast }(\\bigwedge u^{\\ast })\\otimes {g}$ is the projection.",
"We have $[ a^{(i)\\ast }\\otimes {g}, a^{(j)\\ast }\\otimes {g}]\\subset W_{i+j}\\otimes {g}.$ By $d^{\\ast }G(W_{k})=d^{-1}\\circ \\beta (W_{k})\\subset a^{(k)\\ast }$ , we have $d^{\\ast }G\\otimes {\\rm id} ([a^{(i)\\ast }\\otimes {g}, a^{(j)\\ast }\\otimes {g}])\\subset a^{(i+j)\\ast }\\otimes {g}.$ This implies $\\phi _{r}(t)\\in a^{(r)\\ast }\\otimes {g}$ and we have $\\phi (t)=\\phi _{1}(t)+\\dots +\\phi _{\\nu }(t).$ By Lemma REF , we have $ {\\mathcal {H}}^{2}(\\bigwedge {n}^{\\ast })\\subset {\\rm Ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }} \\subset \\bigoplus _{l\\le \\nu +1, }W_{l}$ and hence $ H(\\bigwedge ^{2} {n}^{\\ast }\\otimes {g})\\subset {\\rm Ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}\\otimes {g}\\subset \\bigoplus _{l\\le \\nu +1, }W_{l}\\otimes {g}.$ Since we have $ [\\phi _{i}(t),\\phi _{j}(t)]\\in W_{i+j}\\otimes {g}$ by $\\phi _{r}(t)\\in a^{(r)\\ast }\\otimes {g}$ , we have $H[\\phi _{i}(t),\\phi _{j}(t)]=0$ for $\\nu +1<i+j$ .",
"Hence $H[\\phi (t),\\phi (t)]=0$ are polynomial equations of degree at most $\\nu +1$ .",
"Complex analogy Complex parallelizable solvmanifolds Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group.",
"Consider the Lie algebra ${g}_{1,0}$ (resp.",
"${g}_{0,1}$ ) of the left-invariant holomorphic (resp.",
"anti-holomorphic) vector fields on $G$ .",
"Let $N$ be the nilradical of $G$ .",
"We can take a simply connected complex nilpotent subgroup $C\\subset G$ such that $G=C\\cdot N$ (see [3]).",
"Since $C$ is nilpotent, the map $C\\ni c \\mapsto ({\\rm Ad}_{c})_{s}\\in {\\rm Aut}({g}_{1,0})$ is a homomorphism where $({\\rm Ad}_{c})_{s}$ is the semi-simple part of ${\\rm Ad}_{s}$ .",
"We have a basis $X_{1},\\dots ,X_{n}$ of ${g}_{1,0}$ such that $({\\rm Ad}_{c})_{s}={\\rm diag} (\\alpha _{1}(c),\\dots ,\\alpha _{n}(c))$ for $c\\in C$ .",
"Let $x_{1},\\dots , x_{n}$ be the basis of ${g}^{\\ast }_{1,0}$ which is dual to $X_{1},\\dots ,X_{n}$ .",
"Theorem 4.1 ([11]]) Suppose $G$ has a lattice $\\Gamma $ .",
"Let $B^{\\ast }_{\\Gamma }$ be the subcomplex of $(A^{0,\\ast }(G/\\Gamma ),\\bar{\\partial }) $ defined as $B^{\\ast }_{\\Gamma }=\\left\\langle \\frac{\\bar{\\alpha }_{I}}{\\alpha _{I} }\\bar{x}_{I}{\\Big \\vert }\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1\\right\\rangle $ where for a multi-index $I=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ we write $x_{I}=x_{i_{1}}\\wedge \\dots \\wedge x_{i_{p}}$ , and $\\alpha _{I}=\\alpha _{i_{1}}\\cdots \\alpha _{i_{p}}$ .",
"Consider the nilshadow $u$ of the $-Lie algebra $ g$.Then we have:\\begin{itemize}\\item The inclusion B^{\\ast }_{\\Gamma }\\subset A^{0,\\ast }(G/\\Gamma ) induces an isomorphism in cohomology.\\item B^{\\ast }_{\\Gamma } can be regarded as a sub-DGA of \\bigwedge u^{\\ast }.\\end{itemize}$ It is known that a simply connected solvable Lie group $G$ admitting a lattice $\\Gamma $ is unimodular.",
"Hence we have $\\alpha _{1}\\cdots \\alpha _{n}=1$ .",
"For a multi-index $I\\subset \\lbrace 1,\\dots , n\\rbrace $ and its complement $I^{\\prime }=\\lbrace 1,\\dots , n\\rbrace -I$ , if $\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1$ then $\\left(\\frac{\\bar{\\alpha }_{I^{\\prime }}}{\\alpha _{I^{\\prime }}}\\right)_{ \\vert _{\\Gamma }}=1$ .",
"Thus the DGA $B^{\\ast }_{\\Gamma }$ as in Theorem REF is of PD-type.",
"Deformations of holomorphic vector bundles For a compact complex manifold $(M,J)$ and a holomorphic vector bundle $E$ over $M$ , consider $L^{\\ast }=A^{0,\\ast }(M, {\\rm End}(E))$ the differential graded Lie algebra of differential forms of $(0,\\ast )$ -type with values in the holomorphic vector bundle ${\\rm End}(E)$ with the Dolbeault operator induced by the holomorphic structure on $E$ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $E$ (see [6] and [7]).",
"Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group with a lattice $\\Gamma $ .",
"For the Lie algebra ${gl}_{n}($ of complex valued $n\\times n$ matrices, we consider the DGLA $L^{\\ast }=A^{0,\\ast }(G/\\Gamma )\\otimes {gl}_{n}($ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $G/\\Gamma \\times {n}$ near the trivial holomorphic structure.",
"As an analytic germ, the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ is an invariant under quasi-isomorphisms between analytic DGLAs.",
"Hence by Theorem REF , considering the DGLA $\\overline{L^{\\ast }}=B^{\\ast }_{\\Gamma }\\otimes {gl}_{n}($ , the analytic germ ${\\mathcal {K}}(L^{\\ast })$ is equivalent to ${\\mathcal {K}}(\\overline{L^{\\ast }})$ .",
"As Section REF , the analytic germ ${\\mathcal {K}}(\\overline{L^{\\ast }})$ is equivalent to the analytic germ $(F(B^{\\ast }_{\\Gamma }, {gl}_{n}(), 0)$ .",
"Hence we have the following theorem.",
"Theorem 4.2 Let $G$ be a simply connected complex solvable Lie group with a lattice $\\Gamma $ and ${g}$ the $-Lie algebra of $ G$.We consider the nilshadow $ u$ of $ g$.Then the analytic germ which represents the deformation functor for deformations of holomorphic structures on $ G/n$ near the trivial holomorphic structure is linearly embedded in the analytic germ $ (F(u, gln(), 0)$.$ Moreover, suppose that the Lie algebra $u$ is $\\nu $ -step naturally graded.",
"Then such analytic germ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"Acknowledgements This research is supported by JSPS Research Fellowships for Young Scientists."
],
[
"Nilshadows and cohomology of solvmanifolds",
"Let ${g}$ be a solvable $K$ -Lie algebra for $K=\\mathbb {R}$ or $.Let $ n$ be the nilradical of $ g$.There exists a subvector space (not necessarily Lie algebra) $ V$ of $ g$ so that$ g=Vn$ as the direct sum of vector spaces and for any $ A,BV$ $ (adA)s(B)=0$ where $ (adA)s$ is the semi-simple part of $ adA$ (see \\cite [Proposition I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.1.1] {DER}).We define the map $ ads:gD(g)$ as$ adsA+X=(adA)s$ for $ AV$ and $ Xn$.Then we have $ [ads(g), ads(g)]=0$ and $ ads$ is linear (see \\cite [Proposition I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.1.1] {DER}).Since we have $ [g,g]n$, the map $ ads:gD(g)$ is a representation and the image $ ads(g)$ is abelian and consists of semi-simple elements.Let $ g =Im adsg$and$$u=\\lbrace X-{\\rm ad}_{sX}\\in \\bar{{g}} \\vert X\\in {g}\\rbrace .$$Then we have $ [g,g]nu$ and $ u$ is the nilradical of $ g$ (see \\cite {DER}).Hence we have $ g= Im adsu$.It is known that the structure of the Lie algebra $ u$ is independent of a choice of a subvector space $ V$ (see \\cite [Corollary I\\hspace{-1.00006pt}I\\hspace{-1.00006pt}I.3.6]{DER} ).$ Lemma 2.1 ([9]) Suppose ${g}=\\mathbb {R}^{k}\\ltimes _{\\phi } {n}$ such that $\\phi $ is a semi-simple action and ${n}$ is nilpotent.",
"Then the nilshadow $u$ of ${g}$ is the direct sum $\\mathbb {R}^{k}\\oplus {n}$ .",
"Let $G$ be a simply connected solvable Lie group with the $\\mathbb {R}$ -Lie algebra $g$ .",
"We denote by ${\\rm Ad}_{s}:G\\rightarrow {\\rm Aut}({g})$ the extension of ${\\rm ad}_{s}$ .",
"Then ${\\rm Ad}_{s}(G)$ is diagonalizable.",
"Let $X_{1},\\cdots ,X_{n}$ be a basis of ${g}\\otimes { such that {\\rm Ad}_{s} is represented by diagonal matrices.Then we have {\\rm Ad}_{sg}X_{i}=\\alpha _{i}(g)X_{i} for characters \\alpha _{i} of G.Let x_{1},\\dots ,x_{n} be the dual basis of X_{1},\\dots ,X_{n}.",
"}We suppose $ G$ has a lattice $$.Then we consider the sub-DGA $ A$ of the de Rham complex $ A(G/) which is given by $A^{p}_{\\Gamma }=\\left\\langle \\alpha _{I} x_{I} {\\Big \\vert } \\begin{array}{cc}I\\subset \\lbrace 1,\\dots ,n\\rbrace ,\\\\ (\\alpha _{I})_{\\vert _{\\Gamma }}=1 \\end{array}\\right\\rangle .$ where for a multi-index $I=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ we write $x_{I}=x_{i_{1}}\\wedge \\dots \\wedge x_{i_{p}}$ , and $\\alpha _{I}=\\alpha _{i_{1}}\\cdots \\alpha _{i_{p}}$ .",
"Theorem 2.2 ([9]) Let $G$ be a simply connected solvable Lie group with a lattice $\\Gamma $ .",
"Then we have : The inclusion $A^{\\ast }_{\\Gamma }\\subset A^{\\ast }(G/\\Gamma )\\otimes induces an isomorphism in cohomology.\\item $ A$ can be regarded as a sub-DGA of $ u.",
"We explain the second assertion more precisely.",
"We consider the subspace $\\tilde{u}=\\langle \\alpha _{1}^{-1}X_{1},\\dots , \\alpha _{n}^{-1}X_{n}\\rangle $ of the space of complex valued vector fields on $G$ .",
"Then $\\tilde{u}=\\langle \\alpha _{1}^{-1}X_{1},\\dots , \\alpha _{n}^{-1}X_{n}\\rangle $ is a Lie sub-algebra of the Lie algebra of vector fields and the map $\\tilde{u}\\ni \\alpha _{i}^{-1}X_{i}\\mapsto X_{i}-{\\rm ad}_{sX_{i}}\\in u\\otimes is a Lie algebra isomorphismwhere $ $ is the nilshadow of $ g$ (see \\cite [Proof of Lemma 5.2]{K2}).$ Example 1 Let ${g}$ be a 4-dimensional Lie algebra such that ${g}=\\langle T,X,Y,Z\\rangle $ $[T,X]=X$ , $[T,Y]=-Y$ , $[X,Y]=Z$ .",
"Then we have the splitting ${g}=\\langle T\\rangle \\ltimes \\langle X,Y,Z\\rangle $ such that $\\langle X,Y,Z\\rangle $ is the three dimensional real Heisenberg Lie algebra $h(3)$ and the action of $\\langle T\\rangle $ is semi-simple.",
"Hence by Lemma REF , the nilshadow $u$ of ${g}$ is given by $u=\\mathbb {R}\\oplus h(3)$ .",
"Hence as similar to [6], the analytic germ $(F(\\bigwedge u^{\\ast }, a), 0)$ is equivalent to a cubic cone.",
"Consider the simply connected solvable Lie group $G$ whose Lie algebra is ${g}$ .",
"Then $G$ has a lattice $\\Gamma $ [13].",
"We can easily show that the DGA $A^{\\ast }(G/\\Gamma )$ is formal and hence the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most 2.",
"Hence $(R(\\Gamma , A),{\\bf 1})$ is linearly embedded in the analytic germ $(F(\\bigwedge u^{\\ast }, a), 0)$ but its singularity is different from $(F(\\bigwedge u^{\\ast }, a), 0)$ .",
"By Lemma REF , we give one more corollary of Theorem REF .",
"Corollary 2.3 Let ${g}=\\mathbb {R}^{k}\\ltimes _{\\phi } {n}$ such that $\\phi $ is a semi-simple action and ${n}$ is a $\\nu $ -step naturally graded nilpotent Lie algebra.",
"Consider the simply connected solvable Lie group $G$ whose Lie algebra is ${g}$ .",
"Suppose $G$ has a lattice $\\Gamma $ .",
"Then the analytic germ $(R(\\Gamma , A),{\\bf 1})$ at the trivial representation ${\\bf 1}$ is cut out by polynomial equations of degree at most $\\nu +1$ ."
],
[
"Finite-dimensional DGAs of Poincaré duality type",
"Let $A^{\\ast }$ be a finite-dimensional graded commutative $-algebra.\\begin{definition}[\\cite {KSP}]A^{\\ast } is of Poincaré duality type (PD-type) if the following conditions hold:\\begin{itemize}\\item A^{\\ast <0}=0 and A^{0}= where 1 is the identity element of A^{\\ast }.\\item For some positive integer n, A^{\\ast >n}=0 and A^{n}=v̏ for v\\ne 0.\\item For any 0<i<n the bi-linear map A^{i}\\times A^{n-i}\\ni (\\alpha ,\\beta )\\mapsto \\alpha \\cdot \\beta \\in A^{n} is non-degenerate.\\end{itemize}\\end{definition}$ Suppose $A^{\\ast }$ is of PD-type.",
"Let $h$ be a Hermitian metric on $A^{\\ast }$ which is compatible with the grading.",
"Take $v\\in A^{n}$ such that $h(v,v)=1$ .",
"Define the $-anti-linear map $ : AiAn-i$ as $ =h(,)v$.$ Definition 3.1 ([10]) A finite-dimensional DGA $(A^{\\ast },d)$ is of PD-type if the following conditions hold: $A^{\\ast }$ is a finite-dimensional graded $-algebra of PD-type.\\item $ dAn-1=0$ and $ dA0=0$.$ Let $(A^{\\ast },d)$ be a finite-dimensional DGA of PD-type.",
"Denote $d^{\\ast }=-\\bar{\\ast }d\\bar{\\ast }$ .",
"Lemma 3.2 ([10]) We have $h(d\\alpha , \\beta )=h(\\alpha ,d^{\\ast }\\beta )$ for $\\alpha \\in A^{i-1}$ and $\\beta \\in A^{i}$ .",
"Define $\\Delta =dd^{\\ast }+d^{\\ast }d$ .",
"and ${\\mathcal {H}}^{\\ast }(A)=\\ker \\Delta $ .",
"By Lemma REF and finiteness of the dimension of $A^{\\ast }$ , we can easily show the following lemma.",
"Lemma 3.3 ([10]) We have the Hodge decomposition $A^{r}={\\mathcal {H}}^{r}(A)\\oplus \\Delta (A^{r})={\\mathcal {H}}^{r}(A)\\oplus d(A^{r-1})\\oplus d^{\\ast }(A^{r+1}).$ By this decomposition, the inclusion ${\\mathcal {H}}^{\\ast }(A)\\subset A^{\\ast }$ induces a isomorphism ${\\mathcal {H}}^{p}(A)\\cong H^{p}(A)$ of vector spaces.",
"We denote by $H$ the projection $H: A^{p} \\rightarrow {\\mathcal {H}}^{p}(A)$ and define the operator $G$ as the composition $ \\Delta ^{-1}_{\\vert \\Delta (A^{p})}\\circ ({\\rm id }-H)$ .",
"Let $\\beta :A^{\\ast }\\rightarrow dA^{\\ast -1}$ be the projection for the decomposition $A^{r}={\\mathcal {H}}^{r}(A)\\oplus d(A^{r-1})\\oplus d^{\\ast }(A^{r+1}).$ The restriction map $d: d^{\\ast }(A^{\\ast })\\rightarrow d(A^{\\ast -1})$ is an isomorphism.",
"Take the inverse $d^{-1}:d(A^{\\ast -1})\\rightarrow d^{\\ast }(A^{\\ast })$ .",
"Consider the map $d^{\\ast }G:A^{\\ast }\\rightarrow A^{\\ast -1}$ .",
"Then for $\\omega \\in {\\mathcal {H}}^{r}(A)$ , $d^{\\ast }x\\in d^{\\ast }(A^{r})$ and $d^{\\ast }y \\in d^{\\ast }(A^{r+1})$ , we have $d^{\\ast }G(\\omega +dd^{\\ast }x+d^{\\ast }y)=d^{\\ast }(dd^{\\ast })^{-1}dd^{\\ast }x=d^{\\ast }x.$ Hence we have $d^{\\ast }G=d^{-1}\\circ \\beta $ .",
"Kuranishi spaces of finite-dimensional DGLAs Let $L^{\\ast }$ be a finite-dimensional DGLA with a differential $d$ .",
"Consider the splitting $d(L^{p})\\rightarrow L^{p}$ for the short exact sequence ${0[r]& {\\rm ker} \\,d_{\\vert _{L^{p}}} [r]& L^{p}[r]^d&d(L^{p})[r]&0}$ and the splitting $H^{p}(L^{\\ast })\\rightarrow {\\rm ker} \\,d_{\\vert _{L^{p}}}$ for the short exact sequence ${0[r]& d(L^{p-1})[r]& {\\rm ker} \\,d_{\\vert _{L^{p}}}[r]&H^{p}(L^{\\ast })[r]&0.",
"}$ Denote by $\\mathcal {A}^{p}$ and $\\mathcal {H}^{p}$ the images of the splittings $d(L^{p})\\rightarrow L^{p}$ and $H^{p}(L^{\\ast })\\rightarrow {\\rm ker} \\,d_{\\vert _{L^{p}}}$ respectively.",
"Then we have $L^{p}= {\\mathcal {H}}^{p}\\oplus d(L^{p-1})\\oplus \\mathcal {A}^{p}.$ Consider the projections $\\beta ^{\\ast }: L^{\\ast }\\rightarrow d(L^{\\ast -1})$ , $H: L^{\\ast }\\rightarrow \\mathcal {H}^{\\ast }$ and $\\alpha ^{\\beta }: L^{\\ast }\\rightarrow \\mathcal {A}^{\\ast }$ .",
"Since the restriction $ d : \\mathcal {A}^{p} \\rightarrow d(L^{p})$ is an isomorphism, we have the inverse $d^{-1}: d(L^{p})\\rightarrow \\mathcal {A}^{p}$ of $ d : \\mathcal {A}^{p} \\rightarrow d(L^{p})$ .",
"We define $\\delta =d^{-1}\\circ \\beta :L^{p+1}\\rightarrow L^{p}$ .",
"Define the map $F: L^{1}\\rightarrow L^{1}$ as $F(\\zeta )=\\zeta +\\frac{1}{2}\\delta [\\zeta ,\\zeta ].$ Then by the inverse function theorem, on a small ball $B$ in $L^{1}$ , the map $F$ is an analytic diffeomorphism.",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ is defined by ${\\mathcal {K}}(L^{\\ast })=\\lbrace \\eta \\in F(B)\\cap \\mathcal {H}^{1}: H([F^{-1}(\\eta ),F^{-1}(\\eta )])=0\\rbrace .$ It is known that the analytic germ $({\\mathcal {K}}(L^{\\ast }),0)$ is equivalent to the germ at the origin for the variety $\\left\\lbrace \\zeta \\in L^{1}: d\\zeta +\\frac{1}{2}[\\zeta ,\\zeta ]=0, \\,\\, \\delta \\zeta =0\\right\\rbrace $ (see [7]).",
"In particular, if $d(L^{0})=0$ , then ${\\mathcal {K}}(L^{\\ast })$ is equivalent to the germ at the origin for the variety $\\left\\lbrace \\zeta \\in L^{1}: d\\zeta +\\frac{1}{2}[\\zeta ,\\zeta ]=0 \\right\\rbrace .$ Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $L^{1}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}\\delta [\\phi _{s}(t), \\phi _{r-s}(t)].$ Then $F^{-1}$ is given by $\\phi _{1}(t)\\mapsto \\phi (t)$ and the Kuranishi space $ {\\mathcal {K}}(L^{\\ast })$ is an analytic germ in ${m}$ at the origin defined by equations $H\\left([\\phi (t),\\phi (t)]\\right)=0.$ Let $A$ be a finite-dimensional DGA of PD-type and ${g}$ a Lie algebra, and consider the DGLA $A^{\\ast }\\otimes {g}$ .",
"Then we have the Hodge decomposition $A^{\\ast }\\otimes {g}={\\mathcal {H}}^{p}(A)\\otimes {g}\\oplus d(A^{p-1})\\otimes {g}\\oplus d^{\\ast }(A^{p+1})\\otimes {g}$ as above with $\\delta =d^{\\ast } G\\otimes {\\rm id}$ .",
"Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}(A^{\\ast })\\otimes {g}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $A^{1}\\otimes {g}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}d^{\\ast } G\\otimes {\\rm id}[\\phi _{s}(t), \\phi _{r-s}(t)].$ By the above argument we have the following lemma.",
"Lemma 3.4 The analytic germ $(F(A^{\\ast }, g), 0)$ is equivalent to the analytic germ in ${m}$ at the origin defined by equations $H\\left([\\phi (t),\\phi (t)]\\right)=0.$ Nilpotent Lie algebras Let ${n}$ be a $\\nu $ -step nilpotent $K$ -Lie algebra for $K=\\mathbb {R}$ or $.Consider the lower central series$${n}={n}^{(1)}\\supset {n}^{(2)}\\supset \\dots \\supset {n}^{(\\nu )}\\supset {n}^{(\\nu +1)}=\\lbrace 0\\rbrace $$where $ n(i+1)=[n,n(i)]$.Take a subspace $ a(i)$ such that $ n(i)=n(i+1)a(i)$.We have$${n}=a^{(1)}\\oplus a^{(2)}\\oplus \\dots \\oplus a^{(\\nu )}.$$Consider the dual spaces $ n$ and $ a(i)$ of $ n$ and $ a(i)$ respectively.We consider the cochain complex $ n$ of the Lie algebra with the differential $ d$.Then $ n$ is a a finite-dimensional DGA of PD-type.We have$$\\bigwedge {n}^{\\ast } =\\left(\\bigwedge a^{(1)\\ast }\\right)\\wedge \\dots \\wedge \\left(\\bigwedge a^{(\\nu )\\ast }\\right).$$We have$$H^{1}({n})={\\rm ker}\\, d_{\\bigwedge ^{1} {n}^{\\ast }}= a^{(1)\\ast }.$$$ Lemma 3.5 ${\\rm ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}\\subset \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Let $\\sigma \\in {\\rm ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}$ .",
"For a positive integer $k<\\nu $ , we say that $\\sigma $ is $k$ -decomposable if we have a decomposition $\\sigma =\\sigma _{1}+\\sigma _{2}+\\sigma _{3}$ such that: $\\sigma _{1}\\in \\bigoplus _{i+j\\le \\nu +1, i\\le j, k<j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ $\\sigma _{2}\\in \\bigoplus _{ i\\le k} a^{(i)\\ast }\\wedge a^{(k)\\ast }.$ $\\sigma _{3}\\in \\bigoplus _{ i\\le j, j<k} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ If $k\\le \\frac{\\nu +1}{2}$ , then we have $\\sigma \\in \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Consider the case $\\frac{\\nu +1}{2}<k$ .",
"For $X,Y\\in {n}$ and $Z\\in {n}^{(k)}$ , we have $\\sigma _{1}([X,Y],Z)=0$ , $\\sigma _{2}(X,[Y,Z])=0$ , $\\sigma _{2}(Y,[X,Z])=0$ , $\\sigma _{3}([X,Y],Z)=0$ , $\\sigma _{3}(X,[Y,Z])=0$ and $\\sigma _{3}(Y,[X,Z])=0$ .",
"By $d\\sigma =0$ , we have $\\sigma _{2}([X,Y],Z)=\\sigma _{1}(X,[Y,Z])-\\sigma _{1}(Y,[X,Z]).$ Taking $X\\in {n}$ and $Y\\in {n}^{(l-1)}$ such that $\\nu +1<k+l$ , we have $\\sigma _{2}([X,Y],Z)=0.$ Hence for $W\\in {n}^{(l)}$ and $Z\\in {n}^{(k)}$ such that $\\nu +1<k+l$ , we have $\\sigma _{2}(W,Z)=0.$ Thus we have $\\sigma _{2}\\in \\bigoplus _{i+k\\le \\nu +1, i\\le k} a^{(i)\\ast }\\wedge a^{(k)\\ast }.$ Hence taking $\\sigma _{1}^{\\prime }=\\sigma _{1}+\\sigma _{2}$ and $\\sigma _{3}=\\sigma _{2}^{\\prime }+\\sigma _{3}^{\\prime }$ such that $\\sigma _{2}^{\\prime }\\in \\bigoplus _{ i\\le k-1} a^{(i)\\ast }\\wedge a^{(k-1)\\ast }$ and $\\sigma _{3}^{\\prime }\\in \\bigoplus _{ i\\le j, j<k-1} a^{(i)\\ast }\\wedge a^{(j)\\ast },$ by the decomposition $\\sigma =\\sigma _{1}^{\\prime }+\\sigma ^{\\prime }_{2}+\\sigma ^{\\prime }_{3}$ , $\\sigma $ is $(k-1)$ -decomposable.",
"Thus we can say that if $\\sigma $ is $k$ -decomposable and $\\frac{\\nu +1}{2}<k-l-1$ for an integer $l$ , then $\\sigma $ is also $(k-l)$ -decomposable.",
"Take $l$ such that $k-l\\le \\frac{\\nu +1}{2}$ .",
"Then we can say $\\sigma \\in \\bigoplus _{i+j\\le \\nu +1, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Hence it is sufficient to show the above decomposition of $\\sigma $ for $k=\\nu -1$ .",
"This was shown in [1].",
"Hence the Lemma follows.",
"It is known that $[{n}^{(i)},{n}^{(j)}]\\subset {n}^{(i+j)}$ (see [2]) and hence we have $d \\left(a^{(k)\\ast }\\right)\\subset \\bigoplus _{i+j\\le k, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }.$ Definition 3.6 A nilpotent Lie algebra $n$ is called naturally graded if we can take subspaces $a^{(i)}\\subset n$ such that ${n}^{(i)}={n}^{(i+1)}\\oplus a^{(i)}$ and $[a^{(i)},a^{(j)}]\\subset a^{(i+j)}$ for each $i,j$ where ${n}={n}^{(1)}\\supset {n}^{(2)}\\supset \\dots \\supset {n}^{(\\nu )}\\supset {n}^{(\\nu +1)}=\\lbrace 0\\rbrace $ is the lower central series of $n$ .",
"If $n$ is naturally graded, then we have $d \\left(a^{(k)\\ast }\\right)\\subset W_{k}$ where $ W_{k}=\\bigoplus _{i+j= k, i\\le j} a^{(i)\\ast }\\wedge a^{(j)\\ast }$ .",
"Let $g$ be a Hermitian metric on ${n}$ such that the sum ${n}=a^{(1)}\\oplus a^{(2)}\\oplus \\dots \\oplus a^{(\\nu )}$ is an orthogonal direct sum.",
"Then $g$ give a Hermitian metric on the finite-dimensional DGA $\\bigwedge {n}^{\\ast }$ of PD-type.",
"Consider the decomposition $\\bigwedge ^{r}{n}^{\\ast }={\\mathcal {H}}^{r}(\\bigwedge {n}^{\\ast })\\oplus d(\\bigwedge ^{r-1}{n}^{\\ast })\\oplus d^{\\ast }(\\bigwedge ^{r+1}{n}^{\\ast }).$ Then $\\bigwedge ^{2}{n}^{\\ast }=W_{1}\\oplus W_{2}\\oplus \\dots \\oplus W_{2\\nu }$ is an orthogonal direct sum and we have $d^{-1}\\circ \\beta (W_{k})\\subset a^{(k)\\ast }$ by $d(a^{(k)\\ast })\\subset W_{k}$ .",
"Proposition 3.7 Let ${n}$ be a $\\nu $ -step naturally graded nilpotent Lie algebra and $g$ a Lie algebra.",
"Then the analytic germ $(F(\\bigwedge u^{\\ast }, g), 0)$ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"Take a basis $\\zeta _{1},\\dots , \\zeta _{m}$ of ${\\mathcal {H}}^{1}(\\bigwedge u^{\\ast })\\otimes {g}$ .",
"For parameters $t=( t_{i})$ , we consider the formal power series $\\phi (t)=\\sum _{r} \\phi _{r}(t)$ with values in $L^{1}$ given inductively by $\\phi _{1}(t)=\\sum t_{i}\\zeta _{j}$ and $\\phi _{r}(t)=-\\frac{1}{2}\\sum _{s=1}^{r-1}\\delta [\\phi _{s}(t), \\phi _{r-s}(t)].$ By Lemma REF , the analytic germ $(F(\\bigwedge u^{\\ast }, g), 0)$ is equivalent to the analytic germ in ${m}$ at the origine defined by equations $ H\\left([\\phi (t),\\phi (t)]\\right)=0$ where $H:\\bigwedge ^{\\ast }{n}^{\\ast }\\otimes {g}\\rightarrow {\\mathcal {H}}^{\\ast }(\\bigwedge u^{\\ast })\\otimes {g}$ is the projection.",
"We have $[ a^{(i)\\ast }\\otimes {g}, a^{(j)\\ast }\\otimes {g}]\\subset W_{i+j}\\otimes {g}.$ By $d^{\\ast }G(W_{k})=d^{-1}\\circ \\beta (W_{k})\\subset a^{(k)\\ast }$ , we have $d^{\\ast }G\\otimes {\\rm id} ([a^{(i)\\ast }\\otimes {g}, a^{(j)\\ast }\\otimes {g}])\\subset a^{(i+j)\\ast }\\otimes {g}.$ This implies $\\phi _{r}(t)\\in a^{(r)\\ast }\\otimes {g}$ and we have $\\phi (t)=\\phi _{1}(t)+\\dots +\\phi _{\\nu }(t).$ By Lemma REF , we have $ {\\mathcal {H}}^{2}(\\bigwedge {n}^{\\ast })\\subset {\\rm Ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }} \\subset \\bigoplus _{l\\le \\nu +1, }W_{l}$ and hence $ H(\\bigwedge ^{2} {n}^{\\ast }\\otimes {g})\\subset {\\rm Ker}\\, d_{\\bigwedge ^{2} {n}^{\\ast }}\\otimes {g}\\subset \\bigoplus _{l\\le \\nu +1, }W_{l}\\otimes {g}.$ Since we have $ [\\phi _{i}(t),\\phi _{j}(t)]\\in W_{i+j}\\otimes {g}$ by $\\phi _{r}(t)\\in a^{(r)\\ast }\\otimes {g}$ , we have $H[\\phi _{i}(t),\\phi _{j}(t)]=0$ for $\\nu +1<i+j$ .",
"Hence $H[\\phi (t),\\phi (t)]=0$ are polynomial equations of degree at most $\\nu +1$ .",
"Complex analogy Complex parallelizable solvmanifolds Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group.",
"Consider the Lie algebra ${g}_{1,0}$ (resp.",
"${g}_{0,1}$ ) of the left-invariant holomorphic (resp.",
"anti-holomorphic) vector fields on $G$ .",
"Let $N$ be the nilradical of $G$ .",
"We can take a simply connected complex nilpotent subgroup $C\\subset G$ such that $G=C\\cdot N$ (see [3]).",
"Since $C$ is nilpotent, the map $C\\ni c \\mapsto ({\\rm Ad}_{c})_{s}\\in {\\rm Aut}({g}_{1,0})$ is a homomorphism where $({\\rm Ad}_{c})_{s}$ is the semi-simple part of ${\\rm Ad}_{s}$ .",
"We have a basis $X_{1},\\dots ,X_{n}$ of ${g}_{1,0}$ such that $({\\rm Ad}_{c})_{s}={\\rm diag} (\\alpha _{1}(c),\\dots ,\\alpha _{n}(c))$ for $c\\in C$ .",
"Let $x_{1},\\dots , x_{n}$ be the basis of ${g}^{\\ast }_{1,0}$ which is dual to $X_{1},\\dots ,X_{n}$ .",
"Theorem 4.1 ([11]]) Suppose $G$ has a lattice $\\Gamma $ .",
"Let $B^{\\ast }_{\\Gamma }$ be the subcomplex of $(A^{0,\\ast }(G/\\Gamma ),\\bar{\\partial }) $ defined as $B^{\\ast }_{\\Gamma }=\\left\\langle \\frac{\\bar{\\alpha }_{I}}{\\alpha _{I} }\\bar{x}_{I}{\\Big \\vert }\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1\\right\\rangle $ where for a multi-index $I=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ we write $x_{I}=x_{i_{1}}\\wedge \\dots \\wedge x_{i_{p}}$ , and $\\alpha _{I}=\\alpha _{i_{1}}\\cdots \\alpha _{i_{p}}$ .",
"Consider the nilshadow $u$ of the $-Lie algebra $ g$.Then we have:\\begin{itemize}\\item The inclusion B^{\\ast }_{\\Gamma }\\subset A^{0,\\ast }(G/\\Gamma ) induces an isomorphism in cohomology.\\item B^{\\ast }_{\\Gamma } can be regarded as a sub-DGA of \\bigwedge u^{\\ast }.\\end{itemize}$ It is known that a simply connected solvable Lie group $G$ admitting a lattice $\\Gamma $ is unimodular.",
"Hence we have $\\alpha _{1}\\cdots \\alpha _{n}=1$ .",
"For a multi-index $I\\subset \\lbrace 1,\\dots , n\\rbrace $ and its complement $I^{\\prime }=\\lbrace 1,\\dots , n\\rbrace -I$ , if $\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1$ then $\\left(\\frac{\\bar{\\alpha }_{I^{\\prime }}}{\\alpha _{I^{\\prime }}}\\right)_{ \\vert _{\\Gamma }}=1$ .",
"Thus the DGA $B^{\\ast }_{\\Gamma }$ as in Theorem REF is of PD-type.",
"Deformations of holomorphic vector bundles For a compact complex manifold $(M,J)$ and a holomorphic vector bundle $E$ over $M$ , consider $L^{\\ast }=A^{0,\\ast }(M, {\\rm End}(E))$ the differential graded Lie algebra of differential forms of $(0,\\ast )$ -type with values in the holomorphic vector bundle ${\\rm End}(E)$ with the Dolbeault operator induced by the holomorphic structure on $E$ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $E$ (see [6] and [7]).",
"Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group with a lattice $\\Gamma $ .",
"For the Lie algebra ${gl}_{n}($ of complex valued $n\\times n$ matrices, we consider the DGLA $L^{\\ast }=A^{0,\\ast }(G/\\Gamma )\\otimes {gl}_{n}($ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $G/\\Gamma \\times {n}$ near the trivial holomorphic structure.",
"As an analytic germ, the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ is an invariant under quasi-isomorphisms between analytic DGLAs.",
"Hence by Theorem REF , considering the DGLA $\\overline{L^{\\ast }}=B^{\\ast }_{\\Gamma }\\otimes {gl}_{n}($ , the analytic germ ${\\mathcal {K}}(L^{\\ast })$ is equivalent to ${\\mathcal {K}}(\\overline{L^{\\ast }})$ .",
"As Section REF , the analytic germ ${\\mathcal {K}}(\\overline{L^{\\ast }})$ is equivalent to the analytic germ $(F(B^{\\ast }_{\\Gamma }, {gl}_{n}(), 0)$ .",
"Hence we have the following theorem.",
"Theorem 4.2 Let $G$ be a simply connected complex solvable Lie group with a lattice $\\Gamma $ and ${g}$ the $-Lie algebra of $ G$.We consider the nilshadow $ u$ of $ g$.Then the analytic germ which represents the deformation functor for deformations of holomorphic structures on $ G/n$ near the trivial holomorphic structure is linearly embedded in the analytic germ $ (F(u, gln(), 0)$.$ Moreover, suppose that the Lie algebra $u$ is $\\nu $ -step naturally graded.",
"Then such analytic germ is cut out by polynomial equations of degree at most $\\nu +1$ .",
"Acknowledgements This research is supported by JSPS Research Fellowships for Young Scientists."
],
[
"Complex parallelizable solvmanifolds",
"Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group.",
"Consider the Lie algebra ${g}_{1,0}$ (resp.",
"${g}_{0,1}$ ) of the left-invariant holomorphic (resp.",
"anti-holomorphic) vector fields on $G$ .",
"Let $N$ be the nilradical of $G$ .",
"We can take a simply connected complex nilpotent subgroup $C\\subset G$ such that $G=C\\cdot N$ (see [3]).",
"Since $C$ is nilpotent, the map $C\\ni c \\mapsto ({\\rm Ad}_{c})_{s}\\in {\\rm Aut}({g}_{1,0})$ is a homomorphism where $({\\rm Ad}_{c})_{s}$ is the semi-simple part of ${\\rm Ad}_{s}$ .",
"We have a basis $X_{1},\\dots ,X_{n}$ of ${g}_{1,0}$ such that $({\\rm Ad}_{c})_{s}={\\rm diag} (\\alpha _{1}(c),\\dots ,\\alpha _{n}(c))$ for $c\\in C$ .",
"Let $x_{1},\\dots , x_{n}$ be the basis of ${g}^{\\ast }_{1,0}$ which is dual to $X_{1},\\dots ,X_{n}$ .",
"Theorem 4.1 ([11]]) Suppose $G$ has a lattice $\\Gamma $ .",
"Let $B^{\\ast }_{\\Gamma }$ be the subcomplex of $(A^{0,\\ast }(G/\\Gamma ),\\bar{\\partial }) $ defined as $B^{\\ast }_{\\Gamma }=\\left\\langle \\frac{\\bar{\\alpha }_{I}}{\\alpha _{I} }\\bar{x}_{I}{\\Big \\vert }\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1\\right\\rangle $ where for a multi-index $I=\\lbrace i_{1},\\dots ,i_{p}\\rbrace $ we write $x_{I}=x_{i_{1}}\\wedge \\dots \\wedge x_{i_{p}}$ , and $\\alpha _{I}=\\alpha _{i_{1}}\\cdots \\alpha _{i_{p}}$ .",
"Consider the nilshadow $u$ of the $-Lie algebra $ g$.Then we have:\\begin{itemize}\\item The inclusion B^{\\ast }_{\\Gamma }\\subset A^{0,\\ast }(G/\\Gamma ) induces an isomorphism in cohomology.\\item B^{\\ast }_{\\Gamma } can be regarded as a sub-DGA of \\bigwedge u^{\\ast }.\\end{itemize}$ It is known that a simply connected solvable Lie group $G$ admitting a lattice $\\Gamma $ is unimodular.",
"Hence we have $\\alpha _{1}\\cdots \\alpha _{n}=1$ .",
"For a multi-index $I\\subset \\lbrace 1,\\dots , n\\rbrace $ and its complement $I^{\\prime }=\\lbrace 1,\\dots , n\\rbrace -I$ , if $\\left(\\frac{\\bar{\\alpha }_{I}}{\\alpha _{I}}\\right)_{ \\vert _{\\Gamma }}=1$ then $\\left(\\frac{\\bar{\\alpha }_{I^{\\prime }}}{\\alpha _{I^{\\prime }}}\\right)_{ \\vert _{\\Gamma }}=1$ .",
"Thus the DGA $B^{\\ast }_{\\Gamma }$ as in Theorem REF is of PD-type."
],
[
"Deformations of holomorphic vector bundles",
"For a compact complex manifold $(M,J)$ and a holomorphic vector bundle $E$ over $M$ , consider $L^{\\ast }=A^{0,\\ast }(M, {\\rm End}(E))$ the differential graded Lie algebra of differential forms of $(0,\\ast )$ -type with values in the holomorphic vector bundle ${\\rm End}(E)$ with the Dolbeault operator induced by the holomorphic structure on $E$ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $E$ (see [6] and [7]).",
"Let $G$ be a simply connected $n$ -dimensional complex solvable Lie group with a lattice $\\Gamma $ .",
"For the Lie algebra ${gl}_{n}($ of complex valued $n\\times n$ matrices, we consider the DGLA $L^{\\ast }=A^{0,\\ast }(G/\\Gamma )\\otimes {gl}_{n}($ .",
"Then the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ represents the deformation functor for deformations of holomorphic structures on $G/\\Gamma \\times {n}$ near the trivial holomorphic structure.",
"As an analytic germ, the Kuranishi space ${\\mathcal {K}}(L^{\\ast })$ is an invariant under quasi-isomorphisms between analytic DGLAs.",
"Hence by Theorem REF , considering the DGLA $\\overline{L^{\\ast }}=B^{\\ast }_{\\Gamma }\\otimes {gl}_{n}($ , the analytic germ ${\\mathcal {K}}(L^{\\ast })$ is equivalent to ${\\mathcal {K}}(\\overline{L^{\\ast }})$ .",
"As Section REF , the analytic germ ${\\mathcal {K}}(\\overline{L^{\\ast }})$ is equivalent to the analytic germ $(F(B^{\\ast }_{\\Gamma }, {gl}_{n}(), 0)$ .",
"Hence we have the following theorem.",
"Theorem 4.2 Let $G$ be a simply connected complex solvable Lie group with a lattice $\\Gamma $ and ${g}$ the $-Lie algebra of $ G$.We consider the nilshadow $ u$ of $ g$.Then the analytic germ which represents the deformation functor for deformations of holomorphic structures on $ G/n$ near the trivial holomorphic structure is linearly embedded in the analytic germ $ (F(u, gln(), 0)$.$ Moreover, suppose that the Lie algebra $u$ is $\\nu $ -step naturally graded.",
"Then such analytic germ is cut out by polynomial equations of degree at most $\\nu +1$ ."
],
[
"Acknowledgements",
"This research is supported by JSPS Research Fellowships for Young Scientists.",
"This research is supported by JSPS Research Fellowships for Young Scientists."
]
] | 1403.0075 |
[
[
"Metrics without isometries are generic"
],
[
"Abstract We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics."
],
[
"We prove that for any compact manifold of dimension greater than 1, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.",
"Keywords: metrics without isometries; space of pseudo-Riemannian metrics Mathematics Subject Classification (2010): 53C50 Let $V$ be a compact manifold and $\\mathcal {M}_{p,q}$ be the set of smooth pseudo-Riemannian metrics of signature $(p,q)$ on $V$ (we suppose that it is not empty).",
"In [2] D'Ambra and Gromov wrote:“everybody knows that $\\operatorname{Is}(V,g)=\\rm {Id}$ for generic pseudo-Riemannian metrics $g$ on $V$ , for $\\dim (V)\\ge 2$ .” Nevertheless, as far as we know, no proof of this fact is available.",
"The purpose of this short article is to give a proof of this result in the case where $V$ is compact and to precise the meaning of the word generic.",
"Let us recall that it is known since the work of Ebin [3] (see also [4]) that the set of Riemannian metrics without isometries on a compact manifold is open and dense.",
"We prove: Theorem 1 If $V$ is a compact manifold such that $\\dim (V)\\ge 2$ then the set $\\mathcal {G}=\\lbrace g\\in \\mathcal {M}_{p,q}\\,|\\, \\operatorname{Is}(g) =\\rm {Id}\\rbrace $ contains a subset that is open and dense in $\\mathcal {M}_{p,q}$ for the $C^\\infty $ -topology.",
"This result is optimal in the sense that $\\mathcal {G}$ is not always open as we showed in [5].",
"The particularity of the Riemannian case lies in the fact that the natural action of the group of smooth diffeomorphisms of $V$ , denoted by $\\operatorname{Diff}(V)$ , on $\\mathcal {M}_{n,0}$ is proper.",
"Furthermore, Theorem 4.2 of [5] says that when this action is proper then $\\mathcal {G}$ is an open subset of $\\mathcal {M}_{p,q}$ .",
"The idea of proof is therefore to find a big enough subset of $\\mathcal {M}_{p,q}$ invariant by $\\operatorname{Diff}(V)$ on which the action is proper.",
"We have decided to be short rather than self-contained, in particular we are going to use several results from our former work [5].",
"In the following $\\mathcal {M}_{p,q}$ will be endowed with the $C^\\infty $ -topology without further mention of it and by a perturbation we will always mean an arbitrary small perturbation.",
"For any $g\\in \\mathcal {M}_{p,q}$ we denote by $\\operatorname{Scal}_g$ its scalar curvature and by $M_g$ the maximum of $\\operatorname{Scal}_g$ on $V$ .",
"Let $\\mathcal {F}_V$ be the set of pseudo-Riemannian metrics $g$ such that $\\operatorname{Scal}_g^{-1}(M_g)$ contains a (non trivial) geodesic.",
"The big set we are looking for is actually the complement of $\\mathcal {F}_V$ .",
"Proposition 2 The set $\\mathcal {O}_V=\\mathcal {M}_{p,q}\\setminus \\mathcal {F}_V$ is an open dense subset of $\\mathcal {M}_{p,q}$ invariant by the action of $\\operatorname{Diff}(V)$ and the restriction of the action of $\\operatorname{Diff}(V)$ to $\\mathcal {O}_V$ is proper.",
"Proof.",
"The set $\\mathcal {O}_V$ is clearly invariant.",
"Let $g\\in \\mathcal {M}_{p,q}$ and $x_0\\in V$ realizing the maximum of $\\operatorname{Scal}_g$ .",
"It is easy to find a perturbation with arbitrary small support increasing the value of $\\operatorname{Scal}_g(x_0)$ .",
"Repeating these deformation on smaller and smaller neighborhood of $x_0$ we find a perturbation of $g$ such that the maximum of the scalar curvature is realized by only one point (see [3] p. 35 for a similar construction).",
"Hence $\\mathcal {O}_V$ is dense in $\\mathcal {M}_{p,q}$ .",
"Let us see now that $\\mathcal {F}_V$ is closed.",
"Let $g_n$ be a sequence of metrics of $\\mathcal {F}_V$ converging to $g_\\infty $ .",
"For any $n\\in \\mathbb {N}$ there exists a $g_n$ -geodesic $\\gamma _n$ such that $\\operatorname{Scal}_{g_n}$ is constant and equal to $M_{g_n}=\\max _{x\\in V} \\operatorname{Scal}_{g_n}(x)$ on it.",
"As the sequence of exponential maps converges to the exponential map of $g_\\infty $ and as $V$ is compact we see that (up to subsequences) the sequence of geodesics $\\gamma _n$ converges to a $g_\\infty $ -geodesic $\\gamma _\\infty $ .",
"As $\\operatorname{Scal}_{g_n}\\rightarrow \\operatorname{Scal}_{g_\\infty }$ we know that $\\operatorname{Scal}_{g_\\infty }$ is constant along $\\gamma _\\infty $ and its value is necessarily $M_{g_\\infty }$ .",
"Hence $g_\\infty \\in \\mathcal {F}_V$ and $\\mathcal {F}_V$ is closed.",
"Let us suppose that the action of $\\operatorname{Diff}(V)$ on $\\mathcal {M}_{p,q}$ is not proper (otherwise there is nothing to prove).",
"It means (see [5]) that there exists a sequence of metrics $(g_n)_{n\\in \\mathbb {N}}$ converging to $g_\\infty $ and a non equicontinuous sequence of diffeomorphisms $(\\phi _n)_{n\\in \\mathbb {N}}$ such that the sequence of metrics $(\\phi _n^*g_n)$ converges to $g^{\\prime }_\\infty $ .",
"The proposition will follow from the fact that $g_\\infty $ or $g^{\\prime }_\\infty $ have to belong to $\\mathcal {F}_V$ .",
"We first remark that the sequence of linear maps $(D\\phi _n(x_n))_{n\\in \\mathbb {N}}$ lies in $\\operatorname{O}(p,q)$ up to conjugacy by a converging sequence.",
"As the sequence $(\\phi _n)_{n\\in \\mathbb {N}}$ is non equicontinuous, we know by [5] that there exists a subsequence such that $\\Vert D\\phi _{n_k}(x_{n_k})\\Vert \\rightarrow \\infty $ when $k\\rightarrow \\infty $ .",
"We deduce from the $KAK$ decomposition of $\\operatorname{O}(p,q)$ the existence of what are called in [6], see subsection 4.1 therein for details, strongly approximately stable vectors, more explicitly we have: Fact 3 For any sequence $(x_n)_{n\\in \\mathbb {N}}$ of points of $V$ , there exist a sequence $(v_n)_{n\\in \\mathbb {N}}$ such that (up to subsequences): $\\forall n\\in \\mathbb {N},\\ v_n\\in T_{x_n}V$ , $D\\phi _n(x_n)v_n\\rightarrow 0$ , $v_n\\rightarrow v_\\infty \\ne 0$ $\\forall n\\in \\mathbb {N},\\ v_n\\in T_{x_n}V$ , $D\\phi _n(x_n)v_n\\rightarrow 0$ , $v_n\\rightarrow v_\\infty \\ne 0$ Let $(x_n)_{n\\in \\mathbb {N}}$ be a sequence of points of $V$ realizing the maximum of the function $\\operatorname{Scal}_{\\phi _n^*g_n}$ .",
"The manifold being compact, we can assume that this sequence is convergent to a point $x_\\infty $ .",
"We can also assume that the sequence $(\\phi _n(x_n))_{n\\in \\mathbb {N}}$ converges to $y_\\infty $ .",
"Of course $x_\\infty $ (resp.",
"$y_\\infty $ ) realizes the maximum of $\\operatorname{Scal}_{g^{\\prime }_\\infty }$ (resp.",
"$\\operatorname{Scal}_{g_\\infty }$ ).",
"Let $(v_n)_{n\\in \\mathbb {N}}$ be a sequence given by Fact REF .",
"Reproducing the computation p. 471 of [5], we see that the scalar curvature of $g^{\\prime }_\\infty $ is constant along the $g^{\\prime }_\\infty $ -geodesic starting from $x_\\infty $ with speed $v_\\infty $ (by symmetry the scalar curvature of $g_\\infty $ is constant along a geodesic containing $y_\\infty $ ): $&\\operatorname{Scal}_{g^{\\prime }_\\infty }(\\exp _{g^{\\prime }_\\infty }(x_\\infty ,v_\\infty ))-\\operatorname{Scal}_{g^{\\prime }_\\infty }(x_\\infty )&= \\lim _{n\\rightarrow \\infty }\\operatorname{Scal}_{\\phi _n^*g_n}(\\exp _{\\phi _n^*g_n}(x_n,v_n))-\\operatorname{Scal}_{\\phi _n^*g_n}(x_n)\\\\&&=\\lim _{n\\rightarrow \\infty }\\operatorname{Scal}_{g_n}(\\exp _{g_n}(D\\phi _n(x_n,v_n))-\\operatorname{Scal}_{g_n}(\\phi _n(x_n))\\\\&&= \\operatorname{Scal}_{g_\\infty }(y_\\infty )-\\operatorname{Scal}_{g_\\infty }(y_\\infty )=0.$ Hence $g_\\infty $ and $g^{\\prime }_\\infty $ do not belong to $\\mathcal {O}_V$ .$\\Box $ It follows from Theorem 4.2 of [5] and Proposition REF that $\\mathcal {G} \\cap \\mathcal {O}_V$ is open.",
"As $\\mathcal {O}_V$ is dense we just have to show that $\\mathcal {G}$ is dense in $\\mathcal {O}_V$ in order to prove Theorem REF .",
"Let $g$ be a metric in $\\mathcal {O}_V$ , as we saw earlier we can perturb it in such a way that the maximum of $\\operatorname{Scal}_g$ is realized by only one point $p$ .",
"This point is now fixed by isometry.",
"We choose now $U$ an open subset of $V$ that do not contain $p$ in its closure but whose closure is contained in some normal neighborhood $O$ of $p$ .",
"According to [1] by Beig et al., we can perturb again $g$ in such a way that there are no local Killing fields on $U$ .",
"We choose the perturbation in order that $\\operatorname{Scal}^{-1}(M_g)=\\lbrace p\\rbrace $ .",
"The new metric has now a finite isometry group (it is 0-dimensional and compact by Proposition REF as the metric still lies in $\\mathcal {O}_V$ ).",
"Actually, the proof of Proposition REF implies also that the set of germs of local isometries fixing $p$ is itself compact.",
"It means that any isometry of a perturbation of $g$ with support not containing $O$ can send a geodesic $\\gamma _1$ starting from $p$ only on a finite number of geodesics that do not depend on the perturbation.",
"Therefore, it is easy to find a perturbation of the metric along $\\gamma _1$ (with support away from $O$ ) in order to destroy these possibilities.",
"Now, any isometry has to fix pointwise $\\gamma _1$ (we chose a non symmetric perturbation).",
"Repeating this operation for $n=\\dim V$ geodesics $\\gamma _1,\\dots \\gamma _n$ such that the vectors $\\gamma _i^{\\prime }(0)$ span $T_pV$ , we obtain a perturbation of $g$ such that any of its isometries has to be the identity i.e.",
"the perturbed metric is in $\\mathcal {G}$ .",
"Table: NO_CAPTION"
]
] | 1403.0182 |
[
[
"Flag structure for operators in the Cowen-Douglas class"
],
[
"Abstract The explicit description of homogeneous operators and localization of a Hilbert module naturally leads to the definition of a class of Cowen-Douglas operators possessing a flag structure.",
"These operators are irreducible.",
"We show that the flag structure is rigid in the sense that the unitary equivalence class of the operator and the flag structure determine each other.",
"We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class."
],
[
"english english The explicit description of homogeneous operators and localization of a Hilbert module naturally leads to the definition of a class of Cowen-Douglas operators possessing a flag structure.",
"These operators are irreducible.",
"We show that the flag structure is rigid in the sense that the unitary equivalence class of the operator and the flag structure determine each other.",
"We obtain a complete set of unitary invariants which are somewhat more tractable than those of an arbitrary operator in the Cowen-Douglas class.",
"The Cowen-Douglas class $B_n(\\Omega )$ consists of those bounded linear operators $T$ on a complex separable Hilbert space $\\mathcal {H}$ which possess an open set $\\Omega \\subset \\mathbb {C}$ of eigenvalues of constant multiplicity $n$ and admit a holomorphic choice of eigenvectors : $s_1(w), \\ldots , s_n(w)$ , $w\\in \\Omega $ , in other words, there exists holomorphic functions $s_1, \\ldots , s_n: \\Omega \\rightarrow \\mathcal {H}$ which span the eigenspace of $T$ at $w\\in \\Omega $ .",
"The holomorphic choice of eigenvectors $s_1, \\ldots , s_n$ defines a holomorphic Hermitian vector bundle $E_T$ via the map $s: \\Omega \\rightarrow \\mbox{\\rm Gr}(n, \\mathcal {H}),\\,\\, s(w) = \\ker (T-w) \\subseteq \\mathcal {H}.$ In the paper [3], Cowen and Douglas show that there is a one to one correspondence between the unitary equivalence class of the operators $T$ in $B_n(\\Omega )$ and the equivelence classes of the holomorphic Hermitian vector bundles $E_T$ determined by them.",
"They also find a set of complete invariants for this equivalence consisting of the curvature $\\mathcal {K}$ of $E_T$ and a certain number of its covariant derivatives.",
"Unfortunately, these invariants are not easy to compute unless $n$ is 1.",
"Also, it is difficult to determine, in general, when an operator $T$ in $B_n(\\Omega )$ is irreducible, again except in the case $n=1$ .",
"In this case, the rank of the vector bundle is 1 and therefore it is irreducible and so is the operator $T$ .",
"Finding similarity invariants for operators in the class $B_n(\\Omega )$ has been somewhat difficult from the beginning.",
"Counter examples to the similarity conjecture in [3] were given in [1], [2].",
"More recently, significant progress on the question of similarity has been made (cf.",
"[6], [10], [11] ).",
"We isolate a subset of irreducible operators in the Cowen-Douglas class $B_n(\\Omega )$ for which a complete set of tractable unitary invariants is relatively easy to identify.",
"We also determine when two operators in this class are similar.",
"We introduce below this smaller class $\\mathcal {F}B_2(\\Omega )$ of operators in $B_2(\\Omega )$ leaving out the more general definition of the class $\\mathcal {F}B_n(\\Omega ),\\, n>2,$ for now.",
"Definition 1 We let $\\mathcal {F}B_2(\\Omega )$ denote the set of operators $T\\in B_2(\\Omega )$ which admit a decomposition of the form $T=\\begin{pmatrix}T_0 & S \\\\0 & T_1 \\\\\\end{pmatrix}$ for some choice of operators $T_0,T_1\\in \\mathcal {B}_1(\\Omega )$ and a non-zero intertwining operator $S$ between $T_0$ and $T_1$ , that is, $T_0S=ST_1$ .",
"An operator $T$ in $B_2(\\Omega )$ admits a decomposition of the form (cf.",
"[11]) $\\Big ({\\begin{matrix}T_0 & S \\\\0 & T_1 \\\\\\end{matrix}}\\Big )$ for some pair of operators $T_0$ and $T_1$ in ${B}_1(\\Omega )$ .",
"In defining the new class $\\mathcal {F}B_2(\\Omega )$ , we are merely imposing one additional condition, namely that $T_0S=ST_1$ .",
"We show that $T$ is in the class $\\mathcal {F}B_2(\\Omega )$ if and only if there exist a frame $\\lbrace \\gamma _0,\\gamma _1\\rbrace $ of the vector bundle $E_{T}$ such that $\\gamma _0(w)$ and $t_1(w):=\\tfrac{\\partial }{\\partial w}\\gamma _0(w)-\\gamma _1(w)$ are orthogonal for all $w$ in $\\Omega $ .",
"This is also equivalent to the existence of a frame $\\lbrace \\gamma _0,\\gamma _1\\rbrace $ of the vector bundle $E_{T}$ such that $\\tfrac{\\partial }{\\partial w}\\Vert \\gamma _0(w)\\Vert ^2=\\langle \\gamma _1(w),\\gamma _0(w)\\rangle ,\\,w\\in \\Omega $ .",
"Our first main theorem on unitary classification is given below, where we have set $\\mathcal {K}_{T_0}(z) = -\\tfrac{\\partial ^2}{\\partial z \\partial \\bar{z}} \\log \\Vert \\gamma _0(z)\\Vert ^2.$ Theorem 1 Let $T=\\begin{pmatrix}T_0 & S \\\\0 & T_1 \\\\\\end{pmatrix}$ and $\\tilde{T}=\\begin{pmatrix}\\tilde{T}_0 & \\tilde{S} \\\\0 & \\tilde{T}_1 \\\\\\end{pmatrix}$ be two operators in $\\mathcal {F}B_2(\\Omega )$ .",
"Also let $t_1$ and $\\tilde{t}_1$ be non-zero sections of the holomorphic Hermitian vector bundles $E_{T_1}$ and $E_{\\tilde{T}_1}$ respectively.",
"The operators $T$ and $\\tilde{T}$ are equivalent if and only if $\\mathcal {K}_{T_0}=\\mathcal {K}_{\\tilde{T}_0}$ (or $\\mathcal {K}_{T_1}=\\mathcal {K}_{\\tilde{T}_1}$ ) and $\\frac{\\Vert S(t_1)\\Vert ^2}{\\Vert t_1\\Vert ^2}= \\frac{\\Vert \\tilde{S}(\\tilde{t}_1)\\Vert ^2}{\\Vert \\tilde{t}_1\\Vert ^2}$ .",
"In any decomposition $\\Big ({\\begin{matrix}T_0 & S \\\\0 & T_1 \\\\\\end{matrix}} \\Big ),$ of an operator $T\\in \\mathcal {F}B_2(\\Omega ),$ let $t_1$ be a non zero section of holomorphic Hermitian vector bundle $E_{T_1}$ .",
"We assume, without loss of generality, that $S(t_1)$ is a non zero section of $E_{T_0}$ on some open subset of $\\Omega $ .",
"Following the methods of [8], the second fundamental form of $E_{T_0}$ in $E_T$ is easy to compute.",
"It is the $(1,0)$ -form $\\tfrac{\\mathcal {K}_{T_0}(z)}{\\,\\,\\Big (-\\mathcal {K}_{T_0}(z)+\\tfrac{\\Vert t_1(z)\\Vert ^2}{\\big \\Vert S(t_1(z) )\\big \\Vert ^2}\\Big )^{\\!\\!\\stackrel{}{1}/2}}d\\bar{z}.$ Thus the second fundamental form of $E_{T_0}$ in $E_T$ together with the curvature of $E_{T_0}$ is a complete set of invariants for the operator $T$ .",
"The inclusion of the line bundle $E_{T_0}$ in the vector bundle $E_{T}$ of rank 2 is the flag structure of $E_T$ .",
"It is not easy to determine which operators in $B_n(\\Omega )$ are irreducible.",
"We show that the operators in the new class $\\mathcal {F}B_2(\\Omega )$ are always irreducible.",
"Indeed, if we assume $S$ is invertible, then $T$ is strongly irreducible, that is, there is no non-trivial idempotent commuting with $T.$ Recall that an operator $T$ in the Cowen-Douglas class $B_n(\\Omega )$ , up to unitary equivalence, is the adjoint of the multiplication operator $M$ on a Hilbert space $\\mathcal {H}$ consisting of holomorphic functions on $\\Omega ^*:=\\lbrace \\bar{w}: w \\in \\Omega \\rbrace $ possessing a reproducing kernel $K$ .",
"What about operators in $\\mathcal {F}B_2(\\Omega )$ ?",
"A model for these operators is described below.",
"Let $\\gamma =(\\gamma _0,\\gamma _1)$ be a holomorphic frame for the vector bundle $E_T$ , $T\\in \\mathcal {F}B_2(\\Omega )$ .",
"Then the operator $T$ is unitarily equivalent to the adjoint of the multiplication operator $M$ on a reproducing kernel Hilbert space $\\mathcal {H}_{\\Gamma } \\subseteq {\\rm Hol}(\\Omega ^*, \\mathbb {C}^2)$ possessing a reproducing kernel $K_{\\Gamma }:\\Omega ^* \\times \\Omega ^* \\rightarrow \\mathbb {C}^{2\\times 2}$ of the special form that we describe explicitly now.",
"For $z,w\\in \\Omega ^*$ , $K_{\\Gamma }(z,w)&=&\\begin{pmatrix}\\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle &\\langle \\gamma _1(\\bar{w}),\\gamma _0(\\bar{z})\\rangle \\\\\\langle \\gamma _0(\\bar{w}),\\gamma _1(\\bar{z})\\rangle &\\langle \\gamma _1(\\bar{w}),\\gamma _1(\\bar{z})\\rangle \\\\\\end{pmatrix}\\\\&=& \\begin{pmatrix}\\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle &\\frac{\\partial }{\\partial \\bar{w}}\\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle \\\\\\frac{\\partial }{\\partial z} \\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle &\\frac{\\partial ^2}{\\partial z\\partial \\bar{w}} \\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle +\\langle t_1(\\bar{w}),t_1(\\bar{z})\\rangle \\\\\\end{pmatrix},$ where $t_1$ and $\\gamma _0:=S(t_1)$ are frames of the line bundles $E_{T_1}$ and $E_{T_0}$ respectively.",
"It follows that $\\gamma _1(w):=\\tfrac{\\partial }{\\partial w}\\gamma _0(w)-t_1(w)$ and that $t_1(w)$ is orthogonal to $\\gamma _0(w)$ , $w\\in \\Omega $ .",
"Setting $K_0(z,w)=\\langle \\gamma _0(\\bar{w}),\\gamma _0(\\bar{z})\\rangle $ and $K_1(z,w)= \\langle t_1(\\bar{w}),t_1(\\bar{z})\\rangle $ , we see that the reproducing kernel $K_{\\Gamma }$ has the form : $ K_{\\Gamma }(z,w)= \\begin{pmatrix}{K_0}(z,w) & \\frac{\\partial }{\\partial \\bar{w}}{K_0}(z,w) \\\\\\frac{\\partial }{\\partial z}{K_0}(z,w) & \\frac{\\partial ^2}{\\partial z\\partial \\bar{w}}{K_0}(z,w)+{K_1}(z,w) \\\\\\end{pmatrix}.$ This special form of the kernel $K_\\Gamma $ for an operator in the class $\\mathcal {F}B_2(\\Omega )$ entails that a change of frame between any two frames $\\lbrace \\gamma _0,\\gamma _1\\rbrace $ and $\\lbrace \\sigma _0,\\sigma _1\\rbrace $ of vector bundle $E_{T}$ , which has property $\\gamma _0\\perp \\partial \\gamma _0-\\gamma _1$ and $\\sigma _0\\perp \\partial \\sigma _0-\\sigma _1$ , must be induced by a holomorphic $\\Phi :\\Omega \\rightarrow \\mathbb {C}^{2\\times 2}$ of the form $\\Phi =\\Big ( {\\begin{matrix}\\phi & \\phi {\\prime } \\\\0 & \\phi \\\\\\end{matrix}}\\Big )$ for some holomorphic function $\\phi :\\Omega \\rightarrow \\mathbb {C}.$ As an immediate corollary, we see that an unitary operator intertwining two of these operators, represented in the form $T:=\\Big ( {\\begin{matrix} T_0 & S \\\\ 0 & T_1 \\end{matrix}}\\Big )$ and $\\tilde{T}:=\\Big ( {\\begin{matrix} \\tilde{T}_0 & \\tilde{S} \\\\ 0 & \\tilde{T}_1 \\end{matrix}}\\Big )$ , must be diagonal with respect to the implicit decomposition of the two Hilbert spaces $\\mathcal {H}$ and $\\tilde{\\mathcal {H}}.$ As a second corollary, we see that if $T_0=\\tilde{T}_0$ and $T_1=\\tilde{T}_1,$ then the operators $T$ and $\\tilde{T}$ are unitarily equivalent if and only if $\\tilde{S} = e^{i\\theta } S$ for some real $\\theta .$ We now give examples of natural classes of operators that belong to $\\mathcal {F}B_2(\\Omega )$ .",
"Indeed, we were led to the definition of this new class $\\mathcal {F} B_2(\\Omega )$ of operators by trying to understand these examples better.",
"An operator $T$ is called homogeneous if $\\phi (T)$ is unitarily equivalent to $T$ for all $\\phi $ in Möb which are analytic on the spectrum of $T$ .",
"If an operator $T$ is in ${B}_1(\\mathbb {D})$ , then $T$ is homogeneous if and only if $\\mathcal {K}_T(w)=-\\lambda (1-|w|^2)^{-2},$ for some $\\lambda >0$ .",
"A model for all homogeneous operators in $B_n(\\mathbb {D})$ is given in [12].",
"We describe them for $n=2$ .",
"For $\\lambda > 1$ and $\\mu > 0$ , set $K_0(z,w) = (1-z\\bar{w})^{-\\lambda }$ and $K_1(z,w) = \\mu (1-z\\bar{w})^{-\\lambda -2}$ .",
"An irreducible operator $T$ in $B_2(\\mathbb {D})$ is homogeneous if and only if it is unitarily equivalent to the adjoint of the multiplication operator on the Hilbert space $\\mathcal {H}\\subseteq {\\rm Hol}(\\mathbb {D}, \\mathbb {C}^2)$ determined by the positive definite kernel given in equation (REF ).",
"The similarity as well as a unitary classification of homogeneous operators in $B_n(\\mathbb {D})$ were obtained in [12] using non-trivial results from representation theory of semi-simple Lie group.",
"For $n=2$ , this classification is a consequence of Theorem REF .",
"An operator $T$ in $B_1(\\Omega )$ acting on a Hilbert space $\\mathcal {H}$ makes it a module over the polynomial ring via the usual point-wise multiplication.",
"An important tool in the study of these modules is the module tensor product (or, localization).",
"This is the Hilbert module $J \\mathcal {H}^{(k)}_{\\rm loc}$ corresponding to the spectral sheaf $J \\mathcal {H} \\otimes _\\mathcal {P} \\mathbb {C}^k_w$ , where $\\mathcal {P}$ is the polynomial ring and $J:\\mathcal {H} \\rightarrow {\\rm Hol}(\\Omega , \\mathbb {C}^k)$ is the jet map, namely, $Jf = \\sum _{\\ell =0}^{k-1} \\partial ^\\ell f \\otimes \\varepsilon _{\\ell +1},$ $\\varepsilon _1, \\ldots ,\\varepsilon _{k}$ are the standard unit vectors in $\\mathbb {C}^k$ .",
"$\\mathbb {C}^k_w$ is a $k$ - dimensional module over the polynomial ring, the module action on $\\mathbb {C}^k_w$ is via the map $\\mathcal {J}(w),$ see [7] : $(\\mathcal {J}f)(w)=\\left(\\begin{array}{cccc}f(w) & 0 & \\cdots & 0 \\\\\\binom{2}{1}\\partial f(w) & f(w) & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\binom{k}{1}\\partial ^{k-1}f(w) & \\binom{k-1}{1}\\partial ^{k-2}f(w) & \\cdots & f(w) \\\\\\end{array}\\right),$ that is, $(f,v) \\mapsto (\\mathcal {J}f)(w) v,$ $f\\in \\mathcal {P},\\, v\\in \\mathbb {C}^k.$ $J:\\mathcal {H} \\rightarrow {\\rm Hol}(\\Omega , \\mathbb {C}^k)$ is the jet map, namely, $Jf = \\sum _{\\ell =0}^{k-1} \\partial ^\\ell f \\otimes \\varepsilon _{\\ell +1},$ $\\varepsilon _1, \\ldots ,\\varepsilon _{k}$ are the standard unit vectors in $\\mathbb {C}^k$ .",
"$\\mathbb {C}^k_w$ is a $k$ - dimensional module over the polynomial ring, the module action on $\\mathbb {C}^k_w$ is via the map $\\mathcal {J}(w),$ see [7] : $(\\mathcal {J}f)(w)=\\left(\\begin{array}{cccc}f(w) & 0 & \\cdots & 0 \\\\\\binom{2}{1}\\partial f(w) & f(w) & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\binom{k}{1}\\partial ^{k-1}f(w) & \\binom{k-1}{1}\\partial ^{k-2}f(w) & \\cdots & f(w) \\\\\\end{array}\\right),$ that is, $(f,v) \\mapsto (\\mathcal {J}f)(w) v,$ $f\\in \\mathcal {P},\\, v\\in \\mathbb {C}^k.$ We now consider the localization with $k=2$ .",
"If we assume that the operator $T$ has been realized as the adjoint of the multiplication operator on a Hilbert space of holomorphc function possessing a kernel function, say $K$ , then the kernel $JK^{(2)}_{\\rm loc}$ for the localization (of rank 2) given in [7] coincides with $K_\\Gamma $ of equation (REF ).",
"In this case, we have $K_1=K=K_0$ .",
"The operator $T,$ in this case, has the form $\\Big ({\\begin{matrix} T & \\binom{2}{1}\\,I\\\\ 0 & T\\\\\\end{matrix}}\\Big ).$ As is to be expected, using the complete set of unitary invariants given in Theorem REF , we see that the unitary equivalence class of the Hilbert module $\\mathcal {H}$ is in one to one correspondence with that of $J \\mathcal {H}^{(2)}_{\\rm loc}$ .",
"Thus the class $\\mathcal {F}B_2(\\Omega )$ contains two very interesting classes of operators.",
"For $n > 2$ , we find that there are competing definitions.",
"One of these contains the homogeneous operators and the other contains the Hilbert modules obtained from the localization.",
"At this point, we note that most of what is said for the class $\\mathcal {F} B_2(\\Omega )$ remains valid if we assume $T_0$ is in $B_{n}(\\Omega )$ , $n>1$ , instead of $B_1(\\Omega )$ .",
"Although, now we must assume that the operator $S$ has dense range, merely assuming that it is non-zero is not enough.",
"Also, it is no longer possible to describe a complete set of invariants for such an operator as in Theorem REF .",
"We proceed slightly differently to ensure a better understanding.",
"Definition 2 Let $\\mathcal {F}B_n(\\Omega )$ be the set of all operators $T$ in the Cowen-Douglas class $B_n(\\Omega )$ for which we can find operators $T_0,T_1, \\ldots , T_{n-1}$ in $B_1(\\Omega )$ and a decomposition of the form $T=\\begin{pmatrix}T_{0} & S_{0\\,1} &S_{0\\,2}&\\ldots &S_{0\\,n-1}\\\\0&T_{1}&S_{1\\,2}&\\ldots &S_{1\\,n-1} \\\\\\vdots &\\ddots &\\ddots &\\ddots &\\vdots \\\\0&\\ldots &0&T_{n-2}&S_{n-2\\,n-1}\\\\0&\\ldots &\\ldots &0&T_{n-1}\\\\\\end{pmatrix}$ such that none of the operators $S_{i\\,i+1}$ are zero and $T_i S_{i\\,i+1} = S_{i\\,i+1}T_{i+1}.$ If there exists a invertible bounded linear operator $X$ intertwining any two operators, say $T,\\, \\tilde{T}$ in $\\mathcal {F}B_n(\\Omega )$ ($XT = \\tilde{T}X$ ), then we prove that $X$ must be upper triangular with respect to the decomposition mandated in the definition of the class $\\mathcal {F}B_n(\\Omega ).$ It then follows that any unitary operator intertwining these two operators must be diagonal.",
"Thus we see that they are unitarily equivalent if and only there exists unitary operators $U_i: \\mathcal {H}_i \\rightarrow \\tilde{\\mathcal {H}}_i$ , $i=0,1,\\cdots n-1,$ such that $U_i^*\\widetilde{T}_iU_i=T_i$ and $U_iS_{i,j}=\\widetilde{S}_{i,j}U_j,$ $i<j$ .",
"The first of these conditions immediately translates into a condition on the curvature of the line bundles $E_{T_i}.$ The second condition is somewhat more mysterious and is related to a finite number of second fundamental forms inherent in our description of the operator $T.$ In what follows, we make this a little more explicit after making some additional assumptions.",
"Let $T$ be an operator acting on a Hilbert space $\\mathcal {H}.$ Assume that there exists a representation of the form $T= \\begin{pmatrix}T_{0}&S_{0\\,1}&0&\\ldots &0\\\\0&T_{1}&S_{1\\,2}&\\ldots &0\\\\\\vdots &\\ddots &\\ddots &\\ddots &\\vdots \\\\0&\\ldots &0&T_{n-2}&S_{{n-2\\,n-1}}\\\\0&\\ldots &0&0&T_{n-1}\\end{pmatrix}$ for the operator $T$ with respect to some orthogonal decomposition $\\mathcal {H}:=\\mathcal {H}_0\\oplus \\mathcal {H}_1\\oplus \\cdots \\oplus \\mathcal {H}_{n-1}.$ Suppose also that the operator $T_{i}$ is in $B_1(\\Omega ),$ $0\\le i \\le n-1,$ the operator $S_{i-1,i}$ is non zero and $T_{k-1}S_{k-1,k}=S_{k-1,k}T_{k},$ $1\\le i\\le n-1.$ Then we show that the operator $T$ must be in the Cowen-Douglas class $B_n (\\Omega ).$ We can also relate the frame of the vector bundle $E_T$ to those of the line bundles $E_{T_i},$ $i=0,1,\\ldots , n-1.$ Indeed, we show that there is a frame $\\lbrace \\gamma _0,\\gamma _1,\\cdots ,\\gamma _{n-1}\\rbrace $ of $E_{T}$ such that $t_k(w):=\\gamma _k(w)+\\cdots +(-1)^j\\binom{k}{i}\\gamma _{k-j}^{(j)}(w)+\\cdots +(-1)^{k}\\gamma _0^{(k)}(w)$ is a non-zero section of the line bundle $E_{T_k}$ and it is orthogonal to $\\gamma _i(w)$ , $i=0,1,2,\\ldots , k-1.$ We also have $t_{i-1}:=S_{i-1\\,i}(t_i),$ $1 \\le i \\le n-1$ .",
"In this special case, we can extract a complete set of invariants explicitly.",
"Theorem 2 Pick two operators $T$ and $\\tilde{T}$ which admit a decomposition of the form given in (REF ).",
"Find an orthogonal frame $\\lbrace \\gamma _0,t_1,\\cdots ,t_{n-1}\\rbrace $ (resp.",
"$\\lbrace \\tilde{\\gamma }_0,\\tilde{t}_{1},\\cdots ,\\tilde{t}_{n-1}\\rbrace $ ) for the vector bundle ${\\bigoplus \\limits ^{n}_{i=0}E_{T_i}}$ (resp.",
"${\\bigoplus \\limits ^{n}_{i=0}E_{\\tilde{T}_i}}$ ) as above.",
"Then the operators $T$ and $\\tilde{T}$ are unitarily equivalent if and only if $\\mathcal {K}_{T_0} =\\mathcal {K}_{\\tilde{T}_0}\\mbox{and }\\frac{\\Vert S_{i-1\\,i}(t_i)\\Vert ^2}{\\Vert t_{i}\\Vert ^2}=\\frac{\\Vert \\tilde{S}_{i-1\\,i}(\\tilde{t}_i)\\Vert ^2}{\\Vert \\tilde{t}_{i}\\Vert ^2},\\,\\, 1\\le i\\le n-1.$ The operator corresponding to the module action $J\\mathcal {H}^{(k)}_{\\rm loc},$ or for that matter $\\mathcal {H}^{(k)}_{\\rm loc},$ has exactly the form described above while the homogeneous operators (of rank $>2$ ) are not of this form, although they meet the requirements set forth in Definition 2."
]
] | 1403.0257 |
[
[
"Spin disorder in an Ising honeycomb chain cobaltate"
],
[
"Abstract We report on a member of the spin-disordered honeycomb lattice antiferromagnet in a quasi-one-dimensional cobaltate Ba_3Co_2O_6(CO_3)_0.7.",
"Resistivity exhibits as semimetallic along the face-sharing CoO6 chains.",
"Magnetic susceptibility shows strongly anisotropic Ising-spin character with the easy axis along the chain due to significant spin-orbit coupling and a trigonal crystal field.",
"Nevertheless, ^135Ba NMR detects no indication of the long-range magnetic order down to 0.48 K. Marginally itinerant electrons possess large entropy and low-lying excitations with a Wilson ratio R_W = 116, which highlight interplays of charge, spin, and orbital in the disordered ground state."
],
[
"Spin disorder in an Ising honeycomb chain cobaltate Hirokazu Igarashi Yasuhiro Shimizu yasuhiro@iar.nagoya-u.ac.jp Yoshiaki Kobayashi Masayuki Itoh Department of Physics, Graduate School of Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan.",
"We report a new member of the spin-disordered honeycomb lattice antiferromagnet in a quasi-one-dimensional cobaltate Ba$_3$ Co$_2$ O$_6$ (CO$_3$ )$_{0.7}$ .",
"Resistivity exhibits as semimetallic along the face-sharing CoO$_6$ chains.",
"Magnetic susceptibility shows strongly anisotropic Ising-spin character with the easy axis along the chain due to significant spin-orbit coupling and a trigonal crystal field.",
"Nevertheless, $^{135}$ Ba NMR detects no indication of the long-range magnetic order down to 0.48 K. Marginally itinerant electrons possess large entropy and low-lying excitations with a Wilson ratio $R_W$ = 116, which highlight interplays of charge, spin, and orbital in the disordered ground state.",
"72.15.Jf, 75.40.-s, 75.10.Kt, 75.30.Gw Quantum disorder states have attracted material scientists since proposals of residual entropy[1] and resonating valence bond[2] on geometrically frustrated spin lattices.",
"In the absence of frustration, a disorder state is realized due to low-dimensional fluctuations.",
"In dimensions larger than two, a honeycomb lattice has the smallest coordination number and hence largest fluctuations.",
"Realization of the quantum spin liquid on the honeycomb lattice has attracted great theoretical interests.",
"[3], [4] In a strong coupling limit, however, Heisenberg models with nearest neighbor interactions give N${\\rm \\acute{e}}$ el order ground states.",
"[1], [5], [6], [7] Consideration of second and third neighbor exchange interactions induces emergent quantum liquids.",
"[8], [9], [10], [3], [11], [12] Low-lying excitations from the ground state are discussed with respect to gapped[3] or gapless[9], [10] spinon.",
"In the presence of moderate electron correlations and spin-orbit coupling, the Mott transition occurs from a (topological) spin liquid to Dirac semimetal or a topological insulator.",
"[4], [13], [14], [15] Experimental examples of the honeycomb lattice have been recently found in some transition-metal oxides such as Bi$_3$ Mn$_4$ O$_{12}$ (NO$_3$ ),[16] Ba$_3$ CuSb$_2$ O$_9$ ,[17] and Na$_2$ IrO$_3$ .",
"[18] However, the ground state undergoes spin glass for the manganite[19] and a gapped spin-liquid for the cuprate,[20] and a N${\\rm \\acute{e}}$ el order in the iridate.",
"Here we add a new example of the honeycomb lattice antiferromagnet, a quasi-one-dimensional (quasi-1D) cobaltate Ba$_3$ Co$_2$ O$_6$ (CO$_3$ )$_{0.7}$ (Ba326) with gapless low-lying excitations.",
"In contrast to the previous examples, the honeycomb lattice is constructed by itinerant Ising chains for Ba326.",
"The crystal structure belongs to hexagonal $P\\bar{6}$ and consists of CoO$_6$ and CO$_3$ chains running along the $c$ axis,[21], [22] as shown in Fig.",
"1(a).",
"The magnetic CoO$_6$ chains form the honeycomb lattice in the $ab$ plane, and the nonmagnetic CO$_3^{2-}$ chains are located in a tunnel of the honeycomb [Fig.",
"1(b)].",
"The concentration of CO$_3$ ions was determined from the oxygen contents by inert gas fusion-infrared absorption analysis.",
"[22] The incommensurate stoichiometry (possibly disordered) gives a partially-filled conduction band with the nominal valence of Co$^{3.7+}$ ($3d^{5.3}$ ).",
"The chain structure and transport properties of Ba326 are analogous to those of Ca$_3$ Co$_2$ O$_6$ (Ca326)[23] and Ca$_3$ CoRhO$_6$ (Ref.Niitaka) with partially-disordered states on triangular-lattice chains.",
"However, the low-temperature magnetic and electric properties have not been investigated for Ba326.",
"Iwasaki et al.",
"reported the metallic conductivity $\\sigma $ and high Seebeck coefficient $S_{\\rm e}$ above 300 K.[22] The figure of merit $ZT = 0.51 \\times 10^{-5}$ K$^{-1}$ ($Z = \\sigma S_{\\rm e}^2 \\kappa ^{-1}$ , $\\kappa $ : thermal conductivity) is comparable to those of Ca326 and Na$_{0.75}$ CoO$_2$ at 300 K.[22] Figure: (a) Crystal structure of Ba326 consisting of independent chains of CoO 6 _6, closed-shell Ba, and CO 3 _3 along the cc axis.Non-stoichiometric CO 3 _3 occupies 90% in the crystal structure, as determined from the chemical analysis and the X-ray diffraction measurement.The arrows illustrate spins with ferromagnetic correlations along the chain and antiferromagnetic one in the abab plane.",
"(b) The cc-axis projected crystal structure where CoO 6 _6 chains form a honeycomb lattice in the abab plane and CO 3 _3 chains are placed inside honeycomb framework.",
"(c) t 2g t_{2g} orbital occupations of Co 4+ ^{4+} and the spin-orbital levels consisting of three sets of Kramers doublets (ff, gg, hh) in presence of spin-orbit coupling and trigonal field, where a hole is occupied into the highest hh levels.In a trigonal crystal field of the face-sharing CoO$_6$ , a $t_{2g}$ triplet (an effective angular momentum $l$ = 1) splits into a lower lying $|l_z = 0 \\rangle = a_{1g}$ singlet and a higher lying $|l_z = \\pm 1 \\rangle = e^\\prime _g$ doublet.",
"The residual orbital degeneracy for Co$^{4+}$ ($d^5$ ) leads to spin-orbit coupling $\\lambda {\\bf s} \\cdot {\\bf l}$ with a coupling constant $\\lambda $ , comparable to the crystal field splitting ($\\sim $ 0.1 eV) and exchange couplings.",
"[25] Then the spin-orbital levels [Fig.",
"1(c)] are expressed by coherent mixtures of different orbital and spin states.",
"[25] Under anisotropic electron correlations and frustration, the magnetism and transport properties carried by spin-orbital entangled states may have emergent features, as extensively investigated in $5d$ compounds providing a strong $\\lambda $ limit.",
"[13], [14] In this respect, Ba326 is linked to a charge-spin-orbital coupled system on the honeycomb lattice.",
"In this paper, we investigate the ground state of Ba326 through transport, thermodynamic, and magnetic measurements.",
"The charge transport is confined into the $c$ -axis chain and becomes weakly localized in the ground state.",
"The magnetic susceptibility exhibits Curie-Weiss-like temperature dependence with the anisotropic Weiss temperature implying anisotropic exchange interactions between Ising spins.",
"We found no indication of the long-range magnetic order and structural transitions in NMR and specific heat measurements.",
"The reasons are discussed with respect to spin-orbit interactions in the itinerant media.",
"Single crystals of Ba326 were prepared by a K$_2$ CO$_3$ -BaCl$_2$ flux method using a mixture of BaCO$_3$ , Co$_3$ O$_4$ , K$_2$ CO$_3$ , and BaCl$_2$ reagents at 1273 K.[22] The $c$ axis of needle-shape crystals was determined by X-ray diffraction patterns with peaks at $(00l)$ ($l$ : integer).",
"The typical size of the crystal was 1 mm $\\times $ 1 mm $\\times $ 3 mm.",
"To conduct resistivity and thermopower measurements, gold wires were attached to the sample after breaking an oxidized insulating surface.",
"The magnetic susceptibility, NMR, and specific heat were measured for aligned single crystals to gain sensitivity.",
"$^{135}$ Ba NMR measurements were performed in constant magnetic fields $H_0$ = 9.4 T (1.9-200 K) and 8.5 T (0.48-2.5 K) parallel to the $c$ axis.",
"$^{135, 137}$ Ba NQR spectrum was not detected in the frequency range of 6-50 MHz at zero field and hence the nuclear quadrupole frequency $\\nu _Q$ was expected lower than 6 MHz.",
"The spin-echo intensity was not large enough to obtain the nuclear spin-lattice relaxation rate $1/T_1$ due to the small sample amount.",
"Instead, the nuclear spin-spin relaxation rate $1/T_2$ was obtained from the single-exponential decay of integrated spin-echo intensities as a function of pulse interval time $2\\tau $ .",
"Figure: (a) Resistivity measured parallel, ρ ∥ \\rho _\\parallel , and perpendicular, ρ ⊥ \\rho _\\perp , to the cc axis for the identical crystal of Ba326.Inset: the resistivity anisotropy ρ ⊥ \\rho _\\perp /ρ ∥ \\rho _\\parallel plotted against temperature.",
"(b) Magnetic susceptibility under magnetic fields (0.01, 1, 5 T) parallel, χ ∥ \\chi _\\parallel , or (0.1, 1 T) perpendicular, χ ⊥ \\chi _\\perp , to the cc axis for the zero-field-cooling (ZFC, solid curves) and field cooling (FC, dotted curves) measurements (the left-hand axis).χ -1 \\chi ^{-1} at 1 T fits to the Curie-Weiss law (dotted lines) in the TT range 220-300 K (the right-hand axis).Figure 2 shows the temperature $T$ dependence of resistivity and magnetic susceptibility below 300 K. The parallel resistivity $\\rho _\\parallel $ measured along the $c$ axis is conducting with weak $T$ dependence, while the perpendicular one $\\rho _\\perp $ behaves semiconducting down to 5 K. $\\rho _\\parallel $ seems to depend on the current path because of the insulating surface and basically metallic at high temperatures.",
"Resistivity anisotropy ($\\rho _\\perp /\\rho _\\parallel \\sim 10^2$ at 290 K) increases on cooling $T$ ($\\sim 2 \\times 10^3$ at 50 K) [inset of Fig.",
"2(a)] and shows the quasi-1D electronic structure, as expected from the anisotropic crystal structure.",
"Below 100 K, $\\rho _\\parallel $ increases by three orders of magnitude down to 5 K. Since $\\rho _\\perp $ exhibits no anomaly around 100 K, the $\\rho _\\parallel $ increase unlikely comes from a phase transition due to charge and orbital ordering.",
"Considering the incommensurate potential from CO$_3$ ions, weak localization is a possible source of the resistivity increase for the 1D system.",
"However, $\\rho _\\parallel (T)$ does not follow a normal activation type or a variable range hopping model but behaves close to a power law, $\\sim T^{-2.5}$ , below 80 K. Figure: Magnetization of Ba326 as a function of magnetic field parallel to the cc axis at various temperatures.The magnetic susceptibility in Fig.",
"2 (b) also displays highly anisotropic behavior.",
"The parallel susceptibility $\\chi _{\\parallel }$ increases with decreasing $T$ and becomes more than 10 times larger than the perpendicular one $\\chi _{\\perp }$ at 20 K. It manifests the strong Ising nature due to spin-orbit coupling.",
"$\\chi _{\\parallel }^{-1}$ and $\\chi _{\\perp }^{-1}$ show linear $T$ dependence above 200 K with nearly the same slop.",
"Assuming the Curie-Weiss law $\\chi _i^{-1} = (T - \\Theta _i)/C_i^\\prime $ ($i = \\parallel $ , $\\perp $ ) above 200 K, we obtained the Curie constant $C^\\prime _\\parallel $ = 0.48 emu/(Co-mol K) and Weiss temperatures $\\Theta _\\parallel $ = 69 K for $\\chi _{\\parallel }^{-1}$ , and $C^\\prime _{\\perp }$ = 0.46 emu/(Co-mol K) and $\\Theta _\\perp $ = $-100$ K for $\\chi _{\\perp }^{-1}$ .",
"$\\Theta $ should be isotropic in the Heisenberg model and expressed by $\\Theta = (2J_\\parallel + 3J_\\perp )g_i^2/4k_{\\rm B}$ , where $J_\\parallel $ and $J_\\perp $ are exchange couplings parallel and perpendicular to the chain, respectively.",
"The anisotropy likely arises from the trigonal crystal field and spin-orbit coupling, as well as the exchange anisotropy.",
"The effective Curie constant has been evaluated for Co$^{2+}$ as $C^\\prime _i = (N\\mu _{\\rm B}^2 /k_{\\rm B})(g_i^2+B_i k_{\\rm B}T/\\lambda )$ , where $N$ is the Avogadro's number, $g_i$ the thermally averaged g-factor, $\\mu _{\\rm B}$ the Bohr magneton, and $B_i$ the thermally averaged second-order Zeeman coefficient.",
"[26] Postulating the same model for Co$^{4+}$ , the nearly isotropic $C^\\prime _i$ despite the significant spin-orbit coupling is attributable to cancellation of the anisotropy in $g_i^2$ and $B_i k_{\\rm B}T/\\lambda $ .",
"For the low-spin Co$^{4+}$ ($S = 1/2$ ) and Co$^{3+}$ ($S = 0$ ), the effective Curie constant $C^\\prime _i = (N\\mu _{\\rm B}^2 g^2 \\sqrt{S(S+1)}/3k_{\\rm B})$ gives the averaged g-value, $g = 2.7$ .",
"The deviation of $\\chi _{\\parallel }^{-1}$ and $\\chi _{\\perp }^{-1}$ from the Curie-Weiss law below 150 K points to significant short-range correlation and spin-orbit coupling.",
"In the classical Ising spin system with ferromagnetic interactions, the long-range magnetic order occurs at $T_c \\sim \\Theta $ as three-dimensional coupling sets in.",
"Hence there are strong frustration and/or fluctuations against the ordering.",
"Further decreasing $T$ , $\\chi _{\\parallel }$ and $\\chi _{\\perp }$ exhibit no appreciable difference in the zero-field-cooled and field-cooled data at low fields.",
"The result rules out a possible spin glass state.",
"As shown in Fig.",
"3, the magnetization $M$ is nearly linear to the external magnetic field parallel to the chain down to 2 K. A weak field-dependence in $\\chi _{\\parallel }$ below 50 K implies that $M$ begins to saturate at high fields and low temperatures.",
"This robust paramagnetic behavior is surprising for the longitudinal magnetic field parallel to the Ising easy axis, because the analogous compound Ca326 with $S$ = 2 exhibits field-induced ferrimagnetic transitions.",
"[23], [27] Figure: (a) 135 ^{135}Ba NMR spectra for aligned single crystals of Ba326, measured at H 0 H_0 = 9.4 (T>T > 2 K) and 8.5 (T<T < 2 K) T parallel to the cc axis.",
"(b) Knight shift KK versus magnetic susceptibility χ ∥ \\chi _{\\parallel } plot.",
"(c) TT dependence of the linewidth (left axis) defined by the full-width at half maximum intensity and the nuclear spin-spin relaxation rate T 2 -1 T_2^{-1} (right axis).NMR is utilized as a sensitive microscopic probe of magnetic ordering.",
"$^{135}$ Ba NMR spectrum measures transfer hyperfine and dipole fields from Co moments as a resonance shift and broadening.",
"The crystal structure of Ba326 has three Ba sites.",
"In a magnetic field $H_0 \\parallel c$ , we observed a single broad spectrum.",
"The linewidth comes from the three Ba sites with the slightly different hyperfine coupling and electric field gradient under the disordered potential from CO$_3$ .",
"The nuclear quadrupole satellites were not observed in the measured frequency range $\\sim $ 3 MHz.",
"The spectrum exhibits a negative hyperfine shift $K$ on cooling [Fig.",
"4(a)].",
"$K$ well scales to $\\chi _{\\parallel }$ with a hyperfine coupling constant $A$ = $-0.21$ T/$\\mu _{\\rm B}$ [Fig.",
"4(b)].",
"The paramagnetic shift down to 2 K confirms the absence of spontaneous local fields.",
"A slight change in the line shape below 2 K comes from a misalignment of the external field by $\\sim $ 5$^\\circ $ on switching to a $^3$ He cryostat, which does not influence the linewidth defined by full-width at half-maximum.",
"As seen in Fig.",
"4(c), the linewidth is nearly independent of $T$ and biased by the inhomogeneous distribution of the electric field gradient.",
"The weak temperature dependence of 0.2% ($\\sim $ 0.2 T) below 30 K should be a paramagnetic effect proportional to the Knight shift.",
"If it were due to spontaneous magnetic moments, it is no more than 0.1$\\mu _{\\rm B}$ using the obtained $A$ .",
"The temperature dependence of $1/T_2$ is also displayed in Fig.",
"4(c), which is sensitive to spin fluctuations.",
"$1/T_2$ is nearly invariant against $T$ [Fig.",
"4 (c)] without the indication of long-range magnetic ordering.",
"The behavior is distinct from quantum antiferromagnts showing a divergent increase at low temperatures.",
"[28] Figure: (a) Specific heat CC of Ba326 measured at zero field.Inset: C/TC/T plotted against T 2 T^2, where the extrapolation to T=0T = 0 gives the TT-linear coefficient γ\\gamma .",
"(b) Temperature dependence of the magnetic entropy ΔS M \\Delta S_{\\rm M} obtained from the TT-integral of C/TC/T, where the lattice contribution to CC was subtracted by using CC of nonmagnetic Ca 4 _4PtO 6 _6.The dotted lines denote the entropy expected for spin S=1/2S = 1/2, RRln2 (blue), and for extra orbital degrees of freedom (green, see text for the detail).",
"(c) Seebeck coefficient S e S_e measured parallel to the chain in Ba326.A dotted curve shows the data of Na 0.75 _{0.75}CoO 2 _2 (Ref.",
"Terasaki ) for comparison.The specific heat $C$ displays no anomaly due to phase transitions down to 2 K, as displayed in Fig.",
"5(a).",
"The $T$ -linear coefficient $\\gamma $ is evaluated from the $C/T$ intercept plotted against $T^2$ [the inset of Fig.",
"5 (a)].",
"The finite $\\gamma $ indicates low-lying gapless excitations.",
"The anomalous $\\gamma $ has been also observed in the spin-chain system such as Ca326 (Ref.",
"Hardy) and Sr$_5$ Rh$_4$ O$_{12}$ (Ref.",
"Cao) with magnetic ordering.",
"In Ba326 with itinerant and localized spins without ordering, $\\gamma $ measures the charge and spinon density of states.",
"[32], [33] The obtained $\\gamma = 15$ mJ/(Co-mol K$^2$ ) is comparable to that of the thermoelectric cobaltate Na$_{0.75}$ CoO$_2$ , 16 mJ/(Co-mol K$^2$ ), with the enhanced density of states.",
"[34] Considering the large residual $\\chi $ without long-range ordering, local moments are expected to dominate low-energy excitations.",
"The Wilson ratio $R_W = (\\pi ^2/3)(\\chi /\\mu _{\\rm B}^2)/(\\gamma /k_{\\rm B}^2)$ is evaluated as $R_{W \\parallel } = 116$ and $R_{W \\perp } =10$ using $\\chi _{\\parallel }$ = 0.047 and $\\chi _{\\perp } = 0.0043$ emu/mol at 20 K. As known in some spin liquid candidates, $R_W$ is close to unity for organic systems and enhanced in presence of the spin-orbit coupling[32] such as the hyperkagome lattice Na$_4$ Ir$_3$ O$_8$ (Ref.",
"Okamoto) and the diamond lattice FeSc$_2$ S$_4$ (Ref.",
"Fritsch).",
"The enhanced $R_W$ highlights the significant role of spin-orbit coupling in the spinon excitations.",
"[37] From the specific heat data, the magnetic entropy $\\Delta S_{\\rm M}$ was obtained after subtracting the lattice contribution [Fig.",
"5(b)].",
"In a paramagnetic Mott insulator with $S$ = 1/2, $\\Delta S_{\\rm M}$ reaches $\\Delta S_{1/2}$ = $R$ ln($2S+1$ ) = $R$ ln2 = 5.76 ($R$ : the gas constant) with increasing $T$ .",
"We obtained $\\Delta S_{\\rm M}$ far exceeding $\\Delta S_{1/2}$ .",
"The extra contribution is in part attributed to orbital degrees of freedom.",
"Here the spin-orbital degeneracy is at most 6 for Co$^{4+}$ ($d^5$ , $S$ = 1/2) and 18 for Co$^{3+}$ ($d^6$ , $S$ = 1),[38] leading to $\\Delta S_{\\rm M}$ = $R$ (0.7ln6+0.3ln18) = 17.6, as shown in the dotted line in Fig.",
"5(b).",
"Additional valence fluctuations should be also contribute to the excess entropy.",
"Another manifestation of the high entropy is the Seebeck coefficient $S_{\\rm e}$ defined by thermopower per $T$ , as shown in Fig.",
"5(c).",
"The positive $S_{\\rm e}$ points to hole conductivity, as expected for the $3d^{5.3}$ occupation in the $t_{2g}$ orbitals.",
"Similar $T$ dependence was observed in a typical thermoelectric material Na$_{0.75}$ CoO$_2$ (Ref.",
"Terasaki).",
"Since $S_{\\rm e}$ relates to entropy per a conduction electron, the large $S_{\\rm e}$ is consistent with the specific heat.",
"Now we discuss the microscopic origin for the exotic magnetic and transport properties of Ba326.",
"The notable feature is disordering of local moments despite the strong Ising anisotropy.",
"Taking account of the local trigonal distortion along the chain, the $S \\sim 1/2$ local moment comes from the orbital dependent localization.",
"Without spin-orbital coupling, the $t_{2g}$ multiplet consists of $a_{1g} = (yz+zx+xy)/{\\surd 3}$ and $e_g^\\prime = (e^{\\pm i2\\pi /3}yz + e^{\\mp i2\\pi /3}zx + xy)/{\\surd 3}$ orbitals.",
"The $a_{1g}$ orbital elongated along the chain forms the direct $d$ -$d$ overlap and hence a partially-filled 1D conduction band, whereas the orphan $e_g^\\prime $ orbitals can carry local moments.",
"The double-exchange interaction between $e_g^\\prime $ spins via the itinerant $a_{1g}$ orbital causes ferromagnetic correlations along the chain.",
"The direct $d$ -$d$ exchange is expected to be much larger than the interchain superexchange.",
"Along this line, a promising ground state is the N${\\rm \\acute{e}}$ el order for ferromagnetically aligned giant spins on the bipartite honeycomb lattice.",
"[6], [7] To explain the unexpected disorder state, we consider three possible origins.",
"First one is the spin-orbit coupling, as manifested in the Ising anisotropy.",
"It gives rise to the entanglement of the itinerant $a_{1g}$ and localized $e_g^\\prime $ spins.",
"[39] Such a complex admixture may induce Kondo coupling and enhance quantum fluctuations.",
"[33] Indeed, the magnetic susceptibility levels off as the resistivity goes up, analogous to the Kondo system.",
"The second reason is the presence of itinerant electrons that dislike the N${\\rm \\acute{e}}$ el order state.",
"Although the system becomes weakly localized at low temperatures, the itinerancy of $a_{1g}$ spin is robust due to the strong ferromagnetic correlation as seen in the good conductivity.",
"The third one is geometrical frustration arising from the next neighbor interactions $J^\\prime _\\perp $ on the honeycomb lattice.",
"The critical value of $J^\\prime _\\perp $ is evaluated as only $J^\\prime _\\perp = 0.08J_\\perp $ for inducing a spin liquid phase in a Heisenberg system.",
"[12], [11] However, the present system has the marginally itinerant character with the Ising anisotropy, and no theoretical model has been reported so far.",
"Although the ground state of Ba326 is not fully understood yet, the quantum fluctuations likely arise from these complex interplays of spin, charge, and orbital.",
"To give further insight into the physics in Ba326, it will be important to control the ground state by chemical doping or pressure.",
"In conclusion, we have investigated the ground state properties of the thermoelectric cobaltate Ba$_3$ Co$_2$ O$_6$ (CO$_3$ )$_{0.7}$ with the honeycomb lattice.",
"The compound is found to be a rare example of the honeycomb-lattice Ising chain system with the effective spin $S = 1/2$ .",
"The magnetic susceptibility, NMR, and specific heat measurements show no indication of long-range magnetic ordering down to low temperatures.",
"The low-lying excitations with the large Wilson ratio suggest the spin-orbit coupled itinerant system.",
"The result will open further theoretical studies of quantum liquid with moderate electron correlations and spin-orbit coupling.",
"The authors thank Inoue S. for technical assistance, and valuable discussion with M. Tsuchiizu, Y. Motome, N. P. Ong, A. Sondi, and J. Moore.",
"This work was financially supported by the Grants-in-Aid for Scientific Research (No.25610093, 23225005, and 24340080) from JSPS, and the Grant-in-Aid for Scientific Research (No.",
"22014006) on Priority Areas \"Novel State of Matter Induced by Frustration\" from the MEXT."
]
] | 1403.0208 |
[
[
"Direct observation of bulk charge modulations in optimally-doped\n Bi$_{1.5}$Pb$_{0.6}$Sr$_{1.54}$CaCu$_{2}$O$_{8+\\delta}$"
],
[
"Abstract Bulk charge density modulations, recently observed in high critical-temperature ($T_\\mathrm{c}$) cuprate superconductors, coexist with the so-called pseudogap and compete with superconductivity.",
"However, its direct observation has been limited to a narrow doping region in the underdoped regime.",
"Using energy-resolved resonant x-ray scattering we have found evidence for such bulk charge modulations, or soft collective charge modes (soft CCMs), in optimally doped Bi$_{1.5}$Pb$_{0.6}$Sr$_{1.54}$CaCu$_{2}$O$_{8+\\delta}$ (Pb-Bi2212) around the summit of the superconducting dome with momentum transfer $q_{\\parallel}\\sim0.28$ reciprocal lattice units (r.l.u.)",
"along the Cu-O bond direction.",
"The signal is stronger at $T\\simeq T_\\mathrm{c}$ than at lower temperatures, thereby confirming a competition between soft CCMs and superconductivity.",
"These results demonstrate that soft CCMs are not constrained to the underdoped regime, suggesting that soft CCMs appear across a large part of the phase diagram of cuprates and are intimately entangled with high-$T_\\mathrm{c}$ superconductivity."
],
[
"Acknowledgment",
"This work was performed at the ID08 beam line of the ESRF (Grenoble, France).",
"This work is supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract No.",
"DE-AC02-76SF00515; and by the Italian Ministry of University and Research (MIUR) through the Grants PRIN20094W2LAY and PIK-POLARIXS."
]
] | 1403.0061 |
[
[
"Fast pick up technique for high quality heterostructures of bilayer\n graphene and hexagonal boron nitride"
],
[
"Abstract We present a fast method to fabricate high quality heterostructure devices by picking up crystals of arbitrary sizes.",
"Bilayer graphene is encapsulated with hexagonal boron nitride to demonstrate this approach, showing good electronic quality with mobilities ranging from 17 000 cm^2/V/s at room temperature to 49 000 cm^2/V/s at 4.2 K, and entering the quantum Hall regime below 0.5 T. This method provides a strong and useful tool for the fabrication of future high quality layered crystal devices."
],
[
"colorlinks=true, linkcolor=blue, citecolor=blue, filecolor=blue, urlcolor=blue, []Fast pick up technique for high quality heterostructures of bilayer graphene and hexagonal boron nitride P. J. Zomer pj.zomer@gmail.com Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands M. H. D. Guimarães Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands J. C. Brant Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands N. Tombros Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands B. J. van Wees Physics of Nanodevices, Zernike Institute for Advanced Materials, University of Groningen, Groningen, The Netherlands We present a fast method to fabricate high quality heterostructure devices by picking up crystals of arbitrary sizes.",
"Bilayer graphene is encapsulated with hexagonal boron nitride to demonstrate this approach, showing good electronic quality with mobilities ranging from 17 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ at room temperature to 49 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ at 4.2 K, and entering the quantum Hall regime below 0.5 T. This method provides a strong and useful tool for the fabrication of future high quality layered crystal devices.",
"72.80.Vp, 81.05.ue, 73.43.-f graphene, boron nitride, heterostructure, high mobility A critical step for high mobility graphene device fabrication and the rising field of van der Waals heterostructures[1] is marked by the development of polymer based dry transfer methods for two-dimensional (2D) crystals.",
"[2], [3], [4], [5] With these methods, high quality graphene devices on hexagonal boron nitride (h-BN) and more complicated stacks have become generally accessible, but the early methods face a major setback.",
"The method of stacking the crystals one by one typically leaves each transferred crystal contaminated by polymer.",
"To obtain a high quality device, thorough cleaning is required before proceeding with device fabrication or measurement.",
"This cleaning step typically involves several hours of annealing[3], [4], [5] or it may go as far as nanobrooming the entire graphene flake using contact mode atomic force microscopy (AFM).",
"[6], [7] Altogether this makes the fabrication of a multilayer heterostructure not only very time consuming, but also risky as each step may again introduce contaminants to the stack.",
"Figure: (Color online)a) Cross section (not to scale) of the glass slide used to pick up a graphene flake.The pick up process can be repeated several times to create a multilayer heterostructure.b) A stack in the making on PC imaged before deposition on a substrate.",
"BN1 was used to pick up the BLG (black short dashed line) and then BN2 (white long dashed line).",
"The scale bar equals 20 μ\\mu m.c) In the last step the stack is released onto graphite on the substrate at an elevated temperature (∼\\sim 150 ∘ ^{\\circ }C).d) AFM micrograph of the stack in b), deposited on a few layer graphene flake on SiO 2 _2.Two black dotted lines trace the BLG edges, the white dashed line marks the left edge of the few layer graphene flake.The scale bar equals 5 μ\\mu m.This issue has recently been overcome by L. Wang et al.",
"[8], introducing a method that allows for polymer free assembly of layered materials based on van der Waals force.",
"Instead of depositing a 2D crystal, e.g.",
"h-BN, directly on top of another crystal, e.g.",
"graphene, one can use the h-BN to pick up the graphene from the substrate.",
"This can be done because the van der Waals force between the atomically flat h-BN and graphene is stronger than between the graphene or the h-BN and the rough SiO${_2}$ substrate.",
"The power of this method lies in the fact that now the interface between the two crystals has not been contaminated by polymer, and one can directly pick up the next crystal.",
"This way the materials inside the stack not only remain much cleaner, but a stack can also be fabricated considerably faster.",
"One problem when using this method is the reduced capability to pick up graphene flakes larger than the used top h-BN crystal.",
"Therefore one has to etch through the stack before making one-dimensional (1D) contacts to the graphene.",
"While resulting in very high quality devices with electron mean free paths up to 21 ${\\mu }$ m and good electrical contact,[8] this limitation can be problematic for certain device types for which 1D contacts are not desirable, e.g.",
"spintronic devices that include tunnel barriers at the contact interface.",
"[9] In this letter we present a method which allows for fabrication of high quality graphene devices encapsulated in h-BN by successively picking up crystals.",
"The advantage of our approach is that stacks are made without size constraints on the graphene.",
"No cleaning steps are used in the fabrication process, which considerably increases the fabrication speed of a stack from 1 or 2 days down to half an hour.",
"The quality we achieve for h-BN encapsulated BLG devices becomes similar to that of current annealed suspended bilayer graphene devices,[10], [11], [12], [13] while keeping the benefits of having a substrate, for example when including top- and backgates.",
"[14] In our device, we observe quantum Hall levels to start develop below 0.5 T at 4.2 K and the degeneracy of the first Landau level is broken already around 3 T. With this newly developed method, high quality graphene devices are readily accessible without the need for a furnace or etching system.",
"The fabrication of a stack starts with the preparation of a glass slide that is used to pick up other crystals.",
"We first prepare a thin film of polycarbonate (PC, Sigma Aldrich, 6% dissolved in chloroform mixed at HQ Graphene).",
"Using a pipette, the PC is dripped on a microscope slide.",
"Because the PC is difficult to spincoat, a second glass slide is directly put on top of the PC covered slide, spreading out the PC, after which they are immediately separated by sliding the two slides over each other.",
"This results in a reasonably uniform PC film which is then left to harden in air for about 15 minutes.",
"On top of the dried PC film we exfoliate commercially available h-BN (HQ Graphene) by mechanical cleavage using adhesive tape.",
"By optical microscope we select an h-BN crystal suitable to serve as the top layer for the stack.",
"Then we take a new glass slide on which we put a $\\sim $ 4$\\times $ 4$\\times $ 1 mm piece of polydimethylsiloxane (PDMS).",
"The PC with the top layer h-BN is removed from its glass slide using adhesive tape and laid across the PDMS, with the h-BN flake facing up.",
"Now the newly made slide can be used to pick up other crystals.",
"A schematic cross section at this stage of the glass slide, PDMS, PC film and h-BN can be seen in the top half of Fig.REF a.",
"The crystals that are to be picked up in the next step are prepared by exfoliation of graphite (HOPG) or h-BN on separate Si/SiO$_2$ (500 nm) wafers.",
"Before exfoliation the Si/SiO$_2$ substrates are cleaned in a furnace at 400 $^{\\circ }$ C for 5 minutes in air.",
"By optical contrast we select BLG and thin h-BN (5-20 nm) flakes for our stack.",
"The glass slide is mounted in an optical mask aligner with the top layer h-BN flake facing down.",
"The SiO$_2$ , containing BLG, is fixed on the chuck of the mask aligner, this way we can accurately align the h-BN to the BLG (Fig.REF a) and start bringing the PC and the SiO$_2$ in contact.",
"While closely following the progress by an optical microscope, we continue to make contact until the two target crystals are almost in touch, which can be easily distinguished optically.",
"Now we heat the chuck to between 60 and 90 $^{\\circ }$ C. As the SiO$_2$ heats up, the contact area with the PC consequently gradually increases further.",
"When contact is made between the h-BN and BLG, we switch the heater off, which in turn causes the PC film to slowly retract from the substrate as it cools down.",
"When fully retracted, the PC film remains intact on the PDMS and the BLG is picked up.",
"The process just described can easily be repeated to pick up another crystal, in our case h-BN, to form a multilayer heterostructure.",
"An optical micrograph of a stack in the making on the PC layer is shown in Fig.REF b, where we started with BN1 and then subsequently picked up the BLG and BN2.",
"Note that due to the good adhesion between graphene and PC, BN1 does not have to entirely cover the BLG for the pick up to be successful.",
"To release this stack onto a substrate containing graphite that we aim to use as a back gate, we follow the same procedure as for a pick up until the retraction step (Fig.REF c).",
"While still in contact we heat the substrate up to $\\sim $ 150 $^{\\circ }$ C in order to melt the PC onto it.",
"Next, the hot chuck is carefully retracted, releasing the PC from the PDMS and leaving us with the desired stack on the SiO$_2$ substrate.",
"The PC is removed by rinsing in chloroform for 10 minutes.",
"Overall the pick up process has worked for almost 100% of the attempts, excluding manual misalignment.",
"Typically, only a small amount of bubbles is observed in the final stack.",
"[15] Most form just outside the encapsulated graphene area, at the BLG edges, as the AFM image in Fig.REF d shows.",
"Figure: (Color online)a) A finalized device using the stack imaged in Fig.b and d.The black dotted line traces the BLG, the scale bar equals 20 μ\\mu m.b) Encapsulated bilayer graphene square resistance R sq R_{sq} for different temperatures as a function of the backgate voltage.",
"The inset shows the respective conductance G sq G_{sq} as function of charge carrier density nn.The stack is fabricated into an electronic device using standard electron beam lithography and electron beam evaporation techniques.",
"A finished device is shown in Fig.REF a, the contacts are made using of Ti/Au (5/45 nm).",
"From the substrate up, this stack consists of a few layer graphene flake ($\\sim $ 1.5 nm) to be used as backgate, h-BN ($\\sim $ 5.5 nm), BLG and again h-BN ($\\sim $ 16 nm).",
"Since we are interested in the quality of the encapsulated region, we also do not require an additional cleaning step after processing the device.",
"For non-encapsulated graphene such a step would be needed to remove e-beam resist residues which otherwise degrade the electronic quality of the device.",
"The first characterization of the encapsulated BLG is done by measuring the square resistance $R_{sq}$ as function of backgate voltage $V_{bg}$ .",
"The results, measured at room temperature, 77 K and 4.2 K, are shown in Fig.REF b.",
"What can be noted directly, is that the Dirac peak is sharp and very close to zero backgate voltage.",
"This indicates that the device has little intrinsic doping ($\\sim 1.5\\times 10^{11}$ cm$^{-2}$ ) and low inhomogeneity, as expected for graphene on h-BN.",
"[16], [17] Furthermore, the device has proven to be robust.",
"After the measurements were taken at room temperature and 77 K the device was removed from the measurement setup, taken off its chip carrier and rewired, before loading it into a cryostat.",
"This may have caused the small shift in the Dirac point from 0.01 V at 77 K to -0.025 V at 4.2 K, but otherwise this did not notably degrade the device quality.",
"From the backgate dependence of the square resistance we can extract the mobility.",
"First the charge carrier density is calculated using $n=(V-V{_D})\\epsilon _{0} \\epsilon _{r}/e d$ .",
"Here $V$ is the applied backgate voltage, $V_{D}$ the charge neutrality voltage (Dirac point), $\\epsilon _{0}$ is the permittivity of free space, $\\epsilon _{r}$ is the relative permittivity of h-BN, $e$ is the electron charge and $d$ is the h-BN thickness.",
"When subject to a magnetic field perpendicular to the BLG plane, the device exhibits clear quantum Hall levels, shown in Fig.REF a.",
"This provides an alternative way to determine $n$ via the filling factor, expressed as $\\nu =nh/eB$ , where h is Planck's constant and $B$ is the magnetic field.",
"Combining both methods we can determine the ratio $\\epsilon _{r}/d$ for our device, which we find to be $0.52\\pm 0.01$ nm$^{-1}$ (using the data from Fig.REF a).",
"AFM yields an h-BN thickness $d=5.5\\pm 0.2$ nm, hence $\\epsilon _{r}=2.9\\pm 0.2$ .",
"Using $\\mu =1/neR_{sq}$ we find a room temperature mobility of 17 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ at $n\\approx 3.8\\times 10^{11}$ cm$^{-2}$ .",
"For 77 K and 4.2 K we find respectively 30 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ at $n\\approx 1.9\\times 10^{11}$ cm$^{-2}$ and 49 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ at $n\\approx 1.4\\times 10^{10}$ cm$^{-2}$ .",
"The mobility is for all cases determined at the inflection point, where d$R_{sq}$ /d$n$ has an extreme.",
"Figure: (Color online)a) Resistance as function of backgate voltage and magnetic field perpendicular to the BLG plane at 4.2 K. At negative and positive gate voltages additional resistance peaks can be distinguished.b) Traces taken from the data in a) show quantum Hall levels already forming below 0.5 T.c) Close up of the data in a) around the charge neutrality point.The magnetic field for subsequent traces increases in steps of 0.1 T and they are offset with respect to each other by 500 Ω\\Omega for clarity.The measurements reveal the development of the ν=2\\nu =2 plateau already at 3 T.The quantum Hall data in Fig.REF a shows, besides a Landau fan around 0 V, faint additional fans originating at $\\pm $ 1.8 V. These side peaks are likely a consequence of the Moiré superlattice that exists for single and bilayer graphene on h-BN [18], [19], [20], [21], having both the same crystallographic lattice with a mismatch of 1.8%.",
"[22] The wavelength $\\lambda $ of the Moiré pattern can be estimated from $n\\approx 5.2\\times 10^{12}$ cm$^{-2}$ at which the side peaks occur.",
"At this point the Moiré minibands have become fully occupied, which requires 4 electrons per unit cell for valley and spin degeneracy.",
"So using $n=4n_{0}$ with $1/n_{0}=\\sqrt{3}\\lambda ^{2}/2$ as the superlattice unit cell area, we obtain $\\lambda \\sim 9.4$ nm.",
"This corresponds to an angle of $\\sim $ 1.1$^{\\circ }$ between the graphene and boron nitride lattices.",
"[18] The magnetic field data can also be used as an indication of the device quality.",
"Using $\\mu B \\approx 1$ as a condition for the device to enter the quantum Hall regime, a mobility can be determined.",
"[23], [24] We observe the development of quantum Hall at magnetic fields below 0.5 T in Fig.REF b, indicating that the mobility is at least 20 000 cm$^{2}$ V$^{-1}$ s$^{-1}$ .",
"This observation is comparable to what has been achieved for suspended BLG.",
"[10], [11], [12], [13] The filling factors are indicated at $\\nu =24$ and $\\nu =12$ for 0.5 T and 1 T respectively in Fig.REF b.",
"For fields around 1.5 T oscillations are still present at filling factors over $\\nu =192$ .",
"Furthermore it can be noted in Fig.REF c that the degeneracy of the lowest Landau level starts breaking below 3 T, forming plateaus with filling factor $\\nu =2$ .",
"At higher magnetic fields the degeneracy is lifted further, forming first a plateau at $\\nu =3$ around 5 T and later one at $\\nu =1$ just below 8 T. This can also be taken as a demonstration of a good device quality.",
"[12] In conclusion we demonstrated an easy device fabrication method using BLG and h-BN that can be used to build complicated stacks of 2D crystals, opening up many opportunities for material engineering.",
"The main advantage is that the crystal interfaces remain very clean without the need for any cleaning steps, which additionally makes building stacks considerably faster.",
"Since our approach allows for picking up crystals of any size, no etching is required to access any layer in the stack.",
"We showed measurements on a BLG encapsulated in h-BN, fabricated using this method.",
"The quality of this device is comparable to suspended BLG devices, as can be seen from regular charge transport and quantum Hall measurements.",
"With the current need for high quality graphene devices and the trend towards other 2D materials, this method is a strong and important tool for fabrication.",
"We thank L. Wang, I. Meric, C. R. Dean and P. Kim for helpful discussion and we acknowledge B. Wolfs, J. G. Holstein, H. M. de Roosz and H. Adema for their technical assistance.",
"The research leading to these results has received funding from the European Union Seventh Framework Programme under grant agreement n$^{\\circ }$ 604391 Graphene Flagship, the Dutch Foundation for Fundamental Research on Matter (FOM), NWO, NanoNed, the Zernike Institute for Advanced Materials and CNPq, Brazil."
]
] | 1403.0399 |
[
[
"Observational constraints on models of the Universe with time variable\n Gravitational and Cosmological constants along MOG"
],
[
"Abstract The subject of this paper is to investigate the weak regime covariant scalar-tensor-vector gravity (STVG) theory, known as the MOdified gravity (MOG) theory of gravity.",
"First, we show that the MOG in the absence of scalar fields is converted into $\\Lambda(t),G(t)$ models.",
"Time evolution of the cosmological parameters for a family of viable models have been investigated.",
"Numerical results with the cosmological data have been adjusted.",
"We've introduced a model for dark energy (DE) density and cosmological constant which involves first order derivatives of Hubble parameter.",
"To extend this model, correction terms including the gravitational constant are added.",
"In our scenario, the cosmological constant is a function of time.",
"To complete the model,interaction terms between dark energy and dark matter (DM) manually entered in phenomenological form.",
"Instead of using the dust model for DM, we have proposed DM equivalent to a barotropic fluid.",
"Time evolution of DM is a function of other cosmological parameters.",
"Using sophisticated algorithms, the behavior of various quantities including the densities, Hubble parameter, etc.",
"have been investigated graphically.",
"The statefinder parameters have been used for the classification of DE models.",
"Consistency of the numerical results with experimental data of $SneIa+BAO+CMB$ are studied by numerical analysis with high accuracy."
],
[
" A new synthesis of time variable $G,\\Lambda $ models as MOG models",
"All Cosmological data from different sources testify to the fact that our world is made of a substance of negative pressure $73\\%$ ( dark energy (DE) ), missing mass $23\\%$ ( dark matter (DM)) and only $4\\%$ conductive material (baryon matter) [1].",
"DM and DE can have interaction and the interaction of these is not known in the physics.",
"It is not an electromagnetic field and metallic material interaction.",
"Mathematical function is determined phenomenologically because types of interactions is unknown with an overall classification interaction function can be written as $Q=Q(H,\\dot{H},\\rho _m,\\rho _{DE},\\rho _{DM},...)$ .",
"Several models have been proposed to explain the universe's accelerated expansion [4]-[10].",
"The models can be divided into two general groups: the first group of models that are needed to correct the Einstein theory of gravity with a new geometric terms is known as geometric models.",
"The first of these models is $f(R)$ which is obtained by replacing the $R$ Ricci curvature with arbitrary $f(R)$ function[11].",
"The second group of models that are expansion is attributed to exotic fluids with negative pressure.",
"It is believed that exotic fluid is a mimic dark energy equation of state in the present era (for a modern review see [2],[3] ).",
"Both of these models have different applications and important results of these models are derived as alternative cosmological models [12], [13], [14], [15], [16].",
"Several properties of DE have been studied in numerous papers[17], [18], [19].",
"DE can be decay [20] or reconstruct from different theoretical models [21].",
"There is no simple and unique model that can have to describe this exotic energy.",
"Models in which the Scalar-tensor fields used are able to solve such complex issues by simple mathematics to the extend possible [22], [23].",
"So there are very attractive models to study.",
"A scalar-Tensor model is proposed among all the different cosmological models.",
"The model is able to explain the DM and dynamic clusters of galaxies with an additional vector field and relying only baryonic matter [24].",
"This model is known as STVG or MOG.",
"MOG can be seen as a covariant theory with vector-tensor-scalar fields for gravity with the following action: $&&S=-\\frac{1}{16\\pi }\\int \\frac{1}{G}(R+2\\Lambda )\\sqrt{-g}d^4x+S_{\\phi }+S_{M}\\\\&& \\nonumber -\\int \\frac{1}{G}\\Big [\\frac{1}{2}g^{\\alpha \\beta }\\Big (\\nabla _{\\alpha }\\log G\\nabla _{\\beta }\\log G+\\nabla _{\\alpha }\\log \\mu \\nabla _{\\beta }\\log \\mu \\Big )+U_{G}(G)+W_{\\mu }(\\mu )\\Big ]\\sqrt{-g}d^4x.$ The first term of the action is Einstein-Hilbert Lagrangian.",
"The second term is the conventional scalar field and the last term contains a $G$ kinetic energy field that plays the role of the gravitational constant (However, the fields can be considered similar to a time dependent gravitational constant by slowly time varying fields) [25].",
"This action classes are written in covariant forms and are used to investigate the astrophysical phenomena such as rotation curves of galaxies, mass distribution of cosmic clusters or gravitational lenses.",
"The model might be a suitable alternative to $\\Lambda $ CDM model considered [26].",
"In order to understand the role of scalar and vector fields we write the equations of motion for FLRW metric : $ds^2=dt^2-a(t)^2[(1-kr^2)^{-1}dr^2+r^2d\\Omega ^2],\\ \\ d\\Omega ^2=d\\theta ^2+\\sin ^2{\\theta }d\\phi ^2$ Form of the equations can be rewritten as a generalized Friedmann equations as follow [27]: $&&H^2+\\frac{k}{a^2}=\\frac{8\\pi G\\rho }{3}-\\frac{4\\pi }{3}\\left(\\frac{\\dot{G}^2}{G^2}+\\frac{\\dot{\\mu }^2}{\\mu ^2}-\\dot{\\omega }^2-G\\omega \\mu ^2\\phi _0^2\\right)\\nonumber \\\\&&\\qquad {}+\\frac{8\\pi }{3}\\left(\\omega GV_\\phi +\\frac{V_G}{G^2}+\\frac{V_\\mu }{\\mu ^2}+V_\\omega \\right)+\\frac{\\Lambda }{3}+H\\frac{\\dot{G}}{G},\\\\&&\\frac{\\ddot{a}}{a}=-\\frac{4\\pi G}{3}(\\rho +3p)+\\frac{8\\pi }{3}\\left(\\frac{\\dot{G}^2}{G^2}+\\frac{\\dot{\\mu }^2}{\\mu ^2}-\\dot{\\omega }^2-G\\omega \\mu ^2\\phi _0^2\\right)\\nonumber \\\\&&\\qquad {}+\\frac{8\\pi }{3}\\left(\\omega GV_\\phi +\\frac{V_G}{G^2}+\\frac{V_\\mu }{\\mu ^2}+V_\\omega \\right)+\\frac{\\Lambda }{3}+H\\frac{\\dot{G}}{2G}+\\frac{\\ddot{G}}{2G}-\\frac{\\dot{G}^2}{G^2},\\nonumber \\\\&&\\ddot{G}+3H\\dot{G}-\\frac{3}{2}\\frac{\\dot{G}^2}{G}+\\frac{G}{2}\\left(\\frac{\\dot{\\mu }^2}{\\mu ^2}-\\dot{\\omega }^2\\right)+\\frac{3}{G}V_G-V_G^{\\prime }\\nonumber \\\\&&\\qquad {}+G\\left[\\frac{V_\\mu }{\\mu ^2}+V_\\omega \\right]+\\frac{G}{8\\pi }\\Lambda -\\frac{3G}{8\\pi }\\left(\\frac{\\ddot{a}}{a}+H^2\\right)=0,\\\\&&\\ddot{\\mu }+3H\\dot{\\mu }-\\frac{\\dot{\\mu }^2}{\\mu }-\\frac{\\dot{G}}{G}\\dot{\\mu }+G\\omega \\mu ^3\\phi _0^2+\\frac{2}{\\mu }V_\\mu -V^{\\prime }_\\mu =0,\\\\&&\\ddot{\\omega }+3H\\dot{\\omega }-\\frac{\\dot{G}}{G}\\dot{\\omega }-\\frac{1}{2}G\\mu ^2\\phi _0^2+GV_\\phi +V^{\\prime }_\\omega =0.$ Scalar and vector fields interaction terms of the aforementioned classes are self interaction and they are shown by an arbitrary mathematical functions: $V_\\phi (\\phi )$ , $V_G(G)$ , $V_\\omega (\\omega )$ , and $V_\\mu (\\mu )$ .",
"The resulting equations of motion are highly nonlinear and there is no possibility to find analytical solutions.",
"The only possible way to evaluate answer is numerical method.",
"At the same time, we must also determine the shape of the interaction $V_{i}$ .",
"Mathematical differences may be a good solution for finding certain family of potentials.",
"If we consider the $G$ scalar field with a time variable gravitational field (G(t)) and ignore the contributions of the other fields in favor of the G(t), and also due to the cosmological data $\\frac{\\dot{G}}{G}\\ll 1$ , time evolution of G(t) will be the major contribution.",
"In fact, data from the large cosmological confirm our conjecture about just keeping the $G(t)$ , and kinetic part of $G(t)$ can be neglected because: $&&g^{\\alpha \\beta }\\nabla _{\\alpha }\\log G\\nabla _{\\beta }\\log G \\simeq (\\frac{\\dot{G}}{G})^2 \\ll 1.$ Regardless, second-order derivatives of additional fields which introduced additional degrees of freedom and in the absence of additional fields on MOG, with the approximation that the time evolution of the fields is very slowly varying, MOG and Einstein-Hilbert action can be considered as the same.",
"The difference is that now $G(t)$ is a scalar time variable field.",
"Equations of motion are written in the following general form, if we consider small variation of $G(t)$ and $G(t),\\Lambda $ are functions of time [28]: $S\\simeq -\\frac{1}{16\\pi }\\int \\frac{1}{G}(R+2\\Lambda )\\sqrt{-g}d^4x+S_{M}.$ (see for instance [29]) $R_{\\mu \\nu }-\\frac{1}{2}Rg_{\\mu \\nu }\\approx -8 \\pi G(t) \\left[ T_{\\mu \\nu } -\\frac{\\Lambda (t)}{8 \\pi G(t)}g_{\\mu \\nu } \\right],$ Energy-momentum function of matter fields (ordinary or exotic) is proposed as follows: $T_{\\mu \\nu }=\\mathcal {L}_{M}g_{\\mu \\nu }-2\\frac{\\delta \\mathcal {L}_{M}}{\\delta g^{\\mu \\nu }}.$ Cosmological models, which were introduced by the mentioned equations of motion have been investigated several times by different authors [29], [30], [31], [32].",
"But we approach this problem with a more general view.",
"As we have shown, MOG is the limit of weak fields able to induce and introduces a gravitational field $G(t)$ .",
"So, our paper can be considered as a cosmological analysis of MOG in the weak field regime.",
"We are particularly interested to see how cosmological data $SneIa+BAO+CIB$ will constrain our model parameters.",
"Our plan in this paper is: In section II: introducing the cosmological constant and dark model consist of $\\lbrace H,\\dot{H},..\\rbrace $ .",
"In section III: dynamic extraction of the model and additional equation governing $G(t)$ and inference different densities.",
"In section IV, numerical analysis of the equations.",
"In section V, statefinder parameters $(r,s)$ analysis.",
"In section VI, observational constraints.",
"The final section is devoted to the results of references."
],
[
"Toy models",
"A DE model of our interest is described via energy density $\\rho _{D}$ [33]: $\\rho _{D}=\\alpha \\frac{\\ddot{H}}{H}+\\beta \\dot{H}+\\gamma H^{2},$ where $\\beta $ , $\\gamma $ are positive constants, while for $\\alpha $ in light of the time variable scenario, we suppose that $\\alpha (t)=\\alpha _{0}+\\alpha _{1} G(t)+\\alpha _{2} t \\frac{\\dot{G}(t)}{G(t)},$ where $\\alpha _{0}$ , $\\alpha _{1}$ and $\\alpha _{2}$ are positive constants and $G(t)$ is a varying gravitational constant.",
"Its a generalization of Ricci dark energy scenario [34] to higher derivatives terms of Hubble parameter.",
"An interaction term $Q$ between DE and a barotropic fluid $P_{b}=\\omega _{b}\\rho _{b}$ is taken to be $Q=3Hb(\\rho _{b}+\\rho _{D})$ We propose three phenomenological models for DE as the following: The first model is the simplest one, in which we assume that time variable cosmological constant has the same order of energy as the density of DE.",
"$\\Lambda (t)=\\rho _{D},$ In this model, $\\rho _{D}$ is determined using continuity equation with a dissipative interaction term Q. Secondly, generalization of cosmological constant is proposed as a modified Ricci DE model to time variable scenario has an oscillatory form in terms of H. $\\Lambda (t)=\\rho _{b}\\sin ^{3}{(tH)}+\\rho _{D}\\cos {(tH)},$ Note that if we think on trigonometric term as oscillatory term, the amplitudes of the oscillations are assumed to be proportional to the barotropic and DE components.",
"Meanwhile these coefficients satisfy continuity equations.",
"The last toy model is inspired from the small variation of G(t) and a logarithmic term of H. Here, coefficients are written in the forms of barotropic and DE densities .",
"$\\Lambda (t)=\\rho _{b}\\ln {(tH)}+\\rho _{D}\\sin {\\left(t\\frac{\\dot{G}(t)}{G(t)} \\right)}.$ In this model, a time dependent and G variable assumption is imposed.",
"Following the suggested models we will study time evolution and cosmological predictions of our cosmological model.",
"Furthermore, we will compare the numerical results with a package of observational data."
],
[
"Dynamic of models",
"By using the following FRW metric for a flat Universe, $ds^2=-dt^2+a(t)^2\\left(dr^{2}+r^{2}d\\Omega ^{2}\\right),$ field equations (REF ) can be reduced to the following Friedmann equations, $H^{2}=\\frac{\\dot{a}^{2}}{a^{2}}=\\frac{8\\pi G(t)\\rho }{3}+\\frac{\\Lambda (t)}{3},$ and, $\\frac{\\ddot{a}}{a}=-\\frac{4\\pi G(t)}{3}(\\rho +3P)+\\frac{\\Lambda (t)}{3},$ where $d\\Omega ^{2}=d\\theta ^{2}+\\sin ^{2}\\theta d\\phi ^{2}$ , and $a(t)$ represents the scale factor.",
"Energy conservation law $T^{;j}_{ij}=0$ reads as, $\\dot{\\rho }+3H(\\rho +P)=0.$ Combination of (REF ), (REF ) and (REF ) gives the relationship between $\\dot{G}(t)$ and $\\dot{\\Lambda }(t)$ $\\dot{G}=-\\frac{\\dot{\\Lambda }}{8\\pi \\rho }.$ To introduce an interaction between DE and DM (REF ) we should mathematically split it into two following equations $\\dot{\\rho }_{DM}+3H(\\rho _{DM}+P_{DM})=Q,$ and $\\dot{\\rho }_{DE}+3H(\\rho _{DE}+P_{DE})=-Q.$ For the barotropic fluid with $P_{b}=\\omega _{b}\\rho _{b}$ (REF ) will take following form $\\dot{\\rho }_{b}+3H(1+\\omega _{b}-b)\\rho _{b}=3Hb\\rho _{D}.$ Pressure of the DE can be recovered from (REF ) $P_{D}=-\\rho _{D}-\\frac{\\dot{\\rho }_{D}}{3H}-b\\frac{3H^{2}-\\Lambda (t)}{8 \\pi G(t)}.$ Therefore with a fixed form of $\\Lambda (t)$ we will be able to observe behavior of $P_{D}$ .",
"Cosmological parameters of our interest are EoS parameters of DE $\\omega _{D}=P_{D}/\\rho _{D}$ , EoS parameter of composed fluid $\\omega _{tot}=\\frac{P_{b}+P_{D} }{\\rho _{b}+\\rho _{D}},$ deceleration parameter $q$ , which can be written as $q=\\frac{1}{2}(1+3\\frac{P}{\\rho } ),$ where $P=P_{b}+P_{D}$ and $\\rho =\\rho _{b}+\\rho _{D}$ .",
"We have a full system of equations of motion and interaction terms.",
"Now we are ready to investigate cosmological predictions of our model."
],
[
"Numerical analysis of the Cosmological parameters",
"In next sections we fully analyze time evolution of three models of DE.",
"Using numerical integration, we will show that how cosmological parameters $H,G(t),q,w_{\\text{tot}}$ , and time decay rate $\\frac{d\\log G}{dt}$ and densities $\\rho _D,..$ change.",
"We fit parameters like $H_0,$ etc from observational data."
],
[
"Model 1: $\\Lambda (t)=\\rho _{D}$",
"In this section we will consider $\\Lambda (t)$ to be of the form $\\Lambda (t)=\\rho _{D}.$ Therefore for the pressure of DE we will have $P_{D}=\\left( \\frac{b}{8 \\pi G(t)} -1 \\right)\\rho _{D}-\\frac{\\dot{\\rho }_{D}}{3H}-\\frac{3b}{8 \\pi G(t)}H^{2}.$ The dynamics of $G(t)$ we will have $\\frac{\\dot{G}(t)}{G(t)}+\\frac{\\dot{\\rho }_{D}}{3H^{2}-\\rho _{D}}=0.$ Performing a numerical analysis for the general case we recover the graphical behavior of different cosmological parameters.",
"Graphical behavior of Gravitational constant $G(t)$ against time $t$ presented in Fig.",
"(REF ).",
"We see that $G(t)$ is an increasing function.",
"Different plots represent behavior of $G(t)$ as a function of the parameters of the model.",
"For this model with the specific behavior of $G(t)$ for Hubble parameter $H$ gives decreasing behavior over time.",
"It is confirmed by LCDM scenario.",
"From the analysis of the graphical behavior of $\\omega _{tot}$ we made the following conclusion that with $\\alpha _{0}=1$ , $\\gamma =0.5$ , $\\beta =3.5$ , $\\omega _{b}=0.3$ , $b=0.01$ (interaction parameter) and with increasing $\\alpha _{1}$ and $\\alpha _{2}$ we increase the value of $\\omega _{tot}$ for later stages of evolution, while for the early stages, in history, it is a decreasing function.",
"For instance, with $\\alpha _{1}=0.5$ and $\\alpha _{2}=0.5$ (blue line) $\\omega _{tot}$ is a constant and $\\omega _{tot} \\approx -0.9$ (Top left plot in Fig.",
"(REF )).",
"Top right plot of Fig.",
"(REF ) presents graphical behavior of $\\omega _{tot}$ against time as a function of the parameter $b$ characterizing interaction between DE and DM.",
"We see that for the later stages of the evolution the interaction $Q=3hb(\\rho _{b}+\\rho _{D})$ does not play any role.",
"An existence of the interaction can be observed only for relatively early stages of evolution and when $b$ is too much higher than the real values of it estimated from observations.",
"The left-bottom plot shows the decreasing behavior of $\\omega _{tot}$ at early stages of evolution which, while for later stages, becomes a constant.",
"This behavior is observed for $\\alpha _{0}=\\alpha _{1}=\\alpha _{2}=1$ , $\\omega _{b}=0.1$ , $b=0.01$ and for increasing $\\gamma $ and $\\beta $ .",
"With the increase in $\\gamma $ and $\\beta $ , we increase the value of $\\omega _{tot}$ .",
"The right-bottom plot represents behavior as a function of $\\omega _{b}$ .",
"In Fig.REF , the graphical behavior of the deceleration parameter $q$ is observed which is a negative quantity throughout the evolution of the Universe i.e.",
"we have an ever accelerated Universe.",
"Right panel (top and bottom) shows that the behavior of $q$ does not strongly depend upon the interaction parameter $b$ and EoS parameter $\\omega _{b}$ .",
"We also see that $q$ starts its evolution from $-1$ and for a very short period of the history it becomes smaller than $-1$ , but after this $q>-1$ for ever, giving a hope that observational facts can be modeled (for later stages!).",
"Right panel (top and bottom) represents the behavior of $q$ for $\\alpha _{1}=\\alpha _{2}$ and {$\\gamma $ , $\\beta $ } (top and bottom) respectively.",
"With the increase in the values of the parameters, the value of $q$ increases.",
"Some information about $\\omega _{D}$ , $\\Lambda (t)$ and $\\dot{G}(t)/G(t)$ can be found in Appendix.",
"Figure: Behavior of Gravitational constant G(t)G(t) against tt for Model 1.Figure: Behavior of EoS parameter ω tot \\omega _{tot} against tt for Model 1.Figure: Behavior of deceleration parameter qq against tt for Model 1.Figure: Behavior of Hubble parameter H(t)H(t) against tt for Model 1.Figure: Behavior of α\\alpha against tt for Model 1.Figure: Behavior of ω D \\omega _{D} against tt for Model 1.Figure: Behavior of G ˙(t)/G(t)\\dot{G}(t)/G(t) against tt for Model 1."
],
[
"Model 2: $\\Lambda (t)=\\rho _{b}\\sin {(tH)}^{3}+\\rho _{D}\\cos {(tH)}$",
"For the second model we will consider the following phenomenological form of the $\\Lambda (t)$ $\\Lambda (t)=\\rho _{b}\\sin {(tH)}^{3}+\\rho _{D}\\cos {(tH)}.$ Taking into account (REF ) we can write $\\Lambda (t)$ in a different form $\\Lambda (t)=\\left[ 1+\\frac{\\sin (tH)^{3}}{8 \\pi G(t)} \\right]^{-1} \\left( \\frac{3H^{2}}{8 \\pi G(t)}\\sin {(tH)}^{3}-\\rho _{D}(\\sin {(tH)}^{3}-\\cos {(tH)})\\right).$ $\\frac{\\dot{G}(t)}{G(t)}+\\frac{\\dot{\\Lambda }(t)}{3H^{2}-\\Lambda (t)}=0,$ with (REF ) will give us the behavior of $G(t)$ Fig(REF ).",
"We see that $G(t)$ is an increasing-decreasing-increasing function (Top panel and right-bottom plot).",
"The left-bottom plot gives us an information about the behavior of $G(t)$ as a function of $\\gamma $ and $\\beta $ with $\\alpha _{0}=1$ , $\\alpha _{1}=\\alpha _{2}=1.5$ and $\\omega _{b}=0.3$ , $b=0.01$ .",
"We see that with increasing $\\gamma $ and $\\beta $ we are able to change the behavior of $G(t)$ .",
"For instance, with $\\gamma =0.5$ and $\\beta =3.5$ which is a blue line, still preserves the increasing-decreasing-increasing behavior.",
"While for higher values of the parameters, we change the behavior of $G(t)$ compared to the other cases within this model and we have increasing-decreasing behavior.",
"Graphical behavior of $\\omega _{tot}$ can be found in Fig.REF .",
"The behavior of the deceleration parameter $q$ for this model gives us almost the same as for Model 1, where $\\Lambda (t)=\\rho _{D}$ .",
"We also see that with increasing $\\gamma $ and $\\beta $ we increase the value of $q$ (left-bottom plot).",
"The presence of the interaction $Q$ and the barotropic fluid for which EoS parameter $\\omega _{b}<1$ does not leave a serious impact on the behavior of $q$ .",
"This model with this behavior of $q>-1$ can be comparable with the observational facts.",
"Figure: Behavior of Gravitational constant G(t)G(t) against tt Model 2.Figure: Behavior of EoS parameter ω tot \\omega _{tot} against tt for Model 2.Figure: Behavior of deceleration parameter qq against tt for Model 2.Figure: Behavior of Hubble parameter H(t)H(t) versus tt for Model 2.Figure: Behavior of α\\alpha versus tt for Model 2.Figure: Behavior of ω D \\omega _{D} against tt for Model 2.Figure: Behavior of G ˙(t)/G(t)\\dot{G}(t)/G(t) against tt for Model 2."
],
[
"Model 3: $\\Lambda (t)=\\rho _{b}\\ln {(tH)}+\\rho _{D}\\sin {\\left(t\\frac{\\dot{G}(t)}{G(t)}\\right) }$",
"For this model we will consider the following phenomenological form of the $\\Lambda (t)$ $\\Lambda (t)=\\rho _{b}\\ln {(tH)}+\\rho _{D}\\sin {\\left(t\\frac{\\dot{G}(t)}{G(t)}\\right) }.$ Taking into account (REF ) we can write $\\Lambda (t)$ in a different form $\\Lambda (t)=\\left[ 1+\\frac{\\ln (tH)}{8 \\pi G(t)} \\right]^{-1} \\left( \\frac{3H^{2}}{8 \\pi G(t)}\\ln {(tH)}-\\rho _{D}(\\ln {(tH)}-\\sin {\\left(t\\frac{\\dot{G}(t)}{G(t)}\\right) })\\right).$ $\\frac{\\dot{G}(t)}{G(t)}+\\frac{\\dot{\\Lambda }(t)}{3H^{2}-\\Lambda (t)}=0.$ Equation (REF ) with (REF ) will give us the behavior of $G(t)$ .",
"This model also includes several interesting facts about the behavior of the cosmological parameters.",
"After recovering the $G(t)$ we observe that $G(t)$ is an increasing function, and its graphical behavior for the different cases are given in Fig.",
"(REF ).",
"For instance with increasing $\\beta $ and $\\gamma $ with $\\alpha _{0}=\\alpha _{2}=1$ , $\\alpha _{1}=1.5$ , $\\omega _{b}=0.3$ and $b=0.01$ we have the following picture: $\\gamma =0.1$ and $\\beta =2.5$ (a blue line at left-bottom plot) we have a decreasing behavior for $G(t)$ , while for the higher values for $\\gamma $ and $\\beta $ we have increasing behavior for later stages of evolution.",
"With increasing $\\omega _{b}$ we decrease the value of $G(t)$ (right-bottom).",
"We also observe that there is a period in history of the evolution where $G(t)$ can be a constant.",
"With $\\alpha _{0}=\\alpha _{2}=1$ , $\\alpha _{1}=1.5$ , $\\gamma =0.5$ , $\\beta =3.5$ and $\\omega _{b}=0.3$ we see that for non interacting case, when $b=0$ (a blue line at right-top plot) at later stages of evolution $G(t)=const \\approx 1.36$ , while when we include the interaction and increase the value of $b$ , increase in the value of $G(t)$ is observed.",
"Behavior of $G(t)$ from $\\alpha _{0}$ , $\\alpha _{1}$ and $\\alpha _{2}$ can be found at the left-top plot of Fig.",
"(REF ).",
"Other cosmological parameter that we have investigated for this model is a $\\omega _{tot}$ describing interacting DE and DM two component fluid model.",
"From Fig.",
"(REF ) we can make conclusion about the behavior of the parameter.",
"We observe that as a function of $\\alpha _{0}$ , $\\alpha _{1}$ and $\\alpha _{2}$ , while the other parameters are being fixed, we have a decreasing function for the initial stages of evolution, while for the later stages we have a constant value for $\\omega _{tot}$ .",
"With increasing $\\alpha _{1}$ and $\\alpha _{2}$ we will increase $\\omega _{tot}$ and we have a possibility to obtain decreasing-increasing-constant behavior (left-top plot).",
"On the right-top plot we see the role of the interaction $Q$ .",
"Starting with the non interacting case $b=0$ and increasing $b$ we observe the increasing value of $\\omega _{tot}$ .",
"Bottom panel of Fig.REF represents graphical behavior of $\\omega _{tot}$ from $ \\lbrace \\gamma , \\beta \\rbrace $ and $\\omega _{b}$ .",
"The last parameter discussed in this section will be the deceleration parameter $q$ recovered for this specific $\\Lambda (t)$ .",
"Investigating the behavior we conclude that for this model, $\\gamma > 0.1$ and $\\beta >2.5$ should be taken in order to get $q>-1$ ( Fig.",
"(REF ) left-bottom plot).",
"It starts its evolution from $-1$ and then it is strictly $q>-1$ for later stages of evolution.",
"Interaction as well as $\\omega _{b}$ has a small impact on the behavior of $q$ .",
"Left-top plot of Fig.REF represents the behavior of $q$ as a function of $\\alpha _{0}$ , $\\alpha _{1}$ and $\\alpha _{2}$ .",
"As for the other models, additional information about other cosmological parameters of this model can be found in Appendix.",
"Figure: Behavior of Gravitational constant G(t)G(t) against tt Model 3.Figure: Behavior of EoS parameter ω tot \\omega _{tot} against tt for Model 3.Figure: Behavior of deceleration parameter qq against tt for Model 3.Figure: Behavior of Hubble parameter H(t)H(t) against tt for Model 3.Figure: Behavior of α\\alpha against tt for Model 3.Figure: Behavior of ω D \\omega _{D} against tt for Model 3.Figure: Behavior of G ˙(t)/G(t)\\dot{G}(t)/G(t) against tt for Model 3."
],
[
"State finder diagnostic",
"In the framework of GR, Dark energy can explain the present cosmic acceleration.",
"Except cosmological constant many other candidates of dark energy(quintom, quintessence, brane, modified gravity etc.)",
"are proposed.",
"Dark energy is model dependent and to differentiate different models of dark energy, a sensitive diagnostic tool is needed.",
"Since $\\dot{a}>0$ , hence $H>0$ means the expansion of the universe.",
"Also, $\\ddot{a}>0$ implies $q<0$ .",
"Since, the various dark energy models give $H>0$ and $q<0$ , they cannot provide enough evidence to differentiate the more accurate cosmological observational data and the more general models of dark energy.",
"For this aim we need higher order of time derivative of scale factor and geometrical tool.",
"Sahni et.al.",
"[36] proposed geometrical statefinder diagnostic tool, based on dimensionless parameters $(r, s)$ which are function of scale factor and its time derivative.",
"These parameters are defined as $r=\\frac{1}{H^{3}}\\frac{\\dddot{a}}{a} ~~~~~~~~~~~~s=\\frac{r-1}{3(q-\\frac{1}{2})}.$ For $8\\pi G =1$ and $\\Lambda =0$ we can obtain another form of parameters $r$ and $s$ : $r=1+\\frac{9(\\rho +P)}{2\\rho }\\frac{\\dot{P}}{\\dot{\\rho }}, ~~~ s=\\frac{(\\rho +P)}{P}\\frac{\\dot{P}}{\\dot{\\rho }}.$ For the model 3 of our consideration, we presented the $\\lbrace r,s\\rbrace $ in Fig.",
"(REF ) as a function of $\\beta $ and $\\gamma $ .",
"Figure: r-s for model 3. β=2.5\\beta =2.5 and γ=0.1\\gamma =0.1 for the left plot.",
"β=3.5\\beta =3.5 and γ=0.3\\gamma =0.3 for the right plot.",
"α 0 =1.0\\alpha _{0}=1.0, α 1 =1.5\\alpha _{1}=1.5, α 2 =1.0\\alpha _{2}=1.0, ω b =0.3\\omega _{b}=0.3 and b=0.01b=0.01.As we know the pair $\\lbrace r,s\\rbrace =\\lbrace 1,0\\rbrace $ corresponds to the $\\Lambda $ CDM model.",
"It is indicated on our graphs for both models.",
"Further, $\\lbrace 1,0\\rbrace $ which shows the CDM model, is present in our models.",
"But we obsaerve the absence of Einstein static universe due to this fact that our models never mimic the pair $\\lbrace -\\infty ,+\\infty \\rbrace $ .",
"So, our models fit the $\\Lambda CDM$ and CDM perfectly."
],
[
"Observational constraints ",
"To use the $SNIa$ data, we define distance modulus $\\mu $ as a function of the luminosity distance $D_L$ as the following: $\\mu =m-M=5\\log _{10}{D_L},$ Here $D_{L}$ is in the following form: $D_{L}=(1+z)\\frac{c}{H_{0}}\\int _{0}^{z}{\\frac{dz^{\\prime }}{\\sqrt{H(z^{\\prime })}}}.$ Here $m$ and $M$ denote the apparent magnitude and absolute magnitude, respectively.",
"Due to the photon-baryon plasma, Baryonic acoustic oscillations exist in the decoupling redshift $z = 1.090$ .",
"A major for scaling is the following quantity $A=\\frac{\\sqrt{\\Omega _{m0} } }{H(z_{b})^{1/3}} \\left[ \\frac{1}{z_{b}} \\int _{0}^{z_{b}}{\\frac{dz}{H(z)}} \\right]^{2/3}.$ From WiggleZ-data [37] we know that $A = 0.474 \\pm 0.034$ , $0.442 \\pm 0.020$ and $0.424 \\pm 0.021$ at the redshifts $z_{b} =0.44$ , $0.60$ and $0.73$ .",
"The major statistical analysis parameter is: $\\chi ^{2}{(x^{j})}=\\sum _{i}^{n}\\frac{(f(x^{j})_{i}^{t}-f(x^{j})_{i}^{0})^{2}}{\\sigma _{i}},$ Here $f(x^{j})_{i}^{t}$ is the theoretical function of the model's parameters.",
"To conclude the work and model analysis we perform comparison of our results with observational data.",
"SNeIa data allowed us to obtain the following observational constraints for our models.",
"For the Model 1, we found that the best fit can occurred with $\\Omega _{m0}=0.24$ and $H_{0}=0.3$ .",
"For $\\alpha _{0}=0.3$ , $\\alpha _{1}=0.5$ , $\\alpha _{2}=0.4$ and $\\beta =4.0$ , $\\gamma =1.4$ , $\\omega _{b}=0.5$ , while for interaction parameter $b=0.02$ .",
"For the Model 2, we found that the best fit we can obtain with $H_{0}=0.5$ and $\\Omega _{m}=0.4$ .",
"Meanwhile for $\\alpha _{0}=1.0$ , $\\alpha _{1}=1.5$ , $\\alpha _{2}=1.3$ and $\\beta =3.5$ , $\\gamma =0.5$ , $\\omega _{b}=0.3$ , while for interaction parameter $b=0.01$ .",
"Finally we present the results obtained for Model 3, which say that the best fit is possible when $H_{0}=0.35$ and $\\Omega _{m0}=0.28$ .",
"For the parameters $\\alpha _{0}$ , $\\alpha _{1}$ , $\\alpha _{2}$ , $\\beta $ , $\\gamma $ , $\\omega _{b}$ and $b$ we have the numbers $0.7$ ,$1.0$ , $1.2$ , 3, $0.8$ , $0.75$ and $0.01$ respectively.",
"Finally, we would like to discuss the constraints resulted from $SNeIa+BAO+CMB$ [35] .",
"Table: NO_CAPTION Figure: Observational data SneIa+BAO+CMBSneIa+BAO+CMB for distance modulus versus our theoretical results for models 1 and 2.From the graph of luminosity distance versus zm we learn that how $\\mu $ depends on the values of the parameters for different redshifts $z$ .",
"For different values of $\\Omega _M,\\Omega _D=0$ and i the regime of low redshifts $0.001<z<0.01$ , this graph has linearity.",
"For $z>0.4$ the graph has typical form of models with $\\Omega _M$ .",
"Hubble parameter $H$ has a centeral role in the behavior of $\\mu (z)$ for different ranges of $z$ .",
"We can use it to investigate the cosmological parameters."
],
[
"Summary",
"Time varying cosmological models with gravitational and cosmological constant have been studied frequently.",
"Nevertheless, in view of cosmological data, rate of change of G is small.",
"So the first order correction terms are more important.",
"Our approach to $\\Lambda (t),G(t)$ models is slightly different and more general than any other previous work.",
"As a proper generalization of general relativity, scalar-tensor-vector gravity model has been proposed to explain the structure of galaxies and dark matter problem.",
"If we assume small changes in the variation of the scalar fields, MOG model at the level of action becomes equivalent to Einstein-Hilbert model, of course it is necessary that we consider $G(t)$ as a slowly varying scalar field.",
"We proposed three models of generalized Ricci dark energy including $\\Lambda (t), G(t)$ to complete the time evolution of dark energy.",
"Due to the complexity of the model equations, the numerical algorithms with cosmological parameters have been used.",
"Gravitational acceleration region and time evolution of state finder parameters $\\lbrace r,s\\rbrace $ compared with $\\Lambda $ CDM model are numerically studied with high accuracy.",
"We obtained the fit range of data models by comparing the free parameters of dark energy models and cosmological data $SNeIa+BAO+CMB$ .",
"Our model is a model that is consistent with cosmological data while the other theoretical models are not.",
"The authors thank J.W.Moffat for useful comments about MOG."
]
] | 1403.0081 |
[
[
"Origin-Symmetric Bodies of Revolution with Minimal Mahler Volume in\n R^3-a new proof"
],
[
"Abstract Meyer and Reisner had proved the Mahler conjecture for rovelution bodies.",
"In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume.",
"Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler volume."
],
[
"Introduction",
"The well-known Mahler's conjecture (see, e.g.,[11], [18], [29] for references) states that, for any origin-symmetric convex body $K$ in $\\mathbb {R}^n$ , $\\mathcal {P}(K)\\ge \\mathcal {P}(C^n)=\\frac{4^n}{n!}",
",$ where $C^n$ is an $n$ -cube and $\\mathcal {P}(K)=Vol(K)Vol(K^{\\ast })$ , which is known as the Mahler volume of $K$ .",
"For $n=2$ , Mahler [19] himself proved the conjecture, and in 1986 Reisner [26] showed that equality holds only for parallelograms.",
"For $n=2$ , a new proof of inequality (1.1) was obtained by Campi and Gronchi [4].",
"Recently, Lin and Leng [17] gave a new and intuitive proof of the inequality (1.1) in $\\mathbb {R}^2$ .",
"For some special classes of origin-symmetric convex bodies in $\\mathbb {R}^n$ , a sharper estimate for the lower bound of $\\mathcal {P}(K)$ has been obtained.",
"If $K$ is a convex body which is symmetric around all coordinate hyperplanes, Saint Raymond [28] proved that $\\mathcal {P}(K)\\ge 4^n/n!$ ; the equality case was discussed in [20], [27].",
"When $K$ is a zonoid (limits of finite Minkowski sums of line segments), Meyer and Reisner (see, e.g., [12], [25], [26]) proved that the same inequality holds, with equality if and only if $K$ is an $n$ -cube.",
"For the case of polytopes with at most $2n+2$ vertices (or facets) (see, e.g., [2] for references), Lopez and Reisner [15] proved the inequality (1.1) for $n\\le 8$ and the minimal bodies are characterized.",
"Recently, Nazarov, Petrov, Ryabogin and Zvavitch [24] proved that the cube is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the Banach-Mazur distance.",
"Bourgain and Milman [3] proved that there exists a universal constant $c>0$ such that $\\mathcal {P}(K)\\ge c^n \\mathcal {P}(B)$ , which is now known as the reverse Santaló inequality.",
"Very recently, Kuperberg [14] found a beautiful new approach to the reverse Santaló inequality.",
"What's especially remarkable about Kuperberg's inequality is that it provides an explicit value for $c$ .",
"Another variant of the Mahler conjecture without the assumption of origin-symmetry states that, for any convex body $K$ in $\\mathbb {R}^n$ , $\\mathcal {P}(K)\\ge \\frac{(n+1)^{(n+1)}}{(n!",
")^2},$ with equality conjectured to hold only for simplices.",
"For $n=2$ , Mahler himself proved this inequality in 1939 (see, e.g.,[5], [6], [16] for references) and Meyer [21] obtained the equality conditions in 1991.",
"Recently, Meyer and Reisner[23] have proved inequality (1.2) for polytopes with at most $n+3$ vertices.",
"Very recently, Kim and Reisner[13] proved that the simplex is a strict local minimum for the Mahler volume in the Banach-Mazur space of $n$ -dimensional convex bodies.",
"Strong functional versions of the Blaschke-Santaló inequality and its reverse form have been studied recently (see, e.g., [1], [7], [8], [9], [10], [22] ).",
"The Mahler conjecture is still open even in the three-dimensional case.",
"Terence Tao in [30] made an excellent remark about the open question.",
"To state our results, we first give some definitions.",
"In the coordinate plane XOY of $\\mathbb {R}^3$ , let $D=\\lbrace (x,y): -a\\le x\\le a, |y|\\le f(x)\\rbrace ,$ where $f(x)$ ($[-a,a]$ , $a>0$ ) is a concave, even and nonnegative function.",
"An origin-symmetric body of revolution $R$ is defined as the convex body generated by rotating $D$ around the $X$ -axis in $\\mathbb {R}^3$ .",
"$f(x)$ is called its generating function and $D$ is its generating domain.",
"If the generating domain of $R$ is a rectangle (the generating function of $R$ is a constant function), $R$ is called a cylinder.",
"If the generating domain of $R$ is a diamond (the generating function $f(x)$ of $R$ is a linear function on $[-a,0]$ and $f(-a)=0$ ), $R$ is called a bicone.",
"In this paper, we prove that cylinders have the minimal Mahler volume for origin-symmetric bodies of revolution in $\\mathbb {R}^3$ .",
"Theorem 1.1 For any origin-symmetric body of revolution $K$ in $\\mathbb {R}^3$ , we have $\\mathcal {P}(K)\\ge \\frac{4\\pi ^2}{3},$ and the equality holds if and only if $K$ is a cylinder or bicone.",
"Remark 1 In [22], for the Schwarz rounding $\\tilde{K}$ of a convex body $K$ in $\\mathbb {R}^n$ , Meyer and Reisner gave a lower bound for $\\mathcal {P}(\\tilde{K})$ .",
"Especially, for a general body of revolution $K$ in $\\mathbb {R}^3$ , they proved $\\mathcal {P}(K)\\ge \\frac{4^4\\pi ^2}{3^5},$ with equality if and only if $K$ is a cone and $|AO|/|AD|=3/4$ (where, $A$ is the vertex of the cone and $AD$ is the height and $O$ is the Santaló point of $K$ ).",
"The following Theorem $1.2$ is the functional version of the Theorem 1.1.",
"Theorem 1.2 Let $f(x)$ be a concave, even and nonnegative function defined on $[-a,a]$ , $a>0$ , and for $x^{\\prime }\\in [-\\frac{1}{a},\\frac{1}{a}]$ define $f^{\\ast }(x^{\\prime })=\\inf _{x\\in [-a,a]}\\frac{1-x^{\\prime }x}{f(x)}.$ Then, we have $\\left(\\int _{-a}^{a} (f(x))^2 dx\\right)\\left(\\int _{-\\frac{1}{a}}^{\\frac{1}{a}} (f^{\\ast }(x^{\\prime }))^2dx^{\\prime }\\right)\\ge \\frac{4}{3},$ with equality if and if $f(x)=f(0)$ or $f^{\\ast }(x^{\\prime })=1/f(0)$ .",
"Let $C$ be an origin-symmetric convex body in the coordinate plane YOZ of $\\mathbb {R}^3$ and $f(x)$ ($x\\in [-a,a]$ , $a>0$ ) is a concave, even and nonnegative function.",
"A parallel sections homothety body is defined as the convex body $K=\\bigcup _{x\\in [-a,a]}\\lbrace f(x)C+xv\\rbrace ,$ where $v=(1,0,0)$ is a unit vector in the positive direction of the X-axis, $f(x)$ is called its generating function and $C$ is its homothetic section.",
"Applying Theorem 1.2, we prove that among parallel sections homothety bodies in $\\mathbb {R}^3$ , 3-cubes have the minimal Mahler volume.",
"Theorem 1.3 For any parallel sections homothety body $K$ in $\\mathbb {R}^3$ , we have $\\mathcal {P}(K)\\ge \\frac{4^3}{3!",
"},$ and the equality holds if and only if $K$ is a 3-cube or octahedron."
],
[
" Definitions, notation, and preliminaries",
"As usual, $S^{n-1}$ denotes the unit sphere, and $B^n$ the unit ball centered at the origin, $O$ the origin and $\\Vert \\cdot \\Vert $ the norm in Euclidean $n$ -space $\\mathbb {R}^n$ .",
"The symbol for the set of all natural numbers is $\\mathbb {N}$ .",
"Let $\\mathcal {K}^n$ denote the set of convex bodies (compact, convex subsets with non-empty interiors) in $\\mathbb {R}^n$ .",
"Let $\\mathcal {K}^n_o$ denote the subset of $\\mathcal {K}^n$ that contains the origin in its interior.",
"For $u\\in S^{n-1}$ , we denote by $u^{\\perp }$ the $(n-1)$ -dimensional subspace orthogonal to $u$ .",
"For $x$ , $y\\in \\mathbb {R}^n$ , $ x\\cdot y$ denotes the inner product of $x$ and $y$ .",
"Let $\\textrm {int}\\;K$ denote the interior of $K$ .",
"Let $\\textrm {conv}\\;K$ denote the convex hull of $K$ .",
"we denote by $V(K)$ the $n$ -dimensional volume of $K$ .",
"The notation for the usual orthogonal projection of $K$ on a subspace $S$ is $K|S$ .",
"If $K\\in {K}^n_o$ , we define the polar body $K^{\\ast }$ of $K$ by $K^{\\ast }=\\lbrace x\\in \\mathbb {R}^n:~ x\\cdot y \\le 1~, \\forall y\\in K\\rbrace .$ Remark 2 If $P$ is a polytope, i.e., $P={\\rm conv}\\lbrace p_1,\\cdots ,p_m\\rbrace $ , where $p_i$ $(i=1,\\cdots ,m)$ are vertices of polytope $P$ .",
"By the definition of the polar body, we have $P^{\\ast }&=&\\lbrace x\\in \\mathbb {R}^n: x\\cdot p_1\\le 1,\\cdots , x\\cdot p_m\\le 1\\rbrace \\nonumber \\\\&=&\\bigcap _{i=1}^{m}\\lbrace x\\in \\mathbb {R}^n: x\\cdot p_i \\le 1\\rbrace ,$ which implies that $P^{\\ast }$ is an intersection of $m$ closed halfspaces with exterior normal vectors $p_i$ ($i=1,\\cdots ,m$ ) and the distance of hyperplane $\\lbrace x\\in \\mathbb {R}^n: x\\cdot p_i= 1\\rbrace $ from the origin is $1/\\Vert p_i\\Vert $ .",
"Associated with each convex body $K$ in $\\mathbb {R}^n$ is its support function $h_K: \\mathbb {R}^n\\rightarrow [0,\\infty )$ , defined for $x\\in \\mathbb {R}^n$ , by $h_K(x)= \\max \\lbrace y\\cdot x: y\\in K\\rbrace ,$ and its radial function $\\rho _K:\\mathbb {R}^n\\backslash \\lbrace 0\\rbrace \\rightarrow (0,\\infty )$ , defined for $x\\ne 0$ , by $\\rho _K(x)= \\max \\lbrace \\lambda \\ge 0~:~\\lambda x\\in K\\rbrace .$ For $K$ , $L\\in \\mathcal {K}^n$ , the Hausdorff distance is defined by $\\delta (K,L)=\\min \\lbrace \\lambda \\ge 0:~K\\subset L+\\lambda B^n,~L\\subset K+\\lambda B^n\\rbrace .$ A linear transformation (or affine transformation) of $\\mathbb {R}^n$ is a map $\\phi $ from $\\mathbb {R}^n$ to itself such that $\\phi x~=~A x$ (or $\\phi x~=~A x + t$ , respectively), where $A$ is an $n \\times n$ matrix and $t\\in \\mathbb {R}^n$ .",
"It is known that Mahler volume of $K$ is invariant under affine transformation.",
"For $K\\in \\mathcal {K}^n_o$ , if $(x_1,x_2,\\cdots ,x_n)\\in K$ , we have $(\\varepsilon _1x_1,\\cdots ,\\varepsilon _nx_n)\\in K$ for any signs $\\varepsilon _i=\\pm 1$ ($i=1,\\cdots ,n$ ), then $K$ is a 1-unconditional convex body.",
"In fact, $K$ is symmetric with respect to all coordinate planes.",
"The following Lemma 2.1 will be used to calculate the volume of an origin-symmetric body of revolution.",
"Since the lemma is an elementary conclusion in calculus, we omit its proof.",
"Lemma 2.1 In the coordinate plane XOY, let $D=\\lbrace (x,y): a\\le x\\le b, |y|\\le f(x)\\rbrace ,$ where $f(x)$ is a linear, nonnegative function defined on $[a,b]$ .",
"Let $R$ be a body of revolution generated by $D$ .",
"Then $V(R)=\\frac{\\pi }{3}(b-a)\\left[f(a)^2+f(a)f(b)+f(b)^2\\right].$"
],
[
"Main result and its proof",
"In the paper, we consider convex bodies in a three-dimensional Cartesian coordinate system with origin $O$ and its three coordinate axes are denoted by $X$ -axis, $Y$ -axis, and $Z$ -axis.",
"Lemma 3.1 If $K\\in \\mathcal {K}_0^3$ , then for any $u\\in S^2$ , we have $K^{\\ast }\\cap u^{\\perp }=(K|u^{\\perp })^{\\ast }.$ On the other hand, if $K^{\\prime }\\in \\mathcal {K}_0^3$ satisfies $K^{\\prime }\\cap u^{\\perp }=(K|u^{\\perp })^{\\ast }$ for any $u\\in S^2\\cap v_0^{\\perp }$ ($v_0$ is a fixed vector), then, $K^{\\prime }=K^{\\ast }.$ Firstly, we prove (3.1).",
"Let $x\\in u^{\\perp }$ , $y\\in K$ and $y^{\\prime }=y|u^{\\perp }$ , since the hyperplane $u^{\\perp }$ is orthogonal to the vector $y-y^{\\prime }$ , then $ y\\cdot x=(y^{\\prime }+y-y^{\\prime })\\cdot x=y^{\\prime }\\cdot x+(y-y^{\\prime })\\cdot x= y^{\\prime }\\cdot x.$ If $x\\in K^{\\ast }\\cap u^{\\perp }$ , for any $y^{\\prime }\\in K|u^{\\perp }$ , there exists $y\\in K$ such that $y^{\\prime }=y|u^{\\perp }$ , then $x\\cdot y^{\\prime }=x\\cdot y\\le 1$ , thus $x\\in (K|u^{\\perp })^{\\ast }$ .",
"Thus, we have $K^{\\ast }\\cap u^{\\perp }\\subset (K|u^{\\perp })^{\\ast }$ .",
"If $x\\in (K|u^{\\perp })^{\\ast }$ , then for any $y\\in K$ and $y^{\\prime }=y|u^{\\perp }$ , $ x\\cdot y=x\\cdot y^{\\prime }\\le 1$ , thus $x\\in K^{\\ast }$ , and since $x\\in u^{\\perp }$ , thus $x\\in K^{\\ast }\\cap u^{\\perp }$ .",
"Thus, we have $(K|u^{\\perp })^{\\ast }\\subset K^{\\ast }\\cap u^{\\perp }$ .",
"Next we prove (3.3).",
"Let $S^1= S^2\\cap v_0^{\\perp }$ .",
"For any vector $v\\in S^2$ , there exists a $u\\in S^1$ satisfying $v\\in u^{\\perp }$ .",
"Since $K^{\\prime }\\cap u^{\\perp }=(K|u^{\\perp })^{\\ast }$ and $K^{\\ast }\\cap u^{\\perp }=(K|u^{\\perp })^{\\ast }$ , thus $K^{\\prime }\\cap u^{\\perp }=K^{\\ast }\\cap u^{\\perp }$ .",
"Hence, we have $\\rho _{K^{\\prime }}(v)=\\rho _{K^{\\ast }}(v)$ .",
"Since $v\\in S^2$ is arbitrary, we get $K^{\\prime }=K^{\\ast }.$ Lemma 3.2 In the coordinate plane XOY, let $P$ be a 1-unconditional convex body.",
"Let $R$ and $R^{\\prime }$ be two origin-symmetric bodies of revolution generated by $P$ and $P^{\\ast }$ , respectively.",
"Then $R^{\\prime }=R^{\\ast }$ .",
"Let $v_0=\\lbrace 1,0,0\\rbrace $ and $S^1=S^2\\cap v_0^{\\perp }$ , for any $u\\in S^1,$ we have $R|u^{\\perp }=R\\cap u^{\\perp }$ .",
"Since $R^{\\prime }\\cap u^{\\perp }=P^{\\ast }=(R\\cap u^{\\perp })^{\\ast }$ for any $u\\in S^1$ , thus $R^{\\prime }\\cap u^{\\perp }=(R| u^{\\perp })^{\\ast }$ for any $u\\in S^1$ .",
"By Lemma 3.1, we have $R^{\\prime }=R^{\\ast }$ .",
"Lemma 3.3 For any origin-symmetric body of revolution $R$ , there exists a linear transformation $\\phi $ satisfying (i) $\\phi R$ is an origin-symmetric body of revolution; (ii) $\\phi R\\subset C^3=[-1,1]^3,$ where $C^3$ is the unit cube in $\\mathbb {R}^3$ .",
"Let $f(x)$ ($x\\in [-a,a]$ ) be the generating function of $R$ .",
"For vector $v=(1,0,0)$ and any $t\\in [-a,a],$ the set $R\\cap (v^{\\perp }+tv)$ is a disk in the plane $v^{\\perp }+tv$ with the point $(t,0,0)$ as the center and $f(t)$ as the radius.",
"Next, for a $3\\times 3$ diagonal matrix $A=\\begin{bmatrix}b&0&0\\\\0&c&0\\\\0&0&c\\end{bmatrix},$ where $b, c\\in \\mathbb {R}^{+}$ , let $\\phi R=\\lbrace Ax:x\\in R\\rbrace $ , we prove that $\\phi R$ is still an origin-symmetric body of revolution.",
"For $t^{\\prime }\\in [-ab,ab]$ , if $(t^{\\prime },y^{\\prime },z^{\\prime })\\in \\phi R\\cap (v^{\\perp }+t^{\\prime }v)$ , there is $(t,y,z)\\in R\\cap (v^{\\perp }+tv)$ satisfying $t^{\\prime }=bt,\\;\\;y^{\\prime }=cy,\\;\\;z^{\\prime }=cz.$ Hence, we have $\\Vert (t^{\\prime },y^{\\prime },z^{\\prime })-(t^{\\prime },0,0)\\Vert =c\\Vert (t,y,z)-(t,0,0)\\Vert \\le c f(t),$ which implies that $\\phi R\\cap (v^{\\perp }+t^{\\prime }v)\\subset B^{\\prime }$ , where $B^{\\prime }$ is a disk in the plane $v^{\\perp }+t^{\\prime }v$ with $(t^{\\prime },0,0)$ as the center and $cf(t^{\\prime }/b)$ as the radius.",
"On the other hand, if $(t^{\\prime }, y^{\\prime }, z^{\\prime })\\in B^{\\prime }$ , then $\\Vert (t^{\\prime },y^{\\prime },z^{\\prime })-(t^{\\prime },0,0)\\Vert \\le cf(t^{\\prime }/b)$ .",
"Let $t=t^{\\prime }/b$ , $y=y^{\\prime }/c$ and $z=z^{\\prime }/c$ .",
"Noting $t^\\prime \\in [-ab,ab]$ , we have $t\\in [-a,a]$ and $\\Vert (t,y,z)-(t,0,0)\\Vert =\\frac{1}{c}\\Vert (t^{\\prime },y^{\\prime },z^{\\prime })-(t^{\\prime },0,0)\\Vert \\le f(t).$ Hence, we have $(t,y,z)\\in R\\cap (v^{\\perp }+tv)$ , which implies that $(t^{\\prime },y^{\\prime },z^{\\prime })=(bt,cy,cz)\\in \\phi R\\cap (v^{\\perp }+t^{\\prime }v)$ .",
"Thus, $B^{\\prime }\\subset \\phi R\\cap (v^{\\perp }+t^{\\prime }v)$ .",
"Therefore, we have $ \\phi R\\cap (v^{\\perp }+t^{\\prime }v)= B^{\\prime }$ .",
"It follows that $\\phi R$ is an origin-symmetric body of revolution and its generating function is $F(x)=cf(x/b)$ , $x\\in [-ab,ab]$ .",
"Set $b=1/a$ and $c=1/f(0)$ , we obtain $\\phi R\\subset C^3=[-1,1]^3$ .",
"Remark 3 By Lemma 3.3 and the affine invariance of Mahler volume, to prove our theorems, we need only consider the origin-symmetric body of revolution $R$ whose generating domain $P$ satisfies $T\\subset P\\subset Q$ , where $T=\\lbrace (x,y): |x|+|y|\\le 1\\rbrace \\;\\;\\textrm {and}\\;\\;Q=\\lbrace (x,y): \\max \\lbrace |x|,|y|\\rbrace \\le 1\\rbrace .$ In the following lemmas, let $\\triangle ABD$ denote $\\textrm {conv}\\lbrace A, B, D\\rbrace $ , where $A=(-1,1)$ , $B=(0,1)$ and $D=(-1,0)$ .",
"Figure: NO_CAPTIONLemma 3.4 Let $P$ be a 1-unconditional polygon in the coordinate plane $XOY$ satisfying $P\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace O,D,A_2,A_1,B\\rbrace ,$ where $A_1$ lies on the line segment $AB$ and $A_2\\in {\\rm int}\\triangle ABD$ , $R$ the origin-symmetric body of revolution generated by $P$ .",
"Then $ \\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace $ and $ \\mathcal {P}(R)\\ge \\frac{4\\pi ^2}{3},$ where $R_1$ and $R_2$ are origin-symmetric bodies of revolution generated by 1-unconditional polygons $P_1$ and $P_2$ satisfying $P_1\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace O,D,A_2,B\\rbrace $ and $P_2\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace O,D,C,B\\rbrace ,$ respectively, where $C$ is the point of intersection between two lines $A_2D$ and $AB$ .",
"In Figure 3.1, let $A_2=(x_0,y_0)$ and $A_1=(-t,1)$ , then $C=(\\frac{x_0-y_0+1}{y_0},1)\\;\\;\\textrm {and}\\;\\;0\\le t\\le \\frac{-x_0+y_0-1}{y_0}.$ From Remark 2, we can get $P^{\\ast }$ , which satisfies $P^{\\ast }\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace =\\textrm {conv}\\lbrace M,E,D,O,B\\rbrace ,$ where $E$ lies on the line segment $AD$ and $M\\in \\textrm {int} \\triangle ABD$ .",
"Let $F$ be the point of intersection between two lines $EM$ and $AB$ .",
"Let $F_1(t)=\\frac{1}{2}V(R),\\;\\;F_2(t)=\\frac{1}{2}V(R^{\\ast })\\;\\; \\textrm {and}\\;\\;F(t)=F_1(t)F_2(t).$ Firstly, we prove (3.4).",
"The proof consists of three steps for good understanding.",
"First step.",
"We calculate the first and second derivatives of the functions $F(t)$ .",
"Since $EF\\bot OA_2$ and the distance of the line $EF$ from $O$ is $1/\\Vert OA_2\\Vert $ , we have the equation of the line $EF$ $y=-\\frac{x_0}{y_0}x+\\frac{1}{y_0}.$ Similarly, since $BM\\bot OA_1$ and the distance of the line $BM$ from $O$ is $1/\\Vert OA_1\\Vert $ , we get the equation of the line $BM$ $y=tx+1.$ Using equations (3.6) and (3.7), we obtain $M=(x_M,y_M)=\\left(\\frac{1-y_0}{ty_0+x_0},\\frac{x_0+t}{ty_0+x_0}\\right)$ and $E=(x_E,y_E)=\\left(-1, \\frac{x_0+1}{y_0}\\right).$ Noting that $& & P\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace \\nonumber \\\\&=&\\textrm {conv}\\lbrace D,A_2,A_2^{\\prime }\\rbrace \\cup \\textrm {conv}\\lbrace A_1,A_2,A_2^{\\prime },A_1^{\\prime }\\rbrace \\cup \\textrm {conv}\\lbrace O,B,A_1,A_1^{\\prime }\\rbrace ,$ where $A_1^{\\prime }$ and $A_2^{\\prime }$ are the orthogonal projections of points $A_1$ and $A_2$ , respectively, on the $X$ -axis, and applying Lemma 2.1, we have $F_1(t)&=&\\frac{\\pi }{3}y_0^2(x_0+1)+\\frac{\\pi }{3}(-t-x_0)(y_0^2+y_0+1)+\\pi t\\nonumber \\\\&=&\\frac{\\pi }{3}(-y_0^2-y_0+2) t+\\frac{\\pi }{3}(y_0^2-x_0y_0-x_0).$ Thus, we have $F_1^{\\prime }(t)=\\frac{\\pi }{3}(-y_0^2-y_0+2).$ Noting that $& & P^{\\ast }\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace \\nonumber \\\\&=&\\textrm {conv}\\lbrace D,E, M,M^{\\prime }\\rbrace \\cup \\textrm {conv}\\lbrace M,M^{\\prime },O,B\\rbrace ,$ where $M^{\\prime }$ is the orthogonal projection of point $M$ on the $X$ -axis, and applying Lemma 2.1, we obtain $F_2(t)&=&\\frac{\\pi }{3}(x_M-x_E)(y_E^2+y_Ey_M+y_M^2)+\\frac{\\pi }{3}(-x_M)(y_M^2+y_M+1)\\nonumber \\\\&=&\\frac{\\pi }{3}(\\frac{1-y_0}{ty_0+x_0}+1)\\left[(\\frac{x_0+1}{y_0})^2+(\\frac{x_0+1}{y_0})(\\frac{x_0+t}{ty_0+x_0})+(\\frac{x_0+t}{ty_0+x_0})^2\\right]\\nonumber \\\\&~&+\\frac{\\pi }{3}(\\frac{y_0-1}{ty_0+x_0})\\left[(\\frac{x_0+t}{ty_0+x_0})^2+(\\frac{x_0+t}{ty_0+x_0})+1\\right]\\nonumber \\\\&=&\\frac{\\pi }{3} \\frac{\\Delta _1 t^3+\\Delta _2 t^2+\\Delta _3t+\\Delta _4}{y_0^2(ty_0+x_0)^3},$ where $&~&\\Delta _1=y_0^3(x_0^2+3x_0+3),\\nonumber \\\\&~&\\Delta _2=y_0^2(3x_0^3+9x_0^2+9x_0+y_0^3-3y_0+2),\\nonumber \\\\&~&\\Delta _3=3y_0[x_0^4+3x_0^3+3x_0^2+x_0(y_0^3-y_0^2-y_0+1)],\\nonumber \\\\&~&\\Delta _4=x_0^2(x_0^3+3x_0^2+3x_0+2y_0^3-3y_0^2+1).$ Thus, we have $F_2^{\\prime }(t)&=&\\frac{\\pi }{3}\\frac{(3\\Delta _1x_0-\\Delta _2y_0)t^2+(2\\Delta _2x_0-2\\Delta _3y_0)t+(\\Delta _3x_0-3\\Delta _4y_0)}{y_0^2(ty_0+x_0)^4}\\nonumber \\\\&=&\\frac{\\pi }{3}(y_0-1)^2\\frac{-y_0(y_0+2)t^2-2x_0(2y_0+1)t-3x_0^2}{(ty_0+x_0)^4}.$ Then, we have $F^{\\prime }(t)&=&F_1^{\\prime }(t)F_2(t)+F_1(t)F_2^{\\prime }(t)\\nonumber \\\\&=&\\frac{\\pi ^2}{9}\\frac{\\Lambda _1t^4+\\Lambda _2t^3+\\Lambda _3t^2+\\Lambda _4t+\\Lambda _5}{y_0^2(ty_0+x_0)^4},$ where $&~&\\Lambda _1=y_0^4[x_0^2(-y_0^2-y_0+2)+3x_0(-y_0^2-y_0+2)+3(-y_0^2-y_0+2)],\\nonumber \\\\&~&\\Lambda _2=y_0^3[4x_0^3(-y_0^2-y_0+2)+12x_0^2(-y_0^2-y_0+2)+12x_0(-y_0^2-y_0+2)],\\nonumber \\\\&~&\\Lambda _3=y_0^2[6x_0^4(-y_0^2-y_0+2)+18x_0^3(-y_0^2-y_0+2)+18x_0^2(-y_0^2-y_0+2)\\nonumber \\\\&~&\\;\\;\\;\\;\\;\\;\\;\\;\\;+x_0(y_0^5-2y_0^4+8y_0^2-13y_0+6)+(-y_0^6+3y_0^4-2y_0^3)],\\nonumber \\\\&~&\\Lambda _4=y_0[4x_0^5(-y_0^2-y_0+2)+12x_0^4(-y_0^2-y_0+2)+12x_0^3(-y_0^2-y_0+2)\\nonumber \\\\&~&\\;\\;\\;\\;\\;\\;\\;\\;\\;+x_0^2(2y_0^5-4y_0^4+4y_0^3+4y_0^2-14y_0+8)+x_0(-4y_0^6+6y_0^5-2y_0^3)],\\nonumber \\\\&~&\\Lambda _5=x_0^6(-y_0^2-y_0+2)+3x_0^5(-y_0^2-y_0+2)+3x_0^4(-y_0^2-y_0+2)\\nonumber \\\\&~&\\;\\;\\;\\;\\;\\;\\;\\;\\;+x_0^3(y_0^5-2y_0^4+4y_0^3-4y_0^2-y_0+2)+x_0^2(-3y_0^6+6y_0^5-3y_0^4).$ Simplifying the above equation, we get $F^{\\prime }(t)&=&\\frac{\\pi ^2}{9}\\frac{-y_0^2-y_0+2}{y_0^2(ty_0+x_0)^3}\\lbrace (x_0^2+3x_0+3)y_0^3t^3+3x_0(x_0^2+3x_0+3)y_0^2t^2\\nonumber \\\\&~&+[3x_0^4+9x_0^3+9x_0^2+x_0(-y_0^3+3y_0^2-5y_0+3)+y_0^3(y_0-1)]y_0t\\nonumber \\\\&~&+[x_0^5+3x_0^4+3x_0^3+x_0^2\\frac{-y_0^4+y_0^3-3y_0^2+y_0+2}{y_0+2}+x_0\\frac{3y_0^5-3y_0^4}{y_0+2}]\\rbrace .\\nonumber \\\\$ From (3.14), we can get $F^{\\prime \\prime }(t)&=&\\frac{\\pi ^2}{9}\\frac{(4\\Lambda _1x_0-\\Lambda _2y_0)t^3+(3\\Lambda _2x_0-2\\Lambda _3y_0)t^2+(2\\Lambda _3x_0-3\\Lambda _4y_0)t+(\\Lambda _4x_0-4\\Lambda _5y_0)}{y_0^2(ty_0+x_0)^5}\\nonumber \\\\&=&\\frac{\\pi ^2}{9}\\frac{\\Gamma _1t^2+\\Gamma _2t+\\Gamma _3}{y_0^2(ty_0+x_0)^5},$ where $&~&\\Gamma _1=-2x_0y_0^3(y_0^5-2y_0^4+8y_0^2-13y_0+6)-2y_0^6(-y_0^3+3y_0-2),\\nonumber \\\\&~&\\Gamma _2=x_0^2y_0^2(-4y_0^5+8y_0^4-12y_0^3+4y_0^2+16y_0-12)+x_0y_0^5(10y_0^3-18y_0^2+6y_0+2),\\nonumber \\\\&~&\\Gamma _3=x_0^3y_0^2(-2y_0^4+4y_0^3-12y_0^2+20y_0-10)+x_0^2y_0^4(8y_0^3-18y_0^2+12y_0-2).$ Simplifying the above equation, we get $F^{\\prime \\prime }(t)&=&\\frac{\\pi ^2}{9}\\frac{\\frac{\\Gamma _1}{y_0}t+(\\frac{\\Gamma _2}{y_0}-\\frac{x_0\\Gamma _1}{y_0^2})}{y_0^2(ty_0+x_0)^4}\\nonumber \\\\&=&\\frac{\\pi ^2}{9}\\frac{(y_0-1)^2}{(ty_0+x_0)^4}\\lbrace [-2x_0(y_0+2)(y_0^2-2y_0+3)+2y_0^3(y_0+2)]t\\nonumber \\\\&~&+[x_0^2(-2y_0^2-10)+x_0y_0^2(8y_0-2)]\\rbrace .$ Figure: NO_CAPTIONSecond step.",
"We prove that ${\\rm (}i{\\rm )}\\;\\; F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for }(x_0,y_0)\\in \\mathcal {D}_1$ and ${\\rm (}ii{\\rm )}\\;\\; F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\; (x_0,y_0)\\in \\mathcal {D}_2,$ where $\\mathcal {D}_1&=&\\lbrace (x,y): -1\\le x\\le y-1,\\;\\frac{-1+\\sqrt{5}}{2}\\le y\\le 1\\rbrace \\nonumber \\\\&~&\\cup \\lbrace (x,y): -1\\le x\\le \\frac{y^3+2y^2+3y-6}{(2-y)(y+3)},\\;0\\le y\\le \\frac{-1+\\sqrt{5}}{2}\\rbrace \\nonumber \\\\$ and $\\mathcal {D}_2&=&\\lbrace (x,y): \\frac{y^3+2y^2+3y-6}{(2-y)(y+3)}\\le x\\le y-1,\\;0\\le y\\le \\frac{-1+\\sqrt{5}}{2}\\rbrace .\\nonumber \\\\$ In fact, from (3.15), we have that $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0}) &=&\\frac{\\pi ^2}{9y_0^2}\\;G(x_0,y_0),$ where $G(x_0,y_0)=x_0^2(2-y_0)(y_0+3)-x_0(y_0^3+3y_0^2+4y_0-12)-(y_0+2)(y_0^3+3y_0-3).$ Noting that $G(x_0,y_0)$ is a quadratic function of the variable $x_0$ defined on $[-1, y_0-1]$ and $0\\le y_0\\le 1$ , the graph of the quadratic function is a parabola opening upwards.",
"When $x_0=-1$ , we obtain $G(-1,y_0)=-y_0^2(y_0^2+y_0+1)<0.$ When $x_0=y_0-1$ , we have $G(y_0-1,y_0)=-3y_0^2(y_0^2+y_0-1).$ Then we have $G(y_0-1,y_0)\\le 0\\;\\; {\\rm for} \\;\\frac{-1+\\sqrt{5}}{2}\\le y_0\\le 1$ and $G(y_0-1,y_0)\\ge 0\\;\\; {\\rm for} \\;0\\le y_0<\\frac{-1+\\sqrt{5}}{2}.$ When $x_0=\\frac{y_0^3+2y_0^2+3y_0-6}{(2-y_0)(y_0+3)}\\in [-1,y_0-1],$ we have $G(\\frac{y_0^3+2y_0^2+3y_0-6}{(2-y_0)(y_0+3)},y_0)=G(-1,y_0)<0.$ Hence, $G(x_0,y_0)\\le 0,\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_1.$ From (3.20), we have $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;{\\rm for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_1.$ By (3.17), we get $F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})&=&\\frac{\\pi ^2}{9}\\frac{1}{y_0(1-y_0)} H(x_0,y_0),$ where $H(x_0,y_0)=12x_0^2-x_0(4y_0^3+2y_0-12)-2y_0^3(y_0+2).$ Noting that $H(x_0,y_0)$ is a quadratic function of the variable $x_0$ defined on $[-1,y_0-1]$ and the coefficient of the quadratic term is positive, the graph of the quadratic function is a parabola opening upwards.",
"Let $x_0=y_0-1$ , we have $H(y_0-1,y_0)=-6y_0^4-10y_0(1-y_0)\\le 0.$ Let $x_0=\\frac{y_0^3+2y_0^2+3y_0-6}{(2-y_0)(y_0+3)},$ we have $&&H(\\frac{y_0^3+2y_0^2+3y_0-6}{(2-y_0)(y_0+3)},y_0)\\nonumber \\\\&=&\\frac{2y_0^8+4y_0^7+24y_0^6+50y_0^5-38y_0^4-18y_0^3-48y_0^2-72y_0}{(2-y_0)^2(y_0+3)^2}\\nonumber \\\\&\\le & 0.$ From (3.24) and (3.25), we have $H(x_0,y_0)\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Therefore, from (3.22) and $0<y_0<1$ , we have $F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;{\\rm for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Third step.",
"We prove $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace $ .",
"By (3.17), we have $F^{\\prime \\prime }(t)&=&\\frac{\\pi ^2}{9}\\frac{(y_0-1)^2}{(ty_0+x_0)^4}I(t),$ where $I(t)&=&[-2x_0(y_0+2)(y_0^2-2y_0+3)+2y_0^3(y_0+2)] t\\nonumber \\\\&~&+[x_0^2(-2y_0^2-10)+x_0y_0^2(8y_0-2)]$ and $0\\le t\\le \\frac{-x_0+y_0-1}{y_0}.$ Since $-2x_0(y_0+2)(y_0^2-2y_0+3)+2y_0^3(y_0+2)>0,$ $I(t)$ is an increasing function of the variable $t$ .",
"By (3.26), for any $(x_0,y_0)\\in \\mathcal {D}_2,$ we have $F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0.$ From (3.27), we have $I(\\frac{-x_0+y_0-1}{y_0})\\le 0,$ which implies that $I(t)\\le 0$ for any $0\\le t\\le \\frac{-x_0+y_0-1}{y_0}.$ Therefore $F^{\\prime \\prime }(t)\\le 0$ for any $0\\le t\\le \\frac{-x_0+y_0-1}{y_0}.$ It follows that the function $F(t)$ is concave on the interval $[0,\\frac{-x_0+y_0-1}{y_0}],$ which implies $F(t)\\ge \\min \\lbrace F(0),F(\\frac{-x_0+y_0-1}{y_0})\\rbrace .$ Therefore, we have $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace .$ By (3.21), for any $(x_0,y_0)\\in \\mathcal {D}_1$ , we have $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0.$ Now we prove that the inequality (3.4) holds in each of the following situations: $\\textrm {(i)}\\;\\;I(\\frac{-x_0+y_0-1}{y_0})\\le 0;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ $\\textrm {(ii)}\\;\\;I(\\frac{-x_0+y_0-1}{y_0})> 0\\;\\;\\textrm {and}\\;\\;I(0)< 0;$ $\\textrm {(iii)}\\;\\;I(0)\\ge 0.\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;$ We have proved (3.4) in the case (i), and now we prove (3.4) in cases (ii) and (iii).",
"For the case (ii), since $I(t)$ is increasing and by (3.27), there exists a real number $t_0\\in (0,\\frac{-x_0+y_0-1}{y_0})$ satisfying $F^{\\prime \\prime }(t)\\le 0\\;\\;\\textrm {for}\\;\\;t\\in [0,t_0]$ and $F^{\\prime \\prime }(t)> 0\\;\\;\\textrm {for}\\;\\;t\\in (t_0,\\frac{-x_0+y_0-1}{y_0}].$ It follows that $F^{\\prime }(t)$ is decreasing on the interval $[0,t_0]$ and increasing on the interval $(t_0,\\frac{-x_0+y_0-1}{y_0}].$ If $F^{\\prime }(0)\\le 0$ , and since $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0,$ we have $F^{\\prime }(t)\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}],$ which implies that the function $F(t)$ is decreasing and $F(t)\\ge F(\\frac{-x_0+y_0-1}{y_0})\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}].$ Therefore we have $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace =\\mathcal {P}(R_2).$ If $F^{\\prime }(0)>0$ , there exists a real number $t_1\\in (0,\\frac{-x_0+y_0-1}{y_0})$ satisfying $F^{\\prime }(t)>0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,t_1)$ and $F^{\\prime }(t)\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [t_1,\\frac{-x_0+y_0-1}{y_0}],$ which implies that the function $F(t)$ is increasing on the interval $[0,t_1)$ and decreasing on the interval $[t_1,\\frac{-x_0+y_0-1}{y_0}].$ It follows that $F(t)\\ge \\min \\lbrace F(0),F(\\frac{-x_0+y_0-1}{y_0})\\rbrace \\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}].$ We then have $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace .$ For the case (iii), since the function $I(t)$ is increasing, we have $I(t)\\ge 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}].$ Hence, from (3.27), we have $F^{\\prime \\prime }(t)\\ge 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}].$ Therefore, the function $F^{\\prime }(t)$ is increasing on the interval $[0,\\frac{-x_0+y_0-1}{y_0}],$ and since $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0,$ we have $F^{\\prime }(t)\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}],$ which implies that the function $F(t)$ is decreasing on the interval $[0,\\frac{-x_0+y_0-1}{y_0}].$ Therefore, we have $F(t)\\ge F(\\frac{-x_0+y_0-1}{y_0})\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;t\\in [0,\\frac{-x_0+y_0-1}{y_0}],$ which implies that $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace =\\mathcal {P}(R_2).$ Secondly, we prove (3.5).",
"Figure: NO_CAPTIONIn (3.4), if $\\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace =\\mathcal {P}(R_2).$ Let $T=\\lbrace (x,y):|x|+|y|\\le 1\\rbrace $ and $Q=\\lbrace (x,y):\\max \\lbrace |x|,|y|\\rbrace \\le 1\\rbrace .$ Let $R_T$ and $R_Q$ be the origin-symmetric bodies of revolution generated by $T$ and $Q$ , respectively.",
"In (3.4), replacing $R$ , $R_1$ , and $R_2$ , by $R_2$ , $R_T$ , and $R_Q$ , respectively (see (1) of Figure 3.3), we obtain $\\mathcal {P}(R_2)\\ge \\min \\lbrace \\mathcal {P}(R_T),\\mathcal {P}(R_Q)\\rbrace =\\frac{4\\pi ^2}{3}.$ It follows that $\\mathcal {P}(R)\\ge \\mathcal {P}(R_2)\\ge \\frac{4\\pi ^2}{3}.$ In (3.4), if $\\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace =\\mathcal {P}(R_1),$ let $E$ , $F$ be the vertices of $P_1^{\\ast }$ in the second quadrant, where $E$ , $F$ lie on line segments $AD$ and $AB$ , respectively (see (2) of Figure 3.3).",
"Let $P_{DEB}$ be a 1-unconditional polygon satisfying $P_{DEB}\\cap \\lbrace (x,y): x\\le 0,y\\ge 0\\rbrace =\\textrm {conv}\\lbrace E,D,O,B\\rbrace ,$ and let $R_{DEB}$ be an origin-symmetric body of revolution generated by $P_{DEB}$ .",
"In (3.4), replacing $R$ , $R_1$ , and $R_2$ , by ${R_1}^{\\ast }$ , $R_{DEB}$ , and $R_Q$ , respectively (see (3) of Figure 3.3), we have $\\mathcal {P}(R)\\ge \\mathcal {P}(R_1)=\\mathcal {P}({R_1}^{\\ast })\\ge \\min \\lbrace \\mathcal {P}(R_{DEB}),\\mathcal {P}(R_Q)\\rbrace .$ In (3.31), if $\\min \\lbrace \\mathcal {P}(R_{DEB}),\\mathcal {P}(R_Q)\\rbrace =\\mathcal {P}(R_Q),$ we have proved (3.5); if $\\min \\lbrace \\mathcal {P}(R_{DEB}),\\mathcal {P}(R_Q)\\rbrace =\\mathcal {P}(R_{DEB}),$ let ${P_{DEB}}^{\\ast }\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace =\\textrm {conv}\\lbrace G,D,O,B\\rbrace ,$ where $G$ lies on the line segment $AB$ , which is a vertex of ${P_{DEB}}^{\\ast }$ (see (4) of Figure 3.3).",
"In (3.4), replacing $R$ , $R_1$ , and $R_2$ , by ${R_{DEB}}^{\\ast }$ , $R_T$ , and $R_Q$ , respectively, we obtain $\\mathcal {P}({R_{DEB}}^{\\ast })\\ge \\min \\lbrace \\mathcal {P}(R_T),\\mathcal {P}(R_Q)\\rbrace =\\frac{4\\pi ^2}{3}.$ Hence, we have $ \\mathcal {P}(R)\\ge \\mathcal {P}(R_1)\\ge \\mathcal {P}(R_{DEB})\\ge \\frac{4\\pi ^2}{3}.$ Figure: NO_CAPTIONLemma 3.5 Let $P$ be a 1-unconditional polygon in the coordinate plane $XOY$ satisfying $P\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace A_1, A_2, \\cdots , A_{n-1}, D, O, B\\rbrace ,$ where $A_1$ lies on the line segment $AB$ , $A_2,\\cdots , A_{n-1}\\in {\\rm int} \\triangle ABD$ , and the slopes of lines $OA_i$ ($i=1,\\cdots ,n-1$ ) are increasing on $i$ , $R$ the origin-symmetric body of revolution generated by $P$ .",
"Then $ \\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace ,$ where $R_1$ and $R_2$ are origin-symmetric bodies of revolution generated by 1-unconditional polygons $P_1$ and $P_2$ satisfying $P_1\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace A_2, A_3,\\cdots , A_{n-1}, D, O, B\\rbrace $ and $P_2\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace ={\\rm conv}\\lbrace C, A_3,\\cdots , A_{n-1}, D, O, B\\rbrace ,$ respectively, where $C$ is the point of intersection between two lines $A_2A_3$ and $AB$ .",
"In Figure 3.4, let $A_1=(-t,1)$ and $A_2=(x_0,y_0)$ .",
"Let the slope of the line $A_3A_2$ be $k$ , then $\\frac{1-y_0}{-x_0}<k<\\frac{y_0}{x_0+1}$ and the equation of the line $A_3A_2$ is $y-y_0=k(x-x_0).$ In (3.35), let $y=1$ , we get the abscissa of $C$ $x_C=x_0+\\frac{1-y_0}{k}.$ Let $E,\\;F$ and $B$ be the vertices of $P^{\\ast }$ satisfying $BE\\bot OA_1$ and $EF\\bot OA_2$ .",
"Let $I$ be the point of intersection between two lines $EF$ and $AB$ .",
"We have $BE:~y=tx+1$ and $EF:~y=-\\frac{x_0}{y_0}x+\\frac{1}{y_0}.$ Then, we get $I=(\\frac{1-y_0}{x_0},1)$ and $E=(\\frac{1-y_0}{t y_0+x_0},\\frac{t+x_0}{t y_0+x_0}).$ Let $F(t)=\\frac{1}{2}V(R)\\frac{1}{2}V(R^{\\ast })=\\frac{1}{4}\\mathcal {P}(R),$ which is a function of the variable $t$ , where $0\\le t\\le -x_C=\\frac{-x_0k+y_0-1}{k}.$ Our proof has three steps.",
"First step.",
"Calculate $F^{\\prime }(t)$ and $F^{\\prime \\prime }(t)$ .",
"Let $V=\\frac{1}{2}V(R_1)$ and $V^{0}=\\frac{1}{2}V({R_1}^{\\ast })$ , then we obtain $F(t)&=&\\left(V+\\frac{\\pi }{3}(2-y_0-y_0^2)t\\right)\\nonumber \\\\&~&\\times \\left(V^{0}-\\frac{\\pi }{3}\\frac{y_0-1}{x_0}\\left(2-\\frac{t+x_0}{ty_0+x_0}-\\left(\\frac{t+x_0}{t y_0+x_0}\\right)^2\\right)\\right).",
"\\nonumber \\\\$ Therefore, we have $F^{\\prime }(t)&=&\\frac{\\pi }{3}\\frac{(2-y_0-y_0^2)(\\Phi _1t^3+\\Phi _2t^2+\\Phi _3t+\\Phi _4)}{(y_0t+x_0)^3},$ where $&~&\\Phi _1=y_0[-\\frac{\\pi }{3}\\frac{(1-y_0)^2(2y_0+1)}{x_0}+V^0y_0^2],\\nonumber \\\\&~&\\Phi _2=-\\pi (1-y_0)^2(2y_0+1)+3V^0x_0y_0^2,\\nonumber \\\\&~&\\Phi _3=-2\\pi (1-y_0)^2x_0+3V^0x_0^2y_0+(y_0-1)V,\\nonumber \\\\&~&\\Phi _4=V^0x_0^3-\\frac{3x_0(1-y_0)V}{y_0+2}.$ Thus, we have $F^{\\prime \\prime }(t)&=&\\frac{2\\pi }{3}\\frac{(1-y_0)^2}{(t y_0+x_0)^4}J(t),$ where $J(t)&=&(y_0+2)[Vy_0+\\pi x_0(y_0-1)] t\\nonumber \\\\&&+x_0[V(4y_0-1)+\\pi x_0(y_0^2+y_0-2)].$ Second step.",
"We prove that ${\\rm (}i{\\rm )}\\;\\; F^{\\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {or}\\;\\;F^{\\prime \\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for }(x_0,y_0)\\in \\mathcal {D}_1$ and ${\\rm (}ii{\\rm )}\\;\\; F^{\\prime \\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\; (x_0,y_0)\\in \\mathcal {D}_2,$ where $\\mathcal {D}_1$ and $\\mathcal {D}_2$ have been given in (3.18) and (3.19).",
"By (3.39) and (3.40), let $t_0=\\frac{-x_0k+y_0-1}{k},$ we have $F^{\\prime }(t_0)=\\frac{\\pi }{3}(\\Upsilon _1V^0+\\Upsilon _2V+\\Upsilon _3),$ where $&~&\\Upsilon _1=(1-y_0)(y_0+2),\\nonumber \\\\&~&\\Upsilon _2=\\frac{k^2(-x_0k+y_0+2)}{(x_0k-y_0)^3},\\nonumber \\\\&~&\\Upsilon _3=-\\frac{\\pi }{3}\\frac{y_0+2}{x_0(x_0k-y_0)^3}[k^3x_0^3(y_0-1)(-2y_0+3)+3k^2x_0^2y_0(y_0-1)(2y_0-3)\\nonumber \\\\&~&\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;+3kx_0(1-y_0)^3(2y_0+1)+y_0(2y_0+1)(y_0-1)^3].$ Since $k>0$ , $x_0<0$ and $0<y_0<1$ , we have that $\\Upsilon _1\\ge 0$ and $\\Upsilon _2\\le 0$ , thus, as $V$ increases and $V^0$ decreases, $F^{\\prime }(t_0)$ decreases.",
"Let $P_0$ be a 1-unconditional polygon satisfying $P_0\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace =\\textrm {conv}\\lbrace A_2, D, O, B\\rbrace $ and $R_0$ be an origin-symmetric body of revolution generated by $P_0$ .",
"Let $V_0=\\frac{1}{2}V(R_0)$ and ${V_0}^{\\ast }=\\frac{1}{2}V(R_0^{\\ast })$ .",
"In (3.38), let $V=V_0$ and $V^{0}={V_0}^{\\ast }$ , we get a function $F_0(t)$ , which is the same function as $F(t)$ in Lemma 3.4.",
"Since $V\\ge V_0$ and $V^0\\le V_0^{\\ast }$ , we have $F^{\\prime }(\\frac{-x_0k+y_0-1}{k})\\le F_0^{\\prime }(\\frac{-x_0k+y_0-1}{k}).$ Since $\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1},$ we have $0\\le \\frac{-x_0k+y_0-1}{k}\\le \\frac{-x_0+y_0-1}{y_0}.$ In (3.45), let $k=\\frac{y_0}{x_0+1},$ we have $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le F_0^{\\prime }(\\frac{-x_0+y_0-1}{y_0}).$ From Lemma 3.4, we have $F_0^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;(x_0,y_0)\\in \\mathcal {D}_1,$ hence $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;(x_0,y_0)\\in \\mathcal {D}_1.$ If $F^{\\prime \\prime }(t_0)> 0$ , by (3.41), $J(t_0)>0$ , since $x_0<0$ and $0\\le y_0\\le 1$ , $J(t)$ is an increasing linear function, thus $J(t)>0$ for $t\\ge t_0$ , which implies $F^{\\prime \\prime }(t)> 0$ for $t\\ge t_0$ .",
"Thus $F^{\\prime }(t)$ is increasing for $t\\ge t_0$ .",
"Since $F^{\\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0,$ we have $F^{\\prime }(t_0)\\le 0$ .",
"Therefore we have proved (i).",
"Next we prove (ii).",
"Let $G$ be the point of intersection between two lines $AD$ and $A_2A_3$ , then $G=(-1, y_0-k(x_0+1))$ .",
"Let $P_M$ be a 1-unconditional polygon satisfying $P_M\\cap \\lbrace (x,y): x\\le 0,\\;y\\ge 0\\rbrace =\\textrm {conv}\\lbrace A_2, G, D, O, B\\rbrace $ and $R_M$ an origin-symmetric body of revolution generated by $P_M$ .",
"From Lemma 2.1, we have that $\\frac{1}{2}V(R_M)&=&\\frac{\\pi }{3}(x_0+1)[(y_0-k(x_0+1))^2+(y_0-k(x_0+1))y_0+y_0^2]\\nonumber \\\\&~& +\\frac{\\pi }{3}(-x_0)(y_0^2+y_0+1).$ In (3.42), let $V=\\frac{1}{2}V(R_M)$ and $t=\\frac{-x_0k+y_0-1}{k},$ we get a function of the variable $k$ $L(k)&=&\\frac{\\Theta _1k^3+\\Theta _2k^2+\\Theta _3k+\\Theta _4}{k},$ where $&~&\\Theta _1=-\\frac{\\pi }{3}x_0(x_0+1)^3(y_0-1)^2,\\nonumber \\\\&~&\\Theta _2=\\frac{\\pi }{3}(x_0+1)^2y_0(y_0-1)(4x_0y_0-x_0+y_0+2),\\nonumber \\\\&~&\\Theta _3=\\frac{\\pi }{3}(y_0-1)(-5x_0^2y_0^3-9x_0y_0^3-3x_0^2y_0^2-9x_0y_0^2-x_0^2-3y_0^3-6y_0^2),\\nonumber \\\\&~&\\Theta _4=\\frac{\\pi }{3}(y_0-1)(y_0+2)(2x_0y_0^3+3y_0^3-x_0y_0^2+2x_0y_0-3x_0).$ Let $L_1(k)&=&\\Theta _1k^3+\\Theta _2k^2+\\Theta _3k+\\Theta _4.$ Since $k>0$ , to prove $L(k)\\le 0$ , it suffices to prove $L_1(k)\\le 0$ .",
"In the following, we prove $L_1(k)\\le 0$ for $\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}.$ By (3.52), we have $L_1^{\\prime \\prime }(k)&=&6\\Theta _1k+2\\Theta _2.$ Since $L_1^{\\prime \\prime }(\\frac{y_0}{x_0+1})=\\frac{2\\pi }{3}(x_0+1)^3y_0(y_0-1)(y_0+2)\\le 0$ and $\\Theta _1=-\\frac{\\pi }{3}x_0(x_0+1)^3(y_0-1)^2>0,$ then $L_1^{\\prime \\prime }(k)\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}.$ Hence, the function $L_1^{\\prime }(k)$ is decreasing on the interval $[\\frac{1-y_0}{-x_0}, \\frac{y_0}{x_0+1}].$ By (3.52), we have $L_1^{\\prime }(k)&=&3\\Theta _1k^2+2\\Theta _2k+\\Theta _3.$ From (3.54), we have that $L_1^{\\prime }(\\frac{y_0}{x_0+1})&=&\\frac{\\pi }{3}(1-y_0)[x_0^2(2y_0^2+1)+x_0(2y_0^3+4y_0^2)+y_0^3+2y_0^2]\\nonumber \\\\&=&\\frac{\\pi }{3}(1-y_0)\\left[(2y_0^2+1)\\left(x_0+\\frac{y_0^3+2y_0^2}{2y_0^2+1}\\right)^2+\\frac{y_0^2(y_0+2)(1-y_0^3)}{2y_0^2+1}\\right]\\nonumber \\\\&\\ge & 0.$ Therefore $L_1^{\\prime }(k)\\ge 0\\;\\; \\textrm {for}\\;\\;\\textrm {any}\\;\\;\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}.$ It follows that the function $L_1(k)$ is increasing on the interval $[\\frac{1-y_0}{-x_0}, \\frac{y_0}{x_0+1}].$ When $k=\\frac{y_0}{x_0+1},$ we have $R_M=R_0$ and $\\frac{-x_0k+y_0-1}{k}=\\frac{-x_0+y_0-1}{y_0}.$ In Lemma 3.4, for $R=R_0$ , we had proved $F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Hence, $L_1(\\frac{y_0}{x_0+1})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2,$ which implies that $L_1(k)\\le 0$ for any $\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}\\;\\;\\textrm {when}\\; \\;(x_0,y_0)\\in \\mathcal {D}_2.$ It follows that, for $R=R_M$ , $F^{\\prime \\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\;\\textrm {any}\\;\\;\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}$ when $(x_0,y_0)\\in \\mathcal {D}_2.$ In Lemma 3.4, for $R=R_0$ , we know that $F^{\\prime \\prime }(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2,$ from (3.41), which implies that $J(\\frac{-x_0+y_0-1}{y_0})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Since $J(t)$ is an increasing linear function and $\\frac{-x_0k+y_0-1}{k}\\le \\frac{-x_0+y_0-1}{y_0}\\;\\;\\textrm { for}\\;\\;k<\\frac{y_0}{x_0+1},$ we have $J(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2,$ which implies, for $R=R_0$ , that $F^{\\prime \\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0$ for any $\\frac{1-y_0}{-x_0}\\le k\\le \\frac{y_0}{x_0+1}\\;\\;\\textrm {and}\\;\\; (x_0,y_0)\\in \\mathcal {D}_2.$ Therefore, for $V=V(R_0)\\;\\;\\textrm {or}\\;\\;V=V(R_M),$ we have $J(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Since $J(t)=[(y_0+2)y_0t+x_0(4y_0-1)]V+[\\pi x_0(y_0-1)(y_0+2)t-\\pi x_0^2(2-y_0-y_0^2)],\\nonumber \\\\$ which can be considered as a linear function of the variable $V$ , and $V(R_0)<V(R)<V(R_M),$ we have, for any $V=V(R)$ , that $J(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ It follows that $F^{\\prime \\prime }(\\frac{-x_0k+y_0-1}{k})\\le 0\\;\\;\\textrm {for}\\;\\;(x_0,y_0)\\in \\mathcal {D}_2.$ Third step.",
"We prove $\\mathcal {P}(R)\\ge \\min \\lbrace \\mathcal {P}(R_1),\\mathcal {P}(R_2)\\rbrace .$ We omit the proof of this step which is similar to the proof of third step in Lemma 3.4.",
"Lemma 3.6 For any a 1-unconditional polygon $P\\subset [-1,1]^2$ in the coordinate plane $XOY$ satisfying $B, D\\in P$ , let $R$ be an origin-symmetric body of revolution generated by $P$ .",
"Then $\\mathcal {P}(R)\\ge \\frac{4\\pi ^2}{3},$ with equality if and only if $R$ is a cylinder or bicone.",
"Figure: NO_CAPTIONLet $A_1, A_2, \\cdots , A_n$ be the vertices of $P$ contained in the domain $\\lbrace (x,y): x\\le 0, y\\ge 0\\rbrace $ and the slopes of lines $OA_i$ ($i=1,\\cdots n$ ) are increasing on $i$ .",
"Without loss of generality, suppose that the vertex $A_n$ coincides with point $D$ .",
"The vertex $A_1$ satisfies the following two cases: (i) $A_1$ coincides with the point $B$ ; (ii) $A_1$ does not coincide with the point $B$ , but lies on the line segment $BC$ (C is the point of intersection between two lines $A_2A_3$ and $AB$ ).",
"If $R$ satisfies the case (ii), from the Lemma 3.5, we obtain an origin-symmetric body of revolution $R_1$ with smaller Mahler volume than $R$ and its generating domain $P_1$ has fewer vertices than $P$ .",
"If $R$ satisfies the case (i), then its polar body $R^{\\ast }$ satisfies the case (ii).",
"Since $\\mathcal {P}(R)=\\mathcal {P}(R^{\\ast })$ and $P$ has the same number of vertices as $P^{\\ast }$ , from the Lemma 3.5, we can also obtain an origin-symmetric body of revolution $R_1$ with smaller Mahler volume than $R$ and its generating domain $P_1$ has fewer vertices than $P$ .",
"From the above discuss and the proof of (3.5), let $R_0=R$ , we can get a sequence of origin-symmetric bodies of revolution $\\lbrace R_0, R_1, R_2\\cdots , R_N\\rbrace ,$ where $N$ is a natural number depending on the number of vertices of $P$ , satisfying $\\mathcal {P}(R_{i+1})\\le \\mathcal {P}(R_i)$ ($i=0,1,\\cdots , N-1$ ) and $R_N$ is a cylinder or bicone.",
"Therefore, we have $\\mathcal {P}(R)\\ge \\frac{4\\pi ^2}{3},$ with equality if and only if $R$ is a cylinder or bicone.",
"Theorem 3.7 For any origin-symmetric body of revolution $K$ in $\\mathbb {R}^3$ , we have $\\mathcal {P}(K)\\ge \\frac{4\\pi ^2}{3},$ with equality if and only if $K$ is a cylinder or bicone.",
"By Remark 3, without loss of generality, suppose that the generating domain $P$ of $K$ is contained in the square $[-1,1]^2$ and $B, D\\in P$ .",
"Since a convex body can be approximated by a polytope in the sense of the Hausdorff metric (see Theorem 1.8.13 in [29]), hence, for $P$ and any $\\varepsilon >0$ , there is a 1-unconditional polygon $P_{\\varepsilon }$ with $\\delta (P,P_{\\varepsilon })\\le \\varepsilon $ .",
"Let $R_{\\varepsilon }$ be an origin-symmetric body of revolution generated by $P_{\\varepsilon }$ , then $\\delta (K, R_{\\varepsilon })\\le \\varepsilon .$ Thus, there exists a sequence of origin-symmetric bodies of revolution $(R_i)_{i\\in \\mathbb {N}}$ satisfying $\\lim _{i\\rightarrow \\infty }\\delta (R_i, K)=0.$ Since $\\mathcal {P}(K)$ is continuous in the sense of the Hausdorff metric, applying Lemma 3.6, we have $\\mathcal {P}(K)\\ge \\frac{4\\pi ^2}{3},$ with equality if and only if $K$ is a cylinder or bicone.",
"In the following, we will restate and prove Theorem 1.2 and 1.3.",
"Theorem 3.8 Let $f(x)$ be a concave, even and nonnegative function defined on $[-a,a]$ , $a>0$ , and for $x^{\\prime }\\in [-\\frac{1}{a},\\frac{1}{a}]$ define $f^{\\ast }(x^{\\prime })=\\inf _{x\\in [-a,a]}\\frac{1-x^{\\prime }x}{f(x)}.$ Then $\\left(\\int _{-a}^{a} (f(x))^2 dx\\right)\\left(\\int _{-\\frac{1}{a}}^{\\frac{1}{a}} (f^{\\ast }(x^{\\prime }))^2dx^{\\prime }\\right)\\ge \\frac{4}{3},$ with equality if and if $f(x)=f(0)$ or $f^{\\ast }(x^{\\prime })=1/f(0)$ .",
"Let $R$ and $R^{\\prime }$ be origin-symmetric bodies of revolution generated by $f(x)$ and $f^{\\ast }(x^{\\prime })$ , respectively, then their generating domains are $D=\\lbrace (x,y): -a\\le x\\le a, |y|\\le f(x)\\rbrace $ and $D^{\\prime }=\\lbrace (x^{\\prime },y^{\\prime }): -\\frac{1}{a}\\le x^{\\prime }\\le \\frac{1}{a},|y^{\\prime }|\\le f^{\\ast }(x^{\\prime })\\rbrace ,$ respectively.",
"Next, we prove $D^{\\prime }=D^{\\ast }$ .",
"For $(x^{\\prime },y^{\\prime })\\in D^{\\prime }$ and $(x,y)\\in D$ , we have $(x^{\\prime },y^{\\prime })\\cdot (x,y) =x^{\\prime }x+y^{\\prime }y\\le x^{\\prime }x+f^{\\ast }(x^{\\prime })f(x) \\le x^{\\prime }x+\\frac{1-x^{\\prime }x}{f(x)}f(x)=1,$ which implies $(x^{\\prime },y^{\\prime })\\in D^{\\ast }$ .",
"If $(x^{\\prime },y^{\\prime })\\notin D^{\\prime }$ , then either $|x^{\\prime }|>\\frac{1}{a}$ or $|x^{\\prime }|\\le \\frac{1}{a}$ and $|y^{\\prime }|>f^{\\ast }(x^{\\prime })$ .",
"If $x^{\\prime }>\\frac{1}{a}$ (or $x^{\\prime }<-\\frac{1}{a}$ ), then for $(a,0)\\in D$ (or $(-a,0)\\in D$ ), we have $(x^{\\prime },y^{\\prime })\\cdot (a,0)>1\\;\\;(\\textrm {or}\\;\\;(x^{\\prime },y^{\\prime })\\cdot (-a,0)>1),$ which implies $(x^{\\prime },y^{\\prime })\\notin D^{\\ast }$ .",
"If $|x^{\\prime }|\\le \\frac{1}{a}$ and $y^{\\prime }>f^{\\ast }(x^{\\prime })$ (or $y^{\\prime }<-f^{\\ast }(x^{\\prime })$ ), let $f^{\\ast }(x^{\\prime })=\\frac{1-x^{\\prime }x_0}{f(x_0)},$ then for $(x_0, f(x_0))\\in D$ (or $(x_0, -f(x_0))\\in D$ ), we have $(x^{\\prime },y^{\\prime })\\cdot (x_0, f(x_0))>x^{\\prime }x_0+f^{\\ast }(x^{\\prime })f(x_0)=1$ $(\\textrm {or}\\;\\;(x^{\\prime },y^{\\prime })\\cdot (x_0, -f(x_0))>x^{\\prime }x_0+f^{\\ast }(x^{\\prime })f(x_0)=1),$ which implies $(x^{\\prime },y^{\\prime })\\notin D^{\\ast }$ .",
"Hence, we have $D^{\\prime }=D^{\\ast }$ .",
"By Lemma 3.2, we get $R^{\\prime }=R^{\\ast }$ .",
"By Theorem 3.7, we have $\\int _{-a}^{a}(f(x))^2dx\\int _{-\\frac{1}{a}}^{\\frac{1}{a}}(f^{\\ast }(x^{\\prime }))^2dx^{\\prime }=\\frac{1}{\\pi ^2}V(R)V(R^{\\prime })=\\frac{1}{\\pi ^2}\\mathcal {P}(R)\\ge \\frac{4}{3},$ with equality if and if $f(x)=f(0)$ or $f^{\\ast }(x^{\\prime })=1/f(0)$ .",
"By Theorem 3.8, we prove that among parallel sections homothety bodies in $\\mathbb {R}^3$ , 3-cubes have the minimal Mahler volume.",
"Theorem 3.9 For any parallel sections homothety body $K$ in $\\mathbb {R}^3$ , we have $\\mathcal {P}(K)\\ge \\frac{4^3}{3!",
"},$ with equality if and only if $K$ is a 3-cube or octahedron.",
"Let $K=\\bigcup _{x\\in [-a,a]}\\lbrace f(x)C+xv\\rbrace ,$ where $f(x)$ is its generating function and $C$ is homothetic section.",
"Next, for $K^{\\prime }=\\bigcup _{x^{\\prime }\\in [-\\frac{1}{a},\\frac{1}{a}]}\\lbrace f^{\\ast }(x^{\\prime })C^{\\ast }+x^{\\prime }v\\rbrace ,$ where $f^{\\ast }(x^{\\prime })$ is given in (3.62), we prove $K^{\\prime }=K^{\\ast }$ .",
"For any $(x^{\\prime },y^{\\prime },z^{\\prime })\\in K^{\\prime }\\;\\;\\textrm {and}\\;\\; (x,y,z)\\in K,$ we have $(0,y^{\\prime },z^{\\prime })\\in f^{\\ast }(x^{\\prime })C^{\\ast }\\;\\;\\textrm {and}\\;\\;(0,y,z)\\in f(x)C.$ Hence, we have $(0,y^{\\prime },z^{\\prime })\\cdot (0,y,z)\\le f^{\\ast }(x^{\\prime })f(x)\\le \\frac{1-x^{\\prime }x}{f(x)}f(x)=1-x^{\\prime }x.$ It follows that $(x^{\\prime },y^{\\prime },z^{\\prime })\\cdot (x,y,z)=x^{\\prime }x+(0,y^{\\prime },z^{\\prime })\\cdot (0,y,z)\\le 1,$ which implies that $(x^{\\prime },y^{\\prime },z^{\\prime })\\in K^{\\ast }$ .",
"If $(x^{\\prime },y^{\\prime },z^{\\prime })\\notin K^{\\prime }$ , then either $|x^{\\prime }|>\\frac{1}{a}$ or $|x^{\\prime }|\\le \\frac{1}{a}$ and $(0,y^{\\prime },z^{\\prime })\\notin f^{\\ast }(x^{\\prime })C^{\\ast }$ .",
"If $x>\\frac{1}{a}$ (or $x<-\\frac{1}{a}$ ), then for $(a,0,0)\\in K$ (or $(-a,0,0)\\in K$ ), we have $(x^{\\prime },y^{\\prime },z^{\\prime })\\cdot (a,0,0)>1\\;\\;(\\textrm {or}\\;\\;(x^{\\prime },y^{\\prime },z^{\\prime })\\cdot (-a,0,0)>1),$ which implies that $(x^{\\prime },y^{\\prime },z^{\\prime })\\notin K^{\\ast }$ .",
"If $|x^{\\prime }|\\le \\frac{1}{a}$ and $(0,y^{\\prime },z^{\\prime })\\notin f^{\\ast }(x^{\\prime })C^{\\ast }$ , there exists $(0,y,z)\\in C$ such that $(0,y,z)\\cdot (0,y^{\\prime },z^{\\prime })>f^{\\ast }(x^{\\prime }).$ Let $f^{\\ast }(x^{\\prime })=\\frac{1-x^{\\prime }x_0}{f(x_0)}.$ For $(x_0,f(x_0)y,f(x_0)z)\\in K$ we have $&&(x^{\\prime },y^{\\prime },z^{\\prime })\\cdot (x_0,f(x_0)y,f(x_0)z)\\nonumber \\\\&=&x^{\\prime }x_0+f(x_0)(0,y,z)\\cdot (0,y^{\\prime },z^{\\prime })\\nonumber \\\\&>&x^{\\prime }x_0+f(x_0)f^{\\ast }(x^{\\prime })\\nonumber \\\\&=&x^{\\prime }x_0+f(x_0)\\frac{1-x^{\\prime }x_0}{f(x_0)}\\nonumber \\\\&=&1,$ which implies that $(x^{\\prime },y^{\\prime },z^{\\prime })\\notin K^{\\ast }$ .",
"Hence, we have $K^{\\prime }=K^{\\ast }$ .",
"Therefore, we obtain $\\mathcal {P}(K) &=&V(K)V(K^{\\prime })\\nonumber \\\\&=&\\mathcal {P}(C)\\int _{-a}^{a}(f(x))^2dx\\int _{-\\frac{1}{a}}^{\\frac{1}{a}}(f^{\\ast }(x^{\\prime }))^2dx^{\\prime }\\nonumber \\\\&\\ge & \\frac{4^2}{2!}\\frac{4}{3}=\\frac{4^3}{3!",
"},$ with equality if and only if $K$ is a 3-cube or octahedron."
]
] | 1403.0322 |
[
[
"SU(5)xSU(5)' unification and D2 parity: Model for composite leptons"
],
[
"Abstract We study a grand unified SU(5)xSU(5)' model supplemented by D2 parity.",
"The D2 greatly reduces the number of parameters and is important for phenomenology.",
"The model, we present, has various novel and interesting properties.",
"Because of the specific pattern of grand unification symmetry breaking and emerged strong dynamics at low energies, the Standard Model leptons, along with right-handed/sterile neutrinos, come out as composite states.",
"The generation of the charged fermion and neutrino masses are studied within the considered scenario.",
"Moreover, the issues of gauge coupling unification and nucleon stability are investigated in details.",
"Various phenomenological implications are also discussed."
],
[
"Introduction",
"The Standard Model (SM) of electroweak interactions has been a very successful theory for decades.",
"The triumph of this celebrated model occurred thanks to the Higgs boson discovery [1] at CERN's Large Hadron Collider.",
"In spite of this success, several phenomenological and theoretical issues motivate one to think of some physics beyond the SM.",
"Because of renormalization running, the self-coupling of the SM Higgs boson becomes negative at scale near $\\sim 10^{10}$ GeV [2], [3] (with the Higgs mass$\\simeq 126$ GeV), causing vacuum instability (becoming more severe within the inflationary setup; see the discussion in Sec.",
").",
"Moreover, the SM fails to accommodate atmospheric and solar neutrino data [4].",
"The renormalizable part of the SM interactions render neutrinos to be massless.",
"Also, Planck scale suppressed $d=5$ lepton number violating operators do not generate neutrino mass with desirable magnitude.",
"These are already strong motivations to think about the existence of some new physics between electroweak (EW) and Planck scales.",
"Among various extensions of the SM, the grand unification (GUT) [5], [6] is a leading candidate.",
"Unifying all gauge interactions in a single group, at high energies one can deal with a single unified gauge coupling.",
"At the same time, quantization of quark and lepton charges occurs by embedding all fermionic states in unified GUT multiplets.",
"The striking prediction of the grand unified theory is the baryon number violating nucleon decay.",
"This opens the prospect for probing the nature at very short distances.",
"GUTs based on $SO(10)$ symmetry [7] [which includes $SU(2)_L\\times SU(2)_R\\times SU(4)_c$ symmetry [5] as a maximal subgroup] involve right-handed neutrinos (RHNs), which provide a simple and elegant way for neutrino mass generation via the seesaw mechanism [8].",
"In spite of these salient futures, GUT model building encounters numerous problems and phenomenological difficulties.",
"With single scale breaking, i.e., with no new interactions and/or intermediate states between EW and GUT scales, grand unified theories [such as minimal $SU(5)$ and $SO(10)$ ] do not lead to successful gauge coupling unification.",
"Besides this, building GUT with the realistic fermion sector, understanding the GUT symmetry breaking pattern, and avoiding too rapid nucleon decay remain a great challenge.",
"Motivated by these issues, we consider $SU(5)\\times SU(5)^{\\prime }$ GUT augmented with $D_2$ parity (exchange symmetry).",
"The latter, relating two $SU(5)$ gauge groups, reduces the number of parameters, and at and above the GUT scale, one deals with single gauge coupling.",
"The grand unified theories with $SU(5)\\times SU(5)^{\\prime }$ symmetry, considered in earlier works [9], in which at least one gauge factor of the SM symmetry emerges as a diagonal subgroup, have been proven to be very successful for building models with realistic phenomenology.",
"However, to our knowledge, in such constructions the $D_2$ parity has not been applied before.In the second citation of Ref.",
"[9], the exchange symmetry was considered; however, some terms violating this symmetry have been included.",
"The reason could be the prejudice of remaining with extra unwanted chiral matter states in the spectrum.",
"However, within our model due to specific construction, this does not happen, and below the few-TeV scale, surviving states are just of the Standard Model.",
"The $D_2$ parity also plays a crucial role for phenomenology and has interesting implications.",
"By the specific pattern of the $SU(5)\\times SU(5)^{\\prime }$ symmetry breaking and spectroscopy, the successful gauge coupling unification is obtained.",
"Interestingly, within the considered framework, the SM leptons emerge as a composite states, while the quarks are fundamental objects.",
"Lepton mass generation occurs by a new mechanism, finding natural realization within a presented model.",
"Since leptons and quarks have different footing, there is no problem of their mass degeneracy (unlike the minimal SO(10) and $SU(5)$ grand unified theories, which require some extensions [10]).",
"Moreover, along with composite SM leptons, the model involves three families of composite SM singlet fermionic states, which may be identified with RHNs or sterile neutrinos.",
"Thus, the neutrino masses can be generated.",
"In addition, we show that, due to the specific fermion pattern, $d=6$ nucleon decay can be adequately suppressed within the considered model.",
"The model also has various interesting properties and implications, which we also discuss.",
"Since two $SU(5)$ groups will be related by $D_2$ parity, initial states will be doubled, i.e., will be introduced in twins.",
"Because of this, we refer to the proposed $SU(5)\\times SU(5)^{\\prime }\\times D_2$ model as twinification.",
"The paper is organized as follows.",
"In the next section, first we introduce the $SU(5)\\times SU(5)^{\\prime }\\times D_2$ GUT and discuss the symmetry breaking pattern.",
"Then, we present the spectrum of bosonic states.",
"In Sec.",
", considering the fermion sector, we give transformation properties of the GUT matter multiplets under $D_2$ parity and build the Yukawa interaction Lagrangian.",
"The latter is responsible for the generation of quark masses and CKM matrix elements.",
"Because of the specific pattern of the symmetry breaking and strong $SU(3)^{\\prime }$ [originating from $SU(5)^{\\prime }$ gauge symmetry] dynamics, the SM leptons emerge as composite objects.",
"We present a novel mechanism for composite lepton mass generation.",
"Together with the SM leptons, three families of right-handed/sterile neutrinos are composite.",
"We also discuss the neutrino mass generation within our scenario.",
"In Sec.",
"we give details of gauge coupling unification.",
"The issue of nucleon stability is addressed in Sec.",
".",
"Although the GUT scale, within our model, comes out to be relatively low ($\\simeq \\!5\\cdot 10^{11}$ GeV), we show that the $d=6$ baryon number violating operators can be adequately suppressed.",
"This happens to be possible due to the specific pattern of the fermion sector we are suggesting.",
"In Sec.",
"we summarize and discuss various phenomenological constraints and possible implications of the considered scenario.",
"We also emphasize the model's peculiarities and novelties, which open broad prospects for further investigations.",
"Appendix discusses details related to the compositeness and anomaly matching conditions.",
"In Appendix we give details of the gauge coupling unification.",
"In particular, the renormalization group (RG) equations and $b$ factors at various energy intervals are presented.",
"The short-range renormalization of baryon number violating $d=6$ operators is also performed."
],
[
"$SU(5)\\times SU(5)^{\\prime }\\times D_2$ Twinification",
"Let us consider the theory based on $SU(5)\\times SU(5)^{\\prime }$ gauge symmetry.",
"Besides this symmetry, we postulate discrete parity $D_2$ , which exchanges two $SU(5)$ 's.",
"Therefore, the symmetry of the model is $G_{GUT}=SU(5)\\times SU(5)^{\\prime }\\times D_2~.$ As noted, the action of $D_2$ interchanges the gauge fields (in adjoint representations) of $SU(5)$ and $SU(5)^{\\prime }$ , $D_2~:~~~~(A_{\\mu })^a_b\\rightarrow (A_{\\mu }^{\\prime })^{a^{\\prime }}_{b^{\\prime }}~,~~~(A_{\\mu }^{\\prime })^{a^{\\prime }}_{b^{\\prime }}\\rightarrow (A_{\\mu })^a_b~,$ with $(A_{\\mu })^{a}_{b}=\\frac{1}{2}\\sum _{i=1}^{24}A_{\\mu }^{i}(\\lambda ^i)^{a}_{b}$ and $(A_{\\mu }^{\\prime })^{a^{\\prime }}_{b^{\\prime }}=\\frac{1}{2}\\sum _{i^{\\prime }=1}^{24}A_{\\mu }^{^{\\prime }i^{\\prime }}(\\lambda ^{i^{\\prime }})^{a^{\\prime }}_{b^{\\prime }}$ , where $a,b$ and $a^{\\prime },b^{\\prime }$ denote indices of $SU(5)$ and $SU(5)^{\\prime }$ respectively.",
"The $\\lambda ^i, \\lambda ^{i^{\\prime }}$ are corresponding Gell-Mann matrices.",
"Thanks to the $D_2$ , at and above the GUT scale $M_G$ , we have single gauge coupling ${\\alpha _5}=\\alpha _{5^{\\prime }}~.$ Grand unified theories based on product groups allow us to build simple models with realistic phenomenology [9], [11].",
"In our case, as we show below, the EW part [i.e., $SU(2)_w\\times U(1)_Y$ ] of the SM gauge symmetry will belong to the diagonal subgroup of $SU(5)\\times SU(5)^{\\prime }$ .",
"Potential and symmetry breaking For $G_{GUT}$ symmetry breaking and building realistic phenomenology, we introduce the states $H\\sim (5,1)~,~~~\\Sigma \\sim (24,1)~,~~~~~H^{\\prime }\\sim (1,5)~,~~~\\Sigma ^{\\prime } \\sim (1,24)~,~~~\\Phi \\sim (5, \\bar{5})~,$ where in brackets transformation properties under $SU(5)\\times SU(5)^{\\prime }$ symmetry are indicated.",
"$H$ includes SM Higgs doublet $h$ .",
"The introduction of $H^{\\prime }$ is required by $D_2$ symmetry.",
"By the same reason, two adjoints $\\Sigma $ and $\\Sigma ^{\\prime }$ (needed for GUT symmetry breaking) are introduced.",
"The bifundamental state $\\Phi $ will also serve for desirable symmetry breaking.",
"The action of $D_2$ parity on these fields is $D_2~:~~~~~~H_a \\stackrel{\\rightarrow }{_\\leftarrow } H_{a^{\\prime }}^{\\prime }~,~~~~\\Sigma ^a_b \\stackrel{\\rightarrow }{_\\leftarrow } {\\Sigma ^{\\prime }}^{a^{\\prime }}_{b^{\\prime }}~,~~~~\\Phi _a^{b^{\\prime }}\\stackrel{\\rightarrow }{_\\leftarrow } (\\Phi ^\\dag )_{a^{\\prime }}^b ~,$ where we have made explicit the indices of $SU(5)$ and $SU(5)^{\\prime }$ .",
"With Eqs.",
"(REF ), (REF ) and (REF ) one can easily make sure that the kinetic part $|D_{\\mu }H|^2+|D_{\\mu }H^{\\prime }|^2+\\frac{1}{2}\\,\\textup {tr}(D_{\\mu }\\Sigma )^2+\\frac{1}{2}\\,\\textup {tr}(D_{\\mu }\\Sigma ^{\\prime } )^2+|D_{\\mu }\\Phi |^2$ of the scalar field Lagrangian is invariant.",
"The scalar potential, invariant under $G_{GUT}$ symmetry [of Eq.",
"(REF )] is $V=V_{H\\Sigma }+V_{H^{\\prime }\\Sigma ^{\\prime }}+V_{mix}^{(1)}+V_{\\Phi }+V_{mix}^{(2)}~,$ with $V_{H\\Sigma }=-M_{\\Sigma }^2{\\,\\textup {tr}}\\Sigma ^2+\\lambda _1({\\,\\textup {tr}}\\Sigma ^2)^2+\\lambda _2{\\,\\textup {tr}}\\Sigma ^4+H^\\dag \\left(M_H^2-h_1\\Sigma ^2+h_2{\\,\\textup {tr}}\\Sigma ^2\\right)H+\\lambda _H(H^\\dag H)^2~,$ $V_{H^{\\prime }\\Sigma ^{\\prime }}=-M_{\\Sigma }^2{\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2\\!+\\!\\lambda _1({\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2)^2\\!\\!+\\!\\lambda _2{\\,\\textup {tr}}{\\Sigma ^{\\prime }}^4\\!+\\!",
"{H^{\\prime }}^\\dag \\!\\left(M_H^2\\!-\\!h_1{\\Sigma ^{\\prime }}^2\\!+\\!h_2{\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2\\right)\\!\\!H^{\\prime }\\!+\\!\\lambda _H({H^{\\prime }}^\\dag H^{\\prime })^2~,$ $V_{mix}^{(1)}=\\lambda ({\\,\\textup {tr}}\\Sigma ^2)({\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2)+\\tilde{h} \\left(H^\\dag H{\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2+{H^{\\prime }}^\\dag H^{\\prime }{\\,\\textup {tr}}\\Sigma ^2 \\right)+\\hat{h}(H^\\dag H)({H^{\\prime }}^\\dag H^{\\prime })~,$ $V_{\\Phi }=-M_{\\Phi }^2\\Phi ^\\dag \\Phi +\\lambda _{1\\Phi }\\left(\\Phi ^\\dag \\Phi \\right)^2+\\lambda _{2\\Phi }\\Phi ^\\dag \\Phi \\Phi ^\\dag \\Phi ~,$ $V_{mix}^{(2)}=\\mu (H^\\dag \\Phi H^{\\prime }\\!+\\!H\\Phi ^\\dag {H^{\\prime }}^\\dag )\\!+\\!\\frac{\\lambda _{1H\\Phi }}{\\sqrt{25}}(\\Phi ^\\dag \\Phi )\\left[(H^\\dag H)\\!+\\!",
"({H^{\\prime }}^\\dag H^{\\prime })\\right]\\!+\\!\\frac{\\lambda _{2H\\Phi }}{\\sqrt{10}}\\left(\\!H^\\dag \\Phi \\Phi ^\\dag \\!H \\!+\\!",
"{H^{\\prime }}^\\dag \\Phi ^\\dag \\Phi \\!H^{\\prime } \\!\\right)+$ $\\lambda _{1\\Sigma \\Phi }(\\Phi ^\\dag \\Phi )({\\,\\textup {tr}}\\Sigma ^2+{\\,\\textup {tr}}{\\Sigma ^{\\prime }}^2)-\\lambda _{2\\Sigma \\Phi }(\\Phi ^\\dag \\Sigma ^2\\Phi +\\Phi {\\Sigma ^{\\prime }}^2\\Phi ^\\dag )~.$ To make analysis simpler, we have omitted terms with first powers of $\\Sigma $ and $\\Sigma ^{\\prime }$ (such as $H^\\dag \\Sigma H$ , ${H^{\\prime }}^\\dag \\Sigma ^{\\prime } H^{\\prime }$ , etc.)",
"and also cubic terms of $\\Sigma $ and $\\Sigma ^{\\prime }$ .",
"This simplification can be achieved by $Z_2$ discrete symmetry and will not harm anything.",
"The potential terms and couplings in Eqs.",
"(REF ) and (REF ) allow us to have a desirable and self-consistent pattern of symmetry breaking.",
"First, we will sketch the symmetry breaking pattern.",
"Then, we will analyze the potential and discuss the spectrum of bosonic states.",
"We will stick to several stages of the GUT symmetry breaking.",
"At the first step, the $\\Sigma $ develops the vacuum expectation value (VEV)$\\sim M_G$ with $\\langle \\Sigma \\rangle =v_{\\Sigma }{\\rm Diag}\\left(2, 2, 2, -3, -3\\right)~,~~~~v_{\\Sigma }\\sim M_G~.$ This causes the symmetry breaking: $SU(5)\\stackrel{\\langle \\Sigma \\rangle }{_{\\longrightarrow }} SU(3)\\times SU(2)\\times U(1)\\equiv G_{321} .$ We select VEVs of $\\Sigma ^{\\prime } $ and $\\Phi $ much smaller than $M_G$ .",
"As it will turn out, the phenomenologically preferred scenario is $\\langle \\Sigma ^{\\prime } \\rangle \\sim 4\\cdot 10^{6}$ GeV and $\\langle \\Phi \\rangle \\sim 8\\cdot 10^{4}$ GeV.",
"With $\\langle \\Sigma ^{\\prime } \\rangle =v_{\\Sigma ^{\\prime }}{\\rm Diag}\\left(2, 2, 2, -3, -3\\right)~,$ the breaking $SU(5)^{\\prime }\\stackrel{\\langle \\Sigma ^{\\prime } \\rangle }{_{\\longrightarrow }} SU(3)^{\\prime }\\times SU(2)^{\\prime }\\times U(1)^{\\prime }\\equiv {G_{321}}^{\\prime }$ is achieved.",
"The last stage of the GUT breaking is done by $\\langle \\Phi \\rangle $ with a direction $\\langle \\Phi \\rangle =v_{\\Phi }\\cdot {\\rm Diag }\\left(0, ~0,~ 0,~ 1,~ 1\\right).$ This configuration of $\\langle \\Phi \\rangle $ breaks symmetries $SU(2)\\times U(1)$ [subgroup of $SU(5)$ ] and $SU(2)^{\\prime }\\times U(1)^{\\prime }$ [subgroup of $SU(5)^{\\prime }$ ] to the diagonal symmetry group: $SU(2)\\times U(1)\\times SU(2)^{\\prime }\\times U(1)^{\\prime }\\stackrel{\\langle \\Phi \\rangle }{_{\\longrightarrow }} \\left[ SU(2)\\times U(1)\\right]_{\\rm diag}~.$ As we see, all VEVs preserve $SU(3)$ and $SU(3)^{\\prime }$ groups arising from $SU(5)$ and $SU(5)^{\\prime }$ respectively.",
"However, unbroken $SU(2)_{\\rm diag}$ is coming (as superposition) partly from $SU(2)\\subset SU(5)$ and partly from $SU(2)^{\\prime }\\subset SU(5)^{\\prime }$ .",
"Similar applies to $U(1)_{\\rm diag}$ ; i.e., it is superposition of two Abelian factors: $U(1)\\subset SU(5)$ and $U(1)^{\\prime }\\subset SU(5)^{\\prime }$ .",
"Now, making the identifications $SU(3)\\equiv SU(3)_c~,~~~~~SU(2)_{\\rm diag}\\equiv SU(2)_w~,~~~~~U(1)_{\\rm diag}\\equiv U(1)_Y$ and taking into account Eqs.",
"(REF ), (REF ), and (REF ), we can see that GUT symmetry is broken as: $G_{GUT}\\rightarrow SU(3)_c\\times SU(2)_w\\times U(1)_Y\\times SU(3)^{\\prime }=G_{SM}\\times SU(3)^{\\prime }~,$ where $G_{SM}=SU(3)_c\\times SU(2)_w\\times U(1)_Y$ denotes the SM gauge symmetry.",
"Because of these, at the intermediate scale $\\mu =M_I (\\sim \\langle \\Phi \\rangle )$ , we will have the matching conditions for the gauge couplings, ${\\rm at}~\\mu =M_I:~~~~~~\\frac{1}{g_w^2}=\\frac{1}{g_2^2}+\\frac{1}{{g}_{2^{\\prime }}^2} ~,~~~~~~~\\frac{1}{g_Y^2}=\\frac{1}{g_1^2}+\\frac{1}{{g}_{1^{\\prime }}^2}~,$ where subscripts indicate to which gauge interaction the appropriate coupling corresponds [e.g., $g_{1^{\\prime }}$ is the coupling of $U(1)^{\\prime }$ symmetry, etc.].",
"The extra $SU(3)^{\\prime }$ factor has important and interesting implications, which we discuss below.",
"As was mentioned, while $\\langle \\Sigma \\rangle \\sim M_G$ , the VEVs $\\langle \\Phi \\rangle $ and $\\Sigma ^{\\prime }$ are at intermediate scales $M_I$ and ${M_I}^{\\prime }$ , respectively, $v_{\\Phi }\\sim M_I,~~~~v_{\\Sigma ^{\\prime }}\\sim {M_I}^{\\prime } ~,$ with the hierarchical pattern $M_I\\ll {M_I}^{\\prime }\\ll M_G~.$ Detailed analysis of the whole potential shows that there is true minimum along directions (REF ), (REF ), and (REF ) with $\\langle H\\rangle =\\langle H^{\\prime }\\rangle =0$ .",
"With $\\langle \\Sigma \\rangle \\ne \\langle \\Sigma ^{\\prime } \\rangle $ , the $D_2$ is broken spontaneously.",
"The residual $SU(3)^{\\prime }$ symmetry will play an important role, and the hierarchical pattern of Eq.",
"(REF ) will turn out to be crucial for successful gauge coupling unification (discussed below).",
"The hierarchical pattern (REF ), of the GUT symmetry breaking, makes it simple to minimize the potential and analyze the spectrum.",
"Three extremum conditions, determining $v_{\\Sigma }, v_{\\Sigma ^{\\prime }}$ and $v_{\\Phi }$ along the directions (REF ), (REF ) and (REF ) and obtained from whole potential, are $10(30\\lambda _1+7\\lambda _2)v_{\\Sigma }^2+150\\lambda v_{\\Sigma ^{\\prime }}^2+(10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi })v_{\\Phi }^2=5M_{\\Sigma }^2 ~,$ $150\\lambda v_{\\Sigma }^2+10(30\\lambda _1+7\\lambda _2)v_{\\Sigma ^{\\prime }}^2+(10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi })v_{\\Phi }^2=5M_{\\Sigma }^2~,$ $3(10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi })(v_{\\Sigma }^2+v_{\\Sigma ^{\\prime } }^2)+(4\\lambda _{1\\Phi }+2\\lambda _{2\\Phi })v_{\\Phi }^2=M_{\\Phi }^2~.$ Because of hierarchies (REF ) and (REF ), from the first equation of Eq.",
"(REF ), with a good approximation we obtain $v_{\\Sigma }\\simeq \\frac{M_{\\Sigma }}{\\sqrt{2(30\\lambda _1+7\\lambda _2)}}~.$ Thus, with $2(30\\lambda _1+7\\lambda _2)\\sim 1$ , we should have $M_{\\Sigma }\\approx M_G$ .",
"On the other hand, from the last two equations of Eq.",
"(REF ), we derive $v_{\\Sigma ^{\\prime }}^2\\simeq \\frac{M_{\\Sigma }^2-30\\lambda v_{\\Sigma }^2}{2(30\\lambda _1+7\\lambda _2)}~,~~~v_{\\Phi }^2=\\frac{M_{\\Phi }^2-3(10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi })(v_{\\Sigma }^2+v_{\\Sigma ^{\\prime }}^2)}{4\\lambda _{1\\Phi }+2\\lambda _{2\\Phi }}~.$ To obtain the scales $M_I$ and ${M_I}^{\\prime }$ , according to Eqs.",
"(REF ) and (REF ), we have to arrange (by price of tunings) $M_{\\Sigma }^2-30\\lambda v_{\\Sigma }^2\\approx ({M_I}^{\\prime })^2$ and $M_{\\Phi }^2-3(10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi })(v_{\\Sigma }^2+v_{\\Sigma ^{\\prime }}^2)\\approx M_I^2$ [with $(4\\lambda _{1\\Phi }+2\\lambda _{2\\Phi })\\sim 1$ ].",
"The Spectrum At the first stage of symmetry breaking, the $(X,Y)$ gauge bosons [of $SU(5)$ ] obtain GUT scale masses.",
"They absorb appropriate states (with quantum numbers of leptoquarks) from the adjoint scalar $\\Sigma $ .",
"The remaining physical fragments $(\\Sigma _8, \\Sigma _3, \\Sigma _1)$ [the $SU(3)$ octet, $SU(2)$ triplet, and a singlet, respectively] receive GUT scale masses.",
"These states are heaviest and their mixings with other ones can be neglected.",
"From Eq.",
"(REF ), with Eq.",
"(REF ) we get $M_{\\Sigma _8}^2\\simeq 20\\lambda _2v_{\\Sigma }^2~,~~~M_{\\Sigma _3}^2\\simeq 80\\lambda _2v_{\\Sigma }^2~,~~~M_{\\Sigma _1}^2\\simeq 4M_{\\Sigma }^2~.$ Further, we will not give masses of states that are singlets under all symmetry groups.",
"The mass square of the $SU(3)^{\\prime }$ octet (from $\\Sigma ^{\\prime }$ ) is $M_{{\\Sigma ^{\\prime }}_{8^{\\prime }}}^2=20\\lambda _2v_{\\Sigma ^{\\prime }}^2+\\frac{6}{5}\\lambda _{2\\Sigma \\Phi }v_{\\Phi }^2~.$ The triplet ${\\Sigma ^{\\prime }}_{3^{\\prime }}$ mixes with a real (CP even) $SU(2)_w$ triplet $\\Phi _3$ (from $\\Phi $ ).",
"[Both these states are real adjoints of $SU(2)_w$ .]",
"The appropriate mass squared couplings are $\\frac{1}{2} \\left({\\Sigma ^{^{\\prime }i}}_{3^{\\prime }}, ~ \\Phi _3^i\\right)\\!\\left( \\!\\begin{array}{cc}4M_{\\Sigma ^{\\prime }_{8^{\\prime }}}^2\\!-\\!\\frac{28}{5}\\lambda _{2\\Sigma \\Phi }v_{\\Phi }^2 & 6\\sqrt{2}\\lambda _{2\\Sigma \\Phi }v_{\\Phi }v_{\\Sigma ^{\\prime }} \\\\6\\sqrt{2}\\lambda _{2\\Sigma \\Phi }v_{\\Phi }v_{\\Sigma ^{\\prime }} & 4\\lambda _{2\\Phi }v_{\\Phi }^2 \\\\\\end{array}\\!",
"\\right)\\!\\!\\left(\\!\\!\\!\\begin{array}{c}{\\Sigma ^{^{\\prime }i}}_{3^{\\prime }} \\\\\\Phi _3^i \\\\\\end{array}\\!\\!\\!\\right),$ where $i=1,2,3$ labels the components of the $SU(2)_w$ adjoint.",
"The CP-odd real $SU(2)_w$ triplet from $\\Phi $ is absorbed by appropriate gauge fields after $SU(2)\\times SU(2)^{\\prime }\\rightarrow SU(2)_w$ breaking and becames genuine Goldstone modes.",
"By the VEVs $v_{\\Sigma }$ and $v_{\\Sigma ^{\\prime }}$ , the symmetry $SU(5)\\times SU(5)^{\\prime }\\times D_2$ is broken down to $G_{321}\\times {G_{321}}^{\\prime }$ [see Eqs.",
"(REF ) and (REF )].",
"Thus, between the scales $M_I$ and ${M_I}^{\\prime }$ , we have this symmetry, and the $\\Phi (5, \\bar{5})$ splits into fragments $\\Phi (5, \\bar{5})=\\Phi _{DD^{\\prime }}\\oplus \\Phi _{DT^{\\prime }} \\oplus \\Phi _{TT^{\\prime }} \\oplus \\Phi _{TD^{\\prime }}$ with transformation properties under $G_{321}\\times {G_{321}}^{\\prime }$ given by $G_{321}\\times {G_{321}}^{\\prime }~:&~~~\\Phi _{DD^{\\prime }} \\sim \\left(1, 2, -\\frac{3}{\\sqrt{60}}, 1,2^{\\prime }, \\frac{3}{\\sqrt{60}}\\right),&\\Phi _{DT^{\\prime }}\\sim \\left(1, 2, -\\frac{3}{\\sqrt{60}}, \\bar{3}^{\\prime },1, -\\frac{2}{\\sqrt{60}}\\right),\\nonumber \\\\&~~ \\Phi _{TT^{\\prime }} \\sim \\left(3, 1, \\frac{2}{\\sqrt{60}}, \\bar{3}^{\\prime },1, -\\frac{2}{\\sqrt{60}}\\right),&\\Phi _{TD^{\\prime }}\\sim \\left(3, 1, \\frac{2}{\\sqrt{60}}, 1,2^{\\prime }, \\frac{3}{\\sqrt{60}}\\right).$ The masses of these fragments will be denoted by $M_{DD^{\\prime }}, M_{DT^{\\prime }}, M_{TT^{\\prime }},$ and $M_{TD^{\\prime }}$ , respectively.",
"Since the breaking $G_{321}\\times {G_{321}}^{\\prime }\\rightarrow G_{SM}\\times SU(3)^{\\prime }$ is realized by the VEV of the fragment $\\Phi _{DD^{\\prime }}$ at scale $M_I$ , we take $M_{DD^{\\prime }}\\simeq M_I$ .",
"The state $\\Phi _3$ , participating in Eq.",
"(REF ), emerges from this $\\Phi _{DD^{\\prime }}$ fragment.",
"The remaining three states under $G_{321}\\times SU(3)^{\\prime }$ transform as $G_{321}\\times SU(3)^{\\prime }~:~~\\Phi _{DT^{\\prime }}\\sim \\left(1, 2, -\\frac{5}{\\sqrt{60}}, \\bar{3}^{\\prime }\\right),~~~ \\Phi _{TT^{\\prime }} \\sim \\left(3, 1, 0, \\bar{3}^{\\prime }\\right),~~\\Phi _{TD^{\\prime }}\\sim \\left(3, 2, \\frac{5}{\\sqrt{60}}, 1\\right).$ The mass squares of these fields are given by $M_{DT^{\\prime }}^2=5\\lambda _{2\\Sigma \\Phi }v_{\\Sigma ^{\\prime }}^2~,~~~M_{TT^{\\prime }}^2=5\\lambda _{2\\Sigma \\Phi }(v_{\\Sigma }^2+v_{\\Sigma ^{\\prime }}^2)\\!-\\!2\\lambda _{2\\Phi }v_{\\Phi }^2~,~~~M_{TD^{\\prime }}^2=5\\lambda _{2\\Sigma \\Phi }v_{\\Sigma }^2~.$ With the VEVs toward the directions given in Eqs.",
"(REF ), (REF ), and (REF ), and with the extremum conditions of Eq.",
"(REF ), the potential's minimum is achieved with $30\\lambda _1+7\\lambda _2>0~,~~~\\lambda _2>0~,~~~\\lambda >0~,$ $10\\lambda _{1\\Sigma \\Phi }-3\\lambda _{2\\Sigma \\Phi }>0~,~~~\\lambda _{2\\Sigma \\Phi }>0~,~~~2\\lambda _{1\\Phi }+\\lambda _{2\\Phi }>0~,~~~\\lambda _{2\\Phi }>0~.$ As far as the states $H$ and $H^{\\prime }$ are concerned, they are split as $H\\rightarrow (D_H, T_H)$ and $H^{\\prime }\\rightarrow (D_{H^{\\prime }}, T_{H^{\\prime }})$ , where $D_H, D_{H^{\\prime }}$ are doublets, while $T_H$ and $T_{H^{\\prime }}$ are $SU(3)_c$ and $SU(3)^{\\prime }$ triplets, respectively.",
"Mass squares of these triplets are $M_{T_H}^2=M_{H}^2-4h_1v_{\\Sigma }^2+30(h_2v_{\\Sigma }^2+\\tilde{h}v_{\\Sigma ^{\\prime } }^2)+2\\lambda _{1H\\Phi }v_{\\Phi }^2/\\sqrt{25}~,$ $M_{T_{H^{\\prime }}}^2=M_{H}^2-4h_1v_{\\Sigma ^{\\prime } }^2+30(h_2v_{\\Sigma ^{\\prime } }^2+\\tilde{h}v_{\\Sigma }^2)+2\\lambda _{1H\\Phi }v_{\\Phi }^2/\\sqrt{25}~.$ The states $D_H$ and $D_{H^{\\prime }}$ , under $G_{SM}$ , both have quantum numbers of the SM Higgs doublet.",
"They mix by the VEV $\\langle \\Phi \\rangle $ , and the mass squared matrix is given by $\\left(D_{H}^\\dag , ~ D_{H^{\\prime }}^\\dag \\right)\\!\\left( \\!\\begin{array}{cc}M_{T_H}^2\\!\\!-\\!5h_1v_{\\Sigma }^2\\!+\\!\\lambda _{2H\\Phi }v_{\\Phi }^2/\\sqrt{10} & \\mu v_{\\Phi } \\\\\\mu v_{\\Phi } & M_{T_{H^{\\prime }}}^2\\!\\!-\\!5h_1v_{\\Sigma ^{\\prime } }^2\\!+\\!\\lambda _{2H\\Phi }v_{\\Phi }^2/\\sqrt{10} \\\\\\end{array}\\!",
"\\right)\\!\\!\\left(\\!\\!\\!\\begin{array}{c}D_{H} \\\\D_{H^{\\prime }} \\\\\\end{array}\\!\\!\\!\\right).$ By diagonalization of (REF ), we get two physical states $h$ and $D^{\\prime }$ : $h=\\cos \\theta _hD_H+\\sin \\theta _hD_{H^{\\prime }}~,~~~D^{\\prime }=-\\sin \\theta _hD_H+\\cos \\theta _hD_{H^{\\prime }}~,$ $\\tan 2\\theta _h=\\frac{2\\mu v_{\\Phi }}{M_{T_H}^2-M_{T_{H^{\\prime }}}^2-5h_1(v_{\\Sigma }^2-v_{\\Sigma ^{\\prime } }^2)}~.$ We identify $h$ with the SM Higgs doublet and set its mass square (by fine-tuning) $M_h^2\\sim 100~{\\rm GeV}^2$ .",
"We assume the second doublet $D^{\\prime }$ to be heavy $M_{D^{\\prime }}^2\\gg |M_h|^2$ .",
"For the mixing angle $\\theta _h$ , we also assume $\\theta _h\\ll 1$ .",
"Therefore, according to Eq.",
"(REF ), the SM Higgs mainly resides in $D_H$ (of the $H$ -plet), while $D_{H^{\\prime }}$ (i.e., $H^{\\prime }$ ) includes a light SM doublet with very suppressed weight.",
"The radiative corrections will affect obtained expressions for the masses and VEVs.",
"However, there are enough parameters involved, and one can always get considered symmetry breaking pattern and desirable spectrum.",
"Achieving these will require some fine-tunings.",
"Without addressing here the hierarchy problem and naturalness issues, we will proceed to study various properties and the phenomenology of the considered scenario."
],
[
"$D_2$ symmetry {{formula:80b4e18a-218a-43c2-a231-8215496472e1}} la {{formula:b1f8ab61-fb98-4bad-8d20-2ad888b70da5}} parity",
"We introduce three families of $(\\Psi , F)$ and three families of $(\\Psi ^{\\prime }, F^{\\prime })$ , $3\\times \\left[ \\Psi (10, 1)+F(\\bar{5}, 1)\\right]~,\\hspace{17.07182pt} ~~3\\times \\left[ \\Psi ^{\\prime }(1, \\bar{10})+F^{\\prime }(1, 5)\\right]~,$ where in brackets the transformation properties under $SU(5)\\times SU(5)^{\\prime }$ gauge symmetry are indicated.",
"Here, each fermionic state is a two-component Weyl spinor, in $(\\frac{1}{2},0)$ representation of the Lorentz group.",
"The action of $D_2$ parity on these fields is determined as $D_2~:~~~~~~\\Psi \\stackrel{\\rightarrow }{_\\leftarrow } \\overline{\\Psi ^{\\prime }}\\equiv (\\Psi ^{\\prime })^\\dag ~,~~~{\\bf F}\\stackrel{\\rightarrow }{_\\leftarrow } \\overline{F^{\\prime }}\\equiv (F^{\\prime })^\\dag ~.$ It is easy to verify that, with transformations in Eqs.",
"(REF ) and (REF ), the kinetic part of the Lagrangian ${\\cal L}_{kin}(\\Psi ,F,\\Psi ^{\\prime },F^{\\prime })$ is invariant.The $D_2$ transformation of Eq.",
"(REF ) resembles usual $P$ parity, acting between the electron and positron, within QED.",
"Unlike the QED, the states $(\\Psi , F)$ and $(\\Psi ^{\\prime },F^{\\prime })$ transform under different gauge groups.",
"We can easily write down invariant Yukawa Lagrangian ${\\cal L}_Y+{\\cal L}_{Y^{\\prime }}+{\\cal L}_Y^{mix}$ with ${\\cal L}_Y=\\sum _{n=0}C_{\\Psi \\Psi }^{(n)}\\left(\\frac{\\Sigma }{M_*}\\right)^n\\Psi \\Psi H+\\sum _{n=0}C_{\\Psi F}^{(n)}\\left(\\frac{\\Sigma }{M_*}\\right)^n\\Psi {\\bf F}H^\\dag +{\\rm h.c.}$ ${\\cal L}_{Y^{\\prime }}=\\sum _{n=0}C_{\\Psi \\Psi }^{(n)*}\\left(\\frac{\\Sigma ^{\\prime } }{M_*}\\right)^n\\Psi ^{\\prime } \\Psi ^{\\prime } {H^{\\prime }}^\\dag +\\sum _{n=0}C_{\\Psi F}^{(n)*}\\left(\\frac{\\Sigma ^{\\prime } }{M_*}\\right)^n\\Psi ^{\\prime } {\\bf F}^{\\prime } H^{\\prime } +{\\rm h.c.}$ ${\\cal L}_Y^{mix}=\\lambda _{FF^{\\prime }}F\\Phi F^{\\prime }+\\lambda _{FF^{\\prime }} \\overline{F^{\\prime }}\\Phi ^\\dag \\overline{F}+\\frac{\\lambda _{\\Psi \\Psi ^{\\prime }}}{M}\\Psi (\\Phi ^\\dag )^2\\Psi ^{\\prime } +\\frac{\\lambda _{\\Psi \\Psi ^{\\prime }}}{M}\\overline{\\Psi ^{\\prime }}\\Phi ^2\\overline{\\Psi }~,$ where $M_*, M$ are some cutoff scales.",
"The coupling matrices $\\lambda _{FF^{\\prime }}$ and $\\lambda _{\\Psi \\Psi ^{\\prime }}$ are Hermitian due to the $D_2$ symmetry.",
"The last two higher-order operators in Eq.",
"(REF ), important for phenomenology, can be generated by integrating out some heavy states with mass at or above the GUT scale.",
"For instance, with the scalar state $\\Omega $ in $(\\bar{10},10)$ representation of $SU(5)\\times SU(5)^{\\prime }$ and $D_2$ parity, $\\Omega \\stackrel{\\rightarrow }{_\\leftarrow } \\Omega ^\\dag $ , the relevant terms (of fundamental Lagrangian) will be $\\lambda _{\\Psi \\Psi ^{\\prime }}\\Omega \\Psi \\Psi ^{\\prime } +\\lambda _{\\Psi \\Psi ^{\\prime }} \\Omega ^\\dag \\overline{\\Psi ^{\\prime }}\\!\\!",
"\\cdot \\!\\overline{\\Psi }+\\bar{M}_{\\Omega }(\\Omega \\Phi ^2+\\Omega ^\\dag (\\Phi ^\\dag )^2)+M_{\\Omega }^2\\Omega ^\\dag \\Omega $ .",
"With these couplings, one can easily verify that integration of $\\Omega $ generates the last two operators of Eq.",
"(REF ) (with $M\\approx M_{\\Omega }^2/\\bar{M}_{\\Omega }$ ).",
"Since the $\\Omega $ is rather heavy, its only low-energy implication can be the emergence of these effective operators.",
"Thus, in our further studies, we will proceed with the consideration of Yukawa couplings given in Eqs.",
"(REF )-(REF ).",
"With obvious identifications, let us adopt the following notations for the components from $\\Psi , F$ and $\\Psi ^{\\prime }, F^{\\prime }$ states: $\\Psi =\\lbrace q, u^c, e^c\\rbrace ~,~~~~F=\\lbrace l, d^c\\rbrace ~,$ $\\Psi ^{\\prime } =\\lbrace \\hat{q}, \\hat{u}^c, \\hat{e}^c\\rbrace ~,~~~~F^{\\prime }=\\lbrace \\hat{l}, \\hat{d}^c\\rbrace ~.$ Substituting in Eqs.",
"(REF )-(REF ) the VEVs $\\langle \\Sigma \\rangle , \\langle \\Sigma ^{\\prime }\\rangle $ , and $\\langle \\Phi \\rangle $ , the relevant couplings we obtain are ${\\cal L}_Y\\rightarrow q^TY_Uu^ch+q^TY_Dd^ch^\\dag +e^{cT}Y_{e^cl}lh^\\dag +$ $(C_{qq}qq+C_{u^ce^c}u^ce^c)T_H+(C_{ql}ql+C_{u^cd^c}u^cd^c)T_H^\\dag +{\\rm h.c.}$ ${\\cal L}_{Y^{\\prime }}\\rightarrow C_{\\Psi \\Psi }^{(0)*}(\\frac{1}{2}\\hat{q}\\hat{q}+\\hat{u}^c\\hat{e}^c) T_{H^{\\prime }}^\\dag +C_{\\Psi F}^{(0)*}(\\hat{q}\\hat{l}+\\hat{u}^c \\hat{d}^c)T_{H^{\\prime }}+{\\rm h.c.}+\\cdots $ ${\\cal L}_Y^{mix}\\rightarrow \\hat{l}^TM_{\\hat{l}l}l+e^{cT}M_{e^c\\hat{e}^c}\\hat{e}^c+{\\rm h.c.}~.$ In Eq.",
"(REF ) we have dropped out the couplings with the Higgs doublet because, as we have assumed, $D_{H^{\\prime }}$ includes the SM Higgs doublet with very suppressed weight.",
"Also, we have ignored powers of $\\langle \\Sigma ^{\\prime }\\rangle /M_*$ in comparison with $\\langle \\Sigma \\rangle /M_*$ 's exponents.",
"As we will see, the couplings of $h$ in (REF ) and terms shown in Eqs.",
"(REF ) and (REF ) are responsible for fermion masses and mixings and lead to realistic phenomenology."
],
[
"Fermion masses and mixings: Composite leptons",
"Let us first indicate transformation properties of all matter states, given in Eq.",
"(REF ), under the unbroken $G_{SM}\\times SU(3)^{\\prime }=SU(3)_c\\times SU(2)_w\\times U(1)_Y\\times SU(3)^{\\prime }$ gauge symmetry.",
"Fragments from $\\Psi , F$ transform as $q\\sim (3,2, -\\frac{1}{\\sqrt{60}}, 1)~,~~~u^c\\sim (\\bar{3}, 1, \\frac{4}{\\sqrt{60}}, 1)~,~~~e^c\\sim (1,1,-\\frac{6}{\\sqrt{60}}, 1)$ $l\\sim (1, 2, \\frac{3}{\\sqrt{60}}, 1)~,~~~~~~d^c\\sim (\\bar{3}, 1, -\\frac{2}{\\sqrt{60}}, 1)~,$ while the states from $\\Psi ^{\\prime }, F^{\\prime }$ have the following transformation properties: $\\hat{q}\\sim (1,2, \\frac{1}{\\sqrt{60}}, \\bar{3}^{\\prime })~,~~~\\hat{u}^c\\sim (1, 1, -\\frac{4}{\\sqrt{60}}, 3^{\\prime })~,~~~\\hat{e}^c\\sim (1,1,\\frac{6}{\\sqrt{60}}, 1)$ $\\hat{l}\\sim (1, 2, -\\frac{3}{\\sqrt{60}}, 1)~,~~~~~~\\hat{d}^c\\sim (1, 1, \\frac{2}{\\sqrt{60}}, 3^{\\prime })~.$ In transformation properties of Eq.",
"(REF ), by primes we have indicated triplets and antitriplets of $SU(3)^{\\prime }$ .",
"As we see, transformation properties of quark states in Eq.",
"(REF ) coincide with those of the SM.",
"Therefore, for quark masses and CKM mixings, the first two couplings of Eq.",
"(REF ) are relevant.",
"Since in $Y_{U,D}$ and $Y_{e^cl}$ contribute also higher-dimensional operators, the $Y_U$ is not symmetric and $Y_D\\ne Y_{e^cl}$ .",
"Thus, quark Yukawa matrices can be diagonalized by biunitary transformations $L_u^\\dag Y_UR_u=Y_U^{\\rm Diag}~,~~~~L_d^\\dag Y_DR_d=Y_D^{\\rm Diag}~.$ With these, the CKM matrix (in standard parametrization) is $V_{CKM}=P_1L_u^TL_d^*P_2$ ${\\rm with}~~~P_1\\!=\\!",
"{\\rm Diag}\\!\\left(e^{i\\omega _1}, ~e^{i\\omega _2},~e^{i\\omega _3}\\right)~,~~~~~~~P_2\\!=\\!",
"{\\rm Diag}\\!\\left(e^{i\\rho _1}, ~e^{i\\rho _2},~1\\right)~.$ Composite leptons Turning to the lepton sector, we note that $\\hat{l}$ and $\\hat{e}^c$ have opposite/conjugate transformation properties with respect to $l$ and $e^c$ , respectively.",
"From couplings in Eq.",
"(REF ), we see that these vectorlike states acquire masses $M_{\\hat{l}l}$ and $M_{e^c\\hat{e}^c}$ and decouple .",
"However, within this scenario, composite leptons emerge.",
"The $SU(3)^{\\prime }$ becomes strongly coupled and confines at scale $\\Lambda ^{\\prime } \\sim $ TeV (for details, see Sec.",
").",
"Because of confinement, $SU(3)^{\\prime }$ singlet composite states - baryons ($B^{\\prime }$ ) and/or mesons ($M^{\\prime }$ ) - can emerge.",
"The elegant idea of fermion emergence through the strong dynamics as bound states of more fundamental constituents, was suggested and developed in Refs.",
"[12]-[22].",
"Within our scenario, this idea finds an interesting realization for the lepton states.",
"Formation of composite fermions should satisfy 't Hooft anomaly matching conditionsIn case the chiral symmetry remains unbroken (at least partially) at the composite level.",
"The models avoiding anomaly conditions were suggested in Ref.",
"[19].",
"[14].",
"These give a severe constraint on building models with composite fermions [16], [17], [18], [20], [21], [22].",
"Let us focus on the sector of (three-family) $\\hat{q}, \\hat{u}^c$ and $\\hat{d}^c$ states, which have $SU(3)^{\\prime }$ strong interactions.",
"Ignoring local EW and Yukawa interactions, the Lagrangian of these states possesses global $G_f^{(6)}=SU(6)_L\\times SU(6)_R\\times U(1)_{B^{\\prime }}$ chiral symmetry.",
"Under the $SU(6)_L$ , three families of $\\hat{q}=(\\hat{u}, \\hat{d})$ transform as sextet $6_L$ , while three families of $(\\hat{u}^c, \\hat{d}^c)\\equiv \\hat{q}^c$ form sextet $6_R$ of $SU(6)_R$ .",
"The $U(1)_{B^{\\prime }}$ ($B^{\\prime }$ ) charges of $\\hat{q}$ and $\\hat{q}^c$ are, respectively, $1/3$ and $-1/3$ .",
"Thus, transformation properties of these states under $G_f^{(6)}=SU(6)_L\\times SU(6)_R\\times U(1)_{B^{\\prime }}$ chiral symmetry are $\\hat{q}_{\\alpha } =(\\hat{u}, \\hat{d})_{\\alpha }\\sim (6_L, 1, \\frac{1}{3})~,~~~~~\\hat{q}^c_{\\alpha }=(\\hat{u}^c, \\hat{d}^c)_{\\alpha } \\sim (1, 6_R, -\\frac{1}{3})~,$ where $\\alpha =1,2,3$ is the family index.",
"Because of the strong $SU(3)^{\\prime }$ attractive force, condensates that will break the $G_f^{(6)}$ chiral symmetry can form.",
"The breaking can occur by several steps, and at each step the formed composite states should satisfy anomaly matching conditions.",
"In Appedix , we give a detailed account of these issues and demonstrate that within our scenario three families of $l_0, e^c_0, \\nu ^c_0$ composite states, $(\\hat{q} \\hat{q})\\hat{q} \\sim l_{0\\alpha }=\\!\\!\\left(\\!\\!\\begin{array}{c}\\nu _0 \\\\e_0 \\\\\\end{array}\\!\\!\\right)_{\\alpha }~,~~~~(\\hat{q}^c \\hat{q}^c)\\hat{q}^c =\\left((\\hat{u}^c \\hat{d}^c)\\hat{d}^c,~(\\hat{u}^c \\hat{d}^c)\\hat{u}^c\\right)\\sim l^c_{0\\alpha }\\equiv (\\nu ^c_0, ~e^c_0)_{\\alpha }~,~~~~\\alpha =1,2,3$ emerge.",
"In Eq.",
"(REF ), for combinations $(\\hat{q} \\hat{q})\\hat{q}$ and $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ , the spin-1/2 states are assumed with suppressed gauge and/or flavor indices.",
"For instance, under $(\\hat{q} \\hat{q})\\hat{q}$ we mean $\\epsilon ^{a^{\\prime }b^{\\prime }c^{\\prime }}\\!\\epsilon _{ij}(\\hat{q}_{a^{\\prime }i} \\hat{q}_{b^{\\prime }j})\\hat{q}_{c^{\\prime }k}$ , where $a^{\\prime }, b^{\\prime }, c^{\\prime }=1,2,3$ are $SU(3)^{\\prime }$ indices and $i,j,k=1,2$ stand for $SU(2)_w$ (or $SU(2)_L$ ) indices.",
"Similar applies to the combination $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ .",
"Thus, $(\\hat{q} \\hat{q})\\hat{q}$ and $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ are singlets of $SU(3)^{\\prime }$ .",
"From these, taking into account Eqs.",
"(REF ) and (REF ), it is easy to verify that the quantum numbers of composite states under SM gauge group $G_{SM}=SU(3)_c\\times SU(2)_w\\times U(1)_Y$ are $G_{SM}~:~~~l_0\\sim (1, 2, \\frac{3}{\\sqrt{60}})~,~~~e^c_0\\sim (1, 1, -\\frac{6}{\\sqrt{60}})~,~~~\\nu ^c_0\\sim (1, 1, 0)~.$ As we see, along with SM leptons ($l_0$ and $e^c_0$ ), we get three families of composite SM singlets fermions - $\\nu ^c_0$ .",
"The latter can be treated as composite right-handed/sterile neutrinos in the spirit of Ref.",
"[23].",
"Note that, with this composition, as was expected, the gauge anomalies also vanish (together with the chiral anomaly matching; for details, see Appendix ).",
"Interestingly, the $SU(3)^{\\prime }$ [originating from $SU(5)^{\\prime }$ ] triplet and antitriplets $\\hat{u}^c, \\hat{d}^c$ and $\\hat{q}$ play the role of \"preon\" constituents for the bound-state leptons and right-handed/sterile neutrinos.",
"Moreover, in our scheme the lepton number $L$ is related to the $U(1)_{B^{\\prime }}$ charge as $L=3B^{\\prime }$ .",
"Therefore, \"primed baryon number\" $B^{\\prime }$ [of the $SU(5)^{\\prime }$ ] is the origin of the lepton number.",
"Charged lepton masses Now, we turn to the masses of the charged leptons, which are composite within our scenario.",
"As it turns out, their mass generation does not require additional extension.",
"It happens via integration of the states that are present in the model.",
"As we see from Eq.",
"(REF ), the $SU(5)^{\\prime }$ matter couples with the $SU(3)^{\\prime }$ triplet scalar $T_{H^{\\prime }}$ with mass $M_{T_{H^{\\prime }}}$ .",
"Relevant 4-fermion operators, emerging from the couplings of Eq.",
"(REF ) and by integration of $T_{H^{\\prime }}$ , are ${\\cal L}_{Y^{\\prime }}^{eff}=\\frac{C_{\\Psi \\Psi }^{(0)*}C_{\\Psi F}^{(0)*}}{M_{T_{H^{\\prime }}}^2}\\left[ \\frac{1}{2}(\\hat{q}\\hat{q})(\\hat{q}\\hat{l})+(\\hat{u}^c\\hat{e}^c)(\\hat{u}^c\\hat{d}^c)\\right] +{\\rm h.c.}$ As we see, here appear the combinations $(\\hat{q} \\hat{q})\\hat{q}$ and $(\\hat{u}^c\\hat{d}^c)\\hat{u}^c$ , which according to Eq.",
"(REF ) form composite charged lepton states.",
"We will use the parametrizations $\\frac{1}{2}(\\hat{q}_{\\alpha }\\hat{q}_{\\beta })\\hat{q}_{\\gamma }={\\Lambda ^{\\prime }}^3c_{\\alpha \\beta \\gamma \\delta }l_{0\\delta }~,~~~~~(\\hat{u}^c_{\\alpha }\\hat{d}^c_{\\beta })\\hat{u}^c_{\\gamma }={\\Lambda ^{\\prime }}^3\\bar{c}_{\\alpha \\beta \\gamma \\delta }e^c_{0\\delta }~$ where Greek indices denote family indices and $c, \\bar{c}$ are dimensionless couplings - four index tensors in a family space.",
"The $(l_0, e^c_0)_{\\delta }$ denote three families of composite leptons.",
"Using Eq.",
"(REF ) in Eq.",
"(REF ), we obtain ${\\cal L}_{Y^{\\prime }}^{eff}\\rightarrow \\hat{l}\\hat{\\mu }l_0+e^c_0\\tilde{\\mu }\\hat{e}^c+{\\rm h.c.}$ ${\\rm with}~~~~\\hat{\\mu }_{\\delta ^{\\prime }\\delta }\\equiv \\frac{{\\Lambda ^{\\prime }}^3}{M_{T_{H^{\\prime }}}^2}(C_{\\Psi \\Psi }^{(0)*})_{\\alpha \\beta }(C_{\\Psi F}^{(0)*})_{\\gamma \\delta ^{\\prime }}c_{\\alpha \\beta \\gamma \\delta }~,~~~~~\\tilde{\\mu }_{\\delta \\delta ^{\\prime }}\\equiv \\frac{{\\Lambda ^{\\prime }}^3}{M_{T_{H^{\\prime }}}^2}(C_{\\Psi \\Psi }^{(0)*})_{\\gamma \\delta ^{\\prime }}(C_{\\Psi F}^{(0)*})_{\\alpha \\beta }\\bar{c}_{\\alpha \\beta \\gamma \\delta }~.$ Figure: Diagram responsible for the generation of the charged lepton effective Yukawa matrix.At the next stage, we integrate out the vectorlike states $\\hat{l}, l$ and $e^c, \\hat{e}^c$ , which, respectively, receive masses $M_{\\hat{l}l}$ and $M_{e^c\\hat{e}^c}$ through the coupling in Eq.",
"(REF ).",
"Integrating out these heavy states, from Eqs.",
"(REF ) and (REF ), we get $l\\simeq -\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }l_0~,~~~~e^{cT}\\simeq -e^{cT}_0\\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}}~.$ Substituting these in the $e^{cT}Y_{e^cl}lh^\\dag $ coupling of Eq.",
"(REF ), we see that the effective Yukawa couplings for the leptons are generated: $l_0^TY_Ee^c_0h^\\dagger +{\\rm h.c.}~~~~~~{\\rm with}~~~~ Y_E^T\\simeq \\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}}Y_{e^cl}\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }~.$ The diagram corresponding to the generation of this effective Yukawa operator is shown in Fig.",
"REF .",
"This mechanism is novel and differs from those suggested earlier for the mass generation of composite fermions [22].",
"From the observed values of the Yukawa couplings, we have $|{\\rm Det}Y_E|=\\lambda _e\\lambda _{\\mu }\\lambda _{\\tau }\\approx 1.8\\cdot 10^{-11}$ .",
"On the other hand, natural values of the eigenvalues of $Y_{e^cl}$ can be$\\sim 0.1$ .",
"Thus, $|{\\rm Det}Y_{e^cl}|\\sim 10^{-3}$ .",
"From these and the expression given in Eq.",
"(REF ), we obtain $|{\\rm Det} (\\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}})|\\cdot |{\\rm Det}(\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu })|\\sim 10^{-8}~~,$ the constraint that should be satisfied by two matrices $\\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}}$ and $\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }$ .",
"Neutrino masses Now, we discuss the neutrino mass generation.",
"To accommodate the neutrino data [4], one can use SM singlet fermionic states in order to generate either Majorana- or Dirac-type masses for the neutrinos.",
"Within our model, among the composite fermions, we have SM singlets $\\nu ^c_0$ [see Eqs.",
"(REF ) and (REF )].",
"Here, we stick to the possibility of the Dirac-type neutrino masses, which can be naturally suppressed [23].",
"Because of compositeness, there is no direct Dirac couplings $Y_{\\nu }$ of $\\nu ^c_0$ 's with lepton doublets $l_0$ .",
"Similar to the charged lepton Yukawa couplings, we need to generate $Y_{\\nu }$ .",
"For this purpose, we introduce the $SU(5)\\times SU(5)^{\\prime }$ singlet (two-component) fermionic states $N$ .The number of $N$ states is not limited, but for simplicity we can assume that they are not more than 3.",
"Assigning the $D_2$ parity transformations $N \\stackrel{\\rightarrow }{_\\leftarrow } \\overline{N}$ and taking into account Eqs.",
"(REF ) and (REF ), relevant couplings, allowed by $SU(5)\\times SU(5)^{\\prime }\\times D_2$ symmetry, will be ${\\cal L}_N=C_{FN}FNH+C_{FN}^*F^{\\prime }N{H^{\\prime }}^\\dag -\\frac{1}{2}N^TM_NN+{\\rm h.c.}~~~~{\\rm with}~~~~M_N=M_N^*~.$ These give the following interaction terms: ${\\cal L}_N\\rightarrow C_{FN}lNh+C_{FN}^*\\hat{d}^cNT_{H^{\\prime }}^\\dag -\\frac{1}{2}N^TM_NN+{\\rm h.c.}$ From these and Eq.",
"(REF ), integration of $T_{H^{\\prime }}$ state gives additional affective four-fermion operator $\\frac{C_{\\Psi F}^{(0)*}C_{FN}^*}{M_{T_{H^{\\prime }}}^2}(\\hat{u}^c \\hat{d}^c)(\\hat{d}^c N)+{\\rm h.c.}$ By the parametrization $(\\hat{u}^c_{\\alpha }\\hat{d}^c_{\\beta })\\hat{d}^c_{\\gamma }={\\Lambda ^{\\prime }}^3\\tilde{c}_{\\alpha \\beta \\gamma \\delta }\\nu ^c_{0\\delta }~,$ operators in Eq.",
"(REF ) are given by ${\\cal L}_{N\\nu ^c}^{eff}=N\\mu _{\\nu }\\nu ^c_0+{\\rm h.c.}~~~~{\\rm with}~~~(\\mu _{\\nu })_{\\delta ^{\\prime }\\delta }\\equiv \\frac{{\\Lambda ^{\\prime }}^3}{M_{T_{H^{\\prime }}}^2}(C_{\\Psi F}^{(0)*})_{\\alpha \\beta }(C_{FN}^*)_{\\gamma \\delta ^{\\prime }}\\tilde{c}_{\\alpha \\beta \\gamma \\delta }~.$ Subsequent integration of $N$ states, from Eq.",
"(REF ) and the last term of Eq.",
"(REF ) gives $N\\simeq \\frac{1}{M_N}\\mu _{\\nu }\\nu ^c_0~.$ Substituting this, and the expression of $l$ from Eq.",
"(REF ), in the first term of Eq.",
"(REF ), we arrive at $l_0^TY_{\\nu }\\nu ^c_0h +{\\rm h.c.}~~~~~{\\rm with}~~~~ Y_{\\nu }\\simeq -\\hat{\\mu }^T\\frac{1}{M_{\\hat{l}l}^T}C_{FN}\\frac{1}{M_N}\\mu _{\\nu }~.$ The relevant diagram generating this effective Dirac Yukawa couplings is given in Fig.",
"REF .",
"With $\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }\\sim 10^{-2}$ and $C_{FN}\\sim M_N\\sim \\frac{1}{M_N}\\mu _{\\nu }\\sim 10^{-5}$ , we can get the Dirac neutrino mass $M_{\\nu }^D=Y_{\\nu }\\langle h^{(0)}\\rangle \\sim 0.1$ eV, which is the right scale to explain neutrino anomalies.",
"Note that using Eq.",
"(REF ) in the last term of Eq.",
"(REF ) we also obtain the term $-\\frac{1}{2}{\\nu ^c_0}^TM_{\\nu ^c}\\nu ^c_0$ with $M_{\\nu ^c}\\simeq \\mu _{\\nu }^T\\frac{1}{M_N}\\mu _{\\nu }$ .",
"By proper selection of the couplings $C_{FN}$ and eigenvalues of $M_N$ , the $M_{\\nu ^c}$ can be strongly suppressed.",
"In this case, the neutrinos will be (quasi)Dirac.",
"However, it is possible that some of the species of light neutrinos to be (quasi)Dirac and some of them Majoranas.",
"Detailed studies of such scenarios and their compatibilities with current experiments [24] are beyond the scope of this paper.",
"Figure: Diagram responsible for the generation of the effective Dirac Yukawa matrix for the neutrinos."
],
[
"Gauge coupling unification",
"In this section we will study the gauge coupling unification within our model.",
"We show that the symmetry breaking pattern gives the possibility for successful unification.Possibilities of gauge coupling unification, with the intermediate symmetry breaking pattern and without invoking low-scale supersymmetry, have been studied in Ref.",
"[25].",
"As it turns out, the $SU(3)^{\\prime }$ gauge interaction becomes strongly coupled at scale $\\Lambda ^{\\prime }$ ($\\sim $ few TeV).",
"Thus, below this scale, $SU(3)^{\\prime }$ confines, and all states (including composite ones) are $SU(3)^{\\prime }$ singlets.",
"Therefore, with the masses $M_{\\hat{l}l}^{(\\alpha )}$ and $M_{e^c\\hat{e}^c}^{(\\alpha )}$ ($\\alpha =1,2,3$ ) of vectorlike states $l, \\hat{l}$ and $e^c, \\hat{e}^c$ being above the scale $\\Lambda ^{\\prime }$ , in the energy interval $\\mu =M_Z-\\Lambda ^{\\prime }$ , the states are just those of SM (plus possibly right-handed/sterile neutrinos having no impact on gauge coupling running), and corresponding one-loop $\\beta $ -function coefficients are $(b_Y, b_w, b_c)=\\left(\\frac{41}{10}, -\\frac{19}{6}, -7\\right)$ .",
"Since $\\Lambda ^{\\prime }$ is the characteristic scale of the strong dynamics, it is clear that pseudo-Goldstone and composite states (besides SM leptons) emerging through chiral symmetry breaking and strong dynamics, can have masses below $\\Lambda ^{\\prime }$ (in a certain range).",
"Instead investigating their spectrum and dealing with corresponding threshold effects, we parametrize all these as a single effective $\\Lambda ^{\\prime }$ scale, below which theory is the SM.",
"This phenomenological simplification allows us to proceed with RG analysis.",
"Note, however, that even with taking those kinds of thresholds into account should not harm the success of coupling unification with the price of proper adjustment of the mass scales (given in Table REF and discussed later on).",
"In the energy interval $\\Lambda ^{\\prime } - M_I$ , we have the symmetry $SU(3)_c\\times SU(2)_w\\times U(1)_Y\\times SU(3)^{\\prime }$ , and $SU(3)^{\\prime }$ nonsinglet states (i.e., $\\hat{q}, \\hat{u}^c, \\hat{d}^c$ , $T_{H^{\\prime }}$ , etc.)",
"must be taken into account.",
"As was noted in Sec.",
", we consider hierarchical breaking: $M_I\\ll {M_I}^{\\prime }\\ll M_G$ [see Eqs.",
"(REF ) and (REF )].",
"This choice allows us to have successful unification with confining scale $\\Lambda ^{\\prime }\\sim $ few TeV.One can have unification with $\\langle \\Sigma ^{\\prime }\\rangle =0$ , (i.e., $M_I={M_I}^{\\prime }$ ) and with a modified spectrum.",
"However, with such a choice the value of $\\Lambda ^{\\prime } $ comes out rather large ($\\stackrel{>}{_\\sim }10^5$ GeV).",
"This would also imply the breaking of EW symmetry at a high scale and thus should be discarded from the phenomenological viewpoint.",
"More discussion about this issue is given in Sec.",
".",
"Thus, between the scales $M_I$ and ${M_I}^{\\prime }$ , the symmetry is $G_{321}\\times {G_{321}}^{\\prime }$ [see Eqs.",
"(REF ) and (REF )], and states should be decomposed under these groups [see, for instance, Eqs.",
"(REF ) and (REF )].",
"Since the breaking $G_{321}\\times {G_{321}}^{\\prime }\\rightarrow G_{SM}\\times SU(3)^{\\prime }$ is realized by the VEV of the fragment $\\Phi _{DD^{\\prime }}$ at scale $M_I$ , we take $M_{DD^{\\prime }}\\simeq M_I$ .",
"The remaining three masses, of the fragments coming from $\\Phi $ , can be in a range $\\Lambda ^{\\prime } - M_G$ .",
"Giving more detailed account to these issues in Appendix , below we sketch the main details.",
"Above the scale $M_I$ , all matter states should be included in the RG.",
"Above the scale ${M_I}^{\\prime }$ we have the $SU(5)^{\\prime }$ symmetry, and the fragments $\\Phi _{DD^{\\prime }}, \\Phi _{DT^{\\prime }}$ form the unified $(2,\\bar{5})$ -plet of $G_{321}\\times SU(5)^{\\prime }$ : $(\\Phi _{DD^{\\prime }}, \\Phi _{DT^{\\prime }})\\subset \\Phi _{D\\bar{5}^{\\prime }}$ , while $\\Phi _{TT^{\\prime }}$ and $\\Phi _{TD^{\\prime }}$ states unify in $(3,\\bar{5})$ -plet: $(\\Phi _{TT^{\\prime }}, \\Phi _{TD^{\\prime }})\\subset \\Phi _{T\\bar{5}^{\\prime }}$ .",
"These states, together with the $\\Sigma ^{\\prime }$ -plet, should be included in the RG above the scale ${M_I}^{\\prime }$ .",
"According to Eq.",
"(REF ), at scale $M_I$ , for the EW gauge couplings, we have the boundary conditions $\\alpha _Y^{-1}(M_I)=\\alpha _1^{-1}(M_I)+\\alpha _{1^{\\prime }}^{-1}(M_I)~,~~~\\alpha _w^{-1}(M_I)=\\alpha _2^{-1}(M_I)+\\alpha _{2^{\\prime }}^{-1}(M_I)~.$ The couplings of ${G_{321}}^{\\prime }$ gauge interactions unify and form single $SU(5)^{\\prime }$ coupling at scale ${M_I}^{\\prime }$ : $\\alpha _{1^{\\prime }}({M_I}^{\\prime })=\\alpha _{2^{\\prime }}({M_I}^{\\prime })=\\alpha _{3^{\\prime }}({M_I}^{\\prime })= \\alpha _{5^{\\prime }}({M_I}^{\\prime })~.$ Finally, at the GUT scale $M_G$ , the coupling of $G_{321}$ and $SU(5)^{\\prime }$ unifies: $\\alpha _1(M_G)=\\alpha _2(M_G)=\\alpha _3(M_G)=\\alpha _{5^{\\prime }}(M_G)\\equiv \\alpha _G ~.$ Table: Particle spectroscopy.With solutions (REF ) and (REF ) of RG equations at corresponding energy scales, and taking into account the boundary conditions (REF )-(REF ), we derive $\\left(\\!\\!\\!\\begin{array}{cccc}(b_1^{IG}-b_Y^{ZI}+b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}) , \\!&\\hspace{-5.69054pt} -b_1^{IG} ,&\\hspace{-5.69054pt} (b_{3^{\\prime }}^{II^{\\prime }}-b_{1^{\\prime }}^{II^{\\prime }}) , & \\hspace{-8.53581pt}-2\\pi \\\\(b_2^{IG}-b_w^{ZI}+b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}), \\!&\\hspace{-5.69054pt} -b_2^{IG} , &\\hspace{-5.69054pt} (b_{3^{\\prime }}^{II^{\\prime }}-b_{2^{\\prime }}^{II^{\\prime }}) , &\\hspace{-8.53581pt} -2\\pi \\\\(b_3^{IG}-b_c^{ZI}) , &\\hspace{-5.69054pt} -b_3^{IG} , \\!&\\hspace{-5.69054pt} 0 ,&\\hspace{-8.53581pt} -2\\pi \\\\(b_{5^{\\prime }}^{I^{\\prime }G}-b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}) , \\!&\\hspace{-5.69054pt} -b_{5^{\\prime }}^{I^{\\prime }G} ,&\\hspace{-5.69054pt} (b_{5^{\\prime }}^{I^{\\prime }G}-b_{3^{\\prime }}^{II^{\\prime }}) , &\\hspace{-8.53581pt} -2\\pi \\end{array} \\!\\!\\right)\\!\\!\\left(\\!\\!\\!\\begin{array}{c}\\ln \\frac{M_I}{M_Z} \\\\\\ln \\frac{M_G}{M_Z} \\\\\\ln \\frac{{M_I}^{\\prime }}{M_I} \\\\\\alpha _G^{-1}\\end{array}\\!\\!\\!\\right)\\!\\!=\\!\\!\\left(\\!\\!\\!\\begin{array}{c}2\\pi (\\alpha _{3^{\\prime }}^{-1}(\\Lambda ^{\\prime })\\!-\\!\\alpha _Y^{-1})+b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}\\ln \\frac{\\Lambda ^{\\prime }}{M_Z} \\\\2\\pi (\\alpha _{3^{\\prime }}^{-1}(\\Lambda ^{\\prime })\\!-\\!\\alpha _w^{-1})+b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}\\ln \\frac{\\Lambda ^{\\prime }}{M_Z} \\\\-2\\pi \\alpha _c^{-1}\\\\-2\\pi \\alpha _{3^{\\prime }}^{-1}(\\Lambda ^{\\prime })-b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}\\ln \\frac{\\Lambda ^{\\prime }}{M_Z}\\end{array}\\!\\!\\!\\right),$ where on the right-hand side of this equation the couplings $\\alpha _{Y,w,c}$ are taken at scale $M_Z$ .",
"The factors $b_i^{\\mu _a\\mu _b}$ (like $b_1^{IG}$ , $b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}$ etc.)",
"stand for effective $b$ factors corresponding to the energy interval $\\mu _a-\\mu _b$ and can also include two-loop effects.",
"All expressions and details are given in Appendix .",
"From Eq.",
"(REF ) we can calculate $\\lbrace M_I,~ M_G,~{M_I}^{\\prime } ,~\\alpha _G\\rbrace $ in terms of the remaining inputs.",
"For instance, a phenomenologically viable scenario is obtained when $SU(3)^{\\prime }$ confines at scale $\\Lambda ^{\\prime }\\sim 1$ TeV.",
"Thus, we will take $\\Lambda ^{\\prime }\\sim 1$ TeV and $\\alpha _{3^{\\prime }}^{-1}(\\Lambda ^{\\prime })\\ll 1$ .",
"In Table REF we give selected input mass scales, leading to successful unification with $\\lbrace M_I,~{M_I}^{\\prime } ,~ M_G \\rbrace \\simeq \\lbrace 8.25\\cdot \\!10^4, 4.16\\cdot \\!10^6, 4.95\\cdot \\!10^{11}\\rbrace ~{\\rm GeV}~,~~~\\alpha _G\\simeq 1/31 ~.$ The corresponding picture of gauge coupling running is given in Fig.",
"REF .",
"This result is obtained by solving RGs in the two-loop approximation.",
"More details, including one- and two-loop RG factors at each relevant mass scale, are given in Appendix .",
"Figure: Gauge coupling unification.",
"{Λ ' ,M I ,M I ' ,M G }≃{1800,8.25·10 4 ,4.16·10 6 ,4.95·10 11 }\\lbrace \\Lambda ^{\\prime }, M_I, {M_I}^{\\prime }, M_G \\rbrace \\simeq \\lbrace 1800, 8.25\\cdot \\!10^4, 4.16\\cdot \\!10^6, 4.95\\cdot \\!10^{11}\\rbrace GeV andα G (M G )≃1/31\\alpha _G(M_G)\\simeq 1/31."
],
[
"Nucleon stability",
"In this section we show that, although the GUT scale $M_G$ is relatively low (close to $5\\cdot 10^{11}$ GeV), the nucleon's lifetime can be compatible with current experimental bounds.",
"In achieving this, a crucial role is played by lepton compositeness, because leptons have no direct couplings with $X, Y$ gauge bosons of $SU(5)$ .",
"The baryon number violating $d=6$ operators, induced by integrating out of the $X,Y$ bosons, are $\\frac{g_X^2}{M_X^2}(\\overline{u^c}_a\\gamma _{\\mu }q_b^i)(\\overline{d^c}_c\\gamma ^{\\mu }l^j)\\epsilon ^{abc}\\epsilon _{ij}~,~~~~~~~\\frac{g_X^2}{M_X^2}(\\overline{u^c}_a\\gamma _{\\mu }q_b^i)(\\overline{e^c}\\gamma ^{\\mu }q_c^j)\\epsilon ^{abc}\\epsilon _{ij}~,$ where $g_X$ is the $SU(5)$ gauge coupling at scale $M_X$ (the mass of the $X,Y$ states).",
"According to Eq.",
"(REF ), the states $l, e^c$ contain light leptons $l_0, e^c_0$ .",
"Using this and going to the mass eigenstate basis [with Eqs.",
"(REF ) and (REF )], from Eq.",
"(REF ) we get operators ${\\cal O}_{d6}^{(e^c)}=\\frac{g_X^2}{M_X^2}{\\cal C}^{(e^c)}_{\\alpha \\beta }(\\overline{u^c}\\gamma _{\\mu }u)(\\overline{e^c_{\\alpha }}\\gamma ^{\\mu }d_{\\beta })~,~~~~~~~{\\cal O}_{d6}^{(e)}=\\frac{g_X^2}{M_X^2}{\\cal C}^{(e)}_{\\alpha \\beta }(\\overline{u^c}\\gamma _{\\mu }u)(\\overline{d^c}_{\\beta }\\gamma ^{\\mu }e_{\\alpha })~,$ ${\\cal O}_{d6}^{(\\nu )}=\\frac{g_X^2}{M_X^2}{\\cal C}^{(\\nu )}_{\\alpha \\beta \\gamma }(\\overline{u^c}\\gamma _{\\mu }d_{\\alpha })(\\overline{d^c}_{\\beta }\\gamma ^{\\mu }\\nu _{\\gamma })~,$ with ${\\cal C}^{(e^c)}_{\\alpha \\beta }=(R_u^\\dag L_u^*)_{11}(R_e^\\dag \\tilde{\\mu }^*\\frac{1}{M_{e^c\\hat{e}^c}^*}L_u^*P_1^*V_{CKM})_{\\alpha \\beta }+(R_u^\\dag L_u^*P_1^*V_{CKM})_{1\\beta }(R_e^\\dag \\tilde{\\mu }^*\\frac{1}{M_{e^c\\hat{e}^c}^*}L_u^*)_{\\alpha 1} ~,$ ${\\cal C}^{(e)}_{\\alpha \\beta }=(R_u^\\dag L_u^*)_{11}(R_d^\\dag \\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }L_e^*)_{\\beta \\alpha } ~,$ ${\\cal C}^{(\\nu )}_{\\alpha \\beta \\gamma }=(R_u^\\dag L_u^*P_1^*V_{CKM})_{1\\alpha }(R_d^\\dag \\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }L_e^*)_{\\beta \\gamma }~,$ where in Eq.",
"(REF ) we have suppressed the color indices.",
"Similar to quark Yukawa matrices, the charged lepton Yukawa matrix has been diagonalized by transformation $L_e^\\dag Y_ER_e=Y_E^{\\rm Diag}$ .",
"All fields in Eq.",
"(REF ), are assumed to denote mass eigenstates.",
"We have ignored the neutrino masses (having no relevance for the nucleon decay) and rotated the neutrino flavors $\\nu _0=L_e^*\\nu $ similar to the left-handed charged leptons $e_0=L_e^*e$ .",
"As we will show now, with proper selection of appropriate parameters (such as $\\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}}$ , $\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }$ and/or corresponding entries in some of unitary matrices), appearing in Eq.",
"(REF ), we can adequately suppress nucleon decays within our model.The importance of flavor dependence in $d=6$ nucleon decay was discussed in Refs.",
"[26] and [27].",
"As was shown [27], in specific circumstances, within GUTs one can suppress or even completely rotate away the $d=6$ nucleon decays.",
"Upon the selection of parameters, the constraint (REF ) must be satisfied in order to obtain observed values of charged fermion masses.",
"Introducing the notations $R_u^\\dag L_u^*\\equiv {\\cal U}~,~~~R_d^\\dag \\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }L_e^*\\equiv {\\cal L}~,~~~R_e^\\dag \\tilde{\\mu }^*\\frac{1}{M_{e^c\\hat{e}^c}^*}L_u^*\\equiv {\\cal R}~,$ the couplings in Eq.",
"(REF ) can be rewritten as ${\\cal C}^{(e^c)}_{\\alpha \\beta }={\\cal U}_{11}({\\cal R}P_1^*V_{CKM})_{\\alpha \\beta }+({\\cal U}P_1^*V_{CKM})_{1\\beta }({\\cal R})_{\\alpha 1} ~,$ ${\\cal C}^{(e)}_{\\alpha \\beta }={\\cal U}_{11}{\\cal L}_{\\beta \\alpha } ~,~~~~{\\cal C}^{(\\nu )}_{\\alpha \\beta \\gamma }=({\\cal U}P_1^*V_{CKM})_{1\\alpha }{\\cal L}_{\\beta \\gamma }~.$ Since the matrices ${\\cal U}, {\\cal L}$ and ${\\cal R}$ are not fixed yet, for their structures we will make the selection ${\\cal U}_{11}=0,~~~{\\cal L}=\\left(\\!\\!\\begin{array}{ccc}\\epsilon _1& \\epsilon _2 & \\epsilon _3 \\\\\\times & \\times & \\times \\\\\\times & \\times & \\times \\\\\\end{array}\\!\\!\\right),~~~{\\cal R}=\\left(\\!\\!\\begin{array}{ccc}0& \\times & \\times \\\\0 & \\times & \\times \\\\\\times & \\times & \\times \\\\\\end{array}\\!\\!\\right),$ where $\\times $ stands for some nonzero entry.",
"With this structure we see that for $\\alpha ,\\beta =1,2$ we have ${\\cal C}^{(e^c)}_{\\alpha \\beta }={\\cal C}^{(e)}_{\\alpha \\beta }=0$ , and therefore nucleon decays with emission of the charged leptons do not take place.",
"With one more selection, we will be able to eliminate some nucleon decay modes (but not all) with neutrino emissions.",
"We can impose one more condition, involving ${\\cal U}_{12}$ and ${\\cal U}_{13}$ entries of ${\\cal U}$ , in such a way as to have $({\\cal U}P_1^*V_{CKM})_{11}=0$ .",
"The latter, in expanded form, reads $({\\cal U}P_1^*V_{CKM})_{11}={\\cal U}_{12}e^{-i\\omega _2}V_{cd}+{\\cal U}_{13}e^{-i\\omega _3}V_{td}=0~,~~~ \\Longrightarrow {\\cal U}_{12}e^{-i\\omega _2}=-\\frac{V_{td}}{V_{cd}}{\\cal U}_{13}e^{-i\\omega _3}$ and leads to ${\\cal C}^{(\\nu )}_{12\\gamma }={\\cal C}^{(\\nu )}_{11\\gamma }=0$ .",
"Thus, the decays $p\\rightarrow \\bar{\\nu }\\pi ^+, n\\rightarrow \\bar{\\nu }\\pi ^0, n\\rightarrow \\bar{\\nu }\\eta $ do not take place.",
"Nonvanishing relevant ${\\cal C}^{(\\nu )}$ couplings are ${\\cal C}^{(\\nu )}_{21\\gamma }$ , which, taking into account Eqs.",
"(REF ) and (REF ), are ${\\cal C}^{(\\nu )}_{21\\gamma }=({\\cal U}P_1^*V_{CKM})_{12}\\epsilon _{\\gamma }=\\epsilon _{\\gamma }{\\cal U}_{13}e^{-i\\omega _3}\\frac{V_{ts}V_{cd}-V_{td}V_{cs}}{V_{cd}}\\simeq \\epsilon _{\\gamma }{\\cal U}_{13}e^{-i\\omega _3}\\frac{s_{13}e^{i\\delta }}{V_{cd}}~,$ where in last step we have used standard parametrization of the CKM matrix.",
"Since the matrix ${\\cal U}$ is unitary, due to selection ${\\cal U}_{11}=0$ and the unitarity condition, we will have $|{\\cal U}_{12}|^2+|{\\cal U}_{13}|^2=1$ .",
"With this, by Eq.",
"(REF ) and using central values [28] of CKM matrix elements, we obtain $|{\\cal U}_{12}|\\simeq 0.038, |{\\cal U}_{13}|\\simeq 1$ and $|\\frac{s_{13}}{V_{cd}}|=|\\frac{V_{ub}}{V_{cd}}|\\simeq 1.56\\cdot 10^{-2}$ .",
"These give $|{\\cal C}^{(\\nu )}_{21\\gamma }|\\simeq 1.56\\cdot 10^{-2}|\\epsilon _{\\gamma }|$ .",
"Taking into account all this, for expressions of $p\\rightarrow \\bar{\\nu }K^+$ and $n\\rightarrow \\bar{\\nu }K^0$ decay widths, we obtain [29] $\\Gamma (p\\rightarrow \\bar{\\nu }K^+)\\!\\simeq \\!\\Gamma (n\\rightarrow \\bar{\\nu }K^0)\\!=\\!\\frac{(m_p^2-m_K^2)^2}{32\\pi f_{\\pi }^2m_p^3}\\!\\left(\\!1\\!+\\!\\frac{m_p}{3m_B}(D+3F)\\!\\!\\right)^2\\!\\!\\left(\\!\\frac{g_X}{M_X^2}A_R|\\alpha _H|\\!\\!\\right)^2 \\!\\!\\!\\cdot 2.43\\cdot 10^{-4}\\!\\sum _{\\gamma =1}^3|\\epsilon _{\\gamma }|^2$ where $|\\alpha _H|=0.012~{\\rm GeV}^3$ is a hadronic matrix element and $A_R=A_LA_S^l\\simeq 1.48$ takes into account long- ($A_L\\simeq 1.25$ ) and short-distance ($A_S^l\\simeq 1.18$ ) renormalization effects (see Refs.",
"[30] and [31], respectively.",
"Some details of the calculation of $A_S^l$ , within our model, are given in Appendix REF ).",
"To satisfy current experimental bound $\\tau _p^{exp}(p\\rightarrow \\bar{\\nu }K^+)\\stackrel{<}{_\\sim }5.9\\cdot 10^{33}$ years [32], for $M_X\\simeq 5\\cdot 10^{11}$ GeV and $\\alpha _X\\simeq 1/31$ , we need to have $\\sqrt{|\\epsilon _1|^2+|\\epsilon _3|^2+|\\epsilon _3|^2}\\stackrel{<}{_\\sim } 4.8\\cdot 10^{-6}$ .",
"This selection of parameters is fully consistent with the charged fermion masses.",
"Note, that with Eq.",
"(REF ) there is no conflict with the constraint of Eq.",
"(REF ).",
"We can lower values of $|\\epsilon _{\\gamma }|$ ; however, there is a low bound dictated from this constraint.",
"With $|{\\rm Det} (\\tilde{\\mu }\\frac{1}{M_{e^c\\hat{e}^c}})|\\cdot |{\\rm Det}(\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu })|=|{\\rm Det}({\\cal L})|\\cdot |{\\rm Det}({\\cal R})|\\sim 10^{-8}$ , the lowest value can be $|\\epsilon _{\\gamma }|\\sim 10^{-8}$ , obtained with $|{\\rm Det}({\\cal R})|\\sim 1$ .",
"More natural would be to have $|{\\rm Det}({\\cal R})|\\stackrel{<}{_\\sim } 10^{-2}$ , which suggests $|{\\rm Det}({\\cal L})|\\stackrel{>}{_\\sim } 10^{-6}$ , and therefore $\\sqrt{|\\epsilon _1|^2+|\\epsilon _3|^2+|\\epsilon _3|^2}\\stackrel{>}{_\\sim } \\sqrt{3}\\cdot 10^{-6}$ .",
"This dictates an upper bound for the proton lifetime $\\tau _p=\\tau (p\\rightarrow \\bar{\\nu }K^+)\\stackrel{<}{_\\sim } 5\\cdot 10^{34}$ years and will allow us to test the model in the future [32].",
"Besides $X, Y$ gauge boson mediated operators, there are $d=6$ operators generated by the exchange of colored triplet scalar $T_{H}$ .",
"From the couplings of Eq.",
"(REF ), we can see that the integration of $T_H$ induces baryon number violating $\\frac{1}{M_{T_H}^2}(q^TC_{qq}q)(q^TC_{ql}l)$ and $\\frac{1}{M_{T_H}^2}(u^cC_{u^ce^c}e^c)(u^cC_{u^cd^c}d^c)$ operators, which lead to the couplings $\\frac{1}{M_{T_H}^2}(q^TC_{qq}q)(q^TC_{ql}\\frac{1}{M_{\\hat{l}l}}\\hat{\\mu }l_0)$ and $\\frac{1}{M_{T_H}^2}(u^cC_{u^ce^c}\\frac{1}{M_{e^c\\hat{e}^c}^T}\\tilde{\\mu }^Te^c_0)(u^cC_{u^cd^c}d^c)$ .",
"Couplings $C_{ab}$ appearing in these operators are independent from Yukawa matrices, and proper suppression of relevant terms is possible [similar to the case of couplings in Eq.",
"(REF )], leaving fermion masses and a mixing pattern consistent with experiments.",
"To make a more definite statement about the nucleon lifetime, one has to study in detail the structure of Yukawa matrices.",
"In this respect, extension with flavor symmetries is a motivated framework and can play a crucial role in generating the desirable Yukawa textures [guaranteeing the forms given in Eq.",
"(REF )].",
"Preserving these issues for being addressed elsewhere, let us move to the next section."
],
[
"Various phenomenological constraints and implications",
"In this section we discuss and summarize some peculiarities, phenomenological implications of our model, and constraints needed to be satisfied in order to be consistent with experiments.",
"Also, we list issues opening prospects for further investigations within presented scenario: (i) The discovery of the Higgs boson [1], with mass $\\approx 126$ GeV, revealed that the Standard Model suffers from vacuum instability.",
"Detailed analysis has shown [2] that, due to RG, the Higgs self-coupling becomes negative near the scale $\\sim 10^{10}$ GeV.",
"If the Higgs field is insured to remain in the EW vacuum, the problem perhaps is not as severe.",
"However, with an inflationary universe with the Hubble parameter$\\gg 10^{10}$ GeV (preferred by the recent BICEP2 measurement [33]), the EW vacuum can be easily destabilized by the Higgs's move/tunneling to the \"true\" anti-de Sitter (AdS) vacuum [34].",
"Whether AdS domains take over or crunch depends on the details of inflation, the reheating process, nonminimal Higgs/inflaton couplings, etc.",
"(a detailed overview of these questions can be found in Refs.",
"[35] and [34]).",
"While these and related issues need more investigation, to be on t safe sehide, it is desirable to have a model with positive $\\lambda _h$ at all energy scales (up to the $M_{\\rm Pl}$ ).",
"Since within our model above the $\\Lambda ^{\\prime }$ scale new states appear, this problem can be avoided.",
"As was mentioned in Sec.",
", in our model a light SM doublet $h$ dominantly comes from the $H$ -plet.",
"The coupling $\\lambda _H(H^\\dag H)^2$ gives the self-interaction term $\\lambda _h(h^\\dag h)^2$ (with $\\lambda _h\\approx \\lambda _H$ at the GUT scale).",
"The running of $\\lambda _h$ will be given by $16\\pi ^2\\frac{d}{dt}\\lambda _h=\\beta _{\\lambda _h}^{SM}+\\Delta \\beta _{\\lambda _h}~,$ where $\\beta _{\\lambda _h}^{SM}$ corresponds to the SM part, while $\\Delta \\beta _{\\lambda _h}$ accounts for new contributions.",
"Since the $H$ -plet in the potential (REF ) has additional interaction terms, some of those couplings can help to increase $\\lambda _h$ .",
"For instance, the couplings $\\lambda _{1H\\Phi }, \\lambda _{2H\\Phi }$ , $\\hat{h}$ , etc., contribute as $\\Delta \\beta _{\\lambda _h}\\approx \\frac{(\\lambda _{1H\\Phi })^2}{25}\\left[ 9\\theta (\\mu \\!-\\!M_{TT^{\\prime }})+6\\theta (\\mu \\!-\\!M_{DT^{\\prime }})+6\\theta (\\mu \\!-\\!M_{TD^{\\prime }})+4\\theta (\\mu \\!-\\!M_{DD^{\\prime }})\\right]$ $\\frac{(\\lambda _{2H\\Phi })^2}{10}\\left[3\\theta (\\mu \\!-\\!M_{DT^{\\prime }})+2\\theta (\\mu \\!-\\!M_{DD^{\\prime }}) \\right]+3\\hat{h}^2\\theta (\\mu \\!-\\!M_{T_{H^{\\prime }}})+\\cdots $ Detailed analysis requires numerical studies by solving the system of coupled RG equations (involving multiple couplingsFor methods studying the stability of multifield potentials, see Refs.",
"[3] and [36] and references therein.).",
"While this is beyond the scope of this work, we see that due to positive contributions (see above) into the $\\beta $ function, there is potential to prevent $\\lambda _h$ becoming negative all the way up to the Planck scale.",
"(ii) Since in our model leptons are composite, there will be additional contributions to their anomalous magnetic moment, given by [15] $\\delta a_{\\alpha }\\sim \\left(\\frac{m_{e_{\\alpha }}}{\\Lambda ^{\\prime }}\\right)^2 ~.$ Current experimental measurements [28] of the muon anomalous magnetic moment give $\\Delta a_{\\mu }^{\\rm exp}\\approx 6\\cdot 10^{-10}$ .",
"This, having in mind a possible range $\\sim (1/5-1)$ of an undetermined prefactor in the expression of Eq.",
"(REF ), constrains the scale $\\Lambda ^{\\prime }$ from below: $\\Lambda ^{\\prime }\\stackrel{>}{_\\sim } (1.8-4.3)$ TeV.",
"The selected value of $\\Lambda ^{\\prime }$ , within our model ($\\Lambda ^{\\prime }=1851$ GeV), fits well with this bound.In fact, this new contribution to $a_{\\mu }$ has the potential of resolving a $3-4\\sigma $ discrepancy [28] (if it will persist in the future) between the theory and experiment [37].",
"The value of $\\delta a_e$ is more suppressed (for $\\Lambda ^{\\prime }\\simeq 1.8$ TeV, we get $\\delta a_e\\sim 10^{-13}$ ) and is compatible with experiments ($\\Delta a_e^{\\rm exp}\\approx 2.7\\cdot 10^{-13}$ ).",
"Planned measurements [38] with reduced uncertainties will provide severe constraints and test the viability of the proposed scenario.",
"Similarly, having flavor violating couplings at the level of constituents (i.e., in the sector of $SU(3)^{\\prime }$ fermions $\\hat{q}, \\hat{u}^c, \\hat{d}^c$ ), the new contribution in $e_{\\alpha }\\rightarrow e_{\\beta }\\gamma $ rare decay processes will emerge.",
"For instance, the contribution in the $\\mu \\rightarrow e\\gamma $ transition amplitude will be $\\sim \\lambda _{12}\\frac{m_{\\mu }}{(\\Lambda ^{\\prime })^2}$ , where $\\lambda _{12}$ is (unknown) flavor violating coupling coming from the Yukawa sector of $\\hat{q}, \\hat{u}^c, \\hat{d}^c$ .",
"This gives $Br(\\mu \\rightarrow e\\gamma )\\sim \\lambda _{12}^2(\\frac{M_W}{\\Lambda ^{\\prime }})^4$ , and for $\\Lambda ^{\\prime }\\simeq 1.8$ TeV the constraint $\\lambda _{12}\\stackrel{<}{_\\sim }4\\cdot 10^{-4}$ should be satisfied in order to be consistent with the latest experimental limit $Br^{\\rm exp}(\\mu \\rightarrow e\\gamma )<5.7\\cdot 10^{-13}$ [39].",
"(iii) As was mentioned in Sec.",
"REF (and will be discussed also in Appendix ), the matter sector of $SU(3)^{\\prime }$ symmetry (ignoring EW and Yukawa interactions) possesses $G_f^{(6)}$ chiral symmetry with sextets $6_L\\sim \\hat{q}_{\\alpha }$ and $6_R\\sim \\hat{q}^c_{\\alpha }$ [see Eqs.",
"(REF ) and (REF )].",
"The breaking of this chiral symmetry proceeds by several steps.",
"At the first stage, at scale $\\Lambda ^{\\prime }\\approx 1.8$ TeV, the condensates $\\langle 6_L6_LT_{H^{\\prime }}^\\dag \\rangle \\sim \\langle 6_R6_RT_{H^{\\prime }} \\rangle \\sim \\Lambda ^{\\prime }$ break the $G_f^{(6)}$ .",
"However, these condensates preserve SM gauge symmetry.",
"At the next stage (of chiral symmetry breaking), the condensate $\\langle 6_L6_R\\rangle \\equiv F_{\\pi ^{\\prime }}$ , together with the Higgs VEV $\\langle h\\rangle \\equiv v_h$ , contributes to the EW symmetry breaking.",
"The $F_{\\pi ^{\\prime }}$ denotes the decay constant of the (techni) $\\pi ^{\\prime }$ meson and should satisfy $v_h^2+F_{\\pi ^{\\prime }}^2=(246.2~{\\rm GeV})^2$ .",
"With the light (very SM-like) Higgs boson mainly residing in $h$ and with $F_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim }0.2v_h$ , the $h$ 's signal will be very compatible with LHC data [40].",
"Since the low-energy potential would involve VEVs $\\langle 6_L6_LT_{H^{\\prime }}^\\dag \\rangle , \\langle 6_R6_RT_{H^{\\prime }} \\rangle , F_{\\pi ^{\\prime }}$ and $v_h$ , obtaining mild hierarchy $\\frac{F_{\\pi ^{\\prime }}}{\\Lambda ^{\\prime }}\\stackrel{<}{_\\sim }1/40$ will be possible by proper selection (not by severe fine-tunings) of parameters from perturbative and nonperturbative (effective) potentials.",
"The situation here (i.e., the symmetry breaking pattern, potential (being quite involved because of these VEVs), etc.)",
"will differ from case obtained within QCD with $SU(n)_L\\times SU(n)_R$ chiral symmetry and with the $\\langle n_L\\times n_R\\rangle $ condensate only [41].",
"Moreover, the hierarchy between the confinement scale and the decay constant can have some dynamical origin (see, e.g., Refs.If a conformal window is realized, the value of $F_{\\pi ^{\\prime }}$ can be more reduced [43].",
"[42]).",
"Without addressing these details, our approach is rather phenomenological, with the assumption $F_{\\pi ^{\\prime }}/v_h\\stackrel{<}{_\\sim }0.2$ and $h$ being the Higgs boson (with mass $\\approx 126$ GeV), such that there is allowed a window for a heavier $\\pi ^{\\prime }$ state and the model is compatible with current experiments [44].",
"Models with partially composite Higgs, in which the light Higgs doublet has some ed-mixture of a composite (technipion $\\pi ^{\\prime }$ ) state, with various interesting implications (including necessary constraints, limits, and compatibility with LHC data), were studied in Ref.",
"[40].",
"As mentioned in Sec.",
", it is possible to have unification with the symmetry breaking pattern and the spectrum of intermediate states that give larger values of $\\Lambda ^{\\prime }$ (even with $\\Lambda ^{\\prime }\\sim 10^5$ GeV).",
"However, in such a case, the value of $F_{\\pi ^{\\prime }}$ would be also large, and it would be impossible to bring $F_{\\pi ^{\\prime }}$ to the low value even with fine-tuning.",
"This would mean that the EW symmetry breaking scale would be also large.",
"That is why such a possibility has not been considered.",
"In addition, it is rather generic that the model with composite leptons will be accompanied with excited massive leptons (lepton resonances).",
"Current experiments have placed low bounds on masses of the excited electron and muon to be heavier than $\\sim 1.8$ TeV.",
"This scale is close to the value of $\\Lambda ^{\\prime }$ we have chosen within our model, and will allow us to test the lepton substructure [45] hopefully in the not-far future.",
"Details, related to these issues, deserve separate investigations.",
"(iv) Since the condensate $\\langle 6_L6_R\\rangle =F_{\\pi ^{\\prime }}$ , by some amount, can contribute to the chiral [of the $SU(3)^{\\prime }$ strong sector] and EW symmetry breaking, the scenario shares some properties of hybrid technicolor models with fundamental Higgs states.",
"Moreover, together with technipion $\\pi ^{\\prime }$ , near the $\\Lambda ^{\\prime }$ scale, there will be technimeson states $\\rho _T, \\omega _T$ , etc., with peculiar signatures [46], [47], which can be probed by collider experiments.",
"(v) Because the new states around and above the $\\Lambda ^{\\prime }\\approx 1.8$ TeV scale, there will be additional corrections to the EW precision parameters $T, S, U$ etc.",
"While because strong dynamics near the $\\Lambda ^{\\prime }$ scale, the accurate calculations require some effort, the symmetry arguments provide a good estimate of the additional corrections $\\Delta T, \\Delta S$ , etc.",
"One can easily notice that the isospin breaking effects are suppressed in the sector of additional states.",
"Therefore the mass splittings between doublet components of the additional states will be suppressed (i.e.",
"$\\Delta M\\ll M$ ) and pieces $\\Delta T_f, \\Delta T_s$ of $\\Delta T=\\Delta T_f+\\Delta T_s$ will be given as [48] $\\Delta T_f\\simeq \\frac{N_f}{12\\pi s_W^2}\\left(\\frac{\\Delta M_f}{m_W}\\right)^2~,~~~~~\\Delta T_s\\simeq \\frac{N_s}{24\\pi s_W^2}\\left(\\frac{\\Delta M_s}{m_W}\\right)^2~,$ where subscripts $f$ and $s$ stand for fermions, and scalars, respectively and $N_f, N_s$ account for the multiplicity [or dimension with respect to the group different from $SU(2)_w$ ] of the corresponding doublet state.",
"One can easily verify that within our model in the sector of extra vectorlike $(\\hat{l}+l)_{\\alpha }$ states the mass splitting between doublet components is suppressed as $\\Delta M_{\\hat{l}l}^{(\\alpha )}\\stackrel{<}{_\\sim }\\frac{v_h^2}{M_{\\hat{l}l}^{(\\alpha )}}$ .",
"This, according to Eq.",
"(REF ) and Table REF , gives the negligible contribution: $\\Delta T_{\\hat{l}l}\\stackrel{<}{_\\sim }\\frac{2 \\cdot 2}{12\\pi s_W^2}v_h^4/(m_WM_{\\hat{l}l}^{(1)})^2\\sim 10^{-5}$ .",
"Within the fragments of the scalar $\\Phi $ , the lightest is $\\Phi _{DT^{\\prime }}$ with mass $M_{DT^{\\prime }}\\simeq 8.3$ TeV.",
"Splitting between the doublet components comes from the potential term $\\frac{\\lambda _{2H\\Phi }}{\\sqrt{10}} \\!H^\\dag \\Phi \\Phi ^\\dag \\!H $ , giving $\\Delta M_{DT^{\\prime }}\\simeq \\lambda _{2H\\Phi }v_h^2/(4\\sqrt{10}M_{DT^{\\prime }})$ .",
"This, according to Eq.",
"(REF ), causes enough suppression: $\\Delta T_{DT^{\\prime }}\\simeq \\frac{3}{24\\pi s_W^2}\\lambda _{4H\\Phi }^2v_h^4/(2\\sqrt{10}M_{DT^{\\prime }}m_W)^2\\stackrel{<}{_\\sim }2\\cdot 10^{-5}$ (for $\\lambda _{2H\\Phi }\\stackrel{<}{_\\sim }1.5$ ).",
"As pointed out above, besides the fundamental Higgs doublet ($h$ ), which dominantly includes SM Higgs, there is a composite doublet ($\\pi ^{\\prime }$ - similar to technicolor models) with suppressed VEV - $F_{\\pi ^{\\prime }}$ .",
"Contribution of this extra doublet, into the $T$ parameter, is estimated to be $\\Delta T_{\\pi ^{\\prime }}\\approx \\frac{1}{24\\pi s_W^2}\\!\\left(\\!\\frac{\\Delta M_{\\pi ^{\\prime }}}{m_W}\\!\\right)^2 -\\frac{c_W^2}{4\\pi }c_{\\pi ^{\\prime }}^2\\ln \\frac{M_{\\pi ^{\\prime }}^2}{m_Z^2} ~,$ where the first term is due to the mass splitting $\\Delta M_{\\pi ^{\\prime }}(\\sim v_h^2/(4M_{\\pi ^{\\prime }})$ ) between doublet components of $\\pi ^{\\prime }$ , while second term emerges due to the VEV $\\langle \\pi ^{\\prime }\\rangle =F_{\\pi ^{\\prime }}$ with $c_{\\pi ^{\\prime }}\\approx 2m_Z^2F_{\\pi ^{\\prime }}/(M_{\\pi ^{\\prime }}^2v_h)$ (where $F_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim }0.2v_h$ ).",
"This contribution is also small ($\\Delta T_{\\pi ^{\\prime }}\\approx 2\\cdot 10^{-3}$ ) for $M_{\\pi ^{\\prime }}\\sim 1$ TeV.",
"Since $\\pi ^{\\prime }$ is a composite state, due to the strong dynamics, special care is needed to derive a more accurate result (as was done in Ref.",
"[49] for models with a single composite Higgs performing proper matching at different energy scales).",
"However, since $\\Delta T_{\\pi ^{\\prime }}$ is protected by isospin symmetry, we limit ourselves to the estimates performed here.",
"Moreover, the source of the isospin breaking in the strong $SU(3)^{\\prime }$ sector is $F_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim }0.2v_h$ , causing the mass splitting between composite \"technihadrons\" (denoted collectively as $\\lbrace \\rho ^{\\prime } \\rbrace $ ) of $\\Delta M_{\\rho ^{\\prime }}\\sim F_{\\pi ^{\\prime }}^2/M_{\\rho ^{\\prime }}$ .",
"This, for $M_{\\rho ^{\\prime }}\\sim \\Lambda ^{\\prime }$ , would give the correction $\\Delta T_{\\rho ^{\\prime }}\\sim \\frac{1}{12\\pi s_W^2}F_{\\pi ^{\\prime }}^4/(m_WM_{\\rho ^{\\prime }})^2\\stackrel{<}{_\\sim } 10^{-5}$ .",
"Note that the direct isospin (custodial symmetry) breaking within $\\hat{q}_{\\alpha }$ states is much more suppressed (we have no direct EW symmetry breaking in the Yukawa sector of $\\hat{q}, \\hat{u}^c$ , and $\\hat{d}^c$ states) and thus conclude that within the considered scenario extra corrections to the $T$ parameter are under control.",
"Let us now give the estimate of the additional contributions into the $S$ parameter.",
"Contributions to this parameter from the additional vectorlike $(\\hat{l}+l)_{\\alpha }$ , $(\\hat{e}^c+e^c)_{\\alpha }$ states decouple [50] and are estimated to be $\\Delta S_{\\hat{l}l}\\sim \\Delta S_{\\hat{e}^c e^c}\\stackrel{<}{_\\sim }\\frac{1}{4\\pi }\\frac{v_h^2}{(M_{\\hat{l}l}^{(1)})^2}\\ln \\frac{M_{\\hat{l}l}^{(1)}}{m_{\\tau }}\\sim 10^{-5}$ .",
"The contribution from the scalar $\\Phi _{DT^{\\prime }}$ is $\\Delta S_{DT^{\\prime }}\\simeq \\frac{3}{6\\pi }\\Delta M_{DT^{\\prime }}/M_{DT^{\\prime }}\\simeq \\lambda _{2H\\Phi }v_h^2/(8\\pi \\sqrt{10}M_{DT^{\\prime }}^2)\\stackrel{<}{_\\sim } 2\\cdot 10^{-5}$ , also suppressed, as expected.",
"The contribution of extra (heavy $\\pi ^{\\prime }$ ) composite doublet is $\\Delta S_{\\pi ^{\\prime }}\\approx \\frac{1}{6\\pi }\\frac{\\Delta M_{\\pi ^{\\prime }}}{M_{\\pi ^{\\prime }}}+\\frac{1}{6\\pi }c_{\\pi ^{\\prime }}^2\\ln \\frac{M_{\\pi ^{\\prime }}}{m_h}~,$ where first term is due to the splitting of the doublet components, while second term comes from the VEV $\\langle \\pi ^{\\prime }\\rangle =F_{\\pi ^{\\prime }}$ .",
"With $\\Delta M_{\\pi ^{\\prime }}\\sim v_h^2/(4M_{\\pi ^{\\prime }})$ and $M_{\\pi ^{\\prime }}\\stackrel{>}{_\\sim }1$ TeV Eq.",
"(REF ) gives $\\Delta S_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim } 10^{-3}$ .",
"Similarly suppressed contributions would arise from the techni$-{\\rho ^{\\prime }}$ hadrons: $\\Delta S_{\\rho ^{\\prime }}\\sim \\frac{1}{6\\pi }\\Delta M_{\\rho ^{\\prime }}/M_{\\rho ^{\\prime }}\\sim \\frac{1}{6\\pi }F_{\\pi ^{\\prime }}^2/M_{\\rho ^{\\prime }}^2\\stackrel{<}{_\\sim }4\\cdot 10^{-5}$ (for $M_{\\rho ^{\\prime }}\\sim \\Lambda ^{\\prime }$ ).",
"As far as the contribution from the matter states $\\hat{q}, \\hat{u}^c, \\hat{d}^c$ are concerned, since their masses are too suppressed, in the chiral limit $\\frac{m_f}{m_Z}\\rightarrow 0$ , we can use the expression [48] $\\Delta S_f\\rightarrow \\frac{N_fY_f}{6\\pi }\\left(-2\\ln \\frac{x_1}{x_2}+G(x_1)-G(x_2)\\right)~,$ ${\\rm with}~~~~~~G(x)=-4{\\rm arc\\!\\tanh }\\frac{1}{\\sqrt{1-4x}}~,~~~~~x_i=\\frac{m_{fi}^2}{m_Z^2}~,$ where $m_{f1,2}$ are masses of the components ofthe $f$ fermion with hypercharge $Y_f$ .",
"Verifying that in the limit $x\\rightarrow 0$ the function $G(x)$ goes to $2\\ln x$ , we see that expression for $\\Delta S_f$ in Eq.",
"(REF ) vanishes.",
"Moreover, new contributions to the $U$ parameter are more suppressed.",
"For instance, the contribution due to the $\\pi ^{\\prime }$ is $\\Delta U_{\\pi ^{\\prime }}\\approx \\frac{1}{15\\pi }\\left(\\!\\frac{\\Delta M_{\\pi ^{\\prime }}}{M_{\\pi ^{\\prime }}}\\!\\right)^2 \\!\\!-\\frac{1}{12\\pi }c_{\\pi }^2\\frac{\\Delta M_{\\pi ^{\\prime }}}{M_{\\pi ^{\\prime }}}~,$ which for $M_{\\pi ^{\\prime }}\\sim 1$ TeV, $F_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim }0.2v_h$ becomes $\\Delta U_{\\pi ^{\\prime }}\\stackrel{<}{_\\sim }5\\cdot 10^{-6}$ .",
"All other new contributions to the $U$ are also more suppressed than the corresponding $\\Delta S$ and $\\Delta T$ .",
"This is understandable since $U$ is related to the effective operator with a dimension higher than those of $S$ and $T$ .",
"All these allow us to conclude that new contributions to the EW precision parameters are well below the current experimental bounds [51].",
"(vi) Within the proposed model, spontaneous breaking of two non-Abelian groups $SU(5)\\times SU(5)^{\\prime }$ and discrete $D_2$ parity will give monopole and domain wall solutions, respectively.",
"Since the symmetry breaking scales are relatively low ($\\stackrel{<}{_\\sim }5\\cdot 10^{11}$ GeV), the inflation would not dilute number densities of these topological defects in a straightforward way.",
"Thus, one can think of alternative solutions.",
"For instance, as it was shown in Refs.",
"[52], within models with a certain field content and couplings, it is possible that symmetry restoration cannot happen for arbitrary high temperatures.",
"This would avoid the phase transitions (which usually cause the formation of topological defects).",
"Moreover, by proper selection of the model parameters, it is possible to suppress the thermal production rates of the topological defects (for detailed discussions, see the last two works of Ref.",
"[52]).",
"From this viewpoint, our model with a multiscalar sector and various couplings has potential to avoid domain wall and monopole problems.",
"Thus, it is inviting to investigate the parameter space and see how desirable ranges are compatible with those needed values appearing in Eq.",
"(REF ) (for \"improving\" the running of $\\lambda _h$ ).",
"To cure problems related with topological defects, also other different noninflationary solutions have been proposed [53], and one (if not all) of them could be invoked as well.",
"Certainly, these and other cosmological implications, of the presented scenario, deserve separate investigations.",
"At the end let us note that it would be interesting to build a supersymmetric extension of the considered $SU(5)\\times SU(5)^{\\prime }\\times D_2$ GUT and study related phenomenology.",
"These and related issues will be addressed elsewhere."
],
[
"Acknowledgments",
"I am grateful to K.S.",
"Babu, J. Chkareuli, I. Gogoladze, and S. Raby for useful comments and discussions.",
"The partial support from Shota Rustaveli National Science Foundation (Contracts No.",
"31/89 and No.",
"03/113) are kindly acknowledged.",
"I would like to thank CETUP* (Center for Theoretical Underground Physics and Related Areas), supported by the US Department of Energy under Grant No.",
"DE-SC0010137 and by the US National Science Foundation under Grant No.",
"PHY-1342611, for its hospitality and partial support during the 2013 Summer Program.",
"I also thank Barbara Szczerbinska for providing a stimulating atmosphere in Deadwood during this program."
],
[
"Composite leptons and anomaly matching",
"Here we demonstrate how the composite leptons emerge within our scenario and also discuss anomaly matching conditions.",
"As was noted in Sec.",
"REF , the sector of $\\hat{q}, \\hat{u}^c$ , and $\\hat{d}^c$ states have $G_f^{(6)}$ chiral symmetry [see Eq.",
"(REF )] with the transformation properties of these states given in Eq.",
"(REF ).",
"At scale $SU(3)^{\\prime }$ interaction becomes strong, and the $G_f^{(6)}$ symmetry breaking condensates can be formed.",
"The chiral symmetry breaking can proceed through several steps, and at each level the formed composite states should satisfy anomaly matching conditions [14].",
"The bilinear [$SU(3)^{\\prime }$ -invariant] condensate can be $\\langle 6_L\\times 6_R\\rangle =F_{\\pi ^{\\prime }}$ , with corresponding breaking scale $F_{\\pi ^{\\prime }}$ .",
"As was shown in Ref.",
"[41], with only fundamental states, the chiral symmetry $SU(n)_L\\times SU(n)_R$ will be broken down to the diagonal $SU(n)_{L+R}$ symmetry.",
"Since in our case $F_{\\pi ^{\\prime }}$ also contributes to EW symmetry breaking, we have a bound $F_{\\pi ^{\\prime }}\\stackrel{<}{\\sim }100$ GeV.",
"This scale, in comparison with $\\Lambda ^{\\prime }\\sim $ few$\\times $ TeV, can be ignored at the first stage.",
"Moreover, in our case, light $SU(3)^{\\prime }$ nonsinglet field content is reacher, and the chiral symmetry breaking pattern is also different.",
"Other $SU(3)^{\\prime }$ invariant condensates, including matter bilinears, are $\\langle 6_L6_LT_{H^{\\prime }}^\\dag \\rangle ~~~~~~~{\\rm and} ~~~~~~~\\langle 6_R6_RT_{H^{\\prime }} \\rangle ~.$ Note, that the product of $SU(6)$ sextets gives either symmetric or antisymmetric representations ($6\\times 6=15_A+21_S$ ), but due to $SU(3)^{\\prime }$ contractions, in Eq.",
"(REF ) the antisymmetric 15-plets (i.e.",
"$15_L$ and $15_R$ ) participate.",
"The condensates (REF ) transform as $15_L$ and $15_R$ under $SU(6)_L$ and $SU(6)_R$ , respectively, and therefore break these symmetries.",
"A possible breaking channel is $SU(6)_L\\rightarrow SU(4)_L\\times SU(2)^{\\prime }_L\\equiv G_L^{(4,2)}, ~~~~~~~SU(6)_R\\rightarrow SU(4)_R\\times SU(2)^{\\prime }_R\\equiv G_R^{(4,2)} .$ Indeed, with respect to $G_L^{(4,2)}$ and $G_R^{(4,2)}$ , the $15_L$ and $15_R$ decompose as $SU(6)_L\\rightarrow G_L^{(4,2)} :~~~15_L=(1,1)_L+(6,1)_L+(4,2)_L~,$ $SU(6)_R\\rightarrow G_R^{(4,2)} :~~~15_R=(1,1)_R+(6,1)_R+(4,2)_R~,$ and the VEVs $\\langle (1,1)_L\\rangle $ and $\\langle (1,1)_R\\rangle $ leave $G_L^{(4,2)}\\times G_R^{(4,2)}$ chiral symmetry unbroken.",
"The singlet components ($\\langle (1,1)_L\\rangle $ and $\\langle (1,1)_R\\rangle $ ) from Eq.",
"(REF ) are $\\frac{1}{2}\\langle \\hat{q}\\hat{q}T_{H^{\\prime }}^\\dag \\rangle =\\langle \\hat{u}\\hat{d}T_{H^{\\prime }}^\\dag \\rangle $ and $\\langle \\hat{u}^c\\hat{d}^c T_{H^{\\prime }}\\rangle $ combinations, which leave $G_{SM}$ gauge symmetry unbroken.",
"Therefore, the values of these condensates can be $\\sim $ few$\\cdot $ TeV($\\sim \\Lambda ^{\\prime }$ ) without causing any phenomenological difficulties.",
"Thus, as the first stage of the chiral symmetry breaking, we stick to the channel $G_f^{(6)}\\stackrel{\\Lambda ^{\\prime }}{_{\\longrightarrow }} G_L^{(4,2)}\\times G_R^{(4,2)}\\times U(1)_{B^{\\prime }} ~,$ with $\\langle 6_L6_LT_{H^{\\prime }}^\\dag \\rangle =\\langle \\hat{u}\\hat{d}T_{H^{\\prime }}^\\dag \\rangle \\sim \\Lambda ^{\\prime } ,~~~~\\langle 6_R6_RT_{H^{\\prime }} \\rangle =\\langle \\hat{u}^c\\hat{d}^c T_{H^{\\prime }}\\rangle \\sim \\Lambda ^{\\prime } ~.$ The $SU(6)_{L,R}$ sextets under $G^{(4,2)}_{L,R}$ are decomposed as $6_L=(4,1)_L+(1,2)_L$ and $6_R=(4,1)_R+(1,2)_R$ , respectively.",
"If composite objects are picked up as $(4^{\\prime },1)_{L,R}\\subset [(4,1)_{L,R}]^3$ and $(1,2^{\\prime })_{L,R}\\subset [(1,2)_{L,R}]^3$ , then one can easily check out that the anomalies (of initial and composite states) indeed match and $(4^{\\prime },1)_{L,R}$ and $(1,2^{\\prime })_{L,R}$ can be identified with three families of leptons plus three states of right-handed/sterile neutrinos.",
"For demonstrating all these, it is more convenient to work in a different basis.",
"That would also make it simpler to identify composite states.",
"As it is well known (and in our case turns out more useful), one can describe the $SU(6)$ symmetry (and its representations as well) by its special subgroup (\"S-subgroup\" [54]) $SU(3)_f\\otimes SU(2)\\subset SU(6)$ .",
"In our case, $SU(6)_L\\supset SU(3)_{fL}\\otimes SU(2)_L ~,~~~~SU(6)_R\\supset SU(3)_{fR}\\otimes SU(2)_R ~.$ Under these S subgroups, the sextets decompose as Similar to the description of three-flavor QCD with $(u, d, s)$ spin-1/2 states, either by the sextet of $SU(6)$ or by $(3,2)$ of $SU(3)_f\\times SU(2)_s$ - the Wigner-Weyl realization of the $SU(6)$ chiral symmetry.",
"Here, however, $SU(2)_s$ stands for the spin group and $SU(3)_f$ for the flavor.",
"In our case of Eq.",
"(REF ), $SU(2)$ factors act like isospin rotations relating $\\hat{u}_{\\alpha }$ and $\\hat{d}_{\\alpha }$ and $\\hat{u}^c_{\\alpha }$ with $\\hat{d}^c_{\\alpha }$ , respectively ($\\alpha =1,2,3$ ).",
"$\\hat{q}(6_L)=\\hat{q}(3, 2)_L~,~~~~\\hat{q}^c(6_R)=\\hat{q}^c(3, 2)_R~.$ In these decompositions, $\\hat{q}$ and $\\hat{q}^c$ can be written as matrices, $\\begin{array}{ccc}& {\\begin{array}{ccc}&~~ ~~ {~}_{\\leftarrow ~SU(3)_{fL} ~\\rightarrow }&\\\\\\end{array}}\\\\&{\\!",
"\\hat{q}= \\left(\\begin{array}{ccc}\\end{array}\\hat{u}& ~~\\hat{c} & ~~\\hat{t}\\\\\\hat{d} &~~ \\hat{s} & ~~\\hat{b}\\right.",
"}\\end{array} \\!\\!\\!\\!\\!\\!\\!\\begin{array}{c}\\\\ {~}_{\\uparrow } \\\\ \\!",
"{~}_{SU(2)_L}\\!",
"\\\\ \\vspace{8.5359pt}{~}_{\\downarrow } ~\\end{array}~~~,~~~~~~\\begin{array}{ccc}& {\\begin{array}{ccc}&~~ ~~ {~}_{\\leftarrow ~SU(3)_{fR} ~\\rightarrow }&\\\\\\end{array}}\\\\&{\\!",
"\\hat{q}^c= \\left(\\begin{array}{ccc}\\end{array}\\hat{u}^c& ~~\\hat{c}^c & ~~\\hat{t}^c\\\\\\hat{d}^c &~~ \\hat{s}^c & ~~\\hat{b}^c\\right.",
"}\\end{array} \\!\\!\\!\\!\\!\\!\\!\\begin{array}{c}\\\\ {~}_{\\uparrow } \\\\ \\!",
"{~}_{SU(2)_R}\\!",
"\\\\ \\vspace{8.5359pt}{~}_{\\downarrow } ~\\end{array}~,~$ where schematically actions of $SU(3)$ and $SU(2)$ rotations are depicted.",
"Therefore, transformation properties under the chiral group $G_f^{(3,2)}=SU(3)_{fL}\\otimes SU(2)_L \\times SU(3)_{fR}\\otimes SU(2)_R\\times U(1)_{B^{\\prime }}$ are: $G_f^{(3,2)}~:~~~\\hat{q}\\sim \\left(3_{fL}, ~2_L,~ 1,~ 1,~ \\frac{1}{3}\\right)~,~~~~~~\\hat{q}^c\\sim \\left(1, ~1,~ 3_{fR},~ 2_R,~ -\\frac{1}{3}\\right)~.~$ Relevant anomalies that donot vanish are $A\\left([SU(3)_{fL}]^2\\!\\!\\cdot U(1)_{B^{\\prime }}\\!\\right)\\!=\\!-A\\left([SU(3)_{fR}]^2\\!\\!\\cdot U(1)_{B^{\\prime }}\\!\\right)=\\!",
"1~,$ $A\\left([SU(2)_L]^2\\!\\!\\cdot U(1)_{B^{\\prime }}\\!\\right)\\!=\\!-A\\left([SU(2)_R]^2\\!\\!\\cdot U(1)_{B^{\\prime }}\\!\\right)=\\!",
"\\frac{3}{2}~.$ The anomaly matching condition can be satisfied with the spontaneous breaking of the symmetries $SU(3)_{fL}$ and $SU(3)_{fR}$ down to $SU(2)_{fL}$ and $SU(2)_{fR}$ , respectively.",
"[This happens by condensates (REF ) discussed above.]",
"Thus, the chiral symmetry $G_f^{(3,2)}$ is broken down to $G_f^{(2,2)}$ , where $G_f^{(2,2)}=SU(2)_{fL}\\otimes SU(2)_L \\times SU(2)_{fR}\\otimes SU(2)_R\\times U(1)_{B^{\\prime }}~.$ This breaking is realized, for instance, by the condensates $\\langle \\hat{u}_3\\hat{d}_3T_{H^{\\prime }}^\\dag \\rangle $ and $\\langle \\hat{u}^c_3\\hat{d}^c_3T_{H^{\\prime }} \\rangle $ .",
"Note that with $SU(3)_{fL}\\rightarrow SU(2)_{fL}$ and $SU(3)_{fR}\\rightarrow SU(2)_{fR}$ we will have decompositions $3_{fL}=2_{fL}+1_{fL}$ and $3_{fR}=2_{fR}+1_{fR}$ .",
"At the composite level, the spin-1/2 and $SU(3)^{\\prime }$ singlet combinations $(\\hat{q} \\hat{q})\\hat{q}$ and $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ picked up as $[2^{\\prime }_{fL}+1^{\\prime }_{fL}]$ from $[2_{fL}+1_{fL}]^3$ and $[2^{\\prime }_{fR}+1^{\\prime }_{fR}]$ from $[2_{fR}+1_{fR}]^3$ .",
"Thus, transformations of $(\\hat{q} \\hat{q})\\hat{q}$ and $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ composites under $G_f^{(2,2)}$ areUnder combination $(\\hat{q} \\hat{q})\\hat{q}$ (suppressed gauge/chiral indices), we mean $\\epsilon ^{a^{\\prime }b^{\\prime }c^{\\prime }}\\!\\epsilon _{ij}(\\hat{q}_{a^{\\prime }i} \\hat{q}_{b^{\\prime }j})\\hat{q}_{c^{\\prime }k}$ , where $a^{\\prime }, b^{\\prime }, c^{\\prime }=1,2,3$ are $SU(3)^{\\prime }$ indices and $i,j,k=1,2$ stand for $SU(2)_L/SU(2)_w$ indices.",
"Similar is applied to the combination $(\\hat{q}^c \\hat{q}^c)\\hat{q}^c$ .",
"$G_f^{(2,2)}~:~~~(\\hat{q} \\hat{q})\\hat{q}\\sim \\left([2_{fL}+1_{fL}], ~2_L,~ 1,~ 1,~ 1\\right)~,~~~~~~(\\hat{q}^c \\hat{q}^c)\\hat{q}^c\\sim \\left(1, ~1,~ [2_{fR}+1_{fR}],~ 2_R,~ -1\\right)~.$ These representations will have anomalies that precisely match with those given in Eq.",
"(REF ).",
"Thus, we have three families of $l_0, e^c_0, \\nu ^c_0$ composite states represented in Eq.",
"(REF ), with transformation properties under $G_{SM}$ given in Eq.",
"(REF )."
],
[
"RG equations and $b$ factors",
"In this appendix we discuss details of gauge coupling unification and present one- and two-loop RG coefficients at each relevant energy scale.",
"At the end we calculate short-range renormalization factors $A_S^l$ and $A_S^{e^c}$ for baryon number violating $d=6$ operators.",
"The two-loop RG equation, for gauge coupling $\\alpha _i$ , has the form [55] $\\frac{d}{d\\ln \\mu }\\alpha _i^{-1}=-\\frac{b_i}{2\\pi }-\\frac{1}{8\\pi ^2}\\sum _jb_{ij}\\alpha _j+\\frac{1}{32\\pi ^3}\\sum _fa_i^{f}\\lambda _f^2 ,$ where $b_i$ and $b_{ij}$ account for one- and two-loop gauge contributions, respectively, and $c_i^{f}$ represents the two-loop correction via Yukawa coupling $\\lambda _f$ .",
"For consistency, it is enough to consider the Yukawa coupling RG at the one-loop approximation: $16\\pi ^2\\frac{d}{d\\ln \\mu }\\lambda _f=c_f\\lambda _f^3+\\lambda _f(\\sum _{f^{\\prime }}d_f^{f^{\\prime }}\\lambda _{f^{\\prime }}^2-4\\pi \\sum _ic_f^i\\alpha _i)~.$ RG factors can be calculated using general formulas [55].",
"Since at different energy scales different states appear, these factors also change with energy.",
"For instance, at scale $\\mu $ , the $b_i$ and $b_{ij}$ can be written as $b_i(\\mu )=\\sum _a \\theta (\\mu -M_a)b_i^a$ and $b_{ij}(\\mu )=\\sum _a \\theta (\\mu -M_a)b_{ij}^a$ , where $a$ stands for the state with mass $M_a$ and step function $\\theta (x)=0$ for $x\\le 0$ , and $\\theta (x)=1$ for $x> 0$ .",
"Integration of Eq.",
"(REF ), in energy interval $\\mu _1-\\mu _2$ , gives $\\alpha _i^{-1}(\\mu _2)=\\alpha _i^{-1}(\\mu _1)-\\frac{b_i^{\\mu _1\\mu _2}}{2\\pi }\\ln \\frac{\\mu _2}{\\mu _1}~,$ where an effective $b_i^{\\mu _1\\mu _2}$ factor is given by $b_i^{\\mu _1\\mu _2}=\\!\\!\\left(\\!\\sum _a\\theta (\\mu _2-M_a)b_i^a\\ln \\frac{\\mu _2}{M_a}+\\frac{1}{4\\pi }\\sum _a\\!\\!\\int _{\\mu _1}^{\\mu _2}\\!\\!\\!\\theta (\\mu -M_a)b_{ij}^a\\alpha _jd\\ln \\mu -\\frac{1}{8\\pi ^2}\\int _{\\mu _1}^{\\mu _2}\\!\\!\\!c_i^f\\lambda _f^2d\\ln \\mu \\right)\\!\\frac{1}{\\ln \\frac{\\mu _2}{\\mu _1}} ~.$ The second and third terms in Eq.",
"(REF ) can be evaluated iteratively [56].",
"Although Eq.",
"in (REF ) can be solved numerically (which we do perform for obtaining final results), expressions (REF ) and (REF ) are useful for understanding how unification works.",
"In the energy interval $M_Z-\\Lambda ^{\\prime }$ , we have just SM, while between $\\Lambda ^{\\prime }$ and $M_I$ scales, we have $G_{SM}\\times SU(3)^{\\prime }$ gauge interactions plus additional states.",
"Applying Eq.",
"(REF ) for the couplings $\\alpha _Y, \\alpha _w, \\alpha _c$ , and $\\alpha _{3^{\\prime }}$ , we will have $\\alpha _i^{-1}(M_I)=\\alpha _i^{-1}(M_Z)-\\frac{b_i^{ZI}}{2\\pi }\\ln \\frac{M_I}{M_Z}~,~~~~i=Y, w, c ~,$ $\\alpha _{3^{\\prime }}^{-1}(M_I)=\\alpha _{3^{\\prime }}^{-1}(\\Lambda ^{\\prime })-\\frac{b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}}{2\\pi }\\ln \\frac{M_I}{\\Lambda ^{\\prime }}~,$ where $b_i^{ZI}, b_{3^{\\prime }}^{\\Lambda ^{\\prime }I}$ can be calculated via Eq.",
"(REF ) having appropriate RG factors.",
"Above the scale $M_I$ , we have gauge interactions $G_{321}$ going all the way up to the GUT scale.",
"The ${G_{321}}^{\\prime }$ gauge symmetry appears between scales $M_I$ and ${M_I}^{\\prime }$ , while $SU(5)^{\\prime }$ appears above the ${M_I}^{\\prime }$ scale.",
"Therefore, we will have $\\alpha _i^{-1}(M_G)=\\alpha _i^{-1}(M_I)-\\frac{b_i^{IG}}{2\\pi }\\ln \\frac{M_G}{M_I}~,~~~~i=1, 2, 3 ~,$ $\\alpha _{i^{\\prime }}^{-1}({M_I}^{\\prime })=\\alpha _{i^{\\prime }}^{-1}(M_I)-\\frac{b_{i^{\\prime }}^{II^{\\prime }}}{2\\pi }\\ln \\frac{{M_I}^{\\prime }}{M_I}~,~~~~i^{\\prime }=1^{\\prime }, 2^{\\prime }, 3^{\\prime } ~,$ $\\alpha _{5^{\\prime }}^{-1}(M_G)=\\alpha _{5^{\\prime }}^{-1}({M_I}^{\\prime })-\\frac{b_{5^{\\prime }}^{I^{\\prime }G}}{2\\pi }\\ln \\frac{M_G}{{M_I}^{\\prime }}~.$ From Eqs.",
"(REF ) and (REF ) and taking into account the boundary conditions (REF )-(REF ), we arrive at relations given in Eq.",
"(REF ).",
"The four equations in (REF ) allow us to determine $M_I, {M_I}^{\\prime }, M_G$ and $\\alpha _G$ , in terms of other input mass scales.",
"The latter must be selected in such a way as to get successful unification.",
"This has been done numerically, and results are given in Table REF , Eq.",
"(REF ), and Fig.",
"REF .",
"Now we present all RG $b$ factors needed for writing down RG equations.",
"In the energy interval $\\mu =M_Z-\\Lambda ^{\\prime }$ , the RG factors are just those of the SM: $\\mu =M_Z - \\Lambda ^{\\prime }~:~~~ b_i=\\left(\\frac{41}{10}, -\\frac{19}{6}, -7\\right)~,b_{ij}=\\left(\\begin{array}{ccc}\\frac{199}{50} & \\frac{27}{10} & \\frac{44}{5} \\\\\\frac{9}{10} & \\frac{35}{6} & 12 \\\\\\frac{11}{10} & \\frac{9}{2} & -26 \\\\\\end{array}\\right)~,~(i=Y, w, c)~.$ In the energy interval $\\Lambda ^{\\prime } - M_I$ , we have the symmetry $SU(3)_c\\times SU(2)_w\\times U(1)_Y\\times SU(3)^{\\prime }$ .",
"Also, instead of composite leptons, we have three families of $SU(3)^{\\prime }$ triplets $\\hat{q}, \\hat{u}^c, \\hat{d}^c$ , and vectorlike states $(l, \\hat{l})_{\\alpha }$ and $(e^c, \\hat{e}^c)_{\\alpha }$ ($\\alpha =1,2,3$ ) with masses $M_{\\hat{l}l}^{(\\alpha )}$ and $M_{e^c\\hat{e}^c}^{(\\alpha )}$ , respectively.",
"Moreover, some fragments of $\\Phi (5, \\bar{5})$ [see Eq.",
"(REF )] and $\\Sigma ^{\\prime }_{8^{\\prime }}$ (of $\\Sigma ^{\\prime }$ ) can appear below $M_I$ .",
"Thus, the corresponding $b$ factors in this energy interval are given by $\\mu =\\Lambda ^{\\prime } - M_I :$ $b_Y=\\frac{9}{2}\\!+\\!\\frac{1}{15}\\theta \\!\\left(\\mu -M_{T_{H^{\\prime }}} \\!\\right)\\!+\\!\\frac{2}{5}\\sum _{\\alpha =1}^3\\!\\theta \\!\\left(\\mu \\!-\\!M_{\\hat{l}l}^{(\\alpha )}\\!\\right)\\!+\\!\\frac{4}{5}\\sum _{\\alpha =1}^3\\!\\theta \\!\\left(\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(\\alpha )}\\!\\right)+\\frac{5}{6}\\!\\theta \\!\\left(\\mu \\!-\\!M_{DT^{\\prime }}\\!\\right)+\\frac{5}{6}\\!\\theta \\!\\left(\\mu \\!-\\!M_{TD^{\\prime }}\\!\\right)$ $b_w=-\\frac{7}{6}\\!+\\!\\frac{2}{3}\\sum _{\\alpha =1}^3\\!\\theta \\!\\left(\\mu \\!-\\!M_{\\hat{l}l}^{(\\alpha )}\\!\\right)+\\frac{1}{2}\\!\\theta \\!\\left(\\mu \\!-\\!M_{DT^{\\prime }}\\!\\right)+\\frac{1}{2}\\!\\theta \\!\\left(\\mu \\!-\\!M_{TD^{\\prime }}\\!\\right),$ $b_c=-7+\\frac{1}{3}\\!\\theta \\!\\left(\\mu \\!-\\!M_{TD^{\\prime }}\\!\\right)+\\frac{1}{2}\\!\\theta \\!\\left(\\mu \\!-\\!M_{TT^{\\prime }}\\!\\right),~~$ $b_{3^{\\prime }}=-7\\!+\\!\\frac{1}{6}\\theta \\!\\left(\\mu \\!-\\!M_{T_{H^{\\prime }}}\\!\\right)+\\frac{1}{3}\\!\\theta \\!\\left(\\mu \\!-\\!M_{DT^{\\prime }}\\!\\right)+\\frac{1}{2}\\!\\theta \\!\\left(\\mu \\!-\\!M_{TT^{\\prime }}\\!\\right)+\\frac{1}{2}\\!\\theta \\!\\left(\\mu \\!-\\!M_{8^{\\prime }}\\!\\right)~,$ $\\mu =\\Lambda ^{\\prime } - M_I :~~~b_{ij}=\\left(\\!\\!\\begin{array}{cccc}\\frac{13709}{50} & \\frac{9}{5} & \\frac{44}{5} & \\frac{44}{5} \\\\\\frac{3}{5} & \\frac{91}{3} & 12 & 12 \\\\\\frac{11}{10}& \\frac{9}{2} & -26 & 0 \\\\\\frac{11}{10} & \\frac{9}{2} & 0 & -26 \\\\\\end{array} \\!\\!\\right)\\!+\\sum _a \\theta \\!\\left(\\mu \\!-\\!M_a\\!\\right)b_{ij}^a~,~~~~~(i,j=Y, w, c, 3^{\\prime })~~~{\\rm with:}$ $b_{ij}^{T_{H^{\\prime }}}\\!\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccc}\\frac{4}{75} & 0 & 0 & \\frac{16}{15} \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\\\frac{2}{15} & 0 & 0 & \\frac{11}{3} \\\\\\end{array} \\!\\!\\right)\\!\\!,~~b_{ij}^{DT^{\\prime }}\\!\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccc}\\frac{25}{6} & \\frac{15}{2} & 0 & \\frac{40}{3} \\\\\\frac{5}{2} & \\frac{13}{2} & 0 & 8 \\\\0 & 0 & 0 & 0 \\\\\\frac{5}{3} & 3 & 0 & \\frac{22}{3} \\\\\\end{array} \\!\\!\\right)\\!\\!,~~b_{ij}^{TT^{\\prime }}\\!\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccc}0 & 0 & 0 & 0 \\\\0& 0 & 0 & 0 \\\\0 & 0 & 11 & 8 \\\\0 & 0 & 8 & 11 \\\\\\end{array} \\!\\!\\right)\\!\\!,~~b_{ij}^{TD^{\\prime }}\\!\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccc}\\frac{25}{6} & \\frac{15}{2} & \\frac{40}{3} & 0 \\\\\\frac{5}{2}& \\frac{13}{2} & 8 & 0 \\\\\\frac{5}{2} & 3 & \\frac{22}{3} & 0 \\\\0 & 0 & 0 & 0 \\\\\\end{array} \\!\\!\\right)\\!\\!,$ $b_{ij}^{(l, \\hat{l})_{\\alpha }}\\!\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccc}\\frac{9}{50} & \\frac{9}{10} & 0 & 0 \\\\\\frac{3}{10} & \\frac{49}{6} & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\\\end{array} \\!\\!\\right)\\!\\!,~~b_{ij}^{(e^c, \\hat{e}^c)_{\\alpha }}\\!\\!=\\!\\!",
"{\\rm Diag}\\!\\left(\\!",
"\\frac{36}{25}, 0, 0, 0\\!",
"\\right)\\!",
",~~b_{ij}^{\\Sigma ^{\\prime }_{8^{\\prime }}}\\!\\!=\\!\\!",
"{\\rm Diag}\\!\\left(\\!",
"0, 0, 0, 21\\!",
"\\right)\\!",
"~.$ Between the scales $M_I$ and ${M_I}^{\\prime }$ , the symmetry is $G_{321}\\times {G_{321}}^{\\prime }$ , and all matter states are massless.",
"Also, above the scale $M_I$ , we should include the states $T_{H^{\\prime }}$ and $\\Phi _{DD^{\\prime }}$ as massless and remaining fragments above their mass thresholds.",
"Since $G_{321}$ goes all the way up to the $M_G$ , its one-loop b factors can be determined in the interval $M_I-M_G$ and are given by $\\mu =M_I - M_G :&~ b_1\\!&\\!=\\frac{43}{10}\\!+\\!\\frac{3}{10}\\theta \\!\\left(\\mu -M_{DT^{\\prime }} \\!\\right)\\!+\\!",
"\\frac{1}{5}\\theta \\!\\left(\\mu -M_{TT^{\\prime }} \\!\\right)\\!+\\!",
"\\frac{2}{15}\\theta \\!\\left(\\mu -M_{TD^{\\prime }} \\!\\right),\\nonumber \\\\~ & ~ b_2&\\!=-\\frac{17}{6}\\!+\\!\\frac{1}{2}\\theta \\!\\left(\\mu -M_{DT^{\\prime }} \\!\\right),\\nonumber \\\\~ & ~ b_3&\\!=-7\\!+\\!\\frac{1}{2}\\theta \\!\\left(\\mu -M_{TT^{\\prime }} \\!\\right)\\!+\\!\\frac{1}{3}\\theta \\!\\left(\\mu -M_{TD^{\\prime }} \\!\\right).$ The gauge group ${G_{321}}^{\\prime }$ appears in the interval $M_I-{M_I}^{\\prime }$ , and corresponding one-loop $b$ factors are $\\mu =M_I - {M_I}^{\\prime } :&~ b_{1^{\\prime }}\\!&\\!=\\frac{64}{15}\\!+\\!\\frac{1}{10}\\theta \\!\\left(\\mu -M_{D^{\\prime }} \\!\\right)\\!+\\!\\frac{2}{15}\\theta \\!\\left(\\mu -M_{DT^{\\prime }} \\!\\right)\\!+\\!",
"\\frac{1}{5}\\theta \\!\\left(\\mu -M_{TT^{\\prime }}\\!\\right)\\nonumber \\\\&& +\\!",
"\\frac{3}{10}\\theta \\!\\left(\\mu -M_{TD^{\\prime }}\\!",
"\\right)\\!-\\!",
"\\frac{55}{3}\\theta \\!\\left(\\mu -M_{X^{\\prime }} \\!\\right),\\nonumber \\\\~ & ~ b_{2^{\\prime }}&\\!=-3\\!+\\!\\frac{1}{6}\\theta \\!\\left(\\mu -M_{D^{\\prime }} \\!\\right)\\!+\\!\\frac{1}{2}\\theta \\!\\left(\\mu -M_{TD^{\\prime }} \\!\\right)\\!-\\!",
"11\\theta \\!\\left(\\mu -M_{X^{\\prime }}\\!\\right),\\nonumber \\\\~ & ~ b_{3^{\\prime }}&\\!=-\\frac{41}{6}\\!+\\!\\frac{1}{2}\\theta \\!\\left(\\mu -M_{TT^{\\prime }} \\!\\right)\\!+\\!\\frac{1}{3}\\theta \\!\\left(\\mu -M_{DT^{\\prime }} \\!\\right),$ where terms with $\\theta \\!\\left(\\mu -M_{X^{\\prime }} \\!\\right)$ account for the threshold of $(X^{\\prime },Y^{\\prime })$ gauge bosons of $SU(5)^{\\prime }$ , in case their masses $M_{X^{\\prime }}$ lie slightly below the ${M_I}^{\\prime }$ scale.",
"We will take this effect into account at 1-loop level.",
"The two-loop $b_{ij}$ factors of $G_{321}\\times {G_{321}}^{\\prime }$ form $6\\times 6$ matrices and are determined in the interval $M_I - {M_I}^{\\prime }$ : $\\mu =M_I - {M_I}^{\\prime }:~~b_{ij}=(b^{f}\\!+\\!b^{h}\\!+\\!b^g\\!+\\!b^{T_{H^{\\prime }}}\\!+\\!b^{DD^{\\prime }})_{ij}+\\sum _a \\theta \\!\\left(\\mu \\!-\\!M_a\\!\\right)b_{ij}^a~,~~~~~(i,j=1, 2, 3, 1^{\\prime }, 2^{\\prime }, 3^{\\prime })$ ${\\rm with:}~~b_{ij}^f=3\\left(\\!\\!\\begin{array}{cccccc}\\frac{19}{15} & \\frac{3}{5} & \\frac{44}{15} & 0 & 0 & 0 \\\\\\frac{1}{5} & \\frac{49}{3} & 4 & 0 & 0 & 0 \\\\\\frac{11}{30} & \\frac{3}{2} & \\frac{76}{3} & 0 & 0 & 0 \\\\0 & 0 & 0 & \\frac{19}{15} & \\frac{3}{5} & \\frac{44}{15} \\\\0 & 0 & 0 & \\frac{1}{5} & \\frac{49}{3} & 4 \\\\0 & 0 & 0 & \\frac{11}{30} & \\frac{3}{2} & \\frac{76}{3} \\\\\\end{array} \\!\\!\\right),~~b_{ij}^h=\\left(\\!\\!\\begin{array}{cccccc}\\frac{9}{50} & \\frac{9}{10} & 0 & 0 & 0 & 0 \\\\\\frac{3}{10} & \\frac{13}{6} & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 &0 &0 & 0 \\\\\\end{array} \\!\\!\\right)\\!,$ $b_{ij}^{T_{H^{\\prime }}}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & \\frac{4}{75} & 0 & \\frac{16}{15} \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 &\\frac{2}{15} &0 & \\frac{11}{3} \\\\\\end{array} \\!\\!\\right)\\!,~~b_{ij}^{DD^{\\prime }}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}\\frac{9}{25} & \\frac{9}{5} & 0 & \\frac{9}{25} & \\frac{9}{5} & 0 \\\\\\frac{3}{5} & \\frac{13}{3} & 0 & \\frac{3}{5} & 3 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{9}{25} & \\frac{9}{5} & 0 & \\frac{9}{25} & \\frac{9}{5} & 0 \\\\\\frac{3}{5} & 3 & 0 & \\frac{3}{5} & \\frac{13}{3} & 0 \\\\0 & 0 & 0 &0 &0 & 0 \\\\\\end{array} \\!\\!\\right)\\!,~~b_{ij}^{DT^{\\prime }}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}\\frac{27}{50} & \\frac{27}{10} & 0 & \\frac{6}{25} & 0 & \\frac{24}{5} \\\\\\frac{9}{10} & \\frac{13}{2} & 0 & \\frac{2}{5} & 0 & 8 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{6}{25} & \\frac{6}{5} & 0 & \\frac{8}{75} & 0 & \\frac{32}{15} \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{9}{15} & 3 & 0 &\\frac{4}{15} &0 & \\frac{22}{3} \\\\\\end{array} \\!\\!\\right)\\!,$ $b_{ij}^{TT^{\\prime }}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}\\frac{4}{25} & 0 & \\frac{16}{5} & \\frac{4}{25} & 0 & \\frac{16}{5} \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{2}{5} & 0 & 11 & \\frac{2}{5} & 0 & 8 \\\\\\frac{4}{25} & 0 & \\frac{16}{5} & \\frac{4}{25} & 0 & \\frac{16}{5} \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{2}{5} & 0 & 8 &\\frac{2}{5} &0 & 11 \\\\\\end{array} \\!\\!\\right)\\!,~~b_{ij}^{TD^{\\prime }}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}\\frac{8}{75} & 0 & \\frac{32}{15} & \\frac{6}{25} & \\frac{6}{5} & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\\\frac{4}{15} & 0 & \\frac{22}{3} & \\frac{3}{5} & 3 & 0 \\\\\\frac{6}{25} & 0 & \\frac{24}{5} & \\frac{27}{50} & \\frac{27}{10} & 0 \\\\\\frac{2}{5} & 0 & 8 & \\frac{9}{10} & \\frac{13}{2} & 0 \\\\0 & 0 & 0 &0 &0 & 0 \\\\\\end{array} \\!\\!\\right)\\!,~b_{ij}^{D^{\\prime }}\\!=\\!\\!\\left(\\!\\!\\begin{array}{cccccc}0 & 0 &0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & \\frac{9}{50} & \\frac{9}{10} & 0 \\\\0 & 0 & 0 & \\frac{3}{10} & \\frac{13}{6} & 0 \\\\0 & 0 & 0 &0 &0 & 0 \\\\\\end{array} \\!\\!\\right)\\!,$ $b_{ij}^g=\\!",
"{\\rm Diag}\\!",
"\\left(\\!0, -\\frac{136}{3}, -102, 0, -\\frac{136}{3}, -102\\!\\right)\\!,~~b_{ij}^{\\Sigma ^{\\prime }_{8^{\\prime }}}=\\!",
"{\\rm Diag}\\!",
"\\left(\\!0, 0, 0, 0, 0, 21\\!\\right)\\!.$ In this $M_I-{M_I}^{\\prime }$ energy interval, we have two Abelian factors $U(1)$ and $U(1)^{\\prime }$ and states $\\Phi _i$ (the fragments of $\\Phi $ ) charged under both gauge symmetries.",
"Because of this, the gauge kinetic mixing will be induced [57], [58].",
"Parametrizing the latter as $-\\frac{\\sin \\chi }{2}F_1^{\\mu \\nu }F_{1^{\\prime }\\mu \\nu }$ , and bringing whole gauge kinetic part to the canonical form, one can obtain $\\Phi _i$ 's covariant derivative as [58]: $[\\partial ^{\\mu }+\\frac{i}{2}g_1Q_iA_1^{\\mu }+\\frac{i}{2}(\\bar{g}_{1^{\\prime }}{Q_i}^{\\prime }+g_{11^{\\prime }}Q_i)A_{1^{\\prime }}^{\\mu }]\\Phi _i$ .",
"In this basis $Q_i$ charges are unshifted, and $g_1$ and its RG are unchanged.",
"On the other hand, $\\bar{g}_{1^{\\prime }}=g_{1^{\\prime }}/\\cos \\chi $ and $g_{11^{\\prime }}=-g_1 \\tan \\chi $ .",
"Introducing the ratio $\\delta =g_{11^{\\prime }}/\\bar{g}_{1^{\\prime }}$ , the RGs for $\\bar{\\alpha }_{1^{\\prime }}$ and $\\delta $ will be [58] $\\frac{d}{d\\ln \\mu }(\\bar{\\alpha }_{1^{\\prime }})^{-1}=\\cdots -\\frac{b_1}{2\\pi }\\delta ^2-\\frac{B_{11^{\\prime }}}{\\pi }\\delta ~,~~~~~~\\frac{d}{d\\ln \\mu } \\delta =\\frac{b_1}{2\\pi }\\alpha _1\\delta +\\frac{B_{11^{\\prime }}}{8\\pi ^2}~,$ where $\"\\dots \"$ denote standard one- and two-loop contributions [with form of Eq.",
"(REF )] and $B_{11^{\\prime }}=\\sum _iQ_i{Q_i}^{\\prime }$ is given by $B_{11^{\\prime }}=\\frac{1}{5}\\left[ \\theta (\\mu -M_{DT^{\\prime }})-\\theta (\\mu -M_{DD^{\\prime }})-\\theta (\\mu -M_{TT^{\\prime }})+\\theta (\\mu -M_{TD^{\\prime }})\\right] ~.$ Because of the mass splitting between $\\Phi $ 's fragments, $B_{11^{\\prime }}\\ne 0$ in the interval $M_I-M_{TD^{\\prime }}$ , and therefore $\\delta \\ne 0$ ; i.e., the kinetic mixing is generated.",
"This causes the shift $\\alpha _{1^{\\prime }}^{-1}\\rightarrow \\alpha _{1^{\\prime }}^{-1}+{\\cal O}(\\delta )$ .",
"However, as it turns out, within our model this effect is negligible.",
"We have taken these into account upon numerical studies and got $\\delta (M_I)\\simeq 9.5\\cdot 10^{-3}$ , $\\sin \\chi (M_I)\\simeq -2\\cdot 10^{-2}$ , causing the change of $\\alpha _{1^{\\prime }}^{-1}(M_I)$ by $0.01\\%$ .",
"This has no practical impact on the matching conditions of Eq.",
"(REF ), does not affect the picture of gauge coupling unification and therefore can be safely ignored.",
"Since at and above the scale ${M_I}^{\\prime }$ the ${G_{321}}^{\\prime }$ is embedded in $SU(5)^{\\prime }$ , we will deal with b factors of $G_{321}\\times SU(5)^{\\prime }$ symmetry, and one-loop b factors of $G_{321}$ are given in Eq.",
"(REF ).",
"At energies corresponding to unbroken $SU(5)^{\\prime }$ , the fragments $(\\Phi _{DD^{\\prime }}, \\Phi _{DT^{\\prime }})$ form the unified $(2,\\bar{5})\\equiv \\Phi _{D\\bar{5}^{\\prime }}$ -plet of $G_{321}\\times SU(5)^{\\prime }$ .",
"Similarly, $(T_{H^{\\prime }}, D^{\\prime })\\subset H^{\\prime }$ .",
"Above the scale ${M_I}^{\\prime }$ , these states (together with all fragments of the $\\Sigma ^{\\prime }$ -plet) should be included as massless states.",
"Thus, the one-loop $b$ factor of $SU(5)^{\\prime }$ is given as $\\mu ={M_I}^{\\prime } - M_G~ :~~~b_{5^{\\prime }}=-13\\!+\\!\\frac{1}{2}\\theta \\!\\left(\\mu -M_{T\\bar{5}^{\\prime }} \\!\\right),$ where $M_{T\\bar{5}^{\\prime }}={\\rm max}(M_{TT^{\\prime }}, M_{TD^{\\prime }})$ denotes the mass of the $(3,\\bar{5})$ -plet, which includes $\\Phi _{TT^{\\prime }}$ and $\\Phi _{TD^{\\prime }}$ states: $(\\Phi _{TT^{\\prime }}, \\Phi _{TD^{\\prime }})\\subset \\Phi _{T\\bar{5}^{\\prime }}$ .",
"The two-loop $b_{ij}$ factors, above the scale ${M_I}^{\\prime }$ , form $4\\times 4$ matrices and are $\\mu ={M_I}^{\\prime } - M_G:~~b_{ij}=(b^{f}\\!+\\!b^{h}\\!+\\!b^g\\!+\\!b^{H^{\\prime }}\\!+\\!b^{\\Sigma ^{\\prime }}\\!+\\!b^{D\\bar{5}^{\\prime }})_{ij}+\\theta \\!\\left(\\mu \\!-\\!M_{T\\bar{5}^{\\prime }}\\!\\right)b_{ij}^{T\\bar{5}^{\\prime }}~,~~~~~(i,j=1, 2, 3, 5^{\\prime })$ ${\\rm with}:~b^{f}_{ij}=3\\left(\\!\\!\\begin{array}{cccc}\\frac{19}{15} & \\frac{3}{5} & \\frac{44}{15} & 0 \\\\\\frac{1}{5} & \\frac{49}{3} & 4 & 0 \\\\\\frac{11}{30} & \\frac{3}{2} & \\frac{76}{3} & 0 \\\\0 & 0 & 0 & \\frac{698}{15} \\\\\\end{array} \\!\\!\\right)\\!,~~b^{h}_{ij}=\\left(\\!\\!\\begin{array}{cccc}\\frac{9}{50} & \\frac{9}{10} & 0 & 0 \\\\\\frac{3}{10} & \\frac{13}{6} & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 \\\\\\end{array} \\!\\!\\right)\\!,~~b_{ij}^g=\\!",
"{\\rm Diag}\\!",
"\\left(\\!0, -\\frac{136}{3}, -102, -\\frac{850}{3}\\!\\right)\\!,~~$ $b^{H^{\\prime }}_{ij}=\\frac{97}{15}\\delta _{i5^{\\prime }}\\delta _{j5^{\\prime }} ,~~b^{\\Sigma ^{\\prime }}_{ij}=\\frac{175}{3}\\delta _{i5^{\\prime }}\\delta _{j5^{\\prime }} ,~~b^{D\\bar{5}^{\\prime }}_{ij}=\\left(\\!\\!\\begin{array}{cccc}\\frac{9}{10} & \\frac{9}{2} & 0 & \\frac{72}{5} \\\\\\frac{3}{2} & \\frac{65}{6} & 0 & 24 \\\\0 & 0 & 0 & 0 \\\\\\frac{3}{5} & 3 & 0 & \\frac{194}{15} \\\\\\end{array} \\!\\!\\right)\\!,~~b^{T\\bar{5}^{\\prime }}_{ij}=\\left(\\!\\!\\begin{array}{cccc}\\frac{4}{15} & 0 & \\frac{16}{3} & \\frac{48}{5} \\\\0 & 0 & 0 & 0 \\\\\\frac{2}{3} & 0 & \\frac{55}{3} & 24 \\\\\\frac{2}{5} & 0 & 8 & \\frac{97}{5} \\\\\\end{array} \\!\\!\\right)\\!.$ As far as the Yukawa coupling involving RG factors, $a_i^f, c_f, d_f^{f^{\\prime }}$ , and $c_f^i$ [see Eqs.",
"(REF ) and (REF )], are concerned, within our model only top and \"mirror-top\" Yukawa couplings are large.",
"All other Yukawa interactions are small and can be ignored.",
"Thus, the Yukawa terms $\\lambda _tq_3t^ch$ , $(\\lambda _{\\hat{t}\\hat{b}}\\hat{t}\\hat{b}+\\lambda _{\\hat{t}^c\\hat{\\tau }^c}\\hat{t}^c\\hat{\\tau }^c)T_{H^{\\prime }}$ , and $\\lambda _{\\hat{t}}\\hat{q}_3\\hat{t}^cD^{\\prime }$ are relevant.",
"All these four couplings unify at $M_G$ due to gauge symmetry and $D_2$ parity.",
"For the top Yukawa involved RG factors, in the energy interval $M_Z-M_I$ , we have $a_i^t=\\left(\\frac{17}{10}, \\frac{3}{2}, 2\\right),~~~c_t^i=\\left(\\frac{17}{20}, \\frac{9}{4}, 8\\right),~~~(i=Y, w, c),~~~c_t=\\frac{9}{2}, ~~~d_t^{f^{\\prime }}=0~.$ In energy interval $M_I-M_G$ , with replacement of the indices $(Y, w, c)\\rightarrow (1, 2, 3)$ , the corresponding RG factors will be the same.",
"Since the mass of the state $D^{\\prime }$ is $\\sim {M_I}^{\\prime }$ , the RG with $\\lambda _{\\hat{t}}$ will be relevant above the scale ${M_I}^{\\prime }$ .",
"Within our model, $M_{T_{H^{\\prime }}}\\sim \\Lambda ^{\\prime }$ , and in the RG, the couplings $\\lambda _{\\hat{t}\\hat{b}}$ and $\\lambda _{\\hat{t}^c\\hat{\\tau }^c}$ will be relevant above the scale $\\Lambda ^{\\prime }$ .",
"Between the scales $\\Lambda ^{\\prime }$ and $M_I$ , the mirror matter has EW and $SU(3)^{\\prime }$ interactions.",
"Therefore, we have $\\mu =\\Lambda ^{\\prime }-M_I~:~~~(a_Y, a_w, a_{3^{\\prime }})^{\\hat{t}\\hat{b}}=\\left(\\frac{1}{15}, 2, \\frac{4}{3}\\right),~~~(a_Y, a_w, a_{3^{\\prime }})^{\\hat{t}^c\\hat{\\tau }^c}\\!=\\theta (\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(3)})\\left(\\frac{13}{15}, 0, \\frac{1}{3}\\right),$ $\\hspace{28.45274pt}(c^Y, c^w, c^{3^{\\prime }})_{\\hat{t}\\hat{b}}=\\left(\\frac{1}{10}, \\frac{9}{2}, 8\\right),~~~(c^Y, c^w, c^{3^{\\prime }})_{\\hat{t}^c\\hat{\\tau }^c}\\!=\\theta (\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(3)})\\left(\\frac{13}{5}, 0, 4\\right),$ $\\hspace{85.35826pt} c_{\\hat{t}\\hat{b}}=4,~~ d_{\\hat{t}\\hat{b}}^{\\hat{t}^c\\hat{\\tau }^c}=\\theta (\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(3)}),~~c_{\\hat{t}^c\\hat{\\tau }^c}=3\\theta (\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(3)}),~~d^{\\hat{t}\\hat{b}}_{\\hat{t}^c\\hat{\\tau }^c}=2\\theta (\\mu \\!-\\!M_{e^c\\hat{e}^c}^{(3)}).$ Between $M_I$ and ${M_I}^{\\prime }$ scales, with replacements $(Y, w)\\rightarrow (1^{\\prime }, 2^{\\prime })$ , the corresponding factors will be the same.",
"At and above the scale $M_I$ , the ${G_{321}}^{\\prime }$ is unified in the $SU(5)^{\\prime }$ group, $D^{\\prime }$ should be included in the RG, and three Yukawas unify $\\lambda _{\\hat{t}\\hat{b}}=\\lambda _{\\hat{t}^c\\hat{\\tau }^c}=\\lambda _{\\hat{t}}$ .",
"Thus, dealing with $\\lambda _{\\hat{t}}$ , we will have $\\mu ={M_I}^{\\prime }-M_G~:~~~~a_{5^{\\prime }}^{\\hat{t}}=\\frac{9}{2},~~~~c_{\\hat{t}}=9,~~~~c_{\\hat{t}}^{5^{\\prime }}=\\frac{108}{5},~~~~d_{\\hat{t}}^{f^{\\prime }}=0~.$"
],
[
"Short-range RG factors for $d=6$ operators",
"The baryon number violating $d=6$ operators of Eq.",
"(REF ) involve couplings ${\\cal C}^{(e^c)}$ and ${\\cal C}^{(l)}$ respectively.",
"These couplings run, and in nucleon decay amplitudes, the short-range RG factors $A_S^{l}=\\frac{{\\cal C}^{(l)}(M_Z)}{{\\cal C}^{(l)}(M_X)} ~, ~~~~~~~~~~~A_S^{e^c}=\\frac{{\\cal C}^{(e^c)}(M_Z)}{{\\cal C}^{(e^c)}(M_X)}$ emerge.",
"These factors, having SM gauge interactions and states below the GUT scale, were calculated in Ref.",
"[31].",
"Within our model, calculation can be done similarly.",
"The RG equations for ${\\cal C}^{(l)}$ and ${\\cal C}^{(e^c)}$ , in one-loop approximation, are given by $4\\pi \\frac{d}{dt}{\\cal C}^{(l)}\\!&=&\\!\\!\\!-{\\cal C}^{(l)}\\!\\left[ \\theta (M_I\\!-\\!\\mu )\\!\\left(\\!\\frac{23}{20}\\alpha _Y+\\frac{9}{4}\\alpha _w\\!\\right)+2\\alpha _c\\!+\\!\\theta (\\mu \\!-\\!M_I)\\!\\left(\\!\\frac{23}{20}\\alpha _1+\\frac{9}{4}\\alpha _2\\!\\right)\\right] ,\\nonumber \\\\4\\pi \\frac{d}{dt}{\\cal C}^{(e^c)}\\!&=&\\!\\!\\!-{\\cal C}^{(e^c)}\\!\\left[ \\theta (M_I\\!-\\!\\mu )\\!\\left(\\!\\frac{11}{20}\\alpha _Y+\\frac{9}{4}\\alpha _w\\!\\right)\\!+\\!2\\alpha _c\\!+\\!\\theta (\\mu \\!-\\!M_I)\\!\\left(\\!\\frac{11}{20}\\alpha _1+\\frac{9}{4}\\alpha _2\\!\\right)\\right] .$ Having numerical solutions for the gauge couplings, Eqs.",
"(REF ) can be integrated.",
"Doing so and taking into account Eqs.",
"(REF ), within our model we obtain $A_S^l=1.18$ and $A_S^{e^c}=1.17$ ."
]
] | 1403.0025 |
[
[
"21st Century Ergonomic Education, From Little e to Big E"
],
[
"Abstract Despite intense efforts, contemporary educational systems are not enabling individuals to function optimally in modern society.",
"The main reason is that reformers are trying to improve systems that are not designed to take advantage of the centuries of history of the development of today's societies.",
"Nor do they recognize the implications of the millions of years of history of life on earth in which humans are the latest edition of learning organisms.",
"The contemporary educational paradigm of \"education for all\" is based on a 17th century model of \"printing minds\" for passing on static knowledge.",
"This characterizes most of K-12 education.",
"In contrast, 21st Century education demands a new paradigm, which we call Ergonomic Education.",
"This is an education system that is designed to fit the students of any age instead of forcing the students to fit the education system.",
"It takes into account in a fundamental way what students want to learn -- the concept \"wanting to learn\" refers to the innate ability and desire to learn that is characteristic of humans.",
"The Ergonomic Education paradigm shifts to education based on coaching students as human beings who are hungry for productive learning throughout their lives from their very earliest days."
],
[
"The national education reform movement in the USA (US), which is used here as an example of a highly developed country, has not delivered the results envisioned or hoped for when it was launched by A Nation at Risk (National Comission on Excellence in Education, 1983).",
"Rather than rising levels of national student achievement to meet the needs of the 21st century, the US reform movement has instead yielded a growing burden of responsibilities being piled upon local educators.",
"Current attempts focused on improving student performance have not resulted in significant gains on standardized tests or on graduation rates, especially for the students who traditionally perform poorly.",
"Since A Nation at Risk there has been an increasing crescendo of calls for reform in the US.",
"Yet accompanying each decade of reform have been numerous reports continuing to document the failures of the nation's schools.",
"There has been an endless progression of fads sweeping through the US schools, each buoyed by inflated claims of its power to reshape education for the better.",
"One novelty follows another, typically leaving only broken promises and dashed hopes behind.",
"The parade of failed pseudo-innovations leaves educators and their communities pessimistic about the value of anything new proposed by state or national decision-makers.",
"They are fearful that any new developments and initiatives will again be temporary – here today and gone, if not tomorrow, then in a year or two – eliminated because of lack of commitment or lack of resources or both.",
"Changing priorities, overlapping as well as conflicting educational targets of achievement and well intentioned but poorly structured local, state, and national programs have produced an ineffective, inefficient, and increasingly irrelevant education system.",
"As one looks across the current educational scene it is possible to identify a continuum of initiatives that can be observed in most US schools.",
"This continuum is made up of at least three general categories: chaos, revitalization, and the beginnings of a transformation.",
"Some characteristics of each are summarized below.",
"The first category of the continuum, “chaos,” embraces most reforms entering classrooms from “the outside”.",
"These initiatives may be mandated by politicians, regulators, education administrators, or well-meaning reformers.",
"For example, the initiatives in the US in mathematics, English and science for quasi national standards are creating dilemmas for school districts.",
"Questions being raised include: How do we disseminate the standards?",
"How can we prepare enough teachers quickly enough to teach the new standards?",
"What sort of pedagogy will be most effective and how to implement it?",
"How can teachers prepare their students for the tests that are to measure student learning of the new standards?",
"How do we evaluate teacher effectiveness if we do not use the typical value-added measurements?",
"The reforms have not only sparked chaos in local schools but have actually been counterproductive.",
"They have left parents, local policymakers, and especially teachers leery of investing in new ideas because the likelihood of failure is far greater than of success.",
"Those who recognize the chaos get discouraged and stop fighting it.",
"Meanwhile, newcomers keep emerging certain that the problems of schools are easily solved and they jump in to start a new round of unrealistic reforms.",
"No one knows what should replace the current poorly functioning system or how to phase in a replacement such that genuine improvement takes place despite the innumerable barriers.",
"Typically these new reforms do not result in sustained student achievement.",
"Panaceas such as more subject matter courses in college for teachers, more seat time for students, or subject specialists in classrooms do not address systemic problems of inadequate salaries, poor working conditions, inadequate training in how to teach unmotivated children, or negative impacts of life in current schools on promising young teachers.",
"As Michael Fullan (2007) states “It has become more obvious that the approaches that have been used so far to bring about educational change are not working and cannot work” (p. 299)."
],
[
" Revitalization ",
"But while external reforms are being inflicted on schools and teachers, many educators at the local level are creating a second kind of intervention.",
"They are creating their own innovations to solve the problems they have identified as most pressing in their individual schools and classrooms.",
"The innovations are “revitalization” reforms and have been implemented by the people who created them to meet a personal need locally.",
"They may be limited to a single classroom when a teacher discovers the most effective way to engage the students—techniques that may have been gained through participation in workshops, collaborative conversations with other teachers, or trial and error in the classroom.",
"Local initiatives demonstrate enough success to energize and revitalize local attempts to improve education, even in the face of reforms imposed by outside authorities.",
"However, these reforms typically result in incremental changes and are rarely recognized or widespread.",
"Furthermore, there is typically little research to demonstrate just what makes them successful.",
"There are also some revitalization reforms that are more widespread than a single classroom.",
"They may have been the result of state or nationally funded reforms that have demonstrated success in numbers of schools, even nationally.",
"Although these reforms in schools may not always demonstrate the integrity of the original design, they have had staying power because of the engagement by committed program developers and teachers.",
"Examples of such reforms in the US include: Success for All (Slavin, 2011, 2014), Reading Recovery (Clay, 1993, 2005; Lyons and Pinnell, 2001), Physics by Inquiry (McDermott and the Physics Education Group, 1995), Cooperative Learning (Johnson, Johnson and Holubec.",
"1994; Johnson and Johnson, 2014), National Writing Project (National Writing Project, 2014), and Project Lead the Way (PLTW) (2014).",
"However, many of these reforms rarely surpass inclusion in more than a 1000 of the hundred thousand schools in the US, and even with their limited expansion there is little to ensure integrity with their original design.",
"However, such reforms do provide teachers and schools the opportunity to institute changes in classroom structure and instruction that could lead to more student engagement."
],
[
" Transformation ",
"As locally effective as school or district revitalization reforms may be, they typically are not able to effect state or national change.",
"There need to be sustainable, scalable interventions that enable improvements in learning for all students.",
"Such interventions can be referred to as “transformations” and imply new kinds of structures and standards of performance.",
"Exceptional revitalization innovations currently found in individual teacher's classrooms, in schools led by exceptional principals, and in districts led by exceptional superintendents (our use of the American nomenclature is not important here) may offer promising ideas for transformation.",
"Bold experiments are called for “…that generate new and powerful forces, including, for example, teachers energies and commitments unleashed by altered working conditions and new collective capacities, and students' intellectual labor in collaborating with other students to do the work of learning” (Fullan, 2007, p.299).",
"However, these need to be integrated with other innovations in order to have widespread impact.",
"With time, educational transformation could be accelerated to spread over entire countries and internationally through competition among innovations.",
"Although the details of the classification are tentative and incomplete, it offers some guidelines for assessing programs.",
"It would be desirable to reduce unproductive activities of the kind listed under chaos and make progress towards transformation.",
"For example several of the “given” traditional ways for schooling in the US are only historically contingent patterns from the beginnings of the 20th century, which currently could be challenged.",
"These patterns include: the mandate for twelve years in school, the selection and arrangement of the curriculum, and the academic credit system.",
"Sheppard (2005) specifically points out that the academic credit system became antithetical to the ideas of progressive education as it expanded beyond the administrative and logistical issues for which it was designed.",
"These and other historically contingent patterns that are no longer productive for the 21st century should be open to challenge.",
"However, recognizing that transformations may take many years, shorter-term (three to five years) improvements in education are possible with support for revitalization programs across more schools."
],
[
" Towards Harmony ",
"The current focus for both, national, state or regional funding is on expanded rigorous testing to determine the degree of success in teaching.",
"Unfortunately, what teachers have long known, what gets tested is what gets taught.",
"It does not matter whether the material is actually preparing students for careers or higher education or a fulfilling life.",
"For example, as noted in a 2012 Ohio newspaper the emphasis has been on “test scores above learning, ratings above reasoning, and appearances above academics.” The editorial asks “Is that the best plan we have to develop students who are prepared to succeed in a complicated, competitive global society?” (Cincinnati Enquirer, 2012).",
"And paraphrasing W. Edwards Deming, “...quality can't be achieved through inspection and sorting out defectives, but rather by improving the process...” (Lillrank, 2010).",
"Confidence in the ability to actually reform education continues to erode.",
"Part of the problem is that our education systems are complex and messy.",
"The attempts that have been made to understand those isolated pockets of success that are always selected as examples of what can be accomplished have never been able to account for the inability to bring these to a scale beyond a few thousand schools.",
"The names may stay but the students rarely experience the success of the original model (Education Trust, 1998, 1999; Grissmer and Flanagan, 1998).",
"Research on what impacts student performance continues to be ambiguous.",
"There are significant variations in research design, problems with establishing control groups, and lack of clarity of how teacher preparation and student achievement and performance are defined.",
"And there is little research on the interactions of the many variables that impact the delivery of quality teaching and learning.",
"There is also a lack of historical research on the continuity and change that has impacted education.",
"A new paradigm is needed based on the establishment of principles for learning and rules for teaching—a set of explicit principles that govern the processes of learning and rules that serve as a norm for guiding the process of teaching.",
"These rules and principles are not susceptible to arbitrary decisions and are increasingly recognized as flowing from and incorporating how the brain develops and learns as the product of millions of years of evolution (Fields, 2010; Gladwell, 2008; Dawkins, 2004) and how it can operate most recently in a free society (Dewey, 1997).",
"Some of these principles and rules must support the freedom of all students to learn the way they are capable of learning, help students realize what this means in terms of personal freedom and individual responsibility, and safeguard the freedom to choose, prevent abuses, and respond to changing contexts.",
"The new paradigm of Ergonomic Education proposed below would allow for many competing ideas to fuel students wanting to learn.",
"However, these ideas will only be in harmony as long as there is agreement on the principles of learning and rules of productive teaching studied, recognized, developed and applied by the teachers.",
"With freedom of choice in the hands of the learner and rules of productive teaching applied by the teacher, new knowledge, innovation, and creativity can be fostered to address dramatically changing social and environmental contexts.",
"There are reasons to believe that harmony of new system building and functioning will emerge from the recognition and adoption of the new paradigm of Ergonomic Education."
],
[
" HISTORY CEO ",
"To most people history consists of dates and records of events and individuals – a war, an election, a coup, the birth, life and death of notable personae, and more recently, the lives and contexts of the general population.",
"In contrast to the emphases on specific dates and events, individuals or populations, ongoing processes of change throughout history, have had far more profound impacts on people's daily lives than do most isolated historical events.",
"Historians understand that “the unique and the specific matter as much as the universal; that context and initial conditions matter; that the world is more complex than what is assumed in variable-controlled laboratory experiments; and that predictions are at best problematic” (Staley, 2010, p. 35).",
"However, although messy complexities and idiosyncrasies characterize the study of history, it is still possible to draw inferences that illuminate useful knowledge (Landes, Molkyr and Baumol, 2012, p. 528).",
"Similarities and commonalities in development can be observed in the midst of the differences that separate historical events.",
"These similarities and commonalities can aid in establishing categories and analogies that provide structure for understanding the flow of history.",
"This flow of history encompasses processes of change over extended periods of time.",
"A perspective on the more profound of the ongoing processes of change that have had and continue to have significant impacts are reflected in the phrase: “History CEO”—History Constrains, Enables and Organizes.",
"Significant change processes that span centuries can be identified and have implications for the future.",
"Understanding the driving forces and trajectories of these change processes may help society be better prepared to meet the challenges of the 21st century.",
"Each change process of History CEO, “… in and of itself is indeterminate, always contingent on numerous factors and usually compatible with movement in diverse directions” (Eisenstein, 2005, p. 333).",
"One notable and exceptional example of such diverse paths is the contrast b etween the impact of the printing press and printing in China compared to the impact in Western Europe, particularly in education, over centuries.",
"More than 1300 years ago China developed and used printing on a large scale, but only for a limited time.",
"Initially printing was oriented to religious materials and was encouraged by a woman, a prominent supporter who became an empress.",
"Religious beliefs embraced by the empress enabled the spread of printing.",
"However, soon after her death, printing declined and the process was ignored for two centuries.",
"According to Barrett (2008) “Sheer misogyny was undoubtedly a factor in this, but, dynastic politics, religious rivalries, vested scribal interests, elite snobbery, and even a whiff of xenophobia also played a part to a greater and lesser degree” (Introduction and Acknowledgements).",
"All these conspired to constrain the spread of printing, despite the initial enabling by the empress.",
"In contrast to the history of printing in China, Elizabeth Eisenstein (1980) in her two volume ground-breaking history of printing in early Western Europe has identified a systematic, centuries-long, change and growth for what she has called “the print culture”.",
"Significant and irreversible changes were brought about in Europe following the “invention” of the printing press by Gutenberg (c 1450).",
"In her book The Printing Press as an Agent of Change: Communications and Cultural Change in Early-Modern Europe (Eisenstein, 1983), Eisenstein explains the now-forgotten constraints that were faced by people who had to make do with the oral and scribal culture that existed prior to the introduction of printing.",
"Because books were very scarce, most individuals were “constrained” from gaining knowledge from documented sources of that time.",
"Libraries, which were the repository of written materials, were few and literally geographically out of the range of most individuals.",
"Furthermore, the scribal copying process led to repeated and newly introduced inaccuracies over time.",
"As printed books became more accessible, they enabled the flow of information and the more rapid organization and duplication of data.",
"Multiple copies with identical content were more easily produced, eliminating unending hours of labor and concentration required by scribal copying.",
"The use of books inspired an increasingly widespread variety of activities not possible or even imagined prior to the existence of the printing press.",
"However, it took more than a century and a half after Gutenberg's first publication of Bibles for a “print culture” to emerge (Eisenstein, 2005).",
"The centuries-long build-up of activities associated with books both enabled and organized an increasingly diverse range of applications.",
"Eisenstein's examples of advances in print communications led to changes in religious cultures and significant developments in early science, especially astronomy.",
"The proliferation of books also resulted in a paradox of censorship.",
"Banning certain books constrained the dissemination of ideas (particularly those viewed as threatening to religious canon) in some quarters while simultaneously enabling the printing of those same books in other quarters specifically so that the controversial ideas could be further examined (Eisenstein, 2005, pp.",
"209-285).",
"Although Eisenstein comments that there was undoubtedly an impact on thinking and learning as books became “silent instructors,” carrying their message farther than any public lecture, she does not examine the “print culture” that became embedded in education.",
"Furthermore most educators today do not recognize the considerable significance of the centuries-long impact of the printing press and printing on the history of education.",
"In 1657, two centuries after Gutenberg introduced the printing press, John Amos Comenius, a Czech Moravian teacher, educator, and writer completed his Didactica Magna (The Great Didactic, 1657, originally published in Latin in Amsterdam) “setting forth the whole art of teaching all things to all men” (Comenius,1992, Cover and Title Page).",
"His book advocated universal education for both boys and girls, teaching in the vernacular, the use of textbooks, the development of graded schools, pacing of instruction, and specific roles for universities.",
"In Chapter XXXII Comenius (1992) introduces the “universal and most perfect order of instruction” (pp.",
"287-294).",
"After a brief introduction to the process of printing, Section 5 of the chapter provides analogies of teaching to the printing press where “we might adapt the term ‘typography' and call the new method of teaching `didachography' ” (p. 289).",
"He goes on further to describe the following: “Pursuing this analogy to the art of printing, we will show, by a more detailed comparison, the true nature of this new method of ours, since it will be made evident that knowledge can be impressed on the mind, in the same way that it's concrete form can be printed on paper” (p. 289).",
"“Instead of paper we have pupils whose minds have to be impressed with the symbols of knowledge.",
"Instead of type we have the class books and the rest of the apparatus devised to facilitate the operation of teaching.",
"The ink is replaced by the voice of the master, since this is what conveys information from the books to the mind of the listener; while the press is school discipline, which keeps the pupils up to their work and compels them to learn” (p. 289).Comenius continues his analogy of comparing students to paper that is properly prepared and pressed using ink: “Similarly, the teacher, after he has explained a construction and has shown by examples how easily it can be imitated, asks individual pupils to reproduce what he has said and thus show that they are not merely learners, but actually possessors of knowledge” (p. 293).Section 26 includes discussion of end of year examinations which were to be graded by inspectors.",
"These examinations were necessary to test the students' knowledge ensuring that the “subjects had been properly learned” (Comenius, 1992, p. 293).",
"In 1658, Comenius published Orbis Pictus or Orbis Sensualium Pictus (The Visible World in Pictures) (Comenius, 1887).",
"Each page has an image illustrating something having to do with the natural world such as botany, biology, zoology, religion, human activities.",
"Accompanying each image are descriptors written in the vernacular (German) and Latin.",
"Orbis Pictus became a popular children's textbook and the model for classroom instruction and was translated into many Western European languages by the 1700's.",
"By the mid-19th century Comenius' model of education and instruction had been adopted by most of Europe and the US.",
"It provided a uniform structure that could address the increasing need of educating more and more children with fewer teachers per pupil, providing common materials and common curricula across a nation's cultures.",
"It also met the workplace needs of the technological and manufacturing expansion of the industrial revolution.",
"In 1892, on the three-hundredth anniversary of Comenius' birth, there were many celebrations of his work in the US.",
"In the National Education Association Proceedings for 1893 there were several essays noting Comenius' impact: “We have found in Comenius the source and the forecasting of much that inspires and directs our education now….What is commonplace today was genius three hundred years ago” (Hark, 1893, p. 723).",
"“The place of Comenius in the history of education, therefore, is one of commanding importance.",
"He introduces the whole modern movement in the field of elementary and secondary education.",
"His relation to our present teaching is similar to that held by Copernicus and Newton toward modern science, and Bacon and Descartes toward modern philosophy” (Butler, 1893, p. 728).As the histories of the adoption of Comenius' vision for schooling systems globally are examined, commonalities are expected that illustrate when history constrained, enabled or organized these systems.",
"Today, 121 years since the celebrations in 1892, most education systems and modes of instruction are still embedded in the print culture of 1657.",
"The still widely used practices of relying on textbooks for conveying knowledge and then assessing learning with regurgitation of information via multiple assessments that are not informative is outdated.",
"The technology of the internet has far outpaced the printed textbook as a means for sharing information and gaining static knowledge.",
"And measures of performance are inadequate and require development.",
"We believe the new paradigm of Ergonomic Education will refocus efforts on more productive teaching and learning by taking into account in a fundamental way the reasons humans want to learn and how they benefit from learning.",
"The section, Why e to E, expands on the importance of changing the current educational paradigm to meet the demands of today and be prepared to adapt to the demands of the future.",
"The focus on the print culture illustrates both a paradigm shift that occurred from the scribal culture to the print culture and the persistence of an education culture that has been immune from widespread significant change.",
"In contrast, science, technology, and business are conducted very differently today compared to the 1600's.",
"What can be learned from the constraining, enabling, and organizing principles that have impacted their histories?",
"How might these examples contribute to understanding the challenges of education reform?",
"For example George E. Smith (2009), a historian of science and a philosopher of science, has placed on-line, http://www.stanford.edu/dept/cisst/visitors.html, “Testing Newtonian gravity then and now.” The material reviews the historical process of over three centuries of testing and simultaneous refinement of calculations in Newtonian gravity as it applies to the solar system.",
"He documents the increasing variety of sources of gravitational potentials that have been included, from additional objects discovered in the solar system to later Einstein's corrections required by special and general relativity.",
"But attention to such long-running historical processes as the emergence of the print culture or the testing of Newtonian gravity still seems to be rare.",
"Thomas Hughes (1983), a historian of technology writes in Networks of Power about the expansion of the electric power system from small intercity lighting systems of the late 19th century to the regional networks of the 1930's.",
"He identifies phases of (1) inventor-entrepreneur, (2) technology transfer, (3) system growth and reverse salients, (4) momentum and contingencies, (5) evolving systems and planned new ones.",
"These processes mirror the processes expressed by our History CEO.",
"Hughes framework enables him to compare the development of the electric power systems in the United States, Germany, and England.",
"The comparisons along with their contextual peculiarities illustrate the differences in resources, organizational structures, political climates, economic practices, cultural preferences between regions as well as nations.",
"The emphasis is on the multifaceted complexity of systems and what impacts changes over time.",
"Landes, Molkyr and Baumol (2012) edited The Invention of Enterprise: Entrepreneurship from Ancient Mesopotamia to Modern Times.",
"In contrast to many current books focusing on what could be called “how to be an entrepreneur,” Landes, Molkyr and Baumol wanted to study entrepreneurship and its relationship to economic growth from historical accounts.",
"In 17 chapters this book describes the socio-technological culture in specific periods.",
"Despite variations due to context, commonalities in the development of entrepreneurship across the centuries could be identified and provide useful knowledge that the editors believe have immediate applications (pp.",
"527-528).",
"Peter F. Drucker (1985) also provides a historical account of entrepreneurship in his book Innovation and Entrepreneurship.",
"From an examination of both modern and historical inventions, including inventions of new organizational system structures, Drucker identifies seven sources of innovation (1) the unexpected; (2) incongruities; (3) process need; (4) industry and market structures; (5) demographics; (6) changes in perception; (7) new knowledge.",
"He has identified patterns that again have repeated themselves over time in a variety of contexts.",
"Rather than providing a historical narrative portraying chaotic and unpredictable interactions of many different individuals and events, always in the context of a particular historical setting, the above texts search for patterns that might be connected through space and time.",
"They are particularly interested in patterns that will permit a better understanding of the evolution of historical events that are reflected in today's environment.",
"These patterns may constrain, enable, and/or organize cultures, technologies, political entities, etc.",
"over centuries.",
"History CEO attempts distill these patterns in a way to help understand the predicaments of the current education system.",
"Gaddis (2002) recognizes the importance of patterns in history and of “fitting it all together” (pp.48-49).",
"He does not explicitly address long-running processes of change that could exist in the historical record.",
"However, he introduces many metaphors comparing how one does history to how one does science—particularly the historical sciences of astronomy, geology, and paleontology – all of which deal with very long running processes of change.",
"Significant change processes spanning centuries and multiple disciplines can be identified.",
"History CEO offers a vocabulary for categorizing these processes into a coherent framework.",
"The framework is context dependent, has independent and interdependent variables, is open to positive and negative (Constraining, Enabling, Organizing) feedback loops, and can illustrate the continuity of the change processes.",
"The problems of education are both content specific and interdisciplinary, they are economic and social, and they have historical roots and result from the latest reforms.",
"Comparisons of the processes that dominate the success or failure of socio-technological and socio-cultural systems thus have the potential of providing insights into current education problems and ideas for transformation.",
"However, time and expertise are needed to collect, analyze, and synthesize data.",
"What to do in the meantime?",
"Where to begin?",
"There are sufficient though limited examples of education successes that could provide the seeds of the new paradigm of Ergonomic Education.",
"Engineers, specifically ergonomic/human factor engineers interested in education as a foundation of the structure and function of society could provide the impetus for a new approach for understanding the most pressing problems of education.",
"As problem solvers they have been trained to achieve specific results in complex environments with available resources.",
"Would they be able to analyze the notable education successes and failures and subsequently design and produce a transition process that would lead from the current education system to the Ergonomic Education system of the future?"
],
[
" A ROLE FOR ENGINEERS ",
"In order to address a systems approach to educational reform, Wilson and Barsky (1999) proposed that the six roles of teacher, principal, student, district administrator, consultant, parent and the community, involved in educational change as described by Fullan (1991), needed to be expanded.",
"Wilson and Barsky described a role for engineering research based on an analysis of Reading Recovery (Reading Recovery Council of North America, 2014).",
"Unlike most reforms, Reading Recovery has been successful in the US for over a quarter century.",
"It is an integrated system that has maintained its integrity yet has so far not realized its scale-up potential.",
"It was proposed that the introduction of engineering research would be able to reverse engineer the components contributing to Reading Recovery's success and introduce an R&D mindset that would lead to future expansion.",
"Scientists are concerned with understanding how the natural or biological worlds work, engineers are concerned with problem-solving.",
"Yet both of the disciplines share centuries of development contributing to the knowledge and practices of today.",
"In contrast school improvement research only began a sustained build-up in the 1960's (Fullan, 1991) and school improvement/educational reform appears intellectually fragmented and not systemic.",
"The evidence that an engineering research discipline could significantly impact education reform is based on the experience with discipline-based research in the sciences, particularly physics (Arons, 1990; McDermott and the Physics Education Group, 1996).",
"The recent report by the National Research Council (2012), Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering, emphasized the need for this continuing research which complements the work in the cognitive sciences by introducing specific attention to the disciplines' priorities, knowledges, worldviews and practices.",
"It advocates the adoption of evidence-based teaching practices to improve learning outcomes for undergraduate science and engineering students.",
"However, what constitutes the “evidence” depends on the education paradigm.",
"In order for the Ergonomic Education paradigm to be realized, there must be evidence that it is more successful than the current paradigm in creating an environment of productive learning.",
"The engineering disciplines are poised to accumulate the evidence required through their understanding of historical precedents (e.g,.",
"aircraft manufacturing: Vincenti, 1990; Newman and Vincenti, 2007), knowledge of design (e.g., aircraft evolution from the Wright Brothers to the Boeing 777 and Airbus: Vincenti, 1990; Perrow, 1999; Petroski, 1996, 2006), processes of change (e.g., failures of designs in air transport, bridge building, nuclear power plants: Perrow, 1999; Petroski, 1996, 2006); the role of constraints (Perrow, 1999;Petroski, 2006) and reverse engineering.",
"Ergonomics and Human Factors Engineering in particular have “a unique combination of three fundamental characteristics: (1) they take a systems approach (2) they are design driven and (3) they focus on two closely related outcomes: performance and well-being” (Dul, Bruder, Buckle, Carayon, Falzon, Marras, Wilson and van der Doeler, 2012, p. 4).",
"And they bring a “...holistic, human-centered approach to work systems design that considers physical, cognitive, social, organization, environmental and other relevant factors.” Karwowski (2005, p. 437).",
"Karwowski goes on to point out that organizational ergonomics is concerned with the optimization of socio-technical systems, including their organizational structures, policies and processes.",
"One can ask if the ergonomics/human factors engineering disciplines are willing and interested in accepting the challenge of bringing about a transformation in education which could address the concerns raised by Dul, Bruder, Buckle, Carayon, Falzon, Marras, Wilson and van der Doeler (2012) that “Human factors/ergonomics (HFE) has great potential to contribute to the design of all kinds of systems with people (work systems, product/service systems), but faces challenges in the readiness of its market and in the supply of high-quality applications” (Abstract).",
"However, if the ergonomic engineering system design were applied to change educational systems and showed some promise for overcoming the roadblocks inherent to Comenian design, ergonomics/human factors would certainly get a firm hold in broadly understood contemporary professional culture as a resource of great importance."
],
[
" WHY $\\rm \\bf e$ TO {{formula:a7f3899d-9872-4e19-b17b-d0abb68153a9}} ",
"The issue of education in the 21st century was an important subject of the first STHESCA conference in Kraków in July 2011 (STHESCA, 2011).",
"Preparations included a seminar also held in Kraków in November 2010 that served the purpose of defining a starting point for discussing the subject of science, technology, and higher education in contemporary society (Glazek, 2011).",
"The proposed starting point obtained the working title “e and E.” Lower-case, or small e denotes the contemporary world-wide system of education rooted in the design by Comenius from the 17th century (Comenius, 1992).",
"Upper-case, or big E denotes a system on a par with contemporary needs.",
"The table below gives examples of features of e that, in comparison with features of E, appear already outdated.",
"By the visible contrast, these examples also illustrate that improvement of e does not automatically lead to creation of E. An explanation of the entries in the table is provided below.",
"Table: Examples illustrating differences between systemseand E (Glazek, 2011).Explanations for the entries are providedbelow item by item.",
"System e is focused on teaching subject matter, whereas E is focused on educating a person in the context of a subject.",
"In e, the dominant form of teaching is specified by a curriculum independently of the context of students' lives.",
"In E, a context important to students is a natural stimulus for learning important concepts.",
"In e, students are punished if they do not know, do not understand, or cannot do something, until they fulfill the requirements, even if only superficially.",
"In E, students improve upon what they are good at, and this is how they notice new elements and directions worth studying.",
"In e, teaching subject matter is disconnected from teaching values and building character.",
"Natural sharing of useful information about the world among people in a group in E replaces destructive competition (Bok, 1987) and teaches principles of understanding in making decisions and handling resources.",
"One-size-fits-all testing for grades in e is replaced in E by providing feedback regarding individual progress in skill acquisition.",
"Testing of short-term memorizing “to get credit” in e is replaced in E by assessment of student performance in practice, akin to how skills of all other members of the system E are assessed (Drucker and Maciariello, 2008).",
"Comenius designed the process of teaching students in e as analogous to printing books in a press, while E fulfills contemporary requirements (Drucker, 1993).",
"Reading Recovery (Reading Recovery Council of North America, 2014) has a system for monitoring teachers' work in terms of their students' progress in acquiring skills.",
"This is worth studying as a candidate for use in E; there is no such system in e (Kenneth G. Wilson, personal communication).",
"e functions like a production line ordered according to age, while E accounts for differences among students, enabling them to develop over the lifespan (Drucker, 1993).",
"e becomes outdated and fails, having no system of self-correction, while E is by definition being created so that it changes in agreement with the needs of its clients (Wilson and Daviss, 1994; Wilson and Barsky, 1998).",
"e is based on compulsion, and E on students' will to learn (Glazek and Sarason, 2006) in agreement with the hypothesis (Glazek, 2008) that processes of learning based on will are those that lead to true learning, associated with changes in structure and functioning of the brain and other body parts.",
"In e, the human brain is treated in practice as a device ready for one-time programming, while in E as an organ that grows and changes throughout the lifespan (Fields, 2010; Dawkins, 2004).",
"Ten thousand hours is the amount of time of deliberate practice required for reaching an expert level of performance (Ericsson, 2004; Gladwell, 2008) and a teacher needs this much deliberate practice to become a good teacher in E. The explanation of number 7 in the Table says that a sketch of specifications for system E, congruent with the direction of development of the contemporary world, has already been drawn by Drucker (1993).",
"During Drucker's nearly century-long life he actively studied the practice of management processes involved in the transition of the most-developed countries from domination of a manual work force through instruction to domination of a workforce characterized by mental work based on values, knowledge and skills.",
"Specifications for the emerging system E along with predictable mechanisms of creating, principles of measuring (different from the ones applied in e), and methods of improving E by new generations until e is almost completely eliminated (probably still in the 21st century), form a list of challenges for ergonomic engineers.",
"These will be engineers who out of respect for human factors are willing to engage in redesigning education for the world to have a future.",
"Suppose that inhabitants of the most-advanced countries cease to accept systems of type e and learn in them less and less effectively, while the systems of type e enriched with new knowledge and technology continue to be very effective in developing countries.",
"A question arises: Is not a change from e to E in the leading countries a necessary condition for their continued fulfillment of this role?"
],
[
" ENGINEERING ",
"It is clear from our outline that the task of changing education is not to be completed quickly, even if the change is a necessary condition for continuation of development of democratic social systems.",
"The cohorts of ergonomic engineers will have to engage in such tasks over many generations.",
"To educate the required generations of capable engineers, one has to involve candidates as early as when they are children so that they will later have a chance to excel over time in systemic handling of extremely complex processes of learning.",
"If people continue to apply systems of type e to educate engineers, there is no chance that the alumni will develop the new system E. In order to educate engineers who will possess the needed values, knowledge and skills one has to switch from teaching kids according to the principles of engineering for kids (look, kids, you ought to do this or that) to the principles of engineering by kids (look, kids, you know the problem you face, design a solution).",
"An illustration of the paradigm shift from e to E is provided by the following which incorporates principles of the scouting movement.",
"There is a situation at a scouting camp where one group of scouts is separated from another by a river.",
"They conceive of a bridge and get it built.",
"Instead of imprinting minds in a classroom for engineers, knowledgeable and skillful coaches help students design and build a bridge and de facto some of them become seriously interested in engineering.",
"In contrast, a teacher in a classroom could explain to students who sit at their desks how carpenters work on a bridge.",
"This example does not need further explanation in order to catch the attention of serious human factor thinkers whose intention is to make the human learning as ergonomic as it can be.",
"There exists a great deal of information about the principles and functioning of youth organizations that provide members with the values, knowledge and skills they need.",
"Poland has a particularly rich tradition in this respect because of the long national struggle for freedom.",
"Unfortunately, the available sources are practically solely available in Polish.",
"For completeness, two examples are Janowski (2010) and Kamiński (2013).",
"In addition, contemporary Polish examples of the role that scouting experience can play in producing leaders of great merit include Michał Kulesza, (Wikipedia, 2014) who engaged in scouting in his youth and later designed the democratic government system for Poland after Solidarity took power from communists.",
"An example from the US is provided by Frances Hesselbein (Hesselbein, 2011).",
"She led the Girl Scouts of the US and helped members of the American army leadership understand the principles of education needed for soldiers and officers.",
"One should also note the great need for the education of managers in values, knowledge and skills stressed by Joseph Maciariello (Maciariello and Linkletter, 2011).",
"This concerns not only the management of engineering but quite generally high level management in all types of organizations that comprise society.",
"Maciariello stressed the lack of required education for managers as the key reason for recent world crises.",
"The references provided in this section substantiate a link between the need for Ergonomic Education and the well-being of society on a large scale that actually originates in how kids are educated, with scouting providing examples.",
"These examples are also supported by the dialog between Drucker and Albert Shanker, the late president of the American Federation of Teachers (Drucker, 1990, pp.",
"132-138).",
"The ergonomic principle of designing the environment to fit the user instead of being content with forcing the user to fit the environment provides engineers and engineering education with the opportunity to engage with the big E paradigm.",
"These engineers can become leaders of the new paradigm.",
"The ergonomic perspective of the human condition is here viewed as capable of becoming a basis for the new educational design, provided that the engineers: (1) recognize the principles of productive learning; (2) adhere to the spirit of performance defined by Peter Drucker; (3) incorporate Drucker's principles of management in organizing their new system; and (4) manage the practice of parental support for teacher initiatives in building the new system of engineering by kids.",
"Children and teachers, with support of parents, learn the principles of ergonomic engineering through and in collaboration with qualified engineers.",
"They experience engineering as a vital element of a rational approach to the human condition in a highly developed society.",
"The reason for calling this section Engineering by Kids, is that the students are self-motivated to learn, are carefully informed by their coaches about the progress they make, and are helped by coaches in discovering that they need to learn more."
],
[
" CONCLUSIONS ",
"The paradigm shift from e to E for education is based on the history of science, technology and society (e.g., Hughes, 1983; Smith, 2009) and the millions of years of biological development supporting the current learning abilities of humans (e.g., Dawkins, 2004; Fields, 2010).",
"The new paradigm, big E, brings the current condition of educational chaos to a transition and transformation that recognizes the human condition of wanting to learn.",
"Ergonomic Education focuses on student strengths not weaknesses, it emphasizes the context of learning and not a fixed curriculum, it is not constrained by age-graded classes, and it envisions a self-correcting system for life-long learning.",
"In Ergonomic Education the disciplines are no longer dictated to students but are discovered, practiced and developed by each new generation of students as they develop their strengths both as individuals and as part of a team.",
"It incorporates the concept of productive learning (Glazek and Sarason, 2006) and Drucker's (Drucker and Maciariello, 2008) principles of the spirit of performance and continuity, embodied in the pattern of practice in Engineering by Kids.",
"It is a system that takes a new approach to education by engaging students of any age in building their learning habits through what they want to learn and helping them to learn to benefit from deliberate practice (Ericsson, 2004) as they enter and develop the world of values, knowledge and skills.",
"The Ergonomic Education paradigm shifts to education based on coaching students as human beings who are hungry for productive learning throughout their lives from their very earliest days.",
"All students, regardless of age, are helped to learn in a system formed by competent educators operating within a new organizational structure.",
"Ergonomic engineers are uniquely qualified for designing and building such a functioning system of big E and continuing to make adjustments to improve the system to meet the individual learning needs of students and groups of students as they strive to learn new knowledge and develop social skills.",
"In addition to the direct impact of e to E on education, we propose that this is the path to Drucker's (1993) concept of continuity in a global society of organizations where continuing economic growth, world-wide stability, and democratic leadership can flourish.",
"This is a much more desirable situation than exists in societies which continue to use the Comenian “printing minds” education system to either direct or even limit the opportunities for its citizens to learn.",
"The paradigm of Ergonomic Education, in which the foundations of knowledge, skills of management, and engineering must be combined to build a post-Comenian educational system, appears to be a necessary condition for successful leadership of advanced democratic societies in the world.",
"Acknowledgement: This paper is the result of a long collaboration between the authors and the late Kenneth G. Wilson who died as we were in the midst of formalizing the ideas presented and preparing a series of papers to elaborate on the proposal for a new paradigm for education.",
"REFERENCES Arons, Arnold B.",
"(1990).",
"A Guide to Introductory Physics Teaching.",
"Hoboken, NJ: John Wiley & Sons.",
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]
] | 1403.0281 |
[
[
"Limits to the scope of applicability of extended formulations for LP\n models of combinatorial optimization problems: A summary"
],
[
"Abstract We show that new definitions of the notion of \"projection\" on which some of the recent \"extended formulations\" works (such as Kaibel (2011); Fiorini et al.",
"(2011; 2012); Kaibel and Walter (2013); Kaibel and Weltge (2013) for example) have been based can cause those works to over-reach in their conclusions in relating polytopes to one another when the sets of the descriptive variables for those polytopes are disjoint."
],
[
"Introduction",
"There has been a renewed interest and excellent work in extended formulations (EF's) over the past few years (see Conforti et al.",
"(2010; 2013), and Vanderbeck and Wolsey (2010), for example).",
"Since the seminal paper (Yannakakis (1991)), EF theory has been the single-most important paradigm for deciding the validity of proposed LP models for NP-Complete problems.",
"However, the issue of its scope of applicability has been a largely overlooked issue, leading to the possibility of over-reaching claims (implied or explicitly-stated).",
"The purpose of this paper is to make a contribution towards addressing the issue of delineating the scope of applicability of EF theory.",
"Specifically, we will show that EF theory is not valid for relating the sizes of descriptions of polytopes when the sets of the descriptive variables for those polytopes are disjoint, and that new definitions of the notion of “projection” upon which some of the recent extended formulations works (such as Kaibel (2011), Fiorini et al.",
"(2011, 2012), Kaibel and Walter (2013), and Kaibel and Weltge (2013), for example) have been based can cause those works to over-reach in their conclusions when the sets of the descriptive variables for the polytopes being related are disjoint or can be made so after redundant variables and constraints (with respect to the optimization problem at hand) are removed.",
"It should be noted that the intent of the paper is not to claim the correctness or incorrectness of any particular model that may have been developed in trying to address the “$P=NP$ ” question.",
"Our aim is, strictly, to bring attention to limits to the scope within which EF theory is applicable when attempting to derive bounds on the sizes of linear programming models for combinatorial optimzation problems.",
"In other words, the developments in the paper are not about deciding the correctness/incorrectness of any given LP model, but only about the issue of when such a decision (of correctness/incorrectness) is beyond the scope of EF theory.",
"One of the most fundamental assumptions in EF theory is that the addition of redundant variables and constraints to a given model of an optimization problem at hand does not change the EF relationships for that model.",
"The key point of this paper is to show the flaw in this assumption, which is that it leads to ambiguity and degeneracy/loss of meaningfulness of the notion of EF when the sets of the descriptive variables for the polytopes involved are disjoint.",
"We show that if redundant variables and constraints can be arbitrarily added to the description of a given polytope for the the purpose of establishing EF relationships, then every given mathematical programming model would be an EF of every other mathematical programming model, provided their sets of descriptive variables are disjoint, which would clearly mean a loss of meaningfulness of the notion (of EF).",
"Our developments in this paper were initially motivated by our realization that the “new” definition of EFs (Definition REF ) which was first proposed in Kaibel (2011) and then subsequently used in Fiorini et al.",
"(2011) becomes inconsistent with other and previous/“standard” definitions of EFs (see Definition REF ) when the sets of the descriptive variables of the polytopes being related are disjoint.",
"Comments we received in private communications and also in anonymous reviews on an earlier version of this paper were that our finding of inconsistency was “obvious,” and that all of our developments were “obvious” because of that.",
"Hence, we believe it may be useful to recall at this point that the Fiorini et al.",
"(2011) work (which is based on the “new” definition in question) has been highly-recognized, having received numerous awards.",
"Also, more generally, and perhaps more deeply, Martin's formulation of the Minimum Spanning Tree Problem (MSTP; see section REF of this paper) is cited in almost every EF paper in the current literature as a “normal” EF of Edmonds' formulation of the MSTP (although with the acknowledgement that it “escapes” the results for EF of NP-Complete problems somehow).",
"We argue that these facts highlight the need to bring our notions in this paper to the attention of the Optimization communities in general, and of the EF communities in particular.",
"The plan of the paper is as follows.",
"First, we will review the background definitions in section .",
"Our main result (i.e., the non-validity/non-applicability of EF theory when the sets of descriptive variables are disjoint) is developed in section .",
"In section , we illustrate the discussions of section using the Fiorini et al.",
"(2011; 2012) developments, as well as Martin's (1991) LP formulation of the Minimum Spanning Tree Problem (MSTP).",
"In section , we provide insights into the (correct) meaning/implication of the existence of a linear map between solutions of models, with respect to the task of solving an optimzation problem, when the set of descriptive variables in the models are disjoint.",
"Finally, we offer some concluding remarks in section .",
"The general notation we will use is as follows.",
"Notation 1 $\\mathbb {R}:$ Set of real numbers; $\\mathbb {R}_{\\mathbb {\\nless }}:$ Set of non-negative real numbers; $\\mathbb {N}:$ Set of natural numbers; $\\mathbb {N}_{\\mathbb {+}}:$ Set of positive natural numbers; $``\\mathbf {0}\":$ Column vector that has every entry equal to 0; $``\\mathbf {1}\":$ Column vector that has every entry equal to 1; $(\\cdot )^{T}:$ Transpose of $(\\cdot )$ ; $Conv(\\cdot ):$ Convex hull of $(\\cdot )$ ."
],
[
"Background definitions",
"For the purpose of making the paper as self-contained as possible, we review the basic definitions of extended formulations in this section.",
"Definition 2 (“Standard EF Definition” (Yannakakis (1991); Conforti et al.",
"(2010; 2013))) An extended formulation for a polytope $X$ $\\subseteq $ $\\mathbb {R}^{p}$ is a polyhedron $U$ $=$ $\\lbrace (x,w)$ $\\in $ $\\mathbb {R}^{p+q}$ $:$ $Gx$ $+$ $Hw$ $\\le $ $g\\rbrace $ the projection, $\\varphi _{x}(U)$ $:=$ $\\lbrace x\\in \\mathbb {R}^{p}:$ $(\\exists w\\in \\mathbb {R}^{q}:$ $(x,w) $ $\\in $ $U)\\rbrace ,$ of which onto $x$ -space is equal to $X$ (where $G$ $\\in \\mathbb {R}^{m\\times p},$ $H\\in \\mathbb {R}^{m\\times q},$ and $g\\in \\mathbb {R}^{m}$ ).",
"Definition 3 (“Alternate EF Definition #1” (Kaibel (2011); Fiorini et al.",
"(2011; 2012))) A polyhedron $U$ $=$ $\\lbrace (x,w)$ $\\in $ $\\mathbb {R}^{p+q}$ $:$ $Gx$ $+$ $Hw$ $\\le $ $g\\rbrace $ is an extended formulation of a polytope $X$ $\\subseteq $ $\\mathbb {R}^{p}$ if there exists a linear map $\\pi $ $:$ $\\mathbb {R}^{p+q}$ $\\longrightarrow $ $\\mathbb {R}^{p}$ such that $X$ is the image of $Q$ under $\\pi $ (i.e., $X=\\pi (Q)$ ; where $G\\in \\mathbb {R}^{m\\times p}$ , $H\\in \\mathbb {R}^{m\\times q},$ and $g\\in \\mathbb {R}^{m}$ ).",
"(Kaibel (2011), Kaibel and Walter (2013), and Kaibel and Weltge (2013) refer to $\\pi $ as a “projection.”) Definition 4 (“Alternate EF Definition #2” (Fiorini et al.",
"(2012))) An extended formulation of a polytope $X$ $\\subseteq $ $\\mathbb {R}^{p}$ is a linear system $U$ $=$ $\\lbrace (x,w)$ $\\in $ $\\mathbb {R}^{p+q}$ $:$ $Gx$ $+$ $Hw$ $\\le $ $g\\rbrace $ such that $x\\in X$ if and only if there exists $w\\in \\mathbb {R}^{q}$ such that $(x,w)\\in U.$ (In other words, $U$ is an EF of $X$ if $(x\\in X\\Longleftrightarrow (\\exists $ $w\\in \\mathbb {R}^{q}:(x,w)\\in U))$ ) (where $G$ $\\in \\mathbb {R}^{m\\times p},$ $H\\in \\mathbb {R}^{m\\times q},$ and $g\\in \\mathbb {R}^{m}$ )$.\\medskip $ The purpose of this paper is to point out that the scope of applicability of EF work based on the above definitions is limited to cases in which $U$ cannot be equivalently reformulated (with respect to the task of optimizing linear functions) in terms of the $w$ variables only.",
"For simplicity of exposition, without loss of generality, we will say that $G=\\mathbf {0}$ in the above definitions if there exists a description of $U$ which is in terms of the $w$ -variables only and has the same or smaller complexity order of size.",
"Or, equivalently, without loss of generality, we will say that $G\\ne \\mathbf {0}$ in the above definitions iff the $x$ - and $w$ -variables are required in every valid inequality description of $U$ which has the same or smaller complexity order of size as the description at hand.",
"In particular, if every constraint of $U$ which involves the $x$ -variables is redundant in the description of $U$ , then clearly, every one of those constraints as well as the $x$ -variables themselves can be dropped (without loss, with respect to the task of optimizing linear functions) from the description of $U$ , with the result that $U$ would be stated in terms of the $w$ -variables only.",
"Also, in some cases (all of) the contraints involving the $x$ -variables may become redundant only after other constraints in the description of $U$ are re-written and/or new constraints are added (as exemplified by the case of the minimum spanning tree problem (MSTP) in section REF of this paper).",
"If either of these two cases is applicable, we will say that $G=\\mathbf {0}$ in the above definitions.",
"Otherwise, we will say that $G\\ne \\mathbf {0.",
"}$ Remark 5 The following observations are in order with respect to Definitions REF , REF , and REF : The statement of $U$ in terms of inequality constraints only does not cause any loss of generality, since each equality constraint can be replaced by a pair of inequality constraints.",
"The system of linear equations which specify $\\pi $ in Definition REF must be valid constraints for $X$ and $U$ .",
"Hence, $X$ and $U$ can be respectively extended by adding those constraints to them, when trying to relate $X$ and $U$ using Definition REF .",
"In that sense, Definition REF “extends” Definitions REF and REF .",
"All three definitions are equivalent when $G\\ne \\mathbf {0}$ .",
"However, this is not true when $G=\\mathbf {0,}$ as we will show in section of this paper.",
"In the remainder of this paper, we will use the term “polytope” to refer to the polytope induced by a set of linear inequality constraints or the set of constraints itself, if this is convenient and does not cause ambiguity.",
"$\\square $"
],
[
"Non-applicability and degeneracyconditions for ",
"Our main result will now be developed.",
"Theorem 6 EF developments are not valid for relating the inequality descriptions of $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ in those definitions.",
"Proof.",
"The proof will be in three parts.",
"In Part 1, we will show that when $G=\\mathbf {0}$ , $U$ cannot be an EF of $X$ according to Definition REF .",
"In Part 2, we will show that when $G=\\mathbf {0}$ , $U $ cannot be an EF of $X$ according to Definition REF .",
"In Part 3, we will show that when $G=\\mathbf {0,}$ the EF notion under Definition REF results in the condition that every given polytope is an extended formulation of every other given polytope, provided their sets of descriptive variables are disjoint, which means that the (EF) notion becomes degenerate/meaningless.",
"In the discussion, we will only consider the case in which $U\\ne \\varnothing $ and $X\\ne \\varnothing ,$ since the theorem is trivial when $U=\\varnothing $ or $X=\\varnothing $ .",
"$(i)$ Consider Definition REF .",
"Assume $G=\\mathbf {0.",
"}$ Then, we have: $\\varphi _{x}(U) &=&\\lbrace x\\in \\mathbb {R}^{p}:(\\exists w\\in \\mathbb {R}^{q}:(x,w)\\in U)\\rbrace \\\\&=&\\lbrace x\\in \\mathbb {R}^{p}:(\\exists w\\in \\mathbb {R}^{q}:Hw\\le g\\rbrace \\\\&=&\\mathbb {R}^{p} \\\\&\\ne &X\\text{ \\ (since }X\\text{ is a polytope and thus bounded, whereas }\\mathbb {R}^{p}\\text{ is unbounded).",
"}$ Hence when $G=\\mathbf {0}$ , $U$ cannot be an EF of $X$ according to Definition REF .",
"$(ii)$ Consider Definition REF .",
"Assume $G=\\mathbf {0.",
"}$ Then, we have: $(\\exists w\\in \\mathbb {R}^{q}:\\mathbf {0}x+Hw\\le g)\\Longleftrightarrow (\\exists w\\in \\mathbb {R}^{q}:Hw\\le g)\\Longrightarrow \\left( \\forall x\\in \\mathbb {R}^{p},(x,w)\\in U)\\right) .",
"$ Which implies: $(\\exists w\\in \\mathbb {R}^{q}:\\mathbf {0}x+Hw\\le g)\\nRightarrow x\\in X\\text{\\ (since }X\\ne \\mathbb {R}^{p}\\text{)}.\\text{ } $ From (REF ), the “if and only if” stipulation of Definition REF cannot hold in general.",
"Hence, when $G=\\mathbf {0}$ , $U$ cannot be an EF of $X$ according to Definition REF .$\\medskip $ $(iii)$ Now, consider Definition REF .",
"We will show that the EF notion under this definition becomes degenerate/meaningless when $G=\\mathbf {0}$ .",
"The reasons for this are that a polytope can also be stated in terms of its extreme points (see Rockafellar (1997, pp.",
"153-172), among others), and that a linear map (as stipulated in the definition) could be inferred from this statement without reference to an inequality description of the polytope.",
"The proof consists of a counter-example to the sufficiency of the existence of a linear map, as stipulated in the definition, for implying EF relationships, as stipulated in the definition.",
"Note that if $G=\\mathbf {0}$ in Definition REF , then the linear inequality description of $U$ involves the $w-$ variables only.",
"For the sake of simplicity (but without loss of generality), let $\\overline{U}\\subset $ $\\mathbb {R}^{5}$ be described in the $w$ -variables only as $\\overline{U}=\\lbrace w\\in \\mathbb {R}_{\\nless }^{5}:w_{1}+w_{2}=5;w_{1}-w_{2}=1;w_{3}+w_{4}+w_{5}=0\\rbrace .",
"$ Then, the vertex-description of $\\overline{U}$ is $\\qquad \\overline{U}=\\lbrace w\\in \\mathbb {R}_{\\nless }^{5}:w\\in Conv\\left( \\left\\lbrace (3,2,0,0,0)^{T}\\right\\rbrace \\right) \\rbrace .\\medskip $ Now, let $X$ $\\subset $ $\\mathbb {R}_{\\nless }^{3}$ be specified by its vertex-description as $X=\\lbrace x\\in \\mathbb {R}_{\\nless }^{3}:x\\in Conv(\\left\\lbrace (2,1,5)^{T}\\right\\rbrace )\\rbrace .$ (In other words, $X$ consists of the point in $\\mathbb {R}^{3}$ , $(2,1,5)^{T}$ .)",
"Then, the following are true: $(iii.1)$ $& ((x\\in X)\\text{ \\ and \\ }(w\\in \\overline{U}))\\Longrightarrow x=Aw, \\\\& \\text{where, among other possibilities, }A=\\left[\\begin{tabular}{rrrrr}-1 & 2.5 & 2 & 3 & 4 \\\\1 & -1 & 5 & 6 & 7 \\\\-1 & 4 & 8 & 9 & 10\\end{tabular}\\right] \\text{.}",
"$ Hence, under Definition REF , $\\overline{U}$ is an EF of every one of the infinitely-many possible inequality descriptions of $X $ (since $x=Aw$ in the above is a linear map between $\\overline{U}$ and $X$ ).",
"$(iii.2)$ Similarly, $& ((x\\in X)\\text{ \\ and \\ }(w\\in \\overline{U}))\\Longrightarrow w=Bx, \\\\& \\text{where, among other possibilities, }B=\\left[\\begin{tabular}{rrr}-1 & 1 & 1 \\\\1 & -1 & 0 \\\\3 & 1 & -2 \\\\2 & -11 & 1 \\\\-10 & 30 & 0\\end{tabular}\\right] \\text{.}",
"$ Hence, under Definition REF , every one of the infinitely-many possible inequality descriptions of $X$ is an EF of the inequality descriptions of $\\overline{U}$ as stated in (REF ) (and, in fact, of everyone of the infinitely-many possible inequality descriptions of $\\overline{U}$ ).",
"$(iii.3)$ Clearly, the EF relations based on (REF ) and (REF ) above are degenerate/meaningless, since no meaningful inferences can be made from them in attempting to compare inequality descriptions of $\\overline{U}$ and $X.$ A fundamental notion in extended formulations theory is that the addition of redundant variables and constraints to the inequality description of a polytope does not change the EF relationships for that polytope.",
"We use this fact to generalize the degeneracy/loss of meaningfulness which arises out of Definition REF when $G=\\mathbf {0}$ to Definitions REF and REF , as follows.",
"Theorem 7 Provided redundant constraints and variables can be arbitrarily added to the descriptions of polytopes for the purpose of establishing EF relationships under Definitions REF -REF , the descriptions of any two given non-empty polytopes expressed in disjoint variable spaces can be respectively augmented into being extended formulations of each other.",
"In other words, let $x^{1}\\in \\mathbb {R}^{n_{1}}$ ($n_{1}\\in \\mathbb {N}_{+}$ ) and $x^{2}\\in \\mathbb {R}^{n_{2}}$ ($n_{2}\\in \\mathbb {N}_{+}$ ) be vectors of variables with no components in common.",
"Then, provided redundant constraints and variables can be arbitrarily added to the descriptions of polytopes for the purpose of establishing EF relationships, the inequality-description of every non-empty polytope in $x^{1}$ can be augmented into an EF of the inequality-description of every other non-empty polytope in $x^{2}$ , and vice versa.",
"Proof.",
"The proof is essentially by construction.",
"Let $P_{1}$ and $P_{2}$ be polytopes specified as: $& P_{1}=\\lbrace x^{1}\\in \\mathbb {R}^{n_{1}}:A_{1}x^{1}\\le a_{1}\\rbrace \\ne \\varnothing \\text{ \\ }(\\text{where }A_{1}\\in \\mathbb {R}^{p_{1}\\times n_{1}}\\text{, and }a_{1}\\in \\mathbb {R}^{p_{1}}); \\\\[0.09in]& P_{2}=\\lbrace x^{2}\\in \\mathbb {R}^{n_{2}}:A_{2}x^{2}\\le a_{2}\\rbrace \\ne \\varnothing \\text{ \\ (where }A_{2}\\in \\mathbb {R}^{p_{2}\\times n_{2}}\\text{, and }a_{2}\\in \\mathbb {R}^{p_{2}}).$ Clearly, $\\forall (x^{1},$ $x^{2})\\in P_{1}\\times P_{2},$ $\\forall q\\in \\mathbb {N}_{\\mathbb {+}}$ , $\\forall B_{1}\\in \\mathbb {R}^{q}{}^{\\times n_{1}},$ $\\forall B_{2}\\in \\mathbb {R}^{q\\times n_{2}},$ there exists $u\\in \\mathbb {R}_{\\nless }^{q}$ such that the constraints $B_{1}x^{1}+B_{2}x^{2}-u\\le 0 $ are valid for $P_{1}$ and $P_{2}$ , respectively (i.e., they are redundant for $P_{1}$ and $P_{2},$ respectively).",
"Now, consider : $W:=& \\left\\lbrace (x^{1},x^{2},u)\\in \\mathbb {R}^{n_{1}}\\times \\mathbb {R}^{n_{2}}\\times \\mathbb {R}_{\\nless }^{q}:\\right.",
"\\\\[0.06in]& C_{1}A_{1}x^{1}\\le C_{1}a_{1};\\text{ } \\\\[0.06in]& B_{1}x^{2}+B_{2}x^{1}-u\\le 0; \\\\[0.06in]& \\left.",
"C_{2}A_{2}x^{2}\\le C_{2}a_{2}\\right\\rbrace $ (where: $C_{1}\\in $ $\\mathbb {R}^{p_{1}\\times }{}^{p_{1}}$ and $C_{2}$ $\\in \\mathbb {R}^{p_{2}\\times }{}^{p_{2}}$ are diagonal matrices with positive diagonal entries).",
"Clearly, $W$ augments $P_{1}$ and $P_{2}$ respectively.",
"Hence: $W\\text{ is equivalent to }P_{1},\\text{ and} $ $W\\text{ is equivalent to }P_{2}\\text{ .}",
"$ Also clearly, we have: $\\varphi _{x^{1}}(W)=P_{1}\\text{ \\ (since }P_{2}\\ne \\varnothing ,\\text{ and((\\ref {EF_Thm(c)}) and (\\ref {EF_Thm(d)}) are redundant for }P_{1})),\\text{\\ \\ and } $ $\\varphi _{x^{2}}(W)=P_{2}\\text{ \\ (since }P_{1}\\ne \\varnothing ,\\text{ and((\\ref {EF_Thm(b)}) and (\\ref {EF_Thm(c)}) are redundant for }P_{2})\\text{)}.$ It follows from the combination of (REF ) and (REF ) that $P_{1}$ is an extended formulation of $P_{2}.\\medskip $ It follows from the combination of (REF ) and (REF ) that $P_{2}$ is an extended formulation of $P_{1}.$ Example 8 Let $& P_{1}=\\lbrace x\\in \\mathbb {R}_{\\nless }^{2}:2x_{1}+x_{2}\\le 6\\rbrace ; \\\\[0.06in]& P_{2}=\\lbrace w\\in \\mathbb {R}_{\\nless }^{3}:18w_{1}-w_{2}\\le 23;\\text{ }59w_{1}+w_{3}\\le 84\\rbrace .$ For arbitrary matrices $B_{1},$ $B_{2}$ , $C_{1},$ and $C_{2}$ (of appropriate dimensions, respectively)$;$ say $B_{1}=\\left[\\begin{array}{cc}-1 & 2 \\\\3 & -4\\end{array}\\right] ,$ $B_{2}=\\left[\\begin{array}{ccc}5 & -6 & 7 \\\\-10 & 9 & -8\\end{array}\\right] ,$ $C_{1}=\\left[ 7\\right] ,$ and $C_{2}=\\left[\\begin{array}{cc}2 & 0 \\\\0 & 0.5\\end{array}\\right] ;$ $P_{1}$ and $P_{2}$ can be augmented into extended formulations of each other using $u\\in \\mathbb {R}_{\\nless }^{2}$ and $W$ : $W=& \\left\\lbrace (x,w,u)\\in \\mathbb {R}_{\\nless }^{2+3+2}:\\text{ \\ }\\left[ 7\\right]\\left[\\begin{array}{cc}2 & 1\\end{array}\\right] \\left[\\begin{array}{c}x_{1} \\\\x_{2}\\end{array}\\right] \\le 42\\right.",
"; \\\\& \\left[\\begin{array}{cc}-1 & 2 \\\\3 & -4\\end{array}\\right] \\left[\\begin{array}{c}x_{1} \\\\x_{2}\\end{array}\\right] +\\left[\\begin{array}{ccc}5 & -6 & 7 \\\\-10 & 9 & -8\\end{array}\\right] \\left[\\begin{array}{c}w_{1} \\\\w_{2} \\\\w_{3}\\end{array}\\right] -\\left[\\begin{array}{c}u_{1} \\\\u_{2}\\end{array}\\right] \\le \\left[\\begin{array}{c}0 \\\\0\\end{array}\\right] ; \\\\& \\left.",
"\\left[\\begin{array}{cc}2 & 0 \\\\0 & 0.5\\end{array}\\right] \\left[\\begin{array}{ccc}18 & -1 & 0 \\\\59 & 0 & 1\\end{array}\\right] \\left[\\begin{array}{c}w_{1} \\\\w_{2} \\\\w_{3}\\end{array}\\right] \\le \\left[\\begin{array}{c}46 \\\\42\\end{array}\\right] \\right\\rbrace .\\text{ \\ }$ $\\square $"
],
[
"Application to the Fiorini ",
"Fiorini et al.",
"(2012) is a re-organized and extended version of Fiorini et al.",
"(2011).",
"The key extension is the addition of another alternate defnition of extended formulations (page 96 of Fiorini et al.",
"(2012)) which is recalled in this paper as Definition REF .",
"This new alternate definition is then used to re-arrange “section 5” of Fiorini et al.",
"(2011) into “section 2” and “section 3” of Fiorini et al.",
"(2012).",
"Hence, the developments in “section 5” of Fiorini et al.",
"(2011) which depended on “Theorem 4” of that paper, are “stand-alones” (as “section 3”) in Fiorini et al.",
"(2012), and “Theorem 4” in Fiorini et al.",
"(2011) is relabeled as “Theorem 13” in Fiorini et al.",
"(2012).",
"Claim 9 The developments in Fiorini et al.",
"(2011) are not valid for relating the inequality descriptions of $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ .",
"Proof.",
"Using the terminology and notation of Fiorini et al.",
"(2011), the main results of section 2 of Fiorini et al.",
"(2011) are developed in terms of $Q:=\\lbrace (x,y)\\in \\mathbb {R}^{d+k}$ $\\left| \\text{ }Ex+Fy=g,\\text{ }y\\in C\\right.",
"\\rbrace $ and $P:=\\lbrace x\\in \\mathbb {R}^{d}$ $\\left| \\text{ }Ax\\le b\\right.",
"\\rbrace ,$ with $Q$ (in Fiorini et al.",
"(2011)) corresponding to $U$ in Definitions REF -REF $,$ and $P$ (in Fiorini et al.",
"(2011)) corresponding to $X$ in Definitions REF -REF .",
"Hence, $G=\\mathbf {0}$ in Definitions REF -REF corresponds to $E=\\mathbf {0}$ in Fiorini et al.",
"(2011).",
"Hence, firstly, assume $E=\\mathbf {0}$ in the expression of $Q$ (i.e., $Q:=\\lbrace (x,y)\\in \\mathbb {R}^{d+k}$ $\\left| \\text{ }\\mathbf {0}x+Fy=g,\\text{ }y\\in C\\right.",
"\\rbrace ).$ Then, secondly, consider Theorem 4 of Fiorini et al.",
"(2011) (which is pivotal in that work).",
"We have the following: $(i)$ If $A\\ne \\mathbf {0}$ in the expression of $P\\mathbf {,}$ then the proof of the theorem is invalid since that proof requires setting “$E:=A $ ” (see Fiorini et al.",
"(2011, p. 7)); $(ii)$ If $A=\\mathbf {0,}$ then $P:=\\lbrace x\\in \\mathbb {R}^{d}$ $\\left|\\text{ }\\mathbf {0}x\\le b\\right.",
"\\rbrace .$ This implies that either $P=\\mathbb {R}^{d}$ (if $b\\ge \\mathbf {0}$ ) or $P=\\varnothing $ (if $b\\ngeq \\mathbf {0}$ ).",
"Hence, $P$ would be either unbounded or empty.",
"Hence, there could not exist a non-empty polytope, $Conv(V),$ such that $P=Conv(V)$ (see Fiorini et al.",
"(2011, 16-17), among others).",
"Hence, the conditions in the statement of Theorem 4 of Fiorini et al.",
"(2011) would be ill-defined/impossible.",
"Hence, the developments in Fiorini et al.",
"(2011) are not valid for relating $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ in those definitions.",
"Claim 10 The developments in Fiorini et al.",
"(2012) are not valid for relating the inequality descriptions of $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ .",
"Proof.",
"First, note that “Theorem 13” of Fiorini et al.",
"(2012, p. 101) is the same as “Theorem 4” of Fiorini et al.",
"(2011).",
"Hence, the proof of Claim REF above is applicable to “Theorem 13” of Fiorini et al.",
"(2012).",
"Hence, the parts of the developments in Fiorini et al.",
"(2012) that hinge on this result (namely, from “section 4” onward in Fiorini et al.",
"(2012)) are not valid for relating $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ .",
"Now consider “Theorem 3” of Fiorini et al.",
"(2012) (section 3, page 99).",
"The proof of that theorem hinges on the statement that (using the terminology and notation of Fiorini et al.",
"(2012) which is similar to that in Fiorini et al.",
"(2011)): $Ax\\le b\\Longleftrightarrow \\exists y:E^{\\le }x+F^{\\le }y\\le g^{\\le },\\text{ }E^{=}x+F^{=}y\\le g^{=}.",
"$ Note that $G=\\mathbf {0}$ in Definitions REF -REF would correspond to $E^{\\le }=E^{=}=\\mathbf {0}$ in (REF ).",
"Hence, assume $E^{\\le }=E^{=}=\\mathbf {0}$ in (REF ).",
"Then, clearly, the “if and only if” stipulation of (REF ) cannot be satisfied in general, since $(\\exists y:\\mathbf {0}\\cdot x+F^{\\le }y\\le g^{\\le },\\text{ }\\mathbf {0}\\cdot x+F^{=}y\\le g^{=})\\text{ cannot imply (}Ax\\le b)\\text{ in general.",
"}$ Hence, Theorem 3 of Fiorini et al.",
"(2012) is not valid for relating $U$ and $X$ in Definitions REF -REF , when $G=\\mathbf {0}$ .",
"Hence, the developments in Fiorini et al.",
"(2012) are not valid for relating $U$ and $X$ in Definitions REF -REF when $G=\\mathbf {0}$ in those definitions."
],
[
"The case of the Minimum Spanning Tree Problem: Redundancy matters when “$G=\\mathbf {0}$ ” ",
"The consideration of “$G=\\mathbf {0}$ ” we have introduced in this paper is an important one because, as we have shown, it refines the notion of EFs by separating the case in which the notion is meaningful from the case in which the notion is degenerate and ambiguous.",
"The degeneracy (when “$G=\\mathbf {0}$ ”) stems from the fact that every polytope is potentially an EF of every other polytope, as we have shown in section of this paper.",
"The ambiguity stems from the fact that one would reach contradicting conclusions as to what is/is not an EF of a given polytope, depending on what we do with the redundant constraints and variables which are introduced.",
"This is illustrated in the following example.",
"Example 11 Refer back to the numerical example in Part $(iii)$ of the proof of Theorem REF .",
"An example of an inequality description of $X$ in that numerical example is: $\\overline{X}=\\lbrace x\\in \\mathbb {R}_{\\nless }^{3}:x_{1}-x_{2}+x_{3}=6;$ $x_{1}+x_{2}\\ge 3;$ $x_{1}+x_{3}\\le 7;$ $x_{2}+x_{3}\\ge 6;$ $\\ x_{1}\\le 2\\rbrace $ .",
"(It easy to verify that the feasible set of $\\overline{X}$ is indeed $\\lbrace (2,1,5)^{T}\\rbrace $ .)",
"Let $U^{\\prime }$ denote $\\overline{U}$ augmented with the constraints of the linear map, $x-Aw=0$ .",
"Clearly $U^{\\prime }$ does project to $X$ under the standard definition (Definition REF ), whereas $\\overline{U}$ does not.",
"Hence, the answer to the question of whether or not $\\overline{U}$ is an extended formulation of $X$ under the standard definition depends on what we do with the redundant constraints, $x-Aw=0$ .",
"If these constraints are added to $\\overline{U},$ then $\\overline{U}$ becomes $U^{\\prime }$ , and the answer is “Yes.” If these constraints are left out, the answer is “No.” Hence, the extended formulations relation which is established between $\\overline{U}$ and $X$ under Definition REF is ambiguous (in addition to being degenerate, as we have shown in Theorem REF ).",
"$\\square $ A well-researched case in point for the discussions above is that of the Minimum Spanning Tree Problem (MSTP).",
"Without the refinement brought by the distinction we make between the cases of $G=\\mathbf {0}$ and $G\\ne \\mathbf {0}$ in Definitions REF -REF , the case of the MSTP would mean that it is possible to extend an exponential-sized model into a polynomial-sized one by (simply) adding redundant variables and constraints to it (i.e., augmenting it), which is a clearly-unreasonable/out-of-the-question proposition.",
"To see this, assume (as is normally done in EF work) that the addition of redundant constraints and variables does not matter as far EF relationships are concerned.",
"Since the constraints of Edmonds' model (Edmonds (1970)) are redundant for the model of Martin (1991), one could augment Martin's formulation with these constraints.",
"The resulting model would still be considered a polynomial-sized one.",
"But note that this particular augmentation of Martin's model would also be an augmentation of Edmonds' model.",
"Hence, the conclusion would be that Edmonds' exponential-sized model has been augmented into a polynomial-sized one, which is an impossibility, since one cannot reduce the number of facets of a given polytope by simply adding redundant constraints to the inequality description of that polytope.",
"The distinction we are bringing to attention in this paper explains the paradox, as further detailed below.",
"Example 12 We show that Martin's polynomial-sized LP model of the MSTP is not an EF (in a non-degenerate, meaningful sense) of Edmonds's exponential LP model of the MSTP, by showing that there exists a reformulation of Martin's model which does not require the variables of Edmonds' model (which is essentially the equivalent of having $G=\\mathbf {0}$ in the description of $U$ in Definitions REF -REF ).",
"Using the notation in Martin(1991), i.e.",
": $N:=\\lbrace 1,\\ldots ,n\\rbrace $ (Set of vertices); $E:$ Set of edges; $\\forall S\\subseteq N,$ $\\gamma (S):$ Set of edges with both ends in $S $ .",
"Exponential-sized/“sub-tour elimination” LP formulation (Edmonds (1970)): $(P)$ : $\\left|\\begin{tabular}{ll}\\text{Minimize:} & \\sum \\limits _{e\\in E}c_{e}x_{e} \\\\& \\\\\\text{Subject To:} & \\sum \\limits _{e\\in E}x_{e}=n-1; \\\\& \\\\& \\sum \\limits _{e\\in \\gamma (S)}x_{e}\\le \\left| S\\right| -1; \\ \\ S\\subset E\\ ; \\\\& \\\\& x_{e}\\ge 0 \\ for all e\\in E.\\end{tabular}\\text{ \\ }\\right.",
"$ Polynomial-sized LP reformulation (Martin (1991)): $(Q)$ : $\\left|\\begin{tabular}{ll}\\text{Minimize:} & \\sum \\limits _{e\\in E}c_{e}x_{e} \\\\& \\\\\\text{Subject To:} & \\sum \\limits _{e\\in E}x_{e}=n-1; \\\\& \\\\& z_{k,i,j}+z_{k,j,i}=x_{e}; \\ \\ k=1,\\ldots ,n; \\ e\\in \\gamma (\\lbrace i,j\\rbrace ); \\\\& \\\\& \\sum \\limits _{s>i}z_{k,i,s}+\\sum \\limits _{h<i}z_{k,i,h}\\le 1; \\ \\ k=1,\\ldots ,n; \\ \\ i\\ne k; \\\\& \\\\& \\sum \\limits _{s>k}z_{k,k,s}+\\sum \\limits _{h<k}z_{k,k,h}\\le 0; \\ k=1,\\ldots ,n; \\\\& \\\\& x_{e}\\ge 0 \\ for all e\\in E; \\ \\ z_{k,i,j}\\ge 0 \\ for all k, i,j.\\end{tabular}\\text{ \\ }\\right.",
"\\medskip $ Re-statement of Martin's LP model: For each $e\\in E:$ - Denote the ends of $e$ as $i_{e}$ and $j_{e},$ respectively; - Fix an arbitrary node, $r_{e}$ , which is not incident on $e$ (i.e., $r_{e}$ is such that it is not an end of $e$ ).",
"Then, one can verify that $Q$ is equivalent to: $(Q$ '$)$ : Table: NO_CAPTION$\\left|\\begin{tabular}{ll}\\text{Minimize:} & \\sum \\limits _{e\\in E}c_{e}z_{r_{e},i_{e},j_{e}}+\\sum \\limits _{e\\in E}c_{e}z_{r_{e},j_{e},i_{e}}\\\\& \\\\\\text{Subject To:} & \\sum \\limits _{e\\in E}z_{r_{e},i_{e},j_{e}}+\\sum \\limits _{e\\in E}z_{r_{e},j_{e},i_{e}}=n-1; \\\\& \\\\& z_{k,i_{e},j_{e}}+z_{k,j_{e},i_{e}}=z_{r_{e},i_{e},j_{e}}+z_{r_{e},j_{e},i_{e}}; \\ \\ k=1,\\ldots ,n;\\ \\ \\ e\\in E; \\\\& \\\\& \\sum \\limits _{s>i}z_{k,i,s}+\\sum \\limits _{h<i}z_{k,i,h}\\le 1; \\ \\ i, k=1,\\ldots ,n:i\\ne k; \\\\& \\\\& \\sum \\limits _{s>k}z_{k,k,s}+\\sum \\limits _{h<k}z_{k,k,h}\\le 0; \\ \\ k=1,\\ldots ,n; \\\\& \\\\& z_{k,i,j}\\ge 0 \\ for all k, i, j.\\end{tabular}\\text{ \\ }\\right.",
"$ $\\square \\medskip $ Claim 13 We claim that the reason EF work relating formulation sizes does not apply to the case of the MSTP is that although Martin's model can be made to project to Edmond's model, that projection is degenerate/non-meaningful in the sense we have described in this paper."
],
[
"Alternate/Auxiliary Models",
"In this section, we provide some insights into the meaning of the existence of an affine map establishing a one-to-one correspondence between polytopes when the sets of descriptive variables are disjoint, as brought to our attention in private e-mail communications by Kaibel (2013), and Yannakakis (2013), respectively.",
"The linear map stipulated in Definition REF is a special case of the affine map.",
"Referring back to Definitions REF -REF , we will show in this section that when $G=\\mathbf {0}$ in the expression of $U$ and there exists a one-to-one affine mapping of $X$ onto $U$ , then $U$ is simply an alternate model (a “reformulation”) of $X$ which can be used, in an “auxiliary” way, in order to optimize any linear function of $x$ over $X $ , without any reference to/knowledge of an inequality description of $X$ .",
"Example 14 Let: - $\\ x\\in \\mathbb {R}^{p}$ and $w\\in \\mathbb {R}^{q}$ be disjoint vectors of variables; - $\\ X:=\\lbrace x\\in \\mathbb {R}^{p}:Ax\\le a\\rbrace \\ \\ \\ $ (where $A\\in \\mathbb {R}^{m\\times p}$ , and $a\\in \\mathbb {R}^{m}$ ); - $\\ U:=\\lbrace w\\in \\mathbb {R}^{q}:Dw\\le d\\rbrace $ $\\ \\ $ (where $D\\in \\mathbb {R}^{n\\times q}$ , and $d\\in \\mathbb {R}^{n}$ ); - $\\ L:=\\lbrace (x,w)\\in \\mathbb {R}^{p+q}:x-Cw=b\\rbrace $ (where $C\\in \\mathbb {R}^{p\\times q}$ , and $b\\in \\mathbb {R}^{p}).$ Assume that the non-negativity requirements for $x$ and $w$ are included in the constraints of $X$ and $U$ , respectively, and that $L$ is redundant for $X$ and for $U$ .",
"Then, it is easy to see that the optimization problem, Problem LP$_{1}$ : $\\left|\\begin{tabular}{ll}\\text{Minimize:} & \\alpha ^{T}x \\\\& \\\\\\text{Subject To:} & (x,w)\\in L; \\ w\\in U \\\\& \\\\\\multicolumn{2}{l}{(where \\alpha \\in \\mathbb {R}^{p}).",
"}\\end{tabular}\\text{ \\ }\\right.",
"\\medskip $ is equivalent to the smaller linear program, Problem LP$_{2}$ : $\\left|\\begin{tabular}{ll}\\text{Minimize:} & \\left( \\alpha ^{T}C\\right) w+\\alpha ^{T}b \\\\& \\\\\\text{Subject To:} & w\\in U \\\\& \\\\\\multicolumn{2}{l}{(where \\alpha \\in \\mathbb {R}^{p}).",
"}\\end{tabular}\\text{ \\ }\\right.",
"\\medskip \\medskip $ Hence, if $L$ is the graph of a one-to-one correspondence between the points of $X$ and the points of $U$ (see Beachy and Blair (2006, pp.",
"47-59)), then, the optimization of any linear function of $x$ over $X$ can be done by first using Problem LP$_{\\mathit {2}}$ in order to get an optimal $w,$ and then using Graph $L$ to “retrieve” the corresponding $x$ .",
"Note that the second term of the objective function of Problem LP$_{\\mathit {2}}$ can be ignored in the optimization process of Problem LP$_{\\mathit {2}},$ since that term is a constant.",
"Hence, if $L$ is derived from knowledge of the vertex description of $X$ only, then this would mean that the inequality description of $X$ is not involved in the “two-step” solution process (of using Problem LP$_{\\mathit {2}}$ and then Graph $L$ ), but rather, that only the vertex description of $X$ is involved.",
"$\\ \\ \\square $ Hence, when $G=0$ , the existence of the linear map, $\\pi ,$ stipulated in Definition REF does not imply that $U$ is an EF of $X$ , but rather that $U$ can be used to solve the optimization problem over $X $ without any reference to/knowledge of an inequality description of $X,$ if $\\pi $ is not derived from an inequality description of $X$ .",
"$\\ \\ $"
],
[
"Conclusions",
"We have shown that extended formulations theory aimed at comparing and/or bounding sizes of inequality descriptions of polytopes are not applicable when the set of the descriptive variables for those polytopes are disjoint (i.e., when “$G=\\mathbf {0}$ ”).",
"We have illiustrated our ideas using the Fiorini et al.",
"(2011; 2012) developments, and Martin's (1991) LP formulation of the MSTP, respectively.",
"We have also shown that the “$G=\\mathbf {0}$ ” consideration we have brought to attention explains the existing paradox in EF theory (typified by the case of the MSTP), which is that by simply adding redundant constraints and variables to a model of exponential size one can obtain a model of polynomial size.",
"We believe these constitute important, useful steps towards a more complete definition of the scope of applicability for EF's."
]
] | 1403.0529 |
[
[
"Space time symmetry in quantum mechanics"
],
[
"Abstract New prescription to treat position and time equally in quantum mechanics is presented.",
"Using this prescription, we could successfully derive some interesting formulae such as time-of-arrival for a free particle and quantum tunneling formula.",
"The physical interpretation will be discussed."
],
[
"Introduction",
"Though we can treat time and space symmetric way in relativity, in quantum mechanics the time seems different to other observables: It seems we don't have proper operator for time.",
"A particle detected at one position can be detected at the same position at later time, namely, we encounter the difficulty on orthogonality and normalization and these two measurements do not commute each other.",
"This non commuting property leads us to think about time-of-arrival which means the time that a particle first arrives to a specific position.",
"Allcock [2] tried to build time-of-arrival eigenstates which are orthogonal each other for different time but could not define a consistent time-of-arrival.",
"His study says that because we cannot absorb the particle in an arbitrarily short time, we cannot measure the time-of-arrival at any accuracy.",
"Oppenheim et al[3] insist that using two level detector absorbing a particle in arbitrarily short period time, we can overcome this restrictions.",
"However they found that the limitation on measuring the time-of-arrival with arbitrarily accuracy comes from the clock coupled to the trigger.",
"They show that if we couple the system to a clock to measure the time-of-arrival at which the particle arrives at specific position, then the accuracy of measurement is limited by $\\delta t_A > \\hbar /E_k$ where $\\delta t_A$ is the minimum uncertainty of measuring time of arrival and $E_k$ is the energy of clock.",
"One of interesting approaches to find the time-of-arrival operator has been studied by Grot et al.",
"[1] They tried to develop the time-of-arrival operator for free non-relativistic particle by proper ordering of space time operator in Heisenberg picture, analogous to classical picture.",
"But the eigenstates of time-of-arrival they calculated satisfying eigenvalue equation, did not satisfy the orthogonal condition for different times.",
"They bypassed this difficulty by modifying the time-of-arrival operator so that in the classical limit would not reproduced the time-of-arrival exactly, but would reproduce a quantity arbitrary close to the time-of-arrival.",
"In this article, I will not attempt to develop the time-of-arrival operator nor discuss about dynamical limitation on treating position and time of quantum mechanics in equal manner.",
"Rather it will be focused on how we can put an equal footing on position and time in quantum mechanical evolution.",
"Contrast to other approaches, I assumed that we cannot put an equal footing on both position and time simultaneously.",
"That is, when we treat the position as a quantum operator, we have to treat the time as an evolution parameter.",
"And when we treat the time as a quantum operator, we have to treat the position as an evolution parameter.",
"We will discuss how we can apply this prescription on quantum tunneling process."
],
[
"Prescription",
"In this section notational conventions will be defined in symmetrical way for both position and time.",
"When the time is used as an evolution parameter (TEP), the position is used as a usual quantum observable.",
"When the position is used as an evolution parameter (PEP), the time is used as a usual quantum observable.",
"We can specify any quantum states with one state vector and one evolution parameter."
],
[
"TEP",
"We denote the quantum state $\\psi $ at time $t_1$ by $\\mid \\psi , \\underline{t_1}\\rangle $ where the underline on $\\underline{t_1}$ means that time is the evolution parameter.",
"That is, the first one represents the quantum state and the second one stands for the evolution parameter.",
"We can express (REF ) it shorter by $\\mid \\psi _1\\rangle \\equiv \\mid \\psi , \\underline{t_1}\\rangle $ where the subscript 1 on $\\psi $ means the state is of time $t_1$ .",
"The probability amplitude to find the position $x$ is then $\\langle x \\mid \\psi , \\underline{t_1}\\rangle = \\psi (x,\\underline{t_1})$ where the position $x$ is the usual quantum observable and the time $\\underline{t_1}$ is the evolution parameter.",
"Thus while the operation $\\langle x\\mid x_1\\rangle $ is possible, the operation $\\langle \\underline{t}\\mid \\underline{t_1}\\rangle $ is not possible because both $t$ and $t_1$ are just evolution parameters.",
"By the same reason $\\langle \\underline{t}\\mid \\psi , \\underline{t_1}\\rangle =\\psi (\\underline{t},\\underline{t_1})$ does not make sense.",
"Since this represents the pure evolution process up to $t$ , it must be denoted by $\\langle \\underline{t}\\mid \\psi , \\underline{t_1}\\rangle =\\mid \\psi , \\underline{t}\\rangle $"
],
[
"PEP",
"We denote the quantum state $\\varphi $ at position $x_1$ by $\\mid \\varphi , \\underline{x_1}\\rangle $ where the underline on $\\underline{x_1}$ means that position is the evolution parameter.",
"That is, the first one represents the quantum state and the second one stands for the evolution parameter.",
"We can express (REF ) it shorter by $\\mid \\varphi _1\\rangle \\equiv \\mid \\varphi , \\underline{x_1}\\rangle $ where the subscript 1 on $\\varphi $ means the state is of position $x_1$ .",
"The probability amplitude to find the time $t$ is then $\\langle t \\mid \\varphi , \\underline{x_1}\\rangle = \\varphi (t,\\underline{x_1})$ where the time $t$ is the usual quantum observable and the position $\\underline{x_1}$ is the evolution parameter.",
"Thus while the operation $\\langle t\\mid t_1\\rangle $ is possible, the operation $\\langle \\underline{x}\\mid \\underline{x_1}\\rangle $ is not possible because both $x$ and $x_1$ are just evolution parameters.",
"By the same reason $\\langle \\underline{x}\\mid \\varphi , \\underline{x_1}\\rangle =\\varphi (\\underline{x},\\underline{x_1})$ does not make sense.",
"Since this represents the pure evolution process up to $x$ , it must be denoted by $\\langle \\underline{x}\\mid \\varphi , \\underline{x_1}\\rangle =\\mid \\varphi , \\underline{x}\\rangle $ As we have seen, any quantum state is expressed by one state vector and one evolution parameter as $\\mid \\psi ,\\underline{t}\\rangle $ or $\\mid \\varphi , \\underline{x}\\rangle $ .",
"We cannot specify a quantum state by $\\mid \\psi ,x,\\underline{t}\\rangle $ (two state vectors) or by $\\mid \\psi ,\\underline{x},\\underline{t}\\rangle $ (two evolution parameters).",
"The rule is simple: The state vector $\\psi $ does not operate to the evolution parameter $\\underline{e}$ .",
"So $\\langle \\underline{e}\\mid \\psi \\rangle $ does not work.",
"Only two exceptions are $\\langle \\underline{t}\\mid E_n\\rangle =e^{-iE_nt}$ and $\\langle \\underline{x} \\mid p_n\\rangle =e^{ip_nx}$ where $E_n$ , $p_n$ are the components of identity operator $I=\\sum _n\\mid p_n\\rangle \\langle p_n\\mid =\\sum _n\\mid E_n\\rangle \\langle E_n\\mid $ .",
"If $E$ is not a component of identity operator and $\\langle \\underline{t}\\mid E \\rangle =e^{-iEt}$ contributes only overall phase, then it is physically meaningless.",
"And two state vectors with different evolution parameters do not directly operate each other.",
"For example, two state vectors $\\varphi $ and $\\psi $ in $\\langle \\varphi ,\\underline{t_2}\\mid \\psi ,\\underline{t_1}\\rangle $ do not directly operate each other.",
"In this case, if needed, we may sandwich $I=\\sum _n\\mid E_n\\rangle \\langle E_n\\mid $ between $\\underline{t_2}$ and $\\underline{t_1}$ or we may sandwich $I=\\sum _n\\mid p_n\\rangle \\langle p_n\\mid $ between $\\underline{x_2}$ and $\\underline{x_1}$ .",
"Note in order to use new prescription, it is not good idea to sandwich (REF ) between $$ and $$ / to sandwich (REF ) between $$ and $$ , because it can destroy the information about the Lagrangian of the system."
],
[
"TEP",
"We can express the quantum state $\\psi $ at time $t_1$ by $\\mid \\psi _1,\\underline{t_1}\\rangle = \\mid E_n\\rangle \\langle E_n\\mid \\psi _1,\\underline{t_1}\\rangle $ where the summation is assumed for repeated index $n$ .",
"Thus the probability amplitude to find the state $\\psi _2$ at $t_2$ is $\\langle \\psi _2,\\underline{t_2}\\mid \\psi _1,\\underline{t_1}\\rangle =\\langle \\psi _2,\\underline{t_2}\\mid E_n\\rangle \\langle E_n\\mid \\psi _1,\\underline{t_1}\\rangle $ For example, $\\psi _2=x_2$ , $\\langle x_2,\\underline{t_2}\\mid \\psi _1,\\underline{t_1}\\rangle &=&\\langle x_2,\\underline{t_2}\\mid E_n\\rangle \\langle E_n\\mid \\psi _1,\\underline{t_1}\\rangle \\\\\\langle x_2\\mid \\psi _2,\\underline{t_2}\\rangle &=& \\langle x_2\\mid E_n\\rangle \\langle \\underline{t_2}\\mid E_n\\rangle \\langle E_n\\mid \\psi _1\\rangle \\langle E_n\\mid \\underline{t_1}\\rangle \\\\\\rightarrow \\psi _2(x_2,\\underline{t_2}) &=& \\int \\langle x_2\\mid E\\rangle \\psi _1(E)e^{-iE(t_2-t_1)}{dE}$ where the subscripts 1 and 2 in $\\psi $ mean the states $\\psi $ is of $t_1$ and $t_2$ , we set $\\hbar =1$ ."
],
[
"PEP",
"We can express the quantum state $\\varphi $ at position $x_1$ by $\\mid \\varphi _1,\\underline{x_1}\\rangle = \\mid p_n\\rangle \\langle p_n\\mid \\varphi _1,\\underline{x_1}\\rangle $ The probability amplitude to find the state $\\varphi _2$ at $x_2$ is $\\langle \\varphi _2,\\underline{x_2}\\mid \\varphi _1,\\underline{x_1}\\rangle =\\langle \\varphi _2,\\underline{x_2}\\mid p_n\\rangle \\langle p_n\\mid \\varphi _1,\\underline{x_1}\\rangle $ For example, $\\varphi _2=t_2$ , $\\langle t_2,\\underline{x_2}\\mid \\varphi _1,\\underline{x_1}\\rangle &=&\\langle t_2,\\underline{x_2}\\mid p_n\\rangle \\langle p_n\\mid \\varphi _1,\\underline{x_1}\\rangle \\nonumber \\\\\\langle t_2\\mid \\varphi _2,\\underline{x_2}\\rangle &=& \\langle t_2\\mid p_n\\rangle \\langle \\underline{x_2}\\mid p_n\\rangle \\langle p_n\\mid \\varphi _1\\rangle \\langle p_n\\mid \\underline{x_1}\\rangle \\nonumber \\\\\\rightarrow \\varphi _2(t_2,\\underline{x_2}) &=& \\int \\langle t_2\\mid p\\rangle \\varphi _1(p)e^{ip(x_2-x_1)}dp$"
],
[
"TEP",
"In (REF ) and (REF ), let's use $\\mid x_i,\\underline{t_1}\\rangle \\langle x_i,\\underline{t_1}\\mid $ instead of $ \\mid E_n\\rangle \\langle E_n\\mid $ .",
"$\\mid \\psi _1,\\underline{t_1}\\rangle &=& \\mid x_i,\\underline{t_1}\\rangle \\langle x_i,\\underline{t_1}\\mid \\psi _1,\\underline{t_1}\\rangle =\\int \\mid x,\\underline{t_1}\\rangle \\langle x\\mid \\psi _1\\rangle dx \\\\&=& \\int \\mid x,\\underline{t_1}\\rangle \\langle x\\mid p_n\\rangle \\langle p_n\\mid \\psi _1\\rangle dx=\\int \\mid x,\\underline{t_1}\\rangle \\frac{e^{ip_nx}}{\\sqrt{2\\pi }}\\langle p_n\\mid \\psi _1\\rangle dx$ For the momentum eigenstate $\\psi _1=p_1$ , $\\mid p_1,\\underline{t_1}\\rangle =\\int \\mid x,\\underline{t_1}\\rangle \\frac{e^{ip_1x}}{\\sqrt{2\\pi }}dx$ From (REF ), we can see that a particle in momentum eigenstate evolving from time $\\underline{t_1}$ starts its motion at all equally different positions.",
"() is illustrated in figure REF (a).",
"We can check that if we are more certain about the momentum of a particle, we are less certain about the position of departure at time $\\underline{t_1}$ .",
"This is the fundamental meaning of position-momentum uncertainty relation."
],
[
"PEP",
"In (REF ) and (REF ), let's use $\\mid t_i,\\underline{x_1}\\rangle \\langle t_i,\\underline{x_1}\\mid $ instead of $\\mid p_n\\rangle \\langle p_n\\mid $ .",
"$\\mid \\varphi _1,\\underline{x_1}\\rangle &=& \\mid t_i,\\underline{x_1}\\rangle \\langle t_i,\\underline{x_1}\\mid \\varphi _1,\\underline{x_1}\\rangle =\\int \\mid t,\\underline{x_1}\\rangle \\langle t\\mid \\varphi _1\\rangle dt \\\\&=&\\int \\mid t,\\underline{x_1}\\rangle \\langle t\\mid E_n\\rangle \\langle E_n\\mid \\varphi _1\\rangle dt=\\int \\mid t,\\underline{x_1}\\rangle \\frac{e^{-iE_nt}}{\\sqrt{2\\pi }}\\langle E_n\\mid \\varphi _1\\rangle dt$ For the energy eigenstate $\\varphi _1=E_1$ , $\\mid E_1,\\underline{x_1}\\rangle =\\int \\mid t,\\underline{x_1}\\rangle \\frac{-e^{iE_1t}}{\\sqrt{2\\pi }}dt$ From (REF ), we can see that the particle in energy eigenstate evolving from the position $\\underline{x_1}$ starts its motion at all equally different times.",
"() is illustrated in figure REF (b).",
"We can check that if we are more certain about the energy of a particle, we are less certain about the time of departure at position $\\underline{x_1}$ .",
"This is the fundamental meaning of time-energy uncertainty relation."
],
[
"TEP",
"Let's find the expression for the probability amplitude of quantum state $\\psi _1$ at $\\underline{t_1}$ to be measured $\\psi _2$ at $\\underline{t_2}$ .",
"$\\langle \\psi _2, \\underline{t_2}\\mid \\psi _1,\\underline{t_1}\\rangle &=& \\int \\underbrace{\\langle \\psi _2,\\underline{t_2}\\mid x_2,\\underline{t_2}\\rangle }_{\\langle \\psi _2\\mid x_2\\rangle }\\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\underbrace{\\langle x_1,\\underline{t_1}\\mid \\psi _1, \\underline{t_1}\\rangle }_{\\langle x_1\\mid \\psi _1\\rangle }dx_1 dx_2\\nonumber \\\\&=&\\iint \\langle \\psi _2\\mid x_2\\rangle \\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\langle x_1\\mid \\psi _1\\rangle dx_1 dx_2 $ Equation (REF ) is illustrated in figure REF (a).",
"For example $\\langle x_2,\\underline{t_2}\\mid \\psi , \\underline{t_1}\\rangle &=&\\iint \\langle x_2\\mid x_2^{\\prime }\\rangle \\langle x_2^{\\prime }, \\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\langle x_1\\mid \\psi _1\\rangle dx_1 dx_2^{\\prime }\\\\\\langle x_2\\mid \\psi , \\underline{t_2}\\rangle &=& \\int \\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\langle x_1\\mid \\psi _1\\rangle dx_1\\\\\\rightarrow \\psi (x_2,\\underline{t_2}) &=& \\int \\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\psi (x_1,\\underline{t_1})dx_1$ Take an another example $\\langle p_2,\\underline{t_2} \\mid p_1,\\underline{t_1}\\rangle &= \\iint \\langle p_2\\mid x_2\\rangle \\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\langle x_1\\mid p_1\\rangle dx_1 dx_2\\nonumber \\\\&= \\iint \\frac{e^{-ip_2x_2}}{\\sqrt{2\\pi }}\\langle x_2,\\underline{t_2}\\mid x_1,\\underline{t_1}\\rangle \\frac{e^{ip_1x_1}}{\\sqrt{2\\pi }} dx_1 dx_2$"
],
[
"PEP",
"Let's find the expression for the probability amplitude of quantum state $\\varphi _1$ at $\\underline{x_1}$ to be measured $\\varphi _2$ at $\\underline{x_2}$ .",
"$\\langle \\varphi _2, \\underline{x_2}\\mid \\varphi _1,\\underline{x_1}\\rangle &=& \\int \\underbrace{\\langle \\varphi _2,\\underline{x_2}\\mid t_2,\\underline{x_2}\\rangle }_{\\langle \\varphi _2\\mid t_2\\rangle }\\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\underbrace{\\langle t_1,\\underline{x_1}\\mid \\varphi _1, \\underline{x_1}\\rangle }_{\\langle t_1\\mid \\varphi _1\\rangle }dt_1 dt_2\\nonumber \\\\&=&\\iint \\langle \\varphi _2\\mid t_2\\rangle \\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\langle t_1\\mid \\varphi _1\\rangle dt_1 dt_2 $ Equation (REF ) is illustrated in figure REF (b).",
"For example $\\langle t_2,\\underline{x_2}\\mid \\varphi , \\underline{x_1}\\rangle &=&\\iint \\langle t_2\\mid t_2^{\\prime }\\rangle \\langle t_2^{\\prime }, \\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\langle t_1\\mid \\varphi _1\\rangle dt_1 dt_2^{\\prime }\\\\\\langle t_2\\mid \\varphi , \\underline{x_2}\\rangle &=& \\int \\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\langle t_1\\mid \\varphi _1\\rangle dt_1\\\\\\rightarrow \\varphi (t_2,\\underline{x_2}) &=& \\int \\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\varphi (t_1,\\underline{x_1})dt_1$ Take an another example $\\langle E_2,\\underline{x_2}\\mid E_1,\\underline{x_1}\\rangle &= \\iint \\langle E_2\\mid t_2\\rangle \\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\langle t_1\\mid E_1\\rangle dt_1 dt_2\\nonumber \\\\&= \\iint \\frac{e^{iE_2t_2}}{\\sqrt{2\\pi }}\\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\frac{e^{-iE_1t_1}}{\\sqrt{2\\pi }} dt_1 dt_2 $"
],
[
"Orthogonality of position and time",
"In this section it will be shown how to achieve $\\langle \\: x\\mid x^{\\prime }\\:\\rangle =\\delta (x-x^{\\prime })$ in TEP and $\\langle \\: t\\mid t^{\\prime }\\:\\rangle =\\delta (t-t^{\\prime })$ in PEP.",
"In doing so, we will find the following expression for $\\langle \\: E\\mid p\\:\\rangle $ .",
"$\\langle \\: E\\mid p\\:\\rangle =\\sqrt{\\frac{E}{p_E}}\\delta (\\pm p_E-p) \\qquad \\textrm {in TEP}$ The range of energy is either $[m,\\infty ]$ or $[-m,-\\infty ]$ .",
"The range of momentum is $[-\\infty ,\\infty ]$ .",
"$\\langle \\: E\\mid p\\:\\rangle =\\sqrt{\\frac{p}{E_p}}\\delta (\\pm E_p-E) \\qquad \\textrm {in PEP}$ The range of momentum is either $[0,\\infty ]$ or $[0,-\\infty ]$ .",
"The range of energy is $[-\\infty ,\\infty ]$ except $[-m,m]$ .",
"Where $p_E\\equiv \\sqrt{E^2-m^2}$ and $E_p\\equiv \\sqrt{p^2+m^2}$ for a free particle.",
"Let's check this out."
],
[
"TEP",
"In order to find out the expression of $\\langle \\: E\\mid p\\:\\rangle $ , first verify the orthogonality of momentum, $\\langle \\: p^{\\prime }\\mid p\\:\\rangle =\\int \\langle \\: p^{\\prime }\\mid E\\:\\rangle \\langle \\: E\\mid p\\:\\rangle dE=\\delta (p^{\\prime }-p)$ .",
"Since $EdE=p_Edp_E$ , try $\\langle \\: E\\mid p\\:\\rangle =\\sqrt{E/p_E}\\delta (\\pm p_E-p)$ , where $\\pm $ indicates that when we integrate over $p_E$ , we have to do it for both $+p_E$ and $-p_E$ .",
"$\\langle \\: p^{\\prime }\\mid p\\:\\rangle =\\int \\sqrt{\\frac{E}{p_E}}\\delta (\\pm p_E-p^{\\prime })\\sqrt{\\frac{E}{p_E}}\\delta (\\pm p_E-p)dE$ This is an odd function of $E$ , thus if we integrate over $[-\\infty ,\\infty ]$ , it turns out to be zero.",
"We can fix it by restricting $E$ to either $[m,\\infty ]$ or $[-m,-\\infty ]$ .",
"Then, $\\langle \\: p^{\\prime }\\mid p\\:\\rangle =\\int ^\\infty _m \\frac{E}{p_E}\\delta (\\pm p_E-p^{\\prime })\\delta (\\pm p_E-p)dE$ The sign of $E$ does not specify the sign of $p_E$ .",
"Thus we have to count both positive and negative momentum cases.",
"$\\langle \\: p^{\\prime }\\mid p\\:\\rangle &=\\int ^\\infty _0\\delta (p_E-p^{\\prime })\\delta (p_E-p)dp_E+\\int ^\\infty _0\\delta (p_E+p^{\\prime })\\delta (p_E+p)dp_E\\\\&=\\int ^\\infty _0\\delta (p_E-p^{\\prime })\\delta (p_E-p)dp_E+\\int ^{-\\infty }_0\\delta (-u+p^{\\prime })\\delta (-u+p)d(-u)\\\\&=\\int ^\\infty _{-\\infty }\\delta (p_E-p^{\\prime })\\delta (p_E-p)dp_E=\\delta (p^{\\prime }-p)$ (REF ) with $E$ either $[m,\\infty ]$ or $[-m,-\\infty ]$ ensures the orthogonality of position, $\\langle \\: x\\mid x^{\\prime }\\:\\rangle =\\delta (x-x^{\\prime })$ : $\\langle \\: x\\mid E\\:\\rangle =\\int \\langle \\: x\\mid p\\:\\rangle \\langle \\: p\\mid E\\:\\rangle dp=\\frac{1}{\\sqrt{2\\pi }}\\int e^{ipx}\\sqrt{\\frac{E}{p_E}}\\delta (\\pm p_E-p)dp=\\frac{1}{\\sqrt{2\\pi }}\\sqrt{\\frac{E}{p_E}}e^{\\pm i p_E x}$ $\\langle \\: x\\mid x^{\\prime }\\:\\rangle &=\\int ^\\infty _m\\langle \\: x\\mid E\\:\\rangle \\langle \\: E\\mid x^{\\prime }\\:\\rangle dE=\\frac{1}{2\\pi }\\int ^\\infty _m\\frac{E}{p_E}e^{\\pm ip_E(x-x^{\\prime })}dE\\\\&=\\frac{1}{2\\pi }\\int ^\\infty _{-\\infty } e^{ip_E(x-x^{\\prime })}dp_E=\\delta (x-x^{\\prime })$ If we did not restrict $E$ to either positive or negative values, we couldn't have $\\langle \\: x\\mid x^{\\prime }\\:\\rangle =\\delta (x-x^{\\prime })$ .",
"This is expected because the negative energy particle comes backward in time to be detected at another position at the same time it has already been detected."
],
[
"PEP",
"In order to find out the expression of $\\langle \\: E\\mid p\\:\\rangle $ , first verify the orthogonality of energy, $\\langle \\: E^{\\prime }\\mid E\\:\\rangle =\\int \\langle \\: E^{\\prime }\\mid p\\:\\rangle \\langle \\: p\\mid E\\:\\rangle dp=\\delta (E^{\\prime }-E)$ .",
"Since $EdE=p_Edp_E$ , try $\\langle \\: E\\mid p\\:\\rangle =\\sqrt{p/E_p}\\delta (\\pm E_p-E)$ .",
"$\\langle \\: E^{\\prime }\\mid E\\:\\rangle =\\int \\sqrt{\\frac{p}{E_p}}\\delta (\\pm E_p-E^{\\prime })\\sqrt{\\frac{p}{E_p}}\\delta (\\pm E_p-E)dp$ This is an odd function of $p$ , thus if we integrate over $[-\\infty ,\\infty ]$ , it turns out to be zero.",
"We can fix it by restricting $p$ to either $[0,\\infty ]$ or $[0,-\\infty ]$ .",
"We may fix this problem by making the integrand to an even function of $p$ (i.e., $p\\rightarrow |p|$ ).",
"But as explained at the appendix , this cause another problem.",
"Then, $\\langle \\: E^{\\prime }\\mid E\\:\\rangle =\\int ^\\infty _0 \\frac{p}{E_p}\\delta (\\pm E_p-E^{\\prime })\\delta (\\pm E_p-E)dp$ The sign of $p$ does not specify the sign of $E_p$ .",
"Thus we have to count both positive and negative energy cases.",
"$\\langle \\: E^{\\prime }\\mid E\\:\\rangle &=\\int ^\\infty _0\\delta (E_p-E^{\\prime })\\delta (E_p-E)dE_p+\\int ^\\infty _0\\delta (E_p+E^{\\prime })\\delta (E_p+E)dE_p\\\\&=\\int ^\\infty _0\\delta (E_p-E^{\\prime })\\delta (E_p-E)dE_p+\\int ^{-\\infty }_0\\delta (-u+E^{\\prime })\\delta (-u+E)d(-u)\\\\&=\\int ^\\infty _{-\\infty }\\delta (E_p-E^{\\prime })\\delta (E_p-E)dE_p=\\delta (E^{\\prime }-E)$ (REF ) with $p$ either $[0,\\infty ]$ or $[0,-\\infty ]$ ensures the orthogonality of time, $\\langle \\: t\\mid t^{\\prime }\\:\\rangle =\\delta (t-t^{\\prime })$ : $\\langle \\: p\\mid t\\:\\rangle =\\int \\langle \\: p\\mid E\\:\\rangle \\langle \\: E\\mid t\\:\\rangle dE=\\frac{1}{\\sqrt{2\\pi }}\\int \\sqrt{\\frac{p}{E_p}}\\delta (\\pm E_p-E) e^{iEt} dE=\\frac{1}{\\sqrt{2\\pi }}\\sqrt{\\frac{p}{E_p}}e^{\\pm i E_p t}$ $\\langle \\: t\\mid t^{\\prime }\\:\\rangle &=\\int ^\\infty _0\\langle \\: t\\mid p\\:\\rangle \\langle \\: p\\mid t^{\\prime }\\:\\rangle dp=\\frac{1}{2\\pi }\\int ^\\infty _0\\frac{p}{E_p}e^{\\pm iE_p(t^{\\prime }-t)}dp\\\\&=\\frac{1}{2\\pi }\\int ^\\infty _{-\\infty } e^{iE_p(t^{\\prime }-t)}dE_p=\\delta (t-t^{\\prime })$ If we did not restrict $p$ to either positive or negative values, we couldn't have $\\langle \\: t\\mid t^{\\prime }\\:\\rangle =\\delta (t-t^{\\prime })$ .",
"This is expected because the negative momentum particle comes backward in space to be detected at another time at the same position it has already been detected."
],
[
"Application",
"We have seen how to treat the position and time equally in quantum mechanics especially in evolution process.",
"Let's consider some application of our prescrition."
],
[
"Time-of-arrival",
"We may apply new prescription to the time-of-arrival introduced earlier.",
"By putting (REF ) into (REF ), we can derive the expression of time-of-arrival for a free particle.",
"Then (REF ) turns out $\\varphi _2(t)=\\frac{1}{\\sqrt{2\\pi }}\\int \\sqrt{\\frac{p}{E_p}}e^{\\mp iE_p t+ip(x_2-x_1)}\\varphi _1(p)dp$ where the subscript 1 and 2 in $\\varphi $ stands for the position $x_1$ and $x_2$ .",
"The range of $p$ goes either $[0,\\infty ]$ or $[0,-\\infty ]$ ; $\\mp E_p$ correspond positive and negative energy particle respectively.",
"The negative energy particle evolves in opposite direction to the positive energy particle in time.",
"(REF ) is well consistent with the final result Grot et al[1] derived for a free particle.",
"We have drived (REF ) from $\\langle \\: t_2,\\underline{x_2}\\mid p_n\\:\\rangle \\langle \\: p_n\\mid \\varphi _1,\\underline{x_1}\\:\\rangle $ .",
"We could also derive (REF ) from $\\langle \\: t_2,\\underline{x_2}\\mid t_i,\\underline{x_1}\\:\\rangle \\langle \\: t_i,\\underline{x_1}\\mid \\varphi _1,\\underline{x_1}\\:\\rangle $ .",
"$\\langle \\: t_2,\\underline{x_2}\\mid \\varphi _1,\\underline{x_1}\\:\\rangle &=\\int dt_1\\langle \\: t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\:\\rangle \\langle \\: t_1,\\underline{x_1}\\mid \\varphi _1,\\underline{x_1}\\:\\rangle \\\\\\langle \\: t_2\\mid \\varphi _2,\\underline{x_2}\\:\\rangle &= \\int dt_1\\langle \\: t_2,\\underline{x_2}\\mid p_m\\:\\rangle \\langle \\: p_m\\mid t_1,\\underline{x_1}\\:\\rangle \\langle \\: t_1,\\underline{x_1}\\mid p_n\\:\\rangle \\langle \\: p_n\\mid \\varphi _1,\\underline{x_1}\\:\\rangle \\\\\\varphi _2(t_2)&= \\int dt_1\\langle \\: t_2\\mid p_m\\:\\rangle \\langle \\: p_m\\mid t_1\\:\\rangle \\langle \\: \\underline{x_2}\\mid p_m\\:\\rangle \\langle \\: p_m\\mid \\underline{x_1}\\:\\rangle \\langle \\: t_1\\mid p_n\\:\\rangle \\langle \\: p_n\\mid \\varphi _1\\:\\rangle $ $\\varphi _2(t_2) &=\\iiint dt_1\\Big (\\frac{p_m}{2\\pi E_{p_m}}\\Big )\\sqrt{\\frac{p_n}{2\\pi E_{p_n}}}e^{\\pm i(E_{p_m}-E_{p_n})t_1}e^{\\mp iE_{p_m}t_2} e^{ip_m(x_2-x_1)}\\varphi _1(p_n)dp_m dp_n\\\\&= \\iint \\frac{p_m}{E_{p_m}}\\sqrt{\\frac{p_n}{2\\pi E_{p_n}}}\\delta (E_{p_m}-E_{p_n})e^{\\mp E_{p_m}t_2+ip_m(x_2-x_1)}\\varphi _1(p_n)\\frac{E_{p_m}dE_{p_m}}{p_m}dp_n\\\\&= \\int ^\\infty _0 \\sqrt{\\frac{p_n}{2\\pi E_{p_n}}} e^{\\mp E_{p_n}t_2+ip_n(x_2-x_1)}\\varphi _1(p_n)dp_n$ which reduce to (REF ).",
"Note that if we did not restrict the momentum to either $[0,\\infty ]$ or $[0,-\\infty ]$ , we could not have $e^{ip_n(x_2-x_1)}$ in () from ()."
],
[
"Quantum tunneling",
"Another application is the region of quantum tunneling.",
"(Or inside event horizon.)",
"Thus let's apply the prescription to derive quantum tunneling formula.",
"For $E_1=E_2$ (REF ) becomes $\\langle E_1, \\underline{x_2}\\mid E_1,\\underline{x_1}\\rangle &= \\iint \\langle t_2,\\underline{x_2}\\mid t_1,\\underline{x_1}\\rangle \\frac{1}{2\\pi }e^{iE_1(t_2-t_1)}dt_1 dt_2\\\\&= \\frac{1}{2\\pi }\\int ^\\infty _{-\\infty }\\int ^\\infty _{-\\infty }\\sum _{[x(t)]}\\exp \\Big (i\\int ^{t_2}_{t_1}(L+E_1)dt\\Big ) dt_1 dt_2$ where we have used Feynman kernal.",
"And we can make it simpler by $\\int ^{t_2}_{t_1}(L+E_1)dt=\\int ^{t_2}_{t_1} p\\dot{x}dt=\\int ^{x_2}_{x_1}pdx\\equiv W(x)\\Big |^{x_2}_{x_1}$ where $p\\equiv \\frac{\\partial L}{\\partial \\dot{x}}$ and $W$ stand for the generalized momentum and the Jacobi action respectively.",
"Then finally we have $\\langle E_1, \\underline{x_2}\\mid E_1,\\underline{x_1}\\rangle =\\frac{1}{2\\pi }\\int ^\\infty _{-\\infty }\\int ^\\infty _{-\\infty }\\sum _{[x(t)]} \\exp \\frac{iW(x)\\Big |^{x_2}_{x_1}}{\\hbar } dt_1dt_2$ For a classical object ($W\\gg \\hbar $ ) or for WKB approximation [5], $\\sum _{[x(t)]} \\exp \\Big (\\frac{iW(x)\\Big |^{x_2}_{x_1}}{\\hbar }\\Big )\\sim \\exp \\Big (\\frac{iW(x_\\ell )\\Big |^{x_2}_{x_1}}{\\hbar }\\Big )F(t_2,t_1)$ where $F(t_2,t_1)$ is some function of only $t_2$ and $t_1$ .",
"$x_\\ell $ stands for the least action path satisfying Euler-Lagrange equation $\\frac{\\partial }{\\partial t}\\Big (\\frac{\\partial L}{\\partial \\dot{x}}\\Big )-\\frac{\\partial L}{\\partial x}=0$ Thus $\\langle E_1, \\underline{x_2}\\mid E_1,\\underline{x_1}\\rangle \\simeq \\exp \\Big (\\frac{i}{\\hbar }W(x_\\ell )\\Big |^{x_2}_{x_1}\\Big )$ For tunneling particle, $p^2<0$ , $W=i(\\textrm {Im}W)$ , $\\langle E_1, \\underline{x_2}\\mid E_1,\\underline{x_1}\\rangle \\sim e^{-\\frac{1}{\\hbar }\\textrm {Im}W(x_\\ell )\\Big |^{x_2}_{x_1}}$ The tunneling probability is $P(\\underline{x_1}\\rightarrow \\underline{x_2},E)\\sim e^{-\\frac{2}{\\hbar }\\textrm {Im}W(x_\\ell )\\Big |^{x_2}_{x_1}}$ In (REF ), we have seen that a particle in energy eigenstate departs the initial position $\\underline{x_1}$ at all different times equally.",
"This applies also to the final position $\\underline{x_2}$ in (REF ).",
"Thus it is meaningless to talk about tunneling time of a particle in energy eigenstate; (REF ) reveals this property clearly.",
"We can consider (REF ) as tunneling time for a zero potential, $V=0$ .",
"For an energy eigenstate $\\varphi _1=p_1$ , $|\\varphi _2(t)|^2$ of (REF ) has no time dependence.",
"It makes sense, because the particle in energy eigenstate departs $x_1$ and arrives $x_2$ at all different times equally.",
"We may discuss time-of-arrival or tunneling time only for particles which are not in energy eigenstate."
],
[
"Conclusion",
"We have seen how to put an equal footing on position and time in quantum mechanics.",
"Unlike other approaches, I proposed that we cannot take both position and time as evolution parameters or both as observables.",
"We have to take one as an observable and the other as an evolution parameter; With set of simple prescriptions, we could formulate quantum mechanics in space time symmetric manner.",
"Combining with Feynman path integral, we could understand the fundamental meaning of time-energy uncertainty principle.",
"We could derive the time-of-arrival for a free particle.",
"We could also develop quantum tunneling formula expressed in Jacobi action for classical or WKB limit.",
"This approach may contribute to the development of quantum gravity.",
"One drawback of this prescription is that, for example figure REF (b) suggest that the particle can travel faster than the speed of light or even backward in time.",
"We can fix this problem by just assuming it cannot do it and modifying the integral range in formula for final position.",
"But this is not an elegant way to bypass the problem and it ruins the spirit of space time symmetry we are trying to achieve.",
"Does that mean the prescription for the position as an evolution parameter applies only to stationary case where there is no measurable distinction between past and future?",
"Further study is needed to answer it."
],
[
"Acknowledgments",
"I would like to thank Werner Israel for his support and useful comments on this work."
],
[
"Inconsistency of making $\\langle \\: E\\mid p\\:\\rangle $ even function",
"We saw that (REF ) is an odd function of $p$ .",
"In order to achieve $\\langle \\: E^{\\prime }\\mid E\\:\\rangle =\\delta (E^{\\prime }-E)$ we had to restrict $p$ to either $[0,\\infty ]$ or $[0,-\\infty ]$ .",
"However, we may achieve it by choosing $\\langle \\: E\\mid p\\:\\rangle =\\sqrt{\\frac{|p|}{2E_p}}\\delta (\\pm E_p-E)$ $\\langle \\: p\\mid t\\:\\rangle =\\int \\langle \\: p\\mid E\\:\\rangle \\langle \\: E\\mid t\\:\\rangle dE=\\int ^\\infty _{-\\infty }\\sqrt{\\frac{|p|}{2E_p}}\\delta (\\pm E_p-E)\\frac{e^{iEt}}{\\sqrt{2\\pi }}dE=\\frac{1}{\\sqrt{2\\pi }}\\sqrt{\\frac{|p|}{2E_p}}e^{\\pm iE_pt}$ Then, we can show that (REF ) satisfies $\\langle \\: E^{\\prime }\\mid E\\:\\rangle =\\delta (E^{\\prime }-E)$ and $\\langle \\: t^{\\prime }\\mid t\\:\\rangle =\\delta (t^{\\prime }-t)$ without constraining allowed range of momentum for the same evolution parameter $\\underline{x_1}$ .",
"However, we can also show that (REF ) does not work consistently for $\\langle \\: t_2,\\underline{x_2}\\mid \\varphi _1,\\underline{x_1}\\:\\rangle $ .",
"Using (REF ) to (REF ), we have $\\varphi _2(t)=\\frac{1}{\\sqrt{2\\pi }}\\int ^\\infty _{-\\infty }\\sqrt{\\frac{|p|}{2E_p}}e^{\\mp iE_p t+ip(x_2-x_1)}\\varphi _1(p)dp$ and () changes to $\\varphi _2(t_2) &= \\int ^\\infty _{-\\infty }\\int ^\\infty _0 \\frac{1}{2}\\sqrt{\\frac{|p_n|}{2\\pi 2E_{p_n}}}\\delta (E_{p_m}-E_{p_n})e^{\\mp iE_{p_m}t_2+ip_m(x_2-x_1)}\\varphi _1(p_n)dE_{p_m} dp_n\\\\&+\\int ^\\infty _{-\\infty }\\int ^0_\\infty -\\frac{1}{2}\\sqrt{\\frac{|p_n|}{2\\pi 2E_{p_n}}}\\delta (E_{p_m}-E_{p_n})e^{\\mp iE_{p_m}t_2+ip_m(x_2-x_1)}\\varphi _1(p_n)dE_{p_m} dp_n\\\\&= \\int ^\\infty _{-\\infty }\\int ^\\infty _0 \\sqrt{\\frac{|p_n|}{2\\pi 2E_{p_n}}}\\delta (E_{p_m}-E_{p_n})e^{\\mp iE_{p_m}t_2+ip_m(x_2-x_1)}\\varphi _1(p_n)dE_{p_m} dp_n$ (REF ) is not consistent with (REF ) unless $\\underline{x_2}=\\underline{x_1}$ or $p_m$ and $p_n$ have the same sign.",
"We can also check this inconsistency in classical limit."
]
] | 1403.0459 |
[
[
"Acceleration statistics in thermally driven superfluid turbulence"
],
[
"Abstract New methods of flow visualization near absolute zero have opened the way to directly compare quantum turbulence (in superfluid helium) to classical turbulence (in ordinary fluids such as air or water) and explore analogies and differences.",
"We present results of numerical simulations in which we examine the statistics of the superfluid acceleration in thermal counterflow.",
"We find that, unlike the velocity, the acceleration obeys scaling laws similar to classical turbulence, in agreement with a recent quantum turbulence experiment of La Mantia et al."
],
[
"Acceleration statistics in thermally driven superfluid turbulence Andrew W. Baggaley School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QW, UK Carlo F. Barenghi Joint Quantum Centre Durham-Newcastle, School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK New methods of flow visualization near absolute zero have opened the way to directly compare quantum turbulence (in superfluid helium) to classical turbulence (in ordinary fluids such as air or water) and explore analogies and differences.",
"We present results of numerical simulations in which we examine the statistics of the superfluid acceleration in thermal counterflow.",
"We find that, unlike the velocity, the acceleration obeys scaling laws similar to classical turbulence, in agreement with a recent quantum turbulence experiment of La Mantia et al.",
"67.25.dk (vortices in superfluid helium 4), 47.27.-i (turbulent flows), 47.32.C- (vortex interactions), 47.27.Gs (isotropic and homogeneous turbulence) Turbulence, near omni-present in natural flows, presents an open and difficult problem.",
"It is typically studied, experimentally and theoretically, in a number of fluid media, all of which exhibit continuous velocity fields, e.g.",
"water, air, electrically conducting plasma.",
"However turbulence can also be investigated in a different setting: low-temperature quantum fluids, which exhibit discrete vorticity fields.",
"This quantum turbulence was first studied by Vinen in superfluid helium-4 [1], [2], [3], [4]; later studies have extended it to superfluid helium-3 [5] and atomic Bose-Einstein condensates [6].",
"The motion of quantum fluids is strongly constrained by quantum mechanics; notably the vorticity is concentrated in discrete vortex filaments of fixed circulation $\\kappa $ whose cores have atomic thickness $a_0$ .",
"As first envisaged by Feynman, quantum turbulence consists of a tangle of interacting, reconnecting vortex lines.",
"In helium-4, quantum turbulence can readily be generated in the laboratory, either driving the fluid mechanically, or thermally through an applied heat flux; in this article we shall focus on the latter method, which can be easily described using Landau's two–fluid theory [7].",
"A prototypical experiment consists of a channel which is closed at one end and open to the helium bath at the other end.",
"At the closed end, a resistor inputs a steady flux of heat, $\\dot{Q}$ , into the channel.",
"The heat is carried away from the resistor towards the bath by the normal fluid component, whereas the superfluid component flows towards the resistor to maintain the total mass flux equal to zero.",
"If the relative velocity of superfluid and normal fluid is larger than a small critical value, the laminar counterflow of the two fluids breaks down and a tangle of vortex lines appears, thus limiting the heat conducting properties of helium-4.",
"Recent experiments have made dramatic progress in the ability to visualize the turbulent flow of liquid helium using tracer particles.",
"For example, Bewley et al.",
"[8] detected reconnections of individual vortex lines.",
"Paoletti et al.",
"[9] discovered that in quantum turbulence the velocity statistics are non-Gaussian, in contrast to experimental and numerical studies of classical turbulence which display Gaussian statistics.",
"Follow–up studies argued that this non–classical effect arises from the singular nature of the superfluid vorticity [10], [11].",
"Another important one–point observable is the distribution of turbulent accelerations.",
"In classical turbulence, Mordant et al.",
"[12] found that the acceleration obeys log–normal distributions; they also observed a strong dependence of acceleration on velocity which disagrees with the assumption of local homogeneity [13].",
"In quantum turbulence, accelerations were measured only recently by La Mantia et al. [14].",
"They used tracer particles to extract Lagrangian velocity and acceleration statistics from thermally driven quantum turbulence at a range of temperatures and counterflow velocities.",
"Their results were striking: whilst observing the (now familiar) power–law nature of the one-point velocity statistics, their probability density functions (PDFs) of the acceleration statistics were surprisingly similar to classical results.",
"The physics of the interactions between tracers, superfluid and normal fluid components is complex [15], and what was observed by La Mantia is only the motion of tracers, not of the superfluid itself.",
"To make further progress in this problem here we present superfluid acceleration statistics obtained by direct numerical simulations of thermally driven superfluid turbulence.",
"We model vortex lines [16] as oriented space curves ${\\bf {s}}(\\xi ,t)$ of infinitesimal thickness, where $\\xi $ is arc length and $t$ is time.",
"This vortex filament approach is justified by the large separation of scales between $a_0$ and the typical distance between vortices, $\\ell $ .",
"The governing equation of motion is Schwarz's equation $\\frac{d{\\bf s}}{dt}={\\bf {v}}_s+\\alpha {\\bf {s}}^{\\prime } \\times ({\\bf {v}}_n-{\\bf {v}}_s)-\\alpha ^{\\prime } {\\bf {s}}^{\\prime } \\times \\left({\\bf {s}}^{\\prime } \\times \\left({\\bf {v}}_n-{\\bf {v}}_s\\right)\\right),$ where $t$ is time, $\\alpha $ and $\\alpha ^{\\prime }$ are temperature–dependent friction coefficients [17], ${\\bf {s}}^{\\prime }=d{\\bf {s}}/d\\xi $ is the unit tangent vector at the point ${\\bf {s}}$ , $\\xi $ is arc length, and ${\\bf {v}}_n$ is the normal fluid velocity at the point $\\bf s$ .",
"We work at temperatures comparable to La Mantia's experiment [14]; the relevant friction coefficients are [17] $\\alpha =0.111$ , $\\alpha ^{\\prime }=0.0144$ at $T=1.65~\\rm K$ , $\\alpha =0.142$ , $\\alpha ^{\\prime }=0.0100$ at $T=1.75~\\rm K$ , and $\\alpha =0.181$ , $\\alpha ^{\\prime }=0.0074$ at $T=1.65~\\rm K$ .",
"The superfluid velocity ${\\bf {v}}_s={\\bf {v}}_s^{ext}+{\\bf {v}}_s^{si}$ contains two parts: the superflow induced by the heater, ${\\bf {v}}_s^{ext}$ , and the self-induced velocity of the vortex line at the point ${\\bf {s}}$ , given by the Biot-Savart law [18] ${\\bf {v}}_s^{si} ({\\bf {s}},\\,t)=-\\frac{\\kappa }{4 \\pi } \\oint _{\\cal L} \\frac{({\\bf {s}}-{\\bf {r}}) }{\\vert {\\bf {s}}- {\\bf {r}}\\vert ^3}\\times {\\bf d}{\\bf {r}},$ where $\\cal L$ is the total vortex configuration.",
"The techniques to discretize vortex lines into a variable number of points ${\\bf {s}}_i$ ($i=1,\\ldots , N$ ) held at minimum separation $\\delta /2$ , time-step Eq.",
"(REF ), de–singularize the Biot-Savart integrals Eq.",
"(REF ) and evaluate them via a tree-method (with critical opening angle $0.3$ ) are described in a previous paper [19].",
"Unlike the microscopic Gross-Pitaevskii model, in the vortex filament approach vortex reconnections must be modelled algorithmically.",
"The reconnection algorithm used here is described in [20] and compared to other algorithms in the literature.",
"All numerical simulations are performed in a periodic cube of size ${\\cal D}=0.1\\,$ cm.",
"We take $\\delta =1.6 \\times 10^{-3}\\,$ cm and use time-step of $\\Delta t=10^{-4}\\,$ s comparable to the simulations of Adachi et al. [21].",
"The normal fluid velocity, ${\\bf {v}}_n={\\bf {v}}_n^{ext}$ , driven by the heater, is a prescribed constant flow in the positive $x$ direction.",
"Our simulations are performed in the reference frame of the superflow.",
"We ignore potentially interesting physics arising from boundaries, and any influence of the quantized vortices on the normal fluid, but our model is sufficient for a first study of superfluid acceleration statistics.",
"We present the results of five numerical simulations of counterflow turbulence, three simulations with $v_{ns}=1\\,$ cm/s at temperatures T=1.65, 1.75, 1.85K, and two simulations for T=1.75K at $v_{ns}=0.8\\, , \\, 1.2 \\,$ cm/s.",
"This choice of parameters is motivated by the work of La Mantia [14], but we do not seek direct quantitative comparison with experiments, due to the approximations inherent in our numerical approach and in the measurements (which we discuss later), as well as computational restrictions on the vortex line density that can be simulated.",
"All simulations are initiated with a random configuration of vortex rings which seed the turbulence.",
"As with previous studies, after and initial transient, the vortex line density $L=\\Lambda /V$ (defined as the superfluid vortex length $\\Lambda =\\int _{\\cal L} d\\xi $ in the volume $V={\\cal D}^3$ ) saturates to a quasi-steady state (independent of the initial seed) such that energy input from the driving normal fluid is balanced by dissipation due to friction and vortex reconnections.",
"The intervortex distance is estimated as $\\ell \\approx L^{-1/2}$ .",
"A typical vortex tangle is displayed in Fig.",
"REF .",
"Within the saturated regime we compute velocity and acceleration statistics, using stored velocity information at the discretization points ${\\bf {s}}_i$ via a fourth-order upwind finite–difference scheme $\\mathbf {a}_i^n=\\dfrac{-{\\bf {v}}_i^{n-3}+6{\\bf {v}}_i^{n-2}-18{\\bf {v}}_i^{n-1}+10{\\bf {v}}_i^{n}+3{\\bf {v}}_i^{n+1}}{\\Delta t},$ where $\\mathbf {a}_i^n$ is the acceleration of the $i^{\\rm th}$ vortex point at the $n^{\\rm th}$ time step and ${\\bf {v}}_i^n=d{\\bf {s}}_i^n/dt$ is the velocity of the $i^{\\rm th}$ vortex point at the $n^{\\rm th}$ time step, computed using Eq.",
"(REF ).",
"What we measure thus represents the Lagrangian acceleration of ideal point tracers which are trapped in vortex lines (hence are affected by friction), but are unaffected by Stokes drag.",
"Figure: A snapshot of the vortex configuration(plotted as black space curves) at T=1.75 K, v ns =1 cm /sv_{ns}=1~\\rm cm/s,during the quasi-steady state regime.Vortex line density L=15750 cm -2 L=15750~\\rm cm^{-2},estimated intervortex distance ℓ≈0.008 cm \\ell \\approx 0.008~\\rm cm.First we consider the velocity.",
"PDFs of the velocity components $v_x$ , $v_y$ and $v_z$ of ${\\bf {v}}_i$ from the simulation at T=1.75 K, $v_{ns}=1\\,$ cm/s, which are plotted in Fig.",
"REF .",
"Note the power–law behavior of the tails.",
"Best–fits to the data give PDF$(v) \\propto v^{-3.2}$ ; comparable results are obtained at different $T$ and $v_{ns}$ .",
"The PDF's exponents, close to $-3$ , are the tell–tale signature of quantum turbulence, and can be understood if we consider an isolated straight vortex line (the effect of adding the contributions of many vortices is discussed in ref. [10]).",
"The argument is the following.",
"At the distance $r$ from its axis, the vortex line induces a velocity field $v \\propto 1/r$ .",
"The probability $P(v)dv$ of finding the value $v$ is thus proportional to the area $2 \\pi r dr$ of the annulus between $r$ and $r+dr$ ; therefore $P(v)dv \\sim r dr \\sim (1/v)(dv/v^2) \\sim v^{-3} dv$ , hence ${\\rm PDF}(v) \\sim v^{-3}$ , in agreement with experiments [9] and numerical studies [11].",
"It is also instructive to examine $|{\\bf {v}}_i|$ , the modulus of the velocity ${\\bf {v}}_i$ .",
"Numerical experiments [22] confirm the heuristic argument[23] that counterflow turbulence is featureless (compared with classical turbulence), and the vortex tangle is characterized by the single length scale $\\ell $ .",
"The prominent peak of PDF($|{\\bf {v}}|$ ) displayed in Fig.",
"REF corresponds to the velocity scale $\\kappa / \\ell \\approx 0.13~\\rm cm/s$ , lending further weight to the argument.",
"The mean of the distribution, $\\langle | \\mathbf {v} | \\rangle =0.23~\\rm cm/s$ , is close to the characteristic velocity of a vortex line rotating around another line, $v_{\\ell } = \\kappa /(\\ell /2) \\approx 0.25~\\rm cm/s$ .",
"We turn now the attention to the acceleration.",
"Fig.",
"REF displays statistics of the modulus of the $x$ and $y$ -components ($a_x$ and $a_y$ ) of the acceleration $\\mathbf {a}_i$ , normalized by the corresponding standard deviations ($\\sigma _x$ and $\\sigma _y$ ).",
"The statistics for $a_z$ are indistinguishable from those of $a_y$ .",
"This is not surprising, because $x$ is the longitudinal direction of the counterflow, and the two transversal directions, $y$ and $z$ , are equivalent.",
"It is interesting to notice that the acceleration statistics are not affected by the mild anisotropy of counterflow (for example, at $T=1.75~\\rm K$ and $v_{ns}=1~\\rm cm/s$ , the projected vortex lengths are such that $L_x/L = 0.37, L_y/L = L_z/L = 0.54$ .",
"The results displayed in Fig.",
"REF are computed at fixed temperature ($T=1.75~\\rm K$ ) and varying counterflow velocities $v_{ns}$ (left), and at fixed counterflow velocity ($v_{ns}=1~\\rm cm/s$ ) and varying temperatures (right).",
"In either cases we can fit both a log-normal distribution to the data, and a power law to the tails of the PDF for large accelerations.",
"If we apply the straight vortex line argument to the acceleration $a=v^2/r$ , we find that the probability of the value $a$ is $P(a) da \\sim r dr \\sim (1/a^{1/3}) (da/a^{4/3}) \\sim a^{-5/3} da$ , hence we expect ${\\rm PDF}(a) \\sim a^{-5/3}$ .",
"The exponents shown in Fig.",
"REF are in general more shallow than -5/3.",
"A possible explanation is that vortex reconnections increase the probability of large accelerations.",
"Lognormal distributions The PDF of the log-normal distribution is given by PDF($x$ )=$(1/x\\sqrt{2\\pi }\\sigma )\\exp \\lbrace -(\\ln x-\\mu )^2/2\\sigma ^2\\rbrace $ , where $\\mu $ is the mean of the distribution and $\\sigma ^2$ is the variance.",
"are heavy-tailed (i.e.",
"the tails of the distribution are not exponentially bounded), and show reasonable agreement with the data, as found by La Mantia [14].",
"However it is clear that we observe a power-law scaling for the acceleration statistics in this study, with good agreement to the predicted $-5/3$ scaling.",
"We now consider the mean value $\\langle | \\mathbf {a} | \\rangle $ of the acceleration.",
"The previous argument suggests that the characteristic acceleration of a vortex line rotating around another line is of the order of $a_{\\ell }=v_{\\ell }^2/(\\ell /2)=8 \\kappa ^2/\\ell ^3$ .",
"La Mantia's experiments support this estimate.",
"La Mantia reports that (insets of figure 1 of ref.",
"[14]) that $\\langle | \\mathbf {a} |\\rangle \\approx 3.2$ and $1.9~\\rm cm/s^2$ respectively at $T=1.64~K$ , $\\dot{Q}=586~\\rm W/m^2$ and at $T=1.86~K$ , $\\dot{Q}=595~\\rm W/m^2$ .",
"If we relate the heat flux to the counterflow velocity (via $v_{ns}={\\dot{Q}}/(\\rho _s S T)$ where $S$ and $\\rho _s$ are the specific entropy and the superfluid density), the counterflow velocity to the vortex line density (via $L=\\gamma ^2 v_{ns}^2$ , where $\\gamma $ was calculated by Adachi [21]), and the vortex line density to the characteristic vortex distance (via $\\ell \\approx L^{-1/2}$ ), we find $a_{\\ell } \\approx 3.3$ and $1.1~\\rm cm/s^2$ respectively, in order of magnitude agreement with La Mantia's measurements of $\\langle | \\mathbf {a}| \\rangle $ .",
"The estimate $a_{\\ell }$ also agrees with the numerical simulations.",
"For example, at $T=1.75~\\rm K$ , $v_{ns}=1~\\rm cm/s$ we find $a_{\\ell } \\approx 16~\\rm cm/s^2$ which compares well with mean, median and mode of the computed distribution, which are 72, 35 and $10~\\rm cm/s^2$ respectively.",
"Finally, Fig.",
"REF shows that both velocity and acceleration increase with temperature $T$ (at fixed counterflow velocity $v_{ns}$ ) and with $v_{ns}$ (at fixed $T$ ).",
"La Mantia reports that $\\langle | \\mathbf {a} | \\rangle $ increases with heat flux $\\dot{Q}$ (at fixed $T$ ), but decreases with $T$ (at fixed $\\dot{Q}$ ).",
"There is no disagreement between La Mantia's results and ours.",
"In fact, on one hand we can write $a_{\\ell }=8 \\kappa ^2/\\ell ^2= 8 \\kappa ^2 \\gamma ^2 v_{ns}^3$ : this relation and the fact that $\\gamma $ increases with increasing $T$ [21], explains why in the numerical simulations $\\langle |\\mathbf {a} | \\rangle $ increases with $v_{ns}$ (at fixed $T$ ) and increases with $T$ (at fixed $v_{ns}$ ).",
"On the other hand we can also write $a_{\\ell }=8 \\kappa ^2 (\\gamma {\\dot{Q}}/(\\rho S T))^3$ : this relation accounts for La Mantia's observations that $\\langle | \\mathbf {a}| \\rangle $ increases with $\\dot{Q}$ (at fixed $T$ ) but decreases with $T$ (at fixed $\\dot{Q}$ ) because the quantity $\\gamma /(\\rho _s S T)$ decreases with increasing $T$ [17], [21].",
"Our results shed light onto the complex dynamics of tracer particles.",
"Consider a particle of radius $a_p$ , velocity ${\\bf {v}}_p$ and density $\\rho _p$ which is not trapped into vortices and moves in helium II.",
"Assuming a steady uniform normal fluid, its acceleration is due to Stokes drag and inertial effects [25]: $\\frac{d {\\bf {v}}_p}{dt} =\\frac{9 \\mu _n ({\\bf {v}}_n-{\\bf {v}}_p)}{2 \\rho _0 a_p^2}+\\frac{3\\rho _s}{2 \\rho _0} \\frac{D {\\bf {v}}_s}{Dt},$ where $\\rho $ and $\\mu _n$ are helium's density and viscosity, and $\\rho _0=\\rho _p+\\rho /2$ .",
"The Stokes drag (which pulls the particle along the normal fluid) has magnitude of the order of $9 \\beta \\mu _n v_n/(2 \\rho _0 a_p^2)$ where $v_n=\\rho _s v_{ns}/\\rho $ , $\\beta v_n$ is the average slip velocity and $0<\\beta <1$ ; unfortunately we do not know $\\beta $ and we cannot predict the relative importance of the two contributions to $d{\\bf {v}}_p/dt$ .",
"Temporal variations of ${\\bf {v}}_s$ become important only after the particle has collided with a vortex and triggered Kelvin waves [15], hence, for a free particle, the inertial term (which pulls the particle towards the nearest vortex, effectively a radial pressure gradient) becomes $D {\\bf {v}}_s /DT =\\partial {\\bf {v}}_s \\partial t + ({\\bf {v}}_s \\cdot \\nabla ) {\\bf {v}}_s\\approx ({\\bf {v}}_s \\cdot \\nabla ) {\\bf {v}}_s$ ; its magnitude is of the order of $v_s^2/(\\ell /2) = a_{\\ell }$ (which we interpreted as the acceleration of a particle trapped into a vortex which rotates around another vortex).",
"In La Mantia's experiment $\\rho _0 \\approx 1.9 \\rho $ , so the prefactor in front of the inertial term is of order unity.",
"The order of magnitude agreement between the observed acceleration and our estimate $a_{\\ell }$ suggests that the Stokes term is less important than the inertia term.",
"We can now interpret $a_{\\ell }$ as either the acceleration of a particle trapped into a vortex which rotates around another vortex, of the fluctuating pressure grandient which attracts a free particle to a vortex line.",
"Figure: (Color online) Probability density functions (PDF) of turbulentvelocity components v i v_i (i=x,y,z)i=x,y,z) vs v i /σ i v_i/\\sigma _icomputed from the velocity of the vortex points d𝐬 i /dtd{\\bf {s}}_i/dtfrom the simulation corresponding to Fig.",
"(T=1.75KT=1.75~\\rm K, v ns =1 cm /sv_{ns}=1~\\rm cm/s).",
"(Blue) circles,(red) asterisks and (black) crosses refer respectively to i=xi=x,i=yi=y and i=zi=z components.Gaussian fits, gPDF (v i )=1 2πσ 2 exp (-(v i -μ) 2 /(2σ 2 )){\\rm gPDF}(v_i)=\\frac{1}{\\sqrt{2 \\pi \\sigma ^2}}{\\rm exp}(-(v_i-\\mu )^2/(2 \\sigma ^2)) for each component (i=xi=x:dot-dashed line; i=yi=y: solid line, i=zi=z: solid points)are plotted to emphasize the deviation from Gaussianity.Here σ x =0.1162\\sigma _x=0.1162, σ y =0.1148\\sigma _y=0.1148, and σ z =0.1156\\sigma _z=0.1156.Figure: The probability density function (PDF) of the modulus ofthe velocity |𝐯||\\mathbf {v}|; the dashed (red) lines represents the`characteristic' velocity, κ/ℓ\\kappa /\\ell .The PDF is computed from the data in Fig.",
"(T=1.75KT=1.75~\\rm K, v ns =1 cm /sv_{ns}=1~\\rm cm/s).Figure: PDFs of the xx and yy components of the accelerationof the quantized vortices, scaled by the standard deviation of therelevant component.",
"Left, at fixed temperature, with increasingcounterflow velocity, right, varying the temperature with a fixedcounterflow velocity.",
"Log normal and power law (PDF∼a i -5/3 \\sim a_i^{-5/3}) fits to the data are plotted.Power law fits to the data yieldPDF(|a i |/σ i )∼(a i /σ i ) β (|a_i|/\\sigma _i) \\sim (a_i/\\sigma _i)^\\beta ;β=-1.48\\beta =-1.48, T=1.75 K, v ns =0.8v_{ns}=0.8 \\, cm/s ;β=-1.64\\beta =-1.64, T=1.75 K, v ns =1v_{ns}=1 \\, cm/s ;β=-1.71\\beta =-1.71, T=1.75 K, v ns =1.2v_{ns}=1.2 \\, cm/s ;β=-1.55\\beta =-1.55, T=1.65 K, v ns =1v_{ns}=1 \\, cm/s ;β=-1.58\\beta =-1.58, T=1.75 K, v ns =1v_{ns}=1 \\, cm/s ;β=-1.63\\beta =-1.63, T=1.85 K, v ns =1v_{ns}=1 \\, cm/s.Figure: Dependence of median velocity, 〈|𝐯|〉\\langle {|\\mathbf {v}|}\\rangle ( cm /s\\rm cm/s),and acceleration, 〈|𝐚|〉\\langle {|\\mathbf {a}|}\\rangle ( cm /s 2 \\rm cm/s^2),on temperature TT (left, in K\\rm K)and counterflow velocity v ns v_{ns} (right, in cm /s\\rm cm/s).In each panel, the left axis and (black) circles correspondto acceleration, and the right axis and (red) squarescorrespond to velocity.In conclusion, we have numerically determined the one–point superfluid acceleration statistics in counterflow turbulence, and demonstrated how mean velocity and acceleration scale with counterflow velocity and temperature.",
"The importance of our results springs from the fact that La Mantia did not measure directly the superfluid acceleration or the vortex acceleration, but rather the acceleration of micron–sized solid hydrogen particles, whose dynamics is complex [25], [15].",
"The good agreement between our findings and La Mantia's in terms of acceleration statistics means that this difference is not crucial.",
"We also argue that the probability density function of one–point acceleration statistics should follow a power law distribution, with a $-5/3$ exponent.",
"Our numerical results support these arguments.",
"The results reported by La Mantia did not distinguish between particles which are trapped in vortices (hence move along the imposed superflow) and particles which are free (hence move along the normal fluid).",
"Separate analysis of acceleration statistics of these two groups of particles will be useful.",
"Theoretically, an approach which accounts reasonably well for velocity and acceleration statistics in classical turbulence is the multifractal formalism [26], which in principle could be adapted to model quantum turbulence.",
"C.F.B.",
"is grateful to the EPSRC for grant number EP/I019413/1."
]
] | 1403.0411 |
[
[
"Link-Reliability Based Two-Hop Routing for Wireless Sensor Networks"
],
[
"Abstract Wireless Sensor Networks (WSNs) emerge as underlying infrastructures for new classes of large scale net- worked embedded systems.",
"However, WSNs system designers must fulfill the Quality-of-Service (QoS) requirements imposed by the applications (and users).",
"Very harsh and dynamic physical environments and extremely limited energy/computing/memory/communication node resources are major obstacles for satisfying QoS metrics such as reliability, timeliness and system lifetime.",
"The limited communication range of WSN nodes, link asymmetry and the characteristics of the physical environment lead to a major source of QoS degradation in WSNs.",
"This paper proposes a Link Reliability based Two-Hop Routing protocol for wireless Sensor Networks (WSNs).",
"The protocol achieves to reduce packet deadline miss ratio while consid- ering link reliability, two-hop velocity and power efficiency and utilizes memory and computational effective methods for estimating the link metrics.",
"Numerical results provide insights that the protocol has a lower packet deadline miss ratio and longer sensor network lifetime.",
"The results show that the proposed protocol is a feasible solution to the QoS routing problem in wireless sensor networks that support real-time applications."
],
[
"INTRODUCTION",
"Wireless Sensor Networks (WSNs) form a framework to accumulate and analyze real time data in smart environment applications.",
"WSNs are composed of inexpensive low-powered micro sensing devices called $motes$ [1], having limited computational capability, memory size, radio transmission range and energy supply.",
"Sensors are spread in an environment without any predetermined infrastructure and cooperate to accomplish common monitoring tasks which usually involves sensing environmental data.",
"With WSNs, it is possible to assimilate a variety of physical and environmental information in near real time from inaccessible and hostile locations.",
"Table: Our Results and Comparison with Previous Results for QoS Routing in Wireless Sensor Networks.WSNs have a wide variety of applications in military, industry, environment monitoring and health care.",
"WSNs operate unattended in harsh environments, such as border protection and battlefield reconnaissance hence help to minimize the risk to human life.",
"used of WSNs are used extensively in the industry for factory automation, process control, real-time monitoring of machines, detection of radiation and leakages and remote monitoring of contaminated areas, aid in detecting possible system deterioration and to initiate precautionary maintenance routine before total system breakdown.",
"WSNs are being rapidly deployed in patient health monitoring in a hospital environment, where different health parameters are obtained and forwarded to health care servers accessible by medical staff and surgical implants of sensors can also help monitor a patient’s health.",
"Emerging WSNs have a set of stringent QoS requirements that include timeliness, high reliability, availability and integrity.",
"The competence of a WSN lies in its ability to provide these QoS requirements.",
"The timeliness and reliability level for data exchanged between sensors and control station is of paramount importance especially in real time scenarios.",
"The deadline miss ratio (DMR) [6], defined as the ratio of packets that cannot meet the deadlines should be minimized.",
"Sensor nodes typically use batteries for energy supply.",
"Hence, energy efficiency and load balancing form important objectives while designing protocols for WSNs.",
"Therefore, providing corresponding QoS in such scenarios pose to be a great challenge.",
"Our proposed protocol is motivated primarily by the deficiencies of the previous works (explained in the Section 2) and aims to provide better Quality of Service.",
"This paper explores the idea of incorporating QoS parameters in making routing decisions $i.e.$ ,: (i) reliability (ii) latency and (iii) energy efficiency.",
"Traffic should be delivered with reliability and within a deadline.",
"Furthermore, energy efficiency is intertwined with the protocol to achieve a longer network lifetime.",
"Hence, the protocol is named, Link Reliability based Two-Hop Routing (LRTHR).",
"The protocol proposes the following features.",
"Link reliability is considered while choosing the next router, this selects paths which have higher probability of successful delivery.",
"Routing decision is based on two-hop neighborhood information and dynamic velocity that can be modified according to the required deadline, this results in significant reduction in end-to-end DMR (deadline miss ratio).",
"Choosing nodes with higher residual energy balances, the load among nodes and results in prolonged lifetime of the network.",
"The proposed protocol is devised using a modular design, separate modules are dedicated to each QoS requirement.",
"The link reliability estimation and link delay estimation modules use memory and computational effective methods suitable for WSNs.",
"The node forwarding module is able to make the optimal routing decision using the estimated metrics.",
"We test the performance of our proposed approaches by implementing our algorithms using $ns$ -2 simulator.",
"Our results demonstrates the performance and benefits of LRTHR over earlier algorithms.",
"The rest of the paper is organized as follows: Section 2 gives a review of Related Works.",
"Section 3 and Section 4 explains the Network Model, notations, assumptions and working of the algorithm.",
"Section 5 is devoted to the Simulation and Evaluation of the algorithm.",
"Conclusions are presented in Section 6."
],
[
"RELATED WORK",
"Stateless routing protocols which do not maintain per-route state is a favorable approach for WSNs.",
"The idea of stateless routing is to use location information available to a node locally for routing, i.e., the location of its own and that of its one-hop neighbors without the knowledge about the entire network.",
"These protocols scale well in terms of routing overhead because the tracked routing information does not grow with the network size or the number of active sinks.",
"Parameters like distance to sink, energy efficiency and data aggregation, need to be considered to select the next router among the one-hop neighbors.",
"SPEED (Stateless Protocol for End-to-End Delay) [2] is a well known stateless routing protocol for real-time communication in sensor networks.",
"It is based on geometric routing protocols such as greedy forwarding GPSR (Greedy Perimeter State Routing) [7][8].",
"It uses non-deterministic forwarding to balance each flow among multiple concurrent routes.",
"SPEED combines Medium Access Control (MAC) and network layer mechanism to maintain a uniform speed across the network, such that the delay a packet experiences is directly proportional to its distance to the sink.",
"At the MAC layer, a single hop relay speed is maintained by controlling the drop/relay action in a neighbor feedback loop.",
"Geographic forwarding is used to route data to its destination selecting the next hop as a neighbor from the set of those with a relay speed higher that the desired speed.",
"A back pressure re-routing mechanism is employed to re-route traffic around congested areas if necessary.",
"Lu et al., [9] describe a packet scheduling policy, called Velocity Monotonic Scheduling, which inherently accounts for both time and distance constraints.",
"Sequential Assignment Routing (SAR) [10] is the first routing protocol for sensor networks that creates multiple trees routed from one-hop neighbors of the sink by taking into consideration both energy resources, QoS metric on each path and priority level of each packet.",
"However, the protocol suffers from the overhead of maintaining the tables and states at each sensor node especially when the number of nodes is large.",
"MMSPEED (Multi-path and Multi-SPEED Routing Protocol) [3] is an extension of SPEED that focuses on differentiated QoS options for real-time applications with multiple different deadlines.",
"It provides differentiated QoS options both in timeliness domain and the reliability domain.",
"For timeliness, multiple QoS levels are supported by providing multiple data delivery speed options.",
"For reliability, multiple reliability requirements are supported by probabilistic multi-path forwarding.",
"The protocol provides end-to-end QoS provisioning by employing localized geographic forwarding using immediate neighbor information without end-to-end path discovery and maintenance.",
"It utilizes dynamic compensation which compensates for inaccuracy of local decision as a packet travels towards its destination.",
"The protocol adapts to network dynamics.",
"MMSPEED does not include energy metric during QoS route selection.",
"Chipera et al.,[4](RPAR:Real-Time Power Aware Routing) have proposed another variant of SPEED.",
"Where a node will change its transmission power by the progress towards destination and packet's slack time in order to meet the required velocity; they have not considered residual energy and reliability.",
"Mahapatra et al.,[11] assign an urgency factor to every packet depending on the residual distance and time the packet neesds to travel, and determines the distance the packet needs to be forwarded closer to the destination to meet its deadline.",
"Multi-path routing is performed only at the source node for increasing reliability.",
"Some routing protocols with congestion awareness have been proposed in [12][13].",
"Other geographic routing protocols such as [14-17] deal only with energy efficiency and transmission power in determining the next router.",
"Seada et al.,[18] proposed the PRR (Packet Reception Rate) $\\times $ Distance greedy forwarding that selects the next forwarding node by multiplying the PRR by the distance to the destination.",
"Recent geographical routing protocols have been proposed, such as DARA (Distributed Aggregate Routing Algorithm) [19], GREES (Geographic Routing with Environmental Energy Supply) [20], DHGR (Dynamic Hybrid Geographical Routing) [21] and EAGFS (Energy Aware Geographical Forwarding Scheme) [22].",
"They define either the same combined metric (of all the considered QoS metrics) [2][22][20] or several services but with respect to only one metric [14][13].",
"Sharif et al.,[23] presented a new transport layer protocol that prioritizes sensed information based on its nature while simultaneously supporting the data reliability and congestion control features.",
"Rusli et al.,[24] propose an analytical framework model based on Markov Chain of OR and M/D/l/K queue to measure its performance in term of end-to-end delay and reliability in WSNs.",
"Koulali et al.,[25] propose a hybrid QoS routing protocol for WSNs based on a customized Distributed Genetic Algorithm (DGA) that accounts for delay and energy constraints.",
"Yunbo Wang et al.,[26] investigate the end-to-end delay distribution, they develop a comprehensive cross-layer analysis framework, which employs a stochastic queueing model in realistic channel environments.",
"Ehsan et al.,[27] propose energy and cross-layer aware routing schemes for multichannel access WSNs that account for radio, MAC contention and network constraints.",
"All the above routing protocols are based on one-hop neighborhood information.",
"However, it is expected that multi-hop information can lead to improved performance in many issues including message broadcasting and routing.",
"Spohn et al.,[28] propose a localized algorithm for computing two-hop connected dominating set to reduce the number of redundant broadcast transmissions.",
"An analysis in [29] shows that in a network of $n$ nodes of total of $O(n)$ messages are required to obtain 2-hop neighborhood information and each message has $O(log n)$ bits.",
"Chen et al.,[30] study the performance of 1-hop, 2-hop and 3-hop neighborhood information based routing and propose that gain from 2-hop to 3-hop is relatively minimal, while that from 1-hop to 2-hop based routing is significant.",
"Li et al.,[5] have proposed a Two-Hop Velocity Based Routing Protocol (THVR).",
"The routing choice is decided on the two-hop relay velocity and residual energy, an energy efficient packet drop control is included to enhance packet utilization efficiency while keeping low packet deadline miss ratio.",
"However, THVR does not consider reliability while deciding the route.",
"The protocol proposed in this paper is different from THVR.",
"It considers reliability and uses dynamic velocity that can be altered for each packet as per the desired deadline.",
"It considers energy efficiently and balances the load only among nodes estimated to offer the required QoS."
],
[
"PROBLEM DEFINITION",
"The topology of a wireless sensor network may be described by a graph $G=(N,L)$ , where $N$ is the set of nodes and $L$ is the set of links.",
"The objectives are to, Minimize the deadline miss ratio (DMR).",
"Reduce the end-to-end packet delay.",
"Improve the energy efficiency (ECPP-Energy Consumed Per Packet) of the network."
],
[
"Network Model and Assumptions",
"In our network model, we assume the following: The wireless sensor nodes consists of $N$ sensor nodes and a sink, the sensors are distributed randomly in a field.",
"The nodes are aware of their positions through internal global positioning system (GPS), so each sensor has a estimate of its current position.",
"The $N$ sensor nodes are powered by a non renewable on board energy source.",
"When this energy supply is exhausted the sensor becomes non-operational.",
"All nodes are supposed to be aware of their residual energy and have the same transmission power range.",
"The sensors share the same wireless medium each packet is transmitted as a local broadcast in the neighborhood.",
"The sensors are neighbors if they are in the transmission range of each other and can directly communicate with each other.",
"We assume a MAC protocol, $i.e.$ , IEEE 802.11 which ensures that among the neighbors in the local broadcast range, only the intended receiver keeps the packet and the other neighbors discard the packet.",
"Like all localization techniques, [2][3][31][32][33] each node needs to be aware of its neighboring nodes current state (ID, position, link reliability, residual energy etc), this is done via HELLO messages.",
"Nodes are assumed to be stationary or having low mobility, else additional HELLO messages will be needed to keep the nodes up-to-date about the neighbor nodes.",
"In addition, each node sends a second set of HELLO messages to all its neighbors informing them about its one-hop neighbors.",
"Hence, each node is aware of its one-hop and two-hop neighbors and their current state.",
"The network density is assumed to be high enough to prevent the void situation.",
"Table: CONCLUSIONS"
]
] | 1403.0001 |
[
[
"Scalar-Tensor Gravity Cosmology: Noether symmetries and analytical\n solutions"
],
[
"Abstract In this paper, we present a complete Noether Symmetry analysis in the framework of scalar-tensor cosmology.",
"Specifically, we consider a non-minimally coupled scalar field action embedded in the FLRW spacetime and provide a full set of Noether symmetries for related minisuperspaces.",
"The presence of symmetries implies that the dynamical system becomes integrable and then we can compute cosmological analytical solutions for specific functional forms of coupling and potential functions selected by the Noether Approach."
],
[
"Introduction",
"The discovery of the accelerated expansion of the universe [1], [2], [3], [4], [5], [6], [7], [8] has opened a new path in approaching the cosmological problem.",
"Despite the mounting observational evidences on the existence of the cosmic acceleration, its nature and fundamental origin is still an open question challenging the very foundations of theoretical physics.",
"Usually, the mechanism that is responsible for cosmic acceleration is attributed to new physics which is based either on a modified theory of gravity or on the existence of some sort of dark energy which is associated with new fields in nature (see [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and references therein).",
"From the mathematical viewpoint, in order to study the cosmological features of a particular “dark energy” model, it is essential to specify the covariant Einstein-Hilbert action of the model and find out the corresponding energy-momentum tensor.",
"This methodology provides an elegant way to deal with dark energy in cosmology.",
"Within this framework, the standard view of the classical scalar field dark energy can be generalized considering scalar-tensor theories of gravity in which the scalar field $\\psi $ is non-minimally coupled to the Ricci scalar $R$ .",
"Generally, any theory of gravity that is not simply linear in the Ricci scalar can be reduced to a scalar-tensor one, implying that among the modified gravity models the scalar-tensor theory of gravity is one of the most general case that contains also other alternatives (for a review see [31]).",
"As an example, the $f(R)$ -gravity can be seen as a particular case of scalar-tensor gravity obeying the following criteria: (a) the scalar field is non-minimally coupled to the Ricci scalar and (b) a self-interacting potential is present while there is no kinetic term.",
"In this specific case, the scalar field is $\\psi =f^{\\prime }(R)$ which is the first derivative of $f(R)$ function with respect to $R$ .",
"In general, large classes of alternative theories of gravity, non-linear in the curvature invariants or non-minimally coupled in the Jordan frame, can be reduced to general relativity plus scalar field(s) in the Einstein frame [12].",
"In a recent paper by the same authors [32], conformally related metrics and Lagrangians, in the framework of scalar-tensor cosmology, have been studied.",
"In particular, it has been proven that the field equations of two conformally related Lagrangians are also conformally related if the corresponding Hamiltonian vanishes.",
"This is an important feature strictly related to the energy conditions of the theory.",
"Also, it has been shown that to every non-minimally coupled scalar field, we can associate a unique minimally coupled scalar field in a conformally related space with an appropriate potential.",
"The existence of such a connection can be used in order to study the dynamical properties of the various cosmological models, since the field equations of a non-minimally coupled scalar field can be reduced, at conformal level, to the field equations of the minimally coupled scalar field.",
"With the current work, we complete our previous program on scalar-tensor theories by calculating the corresponding Noether point symmetries as well as the related analytical solutions.",
"It is interesting to mention that Noether point symmetries have gained a lot of attention in cosmology (see [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47]), since they can be used as a selection criterion in order to discriminate the dark energy models, including those of modified gravity [44] as well as to provide analytical solutions.",
"Such a program started in [34] where inflationary models have been considered.",
"The paradigm can be shortly summarized as follows.",
"The existence of a Noether symmetry selects the forms of non-minimal coupling and potential in general scalar-tensor theories of gravity.",
"As a consequence, the related dynamical system which results reduced because every symmetry is related to a first integral of motion.",
"In most cases, the presence of such integrals of motion allows to find out general solutions for dynamics.",
"It is important to stress that by choosing particular classes of metrics, one reduces the field theory to a point-like one.",
"From a cosmological viewpoint, this means that we are considering dynamical systems defined on minisuperspaces.",
"These finite-dimensional dynamical systems are extremely interesting in Quantum Cosmology (see [43] for a discussion).",
"This consideration is important since allows one to deal with Noether Symmetry Approach both in early and late cosmology.",
"Some remarks are important at this point to relate the Noether Symmetry Approach to the presence of conserved physical quantities.",
"Generally speaking, in modified gravitational theories, where the Birkhoff theorem is not guaranteed, the Noether approach can provide a useful tool towards describing the global dynamics [48], through the first integrals of motion.",
"Moreover, besides the technical possibility of reducing the dynamical system, the first integrals of motion give always rise to conserved currents that are not only present in physical space-time but also in configuration spaces (see the discussion in [32] and [34]).",
"While in space-time such currents are linear momentum, angular momentum etc.",
"in configuration space the conserved quantities emerge as relations among dynamical variables, in particular, among their functions as couplings and self-interaction potentials.",
"For example, as discussed in Capozziello & Ritis [49], the presence of Noether symmetries in scalar-tensor gravity gives rise to an effective cosmological constant and gravitational asymptotic freedom behaviours induced by potentials and couplings.",
"This means that, while in the standard spacetime the Noether charges are directly related to conserved observable quantities, in the configuration space (minisuperspace), they are present as ”selection rules” for potentials and coupling functions which are capable of assigning realistic dynamics.",
"In the present work, we complete the program started in [34] and [32], discussing the general structure of scalar-tensor cosmological models compatible with the existence of Noether symmetries.",
"Moreover, the current work can be seen as a natural continuation of our previous works [44].",
"The layout of the paper is the following.",
"In Sec.",
"2, we present the main ingredients of the dynamical problem under study.",
"In Secs.",
"3 and 4 we provide the Noether point symmetries as well as the corresponding analytical solutions for the two classes of models considered.",
"We draw our conclusions in Sec.",
"5."
],
[
"The Minisuperspace and the dynamical system",
"In the context of scalar-tensor cosmology, let us consider a scalar field $\\psi $ (non-minimally) interacting with the gravitational field.",
"In this framework, the field equations can be derived from the following general action $S=\\int dt dx^{3}\\sqrt{-g}\\left[ F\\left( \\psi ,R \\right) +\\frac{\\varepsilon }{2}g_{ij}\\psi ^{;i}\\psi ^{;j}-V\\left( \\psi \\right) \\right] +S_{m}$ where $\\varepsilon =\\pm 1$ , $\\psi $ denotes the scalar field, $V(\\psi )$ is the self-interaction potential, $F(\\psi ,R)$ is the coupling function, $R$ is the Ricci scalar and $S_{m}$ is the matter action.",
"The parameter $\\varepsilon $ indicates if we are dealing with a regular scalar field or a ghost field.",
"Assuming a spatially flat FRW space-time $ds^{2}=-dt^{2}+a^{2}\\left( t\\right) \\delta _{ij}dx^{i}dx^{j} \\;,$ the infinite degrees of freedom of the field theory reduce to a finite number.",
"In this specific case, the minisuperspace is a 2-dimensional configuration space defined by the variables ${\\cal Q}=\\lbrace \\psi ,a\\rbrace $ .",
"The tangent space on which dynamics is defined is ${\\cal TQ}=\\lbrace \\psi ,\\dot{\\psi }, a, \\dot{a}\\rbrace $ where the dot indicates the derivative with respect to the cosmic time which is the natural affine parameter for the problem.",
"Of course, if we consider $F(\\psi ,R)=R$ then the action (REF ) boils down to the nominal, minimally coupled, scalar field dark energy.",
"On the other hand, the $f(R)$ modified gravity is fully recovered for $F(\\psi ,R)=f(R)$ and in the absence of the kinetic term in the action.",
"In this study, we consider the case where the coupling function is proportional to $R$ , $F(\\psi ,R)=F(\\psi )R$ .",
"Due to the fact that almost every dynamical system is described by a corresponding Lagrangian, below we apply such ideas to the scalar field cosmology.",
"Indeed the corresponding Lagrangian and the Hamiltonian (total energy density) of the field equations are $L=6F\\left( \\psi \\right) a\\dot{a}^{2}+6F_{\\psi }\\left( \\psi \\right) a^{2}\\dot{a}\\dot{\\psi }+\\frac{\\varepsilon }{2}a^{3}\\dot{\\psi }^{2}-a^{3}V\\left(\\psi \\right)$ $E=6F\\left( \\psi \\right) a\\dot{a}^{2}+6F_{\\psi }\\left( \\psi \\right) a^{2}\\dot{a}\\dot{\\psi }+\\frac{\\varepsilon }{2}a^{3}\\dot{\\psi }^{2}+a^{3}V\\left( \\psi \\right) \\;.",
"$ Note that the Lagrangian (REF ) is autonomous, hence the Hamiltonian $E$ is a constant of motion (see also the discussion in [32]).",
"This constant corresponds to the trivial Noether point symmetry $\\partial _{t}$ (first integral of motion).",
"Using the 00 component of the conservation equation $T_{;\\mu }^{\\nu \\mu }=0~$ we find that the Hamiltonian $E$ is related to the matter density $\\rho _{m}$ as ${\\displaystyle \\rho _{m}=\\frac{\\left|E\\right|}{a^{3}}}$ .",
"Following the technique described in [50], [51], [44] it is essential to split the Lagrangian (REF ) in the kinematic part, which defines the kinematic metric (hereafter KM), and the remaining part which we consider to be the potential.",
"Indeed the kinematic metric is written as $ds_{KM}^{2}=12F\\left( \\psi \\right) a\\dot{a}^{2}+12F_{\\psi }\\left( \\psi \\right) a^{2}\\dot{a}\\dot{\\psi }+\\varepsilon a^{3}\\dot{\\psi }^{2} \\;.$ The above metric is not the FRW metric of the background space-time but a metric defined on the tangent space ${\\cal TQ}$ .",
"It is related to the minisuperspace configuration metric in the two dimensional space $\\lbrace a,\\psi \\rbrace $ .",
"The corresponding Ricci scalar of the metric $ds_{KM}^{2}$ is computed to be: $R_{KM}=\\frac{\\varepsilon }{4a^{3}}\\frac{\\left( 2F_{\\psi \\psi }F-F_{\\psi }^{2}\\right) }{\\left( \\varepsilon F-3F_{\\psi }^{2}\\right) ^{2}} \\;.$ Obviously, knowing $R_{KM}$ , one can estimate $F(\\psi )$ .",
"If we assume that the curvature $R_{KM}$ is constant then Eq.",
"(REF ) implies that $R_{KM}\\equiv 0$ (due to the presence of $a$ in the denominator) and the minisuperspace is flat.",
"We realize that we need to consider the following two cases: (A) the minisuperspace $\\lbrace a,\\psi \\rbrace $ is maximal symmetric (flat $R_{KM}\\equiv 0$ ) and (B) the case where the minisuperspace is not necessarily flat but it is conformally flat, because all two dimensional spaces are conformally flat.",
"In the following, we consider these two situations in detail."
],
[
"The case of maximally symmetric $\\lbrace a,\\psi \\rbrace $ minisuperspace",
"In this case using the condition $R_{KM}=0$ , Eq.",
"(REF ) reduces to $2F_{\\psi \\psi }F-F_{\\psi }^{2}=0 $ and then a solution is $F\\left( \\psi \\right) =-\\frac{F_{0}\\varepsilon }{12}\\left( \\psi +\\psi _{0}\\right) ^{2} $ where $F_{0}\\varepsilon >0$ .",
"In order to determine the homothetic algebra of the kinematic metric (REF ), we write it in a more familiar form.",
"Actually, in Tsamparlis et al.",
"[32] we introduced the conformal variables $A$ , $\\Psi $ and ${\\cal N}$ by the relations $A=\\sqrt{-2F}a $ $d\\Psi =\\sqrt{\\left( \\frac{3\\varepsilon F_{\\psi }^{2}-F}{2F^{2}}\\right) }~d\\psi $ ${\\cal N}=\\frac{1}{\\sqrt{-2F}} $ with $F\\left( \\Psi \\right)<0$ .",
"In the new variables, the kinematic metric (REF ) and the Lagrangian (REF ) become $ds_{KM}^{2}=N^{2}\\left( \\Psi \\right) \\left[ -3A\\dot{A}^{2}+\\frac{\\varepsilon }{2}A^{3}\\dot{\\Psi }^{2}\\right]$ $L=N^{2}\\left( \\Psi \\right) \\left[ -3A\\dot{A}^{2}+\\frac{\\varepsilon }{2}A^{3}\\dot{\\Psi }^{2}\\right] -A^{3}\\bar{V}\\left( \\Psi \\right) $ where $N^{2}={\\cal N}$ , $\\bar{V}\\left( \\Psi \\right) =N^{6}(\\Psi )V\\left( \\Psi \\right)$ .",
"Also, the coupling function (REF ) takes the form $F\\left( \\Psi \\right) =-\\frac{\\varepsilon F_{0}}{12}{\\rm e}^{\\sqrt{6\\varepsilon }|k|\\Psi }$ where $|k|=\\frac{1}{3}\\sqrt{\\frac{|F_{0}|}{|1+\\varepsilon F_{0}|}} \\;.$ Notice, that the inequality $F_{0}\\varepsilon >0$ is satisfied either for $\\varepsilon =+1$ with $F_{0}>0$ or for $\\varepsilon =-1$ with $-1<F_{0}<0$ .",
"We would like to mention here that in the case of $\\varepsilon =-1$ with $F_{0}<-1$ one has to replace $\\Psi $ with $i\\Psi $ .",
"We further simplify the above calculations by introducing a new coordinate system $(r,\\theta )$ defined as $r=\\sqrt{\\frac{8}{3}}A^{\\frac{3}{2}}~,~\\theta =\\sqrt{\\frac{3\\varepsilon }{8}}~\\Psi \\;.",
"$ Inserting the above variables into Eq.",
"(REF ), we immediately obtain $ds_{KM}^{2}=N^{2}\\left( \\theta \\right) \\left( -dr^{2}+r^{2}d\\theta ^{2}\\right) $ which is directly related to the flat 2D Lorentzian space with metric $ds^{2}=-dr^{2}+r^{2}d\\theta ^{2}$ with the conformal factor $N\\left( \\theta \\right)$ $N^{2}(\\theta )=N_{0}^{2}{\\rm e}^{\\mp 2|k|\\theta } \\;\\;\\;N_{0}^{2}=(\\frac{6}{\\varepsilon F_{0}})^{1/2} \\;.$ Finally, the Lagrangian takes a rather simple form $L=N^{2}\\left( \\theta \\right) \\left( -\\frac{1}{2}\\dot{r}^{2}+\\frac{1}{2}r^{2}\\dot{\\theta }^{2}\\right) -r^{2}V\\left( \\theta \\right) .",
"$ Armed with the above expressions, we can deduce the homothetic algebra of the metric from well known previous results (see [50], [51], [44])."
],
[
"Searching for Noether point symmetries",
"Let us determine now all the potentials $V\\left( \\psi \\right) $ for which the above dynamical system admits Noether point symmetries beyond the trivial one $\\partial _{t}$ related to the energy.",
"Subsequently, we shall use the resulting Noether integrals in order to find out analytical solutions.",
"For $\\left|k\\right|\\ne 1$ the homothetic algebra consists of the gradient Killing vectors (KVs) $K^{1} &=&\\frac{e^{\\left( 1-k\\right) \\theta }r^{k}}{N_{0}^{2}}\\left(-\\partial _{r}+\\frac{1}{r}\\partial _{\\theta }\\right) ~,~S_{1}\\left( r,\\theta \\right) =\\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{\\left( k+1\\right) } \\\\K^{2} &=&\\frac{e^{-\\left( 1+k\\right) \\theta }r^{-k}}{N_{0}^{2}}\\left(\\partial _{r}+\\frac{1}{r}\\partial _{\\theta }\\right) ,~S_{2}\\left( r,\\theta \\right) =\\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1}$ (for $\\left|k\\right|=1$ see Appendix A) the non-gradient KV $K^{3}=r\\partial _{r}-\\frac{1}{k}\\partial _{\\theta }$ and the gradient homothetic vectors (HV) $H^{i}=\\frac{1}{N_{0}^{2}\\left( k^{2}-1\\right) }\\left( -r\\partial _{r}+k\\partial _{\\theta }\\right) ~,~H\\left( r,\\theta \\right) =\\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)}.$ Specifically, we ask the question: are there potentials that can provide non-trivial Noether point symmetries and consequently first integrals of motion?",
"Below we present all possible cases: First of all, by using the gradient KV $K^{1}$ , we find a) for $V\\left( \\theta \\right) =V_{0}e^{2\\theta }$ , we have the Noether symmetries $K^{1},~tK^{1}$ with Noether integrals $I_{1}=\\frac{d}{dt}\\left( \\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{\\left(k+1\\right) }\\right)$ $I_{2}=t\\frac{d}{dt}\\left( \\frac{r^{1+k}e^{\\left(1+k\\right) \\theta }}{\\left( k+1\\right) }\\right) -\\left( \\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{\\left( k+1\\right) }\\right)$ b) for $V\\left( \\theta \\right) =V_{0}e^{2\\theta }-\\frac{mN_{0}^{2}}{2\\left(k^{2}-1\\right) }e^{2k\\theta }$ , we obtain the Noether symmetries $e^{\\pm \\sqrt{m}t}K^{1}$ , where $m=$ constant, with Noether integrals $I_{\\pm }^{\\prime }=e^{\\pm \\sqrt{m}t}\\left[ \\frac{d}{dt}\\left( \\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{\\left( k+1\\right) }\\right) \\mp \\sqrt{m}\\left( \\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{\\left( k+1\\right) }\\right) \\right] $ From the above Noether integrals, we construct the time independent first integral $I_{K^{1}}=I_{+}I_{-}.$ The gradient KV $K^{2}$ produces the Noether symmetries for the following potentials a) for $V\\left( \\theta \\right) =V_{0}e^{-2\\theta }$ ,we have the Noether symmetries $K^{1},~tK^{1}$ with Noether integrals $J_{1}=\\frac{d}{dt}\\left( \\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1}\\right) ~,$ $J_{2}=t\\frac{d}{dt}\\left( \\frac{r^{1-k}e^{-\\left( 1-k\\right)\\theta }}{k-1}\\right) -\\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1}$ b) for $V\\left( \\theta \\right) =V_{0}e^{-2\\theta }-\\frac{mN_{0}^{2}}{2\\left(k^{2}-1\\right) }e^{2k\\theta }$ , we have the Noether symmetries $e^{\\pm \\sqrt{m}t}K^{2}$ $m=$ constant, with Noether integrals $J_{\\pm }^{^{\\prime }}=e^{\\pm \\sqrt{m}t}\\left[ \\frac{d}{dt}\\left( \\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1}\\right) \\mp \\sqrt{m}\\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1}\\right] $ Combining the latter Noether integrals, we construct the time-independent first integral $J_{K^{2}}=J_{+}^{\\prime }J_{-}^{\\prime }.$ The non gradient KV $K^{3}$ produces a Noether symmetry for the potential $V\\left( \\theta \\right) =V_{0}e^{2k\\theta }$ with Noether integral $I_{3}=\\frac{re^{2k\\theta }}{k}\\left( k\\dot{r}+r\\dot{\\theta }\\right) .$ The gradient HV produces the following Noether symmetries for the following potentials a) for $V\\left( \\theta \\right) =V_{0}e^{-2\\frac{\\left( k^{2}-2\\right) }{k}\\theta }$ , $k^{2}-2\\ne 0$ we have the Noether symmetries $2t\\partial _{t}+H^{i}~,~t^{2}\\partial _{t}+tH^{i}$ with Noether integrals $I_{H_{1}}=2tE-\\frac{d}{dt}\\left( \\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)}\\right) ~,~$ $I_{H_{2}}=t^{2}E-t\\frac{d}{dt}\\left( \\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)}\\right) +\\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)} \\,.",
"$ We note that in this case the system is the Ermakov-Pinney dynamical system [52] and admits the Noether symmetry algebra $sl(2,R)$ .",
"b) For $V\\left( \\theta \\right) =V_{0}e^{-2\\frac{\\left( k^{2}-2\\right) }{k}\\theta }-\\frac{N_{0}^{2}m}{k^{2}-1}e^{2k\\theta }~$ , $k^{2}-2\\ne 0$ we have the Noether symmetries $\\frac{2}{\\sqrt{m}}e^{\\pm \\sqrt{m}t}\\partial _{t}\\pm e^{\\pm \\sqrt{m}t}H^{i}$ , $m=$ constant with Noether integrals $I_{\\pm }=e^{\\pm 2\\sqrt{m}t}\\left[ \\frac{1}{\\sqrt{m}}E\\mp \\frac{d}{dt}\\left(\\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)}\\right) +2\\sqrt{m}\\left( \\frac{1}{2}\\frac{r^{2}e^{2k\\theta }}{(k^{2}-1)}\\right) \\right] $ This is also the Ermakov-Pinney dynamical system with a linear oscillator.",
"Therefore it admits the Ermakov - Pinney invariant which we may construct with the use of the dynamical Noether symmetries or with the use of the corresponding Killing Tensor.",
"Lastly, the case $V\\left( \\theta \\right) =0$ corresponds to the free particle (see [50])."
],
[
"Analytical solutions",
"Using the above Noether symmetries and the corresponding integral of motions, we can fully solve the dynamical problem of the scalar tensor cosmology.",
"In order to simplify the analytical solutions, we consider the new variables $x=S_{1}(r,\\theta )=\\frac{r^{1+k}e^{\\left( 1+k\\right) \\theta }}{k+1}~,~y=S_{2}(r,\\theta )=\\frac{r^{1-k}e^{-\\left( 1-k\\right) \\theta }}{k-1} $ and the inverse transformation is $\\theta &=&\\frac{1}{2|k^{2}-1| }\\ln \\left[ \\frac{|k^{2}-1|^{1-k}}{\\left( k-1\\right) ^{2}}\\frac{x^{1-k}}{y^{1+k}}\\right] \\\\r &=&\\sqrt{|k^{2}-1| xy}\\left[ \\frac{|k^{2}-1|^{1-k}}{\\left( k-1\\right) ^{2}}\\frac{x^{1-k}}{y^{1+k}}\\right] ^{\\frac{k}{2\\left(k^{2}-1\\right) }} \\;.",
"$ We find that in the new coordinates $(x,y)$ , the Lagrangian (REF ) takes the form $L\\left( x,y,\\dot{x},\\dot{y}\\right) =\\epsilon _{k}\\frac{N_{0}^{2}}{2}\\dot{x}\\dot{y}-U\\left( x,y\\right) $ where $U\\left( x,y\\right) =r^{2}V\\left( \\theta \\right)$ and $\\epsilon _{k}=+1$ for $|k|>1$ ($\\epsilon _{k}=-1$ for $|k|<1$ ).",
"Note that $V\\left(\\theta \\right)$ are the potentials which have been presented in the previous section.",
"We would like to stress that the solution of the field equations for each potential is a formal and lengthy operation which adds nothing but unnecessary material to the matter.",
"What is interesting of course is the final answer for each case and this is what we show in a compact presentation below.",
"Specifically, the analytical solutions can be categorized into seven separate cases The first class is $U_{1}\\left( x,y\\right) =V_{0}r^{2}e^{2k\\theta }=V_{0}|k^{2}-1|xy$ $x\\left( t\\right) &=&x_{1}{\\rm Sinn} \\left( \\omega t\\right) +x_{2}{\\rm Coss} \\left(\\omega t\\right) \\\\y\\left( t\\right) &=&y_{1}{\\rm Sinn} \\left( \\omega t\\right) +y_{2}{\\rm Coss} \\left(\\omega t\\right) $ where $(\\mathrm {Sinn}\\omega ,\\mathrm {Coss}\\omega )=\\left\\lbrace \\begin{array}[c]{cc}(\\mathrm {sin}\\omega ,\\mathrm {cos}\\omega ) & \\mbox{$|k|>1$}\\\\(\\mathrm {sinh}\\omega ,\\mathrm {cosh}\\omega ) & \\mbox{$\\;\\;\\;|k|<1$}\\end{array}\\right.",
"$ $\\omega ^{2}=\\frac{2V_{0}|k^{2}-1|}{N_{0}^{2}}$ and the Hamiltonian is $\\ $ $E=V_{0}|k^{2}-1| \\left( x_{1}y_{1}+\\epsilon _{k}x_{2}y_{2}\\right)$ $U_{2}\\left( x,y\\right) =V_{0}r^{2}e^{2\\theta }=V_{0}\\left( k+1\\right) ^{\\frac{2}{1+k}}~x^{\\frac{^{2}}{1+k}}$ , as long as $k\\ne -3$ we have $x\\left( t\\right) &=&x_{1}t+x_{2} \\\\y\\left( t\\right) &=&-\\epsilon _{k}\\frac{2\\bar{V}\\left( k+1\\right) \\left(x_{1}t+x_{2}\\right) ^{\\left( 1+\\frac{2}{1+k}\\right) }}{x_{1}^{2}\\left(3+k\\right) N_{0}^{2}}+y_{1}t+y_{2}\\nonumber \\\\& &.$ where $\\bar{V}=V_{0}\\left( k+1\\right) ^{\\frac{2}{1+k}}$ and the Hamiltonian is $E=\\epsilon _{k}\\frac{y_{1}x_{1}N_{0}^{2}}{2}.$ If $k=-3$ then $y(t)$ becomes $y\\left( t\\right) =-2\\frac{\\bar{V}}{N_{0}^{2}x_{1}^{2}}\\ln \\left(x_{1}t+x_{2}\\right) +y_{1}t+y_{2}.",
"$ $U_{3}\\left( x,y\\right) =V_{0}r^{2}e^{-2\\theta }=V_{0}|k-1|^{\\frac{2}{1-k}}y^{\\frac{2}{1-k}}$ When $k\\ne 3$ $x\\left( t\\right) &=&\\frac{2\\bar{V}|k-1| \\left(y_{1}t+y_{2}\\right) ^{1+\\frac{2}{k-1}}}{y_{1}^{2}\\left( k-3\\right) N_{0}^{2}}+x_{1}t+x_{2} \\\\y\\left( t\\right) &=&y_{1}t+y_{2} $ where $\\bar{V}=V_{0}|k-1|^{\\frac{2}{1-k}}$ and the Hamiltonian is $E=\\epsilon _{k}\\frac{y_{1}x_{1}N_{0}^{2}}{2}.$ In this context if $k=3$ then $x(t)$ takes the form $x\\left( t\\right) =-\\frac{2\\bar{V}}{N_{0}^{2}y_{1}^{2}}\\ln \\left(y_{1}t+y_{2}\\right) +x_{1}t+x_{2}.",
"$ $U_{4}\\left( x,y\\right) =V_{0}r^{2}e^{2\\theta }+mr^{2}e^{2k\\theta }=\\bar{V}_{0}~x^{\\frac{^{2}}{1+k}}+\\bar{m}xy$ , in this class we find $x\\left( t\\right)=x_{1}{\\rm Sinn} \\left( \\omega t+\\omega _{0}\\right) \\\\y\\left( t\\right)={\\rm Coss} \\left( \\omega t+\\omega _{0}\\right) \\left( y_{1}+2\\epsilon _{K}\\frac{\\omega }{\\bar{m}}\\int \\frac{E-x_{1}\\bar{V}_{0}{\\rm Sinn} \\left( \\omega t+\\omega _{0}\\right) ^{\\frac{2}{1+k}}}{x_{1}\\left( {\\rm Coss} \\left( \\omega t+\\omega _{0}\\right) +1\\right) }dt\\right)$ where $\\bar{V}_{0}=V_{0}\\left( k+1\\right) ^{\\frac{2}{1+k}}~,~\\bar{m}=m|k^{2}-1|$ , $\\omega ^{2}=\\frac{2\\bar{m}}{N_{0}^{2}}$ and $E=y_{2}$ .",
"similarly for $U_{5}\\left( x,y\\right)=r^{2}e^{-2\\theta }+mr^{2}e^{2k\\theta }=\\bar{V}_{0}y^{\\frac{2}{1-k}}+\\bar{m}xy$ we obtain $x\\left( t\\right)={\\rm Coss} \\left( \\omega t+\\omega _{0}\\right) \\left( x_{1}+2\\epsilon _{k}\\frac{\\omega }{\\bar{m}}\\int \\frac{x_{2}-y_{1}\\bar{V}_{0}{\\rm Sinn} \\left( \\omega t+\\omega _{0}\\right) ^{\\frac{2}{1-k}}}{y_{1}\\left( {\\rm Coss} \\left( \\omega t+\\omega _{0}\\right) +1\\right) }dt\\right) \\\\y\\left( t\\right) =y_{1}{\\rm Sinn} \\left( \\omega t+\\omega _{0}\\right)$ where $\\bar{V}_{0}=V_{0}|k-1|^{\\frac{2}{1-k}}$ and $E=x_{2}$ .",
"$U_{6}\\left( x,y\\right) =V_{0}r^{2}e^{-2\\frac{\\left( k^{2}-2\\right) }{k}\\theta }+mr^{2}e^{2k\\theta }=\\bar{V}_{0}\\frac{1}{y^{2}}\\left( \\frac{x}{y}\\right) ^{\\frac{2}{k}-1}+\\bar{m}xy$ with $\\bar{V}_{0}=V_{0}\\frac{|k^{2}-1|^{\\frac{2}{k}-1}}{|k-1|^{\\frac{4}{k}}}$ .",
"The current dynamical system is the so called Ermakov-Pinney system.",
"To solve this dynamical problem, it is convenient to go to the following coordinates $(x,y)=(ze^{w},ze^{-w})$ .",
"In this coordinate system we recover the Ermakov-Pinney equation: $\\ddot{z}+2\\epsilon _{k}\\bar{m}z+\\epsilon _{k}N_{0}^{2}\\frac{J_{EL}}{z^{3}}=0$ where $J_{EL}=z^{4}\\dot{w}^{2}-2\\epsilon _{k}\\frac{\\bar{V}_{0}}{N_{0}^{2}}e^{\\frac{4}{k}w}$ is the Ermakov invariant.",
"The solution of the above differential equation is $z\\left( t\\right) &=&\\left[ l_{0}z_{1}\\left( t\\right) +l_{1}z_{2}\\left(t\\right) +l_{3}\\right]^{\\frac{1}{2}} \\\\e^{\\frac{4}{k}w\\left( t\\right) } &=&-\\epsilon _{k}\\frac{N_{0}^{2}J_{EL}}{2\\bar{V}_{0}}\\left[ 1-\\tanh ^{2}\\left( \\frac{2\\sqrt{J_{EL}}}{k}\\left( \\int \\frac{dt}{z^{2}\\left( t\\right) }+l_{4}\\right) \\right)\\right] $ where $z_{1,2}\\left( t\\right) $ are solutions of the differential equation $\\ddot{z}+2\\epsilon _{k}\\bar{m}z=0$ and $l_{0-4}$ are constants.",
"Lastly, $U_{7}(x,y)=0$ is the free particle system, a solution of which is $x\\left( t\\right) &=&x_{1}t+x_{2}~,~y\\left( t\\right) =y_{1}t+y_{2}$ with $E=\\epsilon _{k}\\frac{N_{0}^{2}}{2}x_{1}y_{1}$ ."
],
[
" The case of 2d conformally-flat metric ",
"In this case the kinetic metric (REF ) is non-flat (i.e.",
"$R_{KM}\\ne 0)$ but, of course, it is conformally flat being a two dimensional metric.",
"Its conformal algebra is infinity dimensional; however it has a closed subalgebra consisting of the following vectors (this is the special conformal algebra of $M^{2}$ ): $X^{1} &=&\\cosh \\theta \\partial _{r}-\\frac{1}{r}\\sinh \\theta \\partial _{\\theta }~~,~X^{2}=\\sinh \\theta \\partial _{r}-\\frac{1}{r}\\cosh \\theta \\partial _{\\theta } \\\\X^{3} &=&\\partial _{\\theta }~~\\ ,~X^{4}=r\\partial _{r}~~,~X^{5}=\\frac{1}{2}r^{2}\\cosh \\theta \\partial _{r}+\\frac{1}{2}r\\sinh \\theta \\partial _{\\theta }\\\\X^{6} &=&\\frac{1}{2}r^{2}\\sinh \\theta \\partial _{r}+\\frac{1}{2}r\\cosh \\theta \\partial _{\\theta } \\,.$ We remind the reader that the variables $r$ and $\\theta $ are defined in Eq.",
"(REF ).",
"Writing $L_{X^{I}}g_{ij}=2C_{I}\\left( r,\\theta \\right) g_{ij}$ we find the conformal factors of the CKVs $X^{I}$ $I=1,...6$ above in terms of the the conformal function.",
"The result is: $C_{1}\\left( r,\\theta \\right)=-\\frac{1}{r}\\sinh \\theta \\frac{N_{,\\theta }}{N},\\;\\;C_{2}\\left( r,\\theta \\right) =-\\frac{1}{r}\\cosh \\theta \\frac{N_{,\\theta }}{N}$ $C_{3}\\left( r,\\theta \\right)=\\frac{N_{,\\theta }}{N},\\;\\;C_{4}\\left(r,\\theta \\right) =1$ $C_{5}\\left( r,\\theta \\right) =\\frac{r}{2}\\left( \\frac{2N\\cosh \\theta +\\sinh \\theta N_{\\theta }}{N}\\right)$ $C_{6}\\left( r,\\theta \\right)=\\frac{r}{2}\\left( \\frac{2N\\sinh \\theta +\\cosh \\theta N_{\\theta }}{N}\\right).$ We would like to remind the reader that the coupling function $N(\\theta )$ does not obey Eq.",
"(REF ), otherwise the kinetic metric of the Lagrangian (REF ) is flat ($R_{KM}$ vanishes) and we return to Sec.",
"3.",
"The latter means that the vectors $X^{I}$$I=1,...6$ , except the $I=4$ , are proper CKVs therefore they do not give (if proper) a Noether point symmetry.",
"The vector $X_{4}$ is a non-gradient HV which also does not produce a Noether point symmetry.",
"Therefore, according to theorem in [50], [51], only Killing vectors are possible to serve as Noether symmetries.",
"Killing vectors do not exist in general but only for special forms of the conformal function $N(\\theta )$ .",
"Each of such forms of $N(\\theta )$ results in a potential $V(\\theta )\\ $ hence in a scalar field potential which admits Noether point symmetries.",
"In the following, we shall determine the possible $N(\\theta )$ forms which lead to a KV and give the corresponding Noether point symmetry and the corresponding Noether integral which will be used for the solution of the field equations."
],
[
" Searching for Noether symmetries",
" If $N\\left( \\theta \\right) =\\frac{N_{0}}{\\cosh 2\\theta -1}$ then $X^{5}$ is a non-gradient KV and a Noether symmetry of the Lagrangian (REF ) for the potential $V\\left( \\theta \\right) =\\frac{V_{0}}{\\cosh 2\\theta -1}~\\text{or }V\\left( \\theta \\right) =0 \\,.",
"$ The corresponding Noether integral is $I_{X^{5}}=\\frac{N_{0}^{2}r^{2}}{\\left( \\cosh 2\\theta -1\\right) ^{2}}\\left( r\\dot{\\theta }\\sinh \\theta -\\dot{r}\\cosh \\theta \\right).",
"$ If $N\\left( \\theta \\right) =\\frac{N_{0}}{\\cosh 2\\theta +1}$ then $X^{6}$ is a non gradient KV, $X^{6}$ and a Noether symmetry for the Lagrangian (REF ) if $V\\left( \\theta \\right) =\\frac{V_{0}}{\\cosh 2\\theta +1}~\\text{or }V\\left( \\theta \\right) =0 \\,.$ The corresponding Noether integral is $I_{X^{6}}=\\frac{N_{0}^{2}r^{2}}{\\left( \\cosh 2\\theta +1\\right) ^{2}}\\left( r\\dot{\\theta }\\cosh \\theta -\\dot{r}\\sinh \\theta \\right)\\,.$ If $N\\left( \\theta \\right) =\\frac{N_{0}}{\\cosh ^{2}\\left( \\theta +\\theta _{0}\\right) }$ then the linear combination $X^{56}=c_{1}X^{5}+c_{2}X^{6}$ where $c_{1}=\\sinh \\left( \\theta _{0}\\right) $ and $c_{2}=\\cosh \\left( \\theta _{0}\\right) $ .",
"$X^{56}$ is a Noether symmetry for the Lagrangian (REF ) if $V\\left( \\theta \\right) =\\frac{V_{0}}{\\cosh ^{2}\\left( \\theta +\\theta _{0}\\right) }~\\text{or }V\\left( \\theta \\right) =0 $ with Noether integral $I_{X^{56}}=\\frac{N_{0}^{2}r^{2}}{\\cosh ^{4}\\left( \\theta +\\theta _{0}\\right)}\\left[ r\\dot{\\theta }\\cosh \\left( \\theta +\\theta _{0}\\right) -\\dot{r}\\sinh \\left( \\theta +\\theta _{0}\\right) \\right]$ Obviously the third case is the most general situation and it contains cases 1 and 2 (and the trivial case) as special cases.",
"Therefore, in the following, we look for analytic solutions for the vector $X^{56}$ only.",
"We recall that $\\frac{1}{\\sqrt{-2F\\left( \\theta \\right) }}=N^{2}\\left(\\theta \\right) $ from which follows: $~F\\left( \\theta \\right) =-\\frac{1}{2N_{0}^{4}}\\cosh ^{8}\\left( \\theta +\\theta _{0}\\right) ~,N_{0}\\in \\mathbb {R}.",
"$ We may consider $\\theta _{0}=0$ (e.g.",
"by introducing the new variable $\\Theta =\\theta +\\theta _{0}).$ For the potential (REF ) Lagrangian (REF ) becomes $L=\\frac{N_{0}^{2}}{\\cosh ^{4}\\theta }\\left( -\\frac{1}{2}\\dot{r}^{2}+\\frac{1}{2}r^{2}\\dot{\\theta }^{2}\\right) -r^{2}\\frac{V_{0}}{\\cosh ^{2}\\theta }$ and the Hamiltonian $E=\\frac{N_{0}^{2}}{\\cosh ^{4}\\theta }\\left( -\\frac{1}{2}\\dot{r}^{2}+\\frac{1}{2}r^{2}\\dot{\\theta }^{2}\\right) +r^{2}\\frac{V_{0}}{\\cosh ^{2}\\theta }.$ The Euler-Lagrange equations provide the equations of motion: $\\ddot{r}+r\\dot{\\theta }^{2}-4\\tanh \\theta ~\\dot{r}\\dot{\\theta }-2\\frac{V_{0}}{N_{0}^{2}}r\\cosh ^{2}\\theta =0 $ $\\ddot{\\theta }-2\\tanh \\theta ~\\left( \\frac{1}{r^{2}}\\dot{r}^{2}+\\dot{\\theta }^{2}\\right) +\\frac{2}{r}\\dot{r}\\dot{\\theta }-2\\frac{V_{0}}{N_{0}^{2}}\\cosh \\theta \\sinh \\theta =0 $ and the Noether integral $I$ for $\\theta _{0}=0$ becomes: $I=\\frac{N_{0}^{2}r^{2}}{\\cosh ^{4}\\left( \\theta +\\theta _{0}\\right)}\\left[ r\\dot{\\theta }\\cosh \\theta -\\dot{r}\\sinh \\theta \\right] .",
"$ In order to proceed with the solution of the system of equations (REF ), (REF ) we change to the coordinates $x,y~$ which we define by the relations $r=\\frac{x}{\\sqrt{1-x^{2}y^{2}}}~,~\\theta =\\arctan h\\left( xy\\right) .$ In the coordinates $(x,y)$ the Lagrangian and the Hamiltonian are written as $L=\\frac{N_{0}^{2}}{2}\\left( -\\dot{x}^{2}+x^{4}\\dot{y}\\right) -V_{0}x^{2}$ $E=\\frac{N_{0}^{2}}{2}\\left( -\\dot{x}^{2}+x^{4}\\dot{y}^{2}\\right) +V_{0}x^{2}$ and the Noether integral is $I=x^{4}\\dot{y}.$ In the new variables, the Euler-Lagrange equations read: $\\ddot{x}+2x^{3}\\dot{y}^{2}-\\frac{2V_{0}}{N_{0}^{2}}x &=&0 \\\\\\ddot{y}+\\frac{4}{x}\\dot{x}\\dot{y} &=&0.",
"$ In this context, from the Noether integral, we have $\\dot{y}=\\frac{I}{x^{4}} $ which, upon substitution in the field equations, gives the system: $\\ddot{x}+\\frac{2I^{2}}{x^{5}} -\\frac{2V_{0}}{N_{0}^{2}}x&=&0 \\\\\\frac{N_{0}^{2}}{2}\\left( -\\dot{x}^{2}+\\frac{I^{2}}{x^{4}}\\right)+V_{0}x^{2} &=&E.",
"$ from which we compute $\\dot{x}=\\sqrt{\\frac{I^{2}}{x^{4}}+\\frac{2V_{0}}{N_{0}^{2}}x^{2}-\\frac{2E}{N_{0}^{2}}} $ and the analytical solution $\\int \\frac{dx}{\\sqrt{\\frac{I^{2}}{x^{4}}+\\frac{2V_{0}}{N_{0}^{2}}x^{2}-\\frac{2E}{N_{0}^{2}}}}=t-t_{0}.$ Also, integrating Eq.",
"(REF ), we find $y\\left( t\\right) -y_{0}=\\int \\frac{I}{x^{4}}dt.",
"$ If we consider the special case where $I=0$ then the analytic solution is $x=x_{0}\\sinh \\left( \\frac{\\sqrt{2V_{0}}}{N_{0}}t+x_{1}\\right) ,~y=y_{0}$ with the Hamiltonian constrain $E=-x_{0}^{2}V_{0}.$ Finally, if $V_{0}=0~$ (i.e.",
"free particle) and $I=0$ the analytic solution becomes $x=x_{0}t+x_{1}~,~y=y_{0}$ with Hamiltonian constrain $E=-\\frac{1}{2}x_{0}^{2}N_{0}^{2}$ ."
],
[
"Conclusions",
"In this work we have identified the Noether point symmetries of the equations of motion in the context of scalar-tensor cosmology considering a 2-dimensional minisuperspace ${\\cal Q}=\\lbrace \\psi , a\\rbrace $ .",
"We find that there is a rather large class of hyperbolic and exponential potentials which admit extra (beyond the $\\partial _{t}=0$ ) Noether symmetries which lead to integral of motions.",
"This approach is extremely efficient in physical problems since it can be utilized in order to simplify a given system of differential equations as well as to determine the integrability of the system.",
"Based on the above arguments, we manage to provide general analytical solutions in scalar-tensor cosmologies assuming a FRW spatially flat metric.",
"These solutions can be used in order to compare cosmographic parameters, such as the Hubble expansion rate, the deceleration parameter, snap, jerk and density parameters with observations [53].",
"Such an analysis is in progress and it will be published in a forthcoming paper."
],
[
"Acknowledgments",
"SB acknowledges support by the Research Center for Astronomy of the Academy of Athens in the context of the program “Tracing the Cosmic Acceleration”."
],
[
"Maximally symetric space: the case $|k|=1$",
"In order to complete Sec.3, we provide here the main steps of the Noether algebra in the case where $k=\\pm 1$ .",
"Briefly, we start with the KVs of the kinematic metric $K^{1}=\\frac{1}{N_{0}^{2}}\\frac{e^{-2k\\theta }}{r}\\left(k \\partial _{r}+\\frac{1}{r}\\partial _{\\theta }\\right),~K^{2}=\\frac{1}{N_{0}^{2}}\\left(-kr\\partial _{r}+\\partial _{\\theta }\\right),$ $K^{3}=-r\\left[ \\ln \\left(re^{-k\\theta }\\right) -1\\right] \\partial _{r}+\\ln \\left( re^{-k\\theta }\\right)\\partial _{\\theta }$ where the vectors $K^{1,2}$ are gradient and $K^{3}$ is non-gradient.",
"Also the HV is given by $H^{i}=\\frac{1}{4}r\\left[ 2\\ln \\left( re^{-k\\theta }\\right) +3\\right] \\partial _{r}-\\frac{1}{2}\\left[ \\ln \\left( re^{-k\\theta }\\right)+\\frac{1}{2}\\right]\\partial _{\\theta } \\;.$ Using the theorem in [50], [51] and making some simple calculations (see Sec.",
"3) we find the following results: Noether symmetries generated by the KV $K^{1}$ .",
"a) If $V\\left( \\theta \\right) =V_{0}e^{-2k\\theta }$ then we have the Noether symmetries $K^{1}~,~tK^{1}$ with Noether integrals $I_{1}^{\\prime }=\\frac{d}{dt}\\left(k \\theta -\\ln r\\right),\\;I_{2}^{\\prime }=t\\left[ \\frac{d}{dt}\\left(k\\theta -\\ln r\\right) \\right] -\\left(k \\theta -\\ln r\\right)$ b) If $V\\left( \\theta \\right) =V_{0}e^{-2k\\theta }-\\frac{1}{4}pe^{2k\\theta }\\,$ then we have the Noether symmetries $K^{1}~,~tK^{1}$ with Noether integrals $I_{1}=\\frac{d}{dt}\\left(k \\theta -\\ln r\\right)-pt$ $I_{2}=t\\left[ \\frac{d}{dt}\\left(k \\theta -\\ln r\\right) \\right] -\\left(k \\theta -\\ln r\\right) -\\frac{1}{2}pt^{2}$ Noether symmetries generated by the KV $K^{2}$ .",
"a) If $V\\left( \\theta \\right) =V_{0}e^{2k\\theta }~$ then we have the extra Noether symmetries $K^{2}~,~tK^{2}$ with Noether integrals $J_{1}=\\left[ \\frac{d}{dt}\\left( \\frac{1}{2}e^{2k\\theta }r^{2}\\right) \\right],~J_{2}=t\\left[ \\frac{d}{dt}\\left( \\frac{1}{2}e^{2k\\theta }r^{2}\\right) \\right] -\\frac{1}{2}e^{2k\\theta }r^{2} $ b) If $V\\left( \\theta \\right) =\\left( V_{0}e^{2k\\theta }-\\frac{m}{2}k\\theta e^{2k\\theta }\\right) $ , then we have the Noether symmetries $e^{\\pm \\sqrt{m}t}K^{2}$ with Noether integrals $J_{1,2}^{\\prime }=e^{\\pm \\sqrt{m}t}\\left( \\left[ \\frac{d}{dt}\\left( \\frac{1}{2}e^{2\\theta }r^{2}\\right) \\right] \\mp \\frac{\\sqrt{m}}{2}e^{2\\theta }r^{2}\\right)$ If $V\\left( \\theta \\right) =0$ then the system becomes the free particle (see [50]).",
"To this end the corresponding analytical solutions can be found utilizing the above integrals the arguments of Sec.",
"3 and the new coordinates $(u,v)=(k\\theta -{\\rm ln}r,\\frac{1}{2}{\\rm e}^{2k\\theta }r^{2})$ ."
]
] | 1403.0332 |
[
[
"Comment on \"Experimental Test of Error-Disturbance Uncertainty Relations\n by Weak Measurement\""
],
[
"Abstract In this comment on the paper by F. Kaneda, S.-Y.",
"Baek, M. Ozawa and K. Edamatsu [Phys.",
"Rev.",
"Lett.",
"112, 020402, 2014, arXiv:1308.5868], we point out that the claim of having refuted Heisenberg's error-disturbance relation is unfounded since it is based on the choice of unsuitable and operationally problematical quantifications of measurement error and disturbance.",
"As we have shown elsewhere [PRL 111, 160405, 2013], for appropriate choices of operational measures of error and disturbance, Heisenberg's heuristic relation can be turned into a precise inequality which is a rigorous consequence of quantum mechanics."
],
[
"Comment on “Experimental Test of Error-Disturbance Uncertainty Relations by Weak Measurement” Paul Busch paul.busch@york.ac.uk Department of Mathematics, University of York, York, United Kingdom Pekka Lahti pekka.lahti@utu.fi Turku Centre for Quantum Physics, Department of Physics and Astronomy, University of Turku, FI-20014 Turku, Finland Reinhard F. Werner reinhard.werner@itp.uni-hannover.de Institut für Theoretische Physik, Leibniz Universität, Hannover, Germany This paper [1] is the latest in a long series of theoretical and experimental works purporting to have demonstrated a violation of Heisenberg's so-called error-disturbance uncertainty relation.",
"This claim, which originated with theoretical work of Ozawa around 2002, has stirred a considerable media hype since 2012, which would be justified if the claim were correct.",
"But it is not, for the following reasons.",
"One has to recall at this point that Heisenberg's original discussion of the $\\gamma $ -ray microscope, in keeping with the aims he states in his introduction, is intended only as a heuristic tool.",
"He is intentionally vague, and states his relations not as inequalities but using a mathematically unexplained tilde.",
"Although he promises a proof to be given later in the paper on the basis of his commutation relations he never gave one, nor ever stated his relation in a form sufficiently precise to even begin thinking of a proof.",
"This is why finding precise quantum mechanical counterparts of Heisenberg's heuristic error and disturbance is not at all straightforward.",
"But it is possible: We have given a natural definition of error and disturbance for which we proved an inequality of the usual form [2], [3].",
"This puts the error-disturbance tradeoff on the same level of quantitative rigour and generality as the usual uncertainty relation for the variances of position and momentum in the same state.",
"What Ozawa calls “Heisenberg's error-disturbance relation” – inequality (1) in [1] – is superficially of the same form, but he chooses different formal definitions of error and disturbance, which can be traced to the work of Arthurs and Kelly in the 60s [4].",
"We have shown [5], [6] that these definitions have serious conceptual deficiencies undermining their interpretation as error and disturbance.",
"Moreover, it has been shown by simple counterexamples [7], long before Ozawa, that with these definitions a general uncertainty relation does not hold.",
"This unspectacular observation has now repeatedly been verified experimentally with [1] the latest in the series.",
"Similarly, groups in Toronto [8] and Brisbane [9] have reported experiments violating the inequality that Ozawa wrongly attributes to Heisenberg.",
"[In contrast to the Toronto group, the Brisbane researchers have not adopted this attribution.]",
"But these experimental results do not help to refute Heisenberg's heuristics or the general idea of a quantitative error-disturbance tradeoff.",
"They only show that the definitions for error and disturbance chosen by Ozawa, in addition to their intrinsic problems, are not suitable for the task of expressing such a tradeoff.",
"Ozawa has provided some additional terms, which turn his false inequality (1) into a correct one (Eq.",
"(3) of [1]).",
"This inequality has recently been tightened in an interesting paper by Branciard [10].",
"Both of these inequalities have been confirmed in some of the experiments mentioned.",
"In a recent letter [11] Ozawa and his coauthors have also changed to a state-independent definitions more like ours.",
"These are positive contributions to the research field of rigorous measurement uncertainty relations.",
"We hope that in the future this field can concentrate on its scientific agenda and leave behind the misrepresentations by Ozawa's “refuting Heisenberg” campaign."
]
] | 1403.0367 |
[
[
"Noether Symmetries of Some Homogeneous Universe Models in Curvature\n Corrected Scalar-Tensor Gravity"
],
[
"Abstract We explore Noether gauge symmetries of FRW and Bianchi I universe models for perfect fluid in scalar-tensor gravity with extra term $R^{-1}$ as curvature correction.",
"Noether symmetry approach can be used to fix the form of coupling function $\\omega(\\phi)$ and the field potential $V(\\phi)$.",
"It is shown that for both models, the Noether symmetries, the gauge function as well as the conserved quantity, i.e., the integral of motion exist for the respective point like Lagrangians.",
"We determine the form of coupling function as well as the field potential in each case.",
"Finally, we investigate solutions through scaling or dilatational symmetries for Bianchi I universe model without curvature correction and discuss its cosmological implications."
],
[
"Introduction",
"The existence of dark energy (DE) and its role on the expansion history of the universe has become a center of interest for the researchers.",
"It is a mysterious type of energy having negative pressure that is believed to be a basic reason for the rapid expanding behavior of the universe [1], [2].",
"For the description of its cryptical nature, there are mainly two kinds of efforts: modified matter approach like quintessence, Chaplygin gas, phantom, quintom, tachyon etc.",
"[3] and the modified gravity (due to some extra degrees of freedom) including $f(R)$ gravity, scalar-tensor theory, $f(T)$ gravity etc.",
"[4].",
"Although the modified matter approach has many novel features but this is not fully free from ambiguities.",
"The modified gravity approach is considered to be more appropriate in this respect.",
"The dominant presence of DE in the universe leads to numerous theoretical problems like cosmic coincidence and fine-tuning problems [5].",
"Scalar-tensor theories are proved to be important efforts in the investigation of DE problem as well as various cosmic issues like the early and late time behavior of the universe and inflation [6].",
"The phenomenon of cosmic acceleration can be better described by introducing some sub-dominant terms of geometric origin like inverse of the Ricci scalar in the Einstein-Hilbert action.",
"The simplest action with such modification is defined as [7] $S=\\frac{1}{8\\pi G}\\int \\sqrt{-g}(R-\\frac{\\mu _0^4}{R})d^4x,$ where $R$ is the Ricci scalar, $G$ is the gravitational constant and $\\mu _0$ is an arbitrary non-zero constant.",
"In order to be consistent with observations and physical constraints, the action of scalar-tensor theories, in particular, Brans-Dicke (BD) theory can be modified in the following form [8] $S=\\int [\\phi (R-\\frac{\\mu _0^4}{R})+\\frac{\\omega (\\phi )}{\\phi }g^{\\mu \\nu }\\nabla _\\mu \\phi \\nabla _\\nu \\phi -V(\\phi )+L_m]\\sqrt{-g}d^4x,$ where $\\phi $ is the scalar field, $\\omega (\\phi )$ is the BD coupling function, $V(\\phi )$ is the field potential, $L_m$ is the matter part of the Lagrangian and $\\nabla _\\mu $ indicates the covariant derivative.",
"In the cosmological contexts, there are two types of Noether symmetry techniques available in literature [9].",
"Symmetries which are obtained by setting the Lie derivative of the Lagrangian to zero are called Noether symmetries.",
"The second technique is related with the more general symmetries known as Noether gauge symmetries (containing the Noether symmetries as a subcase) which involve non-zero gauge function.",
"Noether symmetries have many significant applications in cosmology and theoretical physics.",
"In particular, the existence of Noether symmetries leads to a specific form of coupling function and the field potential in scalar-tensor theories.",
"Physically, symmetries lead to the existence of conserved quantities while on mathematical grounds, these reduce dynamics of the system due to the presence of cyclic variables [10].",
"Using Noether symmetry technique, the homogeneous universe models like FRW and Bianchi models have been discussed in $f(R)$ and scalar-tensor gravity [11].",
"Motavali and Golshani [12] explored the form of coupling function and the field potential for FRW universe model using Noether symmetries.",
"Camci and Kucukakca [13] evaluated Noether symmetries for Bianchi I, III and Kantowski-Sachs spacetimes and discussed some field potentials.",
"In recent papers [14], we have explored approximate Lie and Noether symmetries of some black holes and colliding plane waves in the framework of GR.",
"Kucukakca and Camci [15] have obtained the function $f(R)$ and the scale factor using Noether symmetry approach in Palatini $f(R)$ theory.",
"Capozziello et al.",
"[16] have discussed non-static spherically symmetric solutions in $f(R)$ gravity via Noether symmetry analysis.",
"Shamir et al.",
"[17] have investigated Noether symmetries and the respective conserved quantities for FRW and general static spherically symmetric spacetimes in $f(R)$ gravity.",
"Jamil et al.",
"[18] have discussed Noether gauge symmetries and the respective conserved quantities with different forms of potential for Bianchi I (BI) universe model in generalized Saez-Ballester scalar-tensor gravity.",
"Kucukakca et al.",
"[19] have explored BI universe model through Noether symmetry analysis with degeneracy condition of the Lagrangian and concluded that their results are consistent with the observations.",
"Motavali et al.",
"[8] calculated the Noether symmetries of the Lagrangian with an extra curvature term for FRW universe model.",
"Consider the point transformations (invertible transformations of “generalized positions\") that depend only upon one infinitesimal parameter $\\sigma $ , i.e., $Q^i=Q^i(q^j,\\sigma )$ which can generate one-parameter Lie group [14], [16], [18].",
"The vector field with unknowns $\\alpha ^i$ defined by $\\nonumber \\textbf {X}=\\alpha ^i(q^j)\\frac{\\partial }{\\partial q^i}+[\\frac{d}{d\\lambda }(\\alpha ^i(q^j))]\\frac{\\partial }{\\partial \\dot{q}^i}$ is said to be a Noether symmetry for the dynamics derived by the Lagrangian if it leaves the Lagrangian invariant, that is, $L_X\\mathcal {L}=0$ .",
"In this case, the Euler-Lagrange equations and and the constant of motion can be written as $\\frac{d}{d\\lambda }(\\frac{\\partial \\mathcal {L}}{\\partial \\dot{q^i}})-\\frac{\\partial \\mathcal {L}}{\\partial q^i}=0,\\quad \\vartheta =\\alpha ^i\\frac{\\partial \\mathcal {L}}{\\partial \\dot{q^i}}.$ Noether gauge symmetries are the generalization of Noether symmetries (as it is expected that they contain some extra symmetries).",
"Consider a vector field X as $\\textbf {X}=\\tau (t,q^{i})\\frac{\\partial }{\\partial t}+\\eta ^{j}(t,q^{i})\\frac{\\partial }{\\partial q^{j}}$ and its first-order prolongation is defined as $\\nonumber \\textbf {X}^{[1]}=\\textbf {X}+(\\eta ^{j}_{,t}+\\eta ^{j}_{,i}\\dot{q}^{i}-\\tau _{,t}\\dot{q}^{j}-\\tau _{,i}\\dot{q}^{i}\\dot{q}^{j})\\frac{\\partial }{\\partial \\dot{q}^{j}}.$ Here $\\tau $ and $\\eta ^j$ are the unknown functions to be determined and $t$ is the affine parameter.",
"The vector field $\\textbf {X}$ is said to be Noether gauge point symmetry of the Lagrangian $\\mathcal {L}(t,q^{i},\\dot{q}^{i})$ , if there exists a function (known as gauge term) $G(t,q^{i})$ such that the following condition is satisfied $\\textbf {X}^{[1]}L+(D_{t}\\tau )L=D_{t}G;\\quad D_{t}=\\frac{\\partial }{\\partial t}+\\dot{q}^{i}\\frac{\\partial }{\\partial q^{i}}.$ Here $D_t$ is the total derivative operator.",
"There are two physical frames available in literature: Einstein and Jordan frames which are related with each other by a conformal transformation ($\\widetilde{g}=e^{2\\Omega }g$ ).",
"It is argued that both these frames are equivalent on mathematical as well as physical grounds in the classical gravity regime where the conformal mapping is well defined.",
"The compatibility of the Noether symmetries and the conformal transformations have been discussed in literature [20].",
"It is proved that the Noether point symmetry if exists, it remains preserved under the conformal transformations.",
"In the present paper, we evaluate Noether gauge symmetries of the non-vacuum point like Lagrangian for FRW universe model and then extend to locally rotationally symmetric (LRS) BI universe model, the simplest generalization of FRW universe.",
"The paper is designed in the following manner.",
"In section 2, we evaluate Noether gauge symmetries for FRW universe model with correction term.",
"Section 3 provides Noether as well as Noether gauge symmetries for BI universe model with correction term.",
"In section 4, we discuss BI solutions using scaling or dilatation symmetries without correction term.",
"Finally, we present an outlook in the last section."
],
[
"Noether Gauge Symmetries for FRW Universe Model",
"For the sake of simplicity, we take $\\phi =\\varphi ^2$ and $\\mu _0^4=-\\mu $ .",
"Thus the action for scalar-tensor gravity with extra curvature term (REF ) can be written as $S=\\int [\\varphi ^2(R+\\frac{\\mu }{R})+4\\omega (\\varphi )g^{\\mu \\nu }\\nabla _\\mu \\varphi \\nabla _\\nu \\varphi -V(\\varphi )+L_m]\\sqrt{-g}d^4x.$ The homogeneous, non-flat FRW universe model is given by $ds^2=dt^2-a^2(t)[\\frac{dr^2}{1-kr^2}+r^2(d\\theta ^2+\\sin ^2\\theta d\\varphi ^2)],$ where $a(t)$ is the scale factor and $k(=0,\\pm 1)$ is the curvature index.",
"The matter part of the Lagrangian is described by the perfect fluid $T_{\\mu \\nu }=(\\rho +P)u_\\mu u_\\nu -Pg_{\\mu \\nu },$ where $\\rho ,~P$ and $u_\\mu $ denote the energy density, pressure and four velocity, respectively.",
"The equation of state (EoS) for perfect fluid is $P=\\epsilon \\rho $ , where $\\epsilon $ is the EoS parameter.",
"The energy conservation leads to $\\rho =\\rho _0a^{-3(1+\\epsilon )}$ and hence the pressure becomes $P=\\epsilon \\rho _0a^{-3(1+\\epsilon )}$ .",
"We are interested in the Noether gauge symmetries (non-zero guage function) of FRW model with perfect fluid matter contents.",
"For this purpose, the point like Lagrangian constructed by the partial integration of the action (REF ) is [8] $\\nonumber \\mathcal {L}&=&2a^3\\varphi ^2\\mu q+6(\\mu q^2-1)(2a^2\\varphi \\dot{a}\\dot{\\varphi }+\\varphi ^2a\\dot{a}^2)+12\\mu \\varphi ^2a^2q\\dot{a}\\dot{q}-6ka\\varphi ^2(\\mu q^2\\\\&-&1)+a^3(4\\omega (\\varphi )\\dot{\\varphi }^2-V(\\varphi ))+\\rho _0\\epsilon a^{-3\\epsilon }.$ In this case, the configuration space is given by $(t,a,\\varphi ,q)$ , consequently the Lagrangian is defined as $\\mathcal {L}:TQ\\rightarrow \\mathbb {R}$ , where $TQ=(t,a,\\varphi ,q,\\dot{a},\\dot{\\varphi },\\dot{q})$ is the respective tangent space and $\\mathbb {R}$ is the set of real numbers.",
"The first-order prolongation of the symmetry generator is given by $\\nonumber X^{[1]}&=&\\tau (t,a,\\varphi ,q)\\frac{\\partial }{\\partial t}+\\alpha (t,a,\\varphi ,q)\\frac{\\partial }{\\partial a}+\\beta (t,a,\\varphi ,q)\\frac{\\partial }{\\partial \\varphi }+\\gamma (t,a,\\varphi ,q)\\frac{\\partial }{\\partial q}\\\\\\nonumber &+&\\alpha _{t}(t,a,\\varphi ,q)\\frac{\\partial }{\\partial \\dot{a}}+\\beta _{t}(t,a,\\varphi ,q)\\frac{\\partial }{\\partial \\dot{\\varphi }}+\\gamma _{t}(t,a,\\varphi ,q)\\frac{\\partial }{\\partial \\dot{q}},$ where $\\alpha ,~\\beta $ and $\\gamma $ are unknown functions to be determined.",
"Moreover, $\\nonumber \\alpha _{t}=D_{t}\\alpha -\\dot{a}D_{t}\\tau ,\\quad \\beta _{t}=D_{t}\\beta -\\dot{\\phi }D_{t}\\tau , \\quad \\gamma _{t}=D_{t}\\gamma -\\dot{q}D_{t}\\tau ,$ where $\\nonumber D_{t}=\\frac{\\partial }{\\partial t}+\\dot{a}\\frac{\\partial }{\\partial a}+\\dot{\\varphi }\\frac{\\partial }{\\partial \\varphi }+\\dot{q}\\frac{\\partial }{\\partial q}.$ Substituting these values with Eq.",
"(REF ) in Eq.",
"(REF ), we get the following system of determining equations $&&\\tau _{q}=0,\\quad \\tau _{a}=0,\\quad \\tau _{\\varphi }=0,\\\\\\nonumber &&12(\\mu q^2-1)a^2\\varphi \\alpha _{t}+8a^3\\omega (\\varphi )(t)\\beta _{t}+\\tau _{\\varphi }(2a^3\\varphi ^2\\mu q-a^3V(\\varphi )\\\\&&-6ka\\varphi ^2(\\mu q^2-1))=G_{\\varphi },\\\\\\nonumber &&12(\\mu q^2-1)a\\varphi ^2\\alpha _{t}+12(\\mu q^2-1)a^2\\varphi \\beta _{t}+12\\mu qa^2\\varphi ^2\\gamma _{t}+(2a^3\\varphi ^2\\mu q\\\\&&-a^3V(\\varphi )-6ka\\varphi ^2(\\mu q^2-1))\\tau _{a}=G_{a},\\\\\\nonumber && 12(\\mu q^2-1)a^2\\varphi ^2\\alpha _{t}+(2a^3\\varphi ^2\\mu q-a^3V(\\varphi )-6ka\\varphi ^2(\\mu q^2-1))\\tau _{q}=G_{q},\\\\\\\\\\nonumber &&6a^2\\varphi ^2\\mu q\\alpha -6k\\varphi ^2(\\mu q^2-1)\\alpha -3a^2\\alpha V(\\varphi )+4a^3q\\mu \\varphi \\beta -12ka\\beta \\varphi (\\mu q^2\\\\\\nonumber &&-1)-a^3\\beta V^{\\prime }(\\varphi )+2a^3\\mu \\varphi ^2\\gamma -12k\\mu \\varphi ^2qa\\gamma +(2a^3q\\mu \\varphi ^2-6ka\\varphi ^2(\\mu q^2\\\\&&-1)-a^3V(\\varphi ))\\tau _{t}-3\\rho _0\\epsilon ^2a^{-(1+3\\epsilon )}\\alpha =G_{t},\\\\\\nonumber &&24a\\alpha \\varphi (\\mu \\mu q^2-1)+12a^2\\beta (\\mu q^2-1)+24a^2\\varphi \\mu q\\gamma +12(\\mu q^2-1)a^2\\varphi \\alpha _{a}\\\\\\nonumber &&+12(\\mu q^2-1)a\\varphi ^2\\alpha _{\\varphi }+12a\\varphi (\\mu q^2-1)\\beta _{\\varphi }+12\\mu qa^2\\varphi ^2\\gamma _{\\varphi }+8a^3\\omega (\\varphi )\\beta _{a}\\\\&&-12(\\mu q^2-1)a^2\\varphi \\tau _{t}=0,\\\\\\nonumber &&6(\\mu q^2-1)\\alpha \\varphi ^2+12a\\beta \\varphi (\\mu q^2-1)+12\\mu qa\\varphi ^2\\gamma +12(\\mu q^2-1)a\\varphi ^2\\alpha _{a}\\\\&&+12(\\mu q^2-1)a^2\\varphi \\beta _{a}+12\\mu qa^2\\varphi ^2\\gamma _{a}-6a\\varphi ^2(\\mu q^2-1)\\tau _{t}=0,\\\\\\nonumber &&24\\mu qa\\varphi ^2\\alpha +24\\mu qa^2\\varphi \\beta +12\\mu a^2\\varphi ^2\\gamma +12\\mu qa^2\\varphi ^2\\alpha _{a}+12a\\varphi ^2(\\mu q^2-1)\\alpha _{q}\\\\&&+12a^2\\varphi (\\mu q^2-1)\\beta _{q}+12\\mu qa^2\\varphi ^2\\gamma _{q}-12\\mu qa^2\\varphi ^2\\tau _{t}=0,\\\\\\nonumber &&12a^2\\omega (\\varphi )\\alpha +4a^3\\beta \\omega ^{\\prime }(\\varphi )+12a^2\\varphi (\\mu q^2-1)\\alpha _{\\varphi }+8a^3\\omega (\\varphi )\\beta _{\\varphi }\\\\&&-4a^3\\omega (\\varphi )\\tau _{t}=0,\\\\ &&12\\mu qa^2\\varphi ^2\\alpha _{\\varphi }+12(\\mu q^2-1)a^2\\varphi \\alpha _{q}+8a^3\\omega (\\varphi )\\beta _{q}=0,\\\\&& 12\\mu qa^2\\varphi ^2\\alpha _{q}=0,$ where $G=G(t,a,\\varphi ,q)$ .",
"This is a system of 11 partial differential equations (PDEs) which we solve simultaneously for the unknown functions ($\\tau ,~\\alpha ,~\\beta ,~\\gamma ,~G$ ).",
"The coupling function and the field potential both are also unknown and we specify their forms by the existence of Noether symmetries.",
"Integration of Eq.",
"() implies $\\alpha =\\alpha _1(t,a,\\varphi )$ .",
"Since the above system of PDEs is difficult to solve, therefore we take the ansatze for the functions $\\alpha _1$ and $\\beta $ as $\\alpha _1=\\alpha _0t^{n_1}a^n\\varphi ^m,\\quad \\beta =\\beta _0(q)t^{l_1}a^l\\varphi ^s,$ where $\\beta _0$ is an arbitrary function and $\\alpha _0,~n,~m,~n_1,~l,~l_1,~s$ are the parameters to be determined.",
"Substituting these values in Eq.",
"(), it follows $\\nonumber \\beta _0(q)=-\\frac{3}{4}\\mu (\\frac{m\\alpha _0}{\\omega _0})q^2+c_1, \\quad \\omega (\\varphi )=\\omega _0\\varphi ^{m-s+1},\\quad n=l+1, \\quad n_1=l_1,$ where $c_1$ and $\\omega _0$ are constants.",
"Equation () leads to $\\nonumber \\tau =c_2,\\quad s=m+1,\\quad \\omega _0=1+\\frac{4c_1}{3\\alpha _0},\\quad m=1.$ Consequently, Eq.",
"(REF ) takes the form $\\nonumber \\alpha _1=\\alpha _0a^n\\varphi t^{n_1},\\quad \\beta =(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)\\varphi ^2a^{n-1}t^{n_1}.$ Equation () implies that $\\nonumber \\gamma (t,a,\\varphi ,q)=f(q)a^{n-1}t^{n_1}\\varphi +\\frac{g_1(t,a,\\varphi )}{q},$ where $f(q)=\\frac{3\\alpha _0}{2\\omega _0}(\\frac{\\mu q^3}{4}-\\frac{q}{2})-q\\alpha _0+\\frac{3\\mu \\alpha _0q^3}{8\\omega _0}-qc_1-\\frac{\\alpha _0nq}{2}$ and $g_1$ is an integration function.",
"Inserting these values in Eqs.",
"() and (), it follows that $\\gamma =f(q)a^{n-1}t^{n_1}\\varphi +\\frac{g_3(t)}{aq\\varphi ^2},$ where $g_3$ is an integration function.",
"Moreover, the following constraints should be satisfied $&&nf(q)=\\frac{1-\\mu q^2}{2\\mu q}[{\\alpha _0(1+2n)+2n(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)}],\\\\\\nonumber &&(\\mu q^2-1)[(3+n)\\alpha _0+3(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)]+\\frac{2\\omega _0}{3}(n-1)(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)\\\\&&+3\\mu qf=0.$ Integration of Eq.",
"() yields $\\nonumber G(t,a,\\varphi ,q)=12n_1\\alpha _0\\varphi ^3t^{n_1-1}a^{n+2}(\\frac{\\mu q^3}{3}-q)+h_1(t,a,\\varphi ),$ where $h_1$ is an integration function.",
"Further, Eqs.",
"() and () lead to $\\nonumber G=12n_1\\alpha _0q(\\frac{\\mu q^2}{3}-1)a^{n+2}\\varphi ^3t^{n_1-1}+6a^2\\mu g_{3,t}+h_3(t)$ with the constraints $\\nonumber &&12(\\mu q^2-1)\\alpha _0n_1+8\\omega _0(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)n_1-36n_1\\alpha _0q(\\frac{\\mu q^2}{3}-1)=0,\\\\\\\\\\nonumber &&12(\\mu q^2-1)\\alpha _0n_1+12(\\mu q^2-1)n_1(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)+12\\mu qfn_1\\\\&&=12n_1(n+2)\\alpha _0q(\\frac{\\mu q^2}{3}-1).$ Finally, Eq.",
"() yields $\\nonumber &&6\\mu q\\alpha _0a^{n+2}\\varphi ^3t^{n_1}-6k(\\mu q^2-1)\\alpha _0a^n\\varphi ^3t^{n_1}-3V(\\varphi )\\alpha _0a^{n+2}\\varphi t^{n_1}+4\\mu q(c_1\\\\\\nonumber &&-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)a^{n+2}\\varphi ^3-12k(\\mu q^2-1)(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)a^n\\varphi ^3t^{n_1}-a^{n+2}\\varphi ^2t^{n_1}(c_1\\\\\\nonumber &&-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)\\frac{dV}{d\\varphi }+2\\varphi ^3a^{n+2}f\\mu t^{n_1}+\\frac{2a^2\\mu g_3(t)}{q}-12kq\\mu a^{n}t^{n_1}f\\varphi ^3\\\\\\nonumber &&-12k\\mu g_{3}(t)-3\\rho _0\\epsilon ^2\\alpha _0\\varphi t^{n_1}a^{-(1+3\\epsilon )+n}=12n_1(n_1-1)\\alpha _0q(\\frac{\\mu q^2}{3}-1)\\\\\\nonumber &&a^{n+2}\\varphi ^3t^{n_1-2}+6a^2\\mu g_{3,tt}+h_{3,t}.$ This equation will be satisfied if $n_1=1$ with the following constraints $&&-12k\\mu g_3(t)+2a^2\\mu g_3(t)=6a^2\\mu g_{3,tt}+h_{3,t},\\\\&&-\\frac{3\\alpha _0V(\\varphi )}{\\varphi ^2}-\\frac{1}{\\varphi }\\frac{dV}{d\\varphi }=0,\\\\\\nonumber &&6\\alpha _0q\\mu -6k(\\mu q^2-1)\\alpha _0a^{-2}+4q\\beta _0\\mu -12k(\\mu q^2-1)a^{-2}\\\\&&-12kqfa^{-2}\\mu -\\frac{3\\rho _0\\epsilon ^2a^{-3(1+\\epsilon )\\alpha _0}}{\\varphi ^2}=0.$ Integration of Eqs.",
"(REF ) and () yields $\\nonumber &&g_{3}(t)=c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t),\\\\\\nonumber &&h_3(t)=-12k\\mu \\sqrt{\\frac{\\mu }{3}}[c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)]+c_5,\\\\\\nonumber &&V(\\varphi )=c_6\\varphi ^{3\\alpha _0},$ where $c_3,~c_4,~c_5$ and $c_6$ are constants of integration.",
"Now we can discuss Eq.",
"() for the following two cases, i.e., when $\\epsilon =0$ or $\\alpha _0=0$ .",
"If $\\epsilon =0$ , then pressure becomes zero and matter distribution will be the dust dominated fluid.",
"Moreover, Eq.",
"() leads to the following constraints $\\nonumber &&6q\\alpha _0\\mu +4q\\beta _0\\mu +2f\\mu =0,\\\\ &&-6k(\\mu q^2-1)\\alpha _0-12k(\\mu q^2-1)-12kfq\\mu =0.$ In this case, the solution turns out to be $\\nonumber &&\\alpha =\\alpha _1=\\alpha _0a^n\\varphi t,\\quad \\beta =(c_1-\\frac{3\\alpha _0\\mu q^2}{4\\omega _0})\\varphi ^2a^{n-1}t,\\quad \\tau =c_2,\\quad V=c_6\\varphi ^{3\\alpha _0},\\\\\\nonumber &&\\gamma =(\\frac{3\\alpha _0}{2\\omega _0}(\\frac{\\mu q^3}{4}-\\frac{q}{2})-q\\alpha _0+\\frac{3\\mu \\alpha _0q^3}{8\\omega _0}-qc_1-\\frac{\\alpha _0nq}{2})a^{n-1}t\\varphi \\\\\\nonumber &&+\\frac{c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)}{aq\\varphi ^2},\\\\\\nonumber &&G=12q\\alpha _0(\\frac{\\mu q^2}{3}-1)a^{n+2}\\varphi ^3+\\frac{6a^2\\mu ^{3/2}}{\\sqrt{3}}[c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)-c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)]\\\\\\nonumber &&-12k\\mu \\sqrt{\\frac{\\mu }{3}}[c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)]+c_5.$ Consequently, the symmetry generator is $\\nonumber \\textbf {X}&=&c_2\\frac{\\partial }{\\partial t}+\\alpha _0a^n\\varphi t\\frac{\\partial }{\\partial a}+(c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)\\varphi ^2a^{n-1}t\\frac{\\partial }{\\partial \\varphi }+(f(q)a^{n-1}t\\varphi \\\\\\nonumber &+&\\frac{c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)}{aq\\varphi ^2})\\frac{\\partial }{\\partial q}.$ The corresponding conserved quantity becomes $\\nonumber I&=&c_2(2a^3\\varphi ^2\\mu q+6(\\mu q^2-1)(2a^2\\varphi \\dot{a}\\dot{\\varphi }+\\varphi ^2a\\dot{a}^2)+12\\mu \\varphi ^2a^2q\\dot{a}\\dot{q}-6ka\\varphi ^2(\\mu q^2\\\\\\nonumber &&-1)+a^3(4(1+\\frac{4c_1}{3\\alpha _0})\\dot{\\varphi }^2-c_6\\varphi ^{3\\alpha _0})+(\\alpha _0a^n\\varphi t-c_2\\dot{a})(6(\\mu q^2-1)(2a^2\\varphi \\dot{\\varphi }\\\\\\nonumber &&+2a\\varphi ^2\\dot{a})+12\\mu qa^2\\varphi ^2\\dot{q})+((c_1-\\frac{3\\alpha _0\\mu }{4\\omega _0}q^2)\\varphi ^2a^{n-1}t-c_2\\dot{\\varphi })(12(\\mu q^2-1)a^2 \\\\\\nonumber &&\\varphi \\dot{a}+8a^3\\dot{\\varphi }\\omega _0)+(f(q)a^{n-1}t\\varphi +\\frac{1}{aq\\varphi ^2}(c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t))\\\\\\nonumber &&-c_2\\dot{q})(12\\mu \\varphi ^2a^2q\\dot{a})-12q\\alpha _0(\\frac{\\mu q^2}{3}-1)a^{n+2}\\varphi ^3-\\frac{6a^2\\mu ^{3/2}}{\\sqrt{3}}[c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)\\\\\\nonumber &&-c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)]+12k\\mu \\sqrt{\\frac{\\mu }{3}}[c_{3}\\exp (\\sqrt{\\frac{\\mu }{3}}t)+c_4\\exp (-\\sqrt{\\frac{\\mu }{3}}t)]-c_5.$ For the flat universe, the constraint () restricts $q$ to be constant say $q_0$ as follows $\\nonumber q^2=q_0^2=\\frac{(3+n)\\alpha _0+c_1-2/3(n-1)\\omega _0c_1}{(1-3n)\\mu }.$ Equations (REF ), (REF ), () and (REF ) are four constraints that can be used to restrict the parameters $\\mu ,~\\alpha _0,~c_1$ and $n$ .",
"Notice that $q=1/R$ , where $R$ is the Ricci scalar, which turns out to be constant, i.e., $R=1/q_0$ .",
"This is in agreement with Noether theorem according to which, when a cyclic variable is identified, Noether symmetry appears and, in the present case, the combination $R=1/q_0$ is constant which corresponds to constant scalar curvature solution.",
"Its physical meaning is that the Noether symmetry generator exists for the solutions with constant curvature like de Sitter solutions.",
"It is found that Noether symmetry generator exists for $\\epsilon =0$ and the respective gauge function turns out to be a dynamical quantity.",
"Moreover, the potential is a dynamical quantity given by a power law form while the BD coupling is a constant quantity.",
"The behavior of the field potential depends upon the constant $\\alpha _0$ (for $\\alpha _0>0$ , the field potential behaves as a positive power law while $\\alpha _0<0$ leads to inverse power law potential).",
"Such field potentials have been used to discuss many cosmological issues in literature [22].",
"The existence of Noether gauge symmetries yields the conserved quantity, i.e., Noether charge exists which can be used to reduce the complexity of the Euler-Lagrange equations.",
"For the second case ($\\alpha _0=0$ ), the field potential turns out to be constant, i.e., $V_0=c_6$ and $\\alpha =0$ , also, $\\omega $ is diverging, i.e., $\\omega \\rightarrow \\infty $ , hence we neglect this choice."
],
[
"Noether Gauge Symmetries for LRS Bianchi I Universe Model",
"Here, we calculate the Noether and Noether gauge symmetries of the LRS BI spacetime.",
"The LRS BI universe with scale factors $A$ and $B$ is defined by the line element [21] $ds^2=dt^2-A^2(t)dx^2-B^2(t)(dy^2+dz^2).$ The dynamical constraint evaluated in terms of the Ricci scalar follows $\\nonumber R-2[\\frac{\\ddot{A}}{A}+2\\frac{\\ddot{B}}{B}+\\frac{\\dot{B}^2}{B^2}+2\\frac{\\dot{A}\\dot{B}}{AB}]=0.$ Using the Lagrange multiplier approach, the action can be written as $\\nonumber S&=&\\int [\\varphi ^2(R+\\frac{\\mu }{R})+4\\dot{\\varphi }^2\\omega (\\varphi )-V(\\varphi )+\\chi (R-2\\frac{\\dot{A}}{A}-4\\frac{\\ddot{B}}{B}-2\\frac{\\dot{B}^2}{B^2}\\\\&-&4\\frac{\\dot{A}\\dot{B}}{AB})+L_m](AB^2)d^4x.$ Here $\\chi $ is the Lagrange multiplier parameter.",
"Varying this action with respect to the Ricci scalar, the parameter $\\chi $ turns out to be $\\chi =\\varphi ^2(\\mu R^{-2}-1).$ The matter part of the Lagrangian is described by the perfect fluid (as defined in the previous section).",
"We consider the matter dominated universe for which the matter part of the Lagrangian is given by $\\mathcal {L}_m=\\rho _0(AB^2)^{-1}$ .",
"By substituting the respective values in the action (REF ), it follows $\\nonumber S&=&\\int [2q\\mu AB^2\\varphi ^2+4AB^2\\omega \\dot{\\varphi }^2-AB^2V-\\varphi ^2(\\mu q^2-1)(2A\\dot{B}^2+4B\\dot{A}\\dot{B})\\\\\\nonumber &-&\\varphi ^2(\\mu q^2-1)(2\\ddot{A}B^2+4AB\\ddot{B})+\\rho _0]dt$ The partial integration of this equation provides the canonical point like form of the Lagrangian as $\\nonumber \\mathcal {L}&=&2\\mu qAB^2\\varphi ^2+4AB^2\\omega (\\varphi )\\dot{\\varphi }^2-AB^2V+2\\varphi ^2(\\mu q^2-1)A\\dot{B}^2+4B^2\\varphi (\\mu q^2\\\\\\nonumber &-&1)\\dot{A}\\dot{\\varphi }+4\\mu qB^2\\varphi ^2\\dot{q}\\dot{A}+4B\\varphi ^2(\\mu q^2-1)\\dot{A}\\dot{B}+8AB\\varphi (\\mu q^2-1)\\dot{\\varphi }\\dot{B}\\\\&+&8AB\\mu \\varphi ^2q\\dot{B}\\dot{q}+\\rho _0.$ The Euler-Lagrange equations (REF ) for this Lagrangian become $\\nonumber &&8(\\mu q^2-1)B\\varphi \\dot{B}\\dot{\\varphi }+4(\\mu q^2-1)B^2\\dot{\\varphi }^2+8\\mu \\varphi qB^2\\dot{\\varphi }\\dot{q}+4(\\mu q^2-1)\\varphi B^2\\ddot{\\varphi }\\\\\\nonumber &&+8\\mu qB\\varphi ^2\\dot{q}+8\\mu q\\varphi B^2\\dot{q}\\dot{\\varphi }+4\\mu \\varphi ^2B^2\\dot{q}^2+4\\mu q\\varphi ^2B^2\\ddot{q}+2(\\mu q^2-1)\\varphi ^2\\dot{B}^2\\\\&&+4(\\mu q^2-1)B\\varphi ^2\\ddot{B}-2\\mu qB^2\\varphi ^2-4B^2\\omega (\\varphi )\\dot{\\varphi }^2+B^2V(\\varphi )=0,$ $\\nonumber &&4(\\mu q^2-1)\\varphi ^2A\\ddot{B}+4(\\mu q^2-1)\\varphi ^2B\\ddot{A}+8(\\mu q^2-1)\\varphi AB\\ddot{\\varphi }+8\\mu AB\\varphi ^2q\\ddot{q}\\\\\\nonumber &&+4(\\mu q^2-1)\\varphi ^2\\dot{A}\\dot{B}+8(\\mu q^2-1)B\\varphi \\dot{A}\\dot{\\varphi }+8\\mu \\varphi ^2Bq\\dot{A}\\dot{q}+8(\\mu q^2-1)\\varphi A\\dot{B}\\dot{\\varphi }\\\\\\nonumber &&+8\\mu \\varphi ^2Bq\\dot{A}\\dot{q}+8(\\mu q^2-1)\\varphi A\\dot{B}\\dot{\\varphi }+8\\mu Aq\\varphi ^2\\dot{q}\\dot{B}+32\\mu \\varphi qAB\\dot{q}\\dot{\\varphi }+[8(\\mu q^2\\\\&&-1)AB-8AB\\omega (\\varphi )]\\dot{\\varphi }^2-4q\\mu AB\\varphi ^2+2ABV(\\varphi )=0,\\\\\\nonumber &&4AB^2\\dot{\\varphi }^2\\frac{d\\omega }{d\\varphi }+4A\\varphi (\\mu q^2-1)\\dot{B}^2+AB^2\\frac{dV}{d\\varphi }+8AB\\varphi (\\mu q^2-1)\\ddot{B}+4B^2\\varphi \\\\\\nonumber &&\\times (\\mu q^2-1)\\ddot{A}+8AB^2\\omega (\\varphi )\\ddot{\\varphi }+8B\\varphi (\\mu q^2-1)\\dot{A}\\dot{B}+8B^2\\dot{A}\\dot{\\varphi }\\omega (\\varphi )+16A\\\\&&\\times B\\dot{\\varphi }\\dot{B}\\omega (\\varphi )-4\\mu qAB^2\\varphi =0.$ These equations exhibit dynamics of the spatial components of the Einstein field equations as well as scalar wave equation for BI universe model.",
"Another constraint on the variables can be determined from the Ricci scalar given by $\\nonumber \\frac{1}{q}=2[\\frac{\\ddot{A}}{A}+2\\frac{\\ddot{B}}{B}+\\frac{\\dot{B}^2}{B^2}+2\\frac{\\dot{A}}{A}\\frac{\\dot{B}}{B}].$ The energy function related with the Lagrangian $\\mathcal {L}$ is defined as [8] $\\nonumber E_{\\mathcal {L}}&=&\\dot{A}\\frac{\\partial \\mathcal {L}}{\\partial \\dot{A}}+\\dot{B}\\frac{\\partial \\mathcal {L}}{\\partial \\dot{B}}+\\dot{q}\\frac{\\partial \\mathcal {L}}{\\partial \\dot{q}}+\\dot{\\varphi }\\frac{\\partial \\mathcal {L}}{\\partial \\dot{\\varphi }}-\\mathcal {L}$ Inserting the respective values in this energy function and after simplification, it can be written as $\\nonumber &&\\frac{(\\mu q^2-1)\\varphi ^2}{6}\\frac{\\dot{B}^2}{B^2}+\\omega (\\varphi )\\dot{\\varphi }^2+\\frac{(\\mu q^2-1)\\varphi \\dot{\\varphi }}{3}\\frac{\\dot{A}}{A}+\\frac{2(\\mu q^2-1)\\varphi \\dot{\\varphi }}{3}\\frac{\\dot{B}}{B}\\\\\\nonumber &&+\\frac{(\\mu q^2-1)\\varphi ^2}{3}\\frac{\\dot{A}}{A}\\frac{\\dot{B}}{B}+\\frac{\\mu \\varphi ^2q\\dot{q}}{3}\\frac{\\dot{A}}{A}+\\frac{2\\mu \\varphi ^2q\\dot{q}}{3}\\frac{\\dot{B}}{B}-\\frac{q\\varphi ^2\\mu }{6}+\\frac{V(\\varphi )}{12}-\\frac{\\rho _0}{12}=0.\\\\$ This provides the amount of energy in the dynamical system and corresponds to time-time component of the field equations.",
"Consequently, Eqs.",
"(REF )-(REF ) yield the complete set of the field equations for BI universe.",
"Now we check the existence of both Noether and Noether gauge symmetries of point like Lagrangian (REF ).",
"Here, the configuration space for the Lagrangian is defined as $(t,A,B,\\varphi ,q)$ and the respective tangent space is $(t,A,B,\\varphi ,q,\\dot{A}, \\dot{B},\\dot{\\phi },\\dot{q})$ .",
"The first-order prolonged symmetry generator is defined by $\\nonumber \\textbf {X}^{[1]}&=&\\tau \\frac{\\partial }{\\partial t}+\\alpha \\frac{\\partial }{\\partial A}+\\beta \\frac{\\partial }{\\partial B}+\\gamma \\frac{\\partial }{\\partial \\varphi }+\\delta \\frac{\\partial }{\\partial q}+\\alpha _{t}\\frac{\\partial }{\\partial \\dot{A}}+\\beta _{t}\\frac{\\partial }{\\partial \\dot{B}}+\\gamma _{t}\\frac{\\partial }{\\partial \\dot{\\varphi }}+\\delta _{t}\\frac{\\partial }{\\partial \\dot{q}},\\\\\\nonumber $ where $\\tau ,~\\alpha ,~\\beta ,~\\gamma $ and $\\delta $ are unknown functions to be determined.",
"Moreover, $\\nonumber &&\\alpha _{t}=D_{t}\\alpha -\\dot{A}D_{t}\\tau ,\\quad \\beta _{t}=D_{t}\\beta -\\dot{B}D_{t}\\tau , \\quad \\gamma _{t}=D_{t}\\gamma -\\dot{\\varphi }D_{t}\\tau ,\\\\\\nonumber &&\\delta _{t}=D_{t}\\delta -\\dot{q}D_{t}\\tau .$ In this configuration, the total derivative operator $D_{t}$ is $\\nonumber D_{t}=\\frac{\\partial }{\\partial t}+\\dot{A}\\frac{\\partial }{\\partial A}+\\dot{B}\\frac{\\partial }{\\partial B}+\\dot{\\varphi }\\frac{\\partial }{\\partial \\varphi }+\\dot{q}\\frac{\\partial }{\\partial q}.$ Using all these values in Eq.",
"(REF ), the system of determining equations will beome $&&\\tau _{q}=0,\\quad \\tau _{\\varphi }=0,\\quad \\tau _{A}=0,\\quad \\tau _{B}=0,\\\\\\nonumber &&B^2\\alpha (2q\\mu \\varphi ^2-V(\\varphi ))+AB\\beta (4q\\mu \\varphi ^2-2V(\\varphi ))+AB^2\\gamma (4q\\mu \\varphi -V^{\\prime }(\\varphi ))\\\\&&+2AB^2\\mu \\varphi ^2\\delta -\\epsilon ^2\\rho _0(AB^2)^{-(1+\\epsilon )}(B^2\\alpha +2AB\\beta )=G_{t},\\\\&&4B^2q\\varphi ^2\\mu \\alpha _{t}+8ABq\\mu \\varphi ^2\\beta _{t}=G_{q},\\\\\\nonumber &&4\\varphi ^2(\\mu q^2-1)B\\alpha _{t}+4(\\mu q^2-1)A\\varphi ^2\\beta _{t}+8AB\\varphi (\\mu q^2-1)\\gamma _{t}\\\\ &&+8AB\\varphi ^2\\mu q\\delta _{t}=G_{B},\\\\ &&4\\varphi ^2(\\mu q^2-1)B\\beta _{t}+4\\mu B^2\\varphi ^2q\\delta _{t}+4(\\mu q^2-1)\\varphi B^2\\gamma _{t}=G_{A},\\\\ &&8AB\\varphi (\\mu q^2-1)+8AB^2\\omega (\\varphi )\\gamma _{t}+4(\\mu q^2-1)\\varphi B^2\\alpha _{t}=G_{\\varphi },\\\\&&4\\varphi ^2(\\mu q^2-1)B\\beta _{A}+4\\mu B^2\\varphi ^2q\\delta _{A}+4(\\mu q^2-1)\\varphi B^2\\gamma _{A}=0,\\\\&&4B^2\\mu \\varphi ^2q\\alpha _{q}+8AB\\phi ^2\\mu q\\beta _{q}=0,\\\\\\nonumber &&4B^2\\alpha \\omega (\\varphi )+8AB\\omega (\\varphi )\\beta +4AB^2\\omega ^{\\prime }(\\varphi )\\gamma +4(\\mu q^2-1)\\varphi B^2\\alpha _{\\varphi }\\\\&&+8AB(\\mu q^2-1)\\varphi \\beta _{\\varphi }+8AB^2\\omega (\\varphi )\\gamma _{\\varphi }-8AB^2\\omega (\\varphi )\\tau _{t}=0,\\\\\\nonumber &&2\\alpha \\varphi ^2(\\mu q^2-1)+4A\\varphi \\gamma (\\mu q^2-1)+4q\\mu \\varphi ^2A\\delta +4\\varphi ^2(\\mu q^2-1)B\\alpha _{B}$ $\\nonumber &&+4\\varphi ^2(\\mu q^2-1)A\\beta _{B}-4\\varphi ^2(\\mu q^2-1)A\\tau _{t}+8ABq\\mu \\varphi ^2\\delta _{B}\\\\&&+8AB(\\mu q^2-1)\\varphi \\gamma _{B}=0,\\\\\\nonumber &&4\\varphi ^2(\\mu q^2-1)\\beta +8\\varphi (\\mu q^2-1)B\\gamma +8\\mu qB\\varphi ^2\\delta +4\\varphi ^2(\\mu q^2-1)B\\alpha _{A}\\\\\\nonumber &&-4\\varphi ^2(\\mu q^2-1)B\\tau _{t}+4\\varphi ^2(\\mu q^2-1)A\\beta _{A}+4\\varphi ^2(\\mu q^2-1)B\\beta _{B}-4\\varphi ^2\\\\\\nonumber &&\\times (\\mu q^2-1)B\\tau _{t}+8ABq\\varphi ^2\\mu \\delta _{A}+4B^2q\\mu \\varphi ^2\\delta _{B}+8AB(\\mu q^2-1)\\varphi \\gamma _{A}\\\\&&+4(\\mu q^2-1)B^2\\varphi \\gamma _{B}=0,\\\\\\nonumber &&8B\\mu \\varphi ^2q\\beta +8qB\\mu \\varphi \\gamma +4B^2\\mu \\varphi ^2\\delta +4B^2\\mu \\varphi ^2q\\alpha _{A}-8B^2\\mu \\varphi ^2q\\tau _{t}+8AB\\\\\\nonumber &&\\times \\varphi ^2q\\mu \\beta _{A}+4B\\varphi ^2(\\mu q^2-1)\\beta _{q}+4B^2\\mu \\varphi ^2q\\delta _{q}+4(\\mu q^2-1)B^2\\varphi \\gamma _{q}=0,\\\\&&8B\\varphi (\\mu q^2-1)\\beta +4(\\mu q^2-1)B^2\\gamma +8qB^2\\mu \\varphi \\delta +4(\\mu q^2-1)B^2\\varphi \\alpha _{A}\\\\\\nonumber &&-8(\\mu q^2-1)B^2\\varphi \\tau _{t}+8AB\\varphi (\\mu q^2-1)\\beta _{A}+4\\varphi ^2(\\mu q^2-1)B\\beta _{\\varphi }+4B^2q\\mu \\\\&&\\varphi ^2\\delta _{\\varphi }+8AB^2\\omega (\\varphi )\\gamma _{A}+4B^2(\\mu q^2-1)\\varphi \\gamma _{\\varphi }=0,\\\\\\nonumber &&8Bq\\mu \\varphi ^2\\alpha +8Aq\\mu \\varphi ^2\\beta +16qAB\\mu \\varphi \\gamma +8AB\\mu \\varphi ^2\\delta +4B^2q\\mu \\varphi ^2\\alpha _{B}\\\\\\nonumber &&+4B\\varphi ^2(\\mu q^2-1)\\alpha _{q}+8ABq\\mu \\varphi ^2\\beta _{B}+4\\varphi ^2A(\\mu q^2-1)\\beta _{q}-16ABq\\mu \\varphi ^2\\tau _{t}\\\\&&+8ABq\\mu \\varphi ^2\\delta _{q}+8AB\\varphi (\\mu q^2-1)\\gamma _{q}=0,\\\\\\nonumber &&8B\\varphi (\\mu q^2-1)\\alpha +8A\\varphi (\\mu q^2-1)\\beta +8AB(\\mu q^2-1)\\gamma +16ABq\\mu \\varphi \\delta \\\\\\nonumber &&+4(\\mu q^2-1)B^2\\varphi \\alpha _{B}+4(\\mu q^2-1)B\\varphi ^2\\alpha _{\\varphi }+8AB(\\mu q^2-1)\\varphi \\beta _{B}+4A\\\\\\nonumber &&\\times (\\mu q^2-1)\\varphi ^2\\beta _{\\varphi }-16AB(\\mu q^2-1)\\varphi \\tau _{t}+8ABq\\mu \\varphi ^2\\delta _{\\varphi }+8AB^2\\omega (\\varphi )\\gamma _{B}\\\\&&+8AB(\\mu q^2-1)\\varphi \\gamma _{\\varphi }=0,\\\\\\nonumber &&4B^2q\\mu \\varphi ^2\\alpha _{\\varphi }+4(\\mu q^2-1)B^2\\varphi \\alpha _{q}+8ABq\\mu \\varphi ^2\\beta _{\\varphi }+8AB(\\mu q^2-1)\\varphi \\beta _{q}\\\\&&+8AB^2\\omega (\\varphi )\\gamma _{q}=0.$ Firstly, we calculate the Noether symmetries of Lagrangian that correspond to $\\mathcal {L}_{X}L=0$ and can be found by the system of determining equations (REF )-() with $G=0$ and $\\tau =0$ .",
"In this case, all the functions $\\alpha ,~\\beta ,~\\gamma $ and $\\delta $ are independent of time.",
"Integration of Eq.",
"(REF ) yields $\\tau =\\tau (t)$ .",
"For the sake of simplicity, we take the ansatz for unknowns $\\alpha ,~\\beta $ and $\\gamma $ as $\\alpha =A^aB^b\\varphi ^cq_0(q),\\quad \\beta =A^fB^g\\varphi ^hq_1(q),\\quad \\gamma =A^mB^n\\varphi ^pq_2(q).$ Here $a,b,c,f,g,h,m,n$ and $p$ are parameters to be determined, while $q_0,~q_1$ and $q_2$ are unknown functions of variable $q$ .",
"We would like to find the functions $\\delta ,~V$ and $\\omega $ by requiring the existence of Noether symmetries.",
"From Eq.",
"(), it follows that $\\nonumber a=f+1,\\quad g=b+1,\\quad c=h,\\quad q_0=-2q_1,$ and hence $\\alpha =-2A^{f+1}B^{g-1}\\varphi ^hq_1(q),\\quad \\beta =A^fB^g\\varphi ^hq_1(q),\\quad \\gamma =A^mB^n\\varphi ^pq_2(q).$ Equations () and () imply that $\\omega (\\varphi )=\\frac{c_2}{\\varphi ^{2p}}$ and $q_2=c_1$ , respectively, where $c_1$ and $c_2$ are integration constants.",
"Equation () yields $\\nonumber \\delta =\\frac{(1-\\mu q^2)}{\\mu q}[A^fB^{g-1}\\varphi ^hq_1+c_1A^mB^n\\varphi ^{p-1}]+h_1(B,q,\\varphi ),$ where $h_1$ is an integration function.",
"Further, Eq.",
"(REF ) leads to $h_1=\\frac{h_2(q,\\varphi )}{\\sqrt{B}}$ and $g=2/3$ , hence $\\nonumber &&\\alpha =-2q_1A^{f+1}B^{-1/3}\\varphi ^{h},\\quad \\beta =q_1A^fB^{2/3}\\varphi ^h,\\quad \\gamma =A^mB^n\\varphi ^pc_1,\\\\\\nonumber &&\\delta =\\frac{1-\\mu q^2}{\\mu q}[A^fB^{-1/3}\\varphi ^hq_1+c_1A^mB^n\\varphi ^{p-1}]+\\frac{h_2(q,\\varphi )}{\\sqrt{B}}.$ Equation () implies that $h_2=0$ and $f=-2/3$ .",
"Moreover, Eq.",
"() leads to $m=-2/3,~n=-1/3,~h=-(1+p)$ and $q_1=\\frac{2c_1c_2}{3(1-\\mu q^2)}$ , thus $\\nonumber &&\\alpha =-2(\\frac{2c_1c_2}{3(1-\\mu q^2)})A^{1/3}B^{-1/3}\\varphi ^{-(1+p)},\\quad \\beta =\\frac{2c_1c_2}{3(1-\\mu q^2)}A^{-2/3}B^{2/3}\\varphi ^{-(1+p)},\\\\\\nonumber &&\\gamma =A^{-2/3}B^{-1/3}c_1\\varphi ^p,\\quad \\delta =\\frac{(1-\\mu q^2)}{\\mu q}[A^{-2/3}B^{-1/3}\\varphi ^{-(1+p)}\\frac{2c_1c_2}{3(1-\\mu q^2)}\\\\&&+c_1A^{-2/3}B^{-1/3}\\varphi ^{p-1}].$ Inserting these values in Eq.",
"(), we obtain either $c_1c_2=0$ or $\\mu =0$ .",
"Since $\\mu \\ne 0$ , so $c_1c_2=0$ .",
"When we take $c_1\\ne 0,~c_2=0$ , it follows that $\\nonumber &&\\alpha =0,\\quad \\beta =0,\\quad \\omega =0,\\quad \\gamma =c_1\\varphi ^pA^{-2/3}B^{-1/3},\\\\&&\\delta =\\frac{1-\\mu q^2}{\\mu q}c_1A^{-2/3}B^{-1/3}\\varphi ^{p-1}.$ Equation () leads to $V(\\varphi )=\\frac{c_3\\varphi ^2}{2}+c_4$ and $q=q_0=\\frac{c_3\\pm \\sqrt{c_3^2-8\\mu }}{4\\mu }$ , where $c_3$ and $c_4$ are integration constants.",
"In this case, the symmetry generator follows $\\textbf {X}_1=\\varphi ^pA^{-2/3}B^{-1/3}\\frac{\\partial }{\\partial \\varphi }+\\frac{1-\\mu q^2}{\\mu q}A^{-2/3}B^{-1/3}\\varphi ^{p-1}\\frac{\\partial }{\\partial q}$ which yields only one symmetry and the respective constant of motion is zero, i.e., $I_1=0$ .",
"This shows that for Noether symmetries of the point like Lagrangian exists but there is no non-trivial conserved quantity.",
"For Noether gauge symmetries, we consider the full symmetry generator ($\\tau \\ne 0,~G\\ne 0$ ) in which the unknown functions are dependent on time.",
"Proceeding in the similar way, Eqs.",
"(REF ) and ()-() lead to $\\tau =c_3,\\quad \\alpha =0,\\quad \\beta =0,\\quad \\gamma =c_1t^l\\varphi ^p,\\quad \\delta =\\frac{1-\\mu q^2}{\\mu q}c_1t^l\\varphi ^{p-1},\\quad \\omega =\\frac{c_4}{\\varphi ^{2p}}.$ Equations ()-() yield $l=0$ and $q=q_0$ with $\\nonumber V(\\varphi )=\\frac{1+3q_0^2\\mu }{2q_0}\\varphi ^2, \\quad G=c_5.$ Thus there exist two symmetry generators given by $\\nonumber \\textbf {X}_1=\\frac{\\partial }{\\partial t},\\quad \\textbf {X}_2=\\varphi ^p\\frac{\\partial }{\\partial \\varphi }+(\\frac{1-\\mu q^2}{\\mu q})\\varphi ^{p-1}\\frac{\\partial }{\\partial q}.$ The constant of motion, i.e., the integral of motion can be written as $\\nonumber I=\\tau \\mathcal {L}+(\\alpha -\\dot{A}\\tau )\\frac{\\partial \\mathcal {L}}{\\partial \\dot{A}}+(\\beta -\\dot{B}\\tau )\\frac{\\partial \\mathcal {L}}{\\partial \\dot{B}}+(\\gamma -\\dot{\\varphi }\\tau )\\frac{\\partial \\mathcal {L}}{\\partial \\dot{\\varphi }}+(\\delta -\\dot{q}\\tau )\\frac{\\partial \\mathcal {L}}{\\partial \\dot{q}}-G.$ In this case, these are given by $\\nonumber I_1&=&8AB^2c_4\\dot{\\varphi }\\varphi ^{-p}-\\frac{c_5}{2},\\\\\\nonumber I_2&=&2q_0\\mu AB^2\\varphi ^2-4AB^2\\dot{\\varphi }^2c_4\\varphi ^{-2p}-AB^2\\frac{(1+3\\mu q_0^2)}{2q_0}\\varphi ^2-2(\\mu q_0^2-1)\\varphi ^2\\dot{A}B^2\\\\\\nonumber &-&4(\\mu q_0^2-1)(B^2\\varphi \\dot{A}\\dot{\\varphi }+B\\varphi ^2\\dot{A}\\dot{B}+2AB\\varphi \\dot{B}\\dot{\\varphi })+\\rho _0-\\frac{c_5}{2}.$ It can be concluded that for the point-like Lagrangian of BI universe model, there is only one Noether symmetry generator while two Noether gauge symmetry generators exist.",
"The existence of Noether symmetries allows zero BD coupling function and the quadratic field potential.",
"Such field potentials have widely been used in literature [22] to discuss many cosmological problems in the context of scalar tensor gravity.",
"Since, $\\omega =0$ , therefore these symmetries may correspond to the symmetries of pointlike Lagrangian of BI universe in Palatini $f(R)$ gravity [23].",
"It is found that the Noether charge is zero in the case of Noether symmetries.",
"However, the existence of Noether gauge symmetries leads to dynamical BD coupling parameter (in the form of inverse power law) with quadratic potential.",
"It is observed that the behavior of BD coupling function depends upon the parameter $p$ , for $p>0$ , the BD coupling becomes divergent at $\\varphi =0$ while for $p<0$ , it turns out to be zero there.",
"Moreover, in this case, the gauge function turns out to be constant and the Noether charge, i.e., the conserved quantities exist.",
"The EoS parameter for this configuration is given by $\\nonumber \\omega _\\varphi &=&\\frac{\\omega _{x\\varphi }+2\\omega _{y\\varphi }}{3}=\\frac{1}{3}[3\\mu q_0\\varphi ^2-3/2V(\\varphi )+6\\omega (\\varphi )\\dot{\\varphi }^2-6\\dot{\\varphi }^2(\\mu q_0^2-1)\\\\\\nonumber &-&6\\dot{\\varphi }\\ddot{\\varphi }(\\mu q_0^2-1)-4(\\mu q_0^2-1)\\varphi \\dot{\\varphi }(\\frac{\\dot{A}}{A}+2\\frac{\\dot{B}}{B})][\\mu q_0\\varphi ^2-\\frac{V(\\varphi )}{2}-6\\omega (\\varphi )\\dot{\\varphi }^2\\\\\\nonumber &-&2(\\mu q_0^2-1)\\varphi (\\frac{\\dot{A}}{A}+2\\frac{\\dot{B}}{B})]^{-1},$ where $V$ and $\\omega $ are given in previous cases (found by the Noether symmetry analysis).",
"Following [18], we have tried to plot this expression using Maple software but due to highly non-linear terms present in the field equations with $A,~B$ and $\\varphi $ as unknowns, it is not possible to have the plot of this expression (basically, Maple could not convert the expressions into explicit first-order system of DEs)."
],
[
"Bianchi I Solutions Using Scaling Symmetries",
"In this section, we discuss BI solutions by taking $\\mu =0$ in the Lagrangian with constant BD parameter.",
"In canonical form, the action can be written as $S=\\int \\sqrt{-g}[\\frac{1}{8\\omega }\\varphi ^2R-\\frac{1}{2}g^{\\mu \\nu }\\partial _\\mu \\varphi \\partial _\\nu \\varphi +V_0\\varphi ^2+\\mathcal {L}_m]d^4x.$ Here $\\omega $ is a constant BD parameter and the field potential is taken to be $V=V_0\\varphi ^2$ .",
"Moreover, the matter distribution is taken as the perfect fluid.",
"The corresponding field equations are $&&\\frac{\\varphi ^2}{4\\omega }(2\\frac{\\dot{A}}{A}\\frac{\\dot{B}}{B}+\\frac{\\dot{B}}{B})+\\frac{\\dot{\\varphi }^2}{2}+V_0\\varphi ^2+\\frac{1}{2\\omega }(\\frac{\\dot{A}}{A}+2\\frac{\\dot{B}}{B})\\varphi \\dot{\\varphi }=\\rho ,\\\\\\nonumber &&-\\frac{\\varphi ^2}{4\\omega }(2\\frac{\\ddot{B}}{B}+\\frac{\\dot{B}^2}{B})-\\frac{1}{\\omega }\\frac{\\dot{B}}{B}\\dot{\\varphi }\\varphi -\\frac{1}{2\\omega }\\ddot{\\varphi }\\varphi -V_0\\varphi ^2+(1/2-1/2\\omega )\\dot{\\varphi }^2=P,\\\\\\\\\\nonumber &&-\\frac{\\dot{\\varphi }^2}{4\\omega }(\\frac{\\ddot{B}}{B}+\\frac{\\ddot{A}}{A}+\\frac{\\dot{A}}{A}\\frac{\\dot{B}}{B})-\\frac{1}{2\\omega }(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B})\\dot{\\varphi }-\\frac{1}{2\\omega }\\ddot{\\varphi }\\varphi -V_0\\varphi ^2\\\\&&+(\\frac{1}{2}-\\frac{1}{2\\omega })\\dot{\\varphi }^2=P,\\\\&&\\ddot{\\varphi }+(\\frac{\\dot{A}}{A}+2\\frac{\\dot{B}}{B})\\dot{\\varphi }+[V_0+\\frac{1}{2\\omega }\\lbrace \\frac{\\dot{A}}{A}+2\\frac{\\ddot{B}}{B}+\\frac{\\dot{B}^2}{B^2}+2\\frac{\\dot{A}\\dot{B}}{AB}\\rbrace ]\\varphi =0.$ The equation of continuity is $\\dot{\\rho }+(\\frac{\\dot{A}}{A}+2\\frac{\\dot{B}}{B})(\\rho +P)=0.$ Due to complexity of the system, we use the physical relationship between the scale factors, i.e., $A=B^m;~m\\ne 1$ [21].",
"This condition is originated by the assumption that the ratio of shear scalar to expansion scalar is constant.",
"Consider the power law form for the density $\\rho =\\rho _0(AB^2)^\\varepsilon /3$ , and hence pressure $P=-\\frac{\\varepsilon +3}{3}\\rho $ .",
"Here $\\varepsilon $ is any parameter that acts as an equation of state parameter and its different values can classify different phases of the universe, e.g., $\\varepsilon =-3$ implies matter dominated era and $\\varepsilon =0$ yields dark energy dominated universe.",
"We take dependent variables $H_2=H_2(B)$ and $F=F(B_2)$ with $B$ as independent variable and introduce the following notations in the system of equations (REF )-() $\\nonumber &&F=\\frac{\\dot{\\varphi }}{\\varphi },\\quad H_2=\\frac{\\dot{B}}{B}, \\quad H_1=m\\frac{\\dot{B}}{B},\\quad \\dot{H_2}=BH_2H^{\\prime }_2,\\quad \\dot{H}_1=mBH_2H^{\\prime }_2,\\quad \\\\\\nonumber &&\\frac{\\ddot{B}}{B}=BH_2H^{\\prime }_2+H_2^2,\\quad \\frac{\\ddot{\\varphi }}{\\varphi }=BH_2F^{\\prime }+F^2,$ where prime indicates derivative with respect to $B$ .",
"This leads to $&&H_2^2+(F^2+2V_0)\\frac{2\\omega }{1+2m}+\\frac{2(m+2)}{2m+1}H_2F=\\frac{4\\omega \\rho _0B^{\\varepsilon (m+2)/3}}{(1+2m)\\varphi ^2},$ $&&\\frac{2BH_2}{3}(F^{\\prime }+H^{\\prime }_2)+H_2^2+\\frac{4H_2F}{3}+\\frac{2(2-\\omega )F^2}{3}+\\frac{4V_0\\omega }{3}=-\\frac{4\\omega }{3}\\frac{P}{\\varphi ^2},\\\\\\nonumber &&H_2^2+\\frac{1}{(m^2+m+1)}[BH_2((m+1)H^{\\prime }_2+2F^{\\prime })+2(1+m)H_2F\\\\&&+4V_0\\omega +2(2-\\omega )F^2]=-\\frac{4P\\omega }{(m^2+m+1)\\varphi ^2},\\\\\\nonumber &&BH_2[\\frac{\\omega }{m+2}F^{\\prime }+\\frac{H^{\\prime }_2}{2}]+\\frac{H_2^2}{2}(\\frac{m^2+2m+3}{m+2})+\\omega H_2F\\\\&&+\\frac{\\omega }{m+2}(F^2+2V_0)=0.$ Also, the evolution of energy density is given by $\\nonumber \\dot{\\rho }=\\varepsilon \\rho _0\\frac{(m+2)H_2\\rho }{3}.$ In the above system of five field equations, only three equations are independent.",
"We shall use Eqs.",
"(REF ), (REF ) and () as independent equations.",
"By taking the time derivative of Eq.",
"(REF ), and after some manipulation, it becomes $\\nonumber &&[H_2^2+\\frac{m+2}{1+2m}FH_2]H^{\\prime }_2+\\frac{F^{\\prime }}{1+2m}[2\\omega FH_2+(m+2)H_2^2]=\\frac{\\varepsilon (m+2)H_2^3}{6B}\\\\\\nonumber &&+\\frac{FH_2^2}{B}(\\frac{\\varepsilon (m+2)^2}{3(1+2m)}-1)-\\frac{2F^3\\omega }{B(1+2m)}+(\\varepsilon \\omega -6)\\frac{H_2F^2(m+2)}{3B(1+2m)}\\\\&&+\\frac{2V_0\\omega }{B(1+2m)}[\\frac{\\varepsilon (m+2)H_2}{3}-2F].$ We can write equations for the unknowns $H_2^{\\prime }$ and $F^{\\prime }$ after some manipulation from Eqs.",
"() and (REF ) as follows $\\nonumber &&BH_2\\frac{dH_2}{dB}[\\frac{2(\\omega -1)+m(4\\omega -1)}{2(m+2)(1+2m)}]=H_2^2[\\frac{\\varepsilon \\omega }{6}+\\frac{m^2+2m+3}{2(1+2m)}]+F^2\\omega \\\\\\nonumber &&\\times [\\frac{(\\varepsilon +6)\\omega -3}{3(1+2m)}]+\\omega FH_2[\\frac{\\varepsilon (m+2)^2+3(m-1)}{3(1+2m)}]+\\frac{2V_0\\omega }{3(1+2m)}(\\varepsilon \\omega +3),\\\\\\\\\\nonumber &&BH_2(\\frac{m+2}{2(1+2m)}-\\frac{\\omega }{m+2})\\frac{dF}{dB}=H_2^2[\\frac{\\varepsilon (m+2)^2+6(m^2+2m+3)}{12(m+2)}]$ $\\nonumber &&+\\frac{FH_2}{2}[\\frac{\\varepsilon (m+2)^2}{3(1+2m)}-1+2\\omega +\\frac{m^2+2m+3}{1+2m}]+F^2[\\frac{(m+2)(\\varepsilon \\omega -6)}{6(1+2m)}\\\\&&+\\frac{\\omega }{m+2}+\\frac{\\omega (m+2)}{1+2m}]+V_0\\omega [\\frac{6(1+2m)+\\varepsilon (m+2)^2}{3(1+2m)(m+2)}].$ These form a closed system of equations (two equations involving two unknowns).",
"We adopt the analysis of classical Lie groups [25] to find solution of Eqs.",
"(REF ) and (REF ).",
"Since these equations are quite similar to the equations exhibiting scaling or dilatational symmetries, therefore we assume a vector field $\\nonumber \\textbf {X}=B\\alpha \\frac{\\partial }{\\partial B}+\\beta F\\frac{\\partial }{\\partial F}+\\gamma H_2\\frac{\\partial }{\\partial H_2}$ generated by a scaling group of mappings given by $\\nonumber \\widetilde{B}=\\lambda ^\\alpha B,\\quad \\widetilde{A}=\\lambda ^{m\\alpha }A,\\quad \\widetilde{F}=\\lambda ^\\beta F,\\quad \\widetilde{H}_2=\\lambda ^\\gamma H_2,\\quad \\widetilde{H}_1=m\\lambda ^\\gamma H_2.$ The invariance of Eqs.",
"(REF ) and (REF ) under the above transformation provides two different cases: The scalar field is massive, i.e., $V_0\\ne 0$ which implies that $\\beta =\\gamma =0$ and $\\alpha =1$ .",
"This allows the form of generator given by $\\textbf {X}_1=B\\frac{\\partial }{\\partial B}$ .",
"The scalar field is massless, i.e., $V_0=0$ which yields two choices of the parameters and consequently two symmetry generators exist.",
"(i) $\\beta =\\gamma =0$ with $\\alpha =1$ implies $\\textbf {X}_1=B\\frac{\\partial }{\\partial B}$ (ii) $\\beta =\\gamma =1$ and $\\alpha =0$ , leading to $\\textbf {X}_2=H_2\\frac{\\partial }{\\partial H_2}+F\\frac{\\partial }{\\partial F}$ .",
"In the first case, there is only one symmetry generator with basis of invariants $\\lbrace H_2,F\\rbrace $ .",
"This symmetry ensures that the solution is in the form of invariants given by $F=F(H_2)$ .",
"In order to find the solution, the number of known symmetries should be equal to the order of DE.",
"Since Eqs.",
"(REF ) and (REF ) lead to $\\frac{dF}{dH_2}=K(H_2,F)$ , i.e., first-order differential with no more known symmetries, so the integration in quadratures will not be possible.",
"We construct a solution by imposing the invariance of $F$ and $H_2$ , i.e., $\\frac{dH_2}{dB}=0$ and $\\frac{dF}{dB}=0$ .",
"Equations (REF ) and (REF ) become quadratic equations for $H_2$ and $F$ .",
"The roots of these equations are quite lengthy.",
"To get insights, we choose $m=2,~U_0=2$ and $\\varepsilon =-3$ (matter dominated phase) which yields four roots as $\\nonumber H_{2}&=&\\pm [\\omega (11264-15936\\omega +28912\\omega ^2-4940\\omega ^3-9600\\omega ^4-1600\\omega ^5)\\\\\\nonumber &\\pm &20\\omega ^2(3748096-11585024\\omega +16537312\\omega ^2-11402336\\omega ^3+3126193\\\\\\nonumber &\\times &\\omega ^4-557336\\omega ^5+197920\\omega ^5+92800\\omega ^6+6400\\omega ^7)^{1/2}]^{1/2}[2000\\omega ^5\\\\\\nonumber &+&1400\\omega ^4-83035\\omega ^3+111598\\omega ^2-6512\\omega -15488]^{-1}.$ Likewise, there exist four roots for $F$ given by $\\nonumber F&=&\\pm [\\omega (25696-53008\\omega +29310\\omega ^2-25600\\omega ^3+4000\\omega ^4)\\pm 10\\omega (3748096\\\\\\nonumber &-&24873728\\omega +64116448\\omega ^2-8571569\\omega ^3+65210905\\omega ^4-2517980\\omega ^5\\\\\\nonumber &+&8538400\\omega ^5-1904000\\omega ^6+160000\\omega ^7)^{1/2}]^{1/2}[2000\\omega ^5+1400\\omega ^4-83035\\\\\\nonumber &\\times &\\omega ^3+111598\\omega ^2-6512\\omega -15488]^{-1}.$ Consequently, the scale factor ($H_2=constant$ ) and scalar field ($F=constant$ ) turn out to be $B=B_0\\exp (H_{2i}t)$ and $\\varphi =\\varphi _0\\exp (F_it)$ , respectively.",
"Here $H_{2i}$ and $F_i$ are the roots given above, while $B_0$ and $\\varphi _0$ are present values of the scale factor and the scalar field, respectively.",
"The present values can be restricted by using Eq.",
"(REF ) as follows $\\varphi _0=\\sqrt{\\frac{4\\omega \\rho _0B_0^{-4}}{(5H_{2i}^2+2\\omega F_i^2+8\\omega +8H_{2i}F_i)}}.$ The plots for the scale factor and density function ($\\rho (t)=\\rho _0B_0^{4\\epsilon }e^{4\\epsilon H_{2i}t}/3$ ) are shown in Figure 1.",
"Since the scalar field is massive, therefore $\\omega $ can take any value satisfying $\\omega >-3/2$ to avoid ghost instabilities [23].",
"Figure 1(a) indicates that the scale factor increases for $(-,-)$ and $(+,+)$ roots, while decreases to zero for $(-,+)$ and $(+,-)$ roots.",
"Figure 1(b) shows that the behavior of density is exactly opposite to that of the scale factor.",
"This means that only for $(-,-)$ and $(+,+)$ roots, the constructed model shows expanding behavior with decreasing energy density which is consistent with the recent observations.",
"The scalar field exhibits a similar behavior as shown in Figure 2(a).",
"Figure: Plots (a) and(b) show the scale factor and energy density versus time tt,respectively.",
"Here red and green correspond to (+,+)(+,+) and (-,-)(-,-)roots with ω=1.5\\omega =1.5, respectively, while yellow and purple linesindicate (+,-)(+,-) and (-,+)(-,+) roots with ω=1.2\\omega =1.2, respectively.Figure: Plots (a) and(b) represent the volume of the universe and scalar field versustime tt, respectively.",
"Here red and green correspond to (+,+)(+,+) and(-,-)(-,-) roots with ω=1.5\\omega =1.5, respectively, while yellow andpurple lines indicate (+,+)(+,+) and (-,-)(-,-) roots with ω=1.2\\omega =1.2,respectively.Figure: Plots (a) and(b) show the EoS parameters for scalar field ω φ \\omega _\\phi , versustime.",
"Here red and green lines correspond to (-,+)(-,+) and (+,-)(+,-)roots with ω=200\\omega =200, respectively.For $(-,-)$ and $(+,+)$ roots, the scalar field is expanding, while for $(+,-)$ and $(-,+)$ , the scalar field is contracting with the passage of time.",
"This indicates that the scalar field plays a dominant role in the later phase of cosmic expansion.",
"The volume of the universe is given by $V(t)=B^4=B_0^4\\exp (4H_{2i}t)$ which shows that the universe expands exponentially for different $H_{2i}$ roots as shown in Figure 2(b) (indicating infinite volume in future).",
"Moreover, the average scalar factor, $a=B_0^{4/3}\\exp (\\frac{4H_{2i}t}{3})$ , leads to negative value of the deceleration parameter, which shows the rapid expansion in the universe consistent with the observations.",
"Figure 3 represents the EoS parameter for scalar field $\\omega _\\varphi =p_\\varphi /\\rho _\\varphi $ versus time.",
"These indicate that the universe model lies in the phantom phase for all values of time which is in agreement with the recent rapid expanding behavior of the universe.",
"In the second case, we take $V_0=0$ (massless scalar field).",
"Since there exist two commutating symmetries $X_1$ and $X_2$ , so we have first-order DE $\\frac{dF}{dH_2}=K(F,H_2)$ with one known symmetry.",
"Consequently, the integration in quadrature will be possible and then Eqs.",
"(REF ) and (REF ) yield $\\nonumber &&-\\frac{dH_2}{H_2}=[(\\frac{\\varepsilon \\omega }{6}+\\frac{m^2+2m+3}{2(1+2m)})+G^2\\omega (\\frac{(\\varepsilon +6)\\omega -3}{3(1+2m)})+\\omega G(\\frac{\\varepsilon (m+2)^2}{3(1+2m)}\\\\\\nonumber &&+\\frac{3(m-1)}{3(1+2m)})][(\\frac{2(\\omega -1)+m(4\\omega -1)}{(m+2)^2-2\\omega (1+2m)})\\lbrace (\\frac{\\varepsilon (m+2)^2+6(m^2+2m+3)}{12(m+2)})\\\\\\nonumber &&+G^2(\\frac{(m+2)(\\varepsilon \\omega -6)}{6(1+2m)}+\\frac{\\omega }{m+2}+\\frac{\\omega (m+2)}{1+2m})\\rbrace +G\\lbrace \\frac{2(\\omega -1)+m(4\\omega -1)}{(m+2)^2-2\\omega (1+2m)}\\\\\\nonumber &&\\times (\\frac{\\varepsilon (m+2)^2}{3(1+2m)}-1+2\\omega +\\frac{m^2+2m+3}{1+2m})-(\\frac{\\varepsilon \\omega }{6}+\\frac{m^2+2m+3}{2(1+2m)})\\\\\\nonumber &&-G^2\\omega (\\frac{(\\varepsilon +6)\\omega -3}{3(1+2m)})-\\omega G\\frac{\\varepsilon (m+2)^2+3(m-1)}{3(1+2m)}\\rbrace ]^{-1}dG,$ where $G=\\frac{F}{H_2}$ .",
"The solution of this equation yields directional Hubble parameter in terms of new parameter $G$ .",
"Since the scalar field is massless, so BD coupling parameter should satisfy the range $\\omega \\ge 40,000$ as suggested by the solar system experiments [23].",
"For the present era, its solution can be written as $H_2=H_{2,0}[(G-0.004916)^{0.000024}(G+19512.79)^{0.99995}(G-0.005085)^{0.0000256}],\\\\$ where we have taken $m=0.5,~\\omega =40,000$ and $H_{2,0}$ indicates present value of the parameter $H_2$ .",
"From Eq.",
"(REF ), the scale factor becomes $\\nonumber -\\frac{dB_2}{B_2}&=&\\frac{2(\\omega -1)+m(4\\omega -1)}{2(1+2m)(m+2)}[(\\frac{2(\\omega -1)+m(4\\omega -1)}{(m+2)^2-2\\omega (1+2m)})\\lbrace (\\frac{\\varepsilon (m+2)^2}{12(m+2)}\\\\\\nonumber &+&\\frac{6(m^2+2m+3)}{12(m+2)})+G^2(\\frac{(m+2)(\\varepsilon \\omega -6)}{6(1+2m)}+\\frac{\\omega }{m+2}+\\frac{\\omega (m+2)}{1+2m})\\rbrace \\\\\\nonumber &+&G\\lbrace (\\frac{2(\\omega -1)+m(4\\omega -1)}{(m+2)^2-2\\omega (1+2m)})(\\frac{\\varepsilon (m+2)^2}{3(1+2m)}-1+2\\omega +\\frac{m^2+2m+3}{1+2m})\\\\\\nonumber &-&(\\frac{\\varepsilon \\omega }{6}+\\frac{m^2+2m+3}{2(1+2m)})-G^2\\omega (\\frac{(\\varepsilon +6)\\omega -3}{3(1+2m)})-\\omega G\\frac{\\varepsilon (m+2)^2}{3(1+2m)}\\\\\\nonumber &+&\\frac{3(m-1)}{3(1+2m)}\\rbrace ]^{-1}dG.$ In the present era, for the choice $m=0.5$ and $\\omega =40,000$ , the solution can be written as $\\nonumber B=B_0[(G-0.004916)^{0.000024}(G+19512.79)^{0.99995}(G-0.005085)^{0.0000256}]^{-15999.8}.$ Further, the scalar field can be determined by the relationship $\\frac{d\\varphi }{\\varphi }=G\\frac{da}{a}$ , which follows from $F=\\frac{\\dot{\\varphi }}{\\varphi }=GH$ .",
"The corresponding scalar field takes the form $\\nonumber \\varphi &=&\\varphi _0\\exp (-15999.8G)[(0.00508-G)^{1.3008\\times 10^{-7}}(G+0.004916)^{-1.2008\\times 10^{-7}}\\\\\\nonumber &\\times &(G+19512.8)^{-19511.8}]^{-15999.8}.$ The energy density is given by $\\nonumber \\rho &=&\\rho _0B_0^{-2.5}[(G-0.004916)^{0.000024}(G+19512.79)^{0.99995}(G\\\\\\nonumber &-&0.005085)^{0.0000256}]^{2.5\\times 15999.8}.$ The symbols with 0 subscript indicate the present values.",
"The time related with the solution can be calculated by the expression $t-t_0=\\int \\frac{da}{aH}$ .",
"For particular choice of parameters, it becomes $\\nonumber t-t_0&=&\\int [(0.800207(G+5.62517\\times 10^{-6})(G+0.975609))(\\ln [(G\\\\\\nonumber &+&0.0035)^{0.000012}(G+26655.8)^{0.99997}(G-0.0036)^{0.000012}])((G\\\\\\nonumber &-&0.00508)(G+0.004916)(G+19512.8))^{-1}]dG.$ By inverting this expression, the parameter $G$ can be determined in terms of time and hence the scale factor and the scalar field.",
"The above solutions are parametric solutions in new variable $G$ , i.e., $a=a(G)$ and $\\varphi =\\varphi (G)$ .",
"The above expression can be evaluated numerically and then by the obtained set of data points, we can interpolate the function $G(t)$ .",
"By adopting this procedure, we interpolate the function $G(t)$ using polynomial interpolation and is given by $G(t)=63578.70834t^4-63719.54167t^3+22535.2479t^2-859.1271t+0.0059,\\\\$ where we have used the initial condition $G(0)=0.0059$ .",
"Using this value of $G$ , all the expressions like energy density, scale factors, scalar field and Hubble parameter can be discussed versus time.",
"It can be observed that the obtained model is not free from singularities as the scalar field and scale factor become singular for some particular values of $G$ .",
"Moreover, as $G\\rightarrow 0$ , all these quantities remain finite while as $G\\rightarrow \\infty $ , only the scalar field and the scale factor turn out to be zero.",
"In this case, the deceleration parameter is given by $q=-1-\\frac{3}{m+2}(\\frac{\\dot{H}_2}{H_2^2}),$ where $m=0.5$ ,while $H_2$ and $G$ are given by Eqs.",
"(REF ) and (REF ), respectively.",
"Figure 4 shows that the deceleration parameter remains negative for all values of time which yields the accelerated expansion of the universe model and is well-consistent with the recent observations.",
"Figure: This shows thedeceleration parameter versus time."
],
[
"Summary",
"The modified theories of gravity with action involving positive or negative powers of curvature as an extra term can lead to a better description of phenomenon of initial cosmic inflation and the late-time cosmic acceleration.",
"The main objective of this paper is to evaluate the Noether and Noether gauge symmetries for some homogeneous universe models in the framework of non-vacuum scalar-tensor gravity with inverse curvature correction term, i.e., $R^{-1}$ .",
"For this purpose, we have applied the Noether gauge symmetry analysis to non-flat FRW universe model.",
"Furthermore, we have discussed the Noether and Noether gauge symmetries for BI universe model.",
"In both cases, the matter part of Lagrangian has been taken as perfect fluid.",
"We have constructed the field potential and the coupling function by requiring the existence of Noether symmetries.",
"In the case of FRW universe model, we have a system of 11 PDEs for Noether gauge symmetries of the constructed point like Lagrangian.",
"In literature [8], the Noether symmetries of the same Lagrangian in vacuum has been discussed, where the model with dust matter seem to be more physical, but the Noether symmetries cannot always exist.",
"We have extended this work by exploring more general symmetries, i.e., Noether gauge symmetries by introducing perfect fluid matter part in the Lagrangian.",
"We have found that the Noether symmetry generator exists with non-zero gauge function in matter dominated phase.",
"It is seen that the gauge term turns out to be a dynamical quantity and the integral of motion associated with the dynamics of Lagrangian exist.",
"We have also specified the form of BD coupling function and the field potential.",
"In this case, the BD coupling function turns out to be a constant quantity, while the field potential is given by the power law form.",
"Next, it is shown that the Noether as well as Noether gauge symmetries exist for the flat BI universe model with perfect fluid using point like Lagrangian with curvature corrected term.",
"The existence of Noether symmetry generator allows zero coupling function and quadratic potential with zero integral of motion.",
"The Noether gauge symmetry generators yield the constant gauge function with quadratic potential and variable BD parameter, $\\omega =c_4/\\varphi ^{2p}$ .",
"The behavior of BD coupling parameter is dependent on the parameter $p$ .",
"We have also determined the respective conserved quantities in this case.",
"Finally, we have evaluated the BI solutions using scaling or dilatational symmetries.",
"Since it is difficult to find the BI cosmological solutions in the curvature corrected configuration, so we take $\\mu =0$ in the action, i.e., zero curvature correction term and constant BD coupling parameter.",
"For this purpose, two cases have been taken into account.",
"In the first case, it is seen that both the scale factor and the scalar field evolve exponentially yielding deceleration parameter $q=-1$ which is compatible with the observations and inflationary scenario.",
"Furthermore, the EoS parameter for scalar field turns out to be negative $\\omega _\\varphi <-1$ for $(-,+)$ and $(+,-)$ roots only that confirms the accelerating phase of the universe model.",
"The graphs of scale factor and energy density have also been given.",
"In the second case, there exist two symmetry generators and consequently, the integration in quadrature is possible.",
"By introducing a new parameter $G=F/H_2$ , the forms of scale factor, scalar field, directional Hubble parameters and energy density have been calculated.",
"It is observed that the obtained solution is parametric in the variable $G$ .",
"The relation of new variable $G$ in terms of time has also been given.",
"By solving the function $G$ numerically using polynomial interpolation, all the cosmological parameters can be discussed versus time.",
"In this respect, the plot of the deceleration parameter versus time has been given which shows that the parameter takes negative values for all values of time.",
"This is well-consistent with the current rapid expanding behavior of the universe.",
"It would be interesting to discuss the cylindrically or plane symmetric models using Noether symmetry analysis in the framework of scalar-tensor gravity with curvature correction."
]
] | 1403.0556 |
[
[
"Majorana modes and $p$-wave superfluids for fermionic atoms in optical\n lattices"
],
[
"Abstract We present a simple approach to create a strong $p$-wave interaction for fermions in an optical lattice.",
"The crucial step is that the combination of a lattice setup with different orbital states and $s$-wave interactions can give rise to a strong induced $p$-wave pairing.",
"We identify different topological phases and demonstrate that the setup offers a natural way to explore the transition from Kitaev's Majorana wires to two-dimensional $p$-wave superfluids.",
"We demonstrate how this design can induce Majorana modes at edge dislocations in the optical lattice, and we provide an experimentally feasible protocol for the observation of the non-Abelian statistics."
],
[
"Introduction",
"The quest for realisations of non-Abelian phases of matter, driven by their possible use in fault-tolerant topological quantum computing, has been spearheaded by recent developments in $p$ -wave superconductors.",
"The chiral $p_x + i p_y$ -wave superconductor in two-dimensions exhibiting Majorana modes provides the simplest phase supporting non-Abelian quasiparticles and can be seen as the blueprint of fractional topological order.",
"Alternatively, Kitaev's Majorana wire has emerged as an ideal toy model to understand Majorana modes.",
"Here, we present a way to make the transition from Kitaev's Majorana wires to two-dimensional $p$ -wave superconductors in a system with cold atomic gases in an optical lattice.",
"The main idea is based on an approach to generate $p$ -wave interactions by coupling orbital degrees of freedom with strong $s$ -wave interactions.",
"We demonstrate how this design can induce Majorana modes at edge dislocations in the optical lattice and we provide an experimentally feasible protocol for the observation of the non-Abelian statistics.",
"Candidates for topological phases supporting non-Abelian anyons [1] with potential application in topological quantum computing [2], [3] are found among a variety of systems including superfluid $^3$ He-A [4], the layered superconductor Sr$_2$ Ru O$_4$ [5], the fractional quantum Hall state at $\\nu =5/2$ [6], [7], and superconductor / topological insulator or similar heterostructures [8], [9], [10], [11].",
"Most recently, indium antimonide nano-wires in contact with an $s$ -wave superconductor have shown promising experimental evidence consistent with the presence of the sought-after non-Abelian zero-energy Majorana states [12], [13].",
"However, many questions still ask for a definitive answer.",
"Alongside the tremendous progress in solid-state systems, cold atomic gases provide a different angle when looking at $p$ -wave superconductors.",
"Thanks to their largely different strengths and shortcomings compared to solid-state systems, cold atomic gases might offer solutions to problems that are yet hard to address otherwise.",
"For instance, it is well known that the spatial dimension of a setup can easily be controlled by optical lattices, while Feshbach resonances allow one to tune the interaction strength almost at will [14].",
"Unfortunately, the lifetime of $p$ -wave resonant gases was found to be very limited [15], [16] due to a number of well understood decay channels [17], [18].",
"Identifying realisations of atomic $p$ -wave superfluids with a sufficient lifetime emerged as a central challenge in this field.",
"This led to proposals such as Bose-Fermi mixtures in shallow [19] or deep [20] optical lattices, microwave dressed polar molecules [21], the introduction of synthetic spin-orbit coupling into an $s$ -wave superfluid [22], the quantum Zeno effect [23], or driven dissipation [24].",
"However, the complexity in these proposed setups has so far precluded an experimental realisation.",
"Figure: Lattice setup.Spinless fermions residing at the lattice sites are coupled to a molecular state in the center of each plaquette.The molecular states exhibit a pp-wave symmetry and are doubly degenerate.",
"Anisotropic hoppings t y /t x t_{y}/t_{x}allow for the transition from coupled wires to the 2D isotropic system.Here, we present a simple approach to create a strong $p$ -wave interaction for fermions in an optical lattice.",
"The main idea is based on a resonant coupling from the lattice sites to a molecular state residing in the center of the plaquette in analogy to Ref.",
"[25], [26].",
"The crucial step is that the combination of a lattice setup with different orbital states and $s$ -wave interactions can give rise to a strong induced $p$ -wave pairing; similar ideas have recently been proposed [27].",
"We will demonstrate the appearance of $p$ -wave superfluid phases via this coupling.",
"Moreover, a setup intrinsically based on an optical lattice allows one to naturally explore the transition from a two-dimensional $p$ -wave superfluid to Kitaev's Majorana wire [28].",
"Hence we identify different topological transitions [29], where the combination of the Fermi-surface topology with the symmetry of the $p$ -wave superfluid order parameter gives rise to a rich phase diagram.",
"Most remarkably, we find the appearance of Majorana modes localized at edge dislocations; such edge dislocations correspond to vortices in the phase of the lasers generating the lattice.",
"In combination with another realistic ingredient to modern cold atoms experiments, single site addressability in the lattice [30], [31], we provide a protocol for the observation of the non-Abelian braiding statistics."
],
[
"Effective Hamiltonian",
"We start with the presentation of the Hamiltonian underlying our system.",
"We focus on a setup of spinless fermionic atoms in a quadratic two-dimensional optical lattice.",
"Then, the Hamiltonian is well described by the tight-binding model $H =- \\sum _{\\langle i j\\rangle } t_{i j} c^{\\dag }_{i} c_{j} -\\mu \\sum _{i} c^{\\dag }_{i} c_{i}+ H_{x} +H_{y}.",
"$ Here, $c_{i}^{\\dag }$ and $c_{i}$ denote the fermionic creation (annihilation) operators at lattice site $i$ , while $\\mu $ is the chemical potential fixing the average particle number, and $t_{i j}$ denotes the hopping energy between nearest neighbor sites $\\langle i j \\rangle $ .",
"In order to study the transition from a bulk two-dimensional setup to weakly coupled one-dimensional chains, we allow for an anisotropic hopping $t_{i j}$ , where $t_{ij}=t_{x(y)}$ for hopping along a link in the x-(y-)direction, respectively.",
"The interaction between the fermions is driven by resonant couplings $H_{x(y)}$ to two distinct lattice bound states $X_p$ and $Y_p$ residing in the center of each plaquette as shown in Fig.",
"REF ; similar setups for bosonic atoms have been previously proposed [25].",
"For spinless fermions on the lattice sites, these bound states must exhibit an odd parity symmetry for a non-vanishing interaction, which in our situation is a two-fold degenerate $p$ -wave symmetry.",
"Then, the coupling Hamiltonians reduce to $H_{x} & =& \\gamma \\sum _{p}X^{\\dag }_{p} X_{p} + g \\sum _{p} \\left[ X^{\\dag }_{p} \\left( c_{2} c_{3} - c_{4} c_{1}\\right)+ {\\rm h.c.} \\right], \\nonumber \\\\H_{y} &=& \\gamma \\sum _{p} Y^{\\dag }_{p} Y_{p} + g \\sum _{p} \\left[ Y^{\\dag }_{p} \\left( c_{1} c_{2} - c_{3} c_{4}\\right)+ {\\rm h.c.} \\right],$ where the summation $\\sum _{p}$ runs over all plaquettes.",
"The four lattice sites surrounding each plaquette are labelled as shown in Fig.",
"REF .",
"The couplings to the lattice bound states respect the $p$ -wave symmetry with coupling strength $g$ , while the detuning from resonance is given by $\\gamma $ .",
"The latter quantity also includes the chemical potential $\\gamma = \\hbar \\omega - 2 \\mu $ with $\\hbar \\omega $ the energy difference between the molecular state and two free fermions.",
"The most crucial part is the possibility to induce a strong $p$ -wave interaction by the combination of orbital degrees of freedom and $s$ -wave interactions.",
"Here, we provide a sketch of this fundamental idea (for details we refer to the Appendix): The two-particle states $X_{p}$ and $Y_{p}$ in the center of the plaquette consist of two orbital states in the optical lattice forming a repulsively bound state [32].",
"In order for these lattice molecules to fulfill the $p$ -wave symmetry, we choose the lowest and the first excited state in the lattice confining the atoms in the center of the plaquette.",
"Furthermore, the two fermions in the two orbital states have to be in different hyperfine states in order to profit from a stable $s$ -wave interaction which can be tuned by conventional Feshbach resonances [33].",
"This requires the coupling to the plaquette states to induce transitions between hyperfine states.",
"To summarize: the $s$ -wave interaction leads to the formation of repulsively bound pairs, while the orbital degree of freedom is responsible for the $p$ -wave character of these lattice bound molecules.",
"It is via this mechanism that the optical lattice breaks rotational symmetry and couples to states with different orbital symmetry allowing for the conversion of $s$ -wave to $p$ -wave interactions.",
"Figure: Topological phases and mean-field phase diagram.a, Topological phase diagram: We can distinguish between three different topological regions.However, the topological indices also depend on the order parameter of the superfluid (see main text).The different phases are denoted as SF ν:ν x ν y _{\\nu :\\nu _{x}\\nu _{y}} for time-reversal invariant andcSF ν:ν x ν y _{\\nu :\\nu _{x}\\nu _{y}} for chiral superfluids with ν\\nu the strong topological index and ν x,y \\nu _{x,y} the weak ones.b, Mean-field phase diagram for γt x /g 2 =1.5\\gamma t_{x}/g^2 = 1.5:For the rotationally symmetric setup with t x =t y t_{x}=t_{y} the ground state is given by a p x +ip y p_x+i p_y superfluid.While for strong anisotropy t x ≠t y t_{x} \\ne t_{y} a pure p x p_{x} or p y p_{y} superfluid order parameter dominates.The grey dots mark the points, where the gap parameters are too small for a convergence of the numerical calculations,and therefore no superfluid phase is accessible for experimentally realistic temperatures.c-d shows the gap parameters Δ x \\Delta _{x}, Δ y \\Delta _{y}, and (|Δ x |-|Δ y |)/(|Δ x |+|Δ y |)(|\\Delta _{x}|-|\\Delta _{y}|)/(|\\Delta _{x}|+|\\Delta _{y}|) along a cut through the phase diagramat t y =t x /2t_{y}=t_{x}/2 [see arrow in b], as well as the different topological indices.",
"It is important to stress, that the topological index ν y \\nu _{y}jumps to 1 at a different position, than the vanishing of Δ y \\Delta _{y}, i.e., the topological transitions are essentially decoupled from the mean-field transitions."
],
[
"Mean-field theory",
"We first study the zero temperature phase diagram within mean-field theory.",
"Such a mean-field analysis is well justified as recent numerical calculations in a particle number conserving approach have demonstrated the appearance of a $p$ -wave superfluid exhibiting Majorana modes [26].",
"The authors of Ref.",
"[26] studied a two wire setup with a similar interaction between the wires which quantitatively agrees with the mean-field expectations.",
"We tested that the same agreement is also valid for a three-wire setup.",
"As we are here interested in a higher dimensional setup, we expect that the quantitative behavior of the phase diagram is again well captured within mean-field theory.",
"The resonant coupling to the molecular states gives rise to superconducting $p$ -wave pairing via the formation of a Bose-Einstein condensate in the molecular states.",
"Introducing the order parameters $\\Delta _{x} = 4 g \\langle \\sum _{p} X_{p}\\rangle / N $ and $\\Delta _{y} = 4 g \\langle \\sum _{p} Y_{p}\\rangle /N $ for the macroscopic occupation of the molecular state at zero momentum, the Hamiltonian reduces to a quadratic fermionic theory $H= \\frac{1}{2 } \\sum _{{\\bf q}}\\left( \\begin{array}{c} c^{\\dag }_{{\\bf q}} \\\\ c_{-{\\bf q}}\\end{array} \\right)^T \\left(\\begin{array} {c c}\\epsilon _{{\\bf q}} & \\Delta _{\\bf q}\\\\\\Delta _{\\bf q}^{*}& - \\epsilon _{\\bf q}\\end{array} \\right) \\left( \\begin{array}{c} c_{{\\bf q}} \\\\ c^{\\dag }_{-{\\bf q}}\\end{array} \\right) + \\mathcal {F}_{0}.$ Here, $c_{\\bf q} = \\sum _{i} e^{i {\\bf q} {\\bf r}_{i}} c_{i}/\\sqrt{N}$ , and $N$ denotes the number of lattice sites.",
"$\\mathcal {F}_{0}$ accounts for the conventional operator independent parts.",
"Furthermore, the tight-binding dispersion for the fermions reduces to $\\epsilon _{\\bf q} = - 2 \\sum _{\\alpha \\in \\lbrace x,y\\rbrace } t_{\\alpha } \\cos (q_\\alpha a) - \\mu $ , while the gap parameter takes the form of a $p$ -wave superfluid $\\Delta _{\\bf q} = - i \\left[ \\Delta _{x} \\sin (q_{x } a)+\\Delta _{y} \\sin (q_{y } a) \\right],$ where $a$ denotes the lattice spacing.",
"Using a Bogoliubov transformation, we obtain the superfluid excitation spectrum $E_{\\bf q} = \\sqrt{\\epsilon _{\\bf q}^2 + |\\Delta _{\\bf q}|^2}$ , and the ground-state energy per unit cell $\\mathcal {F}(\\Delta _{x},\\Delta _{y}) = \\int \\frac{\\mathrm {d}{\\bf q}}{v_{0}} \\frac{ \\epsilon _{\\bf q} - E_{\\bf q}}{2} + \\frac{\\gamma }{16 g^2} \\Big [ |\\Delta _{x}|^2 +|\\Delta _{y}|^2 \\Big ],$ with $v_{0} = (2 \\pi )^2/a^2$ denoting the volume of the first Brillouin zone.",
"The order parameters $\\Delta _{x}$ and $\\Delta _{y}$ are determined by the gap equation minimizing the ground state energy $\\partial _{\\Delta _x} \\mathcal {F}(\\Delta _{x},\\Delta _{y}) = \\partial _{\\Delta _y} \\mathcal {F}(\\Delta _{x},\\Delta _{y}) =0.$ The results of the mean-field theory are shown in Fig.",
"REF b: we find a $p_{x} + i p_{y}$ superfluid for the fully isotropic setup with $t_{x} = t_{y}$ where $\\Delta _{x} = \\pm i \\Delta _{y}$ .",
"In addition to the $U(1)$ symmetry breaking, this phase also breaks time reversal symmetry.",
"For finite interaction strength the $p_{x} + i p_{y}$ superfluid is stable to a small anisotropy in the hopping.",
"Note that the anisotropic behavior is reflected in the order parameter, i.e.",
"$|\\Delta _{x}| \\ne |\\Delta _{y}|$ .",
"However, we denote a $p_{x} + i p_{y}$ superfluid as a phase with a finite order parameter $\\Delta _{x}$ and $\\Delta _{y}$ obeying the fixed phase relation $\\Delta _{x}/\\Delta _{y} = \\pm i |\\Delta _{x}/\\Delta _{y}|$ .",
"For increasing anisotropy $t_{x}\\ne t_{y}$ transitions into a $p_{x}$ ($p_{y}$ ) superfluid can appear, depending on the value of the chemical potential $\\mu $ ."
],
[
"Topological phase transitions",
"In addition to the mean-field transitions, the lattice system also exhibits a series of topological quantum phase transitions beyond those found in the classification of continuum 2D superfluids [34].",
"In the parameter regime, where the superfluid exhibits an excitation gap, the topological properties are characterized by three topological indices [29]: the first denotes the strong topological index given by the Chern number $\\nu $ characterizing the two-dimensional $p_{x}+ i p_{y}$ superfluid, and takes values $\\nu = 0, \\pm 1$ , see Fig.",
"REF a.",
"In addition, the system exhibits two weak topological indices [35], [36], which we denote as $\\nu _{x} = 0,1$ and $\\nu _{y}= 0,1$ .",
"The latter quantities are responsible for the appearance of Majorana modes in Kitaev's Majorana wire [28], and can be finite in $p_{x}$ superfluids as well as chiral $p_{x}+i p_{y}$ superfluids.",
"It is important to stress that the phase boundaries for the topological phase transitions are independent of the strength of the superfluid order parameters, and they only depend on the topology of the Fermi surface.",
"Therefore, we can distinguish between three different regions, see Fig.",
"REF a: region (I) with a closed Fermi surface, region (II) with an open Fermi surface, and finally the strong pairing regime (III), where in absence of interactions the system is in a trivial band insulating (vacuum) state.",
"In the latter region (III), the superfluids exhibit no topological order with $\\nu =\\nu _{x}=\\nu _{y}=0$ ; thus it is not of interest in the following.",
"The combination of the topological indices with the superfluid order parameter allows us now to characterize the different phases.",
"We use the notation SF$_{\\nu :\\nu _{x}\\nu _{y}}$ for time-reversal invariant superfluids and cSF$_{\\nu :\\nu _{x}\\nu _{y}}$ for chiral superfluids.",
"First, we start with the chiral $p_{x} + i p_{x}$ superfluid.",
"Here, we obtain two fundamentally different topological phases, see Fig.",
"REF a: (I) the strong topological superfluids cSF$_{-1:00}$ and cSF$_{1:11}$ with a finite Chern number $\\nu =\\pm 1$ .",
"Within the standard symmetry classification scheme [37], [38], [39], the cSF$_{\\pm 1:\\nu _{x}\\nu _{y}}$ phase is in the symmetry class D (particle-hole symmetry).",
"It is a special property of this phase that the weak indices depend on the chemical potential, i.e., we obtain $\\nu _{x}=\\nu _{y}=1$ for $\\mu >0$ and $\\nu _{x}=\\nu _{y}=0$ for $\\mu <0$ .",
"This property will strongly influence the Majorana modes, see below.",
"In region (II), we find a weak topological superfluid in the symmetry class D (cSF$_{0:01}$ ).",
"On the other hand, for the $p_{x}$ superfluid, we obtain a weak topological superfluid (SF$_{0:01}$ ) in region (II) which belongs to the class BDI, see Fig.",
"REF a.",
"While in the region with closed Fermi surface (I) the superfluid phase becomes gapless without any topological properties.",
"For completeness, we point out that the $p_{y}$ superfluid is gapless in region (I) and (II).",
"The full phase diagram is then obtained by combining the mean-field phase diagram with the topological properties.",
"Its details strongly depend on the strength of the coupling parameters.",
"Here, we are mainly interested in strong couplings with $g^2/\\gamma \\sim t_{x}, t_{y}$ with large superfluid gaps.",
"Most remarkably, we find that for $\\gamma t_{x}/g^2 =1.5$ all of the above discussed topological phases are realized for varying values of $\\mu /t_x$ and $t_{y}/t_{x}$ , see Fig.",
"REF .",
"Figure: Braiding of dislocation based Majorana modes as touchstone of non-Abelian statistics.a, Majorana modes at lattice dislocations: an edge dislocation pair with Burgersvector ±𝐞 y \\pm {\\bf e}_{y} (black arrow) forming a single quantum wire (red line) immersed into the bulksuperfluid.",
"For a finite weak topological index ν y =1\\nu _{y}=1 such a setup generates twoMajorana modes at the end of the wire.",
"The localized wave function of the two Majoranamodes (blue circles) is determined by numerically solving the Bogoliubov-de-Gennes equations.b, Four Majorana modes are generated by the formation of two dislocation pairs, and then spatially separated: first along path 1 and subsequently along path 2.c, The braiding operation is achieved by recombining the Majorana modes along path 3 and then path 4.d, The braiding transforms the initially unoccupied state at each dislocation pair into an occupied fermion mode.The subsequent measurement of the unpaired fermions is a unique signature of the non-Abelian braiding statistics of the Majorana modes."
],
[
"Majorana modes at edge dislocations",
"Associated with the topological index, we expect the appearance of Majorana modes at topological defects in the system.",
"Here, such topological defects can either be vortices or — as a distinct feature of the lattice setup — also lattice dislocations.",
"Generally, we expect the Majorana modes in the vortex core for Chern number $\\nu =\\pm 1$ , while the weak index $\\nu _{x,y}$ gives rise to Majorana modes localized at lattice dislocations with Burgers vector $\\pm {\\bf e}_{x,y}$ .",
"The latter can be easily understood in the limit $t_{y}=0$ , where the system reduces to coupled one-dimensional wires: then, a pair of dislocations corresponds to the inclusion/removal of a one-dimensional wire of finite length into the bulk 2D system, see Fig.",
"REF a.",
"This bulk superfluid induces a $p$ -wave superfluid onto this single chain realizing the ideal toy model of a single Majorana chain [28]; this behavior is in analogy to the proposals for the realisation of Majorana modes in solid state systems [8], [9].",
"The existence of Majorana modes is most conveniently verified using the Bogoliubov-de-Gennes equation, which can be efficiently solved numerically, an example is illustrated in Fig.",
"REF a.",
"In summary, we find for the cSF$_{1:\\nu _{x}\\nu _{y}}$ phase Majorana modes in the core of vortices.",
"Most remarkably, the system also exhibits Majorana modes at edge dislocations ${\\bf e}_{x,y}$ , but only for positive chemical potential $\\mu > 0$ with a finite weak topological index $\\nu _{x}=\\nu _{y}=1$ .",
"Similarly, the cSF$_{0:01}$ and SF$_{0:01}$ only exhibit Majorana modes at edge dislocations with Burgers vector $\\pm {\\bf e}_{y}$ .",
"Nevertheless, the two phases show distinct features due to their different symmetry classification: for the chiral phase cSF$_{0:01}$ in symmetry class D, the topological index is a $\\mathbb {Z}_{2}$ index.",
"Therefore, a pair of double dislocations with Burgers vector $\\pm 2 {\\bf e}_{x, y}$ leads to a hybridization of the Majorana modes, and consequently no ground state degeneracy.",
"In turn, the time reversal symmetric phase SF$_{0:01}$ is in the symmetry class BDI, which gives rise to a $\\mathbb {Z}$ topological index.",
"Consequently, a pair of double dislocations essentially describes a two wire setup and provides four Majorana modes with a four-fold ground state degeneracy.",
"This behavior is well confirmed within the numerical solution of the Bogoliubov-de-Gennes equations."
],
[
"Braiding of non-Abelian anyons and Outlook",
"In cold atomic gases, an edge dislocation corresponds to a vortex in the optical field generating the optical lattice [40].",
"Such edge dislocations are most conveniently generated in a setup with local site addressability [30], [31], [41], where arbitrary shapes of the lattice can be achieved.",
"In combination with a time dependent modulation of the masks generating the lattice, a full spatial and temporal control on edge dislocations is foreseeable in the near future.",
"Such a setup then offers the opportunity for the observation of the non-Abelian statistics of Majorana modes by braiding the dislocations.",
"While the braiding of vortices in a superfluid has previously been predicted for the observation of the non-Abelian statistics [42], [43], such experiments suffer from the difficulty to control a collective degree of freedom such as the superfluid phase, and the problem to insert adiabatically vortices into a superfluid.",
"Here, edge dislocations in the lattice are much more favorable due to the precise and simple control on lattice structures available in cold atomic gases.",
"In the following, we present the protocol for measuring the non-Abelian statistics of the Majorana fermions, see Fig.",
"REF b-d.",
"It is important that all operations are performed adiabatically, i.e., slower than the characteristic time scale given by the superfluid gap.",
"(a) In a first step, we initialize the system by adiabatically creating two dislocation pairs.",
"At each pair, we obtain a single fermionic mode described by the operators $c^{\\dag }_{r,b}$ with a finite energy gap.",
"This fermion mode is unoccupied as at low temperatures all fermions are Cooper paired.",
"The next step separates the two dislocation pairs first along path 1 and then along path 2.",
"This operation splits the fermionic modes into Majorana modes localized at the edge dislocations, and gives rise to a four-fold degenerate ground state of which two are accessible at fixed fermion number parity.",
"However, adiabaticity of the process ensures a well defined initial state with $c_{r,b}|g\\rangle =0$ .",
"(b) Next, we perform the braiding by recombining the two dislocation pairs along path 3 and finally path 4.",
"This process corresponds to moving the two Majorana modes around each other.",
"According to the general non-Abelian braiding rules for Majorana modes [44], this transforms the fermionic operators via $c_{r,b} \\rightarrow c^{\\dag }_{r,b}$ ; here, we drop a phase factor, which is irrelevant for the protocol.",
"As a consequence, the initially unoccupied state becomes occupied by one fermion each, i.e.",
"$c^{\\dag }_{r,b}|g\\rangle =0$ .",
"In a physical picture, the braiding operation takes one Cooper pair from the superfluid condensate and splits it into two fermions with one residing at each dislocation pair.",
"To probe the system one ramps the energy difference of the molecular state to the free fermionic states $\\hbar \\omega $ to negative values, which drives the system into the strong pairing phase with all paired fermions residing in the center of the plaquettes.",
"This procedure is the analogue to the process of forming pairs via a Feshbach resonance [32].",
"Finally, a measurement of the fermionic density on the original lattice sites [30] probes the unpaired fermions in the system.",
"Here, we expect one unpaired fermion at each dislocation pair.",
"In order to test the protocol against induced noise, finite temperature, or violation of adiabaticity, one can test the process against a background measurement with a reversed order of path 3 and 4.",
"Since this process does not braid the two Majorana modes, no unpaired fermions should be present in an ideal experiment.",
"Acknowledgements: We acknowledge support by the Center for Integrated Quantum Science and Technology (IQST) and the Deutsche Forschungsgemeinschaft (DFG) within SFB TRR 21, the Leverhulme Trust (ECF-2011-565), the Newton Trust of the University of Cambridge, the Royal Society (UF120157), SFB FoQus (FWF Project No.",
"F4006-N16), the ERC Synergy Grant UQUAM, SIQS, and Swiss National Science Foundation.",
"GM, SH, CK, and HB thank the Institut d'Etudes Scientifiques Cargèse and CECAM for their hospitality."
],
[
"Appendix",
"Microscopic setup.",
"The fermionic states described by the operators $c_{i}$ ($c_{i}^{\\dag }$ ) reside on the sites of the optical lattice and are in the lowest Bloch band.",
"The design of the interaction requires the coupling of these states to different internal states trapped by an optical lattice with the minima in the center of the plaquettes.",
"Such a setup is most conveniently achieved for cold atomic gases with a metastable $^{3}P_{2}$ state such as $^{87}\\rm {Sr}$ or $^{171}\\rm {Yb}$ .",
"Then, the metastable $^{3}P_{2}$ states are trapped at the sites of the lattice, while the ground state $^1S_{0}$ is trapped in the center of the plaquette for an optical lattice close to the anti-magic wavelength.",
"Therefore, the setup requires only a single two-dimensional optical lattice.",
"In addition, light assisted two-particle losses from the metastable $^{3}P_{2}$ are quenched due to the fermionic statistic.",
"Next, we focus on the state trapped in the center of the plaquette.",
"We are interested in two different hyperfine states in the electronic the ground state $^1S_{0}$ , which will be denoted by a spin index $\\sigma $ with $\\sigma \\in \\lbrace \\downarrow , \\uparrow \\rbrace $ , and a setup with suppressed tunneling between different plaquettes.",
"The lowest lying state exhibits $s$ -wave symmetry and will be denoted as $|0,\\sigma \\rangle _{p} $ , while the the first excited state $|\\alpha ,\\sigma \\rangle _{p}$ with $\\alpha \\in \\lbrace x,y\\rbrace $ is two fold degenerate and exhibits a $p$ -wave symmetry, see Fig.",
"REF b.",
"Figure: Microscopic setup.",
"a, The lattice sites |i〉|i\\rangle are coupled to two different internal states trapped in the center of the plaquette.The first one |0,↑〉 p |0,\\uparrow \\rangle _{p} exhibits an ss-wave orbital symmetry, while the second one |α,↓〉 p |\\alpha ,\\downarrow \\rangle _{p}shows a pp-wave symmetry and is two-fold degenerate.",
"b,Single particle level structure with the relevant transitions (solid lines).",
"The additional transitions (dashed lines) are required for the design of the desired coupling Hamiltonian.",
"c, Energy levels for the two-particle states with the two interfering paths: |ψ〉|\\psi \\rangle describes the state with two fermions on the lattice sites surrounding the plaquette, while |ψ ˜〉=(A p † B αp † -A ¯ p † B ¯ αp † )|ψ〉|\\tilde{\\psi }\\rangle = (A^{\\dag }_{p} B_{\\alpha p}^{\\dag }-\\bar{A}^{\\dag }_{p} \\bar{B}_{\\alpha p}^{\\dag }) |\\psi \\rangle describes the near resonant repulsively bound molecule with pp-wave symmetry.The coupling between the states on the lattice to the center of the plaquette is driven by two Raman transitions.",
"The first Raman transition with detuning $\\delta _{a}$ couples to the state $|0,\\uparrow \\rangle _{p}$ providing the Hamiltonain $H_{a} = w_{a} \\sum _{p}( A^{\\dag }_{p}+ A_{p})$ with $ A^{\\dag }_{p} = a^{\\dag }_{p}\\left(c_{1}+ c_{2} + c_{3} + c_{4}\\right)$ , and the operator $a^{\\dag }_{p}$ creating a fermion in the state $|0,\\uparrow \\rangle _{p}$ .",
"The coupling strength $w_{a}$ accounts for the Rabi frequency as well as the wave function overlap.",
"Note that the form of the coupling is determined by the $s$ -wave symmetry of the state $|0,\\uparrow \\rangle _{p}$ .",
"In analogy, the second Raman transition couples to the states $|\\alpha ,\\downarrow \\rangle _{p}$ with fermionic operators $b^{\\dag }_{\\alpha p}$ ($\\alpha \\in \\lbrace x,y\\rbrace $ ) and the detuning $\\delta _{b}$ .",
"In order to simplify the discussion, we set $\\delta _{b}=-\\delta _{a}$ and $w_{a}=w_{b}$ .",
"Then, the coupling Hamiltonian reduces to $H_{b} = w_{b} \\sum _{p \\alpha }( B^{\\dag }_{\\alpha p}+ B_{\\alpha p}) $ with $B_{x,y p}^{\\dag } = b^{\\dag }_{x,y p}\\left(c_{1}\\pm c_{2} -c_{3}\\mp c_{4}\\right)$ .",
"Note, the different couplings due to the orbital $p$ -wave symmetry of the states $|\\alpha ,\\downarrow \\rangle _{p}$ .",
"The main idea for the design of the interaction is now the fact, that the state $a^{\\dag }_{p} b^{\\dag }_{\\alpha p}|0\\rangle $ with two fermions in the center of the plaquette exhibits a strong onsite interaction $U$ due to the $s$ -wave scattering between two different hyperfine states.",
"Within the rotating frame its energy is given by $\\hbar \\omega =\\delta _{a}+\\delta _{b}+U=U$ , see Fig.",
"REF c. This motivates the introduction of two bosonic molecular states $X^{\\dag }_{p} = a^{\\dag }_{p} b^{\\dag }_{x p}$ and $Y^{\\dag }_{p} = a^{\\dag }_{p} b^{\\dag }_{y p}$ exhibiting orbital $p$ -wave symmetry.",
"For a choice of the detunings with $\\hbar |\\omega | \\ll |\\delta _{a}|$ , we can then adiabatically eliminate all states with a single fermion in the center of the plaquette and arrive at the effective coupling Hamiltonian $H_{c} = \\bar{g}\\sum _{p, \\alpha } \\left[ B^{\\dag }_{\\alpha p}A_{p}^{\\dag } +A_{p} B_{\\alpha p}\\right] \\, ,$ with $\\bar{g}= |w_{a}|^2 U/(U^2-\\delta _{a}^2)$ .",
"Note, that we have omitted additional terms describing an induced hopping of the fermionic operators $c_{i}$ ; these terms will be discussed below.",
"The resonant coupling of the fermionic states $c_{i}$ to the $p$ -wave molecules $X_{p}$ and $Y_{p}$ residing in the center of the plaquette reduces to $B^{\\dag }_{x p} A^{\\dag }_{p}&= & 2 X^{\\dag }_{p} \\left[ c_{2} c_{3} - c_{4} c_{1} + c_{1}c_{3}+ c_{2} c_{4}\\right],\\\\B^{\\dag }_{y p} A^{\\dag }_{p}&= & 2 Y^{\\dag }_{p} \\left[ c_{1} c_{2} - c_{3} c_{4} + c_{1}c_{3}- c_{2} c_{4}\\right].$ This coupling term differs from the desired interaction in Eq.",
"(REF ); the last two terms, describe a second representation of the $p$ -wave symmetry for the coupling.",
"While this coupling Hamiltonian gives rise to interesting $p$ -wave superfluids, it is desirable to suppress these additional coupling terms.",
"In the following, we present a scheme, which completely quenches these terms, while for an experimental realisation it is sufficient to weakly suppress them.",
"The scheme is achieved by an additional transition with opposite detunings but equal coupling strengths, where the phase is spatially varying.",
"The main requirement on the phase is, that the coupling to the state $c_{1}$ exhibits the opposite sign than the coupling to $c_{2}$ , while $c_{1}$ and $c_{3}$ have the same sign.",
"The desired behaviour is achieved employing the principles of Ref.",
"[45], by adding Raman lasers with a contribution of the wave vector $\\mathbf {k}_{\\parallel }$ within the plane of the optical lattice; i.e., $\\mathbf {k}_{\\parallel }= k_0 (\\mathbf {e}_x - \\mathbf {e}_y)$ with $k_0$ the wavelength of the square lattice potential.",
"(By contrast, the Raman lasers for transitions $A_p^\\dag $ , $B_p^\\dag $ must be incident at a right angle to the system.)",
"Then, we obtain the additional coupling terms $ \\bar{A}^{\\dag }_{p} = a^{\\dag }_{p}\\left(c_{1}- c_{2} +c_{3} - c_{4}\\right)$ and $ \\bar{B}_{x,y p}^{\\dag } = b^{\\dag }_{x,y p}\\left(c_{1} \\mp c_{2} - c_{3} \\pm c_{4}\\right)$ .",
"The full coupling Hamiltonian exhibits interference between the two independent excitation channels for the molecules, see Fig.",
"REF c, and reduces to $H_{c} = \\bar{g} \\sum _{p, \\alpha } \\left[ B^{\\dag }_{\\alpha p}A_{p}^{\\dag } - \\bar{B}^{\\dag }_{\\alpha p} \\bar{A}^{\\dag }_{p} + {\\rm h.c.}\\right],$ which reduces to the desired coupling in Eq.",
"(REF ) with $\\hbar \\omega = U$ and $g=4 \\bar{g}=4 |w_{a}|^2 U/(U^2-\\delta _{a}^2)$ .",
"In addition, the induced hopping terms via the single excitation in the center of the plaquette reduces to an additional conventional hopping as in Eq.",
"(REF ) with $t_{x}=t_{y} = 2 |w_{a}|^2 /\\delta _{a}$ .",
"Its interference with the direct hopping allows us to tune the ratios $g/U$ and $t/U$ independently."
]
] | 1403.0593 |
[
[
"Ehrenfest principle and unitary dynamics of quantum-classical systems\n with general potential interaction"
],
[
"Abstract Representation of classical dynamics by unitary transformations has been used to develop unified description of hybrid classical-quantum systems with particular type of interaction, and to formulate abstract systems interpolating between classical and quantum ones.",
"We solved the problem of unitary description of two interpolating systems with general potential interaction.",
"The general solution is used to show that with arbitrary potential interaction between the two interpolating systems the evolution of the so called unobservable variables is decoupled from that of the observable ones if and only if the interpolation parameters in the two interpolating systems are equal."
],
[
"Introduction",
"Koopman-von Neumann (KvN) [1] unitary description of the Liouville equation of classical Hamiltonian dynamical systems was utilized for modeling hybrid quantum-classical systems for the first time by Sherry and Sudarshan [2].",
"They analyzed particular types of interaction between the classical and the quantum parts, and ad hoc prescriptions for definitions of the corresponding Hilbert space operators.",
"It was shown that the pre-measurement process can be modeled as an interaction between a classical apparatus and a quantum system within the unitary framework.",
"Sherry and Sudarshan also analyzed the so called integrity conditions which ought to be satisfied in order that classical variables remain classical during the hybrid unitary evolution in the Heisenberg form.",
"Peres and Terno [3] analyzed consistency of the Koopman-von Neumann-Sudarshan (KNS) hybrid dynamics with the quantum-quantum and the classical-classical limits for the case of linear interaction between harmonic oscillators.",
"Some aspects of the KNS formalism for hybrid system with specific interaction have also been studied in [4].",
"The authors investigated the role of unphysical variables which are called unobservables because they do not influence the evolution of the physical observables of the quantum or the classical part if there is no quantum-classical interaction.",
"It was observed, using particular examples of quantum-classical interaction and specific forms of its Hilbert space description, that the evolution of the unobservable and observable variables become coupled.",
"More recently, KvN formalism and Ehrenfest principle were used to propose a family of abstract unitary systems interpolating between classical system and its quantized counterpart [5].",
"The problem of hybrid dynamics was not analyzed using the interpolating systems.",
"Our goal is to study the same type of questions, but for the most general potential interaction between the classical and the quantum systems.",
"In fact, we shall obtain unitary dynamical equations for two interpolating abstract systems (IAS) with general potential interaction, and use this to show that generally the evolution of the unphysical variables is decoupled from that of the physical ones if and only if the interpolation parameters in the two IAS are equal.",
"In particular, unitary dynamics of hybrid systems with potential interaction in general couples the dynamics of the two types of variables.",
"However, there is one special case in the family of general solutions such that the corresponding quantum-classical potential interaction does not couple the physical and the unphysical variables, and implies other properties consistent with this fact."
],
[
"Interpolating abstract systems and hybrid models",
"Dynamical equations for averages of the basic observables of a classical system and that of its quantized counterpart can be mathematically interpolated by an abstract system that depends on suitable parameter.",
"The first step to achieve this is to rewrite the dynamics of classical and quantum averages using the same mathematical framework.",
"This can be done by rewriting the classical dynamics as a unitary evolution on a suitable Hilbert space, or by rewriting the unitary Schrödinger equation as a (linear) Hamiltonian system on a symplectic manifold.",
"We shall treat here the unitary approach with general potential interaction.",
"Consider an abstract dynamical system with the basic variables $x_j,p_j,\\chi _j,\\pi _j$ (hereafter $j=1,2$ ).",
"Properties of the system, expressed through appropriate algebraic relations between the basic variables, are supposed to depend on parameters $a_j$ .",
"The basic variables satisfy commutation relations $[x_j,p_j]=i\\hbar \\;\\!",
"a_j,\\quad [x_j,\\pi _j]=[\\chi _j,p_j]=i\\hbar ,$ with all other commutators being zero.",
"Let us suppose that the algebra (REF ) is represented by operators acting on a Hilbert space ${\\cal H}$ .",
"Assume that the dynamical variables $x_j,p_j$ are measurable and that their averages in a state $|\\psi \\rangle \\in {\\cal H}$ are computed as $\\langle x_j\\rangle _\\psi =\\langle \\psi |\\hat{x}_j|\\psi \\rangle ,\\quad \\langle p_j\\rangle _\\psi =\\langle \\psi |\\hat{p}_j|\\psi \\rangle .$ Suppose that the dynamics of these averages is given by the Ehrenfest principle $\\frac{d}{dt}\\langle \\psi (t)|\\hat{x}_j|\\psi (t)\\rangle &=\\langle \\psi (t)|\\frac{\\hat{p}_j}{m_j}|\\psi (t)\\rangle , \\\\\\frac{d}{dt}\\langle \\psi (t)|\\hat{p}_j|\\psi (t)\\rangle &=\\langle \\psi (t)|-V_j^{\\prime }(\\hat{x}_j)|\\psi (t)\\rangle ,$ and that the state evolution is unitary $i\\hbar |\\dot{\\psi }\\rangle =\\hat{H}_{I\\!AS}|\\psi \\rangle $ .",
"The corresponding evolution equations for the dynamical variables in the Heisenberg form are $i\\hbar \\,d\\hat{x}_j/dt=[\\hat{x}_j, \\hat{H}_{I\\!AS}]$ and analogously for $\\hat{p}_j,\\hat{\\chi }_j,\\hat{\\pi }_j$ .",
"The operator $\\hat{H}_{I\\!AS}$ is the evolution generator and might depend on all dynamical variables $\\hat{H}_{I\\!AS}=H_{I\\!AS}(\\hat{x}_j,\\hat{p}_j,\\hat{\\chi }_j,\\hat{\\pi }_j)$ .",
"It is not necessarily interpreted as the physical energy.",
"It should be remarked that the relations (REF ) and () are treated as axioms in the general abstract formulation [5], expressing the conservative nature of the dynamics.",
"Following the approach of [5], one can obtain the class of evolution generators yielding () $\\hat{H}_{I\\!AS} &=\\sum _{j=1,2} \\frac{1}{a_j}\\bigg (\\frac{\\hat{p}_j^2}{2m_j}+V_j^{}(\\hat{x}_j)\\bigg )\\nonumber \\\\ &+ F_j(\\hat{x}_j-a_j\\:\\!\\hat{\\chi }_j,\\hat{p}_j-a_j\\:\\!\\hat{\\pi }_j),$ where $F_j$ are arbitrary functions of the indicated arguments.",
"Observe that, consistent with (), there are no terms coupling observables with different subscripts, so that the abstract system (REF ) can be interpreted as a compound system with two noninteracting components.",
"Explicit representation of the operator $\\hat{H}_{I\\!AS}$ depends on the representation space ${\\cal H}$ , and is not important in our analysis.",
"Nevertheless, it should be remarked that the Hilbert space ${\\cal H}$ is determined as a space of an irreducible representation of the algebra (REF ), and is the same space for any value of the parameters $a_j$ .",
"In particular, it is seen that in the case we want to represent two quantum systems, the Hilbert space needed to accommodate (REF ) with $a_1=a_2=1$ is larger than the space $L_2(x_1)\\otimes L_2(x_2)\\equiv L_2(x_1,x_2)$ which is relevant in the standard quantum mechanics without the additional variables $\\chi _j,\\pi _j$ .",
"It can be shown that one irreducible representation of the algebra is provided with the Hilbert space of operators on $L_2(x_1,x_2)$ [6].",
"Thus, the vectors from ${\\cal H}$ can be considered as density matrices or mixed states of the quantum-quantum system [5].",
"Similarly, if the abstract systems represent two classical systems, i.e.",
"when $a_1=a_2=0$ so that $\\hat{x}_j,\\hat{p}_j$ all commute, the interpretation of the state $|\\psi \\rangle $ is that of the amplitude of a probability density $\\rho (x_1,x_2,p_1,p_2)=|\\langle x_1,x_2,p_1,p_2|\\psi \\rangle |^2$ on the corresponding phase space ${\\cal M}(x_1,x_2,p_1,p_2)$ [5].",
"The scalar product in (REF ) coincides with the ensemble average $\\int _{\\cal M} \\rho \\,x_j\\,dM$ or $\\int _{\\cal M} \\rho \\,p_j\\,dM$ .",
"Observe that the classical Hilbert space can be partitioned into equivalence classes $|\\psi \\rangle \\sim e^{i\\phi }|\\psi \\rangle $ , where each class corresponds to a single density $\\rho $ .",
"The evolution equations preserve the equivalence classes because there is no interaction [4].",
"Convenient choices of the arbitrary functions $F_j$ can reproduce the evolution equations for non-interacting classical-classical (C-C) $(a_1=a_2=0)$ , quantum-quantum (Q-Q) $(a_1=a_2=1)$ and classical-quantum systems (C-Q) $(a_1=0$ , $a_2=1)$ .",
"The relevant choice of functions $F_j$ and the corresponding equations can be obtained as the special case from the general equations, that will be given later, with interaction set to zero.",
"For arbitrary $a_1,a_2\\ne 0,1$ the dynamical equations describe the evolution of an abstract system interpolating between the quantum and the classical systems (hence the notation $\\hat{H}_{I\\!AS}$ ).",
"Because there is no interaction between the two systems, the evolution of $\\hat{x}_j,\\hat{p}_j$ is also independent of $\\hat{\\chi }_j,\\hat{\\pi }_j$ .",
"The system has 2+2 degrees of freedom, and each of the degrees of freedom evolves independently of the others.",
"If the abstract system (REF ) is meant to represents two quantum or two classical systems, the variables $\\hat{x}_j,\\hat{p}_j$ are interpreted as physical observables of coordinates and momenta.",
"The variables $\\hat{\\chi }_j,\\hat{\\pi }_j$ , similarly as $\\hat{H}_{I\\!AS}$ , do not represent physical observables.",
"They are dynamically separated from the physical observables and appear because the family of systems (REF ) must interpolate between the classical and the quantum dynamics [5]."
],
[
"IAS with general potential interaction",
"Potential interaction between two quantum systems or between two classical systems appears in the equations of motion in the form of gradients of the corresponding scalar potential.",
"In the extended Hilbert space formalism, which is required for the formulation of the IAS, such potential Q-Q or C-C interaction can be represented by an operator expression in terms of all variables with the role of coordinates $\\hat{W}=W(\\hat{x}_1,\\hat{x}_2,\\hat{\\chi }_1,\\hat{\\chi }_2)$ .",
"We assume that in the dynamical equations for the corresponding momenta $\\hat{W}$ should appear as gradient with respect to the corresponding coordinate.",
"We shall now consider dynamics of two abstract systems with arbitrary values of $a_1,a_2$ and with an arbitrary potential interaction between them.",
"Like in the Q-Q and C-C cases, we demand that the following relations hold $\\frac{d}{dt}\\langle \\Psi (t)|\\hat{x}_j|\\Psi (t)\\rangle &=\\langle \\Psi (t)|\\frac{\\hat{p}_j}{m_j}|\\Psi (t)\\rangle ,\\\\\\frac{d}{dt}\\langle \\Psi (t)|\\hat{p}_j|\\Psi (t)\\rangle &=\\langle \\Psi (t)|-V_j^{\\prime }(\\hat{x}_j)-\\frac{\\partial \\hat{W}}{\\partial \\hat{x}_j}|\\Psi (t)\\rangle .$ Notice that the potential interaction can be completely general.",
"Particular examples of interaction which do not necessarily satisfy () have been assumed in somewhat ad hoc manner and studied in [2], [3], [4].",
"Our goal is to determine the unitary evolution generator $\\hat{H}_{I\\!AS}=H_{I\\!AS}(\\hat{x}_1,\\hat{p}_1,\\hat{x}_2,\\hat{p}_2,\\hat{\\chi }_1,\\hat{\\pi }_1,\\hat{\\chi }_2,\\hat{\\pi }_2)$ such that $i\\hbar \\:\\!|d\\Psi (t)/dt\\rangle =\\hat{H}_{I\\!AS}|\\Psi (t)\\rangle $ holds.",
"The unitary evolution and () give the following relations $\\frac{1}{i\\hbar }[\\hat{x}_j,\\hat{H}_{I\\!AS}]=\\frac{\\hat{p}_j}{m_j},\\quad -\\frac{1}{i\\hbar }[\\hat{p}_j,\\hat{H}_{I\\!AS}]=V_j^{\\prime }(\\hat{x}_j)+\\frac{\\partial \\hat{W}}{\\partial \\hat{x}_j},$ and the related system of partial differential equations (PDEs) for the function $ H_{I\\!AS}$ , $a_j\\frac{\\partial H_{I\\!AS}}{\\partial p_j}+\\frac{\\partial H_{I\\!AS}}{\\partial \\pi _j}&=\\frac{p_j}{m_j},\\\\a_j\\frac{\\partial H_{I\\!AS}}{\\partial x_j}+\\frac{\\partial H_{I\\!AS}}{\\partial \\chi _j}&=V_j^{\\prime }(x_j)+\\frac{\\partial W}{\\partial x_j}.$ The commutation relations (REF ), i.e.",
"the PDEs (), are not consistent for arbitrary choice of the interaction potential $\\hat{W}$ .",
"Jacobi identity $[\\hat{H}_{I\\!AS},[\\hat{p}_1,\\hat{p}_2]]+[\\hat{p}_1,[\\hat{p}_2,\\hat{H}_{I\\!AS}]]+[\\hat{p}_2,[\\hat{H}_{I\\!AS},\\hat{p}_1]]=0$ and the commutation relation $[\\hat{p}_1,\\hat{p}_2]=0$ imply that $[\\hat{p}_1,[\\hat{p}_2,\\hat{H}_{I\\!AS}]]=[\\hat{p}_2,[\\hat{p}_1,\\hat{H}_{I\\!AS}]]$ , so that $\\left[a_1\\frac{\\partial }{\\partial x_1}+\\frac{\\partial }{\\partial \\chi _1},a_2\\frac{\\partial }{\\partial x_2}+\\frac{\\partial }{\\partial \\chi _2}\\right]{ H_{I\\!AS}}=0$ must be satisfied.",
"Invoking the second relation of (), we get the consistency requirement $\\left(a_1\\frac{\\partial }{\\partial x_1}+\\frac{\\partial }{\\partial \\chi _1}\\right)\\frac{\\partial W}{\\partial x_2} -\\left(a_2\\frac{\\partial }{\\partial x_2}+\\frac{\\partial }{\\partial \\chi _2}\\right)\\frac{\\partial W}{\\partial x_1}=0.$ The general solution of (REF ) is $W = \\int _{-\\infty }^\\infty {\\cal W}(x_1+(\\alpha -a_1)\\chi _1,\\,x_2+(\\alpha -a_2)\\chi _2,\\,\\alpha )\\,d\\alpha ,$ where $\\cal W$ is an arbitrary function such that the previous integral is defined.",
"Note that when $a_1\\ne a_2$ , i.e.",
"when the systems are of different type, the interaction potential $\\hat{W}$ will depend on at least one of the unobservables $\\hat{\\chi }_1,\\hat{\\chi }_2$ .",
"This conclusion remains valid in the particular case of hybrid classical-quantum system, where $a_1=0$ corresponds to the classical part and $a_2=1$ is related to the quantum part.",
"Let us stress that this fact is proved here for quite general potential interaction and not just observed for some special choices of the interaction [3], [4].",
"Consider a particular choice of ${\\cal W}\\propto \\delta (\\alpha -a)$ yielding the interaction potential $\\hat{W}=W(\\hat{x}_1+(a-a_1)\\hat{\\chi }_1,\\,\\hat{x}_2+(a-a_2)\\hat{\\chi }_2)$ .",
"The related solution of the PDEs () gives $\\hat{H}_{I\\!AS}&=\\sum _{j=1,2}\\frac{1}{a_j}\\bigg (\\frac{\\hat{p}_j^2}{2m_j}+V^{}_j(\\hat{x}_j)\\bigg )\\nonumber \\\\&+\\frac{1}{a}\\,W(\\hat{x}_1+(a-a_1)\\hat{\\chi }_1,\\,\\hat{x}_2+(a-a_2)\\hat{\\chi }_2)\\nonumber \\\\&+F(\\hat{x}_1-a_1\\:\\!\\hat{\\chi }_1,\\hat{p}_1-a_1\\:\\!\\hat{\\pi }_1,\\hat{x}_2-a_2\\:\\!\\hat{\\chi }_2,\\hat{p}_2-a_2\\:\\!\\hat{\\pi }_2),$ where $F$ is arbitrary real-valued smooth function that commutes with the observables $O(\\hat{x}_1,\\hat{p}_1,\\hat{x}_2,\\hat{p}_2)$ .",
"Let us observe that when the two systems are of the same type, the unobservables do not influence the evolution of the physical observables for the choice $a_1=a_2=a$ .",
"The result (REF ) can be extended, although with some care, to the limit $a\\rightarrow 0$ , which will turn out to be interesting for the hybrid Q-C system.",
"Namely, one can take a part of the function $F$ to be of the suitable form $-\\frac{1}{a}\\,W(\\hat{x}_1-a_1\\hat{\\chi }_1,\\,\\hat{x}_2-a_2\\hat{\\chi }_2)$ that yields in the $a\\rightarrow 0$ limit, $\\hat{H}_{I\\!AS}&=\\sum _{j=1,2}\\frac{1}{a_j}\\bigg (\\frac{\\hat{p}_j^2}{2m_j}+V^{}_j(\\hat{x}_j)\\bigg )\\nonumber \\\\&+\\partial _1 W(\\hat{x}_1-a_1\\hat{\\chi }_1,\\,\\hat{x}_2-a_2\\hat{\\chi }_2)\\,\\hat{\\chi }_1\\nonumber \\\\&+\\partial _2 W(\\hat{x}_1-a_1\\hat{\\chi }_1,\\,\\hat{x}_2-a_2\\hat{\\chi }_2)\\,\\hat{\\chi }_2\\nonumber \\\\&+F(\\hat{x}_1-a_1\\:\\!\\hat{\\chi }_1,\\hat{p}_1-a_1\\:\\!\\hat{\\pi }_1,\\hat{x}_2-a_2\\:\\!\\hat{\\chi }_2,\\hat{p}_2-a_2\\:\\!\\hat{\\pi }_2),$ where $\\partial _j W$ denotes partial derivative of the potential with respect to the $j$ -th argument.",
"The limit of (REF ) when $a_j\\rightarrow 0$ can also be obtained by choosing a part of the function $F$ in the form $-\\frac{1}{a_j}\\big (\\frac{(\\hat{p}_j-a_j\\hat{\\pi }_j)^2}{2m_j}+V^{}_j(\\hat{x}_j-a_j\\hat{\\chi }_j)\\big )$ , as in [5].",
"In particular, this yields the Hamiltonian, as the dynamics generator, of a hybrid classical-quantum system $(a_1\\rightarrow 0$ , $a_2=1)$ $\\hat{H}_{\\rm hyb}&=\\frac{\\hat{p}^{}_1}{m_1}\\,\\hat{\\pi }_1+V_1^{\\prime }(\\hat{x}_1)\\,\\hat{\\chi }_1+\\frac{\\hat{p}_2^2}{2m_2}+V^{}_2(\\hat{x}_2)\\nonumber \\\\&+\\frac{1}{a}\\,W(\\hat{x}_1+a\\hat{\\chi }_1,\\,\\hat{x}_2+(a-1)\\hat{\\chi }_2)\\nonumber \\\\&+F(\\hat{x}_1,\\hat{p}_1,\\hat{x}_2-\\:\\!\\hat{\\chi }_2,\\hat{p}_2-\\:\\!\\hat{\\pi }_2),$ where the first four terms describe non-interacting hybrid system.",
"As already mentioned, the interaction potential depends on at least one of the unobservables $\\hat{\\chi }_1,\\hat{\\chi }_2$ .",
"The appearance of the unphysical variables in the Hamiltonian is not a problem per se, because the Hamiltonian is anyway interpreted as the dynamics generator and not as the physical energy.",
"Additionally, in the purely C-C case $(a_1=a_2=a\\rightarrow 0)$ one gets $\\hat{H}_{c\\text{-}c}&=\\frac{\\hat{p}^{}_1}{m_1}\\,\\hat{\\pi }_1+V_1^{\\prime }(\\hat{x}_1)\\,\\hat{\\chi }_1+\\frac{\\hat{p}^{}_2}{m_2}\\,\\hat{\\pi }_2+V_2^{\\prime }(\\hat{x}_2)\\,\\hat{\\chi }_2\\nonumber \\\\&+\\partial _1 W(\\hat{x}_1,\\hat{x}_2)\\,\\hat{\\chi }_1+\\partial _2 W(\\hat{x}_1,\\hat{x}_2)\\,\\hat{\\chi }_2,$ with the unobservables being present, but not within the arguments of the interaction potential.",
"However, in C-C case the unphysical variables do not appear in the evolution equations of the physical observables.",
"We may remark in passing that $\\hat{W}$ is not interpreted as the potential energy of the hybrid, but as a term in the generator of dynamics corresponding to the potential interaction.",
"However, the crucial property of hybrid Q-C systems is that the equations of motion for the physical and unphysical variables become coupled.",
"Those equations are easily obtained from the generator (REF ).",
"Thus, we have shown that the dynamical equations couple physical and unphysical variables in the case of potential Q-C interaction in general, that is with the Hamiltonian of the general hybrid form (REF ).",
"A very special case of (REF ) is obtained in the limit $a\\rightarrow 0$ with the appropriate choice of the function $F$ yielding the Hamiltonian $\\hat{ H}_{\\rm hyb}&=\\frac{\\hat{p}^{}_1}{m_1}\\,\\hat{\\pi }_1+V_1^{\\prime }(\\hat{x}_1)\\,\\hat{\\chi }_1+\\frac{\\hat{p}_2^2}{2m_2}+V^{}_2(\\hat{x}_2)\\nonumber \\\\&+\\partial _1 W(\\hat{x}_1,\\,\\hat{x}_2-\\hat{\\chi }_2)\\,\\hat{\\chi }_1+\\partial _2 W(\\hat{x}_1,\\,\\hat{x}_2-\\hat{\\chi }_2)\\,\\hat{\\chi }_2,$ with the corresponding equations of motion of the variables $&\\frac{d\\hat{x}_j}{dt} = \\frac{\\hat{p}^{}_j}{m_j},\\\\&\\frac{d\\hat{p}_j}{dt} = -V_j^{\\prime }(\\hat{x}_j)-\\partial _j W(\\hat{x}_1,\\,\\hat{x}_2-\\hat{\\chi }_2),\\\\&\\frac{d}{dt}(\\hat{x}_2-\\hat{\\chi }_2)=0.$ This solution describes the situation when the evolution of the classical system depends on the quantum system only through a constant of motion $\\hat{x}_2-\\hat{\\chi }_2$ .",
"In this very special case of the general hybrid solution, the classical variables see only a quite coarse-grained effect of the quantum evolution.",
"On the other hand, the dynamics of the quantum sector is influenced by the details of the dynamics of the classical physical variables $\\hat{x}_1,\\hat{p}_1$ .",
"In addition, this is the only case of the potential Q-C interaction which satisfies the integrity principle of Sudarshan [2].",
"Namely, the terms $\\partial _j W(\\hat{x}_1,\\,\\hat{x}_2-\\hat{\\chi }_2)$ in this form of the Hamiltonian commute with the momenta $\\hat{p}_1,\\hat{p}_2$ , which assures commutation of the classical variables at different times."
],
[
"Summary",
"We have studied the type of theory of hybrid quantum-classical systems where the evolution is described by unitary transformations on an appropriate Hilbert space.",
"The fact that both classical and quantum mechanics can be formulated on the same Hilbert space makes it possible to introduce a parameter dependent family of abstract systems interpolating between a classical system and its quantized counterpart [5].",
"The variables involved in the formulation of the abstract interpolating model can be divided into two groups, one with the standard physical interpretation and one with no physical interpretation.",
"In the limits of the classical or the quantum system the two groups of variables are dynamically separated.",
"We have studied two such abstract interpolating systems with quite arbitrary potential interaction between them.",
"General solution for the problem of constructing dynamical equations for such a pair of systems is provided for the first time.",
"It is shown that, with the most general type of potential interaction, the dynamics of the two groups of variables is separated if and only if the two abstract interpolating systems have the same value of the interpolation parameter.",
"On the other hand, if the interpolation parameters of the two system are different, the two groups of variables dynamically influence each other.",
"The variables which can be considered as unphysical and cannot be observed in the purely quantum or in the purely classical case, do have an observable effect in the hybrid quantum-classical system.",
"Our results demonstrate this fact for arbitrary potential interaction, in line with the previous special cases [3], [4].",
"Analogous conclusions are obtained in the symplectic approach to the conservative hybrid dynamics [7], [8], and the analogy is worth further investigation.",
"We have also analyzed the particular case of the general solution corresponding to the situation when the classical part is influenced by the quantum part only through a particular combination of the variables from the quantum system that remains constant during the evolution.",
"This, rather special case, is the only possible dynamics of the hybrid system within the framework of unitary dynamics with potential interaction, when the physical and the unphysical variables can be considered as decoupled, and also when the Sudarshan integrity condition of the classical system is satisfied.",
"We acknowledge support of the Ministry of Education and Science of the Republic of Serbia, contracts No.",
"171006, 171017, 171020, 171038 and 45016 and COST (Action MP1006)."
]
] | 1403.0452 |
[
[
"Online Algorithms for Machine Minimization"
],
[
"Abstract In this paper, we consider the online version of the machine minimization problem (introduced by Chuzhoy et al., FOCS 2004), where the goal is to schedule a set of jobs with release times, deadlines, and processing lengths on a minimum number of identical machines.",
"Since the online problem has strong lower bounds if all the job parameters are arbitrary, we focus on jobs with uniform length.",
"Our main result is a complete resolution of the deterministic complexity of this problem by showing that a competitive ratio of $e$ is achievable and optimal, thereby improving upon existing lower and upper bounds of 2.09 and 5.2 respectively.",
"We also give a constant-competitive online algorithm for the case of uniform deadlines (but arbitrary job lengths); to the best of our knowledge, no such algorithm was known previously.",
"Finally, we consider the complimentary problem of throughput maximization where the goal is to maximize the sum of weights of scheduled jobs on a fixed set of identical machines (introduced by Bar-Noy et al.",
"STOC 1999).",
"We give a randomized online algorithm for this problem with a competitive ratio of e/e-1; previous results achieved this bound only for the case of a single machine or in the limit of an infinite number of machines."
],
[
"Introduction",
"Scheduling jobs on machines to meet deadlines is a fundamental area of combinatorial optimization.",
"In this paper, we consider a classical problem in this domain called machine minimization, which is defined as follows.",
"We are given a set of $n$ jobs, where job $j$ is characterized by a release time $r_j$ , a deadline $d_j$ , and a processing length $p_j$ .",
"The algorithm must schedule the jobs on a set of identical machines such that each job is processed for a period $p_j$ in the interval $[r_j, d_j]$ .",
"The goal of the algorithm is to minimize the total number of machines used.",
"The machine minimization problem was previously considered in the offline setting, where all the jobs are known in advance.",
"Garey and Johnson [22], [23] showed that it is NP-hard to decide whether a given set of jobs can be scheduled on a single machine.",
"On the positive side, using a standard LP formulation and randomized rounding [29], one can obtain an approximation factor of $O\\left(\\frac{\\log n}{\\log \\log n}\\right)$ .",
"This was improved by Chuzhoy et al [16] to $O\\left(\\sqrt{\\frac{\\log n}{\\log \\log n}}\\right)$ by using a more sophisticated LP formulation and rounding procedure.",
"This is the best approximation ratio currently known and it is an interesting open question as to whether a constant-factor approximation algorithm exists for this problem.",
"(A constant-factor approximation was claimed in [15], but the analysis is incorrect [14].)",
"In this paper, we consider the online version of this problem, i.e., every job becomes visible to the algorithm when it is released.",
"We evaluate our algorithm in terms of its competitive ratio [9], which is defined as the maximum ratio (over all instances) of the number of machines in the algorithmic solution to that in an (offline) optimal solution.",
"For jobs with arbitrary processing lengths, an information-theoretic lower bound of $\\Omega \\left(\\max (n, \\log \\left(\\frac{p_{\\max }}{p_{\\min }}\\right))\\right)$ (where $p_{\\max }$ and $p_{\\min }$ are respectively the maximum and minimum processing lengths among all jobs) was shown by Saha [30].Saha [30] only mentions the lower bound of $\\Omega \\left(\\log \\left(\\frac{p_{\\max }}{p_{\\min }}\\right)\\right)$ but the same construction also gives a lower bound of $\\Omega (n)$ .",
"We note that matching (up to constants) upper bounds are easy to obtain.",
"An upper bound of $n$ in the competitive ratio is trivially obtained by scheduling every job on a distinct machine.",
"On the other hand, any algorithm with a competitive ratio of $\\alpha $ for the case of unit length jobs can be used as a black box to obtain an algorithm with a competitive ratio of $O(\\alpha \\log \\frac{p_{\\max }}{p_{\\min }})$ for jobs with arbitrary processing lengths.",
"Our main focus in this paper is the online machine minimization problem with unit processing lengths.",
"The previous best competitive ratio for this problem was due to Kao et al [26] who gave an upper bound of 5.2 and a lower bound of 2.09.",
"(Earlier, Shi and Ye [31] had claimed a competitive ratio of 2 for the special case of unit job lengths and equal deadlines, but an error in their analysis was discovered by Kao et al [26].)",
"Saha [30] gave a different algorithm for this problem with a (larger) constant competitive ratio.",
"Our main result in this paper is a complete resolution of the deterministic online complexity of this problem by giving an algorithm that has a competitive ratio of $e$ and a matching lower bound.",
"Theorem 1.1 There is a deterministic algorithm for the online machine minimization problem with uniform job lengths that has a competitive ratio of $e$ .",
"Moreover, no deterministic algorithm for this problem has a competitive ratio less than $e$ .",
"It was brought to our attention that this theorem also follows from the results of [4], who consider energy-minimizing scheduling problems.",
"This result follows from Lemma 4.7 and Lemma 4.8 in [4].",
"To prove this theorem, we first show that the following online algorithm is the best possible: at any point of time, the algorithm schedules available jobs using earliest deadline first on $\\alpha k$ machines, where $k$ is the number of machines in the optimal offline deterministic solution for all jobs that have been released so far and $\\alpha $ is the optimal competitive ratio.",
"Therefore, the main challenge is to find the value of $\\alpha $ .",
"If we under-estimate $\\alpha $ , then this online algorithm is invalid in the sense that it will fail to schedule all jobs.",
"On the other hand, over-estimating $\\alpha $ leads to a sub-optimal competitive ratio.",
"In order to estimate $\\alpha $ , we use an analysis technique that is reminiscent of factor-revealing LPs (see, e.g., Vazirani [33]).",
"However, instead of using the LP per se, we give combinatorial interpretations of the primal and dual solutions.",
"We prove that the earliest deadline first schedule is valid if and only if there exists a fractional schedule satisfying two natural conditions fractional completion and fractional packing.",
"Then, we explicitly construct an online fractional schedule (which corresponds to the optimal dual solution) that uses the same number of machines as the online algorithm.",
"To show a lower bound we explicitly present the offline strategy (which is essentially the optimal primal solution).",
"We also consider the online machine minimization problem where all the deadlines are identical, but the processing lengths of jobs are arbitrary.",
"For this problem, we give a deterministic algorithm with a constant competitive ratio.",
"Theorem 1.2 There is a deterministic algorithm for the online machine minimization problem with uniform deadlines that has a constant competitive ratio.",
"A problem that is closely related to the machine minimization problem is that of throughput maximization.",
"In this problem, every job $j$ has a given weight $w_j$ in addition to the parameters $r_j$ , $p_j$ , and $d_j$ .",
"The goal is now to schedule the maximum total weight of jobs on a given number of machines.",
"If all the job parameters are arbitrary, then this problem has a lower bound of $\\Omega (n)$ in the competitive ratio, even on a single machine where all jobs have unit weight.",
"Initially, the adversary releases a job of processing length equal to its deadline (call it $d$ ) and depending on whether this job is scheduled or not, either releases $d$ unit length jobs with the same deadline, or does not release any job at all.",
"As in the machine minimization problem, we focus on the scenario where all jobs have uniform (wlog, unit) processing length.",
"First, we note that in the unweighted case (i.e., all jobs have unit weight), it is optimal to use the earliest deadline first strategy, where jobs that are waiting to be scheduled are ordered by increasing deadlines and assigned to the available machines in this order.",
"Therefore, we focus on the weighted scenario.",
"Previously, the best online (randomized) algorithms for this problem had a competitive ratio of $\\frac{e}{e-1}$ for the case of a single machine [7], [11], [25].",
"For $k$ machines, Chin et al [11] obtained a competitive ratio of $\\frac{1}{1 - \\left(\\frac{k}{k+1}\\right)^k}$ , which is equal to 2 for $k = 1$ but decreases with increasing $k$ ultimately converging to $\\frac{e}{e-1}$ as $k\\rightarrow \\infty $ .",
"The best known lower bound for randomized algorithms is 1.25 due to Chin and Fung [12], which holds even for a single machine.",
"In this paper, we give an approximation-preserving reduction from the online throughput maximization problem with any number of machines to the online vertex-weighted bipartite matching problem.",
"(We will define this problem formally in section .)",
"Using this reduction and known algorithms for the online vertex-weighted matching problem [1], [20], we obtain a randomized algorithm for the online throughput maximization problem for unit length jobs that has a competitive ratio of $\\frac{e}{e-1}$ independent of the number of machines.",
"Theorem 1.3 There is a randomized algorithm for the online throughput maximization problem that has a competitive ratio of $\\frac{e}{e-1}$ , independent of the number of machines.",
"Related Work.",
"As mentioned previously, the offline version of the machine minimization problem was considered by Chuzhoy et al [16] and Chuzhoy and Codenotti [15].",
"Several special cases of this problem have also been considered.",
"Cieliebak et al [19] studied the problem under the restriction that the length of the time interval during which a job can be scheduled is small.",
"Yu and Zhang [34] gave constant-factor approximation algorithm for two special cases where all jobs have equal release times or equal processing lengths.",
"A related problem is that of scheduling jobs on identical machines where each job has to be scheduled in one among a given set of discrete intervals.",
"For this problem, an approximation hardness of $\\Omega (\\log \\log n)$ was shown by Chuzhoy and Naor [17], even when the optimal solution uses just one machine.",
"The best approximation algorithm known for this problem uses randomized rounding [29] and has an approximation factor of $O\\left(\\frac{\\log n}{\\log \\log n}\\right)$ .",
"The complimentary problem of throughput maximization has a rich history in the offline model.",
"For arbitrary job lengths, the best known approximation ratios for the unweighted and weighted cases are respectively $\\frac{e}{e-1}$ [18] and 2 [6], [8].",
"On the other hand, the discrete version of this problem was shown to be MAX-SNP hard by Spieksma [32].",
"Several variants of this problem have also been explored.",
"E.g., when a machine is allowed to be simultaneously used by multiple jobs, Bar-Noy et al [5] obtained approximation factors of 5 and $\\frac{2e-1}{e-1}$ for the weighted and unweighted cases respectively.",
"In the special case of every job having a single interval, Calinescu et al [10] have a 2-approximation algorithm, which was improved to a quasi-PTAS by Bansal et al [3].",
"As mentioned above, the previous best randomized algorithms for the online throughput maximization problem with unit length jobs were due to Chin et al [11] and Jez [25].",
"For the case of a single machine, Kesselman et al [27] gave a deterministic algorithm with a competitive ratio of 2, which was improved to $\\frac{64}{33} \\simeq 1.939$ by Chrobak et al [13].",
"This was further improved to $2\\sqrt{2} - 1 \\simeq 1.828$ by Englert and Westermann [21] and, in simultaneous work, to $\\frac{6}{\\sqrt{5} + 1} \\simeq 1.854$ by Li et al [28].",
"On the other hand, Andelman et al [2], Chin and Fung [12], and Hajek [24] showed a lower bound of $\\frac{\\sqrt{5}+1}{2} \\simeq 1.618$ on the competitive ratio of any deterministic algorithm for this problem."
],
[
"Optimal Deterministic Algorithm for Unit Jobs",
"In this section we present a deterministic online algorithm with competive ratio $e$ , proving one half of thm:min-unit-optimal.",
"In the next section, we will prove that this algorithm is optimal, completing the other half.",
"The main challenge for the online algorithm is to determine how many machines to open at a given time $t$ .",
"The scheduling policy is easy: since all jobs are unit jobs, we can simply use the earliest deadline first policy.",
"The policy is that if we have $m(t)$ available machines at time $t$ , we should pick $m(t)$ released but not yet completed jobs with the earliest deadlines and schedule them on these $m(t)$ machines (if the total number of available jobs is less than $m(t)$ we schedule all available jobs at time $t$ ).",
"We formally prove that this policy is optimal in Lemma REF .",
"Note that the number of machines used by the offline solution is a constant over time.",
"Consequently, it is really easy to solve the offline problem – we just need to find the optimal $m$ using binary search and then verify that the earliest deadline first schedule is feasible.",
"To state the online algorithm we need to introduce the following notation: Let $\\operatorname{Offline}(t)$ be the offline cost of the solution for jobs released in the time interval $[0,t]$ .",
"That is, we consider the subset of all jobs $J(t)=\\lbrace j: r_j\\le t\\rbrace $ , and let $\\operatorname{Offline}(t)$ be the cost of the optimal offline schedule for $J(t)$ .",
"Note that the algorithm knows all jobs in $J(t)$ at time $t$ , and thus can compute $\\operatorname{Offline}(t)$ .",
"Algorithm $e$ -edf: The online algorithm uses ${\\lceil {e\\,\\operatorname{Offline}(t)} \\rceil }$ machines at time $t$ .",
"It uses the earliest deadline first policy to schedule jobs.",
"Lemma 2.1 Consider a set of jobs $J$ .",
"Suppose we have $m(t)\\in \\mathbb {N}$ machines at time $t\\in \\lbrace 0,\\dots , T\\rbrace $ .",
"The earliest deadline schedule is feasible if and only if for every $d\\in \\lbrace 0,\\dots , T\\rbrace $ , there exists a collection of functions, a fractional solution, $\\lbrace f_j:[0,T]\\rightarrow \\mathbb {R}^+\\rbrace $ satisfying the following conditions (note that $f_j$ 's depend on $d$ ): (Fractional completion) Every job $j$ with $d_j\\le d$ is completed before time $d$ according to the fractional schedule, i.e., $\\int _{r_j}^d f_j(x) dx = 1$ .",
"Note that the fractional solution is allowed to schedule a job $j$ past its deadline $d_j$ (but before $d$ ).",
"(Fractional packing) For every $t \\in [0,T]$ the total number of machines used according to the fractional schedule is at most $m({\\lfloor {t} \\rfloor })$ , i.e., $\\sum _{\\begin{array}{c}j\\in J\\\\d_j\\le d\\end{array}} f_j(t)\\le m({\\lfloor {t} \\rfloor })$ .",
"In one direction – “only if” – this lemma is trivial.",
"If the earliest deadline first schedule is feasaible and uses $m(t)$ machines at time $t$ , then we let $f_j(t)=1$ , if job $j$ is scheduled at time ${\\lfloor {t} \\rfloor }$ ; and $f_j(t)=0$ , otherwise.",
"It is easy to see that $f_j$ 's satisfy conditions both fractional completion and fractional packing conditions.",
"We now assume existence of $f_j$ s that satisfy the conditions above and show that in the earliest deadline first schedule, no job $j$ misses its deadline $d_j$ .",
"Fix $j^*\\in J$ and let $d^*=d_{j^*}$ .",
"Let $\\lbrace f_j\\rbrace $ be the fractional schedule for $d^*$ .",
"Notice that we can assume that $f_j(x)=0$ for $x\\le r_j$ and $x\\ge d^*$ (by simply redefining $f_j$ to be 0 for $x\\le r_j$ and $x\\ge d^*$ ).",
"Let $J^*=\\lbrace j:d_j\\le d^*\\rbrace $ .",
"Denote by $S(t)$ the set of jobs scheduled by the algorithm at one of the first $t$ steps $0,\\dots , t-1$ .",
"We let $S(0)=\\varnothing $ .",
"Proposition 2.2 For every $t\\in \\lbrace 0,\\dots , T\\rbrace $ the following invariant holds: $|S(t)\\cap J^*| \\ge \\sum _{j\\in J^*} \\int _0^t f_j(x) dx.$ We prove this proposition by induction on $t$ .",
"For $t=0$ , the inequality trivially holds, because both sides are equal to 0.",
"Now, we assume that the inequality holds for $t$ , and prove it for $t+1$ .",
"There are two cases.",
"In the first case, $m(t)$ jobs from $J^*$ are scheduled at time $t$ .",
"Then we have: $&|S(t+1)\\cap J^*| = |S(t)\\cap J^*| + m(t) \\ge \\sum _{j\\in J^*} \\int _0^t f_j(x) dx + \\int _t^{t+1} m({\\lfloor {x} \\rfloor }) dx \\\\&\\ge \\sum _{j\\in J^*} \\int _0^t f_j(x) dx + \\int _t^{t+1} \\sum _{j\\in J^*} f_j(x) dx= \\sum _{j\\in J^*} \\int _0^{t + 1} f_j(x) dx,$ where the first equality uses the fact that all machines are busy at time $t$ , the second inequality follows by the inductive hypothesis and the fourth inequality uses the fractional packing condition.",
"In the second case, the number of jobs from $J^*$ scheduled at time $t$ is strictly less than $m(t)$ .",
"This means that all jobs in $J^*$ released by time $t$ are scheduled no later than at time $t$ (note that jobs in $J^*$ have a priority over other jobs according to the earliest deadline policy).",
"In other words, $J^*\\cap J(t)\\subseteq S(t+1)$ .",
"Together with the fact that $S(t+1)\\subseteq J(t)$ this implies that $J^*\\cap S(t+1) = J^*\\cap J(t)$ .",
"On the other hand, for $j\\notin J(t)$ , $r_j\\ge t+1$ and, hence, $\\int _0^{t+1} f_j(x) dx = 0$ .",
"Putting this together, $|J^*\\cap S(t + 1)| = |J^*\\cap J(t)| = \\sum _{j\\in J^*\\cap J(t)} \\int _0^{t+1} f_j(x) dx \\ge \\sum _{j\\in J^*} \\int _0^{t+1} f_j(x) dx,$ where the second equality follows by the fractional completion condition.",
"This finishes the proof of the proposition.",
"By Proposition REF , the number of jobs from $J^*$ scheduled by time $d^*$ is: $|J^* \\cap S(d^*)| \\ge \\sum _{j\\in J^*} \\int _0^{d^*} f_j(x) dx = \\sum _{j\\in J^*} \\int _{r_j}^{d^*} f_j(x) dx = |J^*|.$ Here, we used the fractional completion condition $\\int _{0}^{d^*} f_j(x) dx =1$ .",
"Thus, all jobs in $J^*$ are scheduled before the deadline $d^*$ .",
"Consequently, the job $j^*$ does not miss the deadline $d^*= d_j$ .",
"We now use Lemma REF to prove that the schedule produced by Algorithm $e$ -edf is feasible.",
"Pick an arbitrary deadline $d^*\\in \\lbrace 0,\\dots , T\\rbrace $ , and let $J^* =\\lbrace j:d_j\\le d^*\\rbrace $ .",
"We fractionally schedule every job $j\\in J^*$ in the time interval $[r_j, d^* - (d^*-r_j)/e]$ .",
"We let $f_j(x)={\\left\\lbrace \\begin{array}{ll}\\frac{1}{d^* - x},&\\text{if } x\\in \\big [r_j, d^* - \\frac{(d^*-r_j)}{e}\\big ];\\\\0, &\\text{otherwise}.\\end{array}\\right.",
"}$ Note that the fractional schedule depends on $d^*$ , and possibly $d^* - (d^*-r_j)/e > d_j$ for some $j$ .",
"So the fractional solution may run a job $j$ even after its deadline $d_j$ is passed.",
"Nevertheless, as we show now, functions $f_j$ satisfy the conditions of Lemma REF .",
"First, we check the fractional completion condition.",
"For $j\\in J^*$ , we have: $\\int _{r_j}^{d^*} f_j(x) dx = \\int _{r_j}^{d^*-(d^*-r_j)/e} \\frac{dx}{d^* - x} = \\ln \\frac{d^* - r_j}{(d^*-r_j)/e} = 1.$ We now verify the fractional packing condition.",
"We consider a fixed $t \\in [0,T]$ , and show that this condition holds for this $t$ .",
"Note that $f_j(t)\\ne 0$ if and only if $t \\in [r_j, d^* - \\frac{(d^*-r_j)}{e}]$ , which is equivalent to $d^* - e(d^* - t) \\le r_j \\le t$ .",
"We denote $d^* - e(d^* - t)$ by $r^*$ , and let $\\Lambda = \\lbrace j\\in J^* | r^* \\le r_j\\le t\\rbrace $ .",
"Thus, $f_j(t)\\ne 0$ if and only if $j\\in \\Lambda $ .",
"Then, $\\sum _{j\\in J^*} f_j(t) = \\sum _{j\\in J^*}\\frac{\\mathbf {1} \\big (t\\in [r_j, d^* - \\frac{(d^*-r_j)}{e}]\\big )}{d^*-t} = \\sum _{j\\in \\Lambda } \\frac{1}{d^*-t} = \\frac{|\\Lambda |}{d^* - t}.$ We need to compare the right hand side of (REF ) with $m(t) \\equiv {\\lceil {e \\operatorname{Offline}(t)} \\rceil }$ .",
"By the definition of $\\operatorname{Offline}(t)$ , all jobs in $\\Lambda \\subset J(t)$ can be scheduled on at most $\\operatorname{Offline}(t)$ machines.",
"All jobs in $\\Lambda $ must be completed by time $d^*$ (since $\\forall j\\in \\Lambda \\subset J^*$ , $d_j\\le d^*$ ).",
"The number of completed jobs is bounded by the number of available “machine hours” in the time interval $[r^*,d^*]$ which is $(d^*-r^*)\\times \\operatorname{Offline}(t)$ .",
"Therefore, $|\\Lambda |\\le (d^*-r^*)\\times \\operatorname{Offline}(t)$ , and, since $(d^* - r^*) = e (d^* - t)$ , $\\sum _{j\\in J^*} f_j(t)= \\frac{|\\Lambda |}{d^* - t}\\le \\frac{(d^*-r^*)\\times \\operatorname{Offline}(t)}{d^* - t} = e\\,\\operatorname{Offline}(t).$ This concludes the proof that $f_j$ satisfy the fractional completion and packing conditions, and thus, by Lemma REF , the online schedule is feasible i.e., every job $j$ is completed before its deadline $d_j$ .",
"At every point of time $t$ , Algorithm $e$ -edf uses ${\\lceil {e\\cdot \\operatorname{Offline}(t)} \\rceil }$ machines; $\\operatorname{Offline}(t)$ is a lower bound on the cost of the offline schedule.",
"Hence, the Algorithm $e$ -edf is $e$ competitive."
],
[
"Optimal Deterministic Lower Bound",
"We now prove the second part of Theorem REF , giving a lower bound on the competitive ratio of a deterministic online algorithm.",
"This bound holds even if all deadlines are the same.",
"We present an adversarial strategy that forces any deterministic online algorithm to open $(e-\\varepsilon )\\times \\operatorname{Offline}$ machines.",
"As in the previous section, we let $\\operatorname{Offline}(t)$ to be the offline optimum solution for jobs in $J(t)$ i.e.",
"for jobs released in the time interval $[0,t]$ .",
"Let $\\operatorname{Online}(t)$ be the number of machines used by the online algorithm at time $t$ .",
"Adversary: We fix a sufficiently large number $n$ and $N=n^2$ .",
"At time $t\\in \\lbrace 0,\\dots , T\\rbrace $ , the adversary releases ${\\lfloor {N/(n-t)} \\rfloor }$ unit jobs with deadline $n$ .",
"The stopping time $T$ equals the first $\\tau $ such that $\\operatorname{Online}(\\tau )\\ge e\\,\\operatorname{Offline}(\\tau )$ if such $\\tau $ exists, and $n-1$ otherwise.",
"First we prove the following auxiliary statement.",
"Lemma 3.1 For all $t^*\\in [0,n]$ it holds that $\\operatorname{Offline}(t^*)\\le {\\lceil {N/(e(n-t^*))} \\rceil }$ .",
"We need to show that all jobs in $J(t^*)$ can be scheduled on $m(t^*) = {\\lceil {N/(e(n-t^*))} \\rceil }$ machines.",
"Since all jobs have the same deadline we shall use a greedy strategy: at every point of time we run arbitrary $m(t^*)$ jobs if there are $m(t^*)$ available jobs; and all available jobs otherwise.",
"In the offline schedule the number of machines equals $m(t^*)$ and does not change over time.",
"The number of jobs released by the adversary at time $t$ increases as a function of $t$ .",
"Thus, till a certain integral point of time $s^*$ , the number of available machines is greater than the number of available jobs, and thus all jobs are executed immediately after they are released.",
"After that point of time, $m(t^*)$ machines are completely loaded with jobs until all jobs are processed.",
"The number of jobs released in the time interval $[s^*, t^*]$ is upper bounded by $\\int _{s^*}^{t^*} \\frac{N}{n-x} dx = N \\,\\ln \\frac{n- s^*}{n-t^*}.$ The number of jobs that can be processed in the time interval $[s^*, n]$ is equal to $m(t^*)\\times (n-s^*) \\ge \\frac{N}{e(n-t^*)} \\times (n-s^*).$ Note that $N \\,\\ln \\frac{n - s^*}{(n-t^*)} \\le N\\,\\frac{n - s^*}{e(n-t^*)},$ since for every $x$ , particularly for $x= (n-s^*)/(n-t^*)$ , $\\ln x \\le x/e$ (the minimum of $x/e - \\ln x$ is attained when $x=e$ ).",
"Therefore, all jobs are completed till the deadline $n$ .",
"This completes the proof.",
"$\\Box $ We are now ready to prove the second part of Theorem REF .",
"Consider a run of a deterministic algorithm.",
"We need to show that $\\operatorname{Online}(n)\\ge (e -\\varepsilon ) \\operatorname{Offline}(n)$ .",
"If for some $\\tau $ , $\\operatorname{Online}(\\tau )\\ge (e-\\varepsilon ) \\,\\operatorname{Offline}(\\tau )$ , then we are done: $\\operatorname{Online}(n)\\ge \\operatorname{Online}(\\tau ) \\ge (e-\\varepsilon ) \\ \\operatorname{Offline}(\\tau ) = (e-\\varepsilon ) \\ \\operatorname{Offline}(n)$ (we have $\\operatorname{Offline}(\\tau ) = \\operatorname{Offline}(n)$ since the adversary stops releasing new jobs after time $\\tau $ ).",
"So we assume that $\\operatorname{Online}(t) < (e-\\varepsilon )\\,\\operatorname{Offline}(t)$ for all $t\\in \\lbrace 0,\\dots , n-1\\rbrace $ .",
"By Lemma REF $\\operatorname{Offline}(t)\\le {\\lceil {N/(e(n-t))} \\rceil }\\le N/(e(n-t)) + 1$ .",
"By the assumption $\\operatorname{Online}(t) < (e - \\epsilon ) \\operatorname{Offline}(t) = (1 - \\varepsilon /e) N/(n-t) + O(1)$ .",
"The total number of jobs processed by the online algorithm is upper bounded by $& \\sum _{t=0}^{n-1} \\operatorname{Online}(t) \\le \\int _0^{n-1} \\operatorname{Online}(x) dx + \\operatorname{Online}(n-1)\\\\&\\le \\big (1 - \\frac{\\varepsilon }{e}\\big ) \\Big (\\int _0^{n-1} \\frac{N}{n - x} dx + N + O(n)\\Big )\\le \\big (1 - \\frac{\\varepsilon }{e}\\big ) N\\ln n + N + O(n).$ On the other hand, the total number of jobs released by the adversary is lower bounded by $\\int _{0}^{n-1} \\Big (\\frac{N}{n-x} - 1\\Big )\\,dx = N \\,\\ln n - (n-1).$ We get a contradiction since for a sufficiently large $n$ , expression (REF ) is larger than (REF )."
],
[
"Online Machine Minimization with Equal Deadlines",
"In this section, we give a 16-approximation algorithm for the Online Machine Minimization problem with arbitrary release times and job sizes, but equal deadlines (thus proving Theorem REF ).",
"We assume w.l.o.g.",
"that the common deadline $d = 2^k-1$ .",
"Algorithm.",
"The algorithm splits the time line $[0,d-1]$ into $k$ phases: $[0,{1}{2}(d+1)], [{1}{2}(d+1), {3}{4}(d+1)], \\dots $ .",
"The length of phase $i$ is $\\ell _i = 2^{k-i} = (d+1)/2^i$ ; we denote the beginning of the phase by $a_i$ and the end of the phase by $b_i$ .",
"In any phase $i$ , when a new job $j$ is released, the algorithm classifies it as a short job if the size of the job $s_j\\le \\ell _i/4$ and as a long job otherwise.",
"If $j$ is long, the algorithm opens a new machine and executes $j$ right away; otherwise, the algorithm postpones the execution of job $j$ to the next phase.",
"At the beginning of phase $i$ , the algorithm closes all machines that are idle (closed machines can be reopened later if necessary).",
"Next, it splits the open machines into two pools: those serving long jobs and those serving short jobs according to the following rule.",
"A machine serves long jobs if the remaining length of the job currently being processed on it is at least $\\ell _i/4$ ; otherwise, the machine serves short jobs.",
"Note that some machines in the short jobs pool are actually serving jobs that were long when they were released in a previous phase, but have become short now based on their remaining length.",
"Then, the algorithm schedules all postponed short jobs (that were released in phase $i-1$ ) on machines in the short jobs pool.",
"(Note that machines in the short jobs pool are currently serving long jobs from previous phases that have now become short.",
"When scheduling the postponed short jobs, the algorithm allows these running jobs to complete first.)",
"The algorithm assigns postponed short jobs to machines using an offline greedy algorithm: it picks a postponed job $j$ and schedules it on one of the available machines serving short jobs if that machine can process $j$ before the end of the current phase.",
"If there are no such machines, the algorithm opens a new machine (or reactivates as idle machine) and adds it to the pool serving short jobs.",
"Note that once all postponed jobs are scheduled in the time interval $[a_i, b_i]$ , the algorithm does not assign any new job to the pool serving short jobs since all short jobs released in phase $i$ will be postponed to phase $i+1$ .",
"Analysis.",
"We prove that at every point of time, the number of machines serving short jobs does not exceed $8\\operatorname{Offline}+ 1$ , and the number of machines serving long jobs does not exceed $8\\operatorname{Offline}$ .",
"(Recall that $\\operatorname{Offline}$ denotes the number of machines in an optimal offline solution.)",
"Our proof is by induction on the current phase $i$ .",
"Observe that at the end of phase $i$ , all machines serving short jobs are going to be idle, because all jobs scheduled in this phase must be completed by $b_i$ .",
"These machines will be closed at the beginning of phase $(i+1)$ .",
"Thus, the only machines that may remain open when we transition from phase $i$ to phase $(i+1)$ are machines serving long jobs.",
"The number of such machines is at most $8\\operatorname{Offline}$ by the inductive hypothesis.",
"The next two lemmas show that this property implies that $M_{short}(i+1)$ (resp., $M_{long}(i+1)$ ) — the number of machines serving short jobs (resp., long jobs) in phase $(i+1)$ — is at most $8\\operatorname{Offline}+ 1$ (resp., $8\\operatorname{Offline}$ ).",
"Lemma 4.1 If the number of active machines at the beginning of phase $(i+1)$ is at most $8\\operatorname{Offline}$ , then $M_{short}(i+1) \\le 8\\operatorname{Offline}+ 1$ .",
"At the beginning of phase $(i+1)$ , we schedule all postponed jobs.",
"There are two cases.",
"The first case is that we schedule all postponed jobs on the machines that remained open from the previous phase.",
"As we just argued, the number of such machines is at most $8\\operatorname{Offline}$ .",
"The second case is that we opened some extra machines.",
"This means that every machine serving short jobs (except possibly the last one that we open) finishes processing jobs in phase $(i+1)$ no earlier than time $b_{i+1} - \\ell _{i}/4 = b_{i+1} - \\ell _{i+1}/2$ .",
"Otherwise, if some machine was idle at at time $b_{i+1} - \\ell _{i}/4$ , we would assign an extra short job to this machine instead of opening a new machine (note that the length of any short job is at most $\\ell _{i}/4$ ).",
"Therefore, every machine (but one) is busy for at least half of the time in phase $(i+1)$ .",
"The volume of work they process is lower bounded by $(M_{short}(i+1) - 1)\\times \\ell _{i+1}/2$ .",
"We now need to find an upper bound on the volume of work.",
"Every short job that was released in phase $i$ and got postponed to phase $(i+1)$ must be scheduled by the offline optimal solution in the time interval $[a_i, d]$ (recall that $a_i$ is the beginning of phase $i$ and $d$ is the deadline).",
"A job $j$ that was initially classified as a long job on release but got reclassified as a short job in phase $(i+1)$ must also be partially scheduled by the optimal solution in the time interval $[a_i, d]$ .",
"For such a job, the optimal solution must schedule at least the same amount of work for the time interval $[a_{i+1},d]$ as the online algorithm does for the time interval $[a_{i+1}, b_{i+1}]$ .",
"This follows from the fact that the online algorithm executed job $j$ right after it was released (since it was initially a long job), and thus, no matter how $j$ is scheduled in the optimal solution, the remaining size of $j$ at time $a_{i+1}$ in the optimal solution must be at least that in the online algorithm.",
"Thus, the amount of work done by machines serving short jobs in phase $(i+1)$ in the online schedule does not exceed the amount of work done by all machines in the time interval $[a_i, d]$ in the optimal schedule.",
"The latter quantity is upper bounded by $(d-a_i)\\times \\operatorname{Offline}< 2\\ell _i \\times \\operatorname{Offline}$ .",
"Consequently, $M_{short}(i+1) \\le 8 \\operatorname{Offline}+ 1$ .",
"Lemma 4.2 If the number of active machines at the beginning of phase $(i+1)$ is at most $8\\operatorname{Offline}$ , then $M_{long}(i+1) \\le 8\\operatorname{Offline}$ .",
"Each long job released in phase $(i+1)$ or the remainder of a job that got classified as long in phase $(i+1)$ must have size at least $\\ell _{i+1}/4$ .",
"Therefore, the total volume of these jobs is $\\ell _{i+1}/4 \\times M_{long}(i+1)$ .",
"The same argument as we used for short jobs shows that all this volume must be scheduled by the offline solution in the time interval $[a_{i+1},d]$ .",
"Hence, $(\\ell _{i+1}/4) \\times M_{long}(i+1)\\le 2\\ell _{i+1} \\times \\operatorname{Offline}$ and therefore, $M_{long}(i+1)\\le 8\\operatorname{Offline}$ ."
],
[
"Online Throughput Maximization for Unit Jobs",
"In this section, we will give a reduction of the online throughput maximization problem for unit length jobs to the online vertex-weighted matching problem [1], [20].",
"We will reuse the notation for the throughput maximization problem from the introduction, i.e., a job is characterized by it release time $r_j$ , deadline $d_j$ , and weight $w_j$ .",
"Let us now define the online vertex-weighted matching problem.",
"The input comprises a bipartite graph $G = (U\\cup V, E)$ , where the vertices in $U$ are given offline and have weights $w_u, u\\in U$ associated with them and the vertices in $V$ (and their respective incident edges) appear online.",
"When a vertex in $V$ appears, it must be matched to one of its neighbors in $U$ that has not been matched previously or not matched at all.",
"The goal of the algorithm is to maximize the sum of weights of vertices in $U$ that are eventually matched by the algorithm.",
"Aggarwal et al [1] introduced this problem and obtained a randomized algorithm with a competitive ratio of $\\frac{e}{e-1}$ .",
"An alternative proof of this result was recently obtained by Devanur, Jain, and Kleinberg [20] using a randomized version of the classical primal dual paradigm.",
"Our main contribution is an approximation preserving reduction from the online throughout maximization problem to the online vertex-weighted matching problem.",
"Suppose we are given an instance of the throughput maximization problem.",
"Define an instance of the vertex-weighted matching problem as follows.",
"For every job $j$ , define an offline vertex $u_j\\in U$ with weight $w_j$ .",
"For every time step $t$ , define $k$ online vertices in $V$ , one for each machine.",
"Let $v_{it}$ denote the vertex for machine $i$ in time step $t$ , and add edges between each such online vertex and all offline vertices representing jobs $j$ such that $t\\in [r_j, d_j]$ .",
"Note that the reduction is somewhat counter-intuitive in that online jobs are being mapped to the offline side of the bipartite graph.",
"However, it does produce a valid instance of the online vertex-weighted matching problem since at time $r_j$ , both the offline vertex corresponding to job $j$ and its first set of online neighbors (all online vertices corresponding to time step $r_j$ ) are simultaneously revealed.",
"This is sufficient since every offline vertex in $U$ comes into play in any algorithm only after its first online neighbor in $V$ is revealed.",
"We will now show that there is a 1-1 mapping between solutions of the throughput maximization instance and the vertex-weighted matching instance.",
"Consider any solution to the matching instance.",
"If an offline vertex $u_j$ is matched to a neighbor $v_{it}$ , then we schedule job $j$ at time $t$ on machine $i$ .",
"This is valid since the edge between $u_j$ and $v_{it}$ testifies to the fact that $t\\in [r_j, d_j]$ .",
"Conversely, consider a solution for the throughput maximization problem.",
"If job $j$ is scheduled on machine $i$ at time $t$ , then add the edge between $u_j$ and $v_{it}$ to the matching.",
"First, note that this edge exists since job $j$ could only have been scheduled at some time$t\\in [r_j, d_j]$ ; and second, the edges selected form a matching since no job is scheduled more than once and no machine can have more than one job scheduled on it in the same time step.",
"This completes the reduction and therefore, the proof of Theorem REF ."
]
] | 1403.0486 |
[
[
"Characterizing AGB stars in Wide-field Infrared Survey Explorer (WISE)\n bands"
],
[
"Abstract Since asymptotic giant branch (AGB) stars are bright and extended infrared objects, most Galactic AGB stars saturate the Wide-field Infrared Survey Explorer (WISE) detectors and therefore the WISE magnitudes that are restored by applying point-spread-function fitting need to be verified.",
"Statistical properties of circumstellar envelopes around AGB stars are discussed on the basis of a WISE AGB catalog verified in this way.",
"We cross-matched an AGB star sample with the WISE All-Sky Source Catalog and the Two Mircon All Sky Survey catalog.",
"Infrared Space Observatory (ISO) spectra of a subsample of WISE AGB stars were also exploited.",
"The dust radiation transfer code DUSTY was used to help predict the magnitudes in the W1 and W2 bands, the two WISE bands most affected by saturation, for calibration purpose, and to provide physical parameters of the AGB sample stars for analysis.",
"DUSTY is verified against the ISO spectra to be a good tool to reproduce the spectral energy distributions of these AGB stars.",
"Systematic magnitude-dependent offsets have been identified in WISE W1 and W2 magnitudes of the saturated AGB stars, and empirical calibration formulas are obtained for them on the basis of 1877 (W1) and 1558 (W2) AGB stars that are successfully fit with DUSTY.",
"According to the calibration formulae, the corrections for W1 at 5 mag and W2 at 4 mag are $-0.383$ and 0.217 mag, respectively.",
"In total, we calibrated the W1/W2 magnitudes of 2390/2021 AGB stars.",
"The model parameters from the DUSTY and the calibrated WISE W1 and W2 magnitudes are used to discuss the behavior of the WISE color-color diagrams of AGB stars.",
"The model parameters also reveal that O-rich AGB stars with opaque circumstellar envelopes are much rarer than opaque C-rich AGB stars toward the anti-Galactic center direction, which we attribute to the metallicity gradient of our Galaxy."
],
[
"introduction",
"The asymptotic giant branch (AGB) phase is the final stellar evolutionary stage of intermediate-mass (1 – 8 $M_{\\odot }$ ) stars driven by nuclear burning.",
"Stars in this stage have low surface effective temperatures (below 3000 K) and experience intense mass loss (from $10^{-7}$ to $10^{-4}$ ${M}_{\\odot }{\\rm yr}^{-1}$ ) [11].",
"Heavy elements in the mass outflow from a central star will condense to form dust when the gas temperature drops to the sublimation temperature of the dust grains.",
"Dusty circumstellar envelopes will form at the distance of several stellar radii.",
"Dust grains in the envelopes absorb stellar radiation and re-emit in the infrared.",
"Thus, AGB stars are important infrared sources.",
"The mass-loss process plays an important role in the evolution of AGB stars because it affects the lifetime of the AGB phase and the core-mass of the subsequent post-AGB stars.",
"Statistics of a large sample of AGB stars would help to constrain the evolution of dust envelope.",
"Due to the relative over-abundance between carbon and oxygen, there are two main types of AGB stars: (1) the O-rich with C/O $<1$ and mainly silicate-type grains in the outflow, and (2) C-rich with C/O $>1$ and mainly carbonaceous grains in the envelopes.",
"The different dust compositions of these two types of AGB stars result in different infrared spectral features, which can be used to distinguish the two groups of the stellar objects.",
"The first-generation infrared space-telescope, the Infrared Astronomical Satellite (IRAS; [26]), mapped the sky in two mid-infrared and two far-infrared bands (12, 25 $\\mu $ m and 60, 100 $\\mu $ m, respectively) in the 1980s.",
"It discovered many mass-losing AGB stars in the Milky Way and the Magellanic Clouds.",
"Several dozen AGB stars were observed spectroscopically by the Infrared Space Observatory [16] with the Short-Wave Spectrometer (SWS) onboard, which works in the range of 2.4 $\\mu $ m to 45 $\\mu $ m. These infrared spectra provide tremendous information about dusty circumstellar envelopes around the AGB stars, but they are limited by the sample size.",
"The Galactic Plane Survey of the Spitzer Space Telescope [45] provides high-resolution and sensitive infrared images of Galactic plane at the 3.6, 4.5, 5.8, 8.0, 24, and 70 $\\mu $ m bands.",
"However, it cannot adequately separate the two types of AGB stars in mid-infrared color-color diagrams without filters between 8 $\\mu $ m and 24 $\\mu $ m. Launched on 2009 December 14, the Wide-field Infrared Survey Explorer (WISE) completed an entire sky survey in the 3.4, 4.6, 12 and 22 $\\mu $ m bands (hereafter named W1, W2, W3, and W4, respectively) in 2010.",
"The FWHMs of the point spread functions (PSF) are $6.^{\\prime \\prime }1$ , $6.^{\\prime \\prime }4$ , $6.^{\\prime \\prime }5$ , and $12.^{\\prime \\prime }0$ for the four WISE bands, and the sensitivities ($5 \\sigma $ ) of the point sources are 0.08, 0.11, 1, and 6 mJy [46].",
"The Two Micron All Sky Survey (2MASS; [34]) also mapped the entire sky in three near-infrared, J (1.25 $\\mu $ m), H (1.65 $\\mu $ m) and $\\mathrm {K}_{\\rm s}$ (2.17 $\\mu $ m) bands.",
"They are an important supplement to the mid-infrared WISE data to complete the infrared spectral energy distribution (SED) of AGB stars.",
"Recently, galaxy-wide AGB populations have been identified and studied in nearby galaxies using ground- and space-based near- and mid-infrared data ([30], [4] in the Magellanic Clouds and [14] in M33).",
"While similar research in our galaxy is hindered by foreground extinction, there are some studies on large datasets of Galactic AGB stars, such as the work of [15] in the solar neighborhood.",
"[43], [44] composed a large verified AGB catalog in our galaxy from the literature and investigated infrared colors of AGB stars and their distribution in color-color diagrams.",
"To study an AGB star population, a more time-saving way is constructing a set of model grids to simulate the infrared SED of AGB stars and acquire their physical parameters, such as mass-loss rate and optical depth.",
"Many model grids have been developed to study the evolved-star populations in the local group galaxies [9], [32], [38].",
"All of the models are publicly available.",
"However, the model set of [9] does not have WISE synthetic photometry.",
"The Magellanic Clouds have a lower metallicity than our Galaxy, and [32] and [38] chose a stellar photosphere model with subsolar metallicity as input of a radiative transfer model, which is not the case in our galaxy.",
"Therefore, we used DUSTY [12] to generate a new grid of radiative transfer models to simulate the infrared SEDs of Galactic AGB stars.",
"As AGB stars typically are bright IR objects, most of our AGB samples are brighter than the WISE detector saturation limits and only photometry of unsaturated pixels are available, therefore their WISE magnitudes need to be verified.",
"The main goal of this paper is to study the properties of AGB stars in WISE bands and recalibrate the WISE photometry for sources saturated in WISE bands.",
"We describe our AGB sample, DUSTY algorithm and calibration method in §2.",
"A possible calibrating solution is presented in §3.1.",
"We present our results about the distinct location of the two types of AGB stars in a color-color diagram in §3.2, the model parameter effects on WISE colors in §3.3, and the Galactic longitude distribution in §3.4.",
"Our summary is presented in §4.",
"All the magnitudes mentioned in this paper are Vega magnitudes.",
"[43] composed a catalog of Galactic AGB stars based on reports of the IRAS Low Resolution Spectrograph (LRS; $\\lambda =8$ -22 $\\mu $ m), ISO Short-Wave Spectrometer (SWS), the Near Infrared Spectrometer (NIRS; 1.4-4.0 $\\mu $ m; [25]), radio OH and SiO survey [21], the Midcourse Space Experiment (MSX; [7]) and the 2MASS data.",
"[44] (hereafter Suh11) updated the catalog with SiO maser sources for O-rich AGB stars and additional sources for C-rich AGB stars.",
"The updated catalog contains 3003 O-rich and 1168 C-rich AGB stars with their IRAS colors and has the largest confirmed AGB collection so far.",
"This catalog provides an important entry for the future study of AGB stars.",
"To study the properties of AGB stars in infrared bands, we cross-correlated the AGB catalog with the WISE All Sky Catalog.",
"Because IRAS data have a positional uncertainty ellipse of $45^{\\prime \\prime }$$\\times {9^{\\prime \\prime }}$ , which is much bigger than the position uncertainty in WISE photometry ($\\sim $ $0.^{\\prime \\prime }20$ ), we used coordinates and magnitudes of sources during the cross-correlation to improve the precision.",
"We first used $45^{\\prime \\prime }$ as the searching radius to find matching candidates.",
"In the cases of multiple matches, we selected the WISE source with the highest 12 $\\mu $ m flux.",
"AGB stars are among the brightest sources in the sky at mid-infrared wavelengths.",
"The detection limit of IRAS is higher than that of WISE.",
"All 12 $\\mu $ m IRAS sources needed also to be detected by WISE.",
"However, high flux saturates the WISE detectors and may render them unreliable.",
"A selection criterion of W3 fainter than $-2$ mag was applied to exclude other 70 O-rich and 33 C-rich AGB stars from the sample.",
"This criterion is based on our analysis of WISE magnitudes and ISO SWS spectra of AGB stars, which is discussed in §2.3.",
"The 2MASS observations in J, H and $\\mathrm {K}_{\\rm s}$ bands are included in the WISE catalog.",
"For each WISE source, the associated 2MASS observations were selected with the nearest sources in the 2MASS catalog within $3^{\\prime \\prime }$ .",
"Some of these AGB stars (730 O-rich and 158 C-rich) do not have a 2MASS counterpart, which is perhaps due to heavy circumstellar extinction.",
"This tentative interpretation is supported by the fact that their W3–W4 colors (with medians of 1.874 mag for O-rich and 0.993 mag for C-rich objects), which correspond to an optical depth $\\sim $ 40 at 0.55 $\\mu $ m (according to our discussion in §3.2), are much redder than that of the whole AGB sample (W3–W4=1.314 mag for O-rich and 0.454 mag for C-rich objects).",
"Finally, 2203 (73.6%) O-rich and 958 (83.6%) C-rich AGB stars with 2MASS observations remained.",
"They are our WISE AGB sample.",
"We did not apply interstellar extinction correction to our data because there are no distance estimates.",
"The extinction caused by the interstellar material (ISM) can be included in the extinction of the dust envelope around AGB stars.",
"We expect that the optical depth is overestimated by a factor depending on the distance to the Sun and specific sightlines.",
"The good agreement between DUSTY models and ISO data suggests that our results from estimating the saturated WISE W1 and W2 photometry are not affected by interstellar extinction significantly.",
"To calibrate the WISE photometry, we cross-matched the Suh11 catalog with sources observed by ISO SWS (spectrum data reduced by [33]).",
"To obtain a reliable spectral sample, we excluded several O-rich AGB star spectra that deviate significantly from the typical SEDs of AGB stars and lack 10 $\\mu $ m silicate feature.",
"Spectra with a low signal-to-noise ratio were also excluded.",
"Finally, we obtained 67 O-rich and 44 C-rich AGB stars with reliable ISO mid-infrared spectra.",
"Among them, 66 (49) O-rich and 44 (37) C-rich AGB stars have associated WISE (and 2MASS) observations."
],
[
"Theoretical modeling",
"DUSTY is a radiative transfer code developed by [12] and can be used to model the dusty circumstellar envelope around AGB stars.",
"Many authors have used this code to simulate the infrared emission of AGB stars [31], [23], [10].",
"Taking the advantage of scaling, [12] minimized the number of input parameters by assuming spherical symmetry.",
"The parameters that have to be specified in models are (1) the input radiation field, which can be described by the effective temperature $T_{\\scriptsize {\\textup {eff}}}$ of the cental star; (2) the optical properties of dust grains with a specified chemical composition; (3) the grain-size distribution $n(a)$ ; (4) the dust temperature at the inner boundary of the circumstellar envelope $T_{\\scriptsize {\\textup {in}}}$ ; (5) the relative thickness (radius at the outer boundary over radius at the inner boundary, $r_{\\scriptsize {\\textup {out}}}/r_{\\scriptsize {\\textup {in}}}$ ) of the envelope; (6) envelope density distribution assuming a power law of $\\rho (r) \\propto r^{-\\alpha }$ in this work; (7) the overall optical depth at a reference wavelength $\\tau _{\\lambda }$ , and we assumed a reference wavelength at 0.55 $\\mu $ m. We used DUSTY to generate model templates to simulate the infrared SEDs of AGB stars in our Galaxy.",
"For the radiative field, we took the COMARCS hydrostatic models [1] of AGB star photospheres to represent the C-rich AGB stars and PHOENIX models [18] for O-rich AGB stars.",
"Both models assume metallicity $Z=Z_{\\odot }$ .",
"As noted in [1], most C-rich AGB stars have temperatures lower than 3500 K; the effective temperature of C-rich stellar models in this work ranges from 2500 K to 3500 K with increments of 500 K. For O-rich AGB stars, we used PHOENIX models with stellar effective temperatures between 2100 K and 4500 K in increments of 400 K. The effective temperature increments are larger than in the model grids of [32] and [38], which are 200 and 100 K, respectively, to save calculation time.",
"We explored the effective temperature with increments as small as 100 K and did not find a noticeable increase in the number of AGB stars with good fits.",
"We set the stellar mass for O-rich AGB stars to be 1 $M_{\\odot }$ , which is the only available choice in the PHOENIX model and a better choice than a blackbody SED.",
"As shown in [1] and [18], the surface gravities (log(g/(cm ${\\rm s}^{-2}$ ))) and C/O ratio have a minor effect on near-infrared colors, therefore we only used COMARCS models with log(g)=–0.4 (when the temperature was higher than 3200 K, only log(g)=–0.2 is available) and C/O=1.1, and PHOENIX models with log(g)=–0.5 and C/O=1.1.",
"Suh11 also used DUSTY to simulate the infrared colors of AGB stars.",
"They assumed that each type of AGB circumstellar envelope contains only two types of dust grains.",
"The circumstellar envelopes of O-rich AGB stars only contain silicate and porous corundum $\\textup {Al}_{2}\\textup {O}_3$ .",
"The circumstellar envelopes of C-rich AGB stars contain amorphous carbon (AmC) and SiC.",
"A composition ratio parameter X was defined to specify the number ratio of the two types of dust particles.",
"In O-rich AGB stars, $\\textup {X}=\\textup {silicate}/(\\textup {silicate}+\\textup {Al}_{2}\\textup {O}_3)$ ; in C-rich AGB stars, $\\textup {X}=\\textup {AmC}/(\\textup {AmC}+\\textup {SiC})$ .",
"They found that this assumption can well describe the properties of the envelopes of AGB stars in general.",
"In many previous studies of radiative transfer models of AGB stars, the dust composition ratio X is fixed.",
"For example, [32] used 100% silicate for O-rich AGB stars and [38] used a mixture consisting of 90% AmC and 10% SiC for C-rich AGB stars.",
"[9] introduced three dust composition ratios (X=100%, 40% and 0%) for O-rich and two ratios (X=100% and 85%) for C-rich envelopes.",
"In this work, to save time and for completeness, we fixed the dust ratio to 100% silicate around O-rich AGB stars and adopted five dust composition ratios of AmC and SiC ranging from 100% to 60% for C-rich stars.",
"The optical properties of silicate grains were taken from [27] and the porous corundum $\\textup {Al}_{2}\\textup {O}_3$ from [2].",
"While the optical properties of porous corundum from [2] only cover a wavelength range redward of 7.8 $\\mu $ m, the refraction indices are assumed to be constant at shorter wavelengths, with a value equal to the corresponding end point.",
"We can see from Figure 2 of [2] that the refractive index from [17] and [5] at wavelengths shorter than 7.8 $\\mu $ m is roughly constant.",
"The optical properties of AmC were taken from [41] and SiC from [28].",
"The grain-size distribution was modeled as $n(a) \\propto a^{-q}$ for $a_{min} < a < a_{max}$ (MRN distribution, [24]) with a power-law index q=3.5, $a_{min}=0.005\\,\\mu $ m and $a_{max}=0.25\\,\\mu $ m. In DUSTY models, $T_{in}$ is specified directly instead of the inner radius.",
"We calculated models with $T_{in}$ ranging from 600–1800 K for C-rich and 600–1400 K for O-rich AGB stars with increments of 200 K. Dust grains cannot form at temperatures higher than the sublimation temperature, which is approximately 1400 K (1800 K) for silicate (graphite grains) [29], [37].",
"The ranges of $T_{in}$ are roughly consistent with that of models in [32] and [38].",
"We assumed that the power of the density profile $\\alpha =2$ for the circumstellar envelopes, which is typical for steady circumstellar winds.",
"When $\\alpha =2$ , we find that the variance of the relative thickness of the shell from $10^2$ to $10^5$ does not have a significant impact on the mid-infrared colors at WISE wavelengths.",
"Thus, the relative thickness was fixed at $4 \\times 10^3$ .",
"The optical depth $\\tau $ at 0.55 $\\mu $ m ranges from $10^{-2}$ to $10^2$ and is sampled in logarithmic space, with a finer stepsize at the highest optical depths to ensure an even distribution of models in color-color space.",
"Finally, we calculated 3 (7) values of $T_{\\textup {eff}}$ , 5 (1) values of X, 7 (5) values of $T_{\\textup {in}}$ , and 29 (29) values of $\\tau _{0.55}$ for C-rich (O-rich) AGB stars.",
"This gives a total of 3045 and 1015 models for C-rich and O-rich AGB stars, respectively.",
"The ranges of model parameters are listed in Table 1.",
"Given each set of parameters of $T_{\\scriptsize {\\textup {eff}}}$ , X, $T_{\\scriptsize {\\textup {in}}}$ , $r_{\\scriptsize {\\textup {out}}}/r_{\\scriptsize {\\textup {in}}}$ , $\\alpha $ , $\\tau _{\\scriptsize {\\lambda }}$ , DUSTY computes the emerging SED (from 0.1 $\\mu $ m to 3600 $\\mu $ m) from the envelope.",
"The SED consists of three components: attenuated stellar radiation, scattered radiation, and dust emission.",
"The emerging SED can be convolved with WISE photon-counting relative system response (RSR) curves, $\\lambda R(\\lambda )$ [13], to simulate the WISE photometry: $W_n=-2.5 \\mbox{log} \\left(\\frac{\\int F_{\\lambda } \\lambda R_{\\lambda } d\\lambda }{F_{\\lambda n} b_n} \\right) \\qquad n=1,2,3,4,$ Here $b_n$ is the width (in units of $\\mu $ m) of each WISE band.",
"$F_{\\lambda n}$ are the fluxes at the magnitude zero point in the four bands.",
"Since WISE saturates on Vega, the magnitude zero points are based on fluxes of fainter stars calibrated to the Vega system.",
"We applied 2MASS relative spectral response curves derived by [6] to obtain the simulated 2MASS magnitudes.",
"This procedure was used to simulate the WISE and 2MASS magnitudes for each model template.",
"Table: Parameter range for model templates"
],
[
"WISE photometry and calibration method",
"We applied the WISE RSRs to the ISO spectra to simulate WISE photometry for these ISO objects.",
"Figure 1 compares the observed WISE magnitudes with those simulated from the ISO spectra.",
"In the figure, the 66 O-rich stars are designated as plus signs and the 44 C-rich AGB stars as triangles.",
"Dashed lines indicate equality between the x- and y-axis values.",
"Solid lines in W3 and W4 panels are the best-fit relations given by $W3(ISO) & = & -0.13(\\pm 0.26) \\nonumber \\\\& & +1.00(\\pm 0.00)\\times W3(WISE) \\nonumber \\\\W4(ISO) & = & -0.15(\\pm 0.09) \\nonumber \\\\& & +0.95(\\pm 0.09)\\times W4(WISE).$ We fit the line equation for W3 with a fixed unity slope to safely extrapolate the relation to fainter magnitude.",
"The correlation between ISO and WISE is quite good in the W3 and W4 bands, except that a turning point exists in the W3 band.",
"The correlation only holds for objects fainter than $-2$ mag in W3.",
"That is the reason why we excluded sources with W3 brighter than –2 when we assembled our WISE AGB sample in §2.1.",
"As ISO and WISE observations were taken at different epochs and AGB stars are variable objects even in infrared bands, part of the spread in the W3 and W4 equation is due to the variability of AGB stars.",
"We also compared WISE W3 and W4 bands with IRAS 12 $\\mu $ m and 25 $\\mu $ m data and found they agree well.",
"However, no obvious correlation is observed between WISE data and ISO observations in the W1 and W2 bands.",
"Figure: Comparison between the WISE magnitudes and the ISO spectral observations of AGB stars in WISE W1–W4 bands.",
"Pluses are O-rich and triangles are C-rich AGB stars.",
"Dashed lines indicate equality between the x- and y-axis values, and solid lines are the best-fit linear relations where applicable.It is mentioned in the WISE Explanatory Supplement documentation of the All-Sky Data Release that bright sources will saturate WISE detectors WISE Explanatory Supplement documentation for All-Sky Data Release http://wise2.ipac.caltech.edu/docs/release/allsky/expsup/.",
"The saturation limits are 8.1, 6.7, 3.8, and –0.4 magnitudes for the W1-W4 bands, respectively.",
"For saturated objects, WISE fits the point spread function (PSF) to the unsaturated pixels on the images to recover the saturatured pixels and yield photometry for these objects (we call this PSF-fit photometry).",
"The bright source photometry limits for the WISE 4 bands are 2, 1.5, –3, and –4 mag.",
"When they are brighter than this limit, all the pixels are saturatured and no unsaturated pixels are available to fit to the PSF.",
"Therefore the PSF-fit photometry is quite unreliable for sources brighter than the photometry limit.",
"Because AGB stars typically are bright IR objects, most of our AGB samples are brighter than the saturation limits, and therefore only PSF-fit photometry is available.",
"This comparison suggests that the PSF-fit photometry can be directly calibrated to the ISO synthetic photometry in the W3 and W4 bands and becomes quite unreliable for almost all the considered bright ISO AGB stars in W1 and W2 bands.",
"Our ISO samples are confined to a relatively bright synthetic W3 magnitude range of $W3<0.5$ .",
"To facilitate our calibration down to the WISE saturation limit of 3.8 mag, we assumed that the good correlation between the PSF-fit and ISO synthetic W3 photometry can be extrapolated.",
"To ensure a reliable extrapolation, we adopted the fit with unity slope (Eq.2).",
"To compensate for the loss of the W1 and W2 band data for bright objects like our AGB stars, we developed a calibration strategy based on the use of the DUSTY model template introduced in §2.2.",
"To find the best-fit model to each source, we adopted the following fitting procedure: the goodness of fit was measured by the overall offset between observed and model magnitudes.",
"For each object, each set of model magnitudes was shifted together to obtain the smallest offset from the observed magnitudes.",
"The overall shift is then the average of the differences in the five bands.",
"We assumed that an overall shift smaller than 0.4 mag consistutes a good fit.",
"For comparison, the average root mean square observational uncertainties at these five bands are 0.165 and 0.198 mag for O-rich and C-rich AGB, respectively.",
"We obtained the W3 and W4 magnitudes of the ISO AGB sources by convolving ISO spectra with proper filter-response curves.",
"Figure 2 shows the ISO spectra as a solid line and the associated 2MASS J, H, $\\mathrm {K}_{\\rm s}$ fluxes (squares) for one AGB star of each type.",
"The dotted lines are the fitted model SEDs.",
"The models fit the observations well in the bands of interest.",
"The best-fit model parameters and deviation from observations are displayed in each panel.",
"We note that the best-fit model does not fit the dust resonance features (such as 9.7 $\\mu $ m and 18 $\\mu $ m silicate features) very well, since we only have two synthetic photometry points to sample this part of the SED.",
"There are some other wavelength regions where model spectra do not match the observations, such as the absorption feature at about 5 $\\mu $ m in the spectrum of the selected C-rich AGB star, which is absent from the model SED.",
"The absorption feature at about 5 $\\mu $ m may be the molecular features of $C_3$ at 5.1 $\\mu $ m and/or CO at 4.6 $\\mu $ m [8].",
"The carbon-star spectrum shown in Figure 2 also has deep features at $\\sim $ 3 $\\mu $ m and $\\sim $ 14 $\\mu $ m, which are caused by HCN+$\\textup {C}_2\\textup {H}_2$ [8].",
"Strong $\\textup {C}_2\\textup {H}_2$ features have been observed in carbon stars in nearby galaxies; in the Milky Way's high metallicity it is expected that there is a larger contribution from HCN to these features [23].",
"More complicated DUSTY models with additional dust and molecular species may improve our fits.",
"However, this requires significant efforts on choosing dust compositions for models and beyond the scope of the current work.",
"The difference of the W2 magnitude of this C-rich AGB star between observation and model is 0.11 mag.",
"According to our analysis, this level of uncertainty does not significantly affect our results and our conclusions are still valid.",
"We expect that the model spectra and dust properties can be better constrained with more photometry points in some narrow-band filters that trace the molecular and dust features.",
"Figure: Comparison between photometric/spectroscopic observations andDUSTY best-fit models for AGB stars.",
"The left panel is for a selectedO-rich star and the right panel for a C-rich star.",
"The solid lines are ISOSWS spectra in the range between 2.5 μ\\mu m and 45 μ\\mu m. Thediamonds represent 2MASS J, H, and K s {\\rm K_s}-band fluxes and ISO synthetic photometry for W1, W2, W3, and W4.",
"The dotted lines are the best-fit models described in the text.",
"The molecular features discussed in the text are marked.In total, 45 O-rich AGB and 28 C-rich AGB stars with ISO spectra were successfully modeled.",
"The comparison of W1 and W2 magnitudes between ISO observations and the DUSTY models are shown in Figure 3.",
"Dashed lines indicate equality between the x- and y-axis values.",
"The data points are fitted with a linear equation as $W1(ISO)=-0.21(\\pm 0.07)+1.09(\\pm 0.05)\\times W1(model) \\nonumber \\\\W2(ISO)=-0.15(\\pm 0.06)+1.08(\\pm 0.05)\\times W2(model).$ The correlations of both O-rich and C-rich AGB stars (represented by pluses and triangles, respectively) are very close to the one-to-one equation, which suggests that the DUSTY models agree well with the ISO observations despite the degeneracy of some model parameters and no correction for interstellar extinction.",
"The slight deviation from ISO synthetic photomety may be caused by the molecular features, which are absent in the model spectra.",
"Based on the above results, we calibrated our WISE AGB sample in five steps.",
"First, we calibrated WISE W3 and W4 PSF-fit photometry to ISO synthetic photometry by using Equation 2.",
"Second, we fit the calibrated W3 and W4 and the 2MASS J, H, and $\\mathrm {K}_{\\rm s}$ -band magnitudes and determined the best DUSTY model for each WISE AGB star.",
"Third, we compared the model W1 and W2 magnitudes with the observed ones for our whole sample of AGB stars and identified the magnitude ranges in which good linear correlations hold.",
"Fourth, we assumed that the ISO simulated W1 and W2 magnitudes are reliable as a whole to derive empirical formulas for calibrating the observed W1 and W2 magnitudes to those of the DUSTY model and then to those simulated by ISO.",
"Finally, we applied the obtained calibration formulas to the observed W1 and W2 magnitudes of the objects in the good-correlation magnitude ranges.",
"As a result, 1659 of 2203 O-rich and 835 of 958 C-rich WISE AGB stars meet our good-fit criteria.",
"In total, we calibrated the W1/W2 magnitudes of 2390/2021 AGB stars.",
"The detailed calibration solution is discussed in §3.1.",
"We note that about one fourth (544) of the O-rich AGB stars cannot be fitted by our model grid.",
"We explored the model grid from [32] and found that the situation is not improved.",
"Most of the objects without best-fits tend to have higher W3–W4 colors than fitted ones.",
"Some even have an extreme W3–W4 color with strong deviations from the main distribution of O-rich AGB stars in the color-color diagram and seem to be misclassified as AGB stars or have some extreme properties that we do not know yet.",
"We explored many possible model parameter ranges and did not find a significant improvement.",
"Figure: Comparison between the ISO simulation magnitudes and the simulated magnitudes from the model for AGB stars in the WISE W1 and W2 bands.",
"Pluses are O-rich and triangles are C-rich AGB stars.",
"Dashed lines indicate equality between the x- and y-axis values."
],
[
"WISE calibration solution",
"We show the comparison between the W1 and W2 model magnitudes and the WISE observation magnitudes in Figure 4.",
"A linear relationship seems to exist between the model values and the observations for most sources at the faint end, although these sources are all brighter than the nominal saturation limits.",
"Only a small fraction of the sources at the bright end do not follow the relationships.",
"The features at the bright end are similar to those in Figure 1 when comparing ISO simulation magnitudes with WISE.",
"This also supports the consistency between ISO spectra and DUSTY models.",
"The faint end vertical dashed lines indicate the WISE saturation limits (8 and 6.7 mag for W1 and W2).",
"Vertical dashed lines at the bright end (3 and 2.2 mag for W1 and W2, respectively) indicate the criteria that are visually determined to exclude sources that disagree in the linear relationship.",
"The bright useful limit is consistent with the WISE photometry limit mentioned in §2.3.",
"Twenty data points in the W1 and 32 data points in the W2 panels, which significantly deviate from the linear distribution, were excluded.",
"The correlation coefficients of the data points selected by the criteria and WISE saturation limit are 0.898 and 0.910 in the W1 and W2 bands.",
"To acquire a more realistic relationship between model W1 and W2 and WISE W1 and W2, the uncertainties in the y-axis of the model synthetic magnitudes that are caused by variability of AGB stars need to be specified.",
"We estimated the uncertainties caused by variability with the following three steps: first, we calculated a \"typical\" model SED with typical model parameters ($T_{\\scriptsize {\\textup {eff}}}=3000 K$ , X=1, $T_{\\scriptsize {\\textup {in}}}=1000 K$ and $\\tau _{\\scriptsize {0.55}}=10$ ).",
"Then, for each band of J, H, $\\mathrm {K}_{\\rm s}$ , W3, and W4 of this \"typical\" model SED, we assumed a sinusoidal light curve with different amplitude and different phase.",
"[35] suggested that the variation amplitude decreases with increasing wavelength.",
"Since the variability amplitude of the semiregular variables (SRVs) is typically 0.4–0.8 mag and that of miras in the OGLE-III (Opitcal Graviational Lensing Experiment; [36]) dataset are higher than 1 mag in the I band, we assumed the typical amplitude of variation as 0.7, 0.7, 0.7, 0.4, and 0.4 mag for J, H, $\\mathrm {K}_{\\rm s}$ , W3, and W4 bands, respectively.",
"We sampled each light curve at a random phase to generate a new SED.",
"Finally, we simulated the new SED and compared the new W1 and W2 with the primary W1 and W2 of the \"typical\" model SED.",
"We repeated the second and third steps to find the W1 and W2 model uncertainties, which are 0.522 and 0.480 mag.",
"To determine the linear relation in a more robust way, we used median fitting.",
"The data points were divided into several bins with 150 objects in each bin along the x-axis.",
"For each bin we computed the median x- and y-values and used the median absolute deviation from the median (MADM) to estimate the spread in x- and y-values.",
"Then we fit a line through these new binned data points.",
"The results of the linear equation using median fitting are as follows: $W1(model) & = & 0.28(\\pm 0.83) \\nonumber \\\\& & +0.90(\\pm 0.17)\\times W1(WISE) \\nonumber \\\\W2(model) & = & 0.83(\\pm 0.67) \\nonumber \\\\& & +0.89(\\pm 0.18)\\times W2(WISE).$ The linear relations are plotted as solid lines in the Figure 4.",
"The dotted lines of the unit slope are plotted for reference.",
"The nonunity slope of linear relationships indicate that the offsets of the WISE observations are magnitude-dependent.",
"Although several data points near the bright criteria do not follow the linear relationship, the fitted linear relations are valid for most data points.",
"The dispersion may be caused by the variability of AGB stars and uncertainties in models.",
"The relation between DUSTY models and ISO observations as described in Equation 3 yields our final calibration formulas: $W1(calibrated) & = & 0.10(\\pm 0.91) \\nonumber \\\\& & +0.98(\\pm 0.19)\\times W1(WISE) \\nonumber \\\\W2(calibrated) & = & 0.75(\\pm 0.73)\\nonumber \\\\& & +0.96(\\pm 0.20)\\times W2(WISE).$ The empirical calibration formulas are valid only in the ranges of 3-8 mag for W1 and 2.2-6.7 mag for W2.",
"In our AGB star sample, the W1 and W2 magnitudes of 2390 and 2021 AGB stars are located in these ranges.",
"The median of W(model)–W(PSF) is –0.024 and 0.557 mag for W1 and W2.",
"It can also be deduced from Equation 5 that WISE PSF-fit photometry in general slightly underestimates the W1 flux and overestimates the W2 flux.",
"Several effects may contribute to the magnitude-dependent offsets.",
"First, AGB stars have extended profiles because of their dusty mass-loss winds.",
"The extended profile probably has an exaggerated effect on the PSF magnitudes that are computed using the unsaturated outer part of a saturated star image.",
"Since brighter stars are typically more extended, this effect might contribute to the nonunity slope.",
"Second, variability of AGB stars will introduce uncertainties in model simulation and dispersion in Figure 4, which will reduce the slope.",
"Finally, the flux-dependent shape of the PSF of the WISE detector in the saturated regime might be another contributor to the bright star flux biases, which are discussed in the section on photometric bias in the WISE Explanatory Supplement documentation of the All-Sky Data Release.",
"Since the uncertainties caused by variability of the y-axis are much larger than the uncertainties of observations in the x-axis, variability has a significant influence on the relation in Figure 4.",
"As we mentioned in §2.3, the molecular absorbtion feature in the W1 and W2 bands may be an important reason for the DUSTY model deviation from ISO, some molecules such as CO also have features in the near-infrared band.",
"It is hard to quantify the effect of the molecular feature on the model simulation.",
"However, changes caused by the molecular feature ($\\sim $ 0.1 mag in the W1 and W2 bands) to the infrared SED are much smaller than the change caused by variability ($\\sim $ 0.5 mag in the W1 and W2 bands).",
"Since we have taken variability uncertainties into consideration, adding uncertainties caused by molecular features will not make a noticeable difference to the current results."
],
[
"Separation between C-rich and O-rich AGB",
"We plot the W1$-$ W2 versus W3$-$ W4 color-color diagram of our calibrated WISE sample in Figure 5.",
"It can be seen that the two types of AGB stars are located in different regions in the figure and can be well distinguished in the WISE color-color diagram.",
"A red solid line is plotted to effectively separate the two types of AGB stars.",
"The straight lines successfully separate 87.1% of the O-rich AGB stars and 85.7% of the C-rich AGB stars.",
"We computed the separation line by varying the slope and intercept until the product of the two separation fractions were minimum.",
"This line function is $W1-W2=2.35\\times (W3-W4)-1.24 .$ It can be seen from Figure 5 that some data points lie far from the main distribution of AGB stars and show extremely high W3–W4 colors.",
"They are the sample of objects whose SEDs cannot be successfully fit by our DUSTY model grids.",
"To probe the physical origin of this division, we calculated two sequences of DUSTY models by varying the optical depth while keeping the other three parameters fixed ($T_{\\scriptsize {\\textup {eff}}}=3000$ K, $X=1$ and $T_{\\scriptsize {\\textup {in}}}=1000$ K) for both O-rich and C-rich AGB stars.",
"The optical depth $\\tau _{0.55}$ ranges from 0.01 to 100.",
"The resulting tracks are also plotted as green curves in the left panel of Figure 5.",
"In general, increasing the optical depth, which also indicates a decrease in the dust temperature, causes the data point to move diagonally from left bottom to upper right in the figure.",
"O-rich models differ significantly from C-rich models in that the W1$-$ W2 color increases much more slowly than W3$-$ W4 does with increasing optical depth.",
"Therefore, the distribution of O-rich models moves almost horizontally at the low W1$-$ W2 end.",
"The only difference between these two sequences of models is the dust composition.",
"We also plot the blackbody curve for temperatures from 500 K to 3500 K as the dash-dotted line in the figure.",
"The distribution of carbon stars roughly follows the trend of the blackbody curve, but with slightly bluer W1$-$ W2 or redder W3–W4 colors than the latter.",
"The middle and right panels of Figure 5 show the model SEDs with optical depth equal to 0.01, 1.25, and 40 for O-rich and C-rich AGB stars (middle panel for O-rich and right for C-rich) and WISE W1–W4 response functions.",
"The simulated WISE colors of these four models are indicated as squares in the left panel of Figure 5.",
"The SED of the C-rich star model with an optical depth of 0.01 almost overlaps that of the O-rich model with the same optical depth, since both of them are not affected significantly by dust and are essentially the spectrum of the central star.",
"In contrast, SEDs of the models with an optical depth of 1.25 and 40 are different between the two AGB types, although the SEDs are normalized to the bolometric luminosities of the stars.",
"When $\\tau =1.25$ , the O-rich star model shows negligible emission at the shorter wavelengths and thus is still dominated by the central star emission, while the C-rich star model has begun to be dominated by dust emission at these wavelengths.",
"Conversely, O-rich AGB stars show stronger dust emission at longer wavelengths than C-rich AGB stars.",
"This difference arises because O-rich dust species have much less efficient absorption at shorter wavelengths but more efficient absorption at longer wavelengths than C-rich dust species.",
"This explains that the W1$-$ W2 colors of O-rich model AGB stars are not sensitive to the increase of optical depth in the not very optically thick cases while their W3$-$ W4 colors redden faster than C-rich AGB stars.",
"When the circumstellar envelope becomes very optically thick (with $\\tau =40$ ), the O-rich AGB star model still has bluer W1$-$ W2 and redder W3–W4 colors than the C-rich AGB star model.",
"This is because of the more efficient absorption at longer wavelengths and perhaps also because of the self-absorption feature at 9.8 $\\mu $ m. The C-rich AGB stars are located closer to the blackbody line because the C-rich dust species have a relatively smooth opacity profile."
],
[
"Model parameters",
"From Figure 5, it can be seen that the trend of our WISE AGB sample from left bottom to upper right in the color-color diagram is predominantly caused by changes in optical depth.",
"To analyze the effects of the other three model parameters ($T_{\\scriptsize {\\textup {eff}}}$ , $T_{\\scriptsize {\\textup {in}}}$ and X) on the mid-infrared colors, we took the optical depth tracks in Figure 5 as standard model tracks and constructed new tracks for various values of the other three parameters.",
"The three panels of Figure 6 compare the model tracks produced by varying one of the three other model parameters.",
"Note that not all values of X might be physical.",
"The maximum fractional abundance of SiC in the dust depends on the metallicity as well as on the fraction of Si that condenses into dust.",
"The AmC-to-SiC ratio is also controlled by the dust condensation sequence; depending on whether C or SiC condenses first (see, e.g., Section 5 in [20] and references therein).",
"These model curves are only used to show the range of variation in our models that is caused by changes in the model parameters.",
"Solid lines are the standard model tracks as shown in Figure 5.",
"In the left panel of Figure 6, it can be seen that the change of $T_{\\scriptsize {\\textup {eff}}}$ mainly affects the optically thin objects.",
"This is because the NIR SEDs of optically thin AGB stars are dominated by the central star's radiation.",
"In the middle panel of Figure 6, the change of the dust composition ratio X mainly affects the IR colors of C-rich AGB star models while its effects are quite weak on O-rich AGB star models.",
"This may be because the difference between the extinction curve of the two assumed dust components is steeper in C-rich AGB stars than that in O-rich AGB stars.",
"In the right panel of Figure 6, models with higher $T_{\\scriptsize {\\textup {in}}}$ have fainter (i.e.",
"bluer) W3–W4 and W1–W2 colors, which is consistent with the behavior of the blackbody.",
"The variety of these physical parameters in AGB circumstellar envelopes contributes significantly to the data scatter in the WISE color-color diagram."
],
[
"Implications of the number statistics",
"Because this is compilation of different works, it is hard to quantitatively estimate the completeness of our WISE AGB sample.",
"However, we can determine the completeness from the spatial distribution of the sources.",
"Most of the WISE AGB stars are located at the low Galactic latitude region, as expected.",
"The longitude distributions are shown in Figure 7.",
"The left panel shows the whole sample, while the right two panels show sources with optical depth $\\tau _{0.55} > 10$ and $\\tau _{0.55}<10$ , respectively.",
"It is obvious that more O-rich AGB stars (solid) concentrate toward the direction of the Galactic center, while C-rich AGB stars (dashed) are distributed more evenly along the Galactic longitude.",
"The different longitude distributions of O-rich and C-rich AGB stars are consistent with previous works.",
"For example, [19] found that C-rich AGB stars are located preferentially in the exterior of our Galaxy and O-rich stars tend to reside in the interior of our Galaxy.",
"The Galactic segregation of these two types of AGB stars is usually interpreted as an effect of the metallicity during the formation of C-rich AGB stars.",
"Stellar evolution models of AGB stars show that O-rich AGB stars can evolve into C-rich AGB stars more rapidly at lower metallicity [19].",
"Thus the metallicity gradient in our Galaxy can cause the different longtitude distributions of O-rich and C-rich AGB stars.",
"The consistent spatial distributions between the our sample and other works suggests that the sample might represent a population of AGB stars in the solar neighborhood.",
"This may be because the catalog is built from sources selected randomly in different works.",
"The middle panel of Figure 7 is the longitude distribution of AGB stars with $\\tau _{0.55} > 10$ .",
"Very few high optical depth O-rich AGB stars locate in the direction opposed to the Galactic center.",
"This deficiency of high optical depth object is obvious from comparing the middle panel and the right panel of Figure 7.",
"We can rule out the possibility that the differing Galactic interstellar extinction along different lines of sight could lead to this trend because no significant difference is present in the longitude distribution of low- and high optical depth C-rich AGB stars.",
"The metallcity gradient in our Galaxy may again be a natural explanation because O-rich AGB stars in the exterior of our Galaxy have lower metallicty and will evolve into C-rich AGB stars at earlier evolution times.",
"These O-rich AGB stars may have a relatively lower amount of dust in the circumstellar envelopes and lower optical depth.",
"We reported our findings on AGB star colors in the WISE bands.",
"We found that the WISE W1 and W2 magnitudes of AGB stars do not agree with the spectroscopic measurements from ISO when we compared our sample of ISO simulated W1–W4 magnitudes with WISE observations, which we attribute to the residual bias in the PSF-fit photometry of saturated objects in these two bands.",
"The WISE saturated W3 and W4 magnitudes are directly calibrated based on ISO synthetic photometry.",
"To calibrate the WISE W1 and W2 bands, we resorted to ISO spectra of a subsample of our AGB stars and proved that the radiation transfer code DUSTY can be used to reproduce unbiased W1 and W2 magnitudes of bright stars.",
"Using DUSTY, we successfully developed a calibration method for the observed WISE W1 and W2 bands to lift the residual bias of the PSF-fit photometry.",
"The calibration procedure revealed that WISE may in general have underestimated W1 flux and overestimated W2 flux, and these deviations seem to be magnitude-dependent.",
"Combining the model-calibrated W1 and W2 data with the directly calibrated WISE W3 and W4 magnitudes, we analyzed the W1$-$ W2 vs W3$-$ W4 color-color diagram and found that the two main types of AGB stars, O-rich AGB and C-rich AGB, can be effectively distinguished by their WISE colors.",
"The division is mainly caused by the different extinction efficiencies between silicate-type and carbonaceous grains.",
"The spatial distribution of the AGB sample is consistent with previous work.",
"We also found that O-rich AGB stars with an opaque circumstellar envelope are much rarer toward the anti-Galactic Center direction than C-rich AGB stars, which we attribute to the metallicity gradient of our Galaxy.",
"We thank the referee for thoughtful comments and insightful suggestions, which significantly improved the quality of this paper.",
"This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration and the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation.",
"We are grateful to Martin Cohen, who provided the Vega spectrum to check our magnitude simulation process, and to Edward Wright, who gave us very useful advice about the origin of the magnitude-dependent offset of WISE photometry in the saturated regime.",
"This work is supported by China Ministry of Science and Technology under State Key Development Program for Basic Research (2012CB821800), the National Natural Science Foundation of China (NSFC, Nos.",
"11173056, 11225315 and 11320101002), Chinese Universities Scientific Fund (CUSF) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP, No.",
"20123402110037)."
]
] | 1403.0343 |
[
[
"Investing and Stopping"
],
[
"Abstract In this paper we solve the hedge fund manager's optimization problem in a model that allows for investors to enter and leave the fund over time depending on its performance.",
"The manager's payoff at the end of the year will then depend not just on the terminal value of the fund level, but also on the lowest and the highest value reached over that time.",
"We establish equivalence to an optimal stopping problem for Brownian motion; by approximating this problem with the corresponding optimal stopping problem for a random walk we are led to a simple and efficient numerical scheme to find the solution, which we then illustrate with some examples."
],
[
"Introduction.",
"The fee structure of a hedge fund typically consists of two components, a fixed management feeThis will usually be a relatively low percentage, 2% being common., charged on all assets under management, and a performance feeThis is usually charged at quite a high rate, 20% being common., charged on any gain achieved on the funds invested.",
"The exact contractual agreement has to specify between what dates the gain must have been recorded, what happens to the management fee for funds deposited for part only of a period of reckoning, and many other details, such as any restrictions on investors' freedom to withdraw funds with or without notice periods.",
"We shall simplify the problem here, by assuming that the performance fees are charged at the end of each year on all funds held at the end of the yearIt may be that some funds are withdrawn before the end of the year, and could in principle be liable to pay performance fees, but we shall ignore this, on the grounds that investors would be unlikely to withdraw funds while they were ahead., and the gain is calculated as the increase in value of the funds from the time they were deposited in the the fund, or from the beginning of the year, whichever is later.",
"Thus the baseline for calculating the performance fee resets at the beginning of each year.",
"We shall suppose that the management fee is charged only on the funds still under management at the end of the year; this is a simplification, but as the management fee is typically of smaller magnitude, it is relatively innocent.",
"In principle, the total fees charged at the end of the year by the hedge fund to its clients would depend on the entire history of investments and withdrawals through the year, as well as on the actual performance path of the hedge fund.",
"We shall propose a simplified mechanism for this, which involves some story about how the quantity of assets under management varies as the level of the hedge fund fluctuates, and is explained in detail in Section .",
"This story captures the key features that the AUM rise as the level of the fund rises, and fall as the level falls; that newly-invested funds enter at the current level; and that funds withdrawn will have entered the fund at a level above the current level.",
"The story we tell is not perfect, but has the crucial simplifying property that the fees paid will depend on the level of the fund at the start of the year, at the end of the year, and on the highest and lowest levels attained.",
"This saves us from the need to carry along as a state variable the entire profile of the levels at which the current AUM entered the fund, which then would be impossibly clumsy to work with (compare with the study of Dybvig & Koo [2] on wash sales).",
"In a seminal contribution to this subject, Goetzmann, Ingersoll and Ross [3] provided closed-form solutions to a model which differs from ours fundamentally in that the performance fee is considered to be paid out continuously over time (with a high-water mark provision).",
"This has the undesirable side-effect that at the end of the year, the manager's reward is a function only of the high-water mark.",
"Guasoni and Obłoj [4] followed a similar approach with a continuously paid performance fee, but modelled the manager as a utility optimiser himself, also resulting in (asymptotic) closed-form solutions.",
"Accepting this simplified model, we find ourselves with an optimal control problem for the hedge fund manager, in which the objective is a function of the initial, final, highest and lowest values taken by the controlled process in the year.",
"We shall suppose that the riskless rate is zero, and that the hedge fund manager is in fact just investing in assets which fluctuate but have no drift.",
"This ignores a fund manager's presumed ability to pick winners, time the market, anticipate roll dates, or any other marketing boast; this may be unjust at the level of a single talented manager, but not too far from the situation for the industry as a whole.",
"An alternative justification is that while the assets invested in might have a positive drift, the manager will take expectations under an equivalent measure which removes the drift, as a risk control measure.",
"The level of the fund will therefore evolve in our model as a martingale, which for simplicity we suppose is continuous; the manager can adjust the volatility of the level process by choosing a smaller or larger position in the risky assets, but he cannot affect the drift.",
"Nevertheless, he has an incentive to embrace some risk, as he has a call option interest in the level of the fund, as well as the performance fee incentive.",
"Since any continuous martingale is a time change of Brownian motion, we shall begin our analysis by converting the manager's problem into an optimal stopping problem for a Brownian motion.",
"This is not quite as obvious a step as might at first sight appear, as we explain in Section REF .",
"The next step is to convert the optimal stopping problem for Brownian motion into an optimal stopping problem for a symmetric simple random walk (SSRW), whose value will be close to the value of the original problem; the difference is analyzed and estimated in Section REF .",
"While it would be possible to write down some formulation of the solution to the original continuous problem, it would not be particularly digestible, and there would then be the issue of existence and uniqueness of solutions.",
"Since we do not expect ever to be able to exhibit any closed-form solution, we are forced to numerical methods to gain understanding; and these are naturally discrete in nature.",
"Our estimates allow us to be quite precise about the error committed by the approximation.",
"Finally, this problem can be solved quite efficiently numerically, as we demonstrate in Section REF .",
"We then return in Section to the hedge fund manager's problem, where we state our modelling assumptions on how cash flows into and out of the fund as the level of the fund varies, converting the manager's objective into one of the type studied in Section .",
"We then present numerical solutions of this problem.",
"Section concludes."
],
[
"From investing to stopping.",
"In this Section, we firstly show that the investment problem can be recast as a stopping problem for Brownian motion; then we show that this stopping problem can be approximated by the corresponding stopping problem for SSRW; and finally we explain the algorithm for solving this SSRW stopping problem."
],
[
"The investing/stopping equivalence.",
"We suppose that the level of the fund is $w_0$ at time 0, and evolves as $dw_t = \\theta _t dW_t $ for some previsible process $\\theta $ for which the stochastic integral is defined, where $W$ is a standard Brownian motion.",
"We define $ \\underline{w} _t \\equiv \\inf \\lbrace w_s: s \\leqslant t\\rbrace , \\qquad \\bar{w} _t \\equiv \\sup \\lbrace w_s: s \\leqslant t \\rbrace ,$ and we suppose that the objective of the manager of the fund is $\\sup _\\theta \\, E F( \\underline{w} _1, w_1, \\bar{w} _1)$ for an $F\\!:\\!",
"(-\\infty , w_0]\\times \\mathbb {R} \\times [w_0, \\infty )\\rightarrow \\mathbb {R}$ for which the above expectation is always well-defined (continuous and bounded from above or below, say).",
"The well-known Dubins-Schwarz result says (informally) that any continuous local martingale is a time-change of a Brownian motion.",
"More precisely, if we extend the definition (REF ) of $w$ beyond time 1 by setting $\\theta _t = 1$ for all $t \\ge 1$ , and set $A_t \\equiv \\int _0^t \\theta _s^2\\;ds, \\qquad \\tau _t \\equiv \\inf \\lbrace s: A_s >t\\rbrace , $ then $B_t \\equiv w(\\tau _t)$ defines a Brownian motion relative to the filtration ${\\cal G}_t \\equiv {\\cal F}_{\\tau _t}$ , and each $A_t$ is a ${\\cal G}$ -stopping time.",
"It follows directly that $w_t = B(A_t), \\qquad \\underline{w} _t = \\inf _{0 \\leqslant s\\leqslant A_t} B_s \\equiv \\underline{B}(A_t), \\qquad \\bar{w} _t = \\sup _{0 \\leqslant s \\leqslant A_t} B_s \\equiv \\bar{B}(A_t),$ As a consequence, were it not for the fact that $A_1$ is not in general a stopping time for $B$ , the following result would be trivial.",
"Lemma 1 For a continuous $F$ , bounded from above or below, the equality $\\sup _\\theta \\, E F( \\underline{w} _1, w_1, \\bar{w} _1) = \\sup _{T \\in {\\cal T} }\\, E F(\\underline{B}_T, B_T,\\bar{B}_T) $ is valid, where ${\\cal T}$ denotes the set of stopping times of the Brownian motion $B$ .",
"Remarks.",
"Suppose that $M$ is a continuous martingale which runs like a Brownian motion until some independent exponential random time $T$ , then stands still for one unit of time, and then resumes Brownian motion.",
"The quadratic variation process $[M]$ grows at rate 1 except in the interval $[T,T+1]$ , where it remains constant.",
"It is quite easy to show that $[M]_T$ is a ${\\cal G}$ -stopping time, but it is impossible to discover what $[M]_T = T$ was just by looking at the time-changed Brownian path $B_t = M(\\tau _t)$ .",
"Thus we expect the left-hand side of (REF ) to be at least as big as the right-hand side, but it is not initially obvious that the two sides are the same.",
"Proof of Lemma REF .",
"See Appendix."
],
[
"Approximation by random walk.",
"Thanks to Lemma REF , we are now left to solve an optimal stopping problem for a Brownian motion whose stopping reward is a function of its current value, minimum and maximum, $V =\\sup _{T \\in {\\cal T} }\\, E F(\\underline{B}_T, B_T,\\bar{B}_T).$ Although suppressed in the notation, we think of $V$ as a function of the starting values $X_0 \\equiv (\\underline{B}_0, B_0, \\bar{B}_0)$ .",
"It should come as no surprise that we can approximate $V$ uniformly by a stopping problem for a SSRW $w^on the grid $ B0 + Z$,{\\begin{@align}{1}{-1}V^\\sup _{T \\in {\\hat{ \\cal T}} }\\, E F(\\underline{w}^T,w^T, \\bar{w}^T),\\end{@align}}where $ T$ represents all (discrete) $ w-stopping times.",
"Lemma 2 Let $F$ be uniformly continuous: there is some continuous function $\\psi $ tending to zero at zero such that for all $x, x^{\\prime }$ $|F(x) - F(x^{\\prime })| \\leqslant \\psi ( \\Vert x-x^{\\prime } \\Vert ).\\nonumber $ If the optimization problem is well posed, then $| V^h(X_0) - V(X_0)| \\leqslant \\psi ({3} ).$ Proof of Lemma REF .",
"See Appendix."
],
[
"Solving the random walk stopping problem.",
"Given that we have now replaced the original investment problem with an optimal stopping problem for a SSRW, we are in a position to solve it by numerical meansIt is inconceivable that we may be able to find closed-form solutions, except in some very contrived examples..",
"If we are to follow this route, then we will of course only be able to deal with examples which are finite, and for this reason we are justified in assuming that the random walk will be stopped once it leaves some interval $(w_*,w^*)$ containing $w_0$ .",
"Figure: NO_CAPTIONNow suppose that $w_* = w_0 - m, $ w* = w0+ n for some positive integers $m,n$ , and introduce the notation $F_{jk}(i) = F( w_0 - j w_0 +i w_0 + k,\\quad V_{jk}(i) = V^ w_0 - j w_0 +i w_0 + k\\nonumber $ for $-m \\leqslant -j \\leqslant i \\leqslant k \\leqslant n$ .",
"We have that $V_{jk} \\geqslant F_{jk}$ always, and that if $j = m$ or $k=n$ equality holds, since the random walk must have stopped by the time it reaches those points.",
"We can now solve recursively for the value function $V$ rather as we would solve a dynamic programming problem.",
"We shall have that for $-j < i < k$ $V_{jk}(i) = \\max \\lbrace F_{jk}(i), {\\scriptstyle \\frac{1}{2} } ( V_{jk}(i+1) + V_{jk}(i-1))\\, \\rbrace $ and at the ends of the interval we have $V_{jk}(-j) &=& \\max \\lbrace F_{jk}(-j), {\\scriptstyle \\frac{1}{2} } ( V_{jk}(-j+1) + V_{j+1,k}(-j-1))\\, \\rbrace \\\\V_{jk}(k) &=& \\max \\lbrace F_{jk}(k), {\\scriptstyle \\frac{1}{2} } ( V_{j,k+1}(k+1) + V_{jk}(k-1))\\, \\rbrace .$ The situation is illustrated in Figure A, where we plot the grid of $( \\underline{w} , \\bar{w} )$ pairs, and may imagine that we are looking down on a cube, each point of the form $(w_0 - j w_0 + k$ being the projection down into the plane of points of the form $(w_0-j w_0 + i w_0+k$ , $\\; -j \\leqslant i \\leqslant k$ .",
"At every point of the upper right boundary of the rectangle, where either $j = m$ or $k = n$ , the value function is equal to $F$ and is therefore known.",
"Now we work out the values $V_{m-1, n-1}(i)$ , by solving the optimal stopping problem (REF ) with the boundary conditions (REF ) and ().",
"The two boundary conditions require knowledge of $V$ at $(m-1,n)$ and at $(m,n-1)$ ; but these values are then known, since we know $V=F$ on the solid upper right boundary of the rectangle.",
"Now we calculate the value of $V_{m-2,n-1}$ ; this time, we need to know $V$ at $(m-2,n)$ - where it agrees with $F$ - and at $(m-1,n-1)$ - which we calculated at the first step.",
"Continuing in this fashion, we are able to calculate the values of $V$ at all points of the form $(\\ell , n-1)$ , represented by big dots in Figure A.",
"In like fashion, we can then work out the values of $V$ at all points $(m-1, \\ell )$ marked with diamonds, which gives us the values of $V$ not only at the upper right boundary, but at the the next layer in, depicted by the dots and diamonds in the diagram.",
"But now we have reduced the size of the rectangle by one in each direction, so we can repeat the method just explained to find all the values of $V$ on the dot-dash lines.",
"Proceeding similarly gives us the solution $V$ .",
"Remarks.",
"At each node $(j,k)$ of the rectangle in Figure A we have to solve an optimal stopping problem for random walk in $\\lbrace i : -j-1 \\leqslant i \\leqslant k+1 \\rbrace $ , where the random walk is absorbed at the endpoints $-j-1$ and $k+1$ , with values $V_{j+1,k}(-j-1)$ and $V_{j,k+1}(k+1)$ respectively, and with stopping values $F_{jk}(i)$ at interior points.",
"The value has a geometric interpretation as the least concave majorant of the function defined by the stopping values, and can be calculated rapidly and accurately by policy improvement.",
"It may happen that for a given $(j,k)$ the optimal stopping solution is not to stop in the interior, but otherwise there will be a smallest value $w_0+q \\eta _l(j,k)$ and a largest value $w_0+\\ell \\eta _u(j,k)$ at which $V_{jk}(i) = F_{jk}(i)$ .",
"By convention, we may define $\\eta _l(j,k) =w_0+ (k+1) and $ u(j,k) = w0-(j+1) if the optimal stopping solution does not admit stopping in the interior of the interval.",
"At times $\\tau $ when the random walk reaches a new maximum $w_\\tau = \\bar{w} _\\tau = w_0 + k, it willthereafter continue until either it hits $ w+, or it hits $\\eta _u(j,k)$ , where $ \\underline{w} _\\tau = w_0 - j.",
"If it hits the lowerbarrier $ u(j,k)$ before it hits $ w0+(k+1), then it will stop there for good, unless $\\eta _u(j,k)=w_0 - (j+1), in which case a newminimum has been achieved, and the random walk can continue to move.$"
],
[
"The Hedge Fund Manager's Investing Problem.",
"We return to the problem introduced in Section of the hedge fund manager, who can control the level $w_t$ of the fund through the position $\\theta _t$ in the risky asset.",
"As the level of the fund goes up and down, the assets under management vary.",
"We propose a very simple story for this which allows us to represent the manager's problem in the form (REF ), which can then be solved by the techniques just presented.",
"The basic idea is that there is some $C^1$ non-negative function $\\varphi $ such that at any time $\\tau $ when the level process $w$ is at its running maximum, $w_\\tau = \\bar{w} _\\tau $ , the profile of the basis levels of the assets in the fund should be given by $\\varphi (x) dx, \\qquad 0 \\leqslant x \\leqslant \\bar{w} .$ So in particular, the total assets under management at $\\tau $ would be $\\Phi ( \\bar{w} ) \\equiv \\int _0^{ \\bar{w} } \\varphi (x) \\; dx$ .",
"If we demanded that $\\varphi $ was increasing, this would represent a situation where the more successful the fund, the more people would bring their money to it.",
"What happens as the level of the fund falls back from its running maximum?",
"Investors will take their money out of the fund; as the level rises again, investors will put money in.",
"Now as the level rises again and new money comes into the fund, the basis at which that new money was invested has to be the current level.",
"In order to retain tractability, we shall insist that when money is withdrawn from the fund as the level falls, it is removed only at the current level.",
"This is a restrictive assumption, but we make it nevertheless.",
"So as the level falls, a fraction $(1-p)$ of the assets invested at the current level are removed from the fund, so that in general the profile of basis values in the fund is $\\varphi (x)I_{\\lbrace x \\leqslant w_t\\rbrace } + p \\varphi (x)I_{\\lbrace w_t\\leqslant x \\leqslant \\bar{w} _t\\rbrace },$ which is consistent with (REF ) when $w_t = \\bar{w} _t$ .",
"The assumption we make means that if the level of the fund falls a long way, there will be still quite a lot of assets which came in at a higher level, and have not been taken out yet.",
"This could be understood in terms of the reluctance of investors to realize a loss; investors would be willing to come out at zero gain, and they do in our story, but they would never come out if they would thereby realize a loss.",
"This is not the whole story, because the performance-related part of the manager's fees will be measured relative to the level $w_0$ of the fund at the start of the year.",
"We shall therefore suppose that initially the profile of basis levels is a point mass at $w_0$ of magnitude $\\Phi (w_0)$ , and that if the level falls to $ \\underline{w} $ then the funds $(1-p)(\\Phi (w_0)-\\Phi ( \\underline{w} ))$ which would be removed if the profile $\\varphi $ extended through $(0,w_0)$ will be removed from the atom at zero.",
"Thus when the minimum value of the level is $ \\underline{w} $ , the size of the atom at $w_0$ will be $\\Phi (w_0) - (1-p)(\\Phi (w_0)-\\Phi ( \\underline{w} ))= p \\Phi (w_0) + (1-p) \\Phi ( \\underline{w} ).$ Thus overall the profile of the basis levels will be $\\bigl [\\;\\varphi (x)I_{\\lbrace \\underline{w} _t \\leqslant x \\leqslant w_t\\rbrace } + p \\varphi (x)I_{\\lbrace w_t\\leqslant x \\leqslant \\bar{w} _t\\rbrace }\\; \\bigr ] dx + \\bigl \\lbrace p \\Phi (w_0) + (1-p) \\Phi ( \\underline{w} )\\bigr \\rbrace \\delta _{w_0}.$ Integrating this gives the total assets under management as $\\textrm {AUM} &=& \\Phi (w) - \\Phi ( \\underline{w} ) + p( \\Phi ( \\bar{w} ) - \\Phi (w))+ p \\Phi (w_0) + (1-p) \\Phi ( \\underline{w} )\\nonumber \\\\&=& (1-p)\\Phi (w) + p (\\Phi ( \\bar{w} ) -\\Phi ( \\underline{w} )) + p \\Phi (w_0).$ We shall suppose that there is some constant $\\beta \\in (0,1)$ such that the manager receives $\\textrm {MF}=\\beta \\times \\textrm {AUM}$ as the management fee.",
"The profile (REF ) allows us to calculate the performance component of the manager's reward, which will be $\\textrm {PF} =\\alpha \\bigl [ \\;\\int _{ \\underline{w} }^w (w- x) \\varphi (x) \\; dx + (w- w_0)^+\\bigl \\lbrace p \\Phi (w_0) + (1-p) \\Phi ( \\underline{w} )\\bigr \\rbrace \\; \\bigr ].$"
],
[
"Numerical examples.",
"We suppose that the manager is risk averse, so he tries to maximize $E U(\\textrm {MF + PF}).$ In this first example, we take $U=\\log (x)$ , $\\varphi = \\sqrt{x\\wedge K}$ , $\\alpha =20\\%$ , $\\beta =2\\%$ , $p=0.3$ , $w_0=1$ , $\\Phi (w_0)=1$ and $K=3$ .",
"Figures REF and REF illustrate the resulting payout function $F=U(\\text{MF}+\\text{PF})$ as a function of the three variables $( \\underline{w} , w, \\bar{w} )$ .",
"Since it is decreasing in $ \\underline{w} $ , increasing in $ \\bar{w} $ , as well as S-shaped in $w$ (first convex, then concave), we expect non-trivial results from the stopping problem.",
"It is worth understanding why this should be.",
"For fixed $( \\underline{w} , \\bar{w} )$ , the optimal stopping problem is on the grid $\\lbrace \\underline{w} \\!-\\!h, \\underline{w} , ..., \\bar{w} , \\bar{w} \\!+\\!h\\rbrace $ , where if we stop at $x\\in [ \\underline{w} , \\bar{w} ]$ , we get reward $F( \\underline{w} , x, \\bar{w} )$ , but if we stop at one of the endpoints, we get (at the upper endpoint for example) $F( \\underline{w} , \\bar{w} +h, \\bar{w} +h)$ .",
"This value can be (and in places is) significantly bigger than $F( \\underline{w} , \\bar{w} , \\bar{w} )$ , so we see a picture like Figure B.",
"But if the values at the endpoints are somewhat lower, there can be stopping in the interior.",
"Figure: NO_CAPTIONFigure B illustrates the typical situation for values $( \\underline{w} , \\bar{w} )$ close enough to $(w_0, w_0)$ that it is beneficial to keep going; the set of such values we call the continuation region.",
"Figure C illustrates the situation once $( \\underline{w} , \\bar{w} )$ has moved sufficiently far from $(w_0, w_0)$ .",
"Figure: NO_CAPTIONFigures REF and REF plot out the continuation region and the barriers $\\eta _l, \\eta _u$ for our first example.",
"Various comments are in order (i) If for some $( \\underline{w} , \\bar{w} )$ there is optimal stopping at some interior value, then both $\\eta _l< \\bar{w} +h$ and $\\eta _u> \\underline{w} -h$ .",
"It is not possible to have $\\eta _l< \\bar{w} +h$ and $\\eta _u= \\underline{w} -h$ , or $\\eta _u> \\underline{w} -h$ but $\\eta _l= \\bar{w} +h$ .",
"Note that if $\\eta _l< \\bar{w} +h$ (and therefore $\\eta _u> \\underline{w} -h$ ), we always have $\\eta _l \\le \\eta _u$ .",
"(ii) In the plots computed, the continuation region is a connected set.",
"In general for an optimal stopping problem with stopping reward $g( \\underline{w} , w, \\bar{w} )$ this does not need to happen.",
"(iii) In the plots computed, we have the property that if $( \\underline{w} , \\bar{w} )$ is not in the continuation region, then neither is $(x, y)$ for any $x\\le \\underline{w} ,\\ y\\ge \\bar{w} $ .",
"This means that once $( \\underline{w} , \\bar{w} )$ leaves the continuation region, no further crossing of $[ \\underline{w} , \\bar{w} ]$ will happen.",
"So if we first leave the continuation region by an increase of $ \\bar{w} $ , then $ \\underline{w} $ will not go any lower; we would always choose to stop before that happened.",
"This would be the case of a successful fund which has risen in value; the manager will stop only if the fund level falls for enough from the maximum to endanger the gains and we find that actually only a small fall will trigger stopping.",
"If $( \\underline{w} , \\bar{w} )$ leaves the continuation region by $ \\underline{w} $ falling, we are seeing an unsuccessful fund which has made significant losses.",
"There we see that the stopping barrier is actually quite high; the manager will keep on gambling in the hope of recovering some of the losses and will either gamble to extinction or until enough of the losses have been recovered that he will choose to stop."
],
[
"Conclusions.",
"We have taken the problem of a fund manager whose objective is to maximize the expected utility of his wealth, which is made up of a performance fee and a management fee.",
"Under certain simplifying assumptions, we argue that his objective is a function only of the terminal level of the fund, and the maximum and minimum levels achieved by the fund.",
"A general argument equates the investment problem to a corresponding optimal stopping problem for Brownian motion, which we approximate by discretizing the Brownian motion to a random walk; in this form, the problem can be solved efficiently numerically, and we illustrate the optimal stopping rule with some numerical examples.",
"While stopping problems for Brownian motion based on the value and the running maximum are much studied (see Azéma & Yor for a seminal contribution), there has been less attention to stopping problems involving the value, the running maximum and the running minimum (though see the recent paper of Cox and Obłoj [1] for an important contribution.)",
"The existing literature deals with such questions in the context of finding joint laws for the two (or three) variables $\\bar{B}_\\tau $ , $B_\\tau $ (and $\\underline{B}_\\tau $ ) which are extremal in some senseSee Rogers [5] where the stochastically largest maximum of a martingale whose terminal distribution is specified is shown to be achieved by the Azéma-Yor construction., and the analysis is typically quite detailed.",
"The flavour of the present study is somewhat different however, and we readily turn to numerical methods because the problem is too complicated to be amenable to analysis."
],
[
"Appendix",
"Proof of Lemma REF .",
"First we prove that $\\sup _\\theta \\, E F( \\underline{w} _1, w_1, \\bar{w} _1) \\ge \\sup _{T \\in {\\cal T} }\\, E F(\\underline{B}_T, B_T,\\bar{B}_T).$ If $T \\in {\\cal T}$ , the process $w_t \\equiv B\\biggl ( \\frac{t}{1-t}\\; \\wedge \\; T\\biggr )$ may be represented as $w_t = \\int _0^t I_{ \\lbrace s \\leqslant T^{\\prime } \\rbrace } \\frac{dW_s}{1-s},$ where $T^{\\prime } = T/(1+T)$ and $W$ is the standard Brownian motion defined by $\\int _0^t \\frac{dW_s}{1-s} = B\\biggl ( \\frac{t}{1-t}\\;\\biggr ).$ Accordingly, $( \\underline{w} _1, w_1, \\bar{w} _1) = (\\underline{B}_T, B_T, \\bar{B}_T)$ , proving the first inequality.",
"For the converse inequality, we first notice that $\\sup _\\theta \\, E F( \\underline{w} _1, w_1, \\bar{w} _1) =\\sup _{\\theta \\in {\\cal S}}\\, E F( \\underline{w} _1, w_1, \\bar{w} _1)=\\sup _\\varepsilon \\sup _{\\theta \\in {\\cal S}_\\varepsilon }\\, E F( \\underline{w} _1, w_1, \\bar{w} _1),$ where ${\\cal S}$ is the vector space of simple processes $\\theta = \\sum _{j=0}^n Z_j I_{ \\lbrace t_j < t \\leqslant t_{j+1}\\rbrace }$ for some $ 0 = t_0 < t_1 < \\ldots < t_{n+1} = 1$ , and $Z_j \\in L^\\infty ({\\cal F}_{t_j})$ for all $j$ , and ${\\cal S}_\\varepsilon = \\lbrace \\theta \\in {\\cal S}: |\\theta |\\ge \\varepsilon \\rbrace $ .",
"So it will be sufficient to show that whenever we have some $\\theta \\in {\\cal S}_\\varepsilon $ then there is some Brownian motion $B$ with canonical filtration $({\\cal B}_t)$ and a $({\\cal B}_t)$ -stopping time $T$ such that $E F( \\underline{w} _1, w_1, \\bar{w} _1) = E F(\\underline{B}_T, B_T,\\bar{B}_T).$ Given $\\theta \\in {\\cal S}_\\varepsilon $ of the form (REF ), we form the quadratic variation process $A_t = \\int _0^t \\theta _s^2 \\; ds,$ and define $B_t = w(\\tau _t)$ , where $\\tau $ is the continuous strictly-increasing inverse to $A$ .",
"Next define $T_k = A_{t_k} = \\sum _{j=0}^{k-1} Z_j^2 (t_{j+1} - t_j).$ We claim that for each $k$ , ${\\cal B}_s = {\\cal F}_{\\tau _s}$ for all $0 \\leqslant s \\leqslant T_k$ .",
"It is clear that ${\\cal B}_{T_k} \\subseteq {\\cal F}_{t_k}$ , but we shall prove by induction that equality holds for all $k$ .",
"Evidently equality holds for $k=0$ , since both $\\sigma $ -fields are trivial.",
"Suppose true up to some value of $k$ .",
"Then ${\\cal B}(T_k) ={\\cal F}(t_k)$ , and so $Z_k$ is ${\\cal B}(T_k)$ -measurable.",
"Now for $0\\leqslant s \\leqslant T_{k+1}-T_k$ we have $B_{T_k+s} - B_{T_k} = Z_k\\lbrace \\, w(t_k + sZ_k^{-2}) - w(t_k)\\,\\rbrace $ and therefore we can deduce the path $(w(t_k+u))_{0 \\leqslant u\\leqslant t_{k+1}-t_k}$ from the path of $(B_u)_{0 \\leqslant u \\leqslant T_{k+1}})$ , since $Z_k$ is ${\\cal B}(T_k)$ -measurable.",
"This extends the conclusion out to $k+1$ , and hence for all $k$ , and the equality (REF ) follows from taking $k=n+1$ .",
"Proof of Lemma REF .",
"For 0 we are going to embed a scaled random walk in our Brownian motion.",
"Let therefore $\\sigma ^0=0$ and $\\sigma ^{n+1} = \\inf \\lbrace t>\\sigma ^n; \\ |B_t - B_{\\sigma ^n}| = ,\\quad n \\ge 0.$ Then $w^n=B_{\\sigma ^n},\\ n\\ge 0$ clearly defines a random walk on $B_0 + Z$ satisfying $\\Vert (\\underline{B}_t,B_t, \\bar{B}_t) - ( \\underline{w} ^n,w^n, \\bar{w} ^n)\\Vert \\le {3},$ for $\\sigma ^{n-1} \\le t \\le \\sigma ^{n+1},\\ n \\ge 1$ .",
"Any $w^-stopping time $$ naturally induces an$ B$-stopping time $ =$ with$ w=B$, giving us{\\begin{@align}{1}{-1}V \\ge V^\\end{@align}}Moreover, for any $ B$-stopping time $$,{\\begin{@align*}{1}{-1}\\tau ^\\inf \\lbrace n \\ge 0;\\ \\sigma ^n \\ge \\tau \\rbrace \\end{@align*}}defines a random time with{\\begin{@align*}{1}{-1}| w^{\\tau ^- B_{\\tau } | \\le \\qquad | \\bar{w} ^{\\tau ^- \\bar{B}_{\\tau } | \\le \\qquad | \\underline{w} ^{\\tau ^- \\underline{B}_{\\tau } | \\le }}However, \\tau ^ is not necessarily a w^h-stopping time.",
"Define \\mathcal {G}^h_n=\\sigma (w^h_j: j\\le n) and \\mathcal {G}^h = \\sigma (w^h_j: j \\ge 0).",
"Now let \\pi _n = P[ \\tau \\in (\\sigma ^h_{n}, \\sigma ^h_{n-1}] | \\mathcal {G}^h].",
"Note that{\\begin{@align*}{1}{-1}\\pi _n = P[ \\tau \\in (\\sigma ^h_{n-1}, \\sigma ^h_{n}] | \\mathcal {G}^h_n]\\end{@align*}}since \\mathcal {G}^h_n = \\sigma ( \\xi _j: j=1...n), with \\xi _j IID.",
"Now we give ourselves a uniform random variable U, independent of B and add this to every \\mathcal {G}^h_n to form \\tilde{\\mathcal {G}}^h_n = \\mathcal {G}^h_n \\vee \\sigma (U).",
"If we stop the random walk w^h_n at the \\tilde{\\mathcal {G}}^h-stopping time{\\begin{@align}{1}{-1}S= \\inf \\lbrace n: \\sum _{i=1}^n \\pi _i > U \\rbrace .\\end{@align}}we find that w^h_S has the same law as w^h_{\\tau ^h} and then{\\begin{@align}{1}{-1}V^V - \\psi (h\\sqrt{3}).\\end{@align}}Combining (\\ref {firstineq}) and (\\ref {secondineq}) proves the desired result.",
"}\\end{@align*}}\\begin{thebibliography}{1}\\end{thebibliography}\\bibitem {ay79}J.~Azéma and M.~Yor.\\hspace{1.1pt}Une solution simple au promlème de skorokhod.\\hspace{1.1pt}{\\em Séminaire de probabilités}, 13:90--115, 1979.$ A. M. G. Cox and J. Obłoj.",
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"Preprint, 2010.",
"L. C. G. Rogers.",
"The joint law of the maximum and terminal value of a martingale.",
"Probability Theory and Related Fields, 95:451–466, 1993.",
"Figure: F(w ̲,w,w ¯)F( \\underline{w} , w, \\bar{w} ) for fixed w ̲=0.2 \\underline{w} \\!=\\!0.2: S-shaped in ww and slightly increasing in w ¯ \\bar{w} .Figure: Surface plot of the lower stopping barrier η u \\eta _u that becomes relevant when we leave the continuation region (grey, flat) by increasing w ¯ \\bar{w} ."
]
] | 1403.0202 |
[
[
"Quantum tunneling and evolution speed in an exactly solvable coupled\n double-well system"
],
[
"Abstract Exact analytical calculations of eigenvalues and eigenstates are presented for quantum coupled double-well (DW) systems with Razavy's hyperbolic potential.",
"With the use of four kinds of initial wavepackets, we have calculated the tunneling period $T$ and the orthogonality time $\\tau$ which signifies a time interval for an initial state to evolve to its orthogonal state.",
"We discuss the coupling dependence of $T$ and $\\tau$, and the relation between $\\tau$ and the concurrence $C$ which is a typical measure of the entanglement in two qubits.",
"Our calculations have shown that it is not clear whether the speed of quantum evolution may be measured by $T$ or $\\tau$ and that the evolution speed measured by $\\tau$ (or $T$) is not necessarily increased with increasing $C$.",
"This is in contrast with the earlier study [V. Giovannetti, S. Lloyd and L. Maccone, Europhys.",
"Lett.",
"{\\bf 62} (2003) 615] which pointed out that the evolution speed measured by $\\tau$ is enhanced by the entanglement in the two-level model."
],
[
"Introduction",
"Double-well (DW) potential models have been extensively employed in various fields of quantum physics, in which the tunneling is one of fascinating quantum effects.",
"Although quartic DW potentials are commonly adopted for the theoretical study, one cannot obtain their exact eigenvalues and eigenfunctions of the Schrödinger equation.",
"Then it is necessary to apply various approximate approaches such as perturbation and spectral methods to quartic potential models [1].",
"Razavy [2] proposed the quasi-exactly solvable hyperbolic DW potential, for which one may exactly determine a part of whole eigenvalues and eigenfunctions.",
"A family of quasi-exactly solvable potentials has been investigated [3], [4].",
"The two-level (TL) system which is a simplified model of a DW system, has been employed for a study on qubits which play important roles in quantum information and quantum computation [5].",
"The relation between the entanglement and the speed of evolution of TL systems has been discussed [6], [7], [8], [9].",
"The entanglement in qubits has been studied with the use of uncoupled and coupled TL models [8], [9], [10].",
"In recent years, several experimental studies for coupled TL systems have been reported [11], [12].",
"In contrast to the simplified TL model mentioned above, studies on coupled DW systems are scanty, as far as we are aware of.",
"This is because a calculation of a coupled DW system is much tedious than those of a single DW system and of a coupled TL model.",
"In the present study, we adopt coupled two DW systems, each of which is described by Razavy's potential.",
"One of advantages of our model is that we may exactly determine eigenvalues and eigenfunctions of the coupled DW system.",
"We study the tunneling period $T$ and the orthogonality time $\\tau $ which stands for the time interval for an assumed initial state to develop to its orthogonal state.",
"We investigate the relation between the speed of quantum evolution measured by $\\tau $ and the concurrence which is one of typical measures of entanglement of two qubits.",
"The difference and similarity between results in our coupled DW system and the TL model [6], [7], [8], [9] are discussed.",
"These are purposes of the present paper.",
"The paper is organized as follows.",
"In Sec.",
"II, we describe the calculation method employed in our study, briefly explaining Razavy's potential [2].",
"Exact analytic expressions for eigenvalues and eigenfunctions for coupled DW systems are presented.",
"In Sec.",
"III, with the use of four kinds of initial wavepackets, we perform model calculations of tunneling period $T$ and the orthogonality time $\\tau $ .",
"In Sec.",
"IV, we discuss the relation between the calculated $\\tau $ and the concurrence.",
"Sec.",
"V is devoted to our conclusion.",
"We consider coupled two DW systems whose Hamiltonian is given by $H &=& \\sum _{n=1}^2 \\left[ -\\frac{\\hbar ^2}{2m} \\frac{\\partial ^2}{\\partial x_n^2}+ V(x_n )\\right] - g x_1 x_2,$ with $V(x) &=& \\frac{\\hbar ^2}{2m}\\left[\\frac{\\xi ^2}{8} \\:{\\rm cosh} \\:4x - 4 \\xi \\:{\\rm cosh} \\:2x- \\frac{\\xi ^2}{8}\\right],$ where $x_1$ and $x_2$ stand for coordinates of two distinguishable particles of mass $m$ in double-well systems coupled by an interaction $g$ , and $V(x)$ signifies Razavy's potential [2].",
"The potential $V(x)$ with $\\hbar =m=\\xi =1.0$ adopted in this study is plotted in Fig.",
"1(a).",
"Minima of $V(x)$ locate at $x_s=\\pm 1.38433$ with $V(x_s)=-8.125$ and its maximum is $V(0)=-2.0$ at $x=0.0$ .",
"First we consider the case of $g=0.0$ in Eq.",
"(REF ).",
"Eigenvalues of Razavy's double-well potential of Eq.",
"(REF ) are given by [2] $\\epsilon _0 &=& \\frac{\\hbar ^2}{2m}\\left[ -\\xi -5 -2 \\sqrt{4-2 \\xi +\\xi ^2} \\right], \\\\\\epsilon _1 &=& \\frac{\\hbar ^2}{2m}\\left[ \\xi -5 -2 \\sqrt{4+2 \\xi +\\xi ^2} \\right], \\\\\\epsilon _2 &=& \\frac{\\hbar ^2}{2m}\\left[ -\\xi -5 +2 \\sqrt{4-2 \\xi +\\xi ^2} \\right], \\\\\\epsilon _3 &=& \\frac{\\hbar ^2}{2m}\\left[ \\xi -5 +2 \\sqrt{4+2 \\xi +\\xi ^2} \\right].$ Eigenvalues for the adopted parameters are $\\epsilon _0=-4.73205$ , $\\epsilon _1=-4.64575$ , $\\epsilon _2=-1.26795$ and $\\epsilon _3=0.645751$ .",
"Both $\\epsilon _0$ and $\\epsilon _1$ locate below $V(0)$ as shown by dashed curves in Fig.",
"1(a), and $\\epsilon _2$ and $\\epsilon _3$ are far above $\\epsilon _1$ .",
"In this study, we take into account the lowest two states of $\\epsilon _0$ and $\\epsilon _1$ whose eigenfunctions are given by [2] $\\phi _0(x) &=& A_0 \\; e^{-\\xi \\:{\\rm cosh} \\:2x/4} \\left[3 \\xi \\:{\\rm cosh} \\:x+(4-\\xi +2 \\sqrt{4-2 \\xi +\\xi ^2})\\: {\\rm cosh}\\: 3x \\right], \\\\\\phi _1(x) &=& A_1 \\;e^{-\\xi \\:{\\rm cosh} \\:2x/4} \\left[3 \\xi \\:{\\rm sinh}\\: x+(4+\\xi +2 \\sqrt{4+2 \\xi +\\xi ^2})\\: {\\rm sinh} \\:3x \\right],$ $A_{n}$ ($n=0,1$ ) denoting normalization factors.",
"Figure 1(b) shows the eigenfunctions of $\\phi _0(x)$ and $\\phi _1(x)$ , which are symmetric and anti-symmetric, respectively, with respect to the origin.",
"Figure: (Color online)(a) Razavy's DW potential V(x)V(x) (solid curve),dashed and chain curves expressing eigenvalues of ϵ 0 \\epsilon _0 and ϵ 1 \\epsilon _1,respectively, for ℏ=m=ξ=1.0\\hbar =m=\\xi =1.0 [Eq.()].",
"(b) Eigenfunctions of φ 0 (x)\\phi _0(x) (solid curve) and φ 1 (x)\\phi _1(x) (dashed curve).Figure REF (a) shows the 3D plot of the composite potential $U(x_1, x_2)$ defined by $U(x_1, x_2) &=& V(x_1)+V(x_2) -g x_1 x_2.$ It has four minima of $U(\\pm x_s, \\pm x_s)=-16.25$ and one maximum of $U(0.0, 0.0)=-4.0$ for $g=0.0$ .",
"Solid curves in Fig.",
"REF (b) show contour plots of $U(x_1, x_2)=\\mu $ for $\\mu =-15$ , -10 and -5 with $g=0.0$ .",
"For a comparison, dashed curves shows the result with $g=1.0$ , for which $U(\\pm x_s, \\mp x_s)-U(\\pm x_s, \\pm x_s)=3.8327$ .",
"Dashed curves with $g=1.0$ are slightly different from solid curves with $g=0.0$ .",
"Figure: (Color online)(a) 3D plot of a composite potential U(x 1 ,x 2 )U(x_1, x_2) as functions of x 1 x_1 and x 2 x_2.",
"(b) Contour plots of U(x 1 ,x 2 )=μU(x_1, x_2)=\\mu for μ=-15\\mu =-15, -10-10 and -5-5with g=0.0g=0.0 (solid curves) and g=1.0g=1.0 (dashed curves)."
],
[
"Eigenvalues and eigenstates of the coupled DW system",
"We calculate exact eigenvalues and eigenstates of the coupled two DW systems described by Eq.",
"(REF ).",
"With basis states of $\\phi _0 \\phi _0$ , $\\phi _0 \\phi _1$ , $\\phi _1 \\phi _0$ and $\\phi _1 \\phi _1$ where $ \\phi _n \\phi _k \\equiv \\phi _n(x_1) \\phi _k(x_2)$ , the energy matrix for the Hamiltonian given by Eq.",
"(REF ) is expressed by ${\\cal H} &=& \\left( {\\begin{array}{*{20}c}{2 \\epsilon _0 } & {0 } & {0 } & {-g \\gamma ^2} \\\\{0 } & {\\epsilon _0 + \\epsilon _1 } & {-g \\gamma ^2 } & {0} \\\\{0 } & {-g \\gamma ^2 } & {\\epsilon _0 + \\epsilon _1 } & {0} \\\\{-g \\gamma ^2 } & {0 } & {0 } & {2 \\epsilon _1} \\\\\\end{array}} \\right),$ with $\\gamma &=& \\int _{-\\infty }^{\\infty } \\phi _0(x)\\: x \\: \\phi _1(x)\\:dx=1.13823.$ Eigenvalues of the energy matrix are given by $E_0 &=& \\epsilon -\\sqrt{\\delta ^2+ g^2 \\gamma ^4},\\\\E_1 &=& \\epsilon - g \\gamma ^2, \\\\E_2 &=& \\epsilon + g \\gamma ^2, \\\\E_3 &=& \\epsilon + \\sqrt{\\delta ^2+ g^2 \\gamma ^4},$ where $\\epsilon &=& \\epsilon _1+\\epsilon _0=-9.3778, \\\\\\delta &=& \\epsilon _1-\\epsilon _0=0.0863.$ Corresponding eigenfunctions are given by $\\Phi _0(x_1,x_2) &=& \\cos \\theta \\:\\phi _0(x_1) \\phi _0(x_2)+ \\sin \\theta \\:\\phi _1(x_1) \\phi _1(x_2),\\\\\\Phi _1(x_1,x_2) &=& \\frac{1}{\\sqrt{2}} \\left[ \\phi _0(x_1) \\phi _1(x_2)+ \\phi _1(x_1) \\phi _0(x_2) \\right], \\\\\\Phi _2(x_1,x_2) &=& \\frac{1}{\\sqrt{2}} \\left[- \\phi _0(x_1) \\phi _1(x_2)+ \\phi _1(x_1) \\phi _0(x_2) \\right], \\\\\\Phi _3(x_1,x_2) &=& -\\sin \\theta \\:\\phi _0(x_1) \\phi _0(x_2)+ \\cos \\theta \\: \\phi _1(x_1) \\phi _1(x_2),$ where $\\tan \\:2 \\theta &=& \\frac{g \\gamma ^2}{\\delta }.\\;\\;\\;\\;\\mbox{$\\left(-\\frac{\\pi }{4} \\le \\theta \\le \\frac{\\pi }{4} \\right)$}$ Eigenvalues $E_{\\nu }$ ($\\nu =0$ -3) are plotted as a function of $g$ in Fig.",
"3, which is symmetric with respect to $g=0.0$ .",
"For $g=0.0$ , $E_1$ and $E_2$ are degenerate.",
"We hereafter study the case of $g \\ge 0.0$ .",
"With increasing $g$ , energy gaps between $\\epsilon _0$ and $\\epsilon _1$ and between $\\epsilon _2$ and $\\epsilon _3$ are gradually decreased while that between $\\epsilon _1$ and $\\epsilon _2$ is increased.",
"Figure: (Color online)Eigenvalues E ν E_{\\nu } (ν=0-\\nu =0-3) of a coupled DW system as a functionthe coupling strength gg.The time-dependent wavepacket is expressed by $\\Psi (t) &=& \\Psi (x_1,x_2,t)= \\sum _{\\nu =0}^{3}\\:a_{\\nu } \\: \\Phi _{\\nu }(x_1,x_2) \\:e^{-i E_{\\nu } t/\\hbar },$ where expansion coefficients $a_{\\nu }$ satisfy the relation $\\sum _{\\nu =0}^3 \\vert a_{\\nu } \\vert ^2 &=& 1.$ Expansion coefficients $a_{\\nu }$ may be formally determined for a given initial wavepacket, which requires cumbersome calculations.",
"In this study they are assumed a priori as will be given shortly.",
"The correlation function $\\Gamma (t)$ is defined by $\\Gamma (t) &=& \\vert \\int _{-\\infty }^{\\infty } \\int _{-\\infty }^{\\infty }\\Psi ^*(x_1,x_2, 0) \\:\\Psi (x_1,x_2,t)\\;dx_1\\:dx_2 \\: \\vert , \\\\&=& \\vert \\; \\vert a_0 \\vert ^2 + \\sum _{\\nu =1}^3 \\: \\vert a_{\\nu } \\vert ^2\\:e^{- i \\Omega _{\\nu } t} \\: \\vert ,$ where $\\Omega _{\\nu }=(E_{\\nu }-E_0)/\\hbar $ .",
"The tunneling period $T$ for the initial wavepacket given by Eq.",
"(REF ) is determined by $T &=& \\min _{\\forall \\:t \\:> 0}\\; \\lbrace \\Gamma (t)=1 \\rbrace .$ On the contrary, the orthogonality time $\\tau $ is provided by the time interval such that an initial wavepacket takes to evolve into the orthogonal state [6], [7], [8], [9], $\\tau &=& \\min _{\\forall \\:t \\:> 0}\\; \\lbrace \\Gamma (t)=0 \\rbrace .$ In the case of a simple wavepacket including only two states, e.g.",
"$a_{\\nu }=(1/\\sqrt{2}) \\:(\\delta _{\\nu , 0}+\\delta _{\\nu , \\kappa })$ , the correlation function becomes $\\Gamma (t) &=& \\frac{1}{2} \\vert 1+ e^{-i \\Omega _{\\kappa } t} \\vert = \\sqrt{ \\frac{1+ \\cos \\Omega _{\\kappa } t}{2} },$ for which we easily obtain $T$ and $\\tau $ $T &=& 2 \\tau = \\frac{2 \\pi }{\\Omega _{\\kappa }}.$ In the case of $g=0.0$ where $\\Omega _1=\\Omega _2=\\Omega _3/2$ , Eqs.",
"(REF ) and (REF ) become $T &=& \\min _{\\forall \\:t \\:> 0}\\; \\large \\lbrace \\vert a_0 \\vert ^2+ (\\vert a_1 \\vert ^2+ \\vert a_2 \\vert ^2)\\:z(t)+ \\vert a_3 \\vert ^2 \\:z(t)^2=1 \\large \\rbrace , \\\\\\tau &=& \\min _{\\forall \\:t}\\; \\large \\lbrace \\vert a_0 \\vert ^2+ (\\vert a_1 \\vert ^2+ \\vert a_2 \\vert ^2)\\:z(t) + \\vert a_3 \\vert ^2 z(t)^2 =0 \\large \\rbrace ,$ where $z(t)=e^{- i \\Omega _1 t}$ .",
"Solutions of $T$ and $\\tau $ may be obtainable from roots of respective polynomial equations for $z(t)$ [8], [9].",
"In a general case, however, $T$ and $\\tau $ are obtained by solving Eqs.",
"(REF ) and (REF ) with a numerical method, as will be shown later."
],
[
"Model calculations",
"Dynamical properties of wavepackets A-D with assumed expansion coefficients shown in Table 1, have been studied.",
"We will report results of the case with $g=0.0$ for wavepackets A and B in Sec.",
"III A, and those with $g=0.1$ for wavepackets C and D in Sec.",
"III B.",
"Table: NO_CAPTIONTable 1 Assumed expansion coefficients $a_{\\nu }$ ($\\nu =0$ to 3) for four wavepackets A, B, C and D."
],
[
"Uncoupled double-well system ($g=0.0$ )",
"First we consider the uncoupled DW with $g=0.0$ , for which eigenvalues are $E_0 &=& -9.4641, \\;\\;\\;E_1 = E_2= -9.3778, \\;\\;\\;E_3 = -9.2915,$ leading to $\\Omega _1 &=& \\Omega _2=0.0863, \\;\\;\\;\\Omega _3=0.1726,$ and eigenfunctions are given by $\\Phi _0(x_1,x_2) &=& \\phi _0(x_1) \\phi _0(x_2), \\\\\\Phi _1(x_1,x_2) &=& \\frac{1}{\\sqrt{2}} \\left[ \\phi _0(x_1) \\phi _1(x_2)+ \\phi _1(x_1) \\phi _0(x_2) \\right], \\\\\\Phi _2(x_1,x_2) &=& \\frac{1}{\\sqrt{2}} \\left[- \\phi _0(x_1) \\phi _1(x_2)+ \\phi _1(x_1) \\phi _0(x_2) \\right], \\\\\\Phi _3(x_1,x_2) &=& \\phi _1(x_1) \\phi _1(x_2).$ Figure 4(a), 4(b), 4(c) and 4(d) show eigenfunctions $\\Phi _{\\nu }(x_1,x_2)$ for $\\nu =0$ , 1, 2 and 3, respectively.",
"Figure: (Color online)Eigenfunctions of (a) Φ 0 (x 1 ,x 2 )\\Phi _0(x_1,x_2), (b) Φ 1 (x 1 ,x 2 )\\Phi _1(x_1,x_2), (c) Φ 2 (x 1 ,x 2 )\\Phi _2(x_1,x_2),and (d) Φ 3 (x 1 ,x 2 )\\Phi _3(x_1,x_2) for g=0.0g=0.0.A factorizable product state is expressed by $\\Psi _{prod}&=& \\Psi _{RR}(x_1,x_2) = \\Psi _R(x_1) \\Psi _R(x_2),\\\\&=& \\frac{1}{2} \\left[ \\phi _0(x_1) \\phi _0(x_2)+ \\phi _0(x_1) \\phi _1(x_2)+\\phi _1(x_1) \\phi _0(x_2)+\\phi _1(x_1) \\phi _1 (x_2)\\right], \\\\&=& \\frac{1}{2} \\left[ \\Phi _0(x_1,x_2) +\\Phi _3(x_1,x_2) \\right]+ \\frac{1}{\\sqrt{2}} \\Phi _1(x_1,x_2),$ where magnitude of $\\Psi _R(x_{\\nu })$ $( =[\\phi _0(x_{\\nu })+\\phi _1(x_{\\nu })]/\\sqrt{2} )$ localizes at the right well in the $x_{\\nu }$ axis ($\\nu =1, \\:2$ ).",
"The wavepacket yielding initially the product state given by Eq.",
"() is described by $\\Psi _{A}(x_1,x_2,t) &=& \\frac{1}{2}\\left[ \\Phi _0(x_1,x_2)\\:e^{-i E_0 t/\\hbar } +\\Phi _3(x_1,x_2)\\:e^{-i E_3 t/\\hbar } \\right]+ \\frac{1}{\\sqrt{2}} \\Phi _1(x_1,x_2)\\:e^{-i E_1 t/\\hbar },$ and the relevant correlation function is given by $\\Gamma _A(t) &=& \\vert \\frac{1}{4}\\:\\left( 1+ e^{-i \\Omega _3 t/\\hbar }\\right)+\\frac{1}{2}\\: e^{-i \\Omega _1 t} \\vert ,$ where $\\Omega _1=\\Omega _3/2=0.0863$ .",
"Calculated $\\Gamma _A(t)$ is plotted in Fig.",
"REF (a) which yields the tunneling period of $T=2 \\pi /\\Omega _1=72.81$ and the orthogonality time of $\\tau =T/2=36.40$ .",
"Figures REF (b) and REF (c) will be explained later (Sec.",
"IV B).",
"Figure: (Color online)Correlation functions Γ A (t)\\Gamma _A(t) with (a) g=0.0g=0.0, (b) g=0.1g=0.1 and (c) g=0.2g=0.2for the wavepacket A.Figure: (Color online)Time-dependent magnitudes of |Ψ A (x 1 ,x 2 ,t)| 2 \\vert \\Psi _A(x_1,x_2,t) \\vert ^2at (a) t=0.0t=0.0, (b) t=0.1Tt=0.1 T, (c) t=0.2Tt=0.2 T, (d) t=0.3Tt=0.3 T, (e) t=0.4Tt=0.4 T and (f) t=0.5Tt= 0.5Tfor the wavepacket A [Eq.",
"()] with g=0.0g=0.0 where T=72.81T=72.81.Magnitudes of wavepackets at t=0.6Tt=0.6T, 0.7T0.7T, 0.8T0.8T, 0.9T0.9T and TT arethe same as those at t=0.4Tt=0.4T, 0.3T0.3T, 0.2T0.2T, 0.1T0.1T and 0, respectively.Time-dependent magnitudes of $\\vert \\Psi _A(x_1, x_2, t) \\vert ^2$ are shown in Figs.",
"REF (a)-REF (f).",
"Figure REF (a) shows that the wavepacket initially has the maximum magnitude at the $RR$ side of $(x_1,x_2)=(x_m, x_m)$ with $x_m=1.23534$ near the bottom of the right-side well of $U(x_s, x_s)$ with $x_s=1.38433$ , where $RR$ signifies the right side in the $x_1$ axis and the right side in $x_2$ axis.",
"At $t=0.2 T$ , $\\vert \\Psi _A(x_1, x_2, t) \\vert ^2$ in Fig REF (b) has finite magnitudes near $LL$ , $RL$ and $LR$ sides besides $RR$ one.",
"This implies a tunneling of particles among four bottoms of $U(\\pm x_s, \\pm x_s)$ .",
"The orthogonal state to Eq.",
"() is given by $\\Psi _{LL}(x_1,x_2) &=& \\Psi _L(x_1) \\Psi _L(x_2), \\\\&=& \\frac{1}{2} \\left[ \\phi _0(x_1) \\phi _0(x_2)- \\phi _0(x_1) \\phi _1(x_2)-\\phi _1(x_1) \\phi _0(x_2)+\\phi _1(x_1) \\phi _1 (x_2)\\right], \\\\&=& \\frac{1}{2} \\left[ \\Phi _0(x_1,x_2) +\\Phi _3(x_1,x_2) \\right]- \\frac{1}{\\sqrt{2}} \\Phi _1(x_1,x_2),$ where magnitude of $\\Psi _L(x_{\\nu })$ $( =[\\phi _0(x_{\\nu })-\\phi _1(x_{\\nu })]/\\sqrt{2} )$ localizes at the left well in the $x_{\\nu }$ axis ($\\nu =1, 2$ ).",
"$\\Psi _A(x_1, x_2, t)$ reduces to $\\Psi _{LL}(x_1, x_2)$ at $t=0.5T$ , and it returns to $\\Psi _{RR}(x_1, x_2)$ at $t=T$ .",
"Figure: (Color online)Time-dependent magnitudes of |Ψ B (x 1 ,x 2 ,t)| 2 \\vert \\Psi _B(x_1,x_2,t) \\vert ^2at (a) t=0.0t=0.0, (b) t=0.1Tt=0.1 T, (c) t=0.2Tt=0.2 T, (d) t=0.3Tt=0.3 T, (e) t=0.4Tt=0.4 T and (f) t=0.5Tt=0.5 Tfor the wavepacket B [Eq.",
"()] with g=0.0g=0.0 where T=36.40T=36.40."
],
[
"Wavepacket B: $a_0=1/\\sqrt{2}$ , {{formula:29345528-4e9f-4534-93ef-2f515a066288}} and {{formula:ad36a9ee-1dfd-4dc5-a244-1fee3e20a8b8}}",
"As a typical entangled state which cannot be expressed in a factorized form, we consider the state $\\Psi _{ent}(x_1,x_2) &=&\\frac{1}{\\sqrt{2}} \\left[ \\phi _0(x_1) \\phi _0(x_2)+\\phi _1(x_1) \\phi _1(x_2) \\right], \\\\&=& \\frac{1}{\\sqrt{2}} \\left[ \\Phi _0(x_1,x_2) +\\Phi _3(x_1,x_2) \\right].$ The relevant wavepacket is expressed by $\\Psi _{B}(x_1.",
"x_2, t) &=& \\frac{1}{\\sqrt{2}} \\left[ \\Phi _0(x_1,x_2)\\:e^{-i E_0 t/\\hbar }+\\Phi _3(x_1,x_2)\\:e^{-i E_3 t/\\hbar } \\right],$ and its correlation function is given by $\\Gamma _B(t) &=& \\frac{1}{2}\\vert 1+ e^{-i \\Omega _3 t} \\vert = \\sqrt{ \\frac{1+\\cos \\Omega _3 t}{2} },$ where $\\Omega _3=0.1726$ .",
"The tunneling period becomes $T=2 \\pi /\\Omega _3=36.40$ and the orthogonality time is given by $\\tau =T/2=18.20$ .",
"Figures REF (a)-REF (f) show the time-dependent magnitudes of $\\vert \\Psi _{B}(x_1,x_2) \\vert ^2$ at $0 \\le t \\le T$ .",
"Initially $\\vert \\Psi _{B}(x_1,x_2) \\vert ^2$ has peaks at both $RR$ and $LL$ sides.",
"At $t=0.5 T$ , it reduces to $\\Psi _{ent}^{\\bot }(x_1,x_2) &=&\\frac{1}{\\sqrt{2}} \\left[ \\Phi _0(x_1,x_2) - \\Phi _3(x_1,x_2) \\right],$ which is orthogonal to the assumed initial state given by Eq.",
"(REF ) and which has peaks at both $RL$ and $LR$ sides.",
"Next we study coupled DW systems with $g=0.1$ , for which eigenvalues are $E_0 &=& -9.53347, \\;\\; E_1=-9.50736,\\;\\; E_2=-9.24825,\\;\\; E_3=-9.22213,$ leading to $\\Omega _1&=& 0.02611, \\;\\; \\Omega _2=0.28522, \\;\\;\\Omega _3=0.31134.$ The potential difference between the two bottoms is $U(\\pm x_s, \\mp x_s)-U(\\pm x_s, \\pm x_s)=0.38327$ .",
"Figures REF (a)-REF (d) show eigenfunctions $\\Phi _{\\nu }(x_1,x_2)$ for $\\nu =0-3$ .",
"Figure: (Color online)Eigenfunctions of (a) Φ 0 (x 1 ,x 2 )\\Phi _0(x_1,x_2), (b) Φ 1 (x 1 ,x 2 )\\Phi _1(x_1,x_2), (c) Φ 2 (x 1 ,x 2 )\\Phi _2(x_1,x_2),and (d) Φ 3 (x 1 ,x 2 )\\Phi _3(x_1,x_2) for g=0.1g=0.1."
],
[
"Wavepacket C: $a_0=a_1=1/\\sqrt{2}$ and {{formula:60eb95bd-bf6f-4e11-b283-f5c00b818f97}}",
"With $a_0=a_1=1/\\sqrt{2}$ and $a_2=a_3=0$ , the wavepacket in Eq.",
"(REF ) becomes $\\Psi _C(x_1,x_2,t) &=& \\frac{1}{\\sqrt{2}}\\left[ \\Phi _0(x_1,x_2) \\:e^{-i E_0 t/\\hbar }+\\Phi _1(x_1,x_2) \\:e^{-i E_1 t/\\hbar } \\right],$ whose correlation function is given by $\\Gamma _C(t) &=& \\frac{1}{2} \\vert 1 + e^{-i \\Omega _1 t} \\vert = \\sqrt{ \\frac{1+ \\cos \\Omega _1 t}{2} },$ with $\\Omega _1=0.02611$ .",
"The tunneling period is $T=2 \\pi /\\Omega _1=240.63$ and the orthogonality time is $\\tau =T/2=120.32$ .",
"Figure: (Color online)Time-dependent magnitudes of |Ψ C (x 1 ,x 2 ,t)| 2 \\vert \\Psi _C(x_1,x_2,t) \\vert ^2at (a) t=0.0t=0.0, (b) t=0.1Tt=0.1 T, (c) t=0.2Tt=0.2 T, (d) t=0.3Tt=0.3 T, (e) t=0.4Tt=0.4 T and (f) t=0.5Tt= 0.5 Tfor the wavepacket C [Eq.",
"()] with g=0.1g=0.1 where T=240.63T=240.63.Figure: (Color online)(a) 3D plot of |Ψ C (x 1 ,x m ,t)| 2 \\vert \\Psi _C(x_1,x_m,t) \\vert ^2 as functions of x 1 x_1 and tt with x m =1.23534x_m=1.23534.",
"(b) Time dependence of Γ C (t)\\Gamma _C(t) (solid curve)and |Ψ C (x m ,x m ,t)| 2 \\vert \\Psi _C(x_m,x_m,t) \\vert ^2 (dashed curve) for the wavepacket C (g=0.1g=0.1).Figure: (Color online)3D plot of ρ C (x 1 ,t)\\rho _{C}(x_1, t) as functions of x 1 x_1 and tt for the wavepacket C (g=0.1g=0.1).Figures REF (a)-REF (f) show the time dependence of the magnitude of $\\vert \\Psi _C(x_1,x_2,t) \\vert ^2$ .",
"Figure REF (a) shows that at $t=0$ , the wavepacket given by $\\Psi _{C}(x_1,x_2) &=& \\frac{1}{\\sqrt{2}}\\left[ \\Phi _0(x_1,x_2) +\\Phi _1(x_1,x_2) \\right],$ has the maximum magnitude at the $RR$ side of $(x_1, x_2)=(x_m, x_m)$ .",
"We note that with time development, the magnitude of wavepacket at the initial position at the $RR$ side is decreased while that at the $LL$ side of $(x_1, x_2)=(-x_m, -x_m)$ is increased.",
"At $t=0.5 T$ , $ \\Psi _C(x_1,x_2,t)$ reduces to the state given by $\\Psi _{C}^{\\bot }(x_1,x_2) &=& \\frac{1}{\\sqrt{2}}\\left[ \\Phi _0(x_1,x_2) -\\Phi _1(x_1,x_2) \\right],$ whose magnitude locates at the $LL$ side of $(x_1, x_2)=(-x_m, -x_m)$ , and which is the orthogonal state to Eq.",
"(REF ).",
"This expresses the tunneling of a particle across the potential barrier at the origin of $(x_1, x_2)=(0.0, 0.0)$ .",
"The wavepacket returns to the initial state at $t=T$ .",
"Dynamics of the wavepacket is studied in more detail.",
"We show in Fig.",
"REF (a), the 3D plot of $\\vert \\Psi _C(x_1,x_m,t) \\vert ^2$ as functions of $x_1$ and $t$ .",
"Solid and dashed curves in Fig.",
"REF (b) show time dependences of $\\Gamma (t)$ and $\\vert \\Psi _C(x_m,x_m,t) \\vert ^2$ , respectively, which oscillate with a period of $T=2 \\tau =240.63$ .",
"By using Eqs.",
"(REF )-() and (REF ), we calculate the density probability of the $x_1$ component $\\rho _{C}(x_1, t) &=& \\int _{-\\infty }^{\\infty }\\: \\vert \\Psi _C(x_1, x_2, t) \\vert ^2 \\:dx_2, \\\\&=&\\frac{1}{4} \\left[ (2 \\cos ^2 \\theta +1) \\:\\phi _0(x_1)^2+ (2 \\sin ^2 \\theta +1) \\:\\phi _1(x_1)^2 \\right] \\nonumber \\\\&+& \\frac{1}{\\sqrt{2}}(\\cos \\theta +\\sin \\theta )\\: \\phi _0(x_1)\\phi _1(x_1)\\:\\cos \\Omega _1 t.$ The time-dependent expectation value of $\\langle x_1 \\rangle $ is expressed by $\\langle x_1 \\rangle &=& \\int _{-\\infty }^{\\infty } \\:\\rho _C(x_1, t) \\:x_1 \\:dx_1, \\\\&=& \\frac{\\gamma }{\\sqrt{2}}\\:(\\cos \\theta +\\sin \\theta )\\:\\cos \\Omega _1 t,$ where $\\gamma $ is given by Eq.",
"(REF ).",
"Figure REF shows the 3D plot of $\\rho _{C}(x_1, t)$ .",
"Similar analysis may be made for the component $x_2$ .",
"If we read $x_1 \\rightarrow x_2$ in Fig.",
"REF , it expresses the density probability for the $x_2$ component.",
"A comparison between Figs.",
"REF (a) and REF (a) indicates that $\\Psi _C(x_1, x_2, 0.0)$ is initially similar to $\\Psi _A(x_1, x_2, 0.0)$ , both of which have appreciable magnitudes at the RR site.",
"Nevertheless, their time development is quite different: e.g.",
"$\\Psi _C(x_1, x_2, 0.2T) \\ne \\Psi _A(x_1, x_2, 0.2 T)$ ."
],
[
"Wavepacket D: $a_0=a_1=a_2=a_3=1/2$",
"With $a_0=a_1=a_2=a_3=1/2$ , Eq.",
"(REF ) yields the wavepacket given by $\\Psi _{D}(x_1,x_2, t) &=&\\frac{1}{2} \\sum _{\\nu =0}^3 \\:\\Phi _{\\nu }(x_1,x_2)\\:e^{-i E_{\\nu } t/\\hbar },$ which leads to the correlation function $\\Gamma _D(t) &=& \\frac{1}{4}\\vert 1+ e^{-i \\Omega _1 t}+ e^{-i \\Omega _2 t}+ e^{-i \\Omega _3 t} \\vert ,$ with $\\Omega _1=0.02611$ , $\\Omega _2=0.28522$ and $\\Omega _3=0.31134$ .",
"The time-dependent $\\vert \\Psi _D(x_1,x_2,t) \\vert ^2$ from $t=0$ to $t=0.5 T$ are shown in Figs.",
"REF (a)-REF (f) where $T=242.32$ (below).",
"In order to scrutinize its behavior, we show in Fig.",
"REF (a), the 3D plot of $\\vert \\Psi _D(x_1,x_m,t) \\vert ^2$ as functions of $x_1$ and $t$ .",
"The dashed curve in Fig.",
"REF (b) expresses $\\vert \\Psi _D(x_m,x_m,t) \\vert ^2$ whereas the solid curve shows $C_D(t)$ which is expressed as a superposition of three oscillations with frequencies of $\\Omega _1$ , $\\Omega _2$ and $\\Omega _3$ .",
"Both $C_D(t)$ and $\\vert \\Psi _D(x_m,x_m,t) \\vert ^2$ show rapid and complicated oscillations with zeros of $C_D(t)$ at $t=11.01 \\:(2k+1)$ with $k=0, 1, \\cdots $ .",
"We obtain $T &=& 242.32 \\simeq \\frac{2 \\pi }{E_1-E_0} =240.63, \\\\\\tau &=& 11.01 \\simeq \\frac{\\pi }{E_3-E_1}=11.02.$ The tunneling period $T$ is mainly determined by a energy gap between $E_0$ and $E_1$ , while a small $\\tau $ originates from a large energy gap between $E_1$ and $E_3$ .",
"Figure: (Color online)Time-dependent magnitudes of |Ψ D (x 1 ,x 2 ,t)| 2 \\vert \\Psi _D(x_1,x_2,t) \\vert ^2for (a) t=0t=0, (b) t=0.1Tt=0.1 T, (c) t=0.2Tt=0.2 T, (d) t=0.3Tt= 0.3 T, (e) t=0.4Tt=0.4 T and (f) t=0.5Tt= 0.5 Tfor the wavepacket D [Eq.",
"()] with g=0.1g=0.1 where T=242.32T=242.32.Figure: (Color online)(a) 3D plot of |Ψ D (x 1 ,x m ,t)| 2 \\vert \\Psi _D(x_1,x_m,t) \\vert ^2as functions of x 1 x_1 and tt with x m =1.23534x_m=1.23534.",
"(b) Time dependence of Γ D (t)\\Gamma _D(t) (solid curve)and |Ψ D (x m ,x m ,t)| 2 \\vert \\Psi _D(x_m,x_m,t) \\vert ^2 (dashed curve) for the wavepacket D (g=0.1g=0.1)."
],
[
"Comparison among results of four wavepackets A, B, C and D",
"It has been pointed out [6], [7] that the entanglement enhances the speed of evolution in certain quantum state as measured by the time speed to reach an orthogonal state.",
"The orthogonality time $\\tau $ is shown to be given by [6], [7] $\\tau &\\ge & \\tau _{\\min }\\equiv \\max \\left( \\frac{\\pi \\hbar }{2 E}, \\;\\;\\frac{\\pi \\hbar }{2 \\Delta E} \\right),$ where $E$ and $\\Delta E$ signify expectation and root-mean-square values, respectively, of the energy relative to $E_0$ , $E &=& \\sum _{\\nu }\\: \\vert a_{\\nu }\\vert ^2 \\:(E_{\\nu }-E_0),\\\\\\Delta E &=& \\sqrt{ \\sum _{\\nu }\\: \\vert a_{\\nu }\\vert ^2 \\:(E_{\\nu }-E_0)^2-E^2 }.$ Equations (REF )-() show that the minimum orthogonality time $\\tau _{min}$ depends on the distribution of eigenvalues and the expansion coefficient of wavepackets.",
"Applying Eqs.",
"(REF )-() to our DW model in Eqs.",
"(REF )-(), we have evaluated $E$ , $\\Delta E$ and $\\tau _{min}$ whose results are summarized in the Table 2.",
"We note that $\\tau _{min}$ is determined by $E$ ($< \\Delta E$ ) for the wavepacket C, while it is determined by $\\Delta E$ ($< E$ ) for wavepackets A and D ($E=\\Delta E$ for the wavepacket B).",
"The tunneling period $T$ and the orthogonality time $\\tau $ for four wavepackets A, B, C, and D calculated in the preceding section are summarized in Table 2.",
"It is shown that $\\tau $ in the four wavepackets are in agreement with results of $\\tau \\ge \\tau _{min}$ evaluated by Eqs.",
"(REF )-().",
"$\\tau $ of the entangle wavepacket B is smaller than that of the non-entangled wavepacket A ($g=0.0$ ), which is consistent with an enhancement of $\\tau $ by entanglement in uncoupled qubits [6], [7].",
"Table: NO_CAPTIONTable 2 The tunneling period $T$ [Eq.",
"(REF )], the orthogonality time $\\tau $ [Eq.",
"(REF )], the expectation value of the energy $E$ [Eq.",
"(REF )], the root-mean-square value $\\Delta E$ [Eq.",
"()], the minimum orthogonality time $\\tau _{min}$ [Eq.",
"(REF )], and the concurrence $C$ [Eq.",
"(REF )] in four wavepackets A, B, C and D with couplings $g$ (see Table 1)."
],
[
"Calculation of the concurrence",
"In order to examine the relation between $\\tau _{min}$ and the entanglement, we have calculated the concurrence which is one of typical measures expressing the degree of entanglement.",
"Substituting Eqs.",
"(REF )-() to Eq.",
"(REF ) with $t=0$ , we obtain $\\vert \\Psi \\rangle &=& c_{00} \\vert 0\\;0 \\rangle + c_{01} \\vert 0\\;1 \\rangle + c_{10} \\vert 1\\;0 \\rangle + c_{11} \\vert 1\\;1 \\rangle ,$ with $c_{00} &=& a_0 \\:\\cos \\theta -a_3 \\:\\sin \\theta ,\\;\\;c_{01} = \\frac{1}{\\sqrt{2}}(a_1-a_2), \\nonumber \\\\c_{10} &=& \\frac{1}{\\sqrt{2}}(a_1+a_2),\\;\\; c_{11} = c_0 \\:\\sin \\theta +a_3 \\:\\cos \\theta ,$ where $\\vert k \\;\\ell \\rangle =\\phi _k(x_1) \\phi _{\\ell }(x_2)$ with $k, \\ell =0,1$ .",
"The concurrence $C$ of the state $\\vert \\Psi \\rangle $ given by Eq.",
"(REF ) is defined by [13] $C &=& 2 \\:\\vert c_{00} c_{11}- c_{01} c_{10} \\vert .",
"$ The state given by Eq.",
"(REF ) becomes factorizable if and only if the relation: $c_{00} c_{11}- c_{01} c_{10} =0$ holds.",
"Substituting Eq.",
"(REF ) into Eq.",
"(REF ), we obtain the concurrence $C &=& \\vert (a_0^2-a_3^2) \\sin 2 \\theta + 2 a_0 a_3 \\cos 2\\theta - a_1^2+a_2^2 \\vert .$ By using adopted coefficients in Table 1, we obtain the concurrence for the four wavepackets $C_A &=& \\frac{1}{2} \\;\\vert 1- \\cos 2 \\theta \\vert \\hspace{28.45274pt}\\mbox{(wavepacket A)},\\\\C_B &=& \\vert \\cos 2 \\theta \\vert \\hspace{56.9055pt}\\mbox{(wavepacket B)},\\\\C_C &=& \\frac{1}{2} \\;\\vert 1- \\sin {2 \\theta } \\vert \\hspace{28.45274pt}\\mbox{(wavepacket C)},\\\\C_D &=& \\frac{1}{2} \\; \\vert \\cos {2 \\theta } \\vert \\hspace{42.67912pt}\\mbox{(wavepacket D)},$ which lead to $C_A=0.0$ , $C_B=1.0$ for $g=0.0$ and to $C_C=0.0839$ and $C_D=0.2772$ for $g=0.1$ (see Table 2).",
"Figure: (Color online)The gg dependence of (a) the tunneling period TT and (b) the orthogonality time τ\\tau for wavepacket A (circles), B (triangles), C (inverted triangles) and D (squares).Results of TT for A, C and D are almost identical in (a)."
],
[
"The $g$ dependence of {{formula:919be58b-c9d0-4f41-93f6-2e6fb8e93ca7}} , {{formula:f543e2de-f850-4b4e-b8fe-10875413b96e}} , {{formula:10c6cf30-8a4c-4455-a500-1b132366ef59}} and {{formula:d8158f14-a5f7-4710-bdd8-a0c332a2b140}}",
"So far calculations are reported only for wavepackets A and B with $g=0.0$ and for wavepackets C and D with $g=0.1$ .",
"We have calculated $T$ , $\\tau $ , $\\tau _{min}$ and $C$ , by changing $g$ in a range of $0 \\le g < 0.2$ for four wavepackets A, B, C and D whose expansion coefficients $a_{\\nu }$ ($\\nu =0-3$ ) are given in Table 1.",
"For wavepackets B and C consisting of two terms, it is possible to exactly calculate the tunneling period and the orthogonality time with the use of Eq.",
"(REF ).",
"However, for wavepackets A and D with more than three terms, numerical methods are required for calculations of $T$ and $\\tau $ .",
"Calculated $T$ and $\\tau $ are plotted in Figs.",
"REF (a) and REF (b), respectively.",
"Our calculations show that $T$ and $\\tau $ for the four wavepackets are given by $T_{A} &\\simeq & T_C = \\frac{2 \\pi }{E_1-E_0},\\;\\;\\;T_B = \\frac{2 \\pi }{E_3-E_0},\\:\\:\\:T_D \\simeq \\frac{2 \\pi }{E_1-E_0},\\\\\\tau _{A} &\\simeq & \\tau _C = \\frac{\\pi }{E_1-E_0},\\;\\;\\;\\tau _B = \\frac{\\pi }{E_3-E_0},\\;\\;\\;\\tau _D \\simeq \\frac{\\pi }{E_3-E_1},$ where $E_{\\nu }$ ($\\nu =0 - 3$ ) are $g$ dependent [Eqs.",
"(REF )-()].",
"Figure REF (a) shows that with increasing $g$ , the tunneling period is increased for wavepackets A, C and D while it is decreased for the wavepacket B.",
"This is because a gap of $E_1-E_0$ ($E_3-E_0$ ) is decreased (increased) with increasing $g$ (Fig.",
"3).",
"Due to the similar reason, the orthogonality time for wavepackets A and C are increased with increasing $g$ whereas it is decreased for wavepackets B and D, as shown in Fig.",
"REF (b).",
"Figure: (Color online)The minimum orthogonality time τ min \\tau _{min} (solid curves) and the concurrence CC (dashed curves)as a function of the interaction gg for the wavepackets (a) A, (b) B, (c) C and (d) D,left and right ordinates being for τ min \\tau _{min} and CC, respectively.Figure: (Color online)TT (chain curves), τ\\tau (dashed curve) and τ min \\tau _{min} (solid curves)as a function of the concurrence CC for wavepackets (a) A, (b) B, (c) C and (d) D,TT in (d) being divided by a factor of 20.$g$ dependences of $\\tau _{min}$ and $C$ calculated with the use of Eqs.",
"(REF )-() and Eqs.",
"(REF )-() for the four wavepackets are shown in Figs.",
"REF (a)-REF (d).",
"Figure REF (a) shows that with increasing $g$ for the wavepacket A, the concurrence is increased from a vanishing value of $C=0.0$ at $g=0.0$ while $\\tau _{min}$ is decreased: a kink of $\\tau _{min}$ at $g=0.0366$ is due to a crossover of $\\pi \\hbar /2 E= \\pi \\hbar /2 \\Delta E$ in Eq.",
"(REF ).",
"Figure REF (b) shows that for the wavepacket B, an increase in $g$ induces a decrease in $\\tau _{min}$ and $C$ , the latter being decreased from the maximum concurrence of $C=1.0$ at $g=0.0$ .",
"For the wavepacket C, $\\tau _{min}$ is increased but $C $ is decreased with increasing $g$ , as shown in Fig.",
"REF (c).",
"Figure REF (d) shows that both $\\tau _{min}$ and $C$ are decreased with increasing $g$ for the wavepacket D. Figures REF (a)-REF (d) show $T$ , $\\tau $ and $\\tau _{min}$ as a function of the concurrence $C$ for the four wavepackets.",
"It is shown that for a larger $C$ , $T$ is larger in wavepackets A and B, while it is smaller in wavepackets C and D. We note that for a larger concurrence, $\\tau $ is larger for wavepackets A, B and D, but it is smaller for the wavepacket C. For a larger $\\tau $ , $T$ is larger in wavepackets A, B and C, but it is not the case for the wavepacket D. This fact imposes a question whether the evolution time may be measured by $T$ or $\\tau $ .",
"Furthermore, the speed of quantum evolution measured by either $\\tau $ or $T$ is not necessarily increased when $C$ is increased.",
"This is in contrast with Refs.",
"[6], [7] which claimed that the speed of a development of quantum state is improved by the entanglement.",
"We also note that $\\tau _{min}$ given by Eq.",
"(REF ) provides us with fairly good estimates for lower limits of $\\tau $ for wavepackets B, C and D. However, it does not for the wavepacket A.",
"In order to clarify the point, we show the correlation functions $\\Gamma _A(t)$ with $g=0.0$ , $0.1$ and $0.2$ for the wavepacket A in Figs.",
"REF (a), REF (b) and REF (c), respectively.",
"$\\Gamma _A(t)$ with $g=0.1$ and $0.2$ more rapidly oscillates than that with $g=0.0$ .",
"We obtain $( \\tau , \\tau _{min})=(36.40,\\;25.74)$ , $(121.00,\\;17.28)$ and $(218.77,\\;12.30)$ for $g=0.0$ , 0.1 and 0.2, respectively.",
"With increasing $g$ , $\\tau $ is increased because of a narrowed energy gap of $E_1-E_0$ in Eq.",
"(), whereas $\\tau _{min}$ is decreased by a high-energy contribution of $E_3-E_0$ in Eqs.",
"(REF )-().",
"Although the relation: $\\tau _{min} \\le \\tau $ is actually held, the difference between $\\tau $ and $\\tau _{min}$ is significant with increasing $g$ for the wavepacket A, where $\\tau _{min}$ given by Eqs.",
"(REF )-() is not a good estimate of the lower bound of $\\tau $ determined by Eq.",
"(REF )."
],
[
"Conclusion",
"With the use of an exactly solvable coupled DW system described by Razavy's potential [2], we have studied the dynamics of four wavepackets A, B, C and D (Table 1).",
"Our calculations of tunneling period $T$ and the orthogonality time $\\tau $ yield the followings: (1) Although the relation: $T = 2 \\tau $ holds for wavepackets B and C including two terms, it is not the case in general.",
"In particular for the wavepacket D, $T$ is increased but $\\tau $ is decreased with increasing $g$ (Fig.",
"REF ), and (2) $g$ dependences of $T$ and $\\tau $ considerably depend on a kind of adopted wavepackets (Fig.",
"REF ), and they are increased or decreased with increasing the concurrence, depending on an initial wavepacket (Figs.",
"REF and REF ).",
"A query arises from the item (1) whether the speed of a quantum evolution may be measured by $T$ or $\\tau $ , although it is commonly evaluated by $\\tau $ [6], [7], [8], [9], [10].",
"The item (2) implies that even if the evolution speed is measured by either $\\tau $ or $T$ , it is not necessarily increased by the entanglement.",
"This is in contrast with Refs.",
"[6], [7], [8], [9], [10] which pointed out an enhancement of the evolution speed by the entanglement in TL models.",
"The difference between their results and ours arises from the fact that the coupled DW model has much freedom than the TL model: the latter is a simplified model of the former.",
"It would be interesting to experimentally observed the time-dependent magnitude of $\\vert \\Psi (x_1, x_2, t) \\vert ^2$ , which might be possible with advanced recent technology.",
"In the present study, we do not take into account environmental effects which are expected to play important roles in real DW systems.",
"An inclusion of dissipative effects is left as our future subject.",
"This work is partly supported by a Grant-in-Aid for Scientific Research from Ministry of Education, Culture, Sports, Science and Technology of Japan."
]
] | 1403.0543 |
[
[
"Microscopic Origin and Universality Classes of the Efimov Three-Body\n Parameter"
],
[
"Abstract The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter.",
"We show that the three-body parameter is essentially determined by the zero-energy two-body correlation.",
"As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range.",
"One class involves the universality of the three-body parameter recently found in ultracold atom systems.",
"The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics."
],
[
"3 Microscopic Origin and Universality Classes of the Efimov Three-Body Parameter Pascal Naidon$^{1}$ , Shimpei Endo$^{2}$ , and Masahito Ueda$^{2}$ $\\,^{1}$ RIKEN Nishina Centre, RIKEN, Wakō 351-0198, Japan, $\\,^{2}$ Department of Physics, University of Tokyo, 7-3-1 Hongō, Bunkyō-ku, Tōkyō 113-0033, Japan The low-energy spectrum of three particles interacting via nearly resonant two-body interactions in the Efimov regime is set by the so-called three-body parameter.",
"We show that the three-body parameter is essentially determined by the zero-energy two-body correlation.",
"As a result, we identify two classes of two-body interactions for which the three-body parameter has a universal value in units of their effective range.",
"One class involves the universality of the three-body parameter recently found in ultracold atom systems.",
"The other is relevant to short-range interactions that can be found in nuclear physics and solid-state physics.",
"The Efimov effect is a universal low-energy quantum phenomenon, which was originally predicted in nuclear physics [1] and has rekindled considerable interest since its experimental confirmation with ultracold atoms [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22].",
"It is also expected to occur in solid-state physics [23], [24].",
"This universality stems from the effective three-body attraction that occurs between particles interacting with nearly resonant short-range interactions.",
"As a result of this attraction, three particles may bind even when the interaction is not strong enough to bind two particles.",
"Furthermore, an infinite series of such three-body bound states exists near the unitary point where the interaction is resonant, i.e.",
"where a two-body bound state appears and the $s$ -wave scattering $a$ length diverges.",
"The typical three-body energy spectrum for such systems is represented in Fig.",
"REF in units of inverse length.",
"Near zero energy and large scattering lengths, the three-body spectrum is invariant under a discrete scaling transformation by a universal factor $e^{\\pi /s_{0}}\\approx 22.7$ for identical bosons, where $s_{0}\\approx 1.00624$ characterises the strength of the three-body attraction.",
"A notable consequence of the Efimov effect is the existence of another physical scale beyond the two-body scattering length to fix the low-energy properties of the system.",
"This scale is known as the three-body parameter.",
"In zero-range models, it manifests itself as the necessity to introduce a momentum cutoff or a three-body boundary condition.",
"It can be characterised, for instance, by the scattering length $a_{-}$ at which a trimer appears or by its binding wave number $\\kappa $ at unitarity, as indicated in Fig.",
"REF .",
"Because of the discrete scaling invariance, it is defined up to a power of $e^{\\pi /s_{0}}$ .",
"In this Letter, we will focus on the ground Efimov state, which slightly deviates from the discrete-scaling-invariant structure, but is more easily observed and computed, and still reveals the essence of the physics behind the three-body parameter.",
"Three important questions can be raised concerning the three-body parameter.",
"Is there a simple mechanism that determines the three-body parameter from the microscopic interactions?",
"What is the microscopic length scale which determines the three-body parameter?",
"Finally, if there is such a length scale, what are the conditions for the three-body parameter to be related to that length scale through a universal dimensionless constant, as suggested by experimental observations [15], [17] and recent calculations [25]?",
"This Letter answers these three questions for systems of identical bosons (or three distinguishable fermions with equal mass) where the resonant interaction can be described by a single scattering channel.",
"In ultracold atoms experiments, the interaction is made resonant by using magnetic Feshbach resonances [26].",
"The present results are thus applicable to the case of broad Feshbach resonances, which are dominated by their open channel, but not to narrow Feshbach resonances, which are strongly affected by their closed channel [27].",
"Figure: Schematic Efimov plot: three-body energy E 3 E_{3}scaled as an inverse length as a function of the inverse scatteringlength 1/a1/a.",
"The arrows indicate the scattering length a - a_{-}at which an Efimov trimer state appears, and its binding wave numberκ\\kappa at unitarity (a→∞)a\\rightarrow \\infty ), either of which serves asa measure of the three-body parameter.The question of the physical mechanism setting the three-body parameter was addressed in Refs.",
"[25], [28] for van der Waals interactions, which decay as $1/r^{6}$ at large interparticle distance $r$ .",
"The numerical investigation of Ref.",
"[25] found that in the hyperspherical formalism, in addition to the three-body Efimov attraction at large distances, a three-body repulsion appears at short distances.",
"The distance at which this repulsion appears is comparable to the size of the van der Waals tail of the potential, thus preventing the system from probing the details of the interaction at shorter distances.",
"Therefore, the value of the three-body parameter is set by the van der Waals length associated with that tail.",
"The authors of Ref.",
"[25] remarked that this three-body repulsion is not explained by quantum reflection, as originally suggested in Ref.",
"[29], but attributed it to an increase in kinetic energy due to the squeezing of the hyperangular wave function into a smaller volume caused by the suppression of two-body probability inside the well or the repulsive core of the two-body potential.",
"This point was confirmed and clarified in Ref.",
"[28] where the kinetic energy was shown to originate from an abrupt change of the geometry of the three-particle system caused by the two-particle exclusion in the van der Waals region.",
"At large separation, the system has indeed an elongated geometry due to its Efimov nature, but it must deform to an equilateral configuration to accommodate for the mutual exclusion between the particles.",
"Reference [28] showed that this deformation causes a nonadiabatic increase in kinetic energy that manifests itself as a three-body repulsive barrier.",
"This phenomenon could be reproduced by simple models involving only the knowledge of the pair correlation causing the mutual exclusion between two particles at short separations.",
"One may wonder whether these findings extend to other physical systems.",
"Indeed, the same deformation mechanism is expected to occur in systems for which the two-body interactions tend to suppress the two-body probability at short distance.",
"Thus, pair correlation should provide the essential information that determines the three-body parameter and energy of the three-body system.",
"To investigate the role of pair correlation, we use a simple model that reproduces the pair correlation and can be solved exactly for three particles, and then compare it with full exact calculations.",
"We cannot use a zero-range model because such a model would reproduce the asymptotically free part of the two-body wave function (i.e.",
"the on-shell $T$ -matrix elements), but not its short-range correlation (i.e.",
"the off-shell $T$ -matrix elements).",
"We thus follow the approach introduced in Ref.",
"[28], where the interaction is modelled by a separable potential [30], $\\hat{V}=\\xi \\vert \\chi \\rangle \\langle \\chi \\vert $ , which retains much of the mathematical simplicity of a contact potential, while enabling us to reproduce any pair correlation at zero energy.",
"Indeed, for a given zero-energy two-body $s$ -wave radial wavefunction $u_{0}(r)$ with the asymptotic limit $1-\\frac{r}{a}$ , where $a$ is the scattering length, one can construct a separable potential reproducing this wave function exactly by choosing the following form (in momentum representation): $\\chi (q)=1-q\\int _{0}^{\\infty }dr\\left(1-\\frac{r}{a}-u_{0}(r)\\right)\\sin (qr),$ $\\xi =4\\pi \\left(\\frac{1}{a}-\\frac{2}{\\pi }\\int _{0}^{\\infty }dq\\vert \\chi (q)\\vert ^{2}\\right)^{-1}.$ Table: Three-body properties obtained for variouspotentials considered in Ref.",
", where g 3 g_{3}denotes the factor required to multiply the potential so that theground-state Efimov trimer appears at the three-body threshold, a - a_{-}denotes the scattering length for that factor, and E 3 E_{3} denotesthe energy of the ground-state Efimov trimer at unitarity (a→∞a\\rightarrow \\infty ).The symbols without a prime indicate that the values are taken fromRef.",
", and those with a prime show our resultsbased on the pair correlation using the separable model given by Eqs.",
"()and ().",
"The same units as in Ref.",
"are used.Figure: Binding wave number κ\\kappa of the ground-statetrimer at unitarity (see Fig. )",
"calculatedfrom the separable model in Eqs.",
"() and ()using the zero-energy pair wave function ψ 0 \\psi _{0}, versus its exactvalue, for various two-body potentials.",
"The exact values are takenfrom Ref.",
"for the Lennard-Jones potential (withonly one bound state) and from Ref.",
"for allthe other potentials.",
"The binding wave number is expressed in unitsof the effective range r e r_{e} of each potential, which is calculatedexactly from Eq. ().",
"The shaded area representsthe region of 10% or less deviation from the exact results.This simple prescription reproduces the low-energy two-body physics, in particular the two-body bound state around the unitary limit $a\\rightarrow \\infty $ .",
"We construct separable potentials that reproduce the pair correlation of the various two-body potentials considered in Refs.",
"[25] and [31].",
"The three-body problem for a separable interaction can be cast in the form of an integral equation in momentum space that can easily be solved numerically [32], [33], [28].",
"The results are shown in Table REF where we indicate the values $a_{-}$ and $\\kappa $ for the ground-state trimer.",
"They agree with the exact calculations of Refs.",
"[25] and [31] to within a few percents for each of these potentials.",
"This can be checked in Fig.",
"REF where the value of $\\kappa $ in our model is plotted against its exact value.",
"The method presented here therefore appears as a simple and efficient way to estimate the three-body parameter, and more generally low-energy properties for various kinds of interaction potentials.",
"Now that we have established the connection between the pair correlation and the three-body parameter, we are in a position to ask which length scale in the pair correlation determines the three-body parameter.",
"For most physical interactions, the major effect of pair correlations is to suppress probability at short distance with respect to the free wave.",
"As discussed previously, this creates a three-body repulsion through the nonadiabatic deformation effect.",
"Although the precise shape of the repulsive barrier depends on the particular two-body potential, it should be the length scale associated with the two-body suppression that sets the location of the three-body repulsion, and therefore the three-body parameter.",
"This length scale is given by half the effective range $\\frac{1}{2}r_{e}$ The presence of the the factor $\\frac{1}{2}$ is due to the conventional definition of the effective range., which is the average size of the deviation between the asymptotic and fully correlated probability densities [34], [35]: $\\frac{1}{2}r_{e}=\\int _{0}^{\\infty }dr\\left(\\left(1-\\frac{r}{a}\\right)^{2}-u_{0}(r)^{2}\\right)$ Thus for common interactions which tend to suppress the two-body probability within their range, $r_{e}$ is positive and the three-body parameter, expressed in the dimension of length, is on the order of $\\frac{1}{2}r_{e}$ .",
"Note that the effective range is commonly featured as a term in the low-energy expansion of the scattering phase shift (i.e.",
"the on-shell $T$ -matrix elements).",
"However, we expect that it is not possible to find a connection between the three-body parameter and the effective range from a method which introduces the effective range in this manner, since this expansion concerns only on-shell scattering and does not describe the short-range correlation explicitly Such an approach can be applied, however, to the case of narrow Feshbach resonances (see Ref.",
"[26]) for which the mechanism setting the three-body parameter is different than that of single-channel resonances considered here.. We can now address the final question of whether there are classes of interactions for which the low-energy three-body physics is universally determined.",
"It is clear that if the pair correlation is the same for a certain class of potentials, they must lead to the same three-body parameter.",
"This is indeed the case for potentials with a power-law decaying tail $-C_{n}r^{-n}$ , such as the van der Waals tail $-C_{6}r^{-6}$ relevant to the interaction between ground-state atoms.",
"It is well known that the two-body wave functions in the tail of these potentials are universally described in terms of the length scale $r_{n}=(\\frac{1}{n-2}\\sqrt{mC_{n}}/\\hbar )^{2/(n-2)}$ .",
"If most of the probability amplitude is located in the tail region, which is the case if the short-range region is strongly repulsive or attractive, all these potentials lead to a similar zero-energy pair correlation that is known analytically: $u_{0}(r)=\\Gamma (\\frac{n-1}{n-2})\\sqrt{x}J_{\\frac{1}{n-2}}\\!\\!\\left(2x^{-\\frac{n-2}{2}}\\right)\\\\-\\frac{r_{n}}{a}\\Gamma (\\frac{n-3}{n-2})\\sqrt{x}J_{-\\frac{1}{n-2}}\\!\\!\\left(2x^{-\\frac{n-2}{2}}\\right),$ where $\\Gamma $ and $J_{\\alpha }$ denote the gamma and Bessel functions, and $x=r/r_{n}$ .",
"The universality of the pair correlation is illustrated in Fig.",
"REF for the $8-4$ and Lennard-Jones $(12-6)$ potentials of various depths.",
"This gives a simple explanation of the observed universality of the three-body parameter in atomic systems ranging from light helium [36], [22] to heavy atoms under broad magnetic Feshbach resonances [15], [17].",
"Figure REF shows the binding wave number $\\kappa $ of the ground-state trimer for these power-law decaying potentials, evaluated using our separable potential method.",
"For potential depths supporting more than one two-body bound state, the ground-state trimer is in fact a resonance in the particle-dimer continuum, but it manifests itself simply as a bound state in our model A separable potential has only one two-body bound state.",
"In our model, this state describes the two-body bound state of the original potential which is close to the two-body scattering threshold, while the other bound states of the potential are not included in our model.",
"As a result, the trimer has no lower-lying state to dissociate to, and it appears as a true bound state, rather than a resonance.. One can see that the value of $\\kappa $ remains close to the one obtained from the universal pair correlation.",
"In the particular case of a van der Waals tail, we obtain $a_{-}=-10.86(1)\\, r_{6}$ and $\\kappa =0.187(1)\\, r_{6}^{-1}$ in good agreement with Ref.",
"[25] and experimental observations [17], [37].",
"Since the effective range is related to the van der Waals length $r_{6}$ through $r_{e}=\\frac{4\\pi }{3\\Gamma (3/4)^{2}}r_{6}\\approx 2.78947r_{6}$ , these results correspond to $a_{-}=-7.78(1)\\times (\\frac{1}{2}r_{e})$ and $\\kappa =0.261(1)\\times (\\frac{1}{2}r_{e})^{-1}$ .",
"Figure: Pair correlation at unitarity for potentialsdecaying as power laws -1/r n -1/r^{n}.",
"Top: the 8-4 potential.",
"Bottom:the Lennard-Jones potential.",
"In each graph, the solid curves correspondin order of opacity to potential depths supporting respectively 1,2, and 3 ss-wave bound states, which are obtained by adjusting thevalue of σ\\sigma .",
"The dashed curve represents the universal paircorrelation in Eq.",
"().Figure: Pair correlation at unitarity for potentialsdecaying faster than any power law.",
"Top: the Pöschl-Teller potential;bottom: the Gaussian potential.",
"In each graph, the solid curves correspondin order of opacity to potential depths supporting respectively 1,2, 10, and 120 ss-wave bound states.",
"The dashed lines show the universalpair correlation limit in Eq.",
"().The distance is scaled in units of 1 2r e \\frac{1}{2}r_{e} in the maingraphs, while it is shown in unscaled units of r 0 r_{0} in the insets.Figure: Binding wave number κ\\kappa of theground-state Efimov trimer calculated from the separable model inEqs.",
"() and () for pair correlations correspondingto different two-body interactions, as a function of the depth ofthese potentials as measured by the number of ss-wave two-body boundstates.",
"The dashed lines indicate from top to bottom the values obtainedfor the universal pair correlation in Eq.",
"()with n=4n=4 and n=6n=6, and the universal pair correlation in Eq. (),respectively.",
"This figure shows how the three-body parameter convergesdifferently and to different limits depending on the class of two-bodyinteraction.There is a second class of potentials, which decay faster than any power law, such as the Yukawa potential and other typical nuclear potentials, as well as screened potentials found in solid-state physics.",
"At first glance, the two-body wave functions for these potentials do not seem to exhibit any particular universality.",
"However, if the potential features a deep attraction supporting many bound states, the effective range near unitarity is large.",
"This means that when distances are expressed in units of $\\frac{1}{2}r_{e}$ , there is a sharp drop of probability in the two-body wave function near $r=1$ , as represented in Fig.",
"REF .",
"It can be shown that this rescaled two-body wave function converges to a step function in the limit of strongly attractive potentials The same happens for potentials with finite support, although the effective range does not increase with the number of bound states in this case..",
"In this sense, the three-body parameter is universally determined by the effective range of these potentials, and stems from the universal pair correlation limit: $u_{0}(r)={\\left\\lbrace \\begin{array}{ll}0 & \\mbox{for }r<\\frac{1}{2}r_{e}\\\\1-\\frac{r}{a} & \\mbox{for }r\\ge \\frac{1}{2}r_{e}\\end{array}\\right.",
"}.$ Figure REF shows the trimer binding wave number $\\kappa $ for some of these potentials, namely, the Gaussian potential, the Pöschl-Teller potential with $\\alpha =1$ , the Yukawa potential, the Morse potential with $r_{0}=1$ [31], as well as the neutron-neutron interaction potential in the $\\,^{1}S_{0}$ channel [38].",
"While none of these calculations correspond to a particular physical system, they capture the essence of the Efimov physics occurring in the symmetric channel of nuclear systems, such as the tritium nucleus.",
"Each potential was scaled to reach unitarity, corresponding to different possible depths of the potential.",
"One can see in Fig.",
"REF that as the depth of the potentials is increased, $\\kappa $ converges to the value $\\kappa =0.2190(1)\\times (\\frac{1}{2}r_{e})^{-1}$ obtained for the two-body correlation in Eq.",
"(REF ).",
"The convergence is, however, very slow, because very deep potentials (supporting hundreds of bound states) are required for the pair correlation to approach Eq.",
"(REF ).",
"Finally, one should note that there is a notable exception to these considerations.",
"One might think that the square-well potential, which often lends itself to simple analytical treatments [39], is a useful model potential to investigate the physics of the three-body parameter.",
"However, it turns out to be a special case which does not belong to the two classes discussed above.",
"Even though it decays faster than any power law, it does not belong to the second class because of its absence of tail.",
"In particular, the two-body wave function near unitarity shows no progressive drop of probability in the well, only steady oscillations which get faster as the depth of the well increases, and therefore does not converge to the function in Eq.",
"(REF ).",
"From this we conclude that this potential is not expected to reveal any universality of the three-body parameter.",
"To summarise, we have pointed out how the Efimov three-body parameter is deeply connected to the zero-energy two-body correlation.",
"This allows us to identify the two-body effective range as the relevant length scale setting the three-body parameter for the class of physical interactions which suppress two-body probability at short distance.",
"However, it also shows that, unlike what was suggested in Ref.",
"[25], this suppression of two-body probability does not lead to a single universal value of the three-body parameter in units of the effective range.",
"Indeed we find two qualitatively distinct subclasses of interactions for which the value of the three-body parameter is universally determined.",
"One corresponds to short-range two-body potentials decaying as a power law, relevant to atomic interactions, for which the three-body universality stems from the two-body universality.",
"The other corresponds to very deep two-body potentials decaying faster than any power law, which lead to an abrupt two-body suppression.",
"Typical interactions in nuclear physics decay faster than any power law but support only a few bound states, so that their three-body parameter does not reach this universal limit.",
"In practice, however, one can expect the binding wave number $\\kappa $ to be in the range $0.2\\sim 0.4\\times (\\frac{1}{2}r_{e})^{-1}$ for most physical interactions, and in particular close to $0.35\\times (\\frac{1}{2}r_{e})^{-1}$ for nuclear interactions supporting at most one bound state, as can be seen in Figs.",
"REF and REF .",
"These conclusions are obtained for particles interacting through single-channel two-body interactions, and would not apply in the presence of significant three-body forces, or strongly energy-dependent resonant interactions such as narrow Feshbach resonances [27], [40], [41].",
"P. N. acknowledges support from RIKEN through the Incentive Research Project funding.",
"S. E. acknowledges support from JSPS (Grant No.",
"237049).",
"M. U. acknowledges support by Grants-in-Aid for Scientific Research (Kakenhi No.",
"22340114 and No.",
"22103005) and the Photon Frontier Network Program of MEXT of Japan."
]
] | 1403.0294 |
[
[
"Exploring the Intergalactic Magnetic Field by Means of Faraday\n Tomography"
],
[
"Abstract Unveiling the intergalactic magnetic field (IGMF) in filaments of galaxies is a very important and challenging subject in modern astronomy.",
"In order to probe the IGMF from rotation measures (RMs) of extragalactic radio sources, we need to separate RMs due to other origins such as the source, intervening galaxies, and our Galaxy.",
"In this paper, we discuss observational strategies for the separation by means of Faraday tomography (Faraday RM Synthesis).",
"We consider an observation of a single radio source such as a radio galaxy or a quasar viewed through the Galaxy and the cosmic web.",
"We then compare the observation with another observation of a neighbor source with a small angular separation.",
"Our simulations with simple models of the sources suggest that it would be not easy to detect the RM due to the IGMF of order ~ 1 rad/m/m, an expected value for the IGMF through a single filament.",
"Contrary to it, we find that the RM of at least ~10 rad/m/m could be detected with the SKA or its pathfinders/precursors, if we achieve selections of ideal sources.",
"These results would be improved if we incorporate decomposition techniques such as RMCLEAN and QU-fitting.",
"We discuss feasibility of the strategies for cases with complex Galactic emissions as well as with effects of observational noise and radio frequency interferences."
],
[
"Introduction",
"The intergalactic medium (IGM) in the cosmic web of filaments and clusters of galaxies is expected to be permeated with the intergalactic magnetic field (IGMF) ([72], [53]).",
"The IGMF plays crucial roles in various subjects of astrophysics; propagations of ultra-high energy cosmic-rays and $\\gamma $ -rays ([47], [18], [67], [52], [64], [65], [66]), radio emissions in galaxy clusters ([23], [24]), substructures during cluster mergers ([8], [68]), and configurations of magnetic fields in spiral galaxies ([59]).",
"Seed IGMFs could be generated in inflation, phase transition, and recombination eras ([30], [63], [38]), during cosmic reionization ([28], [43], [74], [6]), and through cosmological shock waves ([41], [50]).",
"The seed fields of any origins could be further amplified through compression and turbulence dynamo in the structure formation ([51], [19], [16], [54]).",
"Also, leakages of magnetic fields and cosmic rays from galaxies should be taken into consideration ([20], [46]).",
"The above diverse processes underline the importance of observational tests for the IGMF.",
"One of a few possible methods to probe cosmic magnetic fields is to utilize Faraday rotation in radio polarimetry ([15], [26], [9], [31]).",
"A rotation of the polarization angle is proportional to the square of the wavelength, and the proportionality constant, rotation measure (RM), provides an integration of magnetic fields with weights of electron densities.",
"This conventional method, however, could work only in the case of observing a single polarized radio source.",
"Otherwise, in cases of multiple emitters along the line-of-sight (LOS), a rotation of the polarization angle draws a complex curve ([12], hereafter BD05), and RM cannot be easily estimated.",
"Moreover, RMs of a few to several tens ${\\rm rad~m^{-2}}$ are usually associated with radio sources ([58], [49]) and the Galaxy ([45], [48]).",
"These RMs are larger than expected RMs through filaments, $\\sim 1-10~{\\rm rad~m^{-2}}$ ([1], [2], [4]), and cannot be easily separated from an observed RM by the conventional method.",
"Therefore, we need to establish alternative methods which allow us to estimate hidden RM components along the LOS.",
"As a method to separate multiple sources and RMs along the LOS, a revolutionary technique, called Faraday RM synthesis or Faraday tomographyWe consider one-dimensional reconstruction in this paper.",
"Although the phrase “tomography\" is generally used as an attempt to reconstruct the actual 3D distribution from observed integrals through the volume, we call this technique as Faraday tomography throughout this paper, foreseeing future 3D imaging of the cosmic web., was first proposed by [13] and extended by BD05.",
"Previous works for the interstellar medium ([55], [56]), the Galaxy ([45]), external galaxies ([36]), and active galactic nuclei ([49]) have demonstrated that the technique is powerful to resolve RM structures along the LOS.",
"It would be thus promising to study the IGMF in eras of wide-band radio polarimetry including Square Kilometer Array (SKA) and its pathfinders/precursors such as Low Frequency Array (LOFAR), Giant Meterwave Radio Telescope (GMRT), and Australia SKA Pathfinder (ASKAP) (see [10], a summary of telescopes therein).",
"In this paper, we discuss observational strategies to probe the IGMF by means of Faraday tomography.",
"We consider frequency coverages and numbers of channels of future observations.",
"Faraday tomography is in general improved by incorporating decomposition techniques such as RMCLEAN ([35]) and QU-fitting ([49]).",
"However, decomposition techniques have their own uncertainties ([22], [42]).",
"Therefore, we concentrate on a standard method of Faraday tomography without any corrections to see its original potential.",
"In fact, the decomposition is powerful for our study, which was addressed in a separate paper ([39]).",
"The rest of this paper is organized as follows.",
"In section 2, we introduce the method of Faraday tomography and describe our model.",
"The results are shown in section 3, and the discussion and summary follow in section 4 and 5, respectively.",
"We first summarize a basic concept of Faraday tomography following a manner described by BD05.",
"The readers who have interests in detailed derivations and improvements of algorithm should refer to recent works ([35], [25], [44], [7]).",
"A fundamental observable quantity is the complex polarized intensity, $P(\\lambda ^2) = Q(\\lambda ^2)+iU(\\lambda ^2)$ , at a given wavelength, $\\lambda $ , where $Q$ and $U$ are the Stokes parameters.",
"$P(\\lambda ^2)$ is given by an integration of intensities along the LOS as ([13]), $P(\\lambda ^2)=\\int _{-\\infty }^{\\infty } F(\\phi ,\\lambda ^2) e^{2i\\phi \\lambda ^2} d\\phi ,$ where $F$ is the Faraday dispersion function (FDF) which is a complex polarized intensity at a given Faraday depth, $\\phi $ .",
"Faraday depth is defined as $\\phi (x)=0.81 \\int _{x}^{0}n_{\\rm e}(x^{\\prime })B_{||}(x^{\\prime })dx^{\\prime }$ in units of ${\\rm rad~m^{-2}}$ , where $n_{\\rm e}$ is the thermal electron density in ${\\rm cm^{-3}}$ , $B_{||}$ is the LOS magnetic field strength in ${\\rm \\mu G}$ , and $x^{\\prime }$ is the LOS physical distance in pc.",
"The FDF should be a function of $\\lambda $ in general (BD05), but, following [13], we assume that all considerable radio sources have similar spectra and neglect wavelength dependence of the FDF.",
"Hence equation (REF ) has the same form as the Fourier transform and the FDF can be formally obtained by the inverse Fourier transform: $F(\\phi )=\\int _{-\\infty }^{\\infty } P(\\lambda ^2)e^{-2i\\phi \\lambda ^2} d\\lambda ^2.$ However, this inversion is not practically perfect, because $P(\\lambda ^2)$ is not defined for negative $\\lambda ^2$ and the coverage for positive $\\lambda ^2$ is limited in real observations.",
"BD05 generalized equation (REF ) by introducing a window function, $W(\\lambda ^2)$ , where $W(\\lambda ^2)=1$ if $\\lambda ^2$ is in observable bands, otherwise $W(\\lambda ^2)=0$ .",
"Observed complex polarized intensity can then be written as $\\tilde{P}(\\lambda ^2)= W(\\lambda ^2)P(\\lambda ^2)= W(\\lambda ^2)\\int _{-\\infty }^{\\infty } F(\\phi ) e^{2i\\phi \\lambda ^2} d\\phi ,$ where the tilde indicates observed or reconstructed quantities.",
"From $\\tilde{P}(\\lambda ^2)$ , we can obtain an approximate reconstruction of the FDF as, $\\tilde{F}(\\phi )= \\int _{-\\infty }^{\\infty } \\tilde{P}(\\lambda ^2) e^{-2i\\phi \\lambda ^2}d\\lambda ^2.$ We define the rotation measure spread function (RMSF): $R(\\phi )=K\\int _{-\\infty }^{\\infty } W(\\lambda ^2) e^{-2i\\phi \\lambda ^2} d\\lambda ^2,$ where $K=\\left(\\int _{-\\infty }^{\\infty } W(\\lambda ^2) d\\lambda ^2 \\right)^{-1},$ is the normalization.",
"Applying the convolution theorem, the approximate FDF, $\\tilde{F}(\\phi )$ , can be written as, $\\tilde{F}(\\phi )= K^{-1} F(\\phi )\\ast R(\\phi )= K^{-1} \\int _{-\\infty }^{\\infty } F(\\phi -\\phi ^{\\prime }) R(\\phi ^{\\prime }) d\\phi ^{\\prime }.$ Therefore, the reconstruction (REF ) is perfect only if the RMSF reduces to the delta function, $R(\\phi )/K \\rightarrow \\delta (\\phi )$ , for a (unphysical) complete observation, i.e.",
"$W(\\lambda ^2) = 1$ for all $\\lambda ^2$ (Eq.",
"REF ).",
"In fact, the RMSF has a finite width (see Figure REF ), so that the data sampling is incomplete and the reconstruction is not perfect.",
"Equations (REF )-(REF ) indicate that the quality of the reconstruction primarily depends on the window function.",
"That is, a wider coverage in $\\lambda ^2$ space improves the reconstruction, as expected from the analogy with Fourier transform.",
"In this paper, we follow the above window-function approach considering the observable bands of SKA and its pathfinders.",
"For other approaches, where the values of non-observed $P(\\lambda ^2)$ are estimated by some assumptions on the properties of the sources (see [13], [27]; BD05)."
],
[
"Promising Target",
"Since the sign of $B_{||}$ in equation (REF ) can be changed, $\\phi (x)$ is not a monotonic function of $x$ in general.",
"Therefore, the FDF, i.e., the distribution of radio sources in $\\phi $ space, does not simply indicate the distribution of radio sources in $x$ space.",
"Nevertheless, the FDF is enough to probe the IGMF, because the IGMF can be identified in $\\phi $ space as demonstrated in this paper.",
"We first introduce general behavior of the FDF.",
"If thermal electrons inducing RMs co-exist with cosmic-ray electrons emitting synchrotron polarizations, the FDF has finite thickness in $\\phi $ space.",
"Also, an accumulation of emissions and RMs within a source results in a thickness, if the sign of $B_{||}$ changes many times and the accumulation behaves like a random walk process.",
"Such a thickness would be natural for the Galaxy and extragalactic radio sources.",
"In addition, if there are magneto-ionic media in front of radio-emitting region, e.g., associated media such as clouds, H$\\alpha $ filaments, and swept IGM by jets, the FDF shifts in $\\phi $ space.",
"The shift is also caused by RMs of discrete intervening galaxies ([71], [11], [32]) and foreground IGMFs in clusters/filaments of galaxies.",
"We probe the IGMF in filaments of galaxies from the shift of the FDF in $\\phi $ space.",
"Thus, the shifts caused by other origins are contaminations.",
"LOSs containing significant RMs of other origins could be avoided as follows.",
"For removing RMs of galaxy clusters, we could exclude sources behind galaxy clusters (e.g., Coma cluster, [45]).",
"The detection limit of current X-ray facilities is enough to substantially exclude RMs of galaxy clusters ([2]).",
"RMs of associated media around sources and RMs of intervening galaxies could have a tight correlation with optical absorber systems and/or could show small fractional polarization due to depolarization ([11], [32]).",
"Thus, we could discard sources with such contaminations.",
"Note that the RM of the IGMF would not affect depolarization, since the IGMF is expected to be smooth enough within the beam size of $\\sim $ arcsecond ([2]).",
"Furthermore, RMs for associated media of distant sources could be small due to a $(1+z)^{-2}$ dilution factor.",
"[32] estimated the dilution and claimed that sources at $z=1$ should only contribute a standard deviation of RMs $\\sim 1.5-3.75~{\\rm rad~m^{-2}}$ .",
"Therefore, we assume that we can select sources toward which the shifts of the FDF caused by RMs of galaxy clusters, the associated media, and intervening galaxies are negligible."
],
[
"Strategy A: Compact Source behind Diffuse Source",
"Hereafter, to make our arguments simple, we mainly describe cases for a pure real $F(\\phi )$ obtained if the intrinsic polarization angle is independent of $\\phi $ .",
"The result for a complex $F(\\phi )$ is discussed in Section .",
"Using selected sources, we consider possible and suitable FDFs to probe the RM due to the IGMF.",
"Figure REF (a) shows an example of the FDF.",
"We consider an observation of a background compact source B (emissions from a radio galaxy or a quasar) viewed through the cosmic web A and a foreground diffuse source A (Galactic emissions).",
"The FDF of the cosmic web A has tiny amplitude, since radio emissions from the IGM are generally much weaker than the others.",
"Therefore, the RM of the cosmic web A can be probed as the “gap\" between FDFs of the two sources.",
"We name this observational strategy as “Strategy A\".",
"The situation arises if all signs of cumulative RMs of the two sources and the cosmic web coincide.",
"Otherwise, some of them would overlap each other in $\\phi $ space.",
"The probability that all signs coincide is $25\\%$ , which is reasonably high to choose such LOSs from multiple observations.",
"Even in the overlapped case, there would be still the gap if the RM of the cosmic web is much larger than those of sources.",
"Therefore, the probability that we find the gap is expected to be larger than 25 %.",
"In this strategy, foreground emissions are necessary to be detected.",
"The intensity of Galactic diffuse emission toward high Galactic latitudes can be scaled as $I \\sim 0.95 \\left(\\frac{f}{1~{\\rm GHz}}\\right)^{-1.5}\\left(\\frac{\\Omega }{1~{\\rm arcmin^2}}\\right)~{\\rm mJy}$ ($f$ is the frequency and $\\Omega $ is the beam size; [29]), and is larger toward lower latitudes.",
"The diffuse emission is thus significant, unless we observe very bright compact sources.",
"In section 3.1, we will demonstrate that the diffuse emission is significant even if the background emission is 100-1000 times brighter.",
"If the source B is a distant source, the FDF of the source B would be Faraday thin (the thickness is small enough, i.e.",
"a delta function).",
"But we keep considering small thickness for compact sources since the thickness as well as the RMSF are notable ambiguities to probe the IGMF, particularly for observations with limited bandwidths.",
"If the source B is Faraday thin, the gap is sharpened and the estimation of the RM due to the IGMF is rather improved."
],
[
"Strategy B: Pair Compact Sources",
"Along with Strategy A, we can consider an extended strategy to probe the IGMF.",
"That is, we observe two compact sources and obtain two FDFs (figures REF a and REF b).",
"We then subtract one FDF from the other to obtain the difference (figure REF c).",
"If the directions of two LOSs are close enough, FDFs of the source A and the cosmic web A are cancelled out each other in the difference, then the difference reveals the gap between two FDFs in $\\phi $ space.",
"And if the sources B and C are both Faraday thin, the gap precisely indicates the RM of the cosmic web B.",
"We name this observational strategy as “Strategy B\".",
"Figure: Root-mean-square value of the RM difference, (Δ RM ) rms =〈| RM (x →+r)- RM (x →)| 2 〉 x → (\\Delta {\\rm RM}) _{\\rm rms} =\\langle \\sqrt{{|{\\rm RM}(\\vec{x}+r)-{\\rm RM}(\\vec{x})|}^2}\\rangle _{\\vec{x}} averaged over the position x →\\vec{x} of the pixels in the RM map, where rr is the separation angle between two pixels.",
"Thick line is for the RM map of the GMF toward high Galactic latitudes (ADPS30, ).",
"Thin lines from the bottom to top are for the RM maps of the IGMF (TS0, ) integrated up to the redshifts of 0.1, 0.3, 0.5, and 1.0, respectively.Figure REF shows how much RM could be different between two LOSs, based on the simulations [2], [3].",
"For example, let us suppose that the RM of the cosmic web B is 7.5 ${\\rm rad~m^{-2}}$ .",
"If the angular separation between the nearby source B and the distant source C is less than $\\sim 1^\\circ $ , the RM difference of the foreground source A could be less than 20% (1.5 ${\\rm rad~m^{-2}}$ ) of the RM of the cosmic web B.",
"The same level of the RM difference is available for the cosmic web A, if the nearby source B is locate at $z=0.3$ and the angular separation is less than $\\sim 0.1^\\circ $ .",
"A RM difference of $\\sim 1.5$ ${\\rm rad~m^{-2}}$ for an angular separation of $\\sim 1^\\circ $ for the GMF is consistent with the gradient of Galactic RM toward high Galactic latitudes, a few ${\\rm rad~m^{-2}}$ per degree ([45]).",
"Note that much smaller-scale structures have been observed toward the Galactic plane ([33], [34])."
],
[
"Prospects",
"Let us consider practical cases for strategies A and B.",
"In Strategy A, if the source B is located at $z=1$ , RMs of associated media would be $\\sim $ a few ${\\rm rad~m^{-2}}$ .",
"Therefore, if the RM of the cosmic web A exceeds $\\sim $ several ${\\rm rad~m^{-2}}$ , it could be predominant on the shift of the FDF of the source.",
"In Strategy B, if the source B is located at $z=0.1$ , the RM difference of the cosmic web A is $\\sim 0.3$ ${\\rm rad~m^{-2}}$ for the angular separation of $\\sim 0.1^\\circ $ .",
"The source B could contain RMs of $\\sim 10$ ${\\rm rad~m^{-2}}$ for associated media.",
"Thus, if the RM of the cosmic web B exceeds $\\sim 15-20$ ${\\rm rad~m^{-2}}$ , it could be predominant on the shift of the FDF of the source C. Such a value is likely if the LOS goes through more filaments than the average between the observer and the sources.",
"Moreover, if we can obtain information of optical absorptions and/or depolarization and can select the sources toward which RMs of associated media and intervening clouds are insignificant, we could identify a much smaller RM of the cosmic web B.",
"We expect that there will be a number of ideal sources for Strategies A and B in millions of candidates to be found in future observations.",
"For example, no absorber systems were detected toward about a half of 84,534 quasars so far [75] and a number of radio sources show large fractional polarization [32].",
"Pair sources are still rare in the largest catalog with the average separation $\\sim 1^\\circ $ ([69]).",
"But the separation will decrease by $\\sim 0.1^\\circ $ with SKA precursors and $\\sim 0.01^\\circ $ with the SKA, so that the number of the pairs would soon increase dramatically.",
"Supposing that optical observations confirm two independent sources B and C, Strategy B could be available even if the sources are in a single beam and are not resolved in space.",
"But if the Faraday depth between two sources is less than the full width at half maximum (FWHM) of the RMSF, Faraday tomography may miss to reconstruct two FDF peaks for some specific intrinsic polarization angles.",
"This phenomenon is known as RM ambiguities ([22], [42]).",
"Thus, Strategy B for unresolved sources is available if the gap caused by the IGMF is sufficiently larger than the FWHM of the RMSF.",
"Such a limitation is practically relaxed if we include low frequency data providing a small FWHM of $O(0.1)~{\\rm rad~m^{-2}}$ .",
"Thanks to progresses of low frequency radio observations such as LOFAR and Murchison Widefield Array (MWA), we would easily access low frequency data and could remove the ambiguity."
],
[
"FDF Model and Calculation",
"Instead of observational data, we use the data calculated from FDF models.",
"We adopt simple FDF models, since the gap appears regardless of detailed profiles of FDFs for sources.",
"The analyses below are rather independent on the shape of the FDF, because it is to be reconstructed by observations.",
"A FDF of a diffuse source is modeled with a function based on the hyperbolic tangent, since the function tends to better reproduce the resultant platykurtic profile expected from the Galaxy models ([62], [70], [3], [40]): $F_{\\rm d}(\\phi )&=& F_{\\rm d0}\\left[\\frac{1}{4}\\left\\lbrace \\tanh {\\left(\\pi \\frac{\\phi - \\phi _{\\rm dw}}{\\phi _{\\rm dw}}\\right)}+ 1\\right\\rbrace \\right.\\nonumber \\\\&\\times &\\left.\\left\\lbrace \\tanh {\\left(-\\pi \\frac{\\phi - \\phi _{\\rm d} - \\phi _{\\rm dw}}{\\phi _{\\rm dw}}\\right)}+ 1\\right\\rbrace \\right],$ where $\\phi _{\\rm dw}$ is the parameter which controls the width of the skirts, and $\\phi _{\\rm d}$ is the position of the back side edge of the FDF in $\\phi $ space.",
"We set $\\phi _{\\rm 0}=0$ for the position of the front side edge, where no generality is lost by this choice since the gap is measured from relative position of the sources in $\\phi $ space.",
"We choose $\\phi _{\\rm dw} = 2.0~{\\rm rad~m^{-2}}$ and the thickness of the FDF $\\phi _{\\rm d} -\\phi _{\\rm 0} = 20~{\\rm rad~m^{-2}}$ as representative values.",
"We normalize FDF intensities of any sources by $F_{\\rm d0}=1$ , since relative amplitude of FDFs is only meaningful while we do not consider observational errors.",
"A FDF of a compact source such as a quasar and a radio galaxy has been approximated by a Gaussian ([13], [25]): $F_{\\rm c}(\\phi )= F_{\\rm c0}\\exp {\\left\\lbrace -\\frac{(\\phi -\\phi _{\\rm c})^2}{2 \\phi _{\\rm cw}^2}\\right\\rbrace },$ where $\\phi _{\\rm c}$ is the position of the FDF center in $\\phi $ space, and $\\phi _{\\rm cw}$ determines the thickness of the FDF.",
"We choose $\\phi _{\\rm cw}=0.2~{\\rm rad~m^{-2}}$ , and the thickness as $\\pm 3\\sigma $ region of the FDF is $2\\times 3 \\phi _{\\rm cw} = 1.2~{\\rm rad~m^{-2}}$ as a reachable value for selected sources (Sections 2.2 and 2.5).",
"Such a value is actually expected for some sources toward high Galactic latitudes (e.g., [58]).",
"We investigate compact sources with $F_{\\rm c0}=0.1-10000$ .",
"A FDF of the cosmic web is modeled with $F_{\\rm IGM}(\\phi ) = 0.$ This means that no emission is considered.",
"As already mentioned, polarized intensities due likely to synchrotron radiations of cosmic-ray electrons in the IGM are generally very small except radio halos and relics in galaxy clusters.",
"Faraday depth of the cosmic web is based on theoretical predictions ([1], [2]); the rms value of RM is $\\sim 1~{\\rm rad~m^{-2}}$ through a single filament at the local universe, and ${\\rm several}-10~{\\rm rad~m^{-2}}$ in integrating filaments up to $z=5$ .",
"[2] also found that some of LOSs going through dense filaments and/or group of galaxies in filaments have RMs of a few tens ${\\rm rad~m^{-2}}$ .",
"Therefore, we consider RM of the cosmic web, $1-30~{\\rm rad~m^{-2}}$ .",
"Note that probing the RM of $O(1)~{\\rm rad~m^{-2}}$ would be fundamentally difficult because of contaminations (sections REF and REF ).",
"But we also test the case to see the capability of Faraday tomography for such a small gap.",
"Figure: Wavelength coverages and the RMSFs for LOFAR (L), GMRT (G), ASKAP (A) observations, combined observations of L+G+A, and the SKA observation.Table: Specifications of radio observatories.The procedure of calculation is as follows.",
"First, we construct the model FDF based on equations (REF ), (REF ) and (REF ).",
"Next, we numerically carry out a Fourier transform of the FDF and synthesize the polarized intensity, $P(\\lambda ^2)$ , according to Equation (REF ).",
"After that, we derive the observable polarized intensity, $\\tilde{P}(\\lambda ^2)$ , by using the window function, $W(\\lambda ^2)$ , in equation (REF ), considering the observable bands of LOFAR, GMRT, ASKAP, and SKA listed in table REF and shown in figure REF .",
"Finally, we numerically carry out a inverse Fourier transform of $\\tilde{P}(\\lambda ^2)$ and obtain the reconstructed FDF, $\\tilde{F}(\\phi )$ , according to equation (REF ).",
"The model FDF is composed of $2.4\\times 10^6$ data points ranging from $-1.5\\times 10^4$ to $1.5\\times 10^4~{\\rm rad~m^{-2}}$ and dividing the $\\phi $ space evenly.",
"The same number of data points are adopted for the polarized intensity data dividing $\\lambda ^2$ space evenly.",
"The data points provide wider $\\lambda ^2$ coverage and at least ten times higher $\\lambda ^2$ resolution than those in the observations, ensuring to minimize numerical errors in the calculation of the mock polarized intensity.",
"We allow that each interferometer has different frequency resolution (number of channels).",
"Here, the frequency resolution is worse than that of the generated data, so that the intensity in each channel is given by averaging the intensities of data points within each channel.",
"Hereafter, we do not take into account observational errors to demonstrate that the quality of reconstruction is mainly determined by frequency coverage.",
"The cases with observational errors are discussed in Section ."
],
[
"Strategy A: Compact Source behind Diffuse Source",
"We first show the results for Strategy A.",
"We define the RM of the gap caused by the cosmic web A as ${\\rm RM_{IGMF}}=\\phi _{f,B}-\\phi _{d,A}.$ Figure: Left panels show the model (gray) and reconstructed (black) Faraday dispersion functions.",
"Right panels show synthesized polarized intensities, where unshadowed regions correspond to wavelength coverages with LOFAR, GMRT, and ASKAP.",
"Panels from the top to bottom show the results for the models with F c0 /F d0 =F_{\\rm c0}/F_{\\rm d0}= 1 and RM IGMF ={\\rm RM_{IGMF}}= 30, 10, and 1 rad m -2 {\\rm rad~m^{-2}}, respectively, for the SKA observation.Figure REF shows model FDFs, synthesized polarized intensities, and reconstructed FDFs with the fixed intensity ratio, $F_{\\rm c0}/F_{\\rm d0}=1$ , and ${\\rm RM_{IGMF}}=$ 30, 10, and 1 ${\\rm rad~m^{-2}}$ , for SKA observations.",
"We can see the gap due to the cosmic web, although its amplitude does not come to zero.",
"This is because the reconstructed FDF of the diffuse source has skirts derived from the RMSF.",
"A slope of the skirt changes at $\\phi \\sim 20$ ${\\rm rad~m^{-2}}$ in the cases with ${\\rm RM_{IGMF}}=$ 30 and 10 ${\\rm rad~m^{-2}}$ .",
"If we regard this change as the edge of the diffuse source, we could estimate ${\\rm RM_{IGMF}}$ from the gap within, at least, a factor of two.",
"On the other hand, for ${\\rm RM_{IGMF}}=$ 1 ${\\rm rad~m^{-2}}$ , it is not easy to recognize the change of the slope, and we need further analyses to remove skirts.",
"The polarized intensity decreases at the longer-wavelengths, $\\lambda ^2\\gtrsim 0.1-1~{\\rm m^2}$ , at which Faraday depolarization (e.g., [60]) happens because the Faraday rotation exceeds $\\pi $ radian.",
"In addition, the emission of the diffuse source interferes with that from the compact source, and results in further depolarization seen as wiggles in ASKAP and GMRT bands.",
"Further general explanations for the behavior of polarized intensity can be obtained in BD05.",
"Figure: Same as Figure but for the models with RM IGMF ={\\rm RM_{IGMF}}= 30 rad m -2 {\\rm rad~m^{-2}} and F c0 /F d0 =F_{\\rm c0}/F_{\\rm d0}= 1000, 100, 10, 1, and 0.1, respectively, from the top to the bottom panels.We next change the intensity ratio, $F_{\\rm c0}/F_{\\rm d0}$ .",
"Figure REF shows the results for ${\\rm RM_{IGMF}}=$ 30 ${\\rm rad~m^{-2}}$ .",
"Clearly, the promising range to find the gap is $F_{\\rm c0}/F_{\\rm d0}=1-10$ .",
"If the compact source is much brighter ($F_{\\rm c0}/F_{\\rm d0}\\gtrsim 100$ ), the gap is substantially buried under the skirt extended from the compact source.",
"On the other hand, if the compact source is much fainter ($F_{\\rm c0}/F_{\\rm d0}\\lesssim 0.1$ ), it may become more difficult to identify the compact source from the skirt extended from the diffuse source.",
"The usefulness of the sources with $F_{\\rm c0}/F_{\\rm d0}=1-10$ can be also understood from behavior of the polarized intensity; we obtain strong wiggles in ASKAP and GMRT bands, which depend on ${\\rm RM_{IGMF}}$ (Figure REF ).",
"This means that observed polarized intensity has significant information about ${\\rm RM_{IGMF}}$ .",
"Actually, if the compact source is too much brighter or too much fainter than the diffuse source, the polarized intensity has a simple monotonic form and the wiggles become faint.",
"It should be noticed that the reconstructed FDFs for $F_{\\rm c0}/F_{\\rm d0} \\sim 100-1000$ show substructures around $\\phi \\sim 0-20$ ${\\rm rad~m^{-2}}$ , which cannot be produced only by a compact source.",
"This indicates that the diffuse foreground emission is still significant, even if it is a few order fainter than the background emission.",
"Actually, the wiggles which have information about ${\\rm RM_{IGMF}}$ can be seen in the polarized intensity.",
"Therefore, these cases would be also promising to find the gap, supported by further analyses to remove the skirt.",
"Figure: Model (gray) and reconstructed (black) Faraday dispersion functions.",
"Each panel shows the result for LOFAR (L), GMRT (G), ASKAP (A) observations, combined observations of L+G, G+A, A+L, L+G+A, and the SKA observation.Figure REF shows the reconstructed FDFs for the LOFAR, GMRT, ASKAP observations, their combinations, and the SKA observation, in the case with $F_{\\rm c0}/F_{\\rm d0}=1$ and ${\\rm RM_{IGMF}}=30~{\\rm rad~m^{-2}}$ .",
"The results clearly demonstrate that the quality of reconstruction mainly depends on frequency coverage of the data; fine RM structures with scales of $\\lesssim {\\rm a~few}~{\\rm rad~m^{-2}}$ are mostly reconstructed with the low frequency data, while broad RM structures with scales of $\\gtrsim {\\rm several}~{\\rm rad~m^{-2}}$ are mainly reconstructed with the mid-frequency data.",
"Consequently, there exist skirts and side lobes derived from the RMSF, and we see a clear gap only for the SKA observation."
],
[
"Strategy B: Pair Compact Sources",
"We next show the results for Strategy B.",
"Figure REF shows the difference between two reconstructed FDFs for various combinations of telescopes and RMs of the cosmic web B, where we assumed a complete subtraction of the source A (the Galaxy).",
"The two sharp peaks associated with the two sources can be reconstructed with the LOFAR observation, although the FDFs have skirts around the peaks and the peak intensities are underestimated by $\\sim 60~\\%$ .",
"The underestimation is slightly improved when we add GMRT and/or ASKAP data, but side lobes with an amplitude of at most $\\sim 40~\\%$ of the reconstructed peak arise.",
"The side lobes could be ascribed to an absence of the data in $\\lambda ^2 \\sim 0.3-0.7~{\\rm m^{2}}$ .",
"The SKA observation nicely reproduces the FDF, though skirts extend to $4.5\\sigma $ level with amplitudes of less than ${\\rm several}~\\%$ of the peak values.",
"We find that the above features do not significantly change within $F_{\\rm c0}/F_{\\rm d0}=100-10000$ .",
"Table: Estimated RM IGMF {\\rm RM_{IGMF}} values.We define the RM of the gap caused by the cosmic web B as ${\\rm RM_{IGMF}}=\\phi _{f,C}-\\phi _{b,B},$ which give the lower-limit of the RM due to the cosmic web B, and the whole of the RM if $\\phi _{\\rm cw}\\rightarrow 0$ .",
"In order to estimate ${\\rm RM_{IGMF}}$ , we refer to the interval of the two peaks seen in the FDF, since we confirmed that the relative Faraday depth between the two peaks are always precisely reconstructed.",
"Here, the interval also includes skirts and side robes caused by the RMSF as well as ${\\rm RM_{c,B}}/2+{\\rm RM_{c,C}}/2$ , where ${\\rm RM_c}$ is the Faraday thickness of the compact source, i.e.",
"$\\phi _b-\\phi _f$ .",
"Instead of estimating the Faraday thicknesses, we simply calculate a half width at half maximum of each peak, and subtract the half widths of the two peaks from the interval.",
"The estimated ${\\rm RM_{IGMF}}$ are listed in Table 2.",
"We find that this method estimates ${\\rm RM_{IGMF}}$ with errors less than $10-25~\\%$ for all the cases shown in Figure REF .",
"Note that the gap is substantially buried under the skirts in the case with ${\\rm RM_{IGMF}}=1~{\\rm rad~m^{-2}}\\sim {\\rm RM_c}$ (not shown), and ${\\rm RM_{IGMF}}$ is not estimated correctly.",
"Therefore, the above accuracy can be obtained, if ${\\rm RM_{IGMF}}\\gg {\\rm RM_c}$ ."
],
[
"Discussion",
"We have presented the cases for a pure real $F(\\phi )$ obtained if the intrinsic polarization angle, $\\chi _0$ , does not depend on $\\phi $ .",
"$\\chi _0$ is, however, determined by structures of magnetic fields, and could be a function of $\\phi $ .",
"In order to see effects of $\\chi _0(\\phi )$ on the reconstruction, we consider a variable $\\chi _0(\\phi )$ in our model.",
"We multiply a phase factor $e^{2i \\chi _0(\\phi )}$ to a real function of $F(\\phi )$ in Equation (REF ), keeping the absolute value of the model FDF to be the same.",
"For $\\chi _0(\\phi )$ , although its general profile is not known, some characteristic behaviors could be understood by using a simple analytic function.",
"We consider a periodic function $\\chi _0(\\phi ) = \\cos (2\\pi \\phi \\times 0.1n)$ for $n=$ 1, 2, 3, and 4, since periodicity is expected from multiple reversals of turbulent magnetic fields.",
"The results with $F_{\\rm c0}/F_{\\rm d0}=1$ and ${\\rm RM_{IGMF}}=10$ ${\\rm rad~m^{-2}}$ are shown in Figure REF .",
"We find that the profiles of the reconstructed FDFs highly depend on $n$ .",
"Nevertheless, the edges of the sources are rather sharp compared with the fiducial model ($n=0$ ).",
"This may be ascribed to the cancellation of polarized emissions at the tails due to a rotation of the intrinsic polarization angle.",
"Therefore, our main simulations could be regarded as conservative cases with largest extension of the skirts, which would be somewhat reduced in realistic situations.",
"Eventually, ${\\rm RM_{IGMF}}$ could be better estimated from the gap between the two sources.",
"A real FDF of the Galaxy would be much more complex.",
"It would have $n\\gg 4$ based on the coherence length of magnetic fields of several tens of pc (see [3], references therein).",
"Even in such a case, our observational strategies would be still available, since the intrinsic polarization angle does not alter the key feature: two sources and the gap between them (Figure REF ).",
"Multiple peaks may become an ambiguity for identifying which peak is an extragalactic origin and which gap is caused by the IGMF.",
"But we could solve the ambiguity, if we carry out an “off-source\" observation and gain the FDF of the Galaxy only.",
"We notice that a real FDF of the Galaxy should depend on Galactic longitude and latitude as well as properties of turbulent magnetic fields such as the driving scale, the Mach number, the plasma $\\beta $ , and so on ([3]).",
"Although considerations of them are beyond the scope of this paper, developing realistic FDFs of the Galaxy must be an important subject to make the detection of the gap more reliable.",
"Realistic FDFs of the Galaxy based on Akahori et al.",
"(2013) will be presented in a separate paper ([40]).",
"Figure: Same as Figure but for the models with F c0 /F d0 =1F_{\\rm c0}/F_{\\rm d0}=1 and RM IGMF =10 rad m -2 {\\rm RM_{IGMF}}=10~{\\rm rad~m^{-2}} and with observational effects: top three panels are for the cases with noise amplitudes of 30, 50, and 100 % of the polarized intensities, and the bottom two panels are for the cases with RFIs on sites X and Y, where unshadowed regions are wavelength coverages for SKA without strong RFIs.Another simplification in this paper was to neglect observational effects.",
"Particularly, it is true that there are significant noise on polarized intensities and some frequencies are probably missing due to radio frequency interferences (RFIs).",
"We demonstrate these effects as follows.",
"For effects of observational noise, we include the noise into the observable polarized intensity, $\\tilde{P}(\\lambda ^2)$ , and get the reconstructed FDF.",
"We consider a random Gaussian noise in each $\\lambda ^2$ domain.",
"The results with noise amplitudes of 30, 50, and 100 % of the polarized intensities for representative cases are shown in top panels of Figures REF and REF .",
"We see that the noise amplitude of 30 % does not dramatically alter the overall profile of the FDF, and the reconstructed FDF would be useful up to the noise amplitude of $\\sim 50~\\%$ .",
"Such a requirement of the noise would limit the sample of radio sources that could be considered.",
"Figure: Same as Figure but for the models with the RM of the cosmic web B, φ f,C -φ f,B =\\phi _{f,C}-\\phi _{f,B}= 10 rad m -2 {\\rm rad~m^{-2}} and with observational effects: top three panels are for the cases with noise amplitudes of 30, 50, and 100 % of the polarized intensities, and the bottom two panels are for the cases with RFIs on sites X and Y.As for effects of RFIs, we discard the data in frequencies where significant RFIs exist, and get the reconstructed FDF.",
"We refer to the recent assessment reporthttp://www.skatelescope.org/wp-content/uploads/2012/06/78g_SKAmon-Max.Hold_.Mode_.Report.pdf of RFIs for the SKA candidate sites, X and Y.",
"The results for representative cases are shown in bottom panels of Figures REF and REF .",
"We can see less-peaked profiles for compact sources due to lack of data in low frequencies around $\\sim 86-108$ MHz and $\\sim 170-270$ MHz.",
"Such a broadening of the FDF for the compact source would produce uncertainties of a few ${\\rm rad~m^{-2}}$ on the estimation of the gap.",
"Reconstructed FDFs have significant skirts and side lobes originating from the RMSF.",
"Such skirts and side lobes are a major ambiguity to probe the IGMF.",
"The issues related to the RMSF could be, however, improved by using decomposition techniques such as RMCLEAN [35] and calibrations of the RMSF by phase correction (BD05) and symmetry assumption [25].",
"Also, wavelet-based tomography [25], [10] would allow better representation of localized structures in the data unlike with decompositions with harmonic functions in the Fourier transform.",
"QU-fitting ([49], [39]) and compressive sampling/sensing [21], [14], [44], [7] would be also promising to probe the gap caused by the IGMF.",
"Another important improvement to get better reconstructions of the FDF is even sampling in $\\lambda ^2$ space.",
"Although we have assumed even sampling in the simulations, observations sample the data evenly in $\\lambda $ space so far.",
"Such data produce unevenly-sampled data in $\\lambda ^2$ space, and cause large numerical artifacts in Fourier transform.",
"In order to minimize numerical errors on the Fourier transform, development of flexible receiver systems which allow us to sample the data evenly in $\\lambda ^2$ space would be a key engineering task for future radio astronomy (e.g., CASPER/ROACHhttps://casper.berkeley.edu/wiki/ROACH)."
],
[
"Summary",
"In order to probe Faraday rotation measure (RM) of the intergalactic magnetic field (IGMF) in the cosmic web from observations of RMs for extragalactic radio sources, we need to separate contributions of other origins of RMs.",
"In this paper, we discussed possible observational strategies to estimate the RM due to the IGMF by means of Faraday tomography (Faraday RM Synthesis).",
"Our quantitative discussion indicated that there are two possible strategies - an observation of a compact source behind a diffuse source (Strategy A) and a comparison between the observation with another observation for a nearby compact source (Strategy B).",
"For the two strategies, we investigated the capability of Faraday tomography in present and future wide-band radio polarimetry.",
"For Strategy A, we confirmed that the RM due to the IGMF can be seen as the relative Faraday depth between the two sources.",
"A promising polarized intensity of the compact source relative to the diffuse source is $\\sim 1-10$ .",
"As for Strategy B, we found that the relative Faraday depth of the compact sources gives a reasonable estimate of the RM with errors less than a few tens percents for LOFAR or SKA observations, if the RM is larger than $10~{\\rm rad~{m^{-2}}}$ .",
"Such an accuracy is expected while the compact sources are $\\sim 100-10000$ times brighter than the diffuse source.",
"Strategy B provides better estimations of the RM, but Strategy A is also important to increase the chance of the estimation.",
"Since we have adopted simple models of radio sources, we discussed more realistic cases for specific situations.",
"We demonstrated that the multiple changes of intrinsic polarization angle within a diffuse source would not make the estimation of the relative Faraday depth between the two sources worse.",
"Multiple peaks may become an ambiguity for identifying the gap caused by the IGMF, but the ambiguity could be solved with an off-source observation.",
"We also considered effects of observational noise and radio frequency interferences, and found that these effects can be practically insignificant for the estimation, at least for the moderate levels of noise and RFI considered here.",
"Our simulations and discussions indicate that it is still not easy to explore RM of $O(1)~{\\rm rad~m^{-2}}$ for the IGMF, because of incompleteness of the reconstruction as well as ambiguities due to RMs associated with the source and intervening clouds.",
"These issues would be, however, improved by decomposition techniques as well as theoretical and observational studies of Faraday depolarization.",
"With the improvements, we would finally reach observations of the IGMF in filaments of galaxies.",
"The authors would like to thank the referee(s) for constructive comments and discussions, B. M. Gaensler, S. P. O'Sullivan, and X. H. Sun for useful comments.",
"T.A.",
"and K.K.",
"acknowledge the supports of the Japan Society for the Promotion of Science (JSPS).",
"T.A.",
"and D.R.",
"were supported by National Research Foundation of Korea through grant 2007-0093860.",
"K.T.",
"was supported by Grants-in-Aid from MEXT of Japan, No.",
"23740179, No.",
"24111710 and No.",
"24340048."
]
] | 1403.0325 |
[
[
"Timed Soft Concurrent Constraint Programs: An Interleaved and a Parallel\n Approach"
],
[
"Abstract We propose a timed and soft extension of Concurrent Constraint Programming.",
"The time extension is based on the hypothesis of bounded asynchrony: the computation takes a bounded period of time and is measured by a discrete global clock.",
"Action prefixing is then considered as the syntactic marker which distinguishes a time instant from the next one.",
"Supported by soft constraints instead of crisp ones, tell and ask agents are now equipped with a preference (or consistency) threshold which is used to determine their success or suspension.",
"In the paper we provide a language to describe the agents behavior, together with its operational and denotational semantics, for which we also prove the compositionality and correctness properties.",
"After presenting a semantics using maximal parallelism of actions, we also describe a version for their interleaving on a single processor (with maximal parallelism for time elapsing).",
"Coordinating agents that need to take decisions both on preference values and time events may benefit from this language.",
"To appear in Theory and Practice of Logic Programming (TPLP)."
],
[
"Introduction",
"Time is a particularly important aspect of cooperative environments.",
"In many “real-life” computer applications, the activities have a temporal duration (that can be even interrupted) and the coordination of such activities has to take into consideration this timeliness property.",
"The interacting actors are mutually influenced by their actions, meaning that $A$ reacts accordingly to the timing and quantitative aspects related to $B$ 's behavior, and vice versa.",
"In fact, these interactions can be often related to quantities to be measured or minimized/maximized, in order to take actions depending from these scores: consider, for example, some generic communicating agents that need to take decisions on a (monetary) cost or a (fuzzy) preference for a shared resource.",
"They both need to coordinate through time-dependent and preference-based decisions.",
"A practical example of such agents corresponds, for example, to software agents that need to negotiate some service-level agreement on a resource, or a service, with time-related side-conditions.",
"For instance, a fitting example is given by auction schemes, where the seller/bidder agents need to agree on a preference for a given prize (e.g., a monetary cost).",
"At the same time, the agents have to respect some timeout and alarm events, respectively representing the absence and the presence of bids for the prize (for instance).",
"The language we present in this paper is well suited for this kind of interactions, as Section shows with examples.",
"The Timed Concurrent Constraint Programming (tccp), a timed extension of the pure formalism of Concurrent Constraint Programming (ccp) [24], has been introduced in [10].",
"The language is based on the hypothesis of bounded asynchrony [25]: computation takes a bounded period of time rather than being instantaneous as in the concurrent synchronous languages ESTEREL [1], LUSTRE [17], SIGNAL [20] and Statecharts [18].",
"Time itself is measured by a discrete global clock, i.e., the internal clock of the tccp process.",
"In [10] the authors also introduced timed reactive sequences, which describe the reaction of a tccp process to the input of the external environment, at each moment in time.",
"Formally, such a reaction is a pair of constraints $\\langle c,d\\rangle $ , where $c$ is the input and $d$ is the constraint produced by the process in response to $c$ .",
"Soft constraints [2], [4] extend classical constraints to represent multiple consistency levels, and thus provide a way to express preferences, fuzziness, and uncertainty.",
"The ccp framework has been extended to work with soft constraints [5], and the resulting framework is named Soft Concurrent Constraint Programming (sccp).",
"With respect to ccp, in sccp the tell and ask agents are equipped with a preference (or consistency) threshold, which is used to determine their success, failure, or suspension, as well as to prune the search; these preferences should preferably be satisfied but not necessarily (i.e.",
"over-constrained problems).",
"We adopt soft constraints instead of crisp ones, since classic constraints show evident limitations when trying to represent real-life scenarios, where the knowledge is not completely available nor crisp.",
"In this paper, we introduce a timed and soft extension of ccp that we call Timed Soft Concurrent Constraint Programming (tsccp), inheriting from both tccp and sccp at the same time.",
"In tsccp, we directly introduce a timed interpretation of the usual programming constructs of sccp, by identifying a time-unit with the time needed for the execution of a basic sccp action (ask and tell), and by interpreting action prefixing as the next-time operator.",
"An explicit timing primitive is also introduced in order to allow for the specification of timeouts.",
"In the first place, the parallel operator of tsccp is first interpreted in terms of maximal parallelism, as in [10].",
"Secondly, we also consider a different paradigm, where the parallel operator is interpreted in terms of interleaving, however assuming maximal parallelism for actions depending on time.",
"In other words, time passes for all the parallel processes involved in a computation.",
"This approach, analogous to that one adopted in [11], is different from that one of [10], [3] (where maximal parallelism was assumed for any kind of action), and it is also different from the one considered in [9], where time does not elapse for timeout constructs.",
"This can be accomplished by allowing all the time-only dependent actions ($\\tau $ -transitions) to concurrently run with at most one action manipulating the store (a $\\omega $ -transition).",
"The paper extends the results in [3] by providing new semantics that allows maximal parallelism for time elapsing and an interleaving model for basic computation steps (see Section ).",
"This new language is called tsccp with interleaving, i.e., tsccp-i, to distinguish it from the version allowing maximal parallelism of all actions.",
"According to the maximal parallelism policy (applied, for example, in the original works as [24] and [27]), at each moment every enabled agent of the system is activated, while in the interleaving paradigm only one of the enabled agents is executed instead.",
"This second paradigm is more realistic if we consider limited resources, since it does not imply the existence of an unbounded number of processors.",
"However, in [10] it is shown that the notion of maximal parallelism of tsccp is more expressive than the notion of interleaving parallelism of other concurrent constraint languages.",
"The presence of maximal parallelism can force the computation to discard some (non-enabled) branches which could became enabled later on (because of the information produced by parallel agents), while this is not possible when considering an interleaving model.",
"Therefore, tsccp is sensitive to delays in adding constraints to the store, whereas this is not the case for ccp and tsccp-i.",
"The rest of the paper is organized as follows: in Section we summarize the most important background notions and frameworks from which tsccp derives, i.e.",
"tccp and sccp.",
"In Section we present the tsccp language, and in Section describes the operational semantics of tscc agents.",
"Section better explains the programming idioms as timeout and interrupt, exemplifies the use of timed paradigms in the tscc language and shows an application example on modeling an auction interaction among several bidders and a single auctioneer.",
"Section describes the denotational semantics for tsccp, and proves the denotational model correctness with the aid of connected reactive sequences.",
"Section explains the semantics for interleaving with maximal parallelism of time-elapsing actions (i.e.",
"the tsccp-i language), while Section describes a timeline for the execution of three parallel agents in tsccp-i.",
"Section describes the denotational semantics of tsccp-i and proves the correctness of the denotational model.",
"Section reports the related work and, at last, Section concludes by also indicating future research.",
"A soft constraint [4], [2] may be seen as a constraint where each instantiation of its variables has an associated value from a partially ordered set which can be interpreted as a set of preference values.",
"Combining constraints will then have to take into account such additional values, and thus the formalism has also to provide suitable operations for combination ($\\times $ ) and comparison ($+$ ) of tuples of values and constraints.",
"This is why this formalization is based on the concept of c-semiring [4], [2], called just semiring in the rest of the paper.",
"A semiring is a tuple $\\langle A,+,\\times ,{\\mathbf {0}},{\\mathbf {1}} \\rangle $ such that: i) $A$ is a set and ${\\mathbf {0}}, {\\mathbf {1}} \\in A$ ; ii) $+$ is commutative, associative and ${\\mathbf {0}}$ is its unit element; iii) $\\times $ is associative, distributes over $+$ , ${\\mathbf {1}}$ is its unit element and ${\\mathbf {0}}$ is its absorbing element.",
"A c-semiring is a semiring $\\langle A,+,\\times ,{\\mathbf {0}},{\\mathbf {1}} \\rangle $ such that: $+$ is idempotent, ${\\mathbf {1}}$ is its absorbing element and $\\times $ is commutative.",
"Let us consider the relation $\\le _S$ over $A$ such that $a \\le _Sb$ iff $a+b = b$ .",
"Then, it is possible to prove that (see [4]): i) $\\le _S$ is a partial order; ii) $+$ and $\\times $ are monotone on $\\le _S$ ; iii) ${\\mathbf {0}}$ is its minimum and ${\\mathbf {1}}$ its maximum; iv) $\\langle A,\\le _S \\rangle $ is a complete lattice (a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum) and, for all $a, b \\in A$ , $a+b = \\mathit {lub}(a,b)$ (where $\\mathit {lub}$ is the least upper bound).",
"Moreover, if $\\times $ is idempotent, then: $+$ distributes over $\\times $ ; $\\langle A,\\le _S \\rangle $ is a complete distributive lattice and $\\times $ its $glb$ (greatest lower bound).",
"Informally, the relation $\\le _S$ gives us a way to compare semiring values and constraints.",
"In fact, when we have $a \\le _Sb$ , we will say that $b$ is better than $a$.",
"In the following, when the semiring will be clear from the context, $a \\le _S b$ will be often indicated by $a \\le b$ ."
],
[
"Constraint System.",
"Given a semiring $S = \\langle A,+,\\times ,{\\mathbf {0}},{\\mathbf {1}} \\rangle $ and an ordered set of variables $V$ over a finite domain $D$ , a soft constraint is a function which, given an assignment $\\eta :V\\rightarrow D$ of the variables, returns a value of the semiring.",
"Using this notation ${\\mathcal {C}}= \\eta \\rightarrow A$ is the set of all possible constraints that can be built starting from $S$ , $D$ and $V$ .",
"Any function in ${\\mathcal {C}}$ involves all the variables in $V$ , but we impose that it depends on the assignment of only a finite subset of them.",
"So, for instance, a binary constraint $c_{x,y}$ over variables $x$ and $y$ , is a function $c_{x,y}: (V\\rightarrow D)\\rightarrow A$ , but it depends only on the assignment of variables $\\lbrace x,y\\rbrace \\subseteq V$ (the support of the constraint, or scope).",
"Note that $c\\eta [v:=d_1]$ means $c\\eta ^{\\prime }$ where $\\eta ^{\\prime }$ is $\\eta $ modified with the assignment $v:=d_1$ (that is the operator $[\\ ]$ has precedence over application).",
"Note also that $c\\eta $ is the application of a constraint function $c:(V \\rightarrow D)\\rightarrow A$ to a function $\\eta :V\\rightarrow D$ ; what we obtain, is a semiring value $c\\eta $ .",
"The partial order $\\le _S$ over ${\\mathcal {C}}$ can be easily extended among constraints by defining $c_1 \\sqsubseteq c_2 \\iff c_1 \\eta \\le c_2 \\eta $ , for each possible $\\eta $ ."
],
[
"Combining and projecting soft constraints.",
"Given the set ${\\mathcal {C}}$ , the combination function $\\otimes : {\\mathcal {C}}\\times {\\mathcal {C}}\\rightarrow {\\mathcal {C}}$ is defined as $(c_1\\otimes c_2)\\eta =c_1\\eta \\times c_2\\eta $ (see also [4], [2], [5]).",
"Informally, performing the $\\otimes $ between two constraints means building a new constraint whose support involves all the variables of the original ones, and which associates with each tuple of domain values for such variables a semiring element which is obtained by multiplying the elements associated by the original constraints to the appropriate sub-tuples.",
"Given a constraint $c \\in {\\mathcal {C}}$ and a variable $v \\in V$ , the projection [4], [2], [5] of $c$ over $V-\\lbrace v\\rbrace $ , written $c\\Downarrow _{(V-\\lbrace v\\rbrace )}$ is the constraint $c^{\\prime }$ s.t.",
"$c^{\\prime }\\eta = \\sum _{d \\in D} c \\eta [v:=d]$ .",
"Informally, projecting means eliminating some variables from the support.",
"This is done by associating with each tuple over the remaining variables a semiring element which is the sum of the elements associated by the original constraint to all the extensions of this tuple over the eliminated variables.",
"We define also a function $\\bar{a}$ [2], [5] as the function that returns the semiring value $a$ for all assignments $\\eta $ , that is, $\\bar{a}\\eta =a$ .",
"We will usually write $\\bar{a}$ simply as $a$ .",
"An example of constants that will be useful later are $\\bar{{\\mathbf {0}}}$ and $\\bar{{\\mathbf {1}}}$ that represent respectively the constraints associating ${\\mathbf {0}}$ and ${\\mathbf {1}}$ to all the assignment of domain values."
],
[
"Solutions.",
"A SCSP [2] is defined as $P= \\langle V, D, C, S \\rangle $ , where $C$ is the set of constraints defined over variables in $V$ (each with domain $D$ ), and whose preference is determined by semiring $S$ .",
"The best level of consistency notion is defined as $\\mathit {blevel}(P) = \\mathit {Sol}(P) \\Downarrow _{\\emptyset }$ , where $Sol(P)= \\bigotimes C$ [2].",
"A problem $P$ is $\\alpha $ -consistent if $\\mathit {blevel}(P) = \\alpha $ [2].",
"$P$ is instead simply “consistent” iff there exists $\\alpha >_S {\\mathbf {0}}$ such that $P$ is $\\alpha $ -consistent.",
"$P$ is inconsistent if it is not consistent.",
"Figure: A SCSP based on a weighted semiring.Example 1 Figure REF shows a weighted SCSP as a graph: the weighted semiring is used, i.e.",
"$\\langle \\mbox{$\\mathbb {R}$}^{+} \\cup \\lbrace \\infty \\rbrace , \\min , \\hat{+},$ $\\infty ,0\\rangle $ ($\\hat{+}$ is the arithmetic plus operation).",
"Variables and constraints are represented respectively by nodes and arcs (unary for $c_1$ -$c_3$ , and binary for $c_2$ ); $D= \\lbrace a, b\\rbrace $ .",
"The solution of the CSP in Figure REF associates a semiring element to every domain value of variables $X$ and $Y$ by combining all the constraints together, i.e.",
"$Sol(P)= \\bigotimes C$ .",
"For instance, for the tuple $\\langle a, a\\rangle $ (that is, $X = Y =a$ ), we have to compute the sum of 1 (which is the value assigned to $X = a$ in constraint $c_1$ ), 5 ($\\langle X = a, Y = a \\rangle $ in $c_2$ ) and 5 ($Y = a$ in $c_3$ ): the value for this tuple is 11.",
"The solution $X = a, Y = b$ is a 7-consistent solution, where 7 corresponds to the blevel of $P$ , i.e., $\\mathit {Sol}(P)\\Downarrow _\\emptyset = 7$ ."
],
[
"Concurrent Constraint Programming over Soft Constraints",
"The basic idea underlying ccp [24] is that computation progresses via monotonic accumulation of information in a constraint global store.",
"Information is produced by the concurrent and asynchronous activity of several agents which can add (tell) a constraint to the store.",
"Dually, agents can also check (ask) whether a constraint is entailed by the store, thus allowing synchronization among different agents.",
"The ccp languages are defined parametrically w.r.t.",
"a given constraint system.",
"The notion of constraint system has been formalized in [26] following Scott's treatment of information systems.",
"Soft constraints over a semiring $S = \\langle \\mathcal {A},+,\\times ,{\\mathbf {0}},{\\mathbf {1}} \\rangle $ and an ordered set of variables $V$ (over a domain $D$ ) have been showed to form a constraint system “à la Saraswat”, thus leading to the definition of Soft Concurrent Constraint Programmingg (sccp) [4], [2], [5].",
"Consider the set ${\\mathcal {C}}$ and the partial order $\\sqsubseteq $ .",
"Then an entailment relation $\\vdash \\subseteq \\wp ({\\mathcal {C}}) \\times {\\mathcal {C}}$ is defined s.t.",
"for each $C \\in \\wp ({\\mathcal {C}})$ and $c \\in {\\mathcal {C}}$ , we have $C\\vdash c \\iff \\bigotimes C \\sqsubseteq c$ (see also [2], [5]).",
"Note that in this setting the notion of token (constraint) and of set of tokens (set of constraints) closed under entailment is used indifferently.",
"In fact, given a set of constraint functions $C_1$ , its closure w.r.t.",
"entailment is a set $\\bar{C_1}$ that contains all the constraints greater than $\\bigotimes C_1$ .",
"This set is univocally representable by the constraint function $\\bigotimes C_1$ .",
"The definition of the entailment operator $\\vdash $ on top of ${\\mathcal {C}}$ , and of the $\\sqsubseteq $ relation, lead to the notion of soft constraint system.",
"It is also important to notice that in [24] it is claimed that a constraint system is a complete algebraic lattice.",
"In the sccp framework, algebraicity is not required [5] instead, since the algebraic nature of the structure ${\\mathcal {C}}$ strictly depends on the properties of the semiringNotice that we do not aim at computing the closure of the entailment relation, but only to use the entailment relation to establish if a constraint is entailed by the current store, and this can be established even if the lattice is not algebraic (that is even if the times operator is not idempotent).. To treat the hiding operator of the language, a general notion of existential quantifier is introduced by using notions similar to those used in cylindric algebras.",
"Consider a set of variables $V$ with domain $D$ and the corresponding soft constraint system ${\\mathcal {C}}$ .",
"For each $x \\in V$ , the hiding function [2], [5] is the function $(\\exists _x c)\\eta =\\sum _{d_i\\in D} c\\eta [x :=d_i]$ .",
"To make the hiding operator computationally tractable, it is required that the number of domain elements in $D$ , having semiring values different from ${\\mathbf {0}}$ , is finite [5].",
"In this way, to compute the sum needed for $(\\exists _x c)\\eta $ , we can consider just a finite number of elements (those different from ${\\mathbf {0}}$ ), since ${\\mathbf {0}}$ is the unit element of the sum.",
"Note that by using the hiding function we can represent the $\\Downarrow $ operator defined in Section REF .",
"In fact, for any constraint $c$ and any variable $x \\subseteq V$ , $c\\Downarrow _{V-x} = \\exists _x c$ [5].",
"To model parameter passing also diagonal elements have to be defined.",
"Consider a set of variables $V$ and the corresponding soft constraint system.",
"Then, for each $x,y \\in V$ , a diagonal constraint is defined as $d_{xy} \\in {\\mathcal {C}}$ s.t., $d_{xy}\\eta [x:= a, y := b]= {\\mathbf {1}}$ if $a=b$ , and $d_{xy}\\eta [x:=a, y := b] = {\\mathbf {0}}$ if $a\\ne b$ [2], [5].",
"[cylindric constraint system [5]] Consider a semiring $S = \\langle \\mathcal {A},+,\\times ,{\\mathbf {0}},{\\mathbf {1}}\\rangle $ , a domain of the variables $D$ , an ordered set of variables $V$ , and the corresponding structure ${\\mathcal {C}}$ .",
"Then, $S_C=\\langle {\\mathcal {C}}, \\otimes ,{\\mathbf {0}},{\\mathbf {1}} , \\exists _x, d_{xy}\\rangle $ , is a cylindric constraint system."
],
[
"Timed Concurrent Constraint Programming",
"A timed extension of ccp, called tccp has been introduced in [10].",
"Similarly to other existing timed extensions of ccp defined in [25], tccp is a language for reactive programming designed around the hypothesis of bounded asynchrony (as introduced in [25]: computation takes a bounded period of time rather than being instantaneous).",
"When querying the store for some information that is not present (yet), a ccp agent will simply suspend until the required information has arrived.",
"In timed applications however often one cannot wait indefinitely for an event.",
"Consider for example the case of a connection to a web service providing some on-line banking facility.",
"In case the connection cannot be established, after a reasonable amount of time an appropriate time-out message has to be communicated to the user.",
"A timed language should then allow us to specify that, in case a given time bound is exceeded (i.e.",
"a time-out occurs), the wait is interrupted and an alternative action is taken.",
"Moreover, in some cases it is also necessary to have a preemption mechanism which allows one to abort an active process $A$ and to start a process $B$ when a specific (abnormal) event occurs.",
"In order to be able to specify these timing constraints tccp introduces a discrete global clock and assumes that ask and tell actions take one time-unit.",
"Computation evolves in steps of one time-unit, so called clock-cycles.",
"Action prefixing is the syntactic marker which distinguishes a time instant from the next one and it is assumed that parallel processes are executed on different processors, which implies that, at each moment, every enabled agent of the system is activated.",
"This assumption gives rise to what is called maximal parallelism.",
"The time in between two successive moments of the global clock intuitively corresponds to the response time of the underlying constraint system.",
"Thus all parallel agents are synchronized by the response time of the underlying constraint system.",
"Since the store is monotonically increasing and one can have dynamic process creation, clearly the previous assumptions imply that the constraint solver takes a constant time (no matter how big the store is), and that there is an unbounded number of processors.",
"However, one can impose suitable restriction on programs, thus ensuring that the (significant part of the) store and the number of processes do not exceed a fixed bound; these restrictions would still allow significant forms of recursion with parameters.",
"Furthermore, a timing construct of the form ${\\bf now}\\ c \\ {\\bf then}\\ A \\ {\\bf else}\\ B $ is introduced in tccp, whose semantics is the following: if the constraint $c$ is entailed by the store at the current time $t$ , then the above agent behaves as $A$ at time $t$ , otherwise it behaves as $B$ at time $t$ .",
"This basic construct allows to derive such timing mechanisms as time-out and preemption [10], [25].",
"The instantaneous reaction can be obtained by evaluating now$c$ in parallel with $A$ and $B$ , within the same time-unit.",
"At the end of this time-unit, the store will be updated by using either the constraint produced by $A$ , or that one produced by $B$ , depending on the result of the evaluation of now$c$ .",
"Clearly, since $A$ and $B$ could contain nested ${\\bf now}\\ {\\bf then}\\ {\\bf else}$ agents, a limit for the number of these nested agents should be fixed.",
"Note that, for recursive programs, such a limit is ensured by the presence of the procedure-call, since we assume that the evaluation of such calls takes one time-unit.",
"In this section we present the tsccp language, which originates from both tccp and sccp.",
"To obtain this aim, we extend the syntax of the cc language with the timing construct now$c$ then$A$ else$B$ (inherited from tccp), and also in order to directly handle the cut level as in sccp.",
"This means that the syntax and semantics of the tell, ask and nowagents have to be enriched with a threshold that is used to check when the agents may succeed, or suspend.",
"Definition 1 (tsccp Language) Given a soft constraint system $\\langle S,D,V\\rangle $ , the corresponding structure ${\\mathcal {C}}$ , any semiring value $a$ , soft constraints $\\phi , \\ c \\in {\\cal C}$ and any tuple of variables $x$ , the syntax of the tsccp language is given by the following grammar: $\\begin{array}{ll}P ::= & F \\text{.}",
"A\\\\F ::= & p(x):: A \\;|\\; F.F\\\\A ::= & {\\bf success} \\;|\\; \\hbox{{\\bf tell}}(c)\\rightarrow _\\phi A\\;|\\;\\hbox{{\\bf tell}}(c)\\rightarrow ^a A \\;|\\; E \\;|\\; A\\parallel A\\;|\\;\\exists x A \\;|\\; p(x) \\;|\\; \\\\& \\Sigma _{i=1}^{n} E_i\\;|\\; {\\bf now}_\\phi \\ c \\ {\\bf then}\\ A \\ {\\bf else}\\ A \\;|\\; {\\bf now}^a \\ c \\ {\\bf then}\\ A \\ {\\bf else}\\ A\\\\E ::= &\\hbox{\\bf ask}(c) \\rightarrow _\\phi A \\;|\\; \\hbox{\\bf ask}(c)\\rightarrow ^a A \\\\\\end{array}$ where, as usual, $P$ is the class of processes, $F$ is the class of sequences of procedure declarations (or clauses), $A$ is the class of agents.",
"In a tsccp process $P=F\\text{.",
"}A$ , $A$ is the initial agent, to be executed in the context of the set of declarations $F$ .",
"The agent success represents a successful termination, so it may not make any further transition.",
"In the following, given an agent $A$ , we denote by $Fv(A)$ the set of the free variables of $A$ (namely, the variables which do not appear in the scope of the $\\exists $ quantifier).",
"Besides the use of soft constraints (see Section REF ) instead of crisp ones, there are two fundamental differences between tsccp and ccp.",
"The first main difference w.r.t.",
"the original cc syntax is the presence of a semiring element $a$ and of a constraint $\\phi $ to be checked whenever an ask or tell operation is performed.",
"More precisely, the level $a$ (respectively, $\\phi $ ) will be used as a cut level to prune computations that are not good enough.",
"The second main difference with respect to ccp (but, this time, also with respect to sccp) is instead the presence of the now$c$ then $A$ else $B$ construct introduced in Section REF .",
"Even for this construct, the level $a$ (or $\\phi $ ) is used as a cut level to prune computations.",
"Action prefixing is denoted by $\\rightarrow $ , non-determinism is introduced via the guarded choice construct $\\Sigma _{i=1}^{n} E_i$ , parallel composition is denoted by $\\parallel $ , and a notion of locality is introduced by the agent $\\exists x A$ , which behaves like $A$ with $x$ considered local to $A$ , thus hiding the information on $x$ provided by the external environment.",
"In the next subsection we formally describe the operational semantics of tsccp.",
"In order to simplify the notation, in the following we will usually write a tsccp process $P=F\\text{.",
"}A$ simply as the corresponding agent $A$ ."
],
[
"An Operational Semantics for ",
"The operational model of tscc agents can be formally described by a transition system $T= ({\\it Conf}, \\longrightarrow )$ where we assume that each transition step takes exactly one time-unit.",
"Configurations in Conf are pairs consisting of a process and of a constraint in ${\\cal C}$ , representing the common store shared by all the agents.",
"The transition relation $\\longrightarrow \\subseteq {\\it Conf} \\times {\\it Conf}$ is the least relation satisfying the rules R1-R17 in Figure REF , and it characterizes the (temporal) evolution of the system.",
"So, $\\langle A,\\sigma \\rangle \\longrightarrow \\langle B,\\delta \\rangle $ means that, if at time $t$ we have the process $A$ and the store $\\sigma $ , then at time $t+1$ we have the process $B$ and the store $\\delta $ .",
"Figure: The transition system for tsccp.Let us now briefly discuss the rules in Figure REF .",
"Here is a brief description of the transition rules: Valued-tell.",
"The valued-tell rule checks for the $a$ -consistency of the Soft Constraint Satisfaction Problem [2] (SCSP) defined by the store $\\sigma \\otimes c$ .",
"A SCSP $P$ is $a$ -consistent if $blevel(P) = a$ , where $blevel(P) =Sol(P) \\Downarrow _{\\emptyset }$ , i.e., the best level of consistency of the problem $P$ is a semiring value representing the least upper bound among the values yielded by the solutions.",
"Rule ${\\bf R1}$ can be applied only if the store $\\sigma \\otimes c$ is $b$ -consistent with $b \\lnot < a$Notice that we use $b \\lnot < a$ instead of $b \\ge a$ because we can possibly deal with partial orders.",
"The same holds also for $\\lnot \\sqsubset $ instead of $\\sqsupseteq $ ..",
"In this case the agent evolves to the new agent $A$ over the store $\\sigma \\otimes c$ .",
"Note that different choices of the cut level $a$ could possibly lead to different computations.",
"Finally, note that the updated store $\\sigma \\otimes c$ will be visible only starting from the next time instant, since each transition step involves exactly one time-unit.",
"Tell.",
"The tell action is a finer check of the store.",
"In this case (see rule R2), a pointwise comparison between the store $\\sigma \\otimes c$ and the constraint $\\phi $ is performed.",
"The idea is to perform an overall check of the store, and to continue the computation only if there is the possibility to compute a solution not worse than $\\phi $ .",
"Note that this notion of tell could be also applied to the classical cc framework: the tell operation would succeed when the set of tuples satisfying constraint $\\phi $ is not a superset of the set of tuples allowed by $\\sigma \\cap c$ .Notice that the $\\otimes $ operator in the crisp case reduces to set intersection.",
"As for the valued tell, the updated store $\\sigma \\otimes c$ will be visible only since the next time instant.",
"In the following, let us use $\\hbox{{\\bf tell}}(c)\\rightarrow A $ and $\\hbox{{\\bf tell}}(c)$ as a shorthand for $\\hbox{{\\bf tell}}(c)\\rightarrow _{\\bar{{\\mathbf {0}}}}A$ and $\\hbox{{\\bf tell}}(c)\\rightarrow _{\\bar{{\\mathbf {0}}}}{\\bf success}$ , respectively.",
"Valued-ask.",
"The semantics of the valued-ask is extended in a way similar to what we have done for the valued-tell action.",
"This means that, to apply the rule R3, we need to check if the store $\\sigma $ entails the constraint $c$ , and also if $\\sigma $ is “consistent enough” w.r.t.",
"the threshold $a$ set by the programmer.",
"Ask.",
"In rule R4, we check if the store $\\sigma $ entails the constraint $c$ , but, similarly to rule R2, we also compare a finer (pointwise) threshold $\\phi $ to the store $\\sigma $ .",
"As for the tell action, let us use $\\hbox{{\\bf ask}}(c) \\rightarrow A$ as a shorthand for $\\hbox{{\\bf ask}}(c) \\rightarrow _{\\bar{{\\mathbf {0}}}}A$ .",
"Parallelism.",
"Rules R5 and R6 model the parallel composition operator in terms of maximal parallelism: the agent $A\\parallel B$ executes in one time-unit all the initial enabled actions of $A$ and $B$ .",
"Considering rule R5 (where maximal parallelism is accomplished in practice), notice that the ordering of the operands in $\\sigma \\otimes \\delta \\otimes \\delta ^{\\prime }$ is not relevant, since $\\otimes $ is commutative and associative.",
"Moreover, for the same two properties, if $\\sigma \\otimes \\delta = \\sigma \\otimes \\gamma $ and $\\sigma \\otimes \\delta ^{\\prime }= \\sigma \\otimes \\gamma ^{\\prime }$ , we have that $\\sigma \\otimes \\delta \\otimes \\delta ^{\\prime }= \\sigma \\otimes \\gamma \\otimes \\gamma ^{\\prime }$ .",
"Therefore the resulting store $\\sigma \\otimes \\delta \\otimes \\delta ^{\\prime }$ is independent from the choice of the constraint $\\delta $ such that $\\langle A,\\sigma \\rangle \\longrightarrow \\langle A^{\\prime }, \\sigma ^{\\prime } \\rangle $ and $\\sigma ^{\\prime }=\\sigma \\otimes \\delta $ (analogously for $\\delta ^{\\prime }$ ).",
"Nondeterminism.",
"According to rule ${\\bf R7}$ , the guarded choice operator gives rise to global non-determinism: the external environment can affect the choice, since ${\\bf ask}(c_j)$ is enabled at time $t$ (and $A_j$ is started at time $t+1$ ) if and only if the store $\\sigma $ entails $c_j$ (and if it is compatible with the threshold too), and $\\sigma $ can be modified by other agents.",
"Valued-now and Now.",
"Rules ${\\bf R8}$ -${\\bf R11}$ show that the agent ${\\bf now}^a \\ c \\ {\\bf then}\\ A\\ {\\bf else}\\ $ $B$ behaves as $A$ or $B$ depending on the fact that $c$ is or is not entailed by the store, provided that the current store $\\sigma $ is compatible with the threshold.",
"Differently from the case of the ask, here the evaluation of the guard is instantaneous: if current store $\\sigma $ is compatible with the threshold $a$ , $\\langle A, \\sigma \\rangle $ ($\\langle B,\\sigma \\rangle $ ) can make a transition at time $t$ and $c$ is (is not) entailed by the store $\\sigma $ , then the agent ${\\bf now}^a \\ c \\ {\\bf then}\\ A\\ {\\bf else}\\ B$ can make the same transition at time $t$ .",
"Moreover, observe that in any case the control is passed either to $A$ (if $c$ is entailed by the current store $\\sigma $ and $\\sigma $ is compatible with the threshold) or to $B$ (in case $\\sigma $ does not entail $c$ and $\\sigma $ is compatible with the threshold).",
"Analogously for the not-valued version, i.e., ${\\bf now}_\\phi \\ c \\ {\\bf then}\\ A\\ {\\bf else}\\ B$ (see rules ${\\bf R12}$ -${\\bf R15}$ ).",
"Finally, we use ${\\bf now}\\ c \\ {\\bf then}\\ A\\ {\\bf else}\\ B$ as a shorthand for the agent ${\\bf now}_{\\bar{{\\mathbf {0}}}}\\ c \\ {\\bf then}\\ A\\ {\\bf else}\\ B$ Hiding variables.",
"The agent $\\exists x A$ behaves like $A$ , with $x$ considered local to $A$ , as show by rule R16.",
"This is obtained by substituting the variable $x$ for a variable $y$ , which we assume to be new and not used by any other process.",
"Standard renaming techniques can be used to ensure this; in rule R16, $A[x/y]$ denotes the process obtained from $A$ by replacing the variable $x$ for the variable $y$ .",
"Procedure-calls.",
"Rule ${\\bf R17}$ treats the case of a procedure-call when the actual parameter equals the formal parameter.",
"We do not need more rules since, for the sake of simplicity, here and in the following we assume that the set F of procedure declarations is closed w.r.t.",
"parameter names: that is, for every procedure-call $p(y)$ appearing in a process F.A, we assume that, if the original declaration for p in F is $p(x) :: A$ , then F contains also the declaration $p(y) :: \\exists x (\\hbox{{\\bf tell}}(d_{{x}{y}})\\parallel A)$ .Here the (original) formal parameter is identified as a local alias of the actual parameter.",
"Alternatively, we could have introduced a new rule treating explicitly this case, as it was in the original ccp papers.",
"Moreover, we assume that if $p(x) :: A \\in F$ , then $Fv(A) \\subseteq x$ .",
"Using the transition system described by (the rules in) Figure REF , we can now define our notion of observables, which considers the results of successful terminating computations that the agent $A$ can perform for each tsccp process $P= F\\text{.",
"}A$ .",
"Here and in the following, given a transition relation $\\longrightarrow $ , we denote by $\\longrightarrow ^*$ its reflexive and transitive closure.",
"Definition 2 (Observables) Let $P= F\\text{.",
"}A$ be a tsccp process.",
"We define ${\\cal O}^{mp}_{io}(P) = \\lbrace \\gamma \\Downarrow _{Fv(A)} \\mid \\langle A, {\\mathbf {1}} \\rangle \\longrightarrow ^*\\langle {\\bf Success}, \\gamma \\rangle \\rbrace $ where ${\\bf Success}$ is any agent which contains only occurrences of the agent ${\\bf success}$ and of the operator $\\parallel $ ."
],
[
"Programming Idioms and Examples",
"We can consider the primitives in Definition REF to derive the soft version of the programming idioms in [10], which are typical of reactive programming.",
"Delay.",
"The delay constructs ${\\bf tell}(c)\\stackrel{t}{\\longrightarrow }_\\phi A$ or ${\\bf ask}(c)\\stackrel{t}{\\longrightarrow }_\\phi A$ are used to delay the execution of agent $A$ after the execution of ${\\bf tell}(c)$ or ${\\bf ask}(c)$ ; $t$ is the number of the time-units of delay.",
"Therefore, in addiction to a constraint $\\phi $ , in tsccp the transition arrow can have also a number of delay slots.",
"This idiom can be defined by induction: the base case is $\\stackrel{0}{\\longrightarrow }_\\phi A \\equiv \\rightarrow _\\phi A$ , and the inductive step is $\\stackrel{n+1}{\\longrightarrow }_\\phi A\\equiv \\rightarrow _\\phi {\\bf tell}({\\bar{{\\mathbf {1}}}})\\stackrel{n}{\\longrightarrow }_{\\bar{{\\mathbf {0}}}} A$ .",
"The valued version can be defined in an analogous way.",
"Timeout.",
"The timed guarded choice agent $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i \\, {\\bf timeout}(m) \\, B$ waits at most $m$ time-units ($m \\ge 0$ ) for the satisfaction of one of the guards; notice that all the ask actions have a soft transition arrow, i.e.",
"$\\rightarrow _i$ is either of the form $\\rightarrow _{\\phi _i}$ or $\\rightarrow ^{a_i}$ , as in Figure REF .",
"Before this time-out, the process behaves just like the guarded choice: as soon as there exist enabled guards, one of them (and the corresponding branch) is nondeterministically selected.",
"After waiting for $m$ time-units, if no guard is enabled, the timed choice agent behaves as $B$ .",
"Timeout constructs can be assembled through the composition of several ${\\bf now}_{\\phi } \\ c \\ {\\bf then}\\ A \\ {\\bf else}\\ B$ primitives (or their valued version), as explained in [10] for the (crisp) tccp language.",
"The timeout can be defined inductively as follows: let us denote by $A$ the agent $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i$ .",
"In the base case, that is $m=0$ , we define $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i \\, {\\bf timeout}(0) \\, B$ as the agent: $\\begin{array}{llll}{\\bf now}_1 \\ c_1 & \\hspace*{-5.69046pt}{\\bf then}\\ A & & \\\\& \\hspace*{-21.33955pt}{\\bf else}\\ (\\ {\\bf now}_2 \\ c_2 &\\hspace*{-5.69046pt}{\\bf then}\\ A & \\\\& & \\hspace*{-21.33955pt}{\\bf else}\\ (\\dots (\\ {\\bf now}_n \\ c_n \\ {\\bf then}\\ A\\ {\\bf else}\\ {\\bf ask}(\\bar{{\\mathbf {1}}}) \\rightarrow \\ B) \\dots ))\\end{array}$ where for $i=1, \\ldots ,n$ , either ${\\bf now}_i = {\\bf now}_{\\phi _i}$ if $\\rightarrow _i$ is of the form $\\rightarrow _{\\phi _i}$ or ${\\bf now}_i = {\\bf now}^{a_i}$ if $\\rightarrow _i$ is of the form $\\rightarrow ^{a_i}$ .",
"Because of the operational semantics explained in rules R8-R11 (see Figure REF ), if a guard $c_{i}$ is true, then the agent $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i$ is evaluated in the same time slot.",
"Otherwise, if no guard $c_i$ is true, the agent $B$ is evaluated in the next time slot.",
"Then, by inductively reasoning on the number of time-units $m$ , we can define $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i \\, {\\bf timeout}(m) \\, B$ as $\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i \\, {\\bf timeout}(0) \\, (\\Sigma _{i=1}^{n} {\\bf ask}(c_i) \\rightarrow _i A_i \\, {\\bf timeout}(m-1) \\, B)\\text{.",
"}$ Watchdog.",
"Watchdogs are used to interrupt the activity of a process on a signal from a specific event.",
"The idiom ${\\bf do}\\; A \\; {\\bf watching}_{\\phi } \\;c$ behaves as $A$ , as long as $c$ is not entailed by the store and the current store is compatible with the threshold; when $c$ is entailed and the current store is compatible with the threshold, the process $A$ is immediately aborted.",
"The reaction is instantaneous, in the sense that $A$ is aborted at the same time instant of the detection of the entailment of $c$ .",
"However, according to the computational model, if $c$ is detected at time $t$ , then $c$ has to be produced at time $t^{\\prime }$ with $t^{\\prime }<t$ .",
"Thus, we have a form of weak preemption.",
"As well as timeouts, also watchdog agents can be defined in terms of the other basic constructs of the language (see Figure REF ).",
"In the following we assume that there exists an (injective) renaming function $\\rho $ which, given a procedure name $p$ , returns a new name $\\rho ( p)$ that is not used elsewhere in the program.",
"Moreover, let us use ${\\bf now}_{\\phi } \\, c \\ {\\bf else}\\ B$ as a shorthand for ${\\bf now}_{\\phi }\\, c \\ {\\bf then}\\ {\\bf success} $ else$ \\ B$ , where we assume that, for any procedure $p$ declared as $p(x) ::A$ , a declaration $\\rho (p)(x) :: {\\bf do}\\; \\rho (A) \\; {\\bf watching}_\\phi \\; c$ is added, where $\\rho (A)$ denotes the agent obtained from $A$ by replacing in it each occurrence of any procedure $q$ by $\\rho (q)$ .",
"The assumption in the case of the $\\exists x A$ agent is needed for correctness.",
"In practical cases, it can be satisfied by suitably renaming the variables associated to signals.",
"In the following $\\rightarrow ^{\\prime }$ is either of the form $\\rightarrow _{\\psi }$ or $\\rightarrow ^{a}$ .",
"Analogously for ${\\bf now}^{\\prime }$ .",
"Figure: Examples of watchdog constructs.The translation in Figure REF can be easily extended to the case of the agent ${\\bf do}\\; A \\; {\\bf watching}_{\\phi } \\;c \\; {\\bf else}\\; B$ , which behaves as the previous watchdog and also activates the process $B$ when $A$ is aborted (i.e., when $c$ is entailed and the current state is compatible with the threshold).",
"In the following we will then use also this form of watchdog.",
"The assumption on the instantaneous evaluation of ${\\bf now}_{\\phi } \\, c $ is essential in order to obtain a preemption mechanism which can be expressed in terms of the ${\\bf now}_{\\phi } \\ {\\bf then}\\ {\\bf else}$ primitive.",
"In fact, if the evaluation of ${\\bf now}_{\\phi } \\ c$ took one time-unit, then this unit delay would change the compositional behavior of the agent controlled by the watchdog.",
"Consider, for example, the agent $A= {\\bf tell}(a)\\rightarrow {\\bf tell}(b)$ , which takes two time-units to complete its computation.",
"The agent $A^t= {\\bf now}\\ c \\ {\\bf else}\\ {\\bf tell}(a)\\rightarrow {\\bf now}\\ c \\ {\\bf else}\\ {\\bf tell}(b)$ (resulting from the translation of ${\\bf do}\\; A \\; {\\bf watching}_{\\bar{{\\mathbf {0}}}} \\;c$ ) compositionally behaves as $A$ , unless a $c$ signal is detected and the current state is compatible with the threshold, in which case the evaluation of $A$ is interrupted.",
"On the other hand, if the evaluation of ${\\bf now}\\ c$ took one time-unit, then $A^t$ would take four time-units and would not behave anymore as $A$ when $c$ is not present.",
"In fact, in this case, the agent $A \\parallel B$ would produce $d$ while $A^t \\parallel B$ would not, where $B$ is the agent ${\\bf ask}(\\bar{{\\mathbf {1}}}) \\rightarrow {\\bf now}\\ a \\ {\\bf then}\\ {\\bf tell}(d)\\ {\\bf else}\\ {\\bf success}$ .",
"The valued version of watchdogs can be defined in an analogous way.",
"Figure: Three (weighted) softconstraints; c 3 =c 1 ⊗c 2 c_{3}= c_{1} \\, \\otimes \\,c_{2}, c 2 ⊢c 1 c_{2} \\vdash c_{1}, c 3 ⊢c 1 c_{3} \\vdash c_{1} and c 3 ⊢c 2 c_{3} \\vdash c_{2}.",
"With this small set of idioms, we have now enough expressiveness to describe complex interactions.",
"For the following examples on the new programming idioms, we consider the Weighted semiring $\\langle \\mathbb {R}^{+} \\cup \\lbrace +\\infty \\rbrace ,min,+,+\\infty ,0 \\rangle $ [2], [4] and the (weighted) soft constraints in Figure REF .",
"We first provide simple program examples in order to explain as more details as possible on how a computation of tsccp agents proceeds.",
"In Section REF we show a more complex example describing the classical actions during a negotiation process; the aim of that example is instead to show the expressivity of the tsccp language, without analyzing its execution in detail.",
"Example 2 (Delay) As a first very simple example, suppose to have two agents $A_1, A_2$ of the form: $A_1:: {\\bf tell}(\\bar{{\\mathbf {1}}}) \\stackrel{2}{\\longrightarrow }\\!^{+\\infty } \\,{\\bf tell}(c_{2}) \\rightarrow \\!^{+\\infty } \\,{\\bf success}$ and $A_2:: {\\bf tell}(\\bar{{\\mathbf {1}}}) \\stackrel{1}{\\longrightarrow }\\!^{+\\infty } \\,{\\bf ask}(c_1)\\rightarrow ^{9} \\, {\\bf success}$ ; their concurrent evaluation in the $\\bar{{\\mathbf {1}}} \\equiv \\bar{0}$ empty store is: $ \\langle ({\\bf tell}(\\bar{0}) \\stackrel{2}{\\longrightarrow }\\!^{+\\infty }\\, {\\bf tell}(c_{2}){\\rightarrow }\\!^{+\\infty } \\,{{\\bf success}}) \\parallel ({\\bf tell}(\\bar{0}) \\stackrel{1}{\\longrightarrow }\\!^{+\\infty } \\, {\\bf ask}(c_1)\\rightarrow ^9 \\,{{\\bf success}}) , \\bar{0} \\rangle \\textit {.",
"}$ The timeline for this parallel execution is described in Figure REF .",
"For the evaluation of ${\\bf tell}$ and ${\\bf ask}$ we respectively consider the rules ${\\bf R1}$ and ${\\bf R3}$ in Figure REF , since both transitions are $a$ -valued.",
"However, both these two actions are delayed: three time-units for the ${\\bf tell}(c_2)$ of $A_1$ (including the first ${\\bf tell}(\\bar{0})$ ), and two time-units for the ${\\bf ask}(c_1)$ of $A_2$ (including the first ${\\bf tell}(\\bar{0})$ ).",
"As explained before, this can be obtained by adding $\\bar{{\\mathbf {1}}}$ to the store with a ${\\bf tell}$ action respectively three, and two times.",
"Therefore, the parallel agent $A_1 \\parallel A_2$ corresponds to: $ ({\\bf tell}(\\bar{0} )\\rightarrow \\!^{+\\infty } \\,{\\bf tell}(\\bar{0} )\\rightarrow \\;^{+\\infty }\\,{\\bf tell}(\\bar{0} )\\rightarrow \\!^{+\\infty } \\,{\\bf tell}(c_2)\\rightarrow \\!^{+\\infty }\\,{\\bf success})\\parallel $ $({\\bf tell}(\\bar{0} )\\rightarrow \\!^{+\\infty }\\,{\\bf tell}(\\bar{0} )\\rightarrow \\!^{+\\infty }\\,{\\bf ask}(c_1) \\rightarrow ^9 \\, {\\bf success})\\textit {.",
"}$ This agent is interpreted by using ${\\bf R5}$ -${\\bf R6}$ in Figure REF in terms of maximal parallelism, i.e., all the actions are executed in parallel.",
"The first two ${\\bf tell}$ of $A_1$ and $A_2$ can be simultaneously executed by using rule ${\\bf R1}$ : the precondition $(\\bar{0} \\otimes \\bar{0}) \\Downarrow _\\emptyset = 0 \\lnot < 9$ of the rule is then satisfied.",
"The store does not change since $\\bar{0} \\otimes \\bar{0} = \\bar{0}$ .",
"At this point, the ${\\bf ask}$ action of $A_2$ is not enabled because $\\bar{0} \\lnot \\vdash c_1$ , that is the precondition $\\sigma \\vdash c_1$ of ${\\bf R3}$ is not satisfied.",
"Therefore, the processor can only be allocated to $A_1$ and, since $(\\bar{0} \\otimes \\bar{0}) \\Downarrow _\\emptyset = 0 \\lnot < +\\infty $ is true (i.e.",
"the precondition of ${\\bf R1}$ is satisfied), at $t = 3$ the computation is in the state: $ \\langle {\\bf tell}(c_{2})\\rightarrow \\!^{+\\infty } \\, {\\bf success}\\parallel {\\bf ask}(c_1) \\rightarrow ^{9} \\, {\\bf success}, \\bar{0} \\rangle \\textit {.",
"}$ Now the ${\\bf tell}$ can be executed because $(\\bar{0} \\otimes c_2) \\Downarrow _\\emptyset = 5 \\lnot < +\\infty $ : therefore, the store becomes equal to $\\bar{0} \\otimes c_2 = c_2$ : $ \\langle {\\bf success}\\parallel {\\bf ask}(c_1) \\rightarrow ^{9}\\, {\\bf success}, c_2 \\rangle \\textit {.",
"}$ At $t = 5$ (see Figure REF ) we can successfully terminate the program: in the store $\\sigma = c_2$ the ${\\bf ask}$ is finally enabled at $t=4$ , according to the two preconditions of rule ${\\bf R3}$ , i.e., $c_2 \\vdash c_1$ and $c_2 \\Downarrow _\\emptyset = 5 \\lnot < 9$ : therefore we have $A_1 \\parallel A_2:: \\langle {\\bf success}\\parallel {\\bf success}, c_2 \\rangle \\textit {.",
"}$ Figure: The timeline of the execution of the A 1 ∥A 2 A_1 \\parallel A_2 parallel agent in Example .Example 3 (Timeout) In this second example we evaluate a timeout construct.",
"Suppose we have two agents $A_1$ and $A_2$ of the form: $A_1 :: (({\\bf ask}(c_1)\\rightarrow \\!^{+\\infty } \\,{\\bf success}) \\,+\\, ({\\bf ask}(c_2)\\rightarrow \\!^{+\\infty }\\,{\\bf success})) \\; {\\bf timeout}(1)$ ${\\bf ask}(c_1)\\rightarrow \\!^{+\\infty } \\,{\\bf success}$ and $A_2 :: {\\bf tell}(\\bar{0})\\stackrel{2}{\\longrightarrow }\\!^{+\\infty } \\,{\\bf tell}(c_3) \\rightarrow \\!^{+\\infty } \\,{\\bf success}$ The description of agent $A_1$ is a shortcut for the following agent, as previously explained in the definition of the timeout: ${\\bf now}^{+\\infty } \\ c_1 \\ {\\bf then}\\ B \\ {\\bf else}\\ ({\\bf now}^{+\\infty } \\ c_2 \\ {\\bf then}\\ B \\ {\\bf else}\\ $ $ ({\\bf ask}(\\bar{1}) \\rightarrow \\ {\\bf now}^{+\\infty } \\ c_1 \\ {\\bf then}\\ B \\ {\\bf else}\\ $ $ ({\\bf now}^{+\\infty } \\ c_2 \\ {\\bf then}\\ B \\ {\\bf else}\\ ({\\bf ask}(\\bar{1}) \\rightarrow {\\bf ask}(c_1)\\rightarrow \\!^{+\\infty } \\,{\\bf success}))))\\textit {.",
"}$ where $B::({\\bf ask}(c_1)\\rightarrow \\!^{+\\infty }\\,{\\bf success}+ {\\bf ask}(c_2)\\rightarrow \\!^{+\\infty } \\,{\\bf success})$ .",
"Their concurrent evaluation in the $\\bar{{\\mathbf {1}}} \\equiv \\bar{0}$ empty store is: $\\begin{array}{ll}\\langle (B \\; {\\bf timeout}(1) \\;{\\bf ask}(c_1)\\rightarrow \\!^{+\\infty } \\,{\\bf success}\\parallel \\\\{\\bf tell}(\\bar{0})\\stackrel{2}{\\longrightarrow }\\!^{+\\infty }\\,{\\bf tell}(c_3)\\rightarrow \\!^{+\\infty } \\,{\\bf success}), \\bar{0}\\rangle \\textit {.",
"}\\end{array}$ The timeline for this parallel execution is given in Figure REF .",
"At $t=0$ the store is empty (i.e., $\\sigma = \\bar{0}$ ), thus both constraints $c_1$ and $c_2$ asked by the nondeterministic choice agent $A_1$ are not entailed.",
"In $A_2$ , the ${\\bf tell}$ of $c_3$ , which would entail both $c_1$ and $c_2$ , is delayed by three time-units: in the first three time-units, ${\\bf tell}(\\bar{0} )\\rightarrow ^{+\\infty }$ is executed according to the delay construct, as shown in Example REF .",
"At $t=2$ the timeout is triggered in $A_1$ , since, according to ${\\bf R1}$ , ${\\bf R6}$ and ${\\bf R9}$ (see Figure REF ), the time elapsing in the timeout construct can be executed together with the delay-${\\bf tell}$ actions of $A_2$ .",
"After the timeout triggering, agent $A_1$ is however blocked, since $c_1$ is not entailed by the current empty store, and the precondition of the ${\\bf ask}$ (rule ${\\bf R3}$ ) is not satisfied.",
"$A_2$ can execute the last delay-${\\bf tell}$ , and then perform the ${\\bf tell}(c_3)$ operation at $t=3$ ; the store becomes $\\sigma = \\bar{0} \\otimes c_3 = c_3$ .",
"This finally unblocks $A_1$ at $t=4$ , since, according to the precondition of rule ${\\bf R3}$ , $\\sigma \\sqsubseteq c_1$ (i.e., $c_3 \\sqsubseteq c_1$ ).",
"Finally, at $t=5$ we have $\\langle {\\bf success}\\parallel {\\bf success}, c_3 \\rangle \\textit {.",
"}$ Figure: The timeline of the execution of the A 1 ∥A 2 A_1 \\parallel A_2 agent in Example .Example 4 (Watchdog) In this example let $A_1 :: {\\bf do}\\; ({\\bf tell}(c_1)\\rightarrow \\!^{+\\infty }\\, {\\bf ask}(c_3)\\rightarrow \\!^{+\\infty } \\,{\\bf success})\\; {\\bf watching}^{+\\infty } (c_2) \\;{\\bf else}$ $ \\; (\\,{\\bf tell}(c_3)\\rightarrow \\!^{+\\infty } \\,{\\bf success}\\, )$ and $ A_2:: {\\bf tell}(c_2)\\rightarrow ^{+\\infty } {\\bf success}\\textit {.",
"}$ We evaluate the following watchguard construct with two agents $A_1$ and $A_2$ in parallel: $ \\langle ( {\\bf do}\\; ({\\bf tell}(c_1)\\rightarrow \\!^{+\\infty }\\, {\\bf ask}(c_3)\\rightarrow \\!^{+\\infty } \\,{\\bf success})\\; {\\bf watching}^{+\\infty } (c_2) \\;{\\bf else}$ $ \\; (\\,{\\bf tell}(c_3)\\rightarrow \\!^{+\\infty } \\,{\\bf success}\\, ) \\parallel {\\bf tell}(c_2)\\rightarrow ^{+\\infty } {\\bf success}), \\bar{0} \\rangle \\textit {.",
"}$ According to Figure REF , agent $A_1$ is translated in the following way, where the agent $B$ is a shorthand for the “else” branch of the watchdog, that is ${\\bf tell}(c_3)\\rightarrow ^{+\\infty } {\\bf success}$ : ${\\bf now}^{+\\infty } \\ c_2 \\ {\\bf then}\\ B \\ {\\bf else}\\ ({\\bf tell}(c_1)\\rightarrow ^{+\\infty } {\\bf now}^{+\\infty } \\ c_2 \\ {\\bf then}\\ B \\ {\\bf else}\\ \\;$ $ ({\\bf ask}(c_3)\\rightarrow ^{+\\infty } \\, {\\bf now}^{+\\infty } \\ c_2 \\ {\\bf then}\\ B \\ {\\bf else}\\ {\\bf success})) \\textit {.}",
"$ The execution timeline for this parallel agent is shown in Figure REF .",
"In the first time-unit we have that $\\sigma = \\bar{0} \\lnot \\sqsubseteq c_2$ , i.e., the store does not imply the guard of the ${\\bf now}^{+\\infty }$ , and therefore the interruption of the watchguard in $A_1$ is not triggered yet.",
"Thus, in the first time-unit, both ${\\bf tell}(c_1)\\rightarrow ^{+\\infty }$ of agent $A_1$ and ${\\bf tell}(c_2)\\rightarrow ^{+\\infty }$ of agent $A_2$ are executed.",
"At time $t=1$ , the interruption of the watchguard is immediately activated (i.e.",
"${\\bf now}^{+\\infty } c_2$ ), since the store is now equal to $c_1 \\otimes c_2= c_3$ and $c_3 \\vdash c_2$ (rule ${\\bf R8}$ in Figure REF ).",
"Therefore, ${\\bf tell}(c_3)\\rightarrow ^{+\\infty }$ of agent $B$ in $A_1$ is executed, while $A_2$ already corresponds to the ${\\bf success}$ agent).",
"Figure: The timeline of the execution of the A 1 ∥A 2 A_1 \\parallel A_2 parallel agent in Example ."
],
[
"An Auction Example",
"In Figure REF we model the negotiation and the management of a generic service offered with a sort of auction: auctions, as other forms of negotiation, naturally need both timed and quantitative means to describe the interactions among agents.",
"We reckon that an auction provides one of the most suitable example where to show the expressivity of the tsccp language, since both time and preference (for a service or an object) are considered.",
"In the following of the description we consider a buyout auction [16], where the auctioneer improves the service and the related consumed resources (or, alternatively, its money price), bid after bid.",
"When one (ore more) of the bidders agrees with the offer, it bids for it and the auction is immediately declared as over.",
"The auctioneer (i.e.",
"$AUCTIONEER$ in Figure REF ) begins by offering a service described with the soft constraint $c_{A_1}$ .",
"We suppose that the cost associated to the soft constraint is expressed in terms of computational capabilities needed to support the execution of the service: e.g., $c_{i} \\sqsubseteq c_{j}$ means that the service described by $c_{i}$ needs more computational resources than $c_{j}$ .",
"By choosing the proper semiring, this load can be expressed as a percentage of the CPU use, or in terms of money, for example; we left this preference generic in the example, since we focus on the interaction among the agents.",
"We suppose that a constraint can be defined over three domains of QoS features: availability, reliability and execution time.",
"For instance, $c_{A_1}$ is defined as $\\mathit {availability}> 95\\% \\wedge \\mathit {reliability}> 99\\% \\wedge \\mathit {execution} \\; \\mathit {time} < 3\\mathit {sec}$ .",
"Clearly, providing a higher availability or reliability, and a lower execution time implies raising the computational resources to support this improvement, thus worsening the preference of the store.",
"Figure: An “auction and management” example for a genericserviceAfter the offer, the auctioneer gives time to the bidders (each of them described with a possibly different agent $BIDDER_i$ in Figure REF ) to make their offer, since the choice of the winner is delayed by $t_{sell}$ time-units (as in many real-world auction schemes).",
"A level $a_A$ is used to effectively check that the global consistency of the store is enough good, i.e., the computational power would not be already consumed under the given threshold.",
"After the winner is nondeterministically chosen among all the bidders asking for the service, the auctioneer becomes a supervisor of the used resource by executing the agent $CHECK$ .",
"Otherwise, if no offer is received within $w_{A}$ time-units, a timeout interrupts the wait and the auctioneer improves the offered service by adding a new constraint: for example, in ${\\bf tell}(c_{A_2})$ , $c_{A_2}$ could be equivalent to $execution \\; time < 1sec$ , thus reducing the latency of the service (from 3 to 1 second) and consequently raising, at the same time, its computational cost (i.e., $\\sigma = c_{A_1} \\otimes c_{A_2}\\sqsubseteq c_{A_1}$ means that we worsen the consistency level of the store).",
"The same offer/wait process is repeated three times in Figure REF .",
"Each of the bidders in Figure REF executes its own task (i.e., $TASK_i$ , left generic since not in the scope of the example), but as soon as the offered resource meets its demand (i.e.",
"$c_{B_i}$ is satisfied by the store: $\\sigma \\sqsubseteq c_{B_i}$ ), the bidder is interrupted and then asks to use the service.",
"The time needed to react and make an offer is modeled with $t_{buy_i}$ : fast bidders will have more chances to win the auction, if their request arrives before the choice of the auctioneer.",
"If one of the bidders wins, then it becomes a user of the resource, by executing $USER_i$ .",
"The agent $USER_i$ uses the service (through the agent$USE\\_SERVICE_i$ , left generic in Figure REF ), but it stops (using agent $STOP_i$ , left generic in Figure REF ) as soon as the service is interrupted, i.e., as the store satisfies ${\\it service=interrupt}$ .",
"On the other side, agent $CHECK$ waits for the use termination, but it interrupts the user if the computation takes too long (more than $w_{C}$ time-units), or if the user absorbs the computational capabilities beyond a given threshold, i.e.",
"as soon as the $c_{check}$ becomes implied by the store (i.e.",
"$\\sigma \\sqsubseteq c_{check}$ ): in fact, $USE\\_SERVICE_i$ could be allowed to ask for more power by “telling” some more constraints to the store.",
"To interrupt the service use, agent $CHECK$ performs a ${\\bf tell}({\\it service=interrupt})$ .",
"All the agents $INIT$ , left generic in Figure REF , can be used to initialize the computation.",
"In order to avoid a heavy notation in Figure REF , we do not show the preference associated to constraints and the consistency check label on the transition arrows, when they are not significative for the example description.",
"Also the $\\phi _{\\mathit {Check}}$ , $\\phi _{\\mathit {Bidder}}$ and $\\phi _{\\mathit {User}}$ thresholds of the watchguard constructs are not detailed.",
"Finally, in the following we model a more refined behaviour of the auctioneer, which accepts the bidding with the highest value, where $\\underline{CHECK}$ , $\\underline{BIDDER_i}$ and $\\underline{USER_i}$ are defined as in Figure REF .",
"Figure: A new “auction and management” example for a genericserviceMany other real-life automated tasks can be modeled with the tsccp language.",
"For example, a quality-driven composition of web services: the agents that represent different web services can add to the store their functionalities (represented by soft constraints) with ${\\bf tell}$ actions; the final store models their composition.",
"The consistency level of the store represents (for example) the total monetary cost of the obtained service, or a value representing the consistency of the integrated functionalities.",
"The reason is that, when we compose the services offered by different providers, we cannot be sure of how much they are compatible.",
"A client wishing to use the composed service can perform an ${\\bf ask}$ with a threshold such that it prevents the client from paying a high price, or having an unreliable service.",
"Softness is also useful to model incomplete service specifications that may evolve incrementally and, in general, for non-functional aspects."
],
[
"The Denotational Model",
"In this section we define a denotational characterization of the operational semantics obtained by following the construction in [10], and by using timed reactive sequences to represent tsccp computations.",
"These sequences are similar to those used in the semantics of dataflow languages [19], imperative languages [8] and (timed) ccp [12], [10].",
"The denotational model associates with a process a set of timed reactive sequences of the form $\\langle \\sigma _1,\\gamma _1\\rangle \\cdots \\langle \\sigma _n,\\gamma _n\\rangle \\langle \\sigma ,\\sigma \\rangle $ where a pair of constraints $\\langle \\sigma _i,\\gamma _i\\rangle $ represents a reaction of the given process at time $i$ : intuitively, the process transforms the global store from $\\sigma _i$ to $\\gamma _i$ or, in other words, $\\sigma _i$ is the assumption on the external environment while $\\gamma _i$ is the contribution of the process itself (which always entails the assumption).",
"The last pair denotes a “stuttering step” in which the agent ${\\bf Success}$ has been reached.",
"Since the basic actions of tsccp are monotonic and we can also model a new input of the external environment by a corresponding tell operation, it is natural to assume that reactive sequences are monotonic.",
"Thus, in the following we assume that each timed reactive sequence $\\langle \\sigma _1,\\gamma _1\\rangle \\cdots \\langle \\sigma _{n-1},\\gamma _{n-1}\\rangle \\langle \\sigma _n,\\sigma _n\\rangle $ satisfies the conditions $\\gamma _i \\vdash \\sigma _i\\hbox{ and } \\ \\sigma _j \\vdash \\gamma _{j-1}, $ for any $i\\in [1,n-1]$ and $j\\in [2,n]$ .",
"The set of all reactive sequences is denoted by ${\\cal S}$ , its typical elements by $s,s_1\\ldots $ , while sets of reactive sequences are denoted by $S,S_1\\ldots $ , and $\\varepsilon $ indicates the empty reactive sequence.",
"Furthermore, the symbol $\\cdot $ denotes the operator that concatenates sequences.",
"In the following, $Process$ denotes the set of tsccp processes.",
"Operationally, the reactive sequences of an agent are generated as follows.",
"Definition 3 (Processes Semantics) We define the semantics ${\\cal R}\\in {\\it Process}\\rightarrow {\\cal P}({\\cal S})$ by $\\begin{array}[t]{ll}{\\cal R}({\\it F\\text{.",
"}A})= & \\lbrace \\langle \\sigma ,\\sigma ^{\\prime }\\rangle \\cdot w \\in {\\cal S} \\mid \\langle {\\it A}, \\sigma \\rangle \\longrightarrow \\langle {\\it B}, \\sigma ^{\\prime } \\rangle \\hbox{ and } w\\in {\\cal R}(\\it F\\text{.}",
"B) \\rbrace \\\\& \\cup \\\\& \\lbrace \\langle \\sigma ,\\sigma \\rangle \\cdot w \\in {\\cal S} \\mid \\langle {\\it A},\\sigma \\rangle \\lnot \\longrightarrow \\hbox{ and }\\\\& \\hspace*{68.28644pt}\\begin{array}[t]{ll} \\mbox{either } {\\it A}\\ne {\\bf Success} \\hbox{ and } w\\in {\\cal R}(F\\text{.",
"}A) \\\\\\mbox{or }{\\it A}={\\bf Success} \\hbox{ and } w\\in {\\cal R}(F\\text{.}",
"A) \\cup \\lbrace \\varepsilon \\rbrace \\rbrace \\textit {.",
"}\\end{array}\\end{array}$ Formally ${\\cal R}$ is defined as the least fixed-point of the operator $\\Phi \\in ({\\it Process}\\rightarrow {\\cal P}({\\cal S})) \\rightarrow {\\it Process}\\rightarrow {\\cal P}({\\cal S})$ defined by [t]lll (I)(F. A)= {,w S A, B, and wI(F. B) } { ,w S A, and [t]ll either ASuccess and wI(F.A) or A=Success and wI(F. A) {}}.",
"The ordering on ${\\it Process}\\rightarrow {\\cal P}({\\cal S})$ is that of (point-wise extended) set-inclusion, and since it is straightforward to check that $\\Phi $ is continuous, standard results ensure that the least fixpoint exists (and it is equal to $\\sqcup _{n\\ge 0} \\Phi ^n(\\bot )$ ).",
"Note that ${\\cal R}(F\\text{.",
"}A)$ is the union of the set of all successful reactive sequences that start with a reaction of $A$ , and the set of all successful reactive sequences that start with a stuttering step of $A$ .",
"In fact, when an agent is blocked, i.e., it cannot react to the input of the environment, a stuttering step is generated.",
"After such a stuttering step, the computation can either continue with the further evaluation of ${\\it A}$ (possibly generating more stuttering steps), or it can terminate if ${\\it A}$ is the ${\\bf Success}$ agent.",
"Note also that, since the ${\\bf Success}$ agent used in the transition system cannot make any move, an arbitrary (finite) sequence of stuttering steps is always appended to each reactive sequence."
],
[
"Correctness",
"The observables ${\\cal O}_{io}^{mp}(P)$ describing the input/output pairs of successful computations can be obtained from ${\\cal R}(P)$ by considering suitable sequences, namely those sequences which do not perform assumptions on the store.",
"In fact, note that some reactive sequences do not correspond to real computations: Clearly, when considering a real computation no further contribution from the environment is possible.",
"This means that, at each step, the assumption on the current store must be equal to the store produced by the previous step.",
"In other words, for any two consecutive steps $\\langle \\sigma _i,\\sigma ^{\\prime }_i\\rangle \\langle \\sigma _{i+1},\\sigma ^{\\prime }_{i+1}\\rangle $ we must have $\\sigma ^{\\prime }_i =\\sigma _{i+1}$ .",
"Thus, we are led to the following.",
"Definition 4 (Connected Sequences) Let $s=\\langle \\sigma _1,\\sigma ^{\\prime }_1\\rangle \\langle \\sigma _2,\\sigma ^{\\prime }_2\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle $ be a reactive sequence.",
"We say that $s$ is connected if $\\sigma _1= {\\mathbf {1}}$ and $\\sigma _i = \\sigma ^{\\prime }_{i-1}$ for each $i$ , $2\\le i\\le n$ .",
"According to the previous definition, a sequence is connected if all the information assumed on the store is produced by the process itself.",
"To be defined as connected, a sequence must also have ${\\mathbf {1}}$ as the initial constraint.",
"A connected sequence $s=\\langle {\\mathbf {1}},\\sigma _1\\rangle \\langle \\sigma _1,\\sigma _2\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle $ represents a tsccp computation of a process $F\\text{.",
"}A$ , where ${\\mathbf {1}}$ is the input constraint and $\\sigma _n \\Downarrow _{Fv(A)}$ is the result.",
"From the above discussion we can derive the following property: [Correctness] For any process $P= F\\text{.",
"}A$ we have ${\\cal O}_{io}^{mp}({ P}) = \\lbrace \\sigma _n \\Downarrow _{Fv(A)}\\mid \\begin{array}[t]{ll} \\mbox{\\rm there exists a connected sequence } s\\in {\\cal R}(P) \\ \\mbox{\\rm such that }\\\\s=\\langle {\\mathbf {1}},\\sigma _1\\rangle \\langle \\sigma _1, \\sigma _2\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle \\rbrace \\textit {.",
"}\\end{array}$ From the close correspondence between the rules of the transition system and the definition of the denotational semantics, we have that $s \\in {\\cal R}(P)$ if and only if $s=\\langle \\sigma _1,\\sigma _1^{\\prime }\\rangle \\langle \\sigma _2,\\sigma _2^{\\prime }\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle $ , $A_1=A$ , $A_n ={\\bf Success}$ and for $i\\in [1,n-1]$ , either $\\langle {A_i},\\sigma _i\\rangle \\longrightarrow \\langle {A_{i+1}},\\sigma ^{\\prime }_i\\rangle $ or $\\langle {A_i},\\sigma _i\\rangle \\lnot \\longrightarrow $ , $A_{i+1}= A_i$ and $\\sigma ^{\\prime }_i= \\sigma _i$ .",
"Then there exists a connected sequence $s\\in {\\cal R}(P)$ if and only if $s=\\langle \\sigma _1,\\sigma _2\\rangle \\langle \\sigma _2,\\sigma _3\\rangle $ $\\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle $ , $A_1=A$ , $\\sigma _1= {\\mathbf {1}}$ , $A_n ={\\bf Success}$ and for $i\\in [1,n-1]$ , $\\langle {A_i},\\sigma _i\\rangle \\longrightarrow \\langle {A_{i+1}},\\sigma _{i+1}\\rangle $ .",
"Therefore, the proof follows by definition of ${\\cal O}_{io}^{mp}(P)$ ."
],
[
"Compositionality of the Denotational Semantics for ",
"In order to prove the compositionality of the denotational semantics, we now introduce a semantics $[\\![F\\text{.",
"}A ]\\!",
"](e)$ , which is compositional by definition and where, for technical reasons, we explicitly represent the environment $e$ that associates a denotation to each procedure identifier.",
"More precisely, assuming that Pvar denotes the set of procedure identifiers, ${\\it Env}={\\it Pvar}\\rightarrow {\\cal P}({\\cal S})$ , with typical element $e$ , is the set of environments.",
"Given $e \\in Env$ , $p \\in {\\it Pvar}$ and $f \\in {\\cal P}({\\cal S})$ , we denote by $e^{\\prime }=e\\lbrace f/p\\rbrace $ the new environment such that $e^{\\prime }(p)=f$ and $e^{\\prime }(p^{\\prime })= e(p^{\\prime })$ for each procedure identifier $p^{\\prime }\\ne p$ .",
"Given a process $F\\text{.",
"}A$ , the denotational semantics $[\\![F\\text{.",
"}A ]\\!",
"]:{\\it Env} \\rightarrow {\\cal P}({\\cal S})$ is defined by the equations in Figure REF , where $\\mu $ denotes the least fixpoint with respect to the subset inclusion of elements of ${\\cal P}({\\cal S})$ .",
"The semantic operators appearing in Figure REF are formally defined as follows; intuitively they reflect the operational behavior of their syntactic counterparts in terms of reactive sequences.In Figure REF the syntactic operator $\\rightarrow _i$ is either of the form $\\rightarrow ^{a_i}$ or $\\rightarrow _{\\phi _i}$ .",
"We first need the following definition.",
"Definition 5 Let $\\sigma , \\phi $ and $c$ be constraints in ${\\cal C}$ and let $a \\in \\mathcal {A}$ .",
"We say that $\\sigma \\succ ^{a}\\, c, $ if $ (\\sigma \\vdash c $ and $\\sigma \\Downarrow _{\\emptyset } \\lnot < a) \\ \\ \\ \\hbox{while } \\ \\ \\ \\sigma \\succ _{\\phi }\\, c, $ if $ (\\sigma \\vdash c$ and $\\sigma \\lnot \\sqsubset \\phi )$ .",
"Definition 6 (Semantic operators) Let $S,S_i$ be sets of reactive sequences, $c,c_i$ be constraints and let $\\succ _i$ be either of the form $\\succ ^{a_i}$ or $\\succ _{\\phi _i}$ .",
"Then we define the operators $\\tilde{tell}$ , $\\tilde{\\sum }$ , $\\tilde{\\parallel }$ , $\\tilde{now}$ and $\\tilde{\\exists } x$ as follows: The (valued) tell operator $\\begin{array}{ll}\\tilde{tell}^a(c, S)= &\\begin{array}[t]{l}\\lbrace s \\in {\\cal S} \\mid s = \\langle \\sigma ,\\sigma \\otimes c\\rangle \\cdot s^{\\prime }, \\ \\sigma \\otimes c\\Downarrow _{\\emptyset } \\lnot <a \\mbox{ and} s^{\\prime } \\in S\\ \\rbrace \\textit {.",
"}\\end{array} \\\\\\end{array}$ $\\begin{array}{ll}\\tilde{tell}_{\\phi }(c, S)= &\\begin{array}[t]{l}\\lbrace s \\in {\\cal S} \\mid s = \\langle \\sigma ,\\sigma \\otimes c\\rangle \\cdot s^{\\prime }, \\ \\sigma \\otimes c \\lnot \\sqsubset \\phi \\mbox{and } s^{\\prime } \\in S\\ \\rbrace \\textit {.",
"}\\end{array} \\\\\\end{array}$ The guarded choice ${\\tilde{\\sum }} _{i=1}^n c_i \\succ _i \\, S_i =\\begin{array}[t]{l}\\lbrace s\\cdot s^{\\prime } \\in {\\cal S} \\mid \\begin{array}[t]{l}s = \\langle \\sigma _1 ,\\sigma _1\\rangle \\cdots \\langle \\sigma _m,\\sigma _m\\rangle ,\\sigma _j {\\lnot \\succ _i }\\,c_i\\\\\\mbox{for each } j\\in [1,m-1], i\\in [1,n],\\\\\\sigma _m \\succ _h\\, c_h \\hbox{ and } s^{\\prime }\\in S_h \\hbox{for an } h\\in [1,n] \\ \\rbrace \\textit {.",
"}\\end{array}\\end{array}$ The parallel composition Let $\\tilde{\\parallel }\\in {\\cal S}\\times {\\cal S}\\rightarrow {\\cal S}$ be the (commutative and associative) partial operator defined as follows: $\\begin{array}{ll}\\langle \\sigma _1,\\sigma _1 \\otimes \\gamma _1\\rangle \\cdots \\langle \\sigma _n,\\sigma _n \\otimes \\gamma _n\\rangle \\langle \\sigma ,\\sigma \\rangle \\ \\tilde{\\parallel }\\ \\langle \\sigma _1,\\sigma _1 \\otimes \\delta _1\\rangle \\cdots \\langle \\sigma _n,\\sigma _n \\otimes \\delta _n\\rangle \\langle \\sigma ,\\sigma \\rangle & =\\\\\\langle \\sigma _1, \\sigma _1 \\otimes \\gamma _1\\otimes \\delta _1\\rangle \\cdots \\langle \\sigma _n, \\sigma _n \\otimes \\gamma _n\\otimes \\delta _n \\rangle \\langle \\sigma ,\\sigma \\rangle \\textit {.",
"}\\end{array}$ We define $S_1\\tilde{\\parallel } S_2$ as the point-wise extension of the above operator to sets.",
"The (valued) now operator $\\begin{array}{ll}\\tilde{now}^a(c, S_{1} , S_{2})=\\lbrace s \\in {\\cal S} \\mid & s = \\langle \\sigma ,\\sigma ^{\\prime } \\rangle \\cdot s^{\\prime }, \\, \\sigma \\Downarrow _{\\emptyset } \\lnot <a \\hbox{ and }\\\\&\\hbox{either }\\sigma \\vdash c \\mbox{ and } s \\in S_{1}\\\\& \\hbox{or } \\sigma \\lnot \\vdash c \\mbox{ and } s \\in S_{2}\\ \\rbrace \\textit {.}",
"\\\\\\end{array}$ $\\begin{array}{ll}\\tilde{now}_{\\phi } (c, S_{1} , S_{2})=\\lbrace s \\in {\\cal S} \\mid & s = \\langle \\sigma ,\\sigma ^{\\prime } \\rangle \\cdot s^{\\prime }, \\sigma \\lnot \\sqsubset \\phi \\hbox{ and }\\\\&\\hbox{either }\\sigma \\vdash c \\mbox{ and } s \\in S_{1}\\\\& \\hbox{or } \\sigma \\lnot \\vdash c \\mbox{ and } s \\in S_{2}\\ \\rbrace \\textit {.}",
"\\\\\\end{array}$ The hiding operator The semantic hiding operator can be defined as follows: $\\begin{array}{ll}{\\bf \\tilde{ \\exists } } x S = \\lbrace s \\in {\\cal S} \\mid & \\mbox{\\rm there exists $ s^{\\prime } \\in S$ such that $s= s^{\\prime }[x/y]$ with $y$ new } \\rbrace \\end{array}$ where $s^{\\prime }[x/y]$ denotes the sequence obtained from $s^{\\prime }$ by replacing the variable $x$ for the variable $y$ , which we assume to be new.To be more precise, we assume that each time that we consider a new application of the operator ${\\bf \\tilde{\\exists } }$ we use a new, different $y$ .",
"As in the case of the operational semantics, this can be ensured by a suitable renaming mechanism.",
"Obviously, the semantic (valued) tell operator reflects the operational behavior of the syntactic (valued) tell.",
"Concerning the semantic choice operator, a sequence in $\\tilde{\\sum } _{i=1}^n c_i\\succ _i \\,S_i$ consists of an initial period of waiting for a store which satisfies one of the guards.",
"During this waiting period, only the environment is active by producing the constraints $\\sigma _j$ , while the process itself generates the stuttering steps $\\langle \\sigma _j,\\sigma _j\\rangle $ .",
"When the store is strong enough to satisfy a guard, that is to entail a $c_h$ and to satisfy the condition on the cut level, the resulting sequence is obtained by adding $s^{\\prime }\\in S_h$ to the initial waiting period.",
"In the semantic parallel operator defined on sequences, we require that the two arguments of the operator agree at each point of time with respect to the contribution of the environment (the $\\sigma _i$ 's), and that they have the same length (in all other cases the parallel composition is assumed being undefined).",
"If $F\\text{.",
"}A$ is a closed process, that is if all the procedure names occurring in $A$ are defined in $F$ , then $[\\![F\\text{.}A]\\!",
"](e)$ does not depend on $e$ , and it will be indicated as $[\\![F\\text{.}A]\\!",
"]$ .",
"Environments in general allow us to define the semantics also of processes that are not closed.",
"The following result shows the correspondence between the two semantics we have introduced and, therefore, it proves the compositionality of ${\\cal R}(F\\text{.",
"}A)$ .",
"From the above discussion we can derive the following property: [Compositionality] If $F\\text{.",
"}A$ is closed then ${\\cal R}(F\\text{.",
"}A) = [\\![F\\text{.",
"}A ]\\!",
"]$ holds.",
"We prove by induction on the complexity of the agent $A$ that $\\begin{array}{ll}[\\![F\\text{.",
"}A ]\\!",
"]=\\lbrace s \\mid & s=\\langle \\sigma _1,\\sigma _1^{\\prime }\\rangle \\langle \\sigma _2,\\sigma _2^{\\prime }\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle , \\\\& A_1=A, A_n ={\\bf Success} \\mbox{ and for } i\\in [1,n-1],\\\\& \\mbox{either } \\langle {A_i},\\sigma _i\\rangle \\longrightarrow \\langle {A_{i+1}},\\sigma ^{\\prime }_i\\rangle \\\\& \\mbox{or } \\langle {A_i},\\sigma _i\\rangle \\lnot \\longrightarrow , \\,A_{i+1}=A_i,\\, \\sigma ^{\\prime }_i= \\sigma _i \\rbrace \\textit {.",
"}\\end{array} $ Then the proof follows by definition of ${\\cal R}(P)$ .",
"When the $P$ is not of the form $F\\text{.",
"}B \\parallel C$ the thesis follows immediately from the close correspondence between the rules of the transition system and the definition of the denotational semantics.",
"Assume now that $P$ is of the form $F\\text{.",
"}B \\parallel C$ .",
"By definition of the denotational semantics, $s \\in [\\![F\\text{.",
"}A ]\\!",
"]$ if and only if $s=\\langle \\sigma _1,\\sigma _1^{\\prime }\\rangle \\langle \\sigma _2,\\sigma _2^{\\prime }\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n}\\rangle $ and there exist $s^{\\prime } \\in [\\![F\\text{.}B]\\!",
"]$ and $s^{\\prime \\prime } \\in [\\![F\\text{.}C]\\!",
"]$ , $\\begin{array}{ll}s^{\\prime }= \\langle \\sigma _1,\\sigma _1 \\otimes \\gamma _1\\rangle \\langle \\sigma _2,\\sigma _2 \\otimes \\gamma _2\\rangle \\cdots \\langle \\sigma _n,\\sigma _n \\rangle \\\\s^{\\prime \\prime }= \\langle \\sigma _1,\\sigma _1 \\otimes \\delta _1\\rangle \\langle \\sigma _2,\\sigma _2 \\otimes \\delta _2\\rangle \\cdots \\langle \\sigma _n,\\sigma _n\\rangle \\end{array}$ such that for each $i\\in [1,n-1]$ , $\\sigma _i^{\\prime } = \\sigma _i \\otimes \\gamma _i \\otimes \\delta _i$ .",
"By inductive hypothesis $s^{\\prime } \\in [\\![F\\text{.}B]\\!",
"]$ and $s^{\\prime \\prime } \\in [\\![F\\text{.}C]\\!",
"]$ if and only if for $i\\in [1,n-1],$ $\\begin{array}{ll}\\mbox{either } \\langle {B_i},\\sigma _i\\rangle \\longrightarrow \\langle {B_{i+1}},\\sigma _i \\otimes \\gamma _i \\rangle ,\\\\\\mbox{ or } \\langle {B_i},\\sigma _i\\rangle \\lnot \\longrightarrow , \\,B_{i+1}=B_i,\\, \\sigma _i \\otimes \\gamma _i = \\sigma _i & \\mbox{and} \\\\\\mbox{either } \\langle {C_i},\\sigma _i\\rangle \\longrightarrow \\langle {{C_{i+1}}},\\sigma _i \\otimes \\delta _i\\rangle , \\\\\\mbox{ or } \\langle {C_i},\\sigma _i\\rangle \\lnot \\longrightarrow , \\,C_{i+1}=C_i,\\, \\sigma _i \\otimes \\delta _i = \\sigma _i \\text{.}",
"\\\\\\end{array}$ $B_1=B$ , $B_n={\\bf Success}$ , $C_1=C$ and $C_n={\\bf Success}$ .",
"Therefore, by Rule R8 and previous observations, we have that (REF ) holds if and only if $B_1\\parallel C_1 =B\\parallel C$ , $B_n\\parallel C_n ={\\bf Success}$ and for $i\\in [1,n-1]$ , $\\begin{array}{ll}\\mbox{either } \\langle B_i \\parallel C_i,\\sigma _i\\rangle \\longrightarrow \\langle B_{i+1} \\parallel C_{i+1},\\sigma _i^{\\prime }\\rangle & \\\\\\mbox{or } \\langle B_i \\parallel C_i,\\sigma _i\\rangle \\lnot \\longrightarrow , \\,A_{i+1}\\parallel B_{i+1}=A_i\\parallel B_{i},\\, \\sigma ^{\\prime }_i= \\sigma _i\\end{array}$ and then the thesis.",
"Figure: The semantics [[F.A]][\\![F\\text{.",
"}A ]\\!",
"](e)."
],
[
"An Interleaving Approach for non-Time-elapsing Actions",
"In this section, we show a different version of the tsccp language: while in tsccp the parallel operator is modeled in terms of maximal parallelism, the same operator can be treated also in terms of interleaving.",
"According to maximal parallelism, at each moment every enabled agent of the system is activated, while in the second paradigm an agent could not be assigned to a “free” processor.",
"Clearly, since we have dynamic process creation, a maximal parallelism approach has the disadvantage that, in general, it implies the existence of an unbound number of processes.",
"On the other hand a naive interleaving semantic could be problematic from the time viewpoint, as in principle the time does not pass for enabled agent which are not scheduled.",
"For the semantics in this section we follow a solution analogous to that one adopted in [11]: we assume that the parallel operator is interpreted in terms of interleaving, as usual, however we must assume maximal parallelism for actions depending on time.",
"In other words, time passes for all the parallel processes involved in a computation.",
"To summarize, in this section we adopt maximal parallelism for time elapsing (i.e.",
"for timeout constructs) and an interleaving model for basic computation steps (i.e.",
"(valued) ask and (valued) tell actions).",
"To distinguish this new approach, we named the resulting language as tsccp-i, i.e., tsccp with interleaving.",
"Time-outs are modeled in tsccp-i by the construct $\\hbox{\\bf askp}_t(c)?_{\\phi } A \\mathit {:} B$ which replaces the ${\\bf now}_{\\phi }\\ c \\ {\\bf then}\\ A \\ {\\bf else}\\ B$ construct of tsccp and directly has time $t$ as one of its parameters, differently from the ${\\bf now}_{\\phi }$ agent.",
"The $\\hbox{\\bf askp}_t$ agent can be interpreted as follows: one is allowed to wait $t$ time-units for the entailment of the constraint $c$ by the store and the subsequent evaluation of the process $A$ ; if this time limit is exceeded, then the process $B$ is evaluated.",
"Analogously for the construct $\\hbox{\\bf askp}_t(c)?^a A \\mathit {:} B$ .",
"Definition 7 (tsccp-i) Given a soft constraint system $\\langle S,D,V\\rangle $ , the corresponding structure ${\\mathcal {C}}$ , any semiring value $a$ , soft constraints $\\phi , c \\in {\\cal C}$ and any tuple of variables $x$ , the syntax of the tsccp-i language is given by the following grammar: $\\begin{array}{ll}P ::= & F \\text{.}",
"A\\\\F ::= & p(x):: A \\;|\\; F.F\\\\A ::= & {\\bf success} \\;|\\; \\hbox{{\\bf tell}}(c)\\rightarrow _\\phi A\\;|\\;\\hbox{{\\bf tell}}(c)\\rightarrow ^a A \\;|\\; E \\;|\\; A\\parallel A\\;|\\;\\exists x A \\;|\\; p(x) \\;|\\; \\\\& \\Sigma _{i=1}^{n} E_i\\;|\\; \\hbox{\\bf askp}_t(c)?_{\\phi } A \\mathit {:} A \\;|\\;\\hbox{\\bf askp}_t(c)?^a A \\mathit {:} A \\\\E ::= &\\hbox{\\bf ask}(c) \\rightarrow _\\phi A \\;|\\; \\hbox{\\bf ask}(c)\\rightarrow ^a A \\\\\\end{array}$ where, as in Definition REF , $P$ is the class of processes, $F$ is the class of sequences of procedure declarations (or clauses), $A$ is the class of agents.",
"As before, in a tsccp-i process $P=F\\text{.",
"}A$ , $A$ is the initial agent, to be executed in the context of the set of declarations $F$ .",
"Analogously to tsccp processes, in order to simplify the notation, in the following we will usually write a tsccp-i process $P=F\\text{.",
"}A$ simply as the corresponding agent $A$ .",
"The operational model of tsccp-i processes can be formally described by a labeled transition system $T= ({\\it Conf}, Label, \\longmapsto )$ , where we assume that each transition step exactly takes one time-unit.",
"Configurations (in) Conf are pairs consisting of a process and a constraint in ${\\cal C}$ representing the common store.",
"$\\mathcal {L} = \\lbrace \\tau ,\\omega \\rbrace $ is the set of labels.",
"We use labels to distinguish “real” computational steps performed by processes which have the control (label $\\omega $ ) from the transitions which model only the passing of time (label $\\tau $ ).",
"So $\\omega $ -actions are those performed by processes that modify the store ($\\bf {tell}$ ), perform a check on the store (${\\bf ask}$ , ${\\bf askp}_t$ ), correspond to exceeding a time-out (${\\bf askp}_0$ ), or perform a choice ($\\Sigma _{i=1}^{n} E_i$ ).",
"On the other hand, $\\tau $ -actions are those performed by time-out processes (${\\bf askp}_t$ ) in case they have not the control.",
"In Figure REF we show the semantics of all the tsccp-i actions, but in the following we describe only the actions whose semantics is different from that one presented in Figure REF (i.e., for tsccp), that is we describe in detail the parallelism and the ${\\bf askp}_t$ agent.",
"The semantics of the other actions of tsccp-i is the same as for tsccp, except for the fact that their transition is labeled with $\\omega $ .",
"Figure: The transition system for tsccp-i.Parallelism Rules Q5 and Q6 in Figure REF model the parallel composition operator in terms of interleaving, since only one basic $\\omega $ -action is allowed for each transition (i.e.",
"for each unit of time).",
"This means that the access to the shared store is granted to one process a time.",
"However, time passes for all the processes appearing in the $\\parallel $ context at the external level, as shown by rule Q5, since $\\tau $ -actions are allowed together with a $\\omega $ -action.",
"On the other hand, a parallel component is allowed to proceed in isolation if (and only if) the other parallel component cannot perform a $\\tau $ -action (rule Q6).",
"To summarize, we adopt maximal parallelism for time elapsing (i.e.",
"$\\tau $ -actions) and an interleaving model for basic computation steps (i.e.",
"$\\omega $ -actions).",
"We have adopted this approach because it seems more adequate to the nature of time-out operators not to interrupt the elapsing of time, once the evaluation of a time-out has started.",
"Clearly one could start the elapsing of time when the time out process is scheduled, rather than when it appears in the top-level current parallel context.",
"This modification could easily be obtained by adding a syntactic construct to differentiate active timeouts from inactive ones, and by accordingly changing the transition system.",
"One could also easily modify the semantics (both operational and denotational) to consider a more liberal assumption which allows multiple ask actions in parallel.",
"Valued-Askp$_t$ The rules ${\\bf Q10}$ -${\\bf Q14}$ in Figure REF show that the time-out process $\\hbox{\\bf askp}_t(c)?^a A \\mathit {:} B$ behaves as $A$ if $c$ is entailed by the store and the store is “consistent enough” with respect to the threshold $a$ in the next $t$ time-units: if $t>0$ and the condition on the store and the cut level are satisfied, then the agent $A$ is evaluated (rule ${\\bf Q10}$ ).",
"If $t>0$ and the condition on the cut level is not satisfied, then the agent $B$ is evaluated (rule ${\\bf Q11}$ ).",
"Finally if $t>0$ , the condition on the cut level is satisfied, but the condition on the store is not satisfied, then the control is repeated at the next time instant and the value of the counter $t$ is decreased (axiom ${\\bf Q12}$ ); note that in this case we use the label $\\omega $ , since a check on the store has been performed.",
"As shown by axiom ${\\bf Q13}$ , the counter can be decreased also by performing a $\\tau $ -action: intuitively, this rule is used to model the situation in which, even though the evaluation of the time-out started already, another (parallel) process has the control.",
"In this case, analogously to the approach in [11] and differently from the approach in [9], time continues to elapse (via $\\tau $ -actions) also for the time-out process (see also the rules Q5 and Q6 of the parallel operator).",
"Axiom ${\\bf Q14}$ shows that, if the time-out is exceeded, i.e., the counter $t$ has reached the value of 0, then the process $\\hbox{\\bf askp}_t(c)?^a A \\mathit {:} B$ behaves as $B$ .",
"Askp$_t$ The rules ${\\bf Q15}$ -${\\bf Q19}$ in Figure REF are similar to rules ${\\bf Q10}$ -${\\bf Q14}$ described before, with the exception that here a finer (pointwise) threshold $\\phi $ is compared to the store $\\sigma $ , analogously to what happens with the ${\\bf tell}$ and ${\\bf ask}$ agents.",
"In the following we provide the definition for the observables of the language, which are clearly based only on $\\omega $ -actions.",
"Definition 8 (Observables for tsccp-i) Let $P= F\\text{.",
"}A$ be a tsccp-i process.",
"We define ${\\cal O}_{io}^{i}(P) = \\lbrace \\gamma \\Downarrow _{Fv(A)} \\mid \\langle A, \\bar{{\\mathbf {1}}} \\rangle \\stackrel{\\omega }{\\longmapsto } ^*\\langle {\\bf Success}, \\gamma \\rangle \\rbrace ,$ where ${\\bf Success}$ is any agent that contains only occurrences of the agent ${\\bf success}$ and of the operator $\\parallel $ ."
],
[
"An Execution Timeline for a ",
"In this section we show a timeline for the execution of three tsccp-i agents in parallel.",
"We consider the three soft constraints shown in Figure REF and the Weighted semiring $\\langle \\mathbb {R}^{+} \\cup \\lbrace +\\infty \\rbrace ,min,+,+\\infty ,0 \\rangle $ [2], [4].",
"Our parallel agent is defined by: $\\begin{array}{ll}A_1 :: & {\\bf askp}_5(c_3)?^{+\\infty } ({\\bf tell}(c_1) \\rightarrow ^{+\\infty } {\\bf success}) \\mathit {:} ({\\bf success}) \\\\A_2 :: & {\\bf tell}(c_1)\\rightarrow ^{+\\infty } {\\bf success}\\\\A_3 ::& {\\bf tell}(c_2)\\rightarrow ^{+\\infty } {\\bf success}\\text{.",
"}\\end{array}$ Their concurrent evaluation in the $\\bar{0}$ empty store is shown in Figure REF .",
"At $t=0$ and $t=1$ the agent $A_1$ can make a $\\tau $ -transition (rule Q13 in Figure REF ), waiting for the elapsing of 1 time-unit.",
"This can be done in parallel with a single other $\\omega $ -action: therefore, the ${\\bf tell}(c_1)$ of agent $A_2$ , and the ${\\bf tell}(c_2)$ of agent $A_3$ cannot run in parallel at the same time, since they are both $\\omega $ -actions.",
"In the execution shown in Figure REF , $A_2$ is executed before $A_3$ (also the opposite is possible, depending on the scheduling), leading to the store $\\sigma = c_1 \\otimes c_2 = c_3$ .",
"At $t=2$ , the guard of ${\\bf askp}_5$ in agent $A_1$ is enabled since $\\sigma \\vdash c_3$ and, therefore, rule Q10 in Figure REF is executed.",
"Finally, at $t=3$ the ${\\bf tell}(c_1)$ action of agent $A_1$ is executed as the last action, and at $t=4$ we have $\\langle {\\bf success}\\parallel {\\bf success}\\parallel {\\bf success}, c_1\\otimes c_2 \\otimes c_1 \\rangle $ .",
"Figure: A timeline for the execution of A 1 ∥A 2 ∥A 3 A_1 \\parallel A_2 \\parallel A_3."
],
[
"Denotational Semantics for ",
"In this section we define a denotational characterization of the operational semantics for tsccp-i.",
"Differently from the denotational semantics for the maximal parallelism version presented in Section.",
"REF , here for computational states we consider triples rather than pairs, as $\\omega $ -actions have to be distinguished from $\\tau $ -actions.",
"This difference leads to a different technical development.",
"Our denotational model for tsccp-i associates with a process a set of timed reactive sequences of the form $\\langle \\sigma _1,\\gamma _1, \\xi _1\\rangle \\cdots \\langle \\sigma _n,\\gamma _n, \\xi _n\\rangle \\langle \\sigma ,\\sigma ,\\omega \\rangle $ .",
"Any triple $\\langle \\sigma _i,\\gamma _i, \\xi _i\\rangle $ represents a reaction (a computation step) of the given process at time $i$ : intuitively, the process transforms the global store from $\\sigma _i$ to $\\gamma _i$ by performing a transition step labeled by $\\xi _{i}$ or, in other words, $\\sigma _i$ is the assumption on the external environment, $\\xi _{i}$ is the label of the performed step while $\\gamma _i$ is the contribution of the process itself (which entails always the assumption).",
"The last pair denotes a “stuttering step”, in which the agent ${\\bf Success}$ has been reached.",
"In the following we will assume that each timed reactive sequence $\\langle \\sigma _1,\\gamma _1, \\xi _1\\rangle \\cdots \\langle \\sigma _{n-1},\\gamma _{n-1},\\xi _{n-1}\\rangle \\langle \\sigma _n,\\sigma _n, \\omega \\rangle $ satisfies the following condition: $\\gamma _i \\vdash \\sigma _i\\hbox{ and } \\ \\sigma _j \\vdash \\gamma _{j-1}, $ for any $i\\in [1,n-1]$ and $j\\in [2,n]$ .",
"The basic idea underlying the denotational model then is that, differently from the operational semantics, inactive processes can always make a $\\tau $ -step, where an inactive process is either suspended (due to the absence of the required constraint in the store) or it is a non-scheduled component of a parallel construct.",
"These additional $\\tau $ -steps, which represent time-elapsing and are needed to obtain a compositional model in a simple way, are then added to denotations as triples of the form $\\langle \\sigma ,\\sigma ,\\tau \\rangle $ .",
"For example, the denotation of the process $\\hbox{{\\bf tell}}(c) \\rightarrow ^a {\\bf success}$ contains all the reactive sequences that have, as first element, a triple $\\langle \\sigma ,\\sigma \\otimes c,\\omega \\rangle $ for any possible initial store $\\sigma $ with $(\\sigma \\otimes c)\\Downarrow _{\\emptyset } \\lnot < a$ , as these represent the action of adding the constraint $c$ to the current store.",
"However, such a denotation contains also sequences where the triple $\\langle \\sigma ,\\sigma \\otimes c,\\omega \\rangle $ (still with $(\\sigma \\otimes c)\\Downarrow _{\\emptyset } \\lnot < a$ ) is preceded by a finite sequence of triples of the form $\\langle \\sigma _1 ,\\sigma _1 ,\\tau \\rangle \\langle \\sigma _2 ,\\sigma _2 ,\\tau \\rangle \\ldots \\langle \\sigma _n ,\\sigma _n ,\\tau \\rangle $ .",
"Such a sequence represents time-elapsing while the process is inactive because some other parallel process is scheduled.",
"The set of all reactive sequences for tsccp-i process is denoted by ${\\cal S}_i$ , its typical elements by $s,s_1\\ldots $ , while sets of reactive sequences are denoted by $S,S_1\\ldots $ and $\\varepsilon $ indicates the empty reactive sequence.",
"The operator $\\cdot $ denotes the operator that concatenates these sequences."
],
[
"Compositionality of the Denotational Semantics for ",
"As in Section REF for the tsccp version, we now introduce a denotational semantics ${\\cal D} (F\\text{.",
"}A)(e)$ which is compositional by definition and where, for technical reasons, we represent explicitly the environment $e$ which associates a denotation to each procedure identifier.",
"More precisely, assuming that Pvar denotes the set of procedure identifier, ${\\it Env}_i={\\it Pvar}\\rightarrow {\\cal P}({\\cal S}_i)$ , with typical element $e$ , is the set of environments.",
"Analogously to Section REF , given $e \\in {\\it Env}_i$ , $p \\in {\\it Pvar}$ and $f \\in {\\cal P}({\\cal S}_i)$ , we denote by $e^{\\prime }=e\\lbrace f/p\\rbrace $ the new environment such that $e^{\\prime }(p)=f$ and $e^{\\prime }(p^{\\prime })= e(p^{\\prime })$ for each procedure identifier $p^{\\prime }\\ne p$ .",
"Before defining formally the denotational semantics, we need to define the operators $\\bar{tell}$ , $\\bar{\\sum }$ , $\\bar{\\parallel }$ , $\\bar{askp}$ and $\\bar{\\exists } x$ , analogous to those given in Section REF for the maximal parallelism language.",
"Definition 9 (Semantic operators for tsccp-i) Let $S,S_i$ be sets of reactive sequences, $c,c_i$ be constraints.",
"Moreover let $\\succ _i$ be either of the form $\\succ ^{a_i}$ or $\\succ _{\\phi _i}$ , defined as in Definition REF .",
"Then we define the operators $\\bar{tell}$ , $\\bar{\\sum }$ , $\\bar{\\parallel }$ , $\\bar{askp}$ and $\\bar{\\exists } x$ as follows: The (valued) tell operator $\\bar{tell}^a: {\\cal C} \\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ($\\bar{tell}_\\phi : {\\cal C} \\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ) is the least function (w.r.t.",
"the ordering induced by $\\subseteq $ ) which satisfies the following equation $\\begin{array}{ll}\\bar{tell}^a(c, S) = &\\lbrace s \\in {\\cal S}_i \\mid s = \\langle \\sigma ,\\sigma \\otimes c,\\omega \\rangle \\cdot s^{\\prime },\\ \\sigma \\otimes c\\Downarrow _{\\emptyset } \\lnot <a \\hbox{ and } s^{\\prime } \\in S\\ \\rbrace \\ \\ \\cup \\\\& \\lbrace s \\in {\\cal S}_i \\mid s = \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } \\hbox{ and } s^{\\prime } \\in \\bar{tell}^a(c, S) \\ \\rbrace \\textit {.",
"}\\end{array}$ $\\begin{array}{ll}\\bar{tell}_{\\phi }(c, S) = &\\lbrace s \\in {\\cal S}_i \\mid s = \\langle \\sigma ,\\sigma \\otimes c,\\omega \\rangle \\cdot s^{\\prime },\\ \\sigma \\otimes c \\lnot \\sqsubset \\phi \\hbox{ and } s^{\\prime } \\in S\\ \\rbrace \\ \\ \\cup \\\\& \\lbrace s \\in {\\cal S}_i \\mid s = \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } \\hbox{ and } s^{\\prime } \\in \\bar{tell}_{\\phi }(c, S) \\ \\rbrace \\textit {.",
"}\\end{array}$ The guarded choice The semantic choice operator ${\\bar{\\sum }} _{i=1}^n : ({\\cal C} \\times \\wp ({\\cal S}_i)) \\times \\cdots \\times ({\\cal C} \\times \\wp ({\\cal S}_i)) \\rightarrow \\wp ({\\cal S}_i)$ is the least function which satisfies the following equation: ${\\bar{\\sum }} _{i=1}^n c_i \\succ _i \\, S_i =\\begin{array}[t]{l}\\lbrace s\\in {\\cal S}_i \\mid \\begin{array}[t]{l}s = \\langle \\sigma ,\\sigma , \\omega \\rangle \\cdot s^{\\prime }, \\\\\\sigma \\succ _h\\, c_h \\hbox{ and } s^{\\prime }\\in S_h \\hbox{for an } h\\in [1,n] \\ \\rbrace \\end{array}\\\\\\cup \\\\\\lbrace s \\in {\\cal S}_i \\mid \\begin{array}[t]{l}s =\\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } \\hbox{ and } \\ s^{\\prime } \\in {\\bar{\\sum }} _{i=1}^n c_i \\succ _i \\, S_i \\rbrace \\textit {.",
"}\\end{array}\\end{array}$ Parallel Composition.",
"Let $\\bar{\\parallel }\\in {\\cal S}_i\\times {\\cal S}_i\\rightarrow {\\cal S}_i$ be the (commutative and associative) partial operator defined by induction on the length of the sequences as follows: $\\begin{array}{l}\\langle \\sigma ,\\sigma ,\\omega \\rangle \\bar{\\parallel } \\langle \\sigma ,\\sigma ,\\omega \\rangle = \\langle \\sigma ,\\sigma ,\\omega \\rangle \\\\\\langle \\sigma ,\\sigma ^{\\prime },x \\rangle \\cdot s \\bar{\\parallel } \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } = \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } \\bar{\\parallel } \\langle \\sigma ,\\sigma ^{\\prime },x \\rangle \\cdot s = \\langle \\sigma ,\\sigma ^{\\prime },x \\rangle \\cdot (s\\bar{\\parallel } s^{\\prime }),\\end{array}$ where $x \\in \\lbrace \\omega ,\\tau \\rbrace $ .",
"We define the operator $S_1\\bar{\\parallel } S_2$ on sets as the image of ${\\cal S}_i\\times {\\cal S}_i$ under the above operator.",
"The (valued) askp operator $\\bar{askp}(t)^a:{\\cal C} \\times \\wp ({\\cal S}_i)\\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ($\\bar{askp(t)}_\\phi :{\\cal C} \\times \\wp ({\\cal S}_i)\\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ), with $t>0$ , is defined as: $\\begin{array}{lll}\\bar{askp}(t)^a(c, S_{1} , S_{2})=& \\lbrace s \\in {\\cal S}_i \\mid & s = \\langle \\sigma ,\\sigma , \\omega \\rangle \\cdot s^{\\prime } \\hbox{ and}\\\\&& \\hbox{either }\\sigma \\succ ^{a} c \\mbox{ and } s \\in S_{1}\\\\&& \\hbox{or } \\sigma \\Downarrow _{\\emptyset } < a \\mbox{ and } s \\in S_{2} \\rbrace \\ \\ \\cup \\\\&\\lbrace s \\in {\\cal S}_i \\mid & s = \\langle \\sigma ,\\sigma , x \\rangle \\cdot s^{\\prime } , \\,s^{\\prime } \\in \\bar{askp}(t-1)^a(c, S_{1} , S_{2}) \\\\&& \\hbox{and either } x=\\tau \\\\&& \\hbox{or } x=\\omega , \\, \\sigma \\lnot \\vdash c \\mbox{ and } \\sigma \\Downarrow _{\\emptyset } \\lnot < a \\ \\rbrace \\textit {.",
"}\\end{array}$ $\\begin{array}{lll}\\bar{askp}(t)_\\phi (c, S_{1} , S_{2})=&\\lbrace s \\in {\\cal S}_i \\mid & s = \\langle \\sigma ,\\sigma ^{\\prime }, \\omega \\rangle \\cdot s^{\\prime } \\hbox{ and}\\\\&& \\hbox{either }\\sigma \\succ _{\\phi } \\,c \\mbox{ and } s \\in S_{1}\\\\&& \\hbox{or } \\sigma \\sqsubset \\phi \\mbox{ and } s \\in S_2 \\rbrace \\ \\ \\cup \\\\&\\lbrace s \\in {\\cal S}_i \\mid & s = \\langle \\sigma ,\\sigma , x \\rangle \\cdot s^{\\prime } , \\,s^{\\prime } \\in \\bar{askp}(t-1)_\\phi (c, S_{1} , S_{2}) \\\\&& \\hbox{and either } x=\\tau \\\\&& \\hbox{or } x=\\omega , \\, \\sigma \\lnot \\vdash c \\mbox{ and } \\sigma \\lnot \\sqsubset \\phi \\ \\rbrace \\textit {.",
"}\\end{array}$ The (valued) askp operator $\\bar{askp}(0)^a:{\\cal C} \\times \\wp ({\\cal S}_i)\\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ($\\bar{askp}(0)_\\phi :{\\cal C} \\times \\wp ({\\cal S}_i)\\times \\wp ({\\cal S}_i) \\rightarrow \\wp ({\\cal S}_i)$ ) is the least function which satisfies the following equation $\\begin{array}{lll}\\bar{askp}(0)^a(c, S_{1} , S_{2})=&\\lbrace s \\in {\\cal S}_i \\mid & \\hbox{either }s = \\langle \\sigma , \\sigma , \\omega \\rangle \\cdot s^{\\prime }\\hbox{ and } s^{\\prime } \\in S_{2} \\\\&& \\hbox{or } s = \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime }\\\\&&\\mbox{and } s^{\\prime } \\in \\bar{askp}(0)^a(c, S_{1} , S_{2}) \\ \\rbrace \\textit {.",
"}\\end{array}$ $\\begin{array}{lll}\\bar{askp}(0)_\\phi (c, S_{1} , S_{2})=&\\lbrace s \\in {\\cal S}_i \\mid & \\hbox{either }s = \\langle \\sigma , \\sigma , \\omega \\rangle \\cdot s^{\\prime }\\hbox{ and } s^{\\prime } \\in S_{2} \\\\&& \\hbox{or } s = \\langle \\sigma ,\\sigma ,\\tau \\rangle \\cdot s^{\\prime } \\\\&& \\mbox{and } s^{\\prime } \\in \\bar{askp}(0)_\\phi (c, S_{1} , S_{2} ) \\ \\rbrace \\textit {.",
"}\\end{array}$ The hiding operator The semantic hiding operator can be defined as follows: $\\begin{array}{ll}{\\bf \\bar{ \\exists } } x S = \\lbrace s \\in {\\cal S}_i \\mid & \\mbox{\\rm there exists $ s^{\\prime } \\in S$ such that $s= s^{\\prime }[x/y]$ with $y$ new } \\rbrace \\end{array}$ where $s^{\\prime }[x/y]$ denotes the sequence obtained from $s^{\\prime }$ by replacing the variable $x$ for the variable $y$ that we assume to be new.As before, we assume that each time that we consider a new applications of the operator ${\\bf \\bar{\\exists } }$ we use a new, different $y$ .",
"It is immediate to see that the previous semantic operators are well defined, that is, the least function which satisfies the equations actually exists and can be obtained by a standard fix-point construction.",
"The $\\bar{tell}$ , $\\bar{\\sum }$ , $\\bar{\\parallel }$ , $\\bar{askp}$ and $\\bar{\\exists } x$ operators have the expected definition, including the mentioned addition of $\\tau $ -steps.",
"In the semantic parallel operator (acting on sequences) we require that at each point of time at most one $\\omega $ -action is present and the two arguments of the operator agree with respect to the contribution of the environment (the first component of the triple).",
"We also require that the two arguments have the same length (in all other cases the parallel composition is assumed being undefined): this is necessary to reflect the passage of time since the $i-th$ element of any sequence corresponds to the given processes action on the $i-th$ time step.",
"Even though we merge point-wise sequences of the same length, this models an interleaving approach for $\\omega $ -actions, because of the previously mentioned addition of $\\tau $ -steps to denotations.",
"Concerning the semantic choice operator, a sequence in $\\bar{\\sum } _{i=1}^n c_i\\succ _i \\,S_i$ consists of an initial period of waiting for a store which satisfies one of the guards.",
"During this waiting period, only the environment is active by producing the constraint $\\sigma $ , while the process itself generates the stuttering steps $\\langle \\sigma ,\\sigma , \\tau \\rangle $ .",
"When the store is strong enough to satisfy a guard, that is to entail a $c_h$ and to satisfy the condition on the cut level, then the resulting sequence is obtained by adding $s^{\\prime }\\in S_h$ to the initial waiting period.",
"We can define the denotational semantics ${\\cal D}$ as follows.",
"Here, $Process_i$ denotes the set of tsccp-i processes.",
"Definition 10 (Processes Semantics) We define the semantics ${\\cal D}\\in {\\it Process_i}\\rightarrow {\\cal P}({\\cal S}_i)$ is the least function with respect to the ordering induced by the set-inclusion, which satisfies the equations in Figure REF Figure: The semantics D(F.A)D( F\\text{.",
"}A)(e) for tsccp-i.Also ${\\cal D}$ is well defined and can be obtained by a fix-point construction.",
"To see this, let us define an interpretation as a mapping $I:Process_i\\rightarrow \\wp ({\\cal S}_i)$ .",
"Then let us denote by ${\\cal I}$ the cpo of all the interpretations (with the ordering induced by $\\subseteq $ ).",
"To the equations in Figure REF , we can then associate a monotonic (and continuous) mapping ${\\cal F}: {\\cal I} \\rightarrow {\\cal I}$ defined by the equations of Figure REF , provided that we replace the symbol ${\\cal D}$ for ${\\cal F}(I)$ , we delete the environment $e$ and that we replace equation ${\\bf F9}$ for the following one: ${\\cal F}(I)( {\\it F\\text{.",
"}p(x)})=I( F\\text{.}",
"ask(\\bar{{\\mathbf {1}}}) \\rightarrow A)\\textit {.",
"}$ Then, one can easily prove that a function satisfies the equations in Figure REF iff it is a fix-point of the function ${\\cal F}$ .",
"Because this function is continuous (on a cpo), well known results ensure us that its least fix-point exists and it equals ${\\cal F}^\\omega $ , where the powers are defined as follows: ${\\cal F}^0 = I_0$ (this is the least interpretation which maps any process to the empty set); ${\\cal F}^ n = {\\cal F} ({\\cal F}^{ n-1})$ and ${\\cal F}^ \\omega = lub \\lbrace {\\cal F}^ n| n\\ge 0\\rbrace $ (where lub is the least upper bound on the cpo ${\\cal I}$ )."
],
[
"Correctness of the Denotational Semantics for ",
"As for the correctness of the denotational semantics presented in Section REF , at each step, the assumption on the current store must be equal to the store produced by the previous step.",
"In other words, for any two consecutive steps $\\langle \\sigma _i,\\sigma ^{\\prime }_i,x_{i}\\rangle \\langle \\sigma _{i+1},\\sigma ^{\\prime }_{i+1}, x_{i+1}\\rangle $ we must have $\\sigma ^{\\prime }_i =\\sigma _{i+1}$ .",
"Furthermore, triples containing $\\tau $ -actions do not correspond to observable computational steps, as these involve $\\omega $ -actions only.",
"Definition 11 (Connected Sequences in tsccp-i) Let $s=\\langle \\sigma _1,\\sigma ^{\\prime }_1, x_{1}\\rangle \\langle \\sigma _2,\\sigma ^{\\prime }_2, x_{2}\\rangle \\cdots \\langle \\sigma _{n},$ $\\sigma _{n},\\omega \\rangle $ be a reactive sequence.",
"We say that $s$ is connected if $\\sigma _1= \\bar{{\\mathbf {1}}} $ , $\\sigma _i = \\sigma ^{\\prime }_{i-1}$ and $x_j=\\omega $ for each $i, j$ , $2\\le i\\le n$ and $1\\le j\\le n-1$ .",
"According to the previous definition, a sequence is connected if all the information assumed on the tuple space is produced by the process itself and only $\\omega $ -actions are involved.",
"To be defined as connected, a sequence must also have $\\bar{{\\mathbf {1}}}$ as the initial constraint.",
"A connected sequence represents a tsccp-i computation, as it will be proved in the remaining of this section.",
"In order to prove the correctness of the denotational semantics, we use a modified transition system $T^{\\prime }$ , where inactive (either suspended or not scheduled) processes can perform $\\tau $ -actions.",
"When considering our notions of observables, we can prove that such a modified transition system is equivalent to the previous one and agrees with the denotational model.",
"The new transition system $T^{\\prime }$ is obtained from the one in Figure REF by deleting rule Q6 and by adding the rules Q0', Q1', Q2', Q3', Q4', Q7', Q8', Q14' and Q19', contained in Figure REF .",
"We denote by $\\Rightarrow $ the relation defined by $T^{\\prime }$ .",
"Figure: The τ\\tau -rules for tsccp-i.The observables induced by the transition system $T^{\\prime }$ are formally defined as follows.",
"Definition 12 Let $P= F\\text{.",
"}A$ be a tsccp-i process.",
"We define ${\\cal O}_{io}^{i^{\\prime }}(P) = \\lbrace \\gamma \\Downarrow _{Fv(A)} \\mid \\langle A, \\bar{{\\mathbf {1}}} \\rangle \\stackrel{\\omega }{\\Rightarrow }\\,\\!\\!^*\\langle {\\bf Success}, \\gamma \\rangle \\rbrace ,$ where ${\\bf Success}$ is any agent which contains only occurrences of the agent ${\\bf success}$ and of the operator $\\parallel $ .",
"Lemma REF shows that the modified transition system agrees with the original one when considering our notion of observables.",
"We first need some definitions and technical lemmata.",
"In the following, given two agents $A$ and $B$ , we say that $A \\simeq B$ if and only if $B$ is obtained from $A$ by replacing an agent of the form $\\exists x A_1$ in $A$ with $A_1 [x/y]$ , where $y$ is new in $A$ .",
"$\\approx $ denotes the reflexive and transitive closure of $ \\simeq $ .",
"The following lemmata hold.",
"Let $F\\text{.",
"}A$ and $F\\text{.",
"}B$ be tsccp-i processes such that $ A \\approx B$ .",
"Then for each store $\\sigma $ and for $x \\in \\lbrace \\omega , \\tau \\rbrace $ $\\langle F\\text{.",
"}A,\\sigma \\rangle \\stackrel{x}{\\longmapsto }\\langle F\\text{.",
"}C,\\sigma ^{\\prime }\\rangle \\mbox{ if and only if }\\langle F\\text{.",
"}B,\\sigma \\rangle \\stackrel{x}{\\longmapsto }\\langle F\\text{.",
"}C,\\sigma ^{\\prime }\\rangle \\textit {.",
"}$ The proof is immediate, by using rule Q9 and by a straightforward inductive argument.",
"From the above Lemma we derive the following corollary: Let $F\\text{.",
"}A$ and $F\\text{.",
"}B$ be tsccp-i processes such that $ A \\approx B$ .",
"Then for each store $\\sigma $ , $\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto }\\,\\!",
"\\!^*\\langle {\\bf Success}, \\gamma \\rangle $ if and only if $\\langle F\\text{.",
"}B, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto }\\,\\!\\!",
"^*\\langle {\\bf Success}, \\gamma \\rangle $ .",
"Let $P= F\\text{.",
"}A$ be a tsccp-i process.",
"Then for each store $\\sigma $ , $\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle F\\text{.",
"}B, \\sigma ^{\\prime }\\rangle $ if and only if $\\sigma = \\sigma ^{\\prime }$ and either $\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\tau }{\\longmapsto }\\langle F\\text{.",
"}C, \\sigma \\rangle $ and $C\\approx B$ or $\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ and $B\\approx A$ .",
"$\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow }\\langle F\\text{.",
"}B, \\sigma ^{\\prime }\\rangle $ if and only if $\\langle F\\text{.",
"}A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto }\\langle F\\text{.",
"}C, \\sigma ^{\\prime }\\rangle $ and $C \\approx B$ .",
"The proof is by induction on the complexity of the agent $A$ .",
"$A$ is of the form $\\bf {success}$ , $\\hbox{{\\bf tell}}(c) \\rightarrow ^{a} A$ , $\\hbox{{\\bf tell}}(c)\\rightarrow _{\\phi } A$ , $\\hbox{{\\bf ask}}(c) \\rightarrow ^{a} A$ , $\\hbox{{\\bf ask}}(c) \\rightarrow _{\\phi } A$ , $\\Sigma _{i=1}^{n}E_i$ , $p(x)$ , $\\hbox{\\bf askp}_0(c)?^a A \\mathit {:} B$ and $\\hbox{\\bf askp}_0(c)?_{\\phi } A \\mathit {:} B$ .",
"The proof is immediate by observing that by the rules in Figure REF , $\\langle A, \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ and by the rules in Figure REF , $\\langle A, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B, \\sigma ^{\\prime } \\rangle $ if and only if $\\langle A, \\sigma \\rangle =\\langle B, \\sigma ^{\\prime } \\rangle $ .",
"$A$ is of the form $\\hbox{\\bf askp}_t(c)?^a A_1 \\mathit {:} A_2$ ($\\hbox{\\bf askp}_t(c)?_{\\phi } A_1 \\mathit {:} A_2$ ), with $t>0$ .",
"The proof is immediate since both the transition systems use the rule Q13 (Q18) of Figure REF .",
"If $A$ is of the form $A_1\\parallel A_2$ .",
"In this case, by definition of the transition system $T^{\\prime }$ and by using rule Q5 of Figure REF , for each store $\\sigma $ , $\\begin{array}{l}\\langle A_1\\parallel A_2, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B_1\\parallel B_2, \\sigma ^{\\prime } \\rangle \\mbox{ if and only if } \\\\\\langle A_1, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B_1, \\sigma ^{\\prime }\\rangle \\mbox{ and }\\langle A_2, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B_2, \\sigma \\rangle \\end{array}$ (the symmetric case is analogous and hence it is omitted).",
"By inductive hypothesis this holds if and only if $\\sigma ^{\\prime }=\\sigma $ and for $i=1,2$ either $\\langle A_i, \\sigma \\rangle \\stackrel{\\tau }{\\longmapsto } \\langle C_i, \\sigma \\rangle $ and $C_i \\approx B_i$ or $\\langle A_i, \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ and $B_i \\approx A_i$ .",
"If there exists $i \\in [1,2]$ such that $\\langle A_i, \\sigma \\rangle \\stackrel{\\tau }{\\longmapsto } \\langle C_i, \\sigma \\rangle $ then the thesis follows by using either rule Q5 or rule Q6.",
"Otherwise $ \\langle A_1\\parallel A_2, \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ .",
"Then the thesis follows since by the previous results ${B_1\\parallel B_2} \\approx {A_1\\parallel A_2}$ .",
"$A$ is of the form $\\exists x A_1$ .",
"By rule Q9 of Figure REF for each store $\\sigma $ , $\\begin{array}{l}\\langle \\exists x A_1, \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B, \\sigma ^{\\prime } \\rangle \\mbox{ if and only if }\\langle A_1[x/y], \\sigma \\rangle \\stackrel{\\tau }{\\Rightarrow }\\langle B, \\sigma ^{\\prime } \\rangle \\end{array}$ By inductive hypothesis this holds if and only if $\\sigma ^{\\prime }=\\sigma $ either $\\langle A_1[x/y], \\sigma \\rangle \\stackrel{\\tau }{\\longmapsto } \\langle C, \\sigma \\rangle $ and $C \\approx B$ or $\\langle A_1[x/y], \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ and $B \\approx A_1[x/y]$ .",
"Therefore, by using rule Q9 of Figure REF and since $ \\exists x A_1 \\approx A_1[x/y]$ , we have that either $\\langle \\exists x A_1, \\sigma \\rangle \\stackrel{\\tau }{\\longmapsto } \\langle C, \\sigma \\rangle $ and $C \\approx B$ or $\\langle \\exists x A_1, \\sigma \\rangle \\stackrel{\\tau }{\\lnot \\longmapsto }$ and $B \\approx \\exists x A_1$ and then the thesis.",
"The proof is analogous to the previous one and hence it is omitted.",
"Let $P= F\\text{.",
"}A$ be a tsccp-i process.",
"Then ${\\cal O}_{io}^{i^{\\prime }} (P) = {\\cal O}_{io}^{i} (P) .$ We prove that there exists a computation $\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow }\\,\\!\\!",
"^*\\langle {\\bf Success}, \\gamma \\rangle $ if and only if there exists a computation $\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto } ^*\\langle {\\bf Success}, \\gamma \\rangle $ .",
"Then the thesis follows by definition of ${\\cal O}_{io}^{i} (P)$ and ${\\cal O}_{io}^{i^{\\prime }} (P).$ The proof is by induction on the length of the computation $\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow }\\,\\!\\!",
"^*\\langle {\\bf Success}, \\gamma \\rangle $ .",
"$n=1)$ In this case $A= {\\bf Success}$ and then the thesis.",
"$n>1)$ In this case $\\begin{array}{lll}\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow } \\,\\!\\!^*\\langle {\\bf Success}, \\gamma \\rangle &\\mbox{iff} & \\\\\\hspace*{28.45274pt}\\mbox{(by definition)} \\\\\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow } \\langle A_1, \\sigma _1 \\rangle \\mbox{ and } \\langle A_1, \\sigma _1 \\rangle \\stackrel{\\omega }{\\Rightarrow }\\,\\!\\!",
"^*\\langle {\\bf Success}, \\gamma \\rangle &\\mbox{iff} & \\\\\\hspace*{28.45274pt}\\mbox{(by inductive hypothesis)} \\\\\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\Rightarrow } \\langle A_1, \\sigma _1 \\rangle \\mbox{ and } \\langle A_1, \\sigma _1 \\rangle \\stackrel{\\omega }{\\longmapsto }\\,\\!\\!",
"^*\\langle {\\bf Success}, \\gamma \\rangle &\\mbox{iff} & \\\\\\hspace*{28.45274pt}\\mbox{(by Point 2 of Lemma~\\ref {lem:cambio})} \\\\\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto } \\langle A_2, \\sigma _1 \\rangle , \\ A_2 \\approx A_1\\mbox{ and } \\langle A_1, \\sigma _1 \\rangle \\stackrel{\\omega }{\\longmapsto } \\,\\!\\!^*\\langle {\\bf Success}, \\gamma \\rangle & \\mbox{iff} &\\\\\\hspace*{28.45274pt}\\mbox{(by Corollary~\\ref {cor:unf})}\\\\\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto } \\langle A_2, \\sigma _1 \\rangle \\mbox{ and } \\langle A_2, \\sigma _1 \\rangle \\stackrel{\\omega }{\\longmapsto } \\,\\!\\!^*\\langle {\\bf Success}, \\gamma \\rangle & \\mbox{iff} &\\\\\\hspace*{28.45274pt}\\mbox{(by definition)} \\\\\\langle A, \\sigma \\rangle \\stackrel{\\omega }{\\longmapsto } \\,\\!\\!^*\\langle {\\bf Success}, \\gamma \\rangle .\\end{array}$ We can now easily prove that, given our definition of ${\\cal D}$ , the modified transition system $T^{\\prime }$ agrees with the denotational model.",
"For any tsccp-i process $P= F\\text{.",
"}A$ we have ${\\cal O}_{io}^{i^{\\prime }}(P) = \\lbrace \\sigma _n \\Downarrow _{Fv(A)}\\mid \\begin{array}[t]{ll} \\mbox{\\rm there exists a connected sequence } s\\in {\\cal D}(P) \\ \\mbox{\\rm such that }\\\\s=\\langle \\sigma _1,\\sigma _2,\\omega \\rangle \\langle \\sigma _2, \\sigma _3,\\omega \\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle \\rbrace \\textit {.",
"}\\end{array}$ We prove by induction on the complexity of the agent $A$ that $\\begin{array}{ll}{\\cal D}(P)=\\lbrace s \\mid & s=\\langle \\sigma _1,\\sigma _1^{\\prime },x_1\\rangle \\langle \\sigma _2,\\sigma _2^{\\prime }, x_2\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle , \\ A_1=A, \\\\& \\mbox{for } i\\in [1,n-1],\\ \\langle {A_i},\\sigma _i\\rangle \\stackrel{x_i}{\\Rightarrow }\\langle {A_{i+1}},\\sigma ^{\\prime }_i\\rangle \\mbox{ and } A_n ={\\bf Success}\\rbrace \\textit {.",
"}\\end{array} $ Then the proof follows by definition of ${\\cal O}_{io}^{i^{\\prime }}(P)$ .",
"When the tsccp-i $P$ is not of the form $F\\text{.",
"}B \\parallel C$ the thesis follows immediately from the close correspondence between the rules of the transition system and the definition of the denotational semantics.",
"Assume now that $P$ is of the form $F\\text{.",
"}B \\parallel C$ .",
"By definition of the denotational semantics, $s \\in {\\cal D}(P)$ if and only if $s=\\langle \\sigma _1,\\sigma _1^{\\prime },x_1\\rangle \\langle \\sigma _2,\\sigma _2^{\\prime }, x_2\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle $ and there exist $s^{\\prime } \\in {\\cal D}(F\\text{.",
"}B)$ and $s^{\\prime \\prime } \\in {\\cal D}(F\\text{.",
"}C)$ , $\\begin{array}{ll}s^{\\prime }=\\langle \\sigma _1,\\kappa _1^{\\prime },x_1^{\\prime }\\rangle \\langle \\sigma _2,\\kappa _2^{\\prime }, x_2^{\\prime }\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle & \\mbox{and} \\\\s^{\\prime \\prime }=\\langle \\sigma _1,\\kappa _1^{\\prime \\prime },x_1^{\\prime \\prime }\\rangle \\langle \\sigma _2,\\kappa _2^{\\prime \\prime }, x_2^{\\prime \\prime }\\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle , &\\end{array}$ such that for each $i\\in [1,n-1]$ , $\\begin{array}{ll} x_i=\\tau \\mbox{ if and only if }& x^{\\prime }_i=x^{\\prime \\prime }_i=\\tau \\mbox{ and in this case }\\sigma _i^{\\prime }=\\kappa _i^{\\prime }=\\kappa _i^{\\prime \\prime }=\\sigma _i, \\\\x_i=\\omega \\mbox{ if and only if }&\\mbox{either } x^{\\prime }_i=\\omega , \\ x^{\\prime \\prime }_i=\\tau , \\ \\kappa _i^{\\prime }=\\sigma _i^{\\prime } \\mbox{ and } \\kappa _i^{\\prime \\prime }=\\sigma _i \\\\& \\mbox{or } x^{\\prime }_i=\\tau , \\ x^{\\prime \\prime }_i=\\omega , \\ \\kappa _i^{\\prime \\prime }=\\sigma _i^{\\prime } \\mbox{ and }\\kappa _i^{\\prime }=\\sigma _i.\\end{array}$ By inductive hypothesis $s^{\\prime } \\in {\\cal D}(F\\text{.",
"}B)$ and $s^{\\prime \\prime } \\in {\\cal D}(F\\text{.",
"}C)$ if and only if $\\begin{array}{ll}\\langle B_i,\\sigma _i\\rangle \\stackrel{x^{\\prime }_i}{\\Rightarrow }\\langle {B_{i+1}},\\kappa ^{\\prime }_i\\rangle \\mbox{ for } i\\in [1,n-1],\\ B_1=B \\mbox{ and } B_n={\\bf Success}, \\\\\\langle C_i,\\sigma _i\\rangle \\stackrel{x^{\\prime \\prime }_i}{\\Rightarrow }\\langle {C_{i+1}},\\kappa ^{\\prime \\prime }_i\\rangle \\mbox{ for } i\\in [1,n-1], \\ C_1=C \\mbox{ and } C_n={\\bf Success}\\text{.",
"}\\end{array}$ Therefore, by Rule R8 and by (REF ), we have that (REF ) holds if and only if $\\begin{array}{ll}\\langle B_i \\parallel C_i,\\sigma _i\\rangle \\stackrel{x_i}{\\Rightarrow }\\langle B_{i+1} \\parallel C_{i+1},\\sigma _{i+1}\\rangle \\mbox{ for } i\\in [1,n-1],& \\\\B_1\\parallel C_1 =B\\parallel C \\mbox{ and }B_n\\parallel C_n ={\\bf Success} &\\end{array}$ and then the thesis.",
"Thus we obtain the following correctness result whose proof is immediate from the previous theorems.",
"[Correctness of tsccp-i] For any tsccp-i process $P= F\\text{.",
"}A$ we have ${\\cal O}_{io}^{i}(P) = \\lbrace \\sigma _n \\Downarrow _{Fv(A)}\\mid \\begin{array}[t]{ll} \\mbox{\\rm there exists a connected sequence } s\\in {\\cal D}(P) \\ \\mbox{\\rm such that }\\\\s=\\langle \\sigma _1,\\sigma _2,\\omega \\rangle \\langle \\sigma _2, \\sigma _3,\\omega \\rangle \\cdots \\langle \\sigma _{n},\\sigma _{n},\\omega \\rangle \\rbrace \\textit {.",
"}\\end{array}$"
],
[
"Related Work",
"By comparing this work with other timed languages using crisp constraints (instead of soft ones as in this paper) as [25], [27], there are three main differences we can find out.",
"First, the computational model of both the languages tcc [27] and default tcc [25] is inspired by that one of synchronous languages: each time interval is identified with the time needed for a ccp process to terminate a computation.",
"Clearly, in order to ensure that the next time instant is reached, the (default) ccp program has to be always terminating; thus, it is assumed that it does not contain recursion.",
"On the other hand, we directly introduce a timed interpretation of the usual programming constructs of ccp by considering the primitive ccp constructs ask and tell as the elementary actions whose evaluation takes one time-unit.",
"Therefore, in our model, each time interval is identified with the time needed for the underlying constraint system to accumulate the tells and to answer the queries (asks) issued at each computation step by the processes of the system.",
"For the definition of our tsccp agents we do not need any restriction on recursion to ensure that the next time instant is reached, since at each moment there are only a finite number of parallel agents, and the next moment in time occurs as soon as the underlying constraint system has responded to the initial actions of all the current agents of the system.",
"A second difference relies in the transfer of information across time boundaries.",
"In [27] and [25], the programmer has to explicitly transfer the (positive) information from a time instant to the next one, by using special primitives that allow one to control the temporal evolution of the system.",
"In fact, at the end of a time interval all the constraints accumulated and all the processes suspended are discarded, unless they are arguments to a specific primitive.",
"On the contrary, no explicit transfer is needed in tsccp, since the computational model is based on the monotonic evolution of the store which is usual in ccp.",
"A third relevant difference is in [27] and [25] the authors present deterministic languages while our language allows for nondeterminism.",
"These three differences also hold between [27] or [25], and the original crisp version of the language, i.e., tccp [10].",
"In [22], the authors generalize the model in [27] in order to extend it with temporary parametric ask operations.",
"Intuitively, these operations behave as persistent parametric asks during a time-interval, but may disappear afterwards.",
"The presented extension goes in the direction of better modeling mobile systems with the use of private channels between the agents.",
"However, also the agents in [22] show a deterministic behavior, instead of our not-deterministic choice.",
"Other timed extension of concurrent constraint programming have been proposed in [21], [23], however these languages, differently from tsccp, do not take into account quantitative aspects; therefore, this achievement represents a very important expressivity improvement with respect to related works.",
"These have been considered by Di Pierro and Wiklicky, who have extensively studied probabilistic ccp (see for example [15]).",
"This language provides a construct for probabilistic choice which allows one to express randomness in a program, without assuming any additional structure on the underlying constraint system.",
"This approach is therefore deeply different from ours.",
"More recently, stochastic ccp has been introduced in [7] to model biological systems.",
"This language is obtained by adding a stochastic duration to the ask and tell primitives, thus it differs from our solutions.",
"In literature we can find other proposals that are related to tuple-based kernel-languages instead of a constraint store, as KLAIM [14] (A Kernel Language for Agents Interaction and Mobility) or SCEL [13] (Software Component Ensemble Language) for instance.",
"These languages are designed to study different properties of systems, as mobility and autonomicity of modeled agents.",
"Their basic specification do not encompass time-based primitives, while mobility features are not present in any of the constraint-based languages reported in this section.",
"The purpose of our language is to model systems where a level of preference and time-sensitive primitives (as a timeout) is required: a good example is represented by agents participating to an auction, as the example given in Section REF .",
"In general, since semiring-based soft constraints allow one to express several quantitative features, our proposal provides a framework which can be instantiated to obtain a variety of specific extensions of ccp."
],
[
"Conclusion and Future Work",
"We have presented the tsccp and tsccp-i in order to join together the expressive capabilities of soft constraints and timing mechanisms in a new programming framework.",
"The agents modeled with these languages are able to deal with time and preference-dependent decisions that are often found during complex interactions.",
"An application scenario can be represented by different entities that need to negotiate generic resources or services, as, for instance, during an auction process.",
"Mechanisms as timeout and interrupt may model the wait for pending conditions or the triggering of some new events.",
"All the tsccp and tsccp-i rules have been formally described by a transition system and, then, also with a denotational characterization of the operational semantics obtained with the use of timed reactive sequences.",
"The resulting semantics has been proved to be compositional and correct.",
"About future work, a first improvement of the presented languages can be the inclusion of a fail agent in the syntax given in Definition REF and Definition REF , and a semantics for the transition rules that lead to a failed computation, in case the guard on the transition rule cannot be enforced due to the preference of the store.",
"In fact, the transition systems we have defined consider only successful computations.",
"If this could be a reasonable choice in a don't know interpretation of the language it will lead to an insufficient analysis of the behavior in a pessimistic interpretation of the indeterminism.",
"At last, we would like to consider other time management strategies (as the one proposed in [28]), and to study how timing and non-monotonic constructs [6] can be integrated together."
]
] | 1403.0461 |
[
[
"Observation of the $X(1840)$ at BESIII"
],
[
"Abstract Observation of the $X(1840)$ in the $3(\\pi^+\\pi^-)$ invariant mass in $J/\\psi\\rightarrow\\gamma 3(\\pi^+\\pi^-)$ at BESIII is reviewed.",
"With a sample of $225.3\\times10^6$ $J/\\psi$ events collected with the BESIII detector at BEPCII, the $X(1840)$ is observed with a statistical significance of $7.6\\sigma$.",
"The mass, width and product branching fraction of the $X(1840)$ are determined.",
"The decay $\\eta^\\prime\\rightarrow 3(\\pi^+\\pi^-)$ is searched for, and the upper limit of the branching fraction is set at the 90% confidence level."
],
[
"Introduction",
"Within the standard model framework, the strong interaction is described by Quantum Chromodynamics (QCD), which suggests the existence of the unconventional hadrons, such as glueballs, hybrid states and multiquark states.",
"The establishment of such states remains one of the main interests in experimental particle physics.",
"Decays of the $J/\\psi $ particle are ideal for the study of the hadron spectroscopy and the searching for the unconventional hadrons.",
"In the decays of the $J/\\psi $ particle, several observations in the mass region 1.8 GeV/c$^2$ - 1.9 GeV/c$^2$ have been presented in different experiments[1][8], such as the $X(p\\bar{p})$[1][3], $X(1835)$[4]$^{,}$[5], $X(1810)$[6]$^{,}$[7] and $X(1870)$[8]."
],
[
"Observation of the $X(1840)$",
"Recently, using a sample of $225.3\\times 10^{6}$ $J/\\psi $ events[9] collected with BESIII detector[10] at BEPCII[11], the decay of $J/\\psi \\rightarrow \\gamma 3(\\pi ^+\\pi ^-)$ was analyzed[12], and the $X(1840)$ was observed in the $3(\\pi ^+\\pi ^-)$ mass spectrum with a statistical significance of $7.6\\sigma $ .",
"Figure: Distribution of the invariant mass of 3(π + π - )3(\\pi ^+\\pi ^-).",
"The dots with errorbars are data; the histogram is phase space events withan arbitrary normalization.The $3(\\pi ^+\\pi ^-)$ invariant mass spectrum is shown in Fig.",
"REF , where the $X(1840)$ can be clearly seen.",
"The parameters of the $X(1840)$ are extracted by an unbinned maximum likelihood fit.",
"In the fit, the background is described by two contributions: the contribution from $J/\\psi \\rightarrow \\pi ^03(\\pi ^+\\pi ^-)$ and the contribution from other sources.",
"The contribution from $J/\\psi \\rightarrow \\pi ^03(\\pi ^+\\pi ^-)$ is determined from MC simulation and fixed in the fit (shown by the dash-dotted line in Fig.",
"REF ).",
"The other contribution is described by a third-order polynomial.",
"The signal is described by a Breit-Wigner function modified with the effects of the detection efficiency, the detector resolution, and the phase space factor.",
"The fit result is shown in Fig.",
"REF .",
"The mass and width of the $X(1840)$ are $M=1842.2\\pm 4.2^{+7.1}_{-2.6}$ MeV/c$^2$ and $\\Gamma =83\\pm 14\\pm 11$ MeV, respectively; the product branching fraction of the $X(1840)$ is $\\mathcal {B}(J/\\psi \\rightarrow \\gamma X(1840))\\times \\mathcal {B}(X(1840)\\rightarrow 3(\\pi ^+\\pi ^-)) = (2.44\\pm 0.36^{+0.60}_{-0.74})\\times 10^{-5}$ .",
"In these results, the first errors are statistical and the second errors are systematic.",
"Figure: The fit of mass spectrum of 3(π + π - )3(\\pi ^+\\pi ^-).",
"The dots with error bars are data;the solid line is the fit result.",
"The dashed line represents all the backgrounds, includingthe background events from J/ψ→π 0 3(π + pi - )J/\\psi \\rightarrow \\pi ^03(\\pi ^+pi^-) (dash-dotted line, fixedin the fit) and a third-order polynomial representing other backgrounds.Figure REF shows the comparisons of the $X(1840)$ with other observations at BESIII[12].",
"The comparisons indicate that at present one can not distinguish whether the $X(1840)$ is a new state or the signal of a $3(\\pi ^+\\pi ^-)$ decay mode of an existing state.",
"Figure: Comparisons of observations at BESIII.",
"The error barsinclude statistical, systematic, and, where applicable, model uncertainties."
],
[
"Search for the decay of $\\eta ^\\prime \\rightarrow 3(\\pi ^+\\pi ^-)$",
"With the same data sample, the decay of $\\eta ^\\prime \\rightarrow 3(\\pi ^+\\pi ^-)$ was searched for[12].",
"The mass spectrum of the $3(\\pi ^+\\pi ^-)$ is shown in Fig.",
"REF , where no events are observed in the $\\eta ^\\prime $ mass region.",
"With the Feldman-Cousins frequentist approach[13], the upper limit of the branching fraction is set to be $\\mathcal {B}(\\eta ^\\prime \\rightarrow 3(\\pi ^+\\pi ^-)) < 3.1\\times 10^{-5}$ at the 90% confidence level, where the systematic uncertainty is taken into account."
],
[
"Summary",
"With a sample of $225.3\\times 10^{6}$ $J/\\psi $ events collected at BESIII, the decay of $J/\\psi \\rightarrow \\gamma 3(\\pi ^+\\pi ^-)$ was analyzed[12].",
"The $X(1840)$ was observed in the $3(\\pi ^+\\pi ^-)$ invariant mass spectrum.",
"The mass, width and product branching fraction of the $X(1840)$ are $M=1842.2\\pm 4.2^{+7.1}_{-2.6}$ MeV/c$^2$ , $\\Gamma =83\\pm 14\\pm 11$ MeV and $\\mathcal {B}(J/\\psi \\rightarrow \\gamma X(1840))\\times \\mathcal {B}(X(1840)\\rightarrow 3(\\pi ^+\\pi ^-)) = (2.44\\pm 0.36^{+0.60}_{-0.74})\\times 10^{-5}$ , respectively.",
"The decay $\\eta ^\\prime \\rightarrow 3(\\pi ^+\\pi ^-)$ was searched for.",
"No events were observed in the $\\eta ^\\prime $ mass region and the upper limit of the branching fraction was set to be $\\mathcal {B}(\\eta ^\\prime \\rightarrow 3(\\pi ^+\\pi ^-)) < 3.1\\times 10^{-5}$ at the 90% confidence level."
]
] | 1403.0167 |
[
[
"Isospin-Violating Dark Matter with Colored Mediators"
],
[
"Abstract In light of positive signals reported by the CDMS-II Si experiment and the recent results of the LUX and SuperCDMS experiments, we study isospin-violating dark matter scenarios assuming that the interaction of the dark matter is mediated by colored particles.",
"We investigate the phenomenology of the model, including collider searches, flavor and CP phenomenology.",
"A minimal possible scenario includes scalar dark matter and new vector-like colored fermions with masses of O(1) TeV as mediators.",
"Such a scenario may be probed at the 14 TeV LHC, while flavor and CP constraints are stringent and severe tuning in the couplings is unavoidable.",
"We also found that, as an explanation of the CDMS-II Si signal, isospin-violating fermionic dark matter models with colored scalar mediators are disfavored by the LHC constraints."
],
[
"Introduction",
"Dark matter (DM), which is expected to be responsible for about $27\\ \\%$ of the mass density of the present universe [1], is still a great mystery to the field of particle physics.",
"Although various cosmological observations have confirmed the existence of DM, its particle-physics properties, such as the mass, strength of its interactions with Standard–Model (SM) particles, and so on, remain fully unknown.",
"Various experiments have been performed to detect direct and indirect signals of DM [2].",
"In recent years, several direct detection experiments (DAMA/LIBRA [3], CoGeNT [4], [5], [6], CRESST [7] and the CDMS-II Si experiment [8]) have found signals that may suggest the existence of light DM with mass around 10 GeV.",
"On the other hand, experiments such as XENON [9], [10], LUX [11], SIMPLE [12], CDMS [13], [14] and SuperCDMS [15], [16] have not found any excess of events that can be interpreted as signals from DM.",
"In particular, the LUX experiment has probed the relevant region of parameters at the highest level of sensitivity and excluded most regions favored by the possible signals of light DM.",
"It has been shown that it is difficult to accommodate positive signals of direct detection experiments and bounds from Xenon-based experiments by considering astrophysical alternatives (e.g., modified halo models) or varying assumptions about the Xenon scintillation efficiencies [17], [18], [19].",
"A scenario that still remains viable in reconciling some of these results is the isospin–violating DM [20], [21], [22], [23], [24], [25], [26].",
"As different types of nuclei are used in different direct detection experiments, isospin–dependent interactions may happen to interfere destructively for a certain type of nuclei, and thus suppress the DM–nucleus scattering cross section.",
"As LUX experiment [11] currently imposes the most stringent bound on DM, one necessarily considers DM that has negligible interaction with the Xenon nucleus.",
"Recent studies after the LUX result [17], [18], [27], [19] have shown that the isospin–violating DM is still compatible with one of positive signals, those of the CDMS-II Si experiment.",
"More recently, SuperCDMS Collaboration reported their first result for the WIMP search using their background rejection capabilities [16].",
"As we shall see, the isospin–violating DM scenario is severely constrained also by SuperCDMS, but there is still a viable region of parameter space.",
"In this paper, we study a minimal extension of the SM with isospin–violating DM, assuming that the isospin–violating interaction of the DM is mediated by colored particles.For recent studies on DM models with colored mediators, see, e.g., Refs.",
"[28], [29], [30], [31], [32].",
"We investigate the phenomenology of the model, including collider searches as well as flavor and CP physics, paying particular attention to the parameter region which is consistent with CDMS–Si, LUX and SuperCDMS results.",
"We show that a minimal viable model includes scalar DM and new colored vector-like fermions with masses of $O(1)$ TeV as mediators.",
"The colored vector–like fermions can be tested at the 14 TeV LHC.",
"On the other hand, the flavor and CP constraints severely restrict the parameters of the model.",
"We also show that fermionic DM models with colored scalar mediators are disfavored by the LHC constraints.",
"The remaining of the paper is organized as follows.",
"In Section , we study effective operators involving SM and DM fields that reproduce the direct detection experimental results.",
"We then study bounds on these operators from collider search and indirect detection.",
"In Section , we introduce a simple model involving only DM and colored mediators as new particles, that can reproduce the effective operators studied in Section .",
"We study the current bound (8 TeV LHC) on the colored mediators and their prospects of discovery for 14 TeV LHC.",
"In Section , we examine flavor and CP constraints on this model.",
"Section is devoted to conclusions.",
"In Appendix , we briefly discuss models of isospin–violating fermionic DM mediated by colored scalars."
],
[
"Direct detection",
"We study the case in which the DM interaction is dominated by spin-independent interaction.It is difficult to interpret the CDMS-Si signal as spin-dependent scattering of isospin-violating DM [33].",
"In the non-relativistic limit, the elastic scattering cross section of DM with a nucleus composed of $Z$ protons and $(A-Z)$ neutrons can be represented as $\\sigma _A &\\simeq \\frac{\\mu _A^2}{4 \\pi m_{\\rm DM}^2}\\left[ f_p Z + f_n (A-Z) \\right]^2,$ where $\\mu _A=m_A m_{\\rm DM}/(m_A + m_{\\rm DM})$ is the reduced mass, $m_A$ is the mass of the nucleus and $m_{\\rm DM}$ is the mass of the DM.",
"$f_n$ and $f_p$ parametrize the coupling between DM and neutron and proton, respectively.",
"Their explicit forms in terms of Lagrangian parameters are shown in the following subsections.",
"An isospin-conserving interaction corresponds to $f_n=f_p$ .",
"If the isospin is violated and the ratio of the couplings satisfy a relation $f_n/f_p\\simeq -Z/(A-Z)$ , the cross section $\\sigma _A$ is suppressed.",
"In particular, the DM–Xenon interaction is suppressed for $f_n/f_p\\simeq -0.7$ .",
"Given the very severe bound from LUX experiment, it is important to include the effects of multiple isotopes [24], which leads to $\\sigma _A &= \\frac{1}{4 \\pi m_{\\rm DM}^2}\\sum _i\\eta _i \\mu _{A_i}^2\\left[ f_p Z + f_n (A_i-Z) \\right]^2,$ where $\\eta _i$ is the natural abundance of the $i$ -th isotope.",
"Results of direct detection experiments are often quoted in terms of “normalized-to-nucleon cross section,\" which is given by $\\sigma _N^{(Z)} &= \\frac{\\mu _p^2}{\\sum _i \\eta _i \\mu _{A_i}^2 A_i^2} \\sigma _{A}=\\frac{\\sum _i \\eta _i \\mu _{A_i}^2 \\left[ Z + (f_n/f_p) (A_i - Z)\\right]^2}{\\sum _i \\eta _i \\mu _{A_i}^2 A_i^2}\\sigma _p\\,.$ In the isospin conserving case, $f_n = f_p$ , this is equal to DM–proton cross section $\\sigma _p$ .",
"The DM–Xenon scattering cross section is minimized for $f_n/f_p\\simeq -0.7$ .",
"Figure: Favored and excluded regions in isospin-violating DM withf n /f p =-0.7f_n/f_p=-0.7.",
"Shaded regions show 68% and 90% confidence levelcontours for a possible signal from the CDMS-Siresult .",
"Black solid, blue dashed, and blue solid linesrepresent the exclusion contours from LUX ,CDMSlite , and the recent SuperCDMS experiments, respectively.The red point represent the benchmark point used in our analysis.In Fig.",
"REF , we show the parameter regions in $(m_{\\rm DM},\\sigma _p)$ plane for $f_n/f_p=-0.7$ , which are favored by CDMS-Si, and excluded by LUX and SuperCDMS.Among many direct detection experimental results, we show in Fig.",
"REF only positive signals from CDMS–Si and bounds from LUX and SuperCDMS.",
"This is because LUX and SuperCDMS give the most stringent constraints on isospin–violating DM with $f_n/f_p=-0.7$ , and only CDMS–Si has significant region of parameters that is not excluded by these bounds.",
"In the following analysis, we consider the following representative point of isospin–violating DM: $m_{\\rm DM}=8~\\text{GeV},\\;\\sigma _p=4\\times 10^{-40}\\text{cm}^2,\\;f_n/f_p = -0.7.$ One can see that this point is marginally allowed by LUX and SuperCDMS, and is favored by CDMS-Si."
],
[
"Effective interactions between quarks and dark matter",
"The particle DM can be a real or complex scalar field $\\phi $ (and $\\phi ^*$ if complex).",
"It can also be a Majorana or a Dirac fermion $\\chi $ .",
"Assuming that the scattering with nucleon is dominated by spin-independent interaction, there exist only six effective operators at the quark level as listed in Table REF .",
"In the present scenario, the energy scales relevant for collider physics, dark matter detection, and CP / flavor physics are different.",
"In our calculation, however, renormalization group effects on the Wilson coefficients are neglected.",
"In this section, we consider only the couplings of DM to up- and down-type quarks, since they give the dominant isospin–violating effects.",
"In the table, we also express the parameters $f_n$ and $f_p$ in terms of the effective couplings $C_q$ , where $B^{(N)}_q = \\langle N | \\bar{q}q | N\\rangle = m_Nf_{T_q^{(N)}}/m_q$ ($N=p, n$ ) are neutron and proton scalar matrix elements.",
"In our numerical analysis, we use the following values: $B_d^{(p)}/B_u^{(p)} = B_u^{(n)}/B_d^{(n)} = 0.80$ [34] and $B_u^{(p)}+B_d^{(p)} = B_u^{(n)}+B_d^{(n)} = 2\\sigma _{\\pi N}/(m_u+ m_d)$ with $\\pi $ -nucleon sigma term $\\sigma _{\\pi N}\\simeq 64$ MeV [34] and light quark mass $(m_u + m_d)/2\\simeq 3.5$ MeV [35].",
"In order to reproduce the DM–nucleon cross section of the representative point in Eq.",
"(REF ), the effective couplings $C_q$ in Table REF for each scenario are determined as: $C_{u}^{\\text{(R,Cs)}}&\\simeq -1.04\\times C_{d}^{\\text{(R,Cs)}} \\simeq (68~\\text{TeV})^{-1},\\\\C_{u}^{\\text{(M,Ds)}}&\\simeq -1.04\\times C_{d}^{\\text{(M,Ds)}} \\simeq (1050~\\text{GeV})^{-2},\\\\C_{u}^{\\text{(Cv,Dv)}}&\\simeq -1.13\\times C_{d}^{\\text{(Cv,Dv)}}\\simeq (720~\\text{GeV})^{-2}.$"
],
[
"LHC bounds on the effective operators",
"The DM-SM effective operator approach applied to collider physics has been useful in complementing direct and indirect probes of DM [36], [37], [38], [39], [40], [41], [42], [43].",
"When the DM production at colliders is accompanied by a jet from initial state radiation, the signature will be a jet (mono–jet) with missing transverse energy (MET).",
"The ATLAS and CMS collaborations have also performed searches on mono–photon plus MET, mono–lepton and mono–$W$ or –$Z$ plus MET.",
"These searches currently provide the most stringent collider bounds on DM [44], [45], [46], [47], [48], [49], [50].",
"To see how severely the isospin–violating DM model is constrained by the LHC data, we have calculated the cross section for these processes.",
"Here, we assume that one of the operators listed in Table REF dominates the signal process.",
"For the mono–$W$ and –$Z$ events, our analysis is based on the ATLAS study given in Ref.",
"[50], which utilizes the hadronic decay modes of $W$ and $Z$ boson.",
"We have generated the signal events using MadGraph 5 [51], assuming the existence of one of the operators given in Table REF .",
"We apply, in accord with [50], the following cuts at the parton level: $p_T^{W, Z} > 250$ GeV, where $p_T^{W, Z}$ is the transverse momentum of $W$ or $Z$ , $|\\eta |^{W, Z} < 1.2$ , where $\\eta ^{W, Z}$ is the pseudo-rapidity, $\\sqrt{y} > 1.2$ , where $\\sqrt{y}={\\rm min}(p_{T1},p_{T2})\\Delta R/m_{\\rm jet}$ , with $p_{Ti} (i=1\\ {\\rm or}\\ 2)$ being the transverse momentum of jet from the decay of $W$ or $Z$ , $\\Delta R$ the distance between jets, and $m_{\\rm jet}$ the calculated mass of the jet.",
"The fiducial efficiency (63 %) has been taken into account as well.",
"Upper bounds on the dimensionful couplings of the effective operators in Table REF are obtained based on the observed upper limits on the cross section at 95 % CL in Ref. [48].",
"We have also calculated the cross section for the mono–jet events.",
"(For the mono–jet bounds on isospin–violating DM, see also [52], [26].)",
"To make a comparison with the ATLAS mono–jet search at 7 TeV [48],Results at 8 TeV [49] do not have significant improvements compared to the limits obtained in [48].",
"we calculate the parton-level cross section with the following cuts on the mono–jet momentum: $p_T > 80$ GeV, $|\\eta | < 2.0$ .",
"The parton-level cross section is multiplied by the signal acceptance.",
"(Here, we also include the efficiency of the detector, which is taken to be 83 % [48].)",
"In [48], the signal acceptance for the cases with ${\\cal O}^{\\text{(R)}}$ , ${\\cal O}^{\\text{(Cs)}}$ , ${\\cal O}^{\\text{(M)}}$ and ${\\cal O}^{\\text{(Ds)}}$ are not presented.",
"For these cases, we use the acceptance for D5 model given in [48].",
"(Notice that the scalar interactions considered in [48], i.e., D1 model, are proportional to the quark masses and the effect of $c$ -quark is important.",
"Thus, we do not use the acceptance of the D1 model in our analysis.)",
"We found that, among several signal regions [48], the one corresponding to $p_T > 350$ GeV (SR3) gives the most stringent bounds to the present model.",
"Comparing our estimations of the cross sections with the observed 95 % CL limit on the “visible cross section” given in [48], we derive upper bounds on the coefficients of the effective operators listed in Table REF .",
"The bounds are given in Table REF .",
"Here, we show the results based on the mono–$W$ and $Z$ events and mono–jet events separately.",
"We can see that the mono–$W$ and –$Z$ processes impose more stringent constraints than the mono-jet process.",
"One of the reasons is that in the isospin–violating DM model, there exists the relative minus sign between the coupling of DM to $u$ - and $d$ -quarks; it results in a constructive interference between two Feynman diagrams for the mono–$W$ production process that greatly enhances the cross section [43].",
"Table: Upper bounds on the coefficients ofthe effective operatorsobtained from mono–WW or –ZZand mono–jet searches.Comparing Eqs.",
"(REF )–() with Table REF , the CDMS-Si point with the vector–type effective operators ${\\cal O}^{\\text{(Cv)}}$ and ${\\cal O}^{\\text{(Dv)}}$ are disfavored.",
"On the other hand, scalar–type interactions, ${\\cal O}^{\\text{(R)}}$ , ${\\cal O}^{\\text{(Cs)}}$ , ${\\cal O}^{\\text{(M)}}$ and ${\\cal O}^{\\text{(Ds)}}$ , are still viable.",
"In the next section we introduce a simple model which can reproduce the effective operators ${\\cal O}^{\\text{(R)}}$ and ${\\cal O}^{\\text{(Cs)}}$ at low energy.",
"(For fermionic DM with effective operators ${\\cal O}^{\\text{(M)}}$ and ${\\cal O}^{\\text{(Ds)}}$ , see Appendix .)",
"Before closing this section, let us comment on the validity of the effective field theory (EFT) approach.",
"The effective operators at low energy are generated by a UV theory, typically by exchanges of heavy mediators.",
"At the LHC, the energy scale of the process can be comparable to or larger than the scale of the UV theory.",
"In such a case, the bound obtained by using EFT may not be valid [53], [54], [55], [28], [29], [30], [31], [56], [32].",
"However, when the effective operators are induced by exchanges of heavy colored mediators, the bound obtained by EFT is typically weaker than the bound obtained by concrete UV models, i.e., EFT gives conservative bounds [32].",
"Thus, we consider the constraints obtained in this subsection as conservative ones, and discuss UV models for the operators which are not disfavored at the level of EFT."
],
[
"Thermal abundance and indirect search",
"Another important check point is the relic abundance.",
"Although we have assumed the correct DM abundance, the thermal relic density in our model is larger than the present DM density.",
"The thermal relic density is determined by the thermally-averaged pair annihilation cross section $\\langle \\sigma _{\\rm ann} v_{\\rm rel} \\rangle $ as $\\Omega _{\\rm thermal} \\simeq 0.2\\times \\left(\\frac{\\langle \\sigma _{\\rm ann} v_{\\rm rel} \\rangle }{1\\ {\\rm pb}}\\right)^{-1}.$ We show $\\langle \\sigma _{\\rm ann} v_{\\rm rel} \\rangle $ in Table REF for the cases where the $s$ -wave annihilation processes dominate.",
"(For other cases, the annihilation is via $p$ -wave processes, with which the cross sections are much smaller.)",
"Substituting the cross sections in the table into Eq.",
"(REF ), we can see that $\\Omega _{\\rm thermal}$ becomes larger than the present density parameter of DM.",
"Thus, we need to consider non-thermal production of DM at $T\\ll m_{\\rm DM}$ in the present scenario.",
"Table: Thermally averaged total pair annihilation crosssections for the cases with the operators 𝒪 (R) {\\cal O}^{\\text{(R)}},𝒪 (Cs) {\\cal O}^{\\text{(Cs)}}, and 𝒪 (Dv) {\\cal O}^{\\text{(Dv)}}.For other cases, the cross sections arepp-wave suppressed, and are much smaller.We also comment on the upper bound on $\\langle \\sigma _{\\rm ann} v_{\\rm rel} \\rangle $ from the observations of Milky-Way satellites by Fermi Large Area Telescope (LAT) [57], [58], [52], [59].",
"With the latest analysis of the $\\gamma $ -ray flux from the satellites [59], the observed upper bound on the pair annihilation cross section into $u\\bar{u}$ or $d\\bar{d}$ final state is about 0.8 pb for $m_{\\rm DM}=10\\ {\\rm GeV}$ .",
"As one can see in Table REF , the annihilation cross section in the present model is an order of magnitude smaller than the Fermi-LAT bound."
],
[
"Colored Mediators of Isospin–Violating Dark Matter",
"As we have seen in the previous section, isospin–violating DM with scalar–type interaction can explain the possible CDMS-Si signal while avoiding the LUX and SuperCDMS constraints as well as the LHC mono–jet and mono–$W$ /$Z$ constraints.",
"In this section we discuss the UV completion of the scalar–type effective couplings.",
"In particular, as mentioned in Introduction, we concentrate on the case that the effective operators are induced by exchanges of heavy colored particles, since they can easily accommodate isospin–violating interactions.",
"For recent studies on other possibilities of isospin–violating DM models, see, for example, [60], [61], [62].",
"As shown in Appendix , fermionic DM models require a light colored scalar with a mass smaller than $O(500)~\\text{GeV}$ , and such a model is already excluded by LHC squark search [63].",
"Thus, in the following discussion, we concentrate on real and complex scalar DM."
],
[
"Model",
"We introduce extra vector-like quarks $Q$ , $U$ , and $D$ , which mediate the coupling between scalar DM and the SM quarks.",
"The matter content and their quantum numbers are summarized in Table REF .",
"(We also list the SM quarks to fix the notation.)",
"We impose a $Z_2$ symmetry to ensure the stability of DM.",
"Table: Quantum numbers of DM, coloredmediators QQ, UU, and DD(and SM quarks q L q_L, u R u_R, and d R d_R).The mass and interaction terms of the new colored fields are given by $-{\\cal L}_{Q,U,D}=&M_Q \\bar{Q}Q+ M_U \\bar{U}U+ M_D \\bar{D}D\\nonumber \\\\& +\\left(\\lambda _Q^i\\; \\phi \\; \\overline{q_L^i} P_R Q+\\lambda _U^i\\; \\phi \\; \\overline{u_R^i} P_L U+\\lambda _D^i\\; \\phi \\; \\overline{d_R^i} P_L D\\right.\\nonumber \\\\&+y_{U_L} H^\\dagger _a \\overline{Q_a} P_L U+y_{D_L} \\epsilon ^{ab} H_a \\overline{Q_b} P_L D\\nonumber \\\\&\\left.+y_{U_R} H^\\dagger _a \\overline{Q_a} P_R U+y_{D_R} \\epsilon ^{ab} H_a \\overline{Q_b} P_R D\\; +\\text{H.c.}\\right).$ The index $i$ stands for the generation of SM quarks.",
"Note that Yukawa couplings between colored mediators and Higgs field is necessary in order to induce a scalar type effective operator between DM and SM quarks, ${\\cal L}_{\\text{eff}}\\sim \\phi \\phi \\bar{q}q$ .",
"Therefore, after the electroweak symmetry is broken, the two up-type colored mediators mix with a mass matrix ${\\cal M_U}=\\left(\\begin{array}{cc}M_U&y_{U_L}^* v\\\\y_{U_R} v&M_Q\\end{array}\\right),$ where $v\\simeq 174$ GeV is the Higgs vacuum expectation value.",
"The two down-type mediators also mix in a similar way.",
"In the case of complex scalar DM, we impose a global U(1) symmetry, where only the DM and colored mediators are charged; the Lagrangian (REF ) has such a symmetry.",
"As we will see, this U(1) symmetry makes phenomenology of complex scalar DM and real one different.",
"In general, DM can couple to all three generations of SM quarks.",
"In addition, there are CP phases of the couplings in Eq.",
"(REF ) which cannot be removed by field redefinitions.",
"These flavor–changing and CP–violating couplings are severely constrained.",
"We will discuss these issues in detail in Section .",
"In this and next subsection, we assume that the couplings to the first generation are dominant, and neglect the effects of the couplings to the second and third generations.",
"(We will omit the generation index $i$ from the coupling constants to the first generation quarks until Section REF .)",
"For the study of the signals at LHC, for simplicity, we take the following parametrization: $& M_Q = M_U = M_D \\equiv M\\,,\\\\& \\lambda _Q = \\lambda _U = \\lambda _D \\equiv \\lambda \\,,\\\\& y_{U_L} = y_{U_R} \\equiv y_U\\,,\\\\& y_{D_L} = y_{D_R} \\equiv y_D\\,,$ where all parameters are taken to be real.",
"Then, the effective coupling constants in Table REF are given by $C_u^{(\\text{R})} = \\frac{2\\lambda ^2 y_U v}{M^2-y_U^2 v^2},~~C_d^{(\\text{R})} = \\frac{2\\lambda ^2 y_D v}{M^2-y_D^2 v^2},~~C_u^{(\\text{Cs})} = \\frac{\\lambda ^2 y_U v}{M^2-y_U^2 v^2},~~C_d^{(\\text{Cs})} = \\frac{\\lambda ^2 y_D v}{M^2-y_D^2 v^2}.$ In our analysis, we take $C_u^{(\\text{R,Cs})} \\simeq (68~\\text{TeV})^{-1}$ (see Eq.",
"(REF )).",
"In Fig.",
"REF , on $(M, \\lambda )$ plane, we show contours on which we obtain $C_u^{(\\text{R,Cs})}=(68~\\text{TeV})^{-1}$ , taking $y_U=0.1$ and 1.",
"As one can see, the masses of the vector-like quarks must be $O(1)~\\text{TeV}$ as far as all the coupling constants are within the perturbative regime.",
"Figure: Contour of the total cross section σ tot \\sigma _{\\rm tot} forthe pair production of colored mediators, at leading order in(M,λ)(M,\\lambda ) plane, for real and complex scalarDM.",
"Blue lines show the contoursof σ tot =0.02\\sigma _{\\rm tot}=0.02, 0.01, and 0.005 pb at s=8\\sqrt{s}=8 TeVfrom the left to right.",
"Red lines show the contours ofσ tot =0.01\\sigma _{\\rm tot}=0.01 and 0.001 pb at s=14\\sqrt{s}=14 TeV from theleft to right.",
"Note that Yukawa couplings y U y_U and y D y_D areadjusted through Eqs.",
"(), in order to reproducethe direct detection cross section.",
"Two black solid lines in each figure showthe contours of y U =0.1y_U=0.1 and 1 from top to bottom."
],
[
"Direct production of colored mediators at LHC",
"Now we are at the position to discuss the LHC constraints/prospects of the colored mediators.",
"In the present scenario, colored mediators are pair-produced at LHC.",
"Here, there is an important difference between the real and complex scalar DM scenarios.",
"In the former case, the processes $pp\\rightarrow {\\cal Q} \\bar{\\cal Q}$ and ${\\cal Q} {\\cal Q}$ both occur, where ${\\cal Q}$ collectively denotes colored mediators while $\\bar{\\cal Q}$ is the anti-particles.",
"In the case of complex scalar DM, on the contrary, $pp\\rightarrow {\\cal Q} {\\cal Q}$ is forbidden, so that the relevant processes are only the pair production of ${\\cal Q}$ and $\\bar{\\cal Q}$ .",
"Notice that the amplitudes with $t$ -channel exchange of DM can enhance the cross section in the present scenario.",
"Once produced, the colored mediators decay into the SM quarks and the DM particle, so the important processes are The process $pp\\rightarrow {\\cal Q}\\phi +j$ also contributes to the events with two jets plus missing energy.",
"Transverse momenta of the emitted jets tend to be smaller than that given by the pair productions in this process.",
"Therefore, the contribution becomes sub-dominant with tighter $p_T$ cuts used in Ref. [63].",
"Since we have neglected these processes, the above bounds are conservative.",
"$pp\\rightarrow \\left\\lbrace \\begin{array}{l}{\\cal Q} \\bar{\\cal Q} \\rightarrow q \\phi ^{(*)}\\; \\bar{q} \\phi ,\\\\{\\cal Q} {\\cal Q} \\rightarrow q \\phi \\; q\\phi ~~~\\mbox{(only for real scalar DM)},\\\\\\bar{\\cal Q} \\bar{\\cal Q} \\rightarrow \\bar{q} \\phi \\; \\bar{q}\\phi ~~~\\mbox{(only for real scalar DM)},\\end{array}\\right.$ where $q$ denotes SM quarks while $\\bar{q}$ is its anti-particle.",
"Thus, the LHC signal is two jets plus missing energy.",
"In Fig.",
"REF , we show the contour of total cross section $\\sigma _{\\rm tot}$ for the pair production of the colored mediators for $\\sqrt{s}=8\\ {\\rm TeV}$ , where $\\sigma _{\\rm tot}$ is calculated by MadGraph 5 [51] at the leading-order and is given by $\\sigma _{\\rm tot} =\\left\\lbrace \\begin{array}{ll}\\sigma (pp\\rightarrow {\\cal Q} \\bar{\\cal Q})+ \\sigma (pp\\rightarrow {\\cal Q} {\\cal Q})+ \\sigma (pp\\rightarrow \\bar{\\cal Q} \\bar{\\cal Q})&:~ \\mbox{real scalar DM},\\\\\\sigma (pp\\rightarrow {\\cal Q} \\bar{\\cal Q})&:~ \\mbox{complex scalar DM}.\\end{array}\\right.$ The di-jet signal with missing energy is studied both at ATLAS and CMS, particularly in the context of supersymmetric (SUSY) models.",
"In the ATLAS analysis [63], a simplified SUSY model is studied, where only first two generation squarks and the lightest neutralino are potentially accessible to LHC while all other SUSY particles (including the gluino) are heavy.",
"In such a model, the lower bound on the common squark mass is 780 GeV, corresponding to the leading-order squark production cross section of 0.013 pb.",
"In general, this value cannot be directly compared with the prediction of the present model because the signal efficiency (i.e., the fraction of signal events which pass the cuts in Ref.",
"[63]) may be different.",
"By using the parton-level analysis with MadGraph 5 [51], we estimated the efficiency for our model as well as that for the simplified SUSY model with a squark mass of 780 GeV.",
"Then, we found that the former is comparable to or larger than the latter.",
"Thus, we translate the ATLAS constraint on the simplified SUSY model to derive a conservative bound on the present model; assuming that $\\sigma _{\\rm tot}$ should be smaller than $\\sim 0.01\\ {\\rm pb}$ and O(1) couplings $\\lambda , y\\lesssim 1$ , $M$ is bounded from below as $M\\gtrsim 1-1.5$ TeV ($1-1.1$ TeV) for real (complex) scalar DM, depending on the coupling $\\lambda $ .",
"Before closing this section, let us discuss the future prospects of the present model.",
"In Fig.",
"REF , we also show the contour of $\\sigma _{\\rm tot}$ at $\\sqrt{s}=14$ TeV.",
"At 14 TeV LHC the sensitivity of the search with two-jets plus missing energy may reach $O$ (0.003) pb and $O$ (0.001) pb or larger, for the integrated luminosities of 300 fb$^{-1}$ and 3000 fb$^{-1}$ , respectively [64].",
"One can see that a large region of the parameter space, possibly above $M\\simeq 3$ TeV (2 TeV) for the case of real (complex) scalar DM, may be covered at 14 TeV LHC."
],
[
"Flavor and CP Constraints",
"In the previous section, we have discussed the LHC phenomenology of the isospin–violating DM model with colored mediators.",
"In the present scenario, the interaction of DM may significantly affect flavor and CP observables, which give very stringent constraints on the present model.",
"We concentrate on the case with scalar DM, since isospin–violating fermionic DM with colored mediators is stringently constrained by the the LHC bounds, as shown in Appendix ."
],
[
"Up- and down-quark masses",
"First we discuss the radiative correction to the SM Yukawa coupling constants in the present model.",
"In particular, we concentrate on the Yukawa coupling constants of up- and down-quarks (which we denote $y_u$ and $y_d$ ) on which the corrections are the most significant.",
"If $y_{U_L}$ or $y_{D_L}$ is non-vanishing, there exist logarithmically-divergent 1-loop contributions to $y_u$ or $y_d$ .",
"Then, the $\\beta $ -functions of the up– and down–quark Yukawa coupling constants become $\\frac{d y_u}{d \\log \\mu } & = \\frac{1}{8\\pi ^2}\\lambda _Q \\lambda _U y_{U_L} + \\cdots ,\\\\\\frac{d y_d}{d \\log \\mu } & = \\frac{1}{8\\pi ^2}\\lambda _Q \\lambda _D y_{D_L} + \\cdots ,$ where $\\mu $ is the renormalization scale and “$\\cdots $ ” are terms proportional to $y_u$ or $y_d$ .",
"The important point is that the $\\beta $ -functions contain terms which are not proportional to the SM Yukawa coupling constants.",
"Consequently, the smallness of $y_u$ and $y_d$ may be affected in particular when the coupling constants in the DM sector are relatively large.",
"As shown in the previous section, the present scenario requires large values of $\\lambda _{Q,U,D}$ and $y_{U,D}$ (cf.",
"Fig.",
"REF ).",
"Thus, we expect significant contribution to the up- and down-quark Yukawa coupling constants from the DM sector.",
"The low-energy values of the Yukawa coupling constants, which are directly related to the up- and down-quark masses, are given by $y_u (\\mu \\ll M_Q) & \\sim y_u (M_*)+ \\frac{1}{8\\pi ^2} \\lambda _Q \\lambda _U y_{U_L}\\log \\frac{M_Q}{M_*} + \\cdots ,\\\\y_d (\\mu \\ll M_Q) & \\sim y_d (M_*)+ \\frac{1}{8\\pi ^2} \\lambda _Q \\lambda _D y_{D_L}\\log \\frac{M_Q}{M_*} + \\cdots ,$ where $M_*$ is the cut-off scale at which the boundary conditions are given.",
"If $\\lambda _{Q,U,D}\\sim y_{U_L,D_L}\\sim 1$ , the second terms in Eqs.",
"(REF ) and () are estimated to be larger than $O(10^{-2})$ , which is much larger than the SM values of those Yukawa coupling constants.",
"In order to realize the Yukawa coupling constants compatible with the up– and down–quark masses, such contributions should be cancelled by $y_{u,d} (M_*)$ , which requires a significant tuning between those two unrelated quantities.",
"For the scenario of isospin–violating DM, in fact, $y_{U_L}$ and $y_{D_L}$ may vanish; in order to generate the operator $\\phi \\phi \\bar{q}q$ , we only need $y_{U_R}$ and $y_{D_R}$ .",
"They also affect the up- and down-quark Yukawa coupling constants.",
"The contributions which are proportional to $y_{U_R}$ and $y_{D_R}$ are finite, and are given by $\\Delta y_u & = \\frac{1}{8 \\pi ^2}\\lambda _Q \\lambda _U y_{U_R}\\frac{M_Q M_U}{M_Q^2 - M_U^2} \\log \\frac{M_U}{M_Q},\\\\\\Delta y_d & = \\frac{1}{8 \\pi ^2}\\lambda _Q \\lambda _U y_{D_R}\\frac{M_Q M_D}{M_Q^2 - M_D^2} \\log \\frac{M_D}{M_Q},$ which are still much larger than the SM values of up- and down-quark Yukawa coupling constant if $\\lambda _{Q,U,D} \\sim y_{U_R,D_R}\\sim 1$ .",
"Thus, the serious tunings of the counter terms of the Yukawa coupling constants are unavoidable in the present model."
],
[
"Electric dipole moment of neutron",
"Next, we consider the electric dipole moment (EDM) of the neutron.",
"If the newly introduced coupling constants have phases, which is the case in general, they become a new source of CP violations.",
"In the present model, the DM sector necessarily couple to the first generation quarks, so the important check point is the neutron EDM.",
"In order to see how large the neutron EDM becomes, we calculate the coefficients of the EDM and chromo-EDM (CEDM) operators of up- and down-quarks: ${\\cal L}_{\\rm (C)EDM} = \\frac{i}{2}\\sum _{f=u,d}\\left[d_f F_{\\mu \\nu } \\bar{f} \\sigma _{\\mu \\nu } \\gamma _5 f+ g_3 \\tilde{d}_f G_{\\mu \\nu }^{(a)} T^a_{\\alpha \\beta }\\bar{f}_\\alpha \\sigma _{\\mu \\nu } \\gamma _5 f_\\beta \\right],$ where $F_{\\mu \\nu }$ and $G_{\\mu \\nu }^{(a)}$ are field-strength tensors of photon and gluon, respectively, $g_3$ is the strong gauge coupling constant, and $T^a_{\\alpha \\beta }$ is the generator for $SU(3)_C$ (with $\\alpha $ and $\\beta $ being color indices, while $a$ being index for the adjoint representation).",
"With the (C)EDMs of quarks being given, the neutron EDM is estimated as [65] $d_n = - 0.12 d_u + 0.47 d_d+ e ( -0.18 \\tilde{d}_u + 0.18 \\tilde{d}_d ).$ (The numerical uncertainties in QCD parameters may change the above formula by $\\sim 10\\ \\%$ [65].",
"The conclusion of this subsection is, however, unaffected by such an uncertainty.)",
"As shown in the previous section, the LHC bounds require that the masses of the colored mediators should be much larger than the Higgs VEV, $M_{Q,U,D} > v$ .",
"In such a case, the coefficients of the (C)EDM operators can be expanded in powers of the Higgs VEV, and we only keep the leading-order terms in $v$ .",
"In the limit of $m_\\phi \\ll M_{Q,U,D}$ (with $m_\\phi $ being the mass of the scalar DM) we obtain $d_u & = \\frac{1}{32 \\pi ^2} \\frac{e Q_U v}{M_Q M_U}\\Im (\\lambda _Q \\lambda _U^* y_{U_R}),\\\\\\tilde{d}_u & = \\frac{1}{32 \\pi ^2} \\frac{v}{M_Q M_U}\\Im (\\lambda _Q \\lambda _U^* y_{U_R}),$ and $d_d$ and $\\tilde{d}_d$ are obtained from $d_u$ and $\\tilde{d}_u$ by replacing the subscripts as $U\\rightarrow D$ .",
"Here, $e$ is the electric charge, $Q_U=\\frac{2}{3}$ , and $Q_D=-\\frac{1}{3}$ .",
"We note here that, at the leading order in $v$ , the contribution proportional to $\\Im (\\lambda _Q \\lambda _U^* y_{U_L})$ vanishes.",
"Taking $M_Q=M_U=M_D$ for simplicity, we obtain $d_n \\simeq \\left[-2.8 \\times 10^{-21} e\\ {\\rm cm}\\times \\Im (\\lambda _Q \\lambda _U^* y_{U_R})+2.5 \\times 10^{-22} e\\ {\\rm cm}\\times \\Im (\\lambda _Q \\lambda _D^* y_{D_R})\\right]\\left( \\frac{1\\ {\\rm TeV}}{M_Q} \\right)^2.$ This should be compared with the present bound on the neutron EDM, which is given by [35] $|d_n| < 0.29 \\times 10^{-25} e\\ {\\rm cm}.$ Thus, the neutron EDM provides a very severe constraint on the complex phases of the couplings, $\\Im (\\lambda _Q \\lambda _{U}^* y_{U_R})\\lesssim O(10^{-5}-10^{-4})$ and $\\Im (\\lambda _Q \\lambda _{D}^* y_{D_R})\\lesssim O(10^{-4}-10^{-3})$ , for $M_Q\\simeq O(1-3)$ TeV."
],
[
"$K$ -{{formula:bd8c3dd3-a76b-4b57-ada7-f89f190041cb}} mixing",
"In the present analysis, we introduced only one set of vector-like fermions (i.e., $Q$ , $U$ , and $D$ ) for minimality.",
"No symmetry forbids their interactions with second- and third-generation quarks.",
"Such interactions in general induce unwanted CP and flavor violations; it is often the case that the $K$ -$\\bar{K}$ mixing parameters, i.e., $\\epsilon _K$ and $\\Delta m_K$ , give stringent constraints.",
"Thus, we consider them in this subsection.",
"The effective $\\Delta S=2$ Hamiltonian can be described as ${\\mathcal {H}}_{\\rm eff} = \\sum _{i=1}^3\\left( C_{L,i} {\\mathcal {Q}}_{L,i}+ C_{R,i} {\\mathcal {Q}}_{R,i} \\right)+ \\sum _{i=4}^5 C_i {\\mathcal {Q}}_i,$ where the operators are ${\\mathcal {Q}}_{L,1} &= ( \\bar{d}_\\alpha \\gamma _\\mu P_L s_\\alpha )( \\bar{d}_\\beta \\gamma ^\\mu P_L s_\\beta ) ,\\\\{\\mathcal {Q}}_{L,2} &= ( \\bar{d}_\\alpha P_L s_\\alpha )( \\bar{d}_\\beta P_L s_\\beta ) ,\\\\{\\mathcal {Q}}_{L,3} &= ( \\bar{d}_\\alpha P_L s_\\beta )( \\bar{d}_\\beta P_L s_\\alpha ),\\\\{\\mathcal {Q}}_4 &= ( \\bar{d}_\\alpha P_L s_\\alpha )( \\bar{d}_\\beta P_R s_\\beta ),\\\\{\\mathcal {Q}}_5 &= ( \\bar{d}_\\alpha P_L s_\\beta )( \\bar{d}_\\beta P_R s_\\alpha ),$ and ${\\mathcal {Q}}_{R,i}=[{\\mathcal {Q}}_{L,i}]_{L\\rightarrow R}$ .",
"We calculate the Wilson coefficients in the present model.",
"As in the case of neutron EDM, we use the mass-insertion approximation and only consider the leading contributions with respect to the insertions of the Higgs VEV.",
"In the case of real scalar DM, sum of the diagrams with no Higgs-VEV insertion vanishes, and the leading contributions are given by $C_{L,2}^{(\\phi : \\rm real)} & = \\frac{1}{16\\pi ^2}\\frac{v^2}{M_Q^4} (\\lambda _D^{s*} \\lambda _Q^{d})^2\\left[y_{D_R}^2 F_0 (x_D, x_\\phi ) +y_{D_R} y_{D_L} F_1 (x_D) +y_{D_L}^2 F_2 (x_D)\\right],\\\\C_{R,2}^{(\\phi : \\rm real)} & = \\frac{1}{16\\pi ^2}\\frac{v^2}{M_Q^4} (\\lambda _Q^{s*} \\lambda _D^{d})^2\\left[y_{D_R}^{*2} F_0 (x_D, x_\\phi ) +y_{D_R}^* y_{D_L}^* F_1 (x_D) +y_{D_L}^{*2} F_2 (x_D)\\right],\\\\C_4^{(\\phi : \\rm real)} & = \\frac{1}{8\\pi ^2}\\frac{v^2}{M_Q^4}\\lambda _Q^{s*} \\lambda _D^{s*}\\lambda _Q^{d} \\lambda _D^{d}\\left[y_{D_R} y_{D_R}^* F_0 (x_D, x_\\phi ) +\\Re (y_{D_R} y_{D_L}^*) F_1 (x_D) +y_{D_L} y_{D_L}^* F_2 (x_D)\\right],$ where $F_0 (x_D, x_\\phi ) &=- \\frac{\\log x_\\phi }{x_D}+ \\frac{-2 x_D^3 + 4 x_D^2 - 4 x_D + (3 x_D-1) \\log x_D + 2}{(x_D-1)^3 x_D},\\\\F_1 (x_D) &=\\frac{-2x_D^2 + 4 x_D \\log x_D + 2}{\\sqrt{x_D} (x_D-1)^3},\\\\F_2 (x_D) &=\\frac{-2 x_D + (x_D+1) \\log x_D+2}{(x_D-1)^3},$ with $x_D\\equiv M_D^2/M_Q^2$ and $x_\\phi \\equiv m_\\phi ^2/M_Q^2$ .",
"(The superscripts $d$ and $s$ of $\\lambda _{Q,D}^{d,s}$ denote the coupling constants to the first and second generations, respectively, cf.",
"Eq.",
"(REF ).)",
"Notice that the above expressions are valid only when $m_\\phi \\ll M_{Q,D}$ .",
"(Other Wilson coefficients vanish at this order.)",
"For complex scalar DM, we obtain $C_{L,1}^{(\\phi : \\rm complex)} = & - \\frac{1}{128 \\pi ^2} \\frac{1}{M_Q^2}(\\lambda _Q^{s*} \\lambda _Q^{d})^2,\\\\C_{R,1}^{(\\phi : \\rm complex)} = & - \\frac{1}{128 \\pi ^2} \\frac{1}{M_D^2}(\\lambda _D^{s*} \\lambda _D^{d})^2,\\\\C_{L,2}^{(\\phi : \\rm complex)} = & \\frac{1}{32\\pi ^2}\\frac{v^2}{M_Q^4} (\\lambda _D^{s*} \\lambda _Q^{d})^2\\left[y_{D_R}^2 F_0 (x_D, x_\\phi ) +y_{D_R} y_{D_L} F_1 (x_D) +y_{D_L}^2 F_2 (x_D)\\right],\\\\C_{R,2}^{(\\phi : \\rm complex)} = & \\frac{1}{32\\pi ^2}\\frac{v^2}{M_Q^4} (\\lambda _Q^{s*} \\lambda _D^{d})^2\\left[y_{D_R}^{*2} F_0 (x_D, x_\\phi ) +y_{D_R}^* y_{D_L}^* F_1 (x_D) +y_{D_L}^{*2} F_2 (x_D)\\right],\\\\C_4^{(\\phi : \\rm complex)} = & \\frac{1}{16\\pi ^2}\\frac{v^2}{M_Q^4}\\lambda _Q^{s*} \\lambda _D^{s*}\\lambda _Q^{d} \\lambda _D^{d}\\nonumber \\\\ &\\left[y_{D_R} y_{D_R}^* F_0 (x_D, x_\\phi ) +\\Re (y_{D_R} y_{D_L}^*) F_1 (x_D) +y_{D_L} y_{D_L}^* F_2 (x_D)\\right],\\\\C_5^{(\\phi : \\rm complex)} = & \\frac{1}{16 \\pi ^2}\\lambda _Q^{s*} \\lambda _D^{s*} \\lambda _Q^{d} \\lambda _D^{d}\\frac{1}{M_Q^2 - M_D^2} \\log \\frac{M_Q}{M_D},$ where we neglected the terms which are higher order in $v$ .",
"(Other Wilson coefficients vanish at this order.)",
"With the Wilson coefficients, we calculate the matrix elements relevant for the study of $K$ -$\\bar{K}$ mixing parameters.",
"Here, our purpose is to obtain semi–quantitative bounds on the model parameters, so we use the vacuum-insertion approximation to evaluate the matrix elements.",
"Then, we obtain [66] $\\langle K | {\\cal H}_{\\rm eff} | \\bar{K} \\rangle = &\\frac{2}{3} (m_K f_K)^2 \\left( C_{L,1} + C_{R,1} \\right)- \\frac{5}{12} \\frac{m_K^2}{m_s^2}(m_K f_K)^2 \\left( C_{L,2} + C_{R,2} \\right)\\nonumber \\\\ &+ \\frac{1}{12} \\frac{m_K^2}{m_s^2}(m_K f_K)^2 \\left( C_{L,3} + C_{R,3} \\right)+ \\left(\\frac{1}{12} + \\frac{1}{2}\\frac{m_K^2}{m_s^2}\\right)(m_K f_K)^2 C_{4}\\nonumber \\\\ &+ \\left(\\frac{1}{4} +\\frac{1}{6} \\frac{m_K^2}{m_s^2}\\right)(m_K f_K)^2 C_{5},$ where $m_K$ is the mass of $K$ , $m_s\\simeq 95\\ {\\rm MeV}$ is the strange-quark mass, and $f_K\\simeq 160\\ {\\rm MeV}$ is the decay constant.",
"With the above matrix element, we estimate the DM sector contributions to the $K$ -$\\bar{K}$ mixing parameters as $|\\epsilon _K^{(\\phi )}| &=\\frac{\\Im \\langle K | {\\cal H}_{\\rm eff} | \\bar{K} \\rangle }{2\\sqrt{2} m_K \\Delta m_K },\\\\\\Delta m_K^{(\\phi )} &=\\frac{1}{m_K}|\\langle K | {\\cal H}_{\\rm eff} | \\bar{K} \\rangle |.$ The numerical values of $\\epsilon _K^{(\\phi )}$ and $\\Delta m_K^{(\\phi )}$ depend on various parameters.",
"Taking $& \\lambda _Q^s = \\lambda _D^2 \\equiv \\lambda ^s,\\\\& y_{D_L} = y_{D_R} \\equiv y_{D},$ as well as the relations given in Eqs.",
"(REF ) – (), for example, we obtain $|\\epsilon _K^{(\\phi : \\rm real)}| \\simeq 6.5\\times 10^3\\times \\Im (\\lambda ^{s*} \\lambda ^d)^2 y_D^2 \\left(\\frac{1~{\\rm TeV}}{M}\\right)^{4} \\left[ 1 + 0.34 \\log \\frac{(M/m_\\phi )}{100} \\right],$ and $|\\epsilon _K^{(\\phi : \\rm complex)}| \\simeq &1.7\\times 10^4\\times \\Im (\\lambda ^{s*} \\lambda ^d)^2 \\left(\\frac{1~{\\rm TeV}}{M}\\right)^{2} \\nonumber \\\\ &+ 3.2\\times 10^3\\times \\Im (\\lambda ^{s*} \\lambda ^d)^2 y_D^2 \\left(\\frac{1~{\\rm TeV}}{M}\\right)^{4} \\left[ 1 + 0.34 \\log \\frac{(M/m_\\phi )}{100} \\right],$ for the cases where $\\phi $ is real and complex, respectively.",
"(Here, we assumed that $y_D$ is real for simplicity.)",
"In addition, with the present choice of parameters, $\\Delta m_K^{(\\phi )} \\simeq 1.5\\times 10^{10}\\ {\\rm sec}^{-1}\\times \\frac{|\\lambda ^{s*} \\lambda ^d|^2}{\\Im (\\lambda ^{s*} \\lambda ^d)^2}|\\epsilon _K^{(\\phi )}|.$ The measured values of the $K$ -$\\bar{K}$ mixing parameters are well explained by the SM, and there exist stringent constraints on the extra contributions to those quantities.",
"Comparing the SM prediction ($\\epsilon _K^{\\rm (SM)} = (1.81\\pm 0.28)\\times 10^{-3}$ [67]) and the experimental value ($\\epsilon _K^{\\rm (exp)} = (2.228\\pm 0.011) \\times 10^{-3}$ [35]), the DM sector contribution to $\\epsilon _K$ is constrained to be $|\\epsilon _K^{(\\phi )}| <9.8\\times 10^{-4}$ .",
"In addition, the experimental value of $\\Delta m_K$ is known to be $\\Delta m_K^{\\rm (exp)}=(0.5293\\pm 0.0009)\\times 10^{10}\\ {\\rm sec}^{-1}$ [35], which we use as an upper bound on $\\Delta m_K^{(\\phi )}$ .",
"Assuming no accidental cancellation among contributions from different Feynman diagrams, the DM sector contributions are likely to become much larger than the upper bounds on those quantities unless some of the coupling constants are much smaller than 1, as indicated by Eqs.",
"(REF ) – (REF )."
],
[
"Conclusion",
"In this paper, we have studied isospin–violating light DM that can explain the possible CDMS-Si signal of light DM, while avoiding the constraints by recent LUX and SuperCDMS experiments.",
"In particular, we considered isospin–violating light DM models with colored mediators.",
"We have shown that a minimal viable model includes scalar DM and new colored vector-like fermions with masses of $O(1)$ TeV as mediators.",
"We investigated the collider searches, flavor and CP phenomenology.",
"The masses of colored mediators are constrained by the 8 TeV LHC results as $M\\gtrsim 1-1.5$ TeV ($1-1.1$ TeV) for real (complex) scalar DM.",
"The 14 TeV LHC may cover a large region of the remaining parameter space.",
"We have also studied flavor and CP constraints on the colored-mediator model for the isospin–violating DM.",
"In such a model, the interaction of quarks with colored mediator and DM should be sizable, which results in large radiative correction to flavor and CP observables.",
"We have studied the effects on the quark masses (in particular, those of up- and down-quarks), EDM of neutron, and the $K$ -$\\bar{K}$ mixing parameters.",
"Radiative corrections to the SM Yukawa couplings from the DM sector are extremely large, and hence fine–tunings are unavoidable.",
"Flavor and CP violating observables also impose severe constraints on the present scenario."
],
[
"Acknowledgment",
"This work was supported by JSPS KAKENHI Grant No.",
"22244021 (K.H., T.M.",
"), No.",
"22540263 (T.M.",
"), No.",
"23104008 (T.M.)",
"and also by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.",
"The work of S.P.L.",
"was supported by the Program for Leading Graduate Schools, MEXT, Japan.",
"The work of Y.Y.",
"has been supported in part by the Ministry of Economy and Competitiveness (MINECO), grant FPA2010-17915, and by the Junta de Andalucía, grants FQM 101 and FQM 6552."
],
[
"Isospin–Violating Fermionic Dark Matter with Colored Scalar Mediators",
"In this appendix, we briefly discuss isospin–violating fermionic DM models with colored scalar mediators.",
"The effective operators $\\mathcal {O}^\\text{(M)}$ and $\\mathcal {O}^\\text{(Ds)}$ in Table REF can be induced by exchanges of colored scalars $\\tilde{Q}_L$ , $\\tilde{Q}_R$ with the following Lagrangian: $L \\supset y_L\\chi q_L\\tilde{Q}_L + y_R\\chi q_R\\tilde{Q}_R+ A H \\tilde{Q}_L^*\\tilde{Q}_R.$ The benchmark point in Eq.",
"(), when interpreted with this Lagrangian, corresponds to $C^{\\text{(M)}}\\simeq &\\frac{y_L y_R A v}{2 m_{\\tilde{Q}}^4}\\simeq \\frac{1}{(1.05\\ \\rm {TeV})^2} ,\\\\C^{\\text{(Ds)}}\\simeq &\\frac{y_L y_R A v}{4 m_{\\tilde{Q}}^4}\\simeq \\frac{1}{(1.05\\ \\rm {TeV})^2} ,$ where we assume all colored scalars have common mass $m_{\\tilde{Q}}$ .",
"Assuming that $y_L$ , $y_R\\lesssim 1$ , and $A \\lesssim m_{\\tilde{Q}}$ for perturbative unitarity condition, the colored scalar mass parameter should be smaller than 460 GeV (360 GeV) for Majorana (Dirac) DM.",
"If the colored scalar is produced at the LHC, it will decay into a SM quark and DM.",
"This collider signature is analogous to that of SUSY models with almost massless neutralino and a very heavy gluino.",
"Such a simplified SUSY model is searched for at the LHC, and the lower limit on the mass of squark is 780 GeV [63].",
"The limit can be directly applied to the current setup, since squark pairs are mainly produced by QCD processes in both models.",
"Hence, as an explanation of the CDMS-Si signal, isospin–violating fermionic DM models with colored scalar mediators are already disfavored by current LHC results."
]
] | 1403.0324 |
[
[
"Two-axis control of a singlet-triplet qubit with an integrated\n micromagnet"
],
[
"Abstract The qubit is the fundamental building block of a quantum computer.",
"We fabricate a qubit in a silicon double quantum dot with an integrated micromagnet in which the qubit basis states are the singlet state and the spin-zero triplet state of two electrons.",
"Because of the micro magnet, the magnetic field difference $\\Delta B$ between the two sides of the double dot is large enough to enable the achievement of coherent rotation of the qubit's Bloch vector about two different axes of the Bloch sphere.",
"By measuring the decay of the quantum oscillations, the inhomogeneous spin coherence time $T_{2}^{*}$ is determined.",
"By measuring $T_{2}^{*}$ at many different values of the exchange coupling $J$ and at two different values of $\\Delta B$, we provide evidence that the micromagnet does not limit decoherence, with the dominant limits on $T_{2}^{*}$ arising from charge noise and from coupling to nuclear spins."
],
[
"Supplemental materials",
"This supplement presents methods used to calibrate the detuning energy (lever-arm $\\alpha $ ) and to convert measurements of time-averaged current through the quantum point contact (QPC) to probabilities of being in the singlet state just after a given pulse sequence has been applied, including data used to extract the spin relaxation time $T_1$ used in the normalization process.",
"Data for the “$\\Delta B$ ” gate and the exchange gate for $\\Delta B=32$ neV are shown here.",
"We present the results of a simulation of the X or “$\\Delta B$ \" gate performed with two different forms for the functional dependence of $J$ on detuning.",
"We also describe the fabrication of the sample and include an image of the micomagnet.",
"We find the conversion between the detuning voltage $V_{\\varepsilon }$ and the detuning energy $\\varepsilon $ from measurements of the charge stability diagram under non-zero source-drain bias, as shown in Fig S1(a).",
"We apply $-200~\\mu $ V between the right dot reservoir and the left reservoir, to raise the Fermi level of the right reservoir 200$~\\mu $ eV higher than that of the left reservoir.",
"By drawing the charge transition lines on top of the stability diagram, as shown in Fig.",
"S1(b), we can measure the shift in gate voltage of charge transitions arising from the 200 $\\mu $ eV potential difference between the two reservoirs.",
"Because of the applied bias, the two triple points turn into triangles.",
"The highlighted points are useful for converting dot energies to gate voltages.",
"Each point has its energy level diagram drawn, as shown in Figs. S1(c-f).",
"Moving from the yellow point to blue point in gate voltage will raise both dot potentials by 200$~\\mu $ eV.",
"Adjusting gate voltages from the yellow point (or blue point) to the green point will create a 200$~\\mu $ eV energy difference between the dots.",
"The detuning direction we used is labeled with a yellow arrow, from the red point to green point, creating 200$~\\mu $ eV energy difference by moving each dot potential in opposite directions by the same amount.",
"The voltage changes measured are 4 mV on LP and $-4$ mV on RP, corresponding to $4\\sqrt{2}$ mV in the detuning direction.",
"Thus the conversion factor is $4\\sqrt{2}$ mV in detuning voltage for each 200$~\\mu $ eV in detuning energy, corresponding to $\\alpha = 35.4~\\mu $ eV/mV.",
"Figure: (a) Charge stability diagram with -200μ~\\mu V right dot reservoir bias voltage applied.",
"Electron occupation numbers are labeled.",
"(b) Same charge stability diagram as (a), with charge transition lines superimposed.A 200μ~\\mu eV potential difference between left and right dot reservoirs shifts the right dot transitions ∼-4\\sim -4 mV on gate RP (see arrows).",
"(c)-(f) Energy level diagrams showing energies for each dot and reservoir correspond to the four positions highlighted in (b).",
"Energy differences are labeled and listed for each case."
],
[
"Method of conversion of the QPC current measurement to probability of being in the singlet state",
"Here we present the methods used to convert measurements of the time-averaged difference in QPC current ($\\Delta I_\\mathrm {QPC}$ ) to probabilities of being in the singlet state just after a given pulse sequence has been applied.",
"The method is similar to the one described in the supplemental material of Ref.",
"[5], except for the pulse sequence used in the extraction of the $\\Delta I_\\mathrm {QPC}$ that corresponds to the one electron change (0,2) to (1,1).",
"All the pulse sequences are generated by a Tektronix AFG3250 pulse generator.",
"The reference lockin signal is a square wave with frequency of either 67 or 111 Hz (red dashed trace in Fig. S2(a)).",
"During one half of a cycle, a pulse train is applied to the gates of the quantum dots (purple trace in Fig. S2(a)).",
"The lockin signal $\\Delta I_\\mathrm {QPC}$ measures the change in the average charge occupation induced by the application of the pulses.",
"The averaging time for each data point is two seconds.",
"To convert the measured $\\Delta I_\\mathrm {QPC}$ to singlet probability $P_\\mathrm {S}$ , we note that the charge state at the end of the pulse is (1,1) for a spin triplet, while it is (0,2) for a spin singlet.",
"If the spin state is a triplet at the end of a pulse, it will relax back to the singlet in a time $T_1$ .",
"Therefore, $P_\\mathrm {S} = 1-\\frac{\\Delta I_\\mathrm {QPC}}{\\Delta I_{1}}\\cdot \\frac{T_{1}}{T_\\mathrm {m}}\\cdot \\left(1-\\exp \\left(-\\frac{T_\\mathrm {m}}{T_{1}}\\right)\\right),$ where $\\Delta I_{1}$ is the value of $\\Delta I_\\mathrm {QPC}$ that corresponds to a one electron change from ((0,2) to (1,1)), and $T_{1}$ is the relaxation time of $T(1,1)$ to $S(0,2)$ .",
"We measure $\\Delta I_{1}$ by sweeping gate voltage along the detuning direction while applying the pulses shown in Fig. S2(a).",
"Fig.",
"S2(b) shows the lockin response as a function of detuning; the maximum change in $\\Delta I_\\mathrm {QPC}$ is $\\Delta I_{1}$ .",
"The spin relaxation time $T_{1}$ for the $T_{-}$ state is extracted by measuring the $S$ -$T_{-}$ oscillation amplitude as a function of $T_\\mathrm {m}$ , the time between successive pulses in the pulse train.",
"Three traces of $S$ -$T_{-}$ oscillations are shown in Fig.",
"S2(c); they demonstrate that the oscillation amplitude decays with increasing $T_\\mathrm {m}$ , as expected.",
"The oscillation amplitude as a function of $T_\\mathrm {m}$ satisfies $\\Delta I_\\mathrm {QPC} = A\\cdot \\frac{T_{1}}{T_\\mathrm {m}}\\cdot \\left(1-\\exp \\left(-\\frac{T_\\mathrm {m}}{T_{1}}\\right)\\right),$ where $A$ is a time-independent coefficient.",
"Fig.",
"S2(d) shows the oscillation amplitude as a function of $T_{\\mathrm {m}}$ ; a fit to Eq.",
"(S2) yields $T_{1} = 9.85\\pm 1.19$ $\\mu $ s. To measure the spin relaxation time $T_{1}$ for the $T_{0}$ state, we measure as a function of $T_\\mathrm {m}$ the value of $\\Delta I_\\mathrm {QPC}$ when we pulse into (1,1) for a time $\\tau _\\mathrm {s}$ significantly longer than the singlet-triplet $T_2^*$ , so that $S$ and $T_{0}$ are completely mixed ($\\tau _\\mathrm {s} > 2T_{2}^{*}$ ).",
"The relaxation time for the $T_0$ state again obeys Eq. (S2).",
"Fig.",
"S1(e) and (f) show measurements of $\\Delta I_\\mathrm {QPC}$ a a function of $T_\\mathrm {m}$ along with the fit to Eq.",
"(S2) used to extract this $T_{1}$ .",
"Figure: Measurements used to determine the relationship between the lockin signal ΔI QPC \\Delta I_\\mathrm {QPC}and the probability of being in the singlet state after application of a given pulse sequence, using Eq. (S1).",
"(a) Pulse sequence used to measure ΔI QPC \\Delta I_\\mathrm {QPC} that corresponds to one electron change from (0,2) to (1,1).",
"The red dashed line indicates the low frequency signal used as the lockin reference.",
"The square pulses shown in purple inside the red dashed line have the same frequency as the actual manipulation pulse used in the experiment.",
"Inset: schematic diagram showing the expected dc value of I QPC I_\\mathrm {QPC} during the 1/2 cycle with pulses applied (red) and the 1/2 cycle with pulses not applied (purple).",
"The black arrow indicates the maximum measured lockin signal ΔI 1 \\Delta I_1 (see panel (b)).",
"(b) ΔI QPC \\Delta I_\\mathrm {QPC} measured as we sweep gate voltage along detuning with the above pulse sequence applied.",
"(c) Three traces of ΔI QPC \\Delta I_\\mathrm {QPC} as a function of pulse width τ s \\tau _s are shown, exhibiting SS-T - T_{-} oscillations.",
"The three traces are acquired with three different values of the time T m T_\\mathrm {m} between successive pulses.",
"The oscillation amplitude decreases significantly with T m T_\\mathrm {m}, indicating that relaxation to (0,2) occurs on a timescale shorter than 20 μ\\mu s.(d) SS-T - T_{-} oscillation amplitude plotted as a function of T m T_\\mathrm {m}.",
"The corresponding value of T 1 T_1 is used to normalize the data shown in Fig.",
"1(h) in the main text.The solid line is a fit to Eq.",
"(S2), which yields the relaxation time T 1 T_{1} shown on the figure.",
"(e),(f) Measurement of the spin relaxation time T 1 T_1 for the T 0 T_0 state.For this measurement, a pulse is applied into (1,1) that is significantly longer than the inhomogeneousdephasing time T 2 * T_2^*, so that the state at the end of the pulse is an equal mixture of SS-T 0 T_0.The decay of ΔI QPC \\Delta I_\\mathrm {QPC} with T m T_\\mathrm {m}, the time between successivepulses, obeys Eq. (S2).",
"The value of T 1 T_{1} extracted by fitting to Eq.",
"(S2) is listed on the plot.",
"T 1 T_{1} from (e),(f) are used to convert ΔI QPC \\Delta I_\\mathrm {QPC} to singlet probability for Fig.",
"2(c) and Fig.",
"3(c), respectively, in the main text."
],
[
"Data for smaller $\\Delta B$ than in main text",
"Fig.",
"S3 reports data showing oscillations in the singlet probability corresponding to the “$\\Delta B$ ” gate and the exchange gate for $\\Delta B=32$ neV.",
"The pulse sequences used here are the same as those shown in Fig.",
"2(b) and Fig.",
"3(a) in the main text.",
"Figure: Data corresponding to the “ΔB\\Delta B” gate and the exchange gate for ΔB=32\\Delta B=32 neV. (a) Singlet probability P S P_{S}, measured as a function of pulse duration and voltage level at pulse tip, V ε p V_\\varepsilon ^{p}.",
"(b) Singlet probability P S P_{S}, measured as a function of pulse duration and pulse level V ε ex V_\\varepsilon ^{ex} in the exchange pulse sequence.",
"The singlet probability is reported in arbitrary units in both (a) and (b)."
],
[
"Simulation of the X or “$\\Delta $ B” gate",
"Fig.",
"S4 reports the results simulations of the “$\\Delta $ B” gate with two different functional forms for the dependence of $J$ on detuning, with the details described in the caption.",
"We find that $J$ appears to vary exponentially as a function of detuning energy, in agreement with previous observations by Dial et al.",
"[18]."
],
[
"Micromagnet fabrication",
"An optical micrograph of the device including the micromagnet is shown in Fig.",
"S5.",
"The micromagnet is 12.64 $\\mu $ m $\\times $ 1.78 $\\mu $ m $\\times $ 242 nm.",
"The magnet was patterned via electron-beam lithography on top of the accumulation gates approximately 1.78 $\\mu $ m to the left and 122 nm above the center of the two quantum dots.",
"The magnet was deposited via electron-beam evaporation with a metal film stack of 2 nm Ti / 20 nm Au / 200 nm Co / 20 nm Au evaporated at approximately 0.3 Å/s.",
"The gold film is intended to help minimize oxidation of the Co film.",
"Figure: “ΔB\\Delta B” gate data compared to simulation results using different functional forms for the dependence of JJ on detuning.",
"(a) Experimentally measured singlet probability P S P_\\mathrm {S} plotted as a function of pulse duration τ s \\tau _{s} and detuning energy ε\\varepsilon at the pulse tip (Fig.",
"2(c) from the main text).",
"Line cuts of data are fit to products of sinusoids and Gaussians, and the resulting maxima from the fits are plotted as colored dots in panels (b) and (c).",
"(b) Simulation of singlet probability P S P_\\mathrm {S} as function of duration τ s \\tau _{s} and detuning energy ε\\varepsilon using J=J 0 exp(-ε/ε 0 )J=J_{0}\\exp (-\\varepsilon /\\varepsilon _{0}), where the best fit is found with ε 0 =62.7μeV\\varepsilon _{0} = 62.7~\\mu eV.",
"(c) Simulation of singlet probability P S P_\\mathrm {S} as function of duration τ s \\tau _{s} and detuning energy ε\\varepsilon using J=ε 2 /4+t c 2 -ε/2J=\\sqrt{\\varepsilon ^{2}/4+t_{c}^{2}}-\\varepsilon /2, where the best fit is found with t c =2.48μeVt_{c} = 2.48~\\mu eV.",
"Oscillation peaks extracted from (a) are plotted on top of (b) and (c), and the comparison suggests that J=J 0 exp(-ε/ε 0 )J=J_{0}\\exp (-\\varepsilon /\\varepsilon _{0}) fits the data well.Figure: Optical micrograph of the device, with the location of the micromagnet marked on the figure."
]
] | 1403.0019 |
[
[
"Cascading Randomized Weighted Majority: A New Online Ensemble Learning\n Algorithm"
],
[
"Abstract With the increasing volume of data in the world, the best approach for learning from this data is to exploit an online learning algorithm.",
"Online ensemble methods are online algorithms which take advantage of an ensemble of classifiers to predict labels of data.",
"Prediction with expert advice is a well-studied problem in the online ensemble learning literature.",
"The Weighted Majority algorithm and the randomized weighted majority (RWM) are the most well-known solutions to this problem, aiming to converge to the best expert.",
"Since among some expert, the best one does not necessarily have the minimum error in all regions of data space, defining specific regions and converging to the best expert in each of these regions will lead to a better result.",
"In this paper, we aim to resolve this defect of RWM algorithms by proposing a novel online ensemble algorithm to the problem of prediction with expert advice.",
"We propose a cascading version of RWM to achieve not only better experimental results but also a better error bound for sufficiently large datasets."
],
[
"Introduction",
"Supervised learning algorithms are provided with instances already classified.",
"In these algorithms, each instance has a label, identifying the class that instance belongs to.",
"In supervised algorithms, the goal is to extract a general hypothesis from a number of labeled instances in order to make predictions about the unlabeled data.",
"Every learning algorithm uses a number of assumptions, therefore, it performs well in some domains; while it does not have appropriate performance in others [1].",
"As a result, combining classifiers is proposed as a new trend to improve the classification performance [2].",
"The paradigm of prediction with expert advice is concerned with converging to the best expert among the ensemble of classifiers with a small misclassification rate during the operation of the algorithm.",
"This concept has been studied extensively in the theoretical machine learning literature [3], [4], and attracts a lot of attentions in practice as well [5], [6].",
"Ensemble methods show outstanding performance in many applications, such as spam detection [7], [8], [9], intrusion detection [10], [11], [12], object tracking [13], [14], and feature ranking [15].",
"Bagging [16] and Boosting [17], [18], which are well-known ensemble methods, rely on a combination of relatively weak models to achieve a satisfactory overall performance than their weak constituents.",
"Online Bagging and Online Boosting[19] are also proposed to handle situations when the entire training data cannot fit into memory, or when the data set is of stream nature [19], [20], [21], [22], [23].",
"While the above-mentioned methods consist of weak learners, the mixture of experts algorithms selects among some learners that are experts in a specific input region [24], [25].",
"Classification of data sequences is also the topic of recent research in the machine learning community.",
"As a pioneering model of online ensemble learning, prediction with expert advice was first introduced in [26], [27], [28] and recent investigations in this area lead to outstanding novel methods [29].",
"Predicting with Expert Advice problem has the primary goal of predicting the label of data with an error rate close to that of the best expert.",
"A simple and intuitive solution to this problem is Halving.",
"Weighted Majority(WM) and its randomized version called randomized weighted majority (RWM) are the most well-known solutions to this problem and presented in [27].",
"These algorithms are based on Halving, but have a better mistake bound dependent on the number of the experts and the error rate of the best expert.",
"Another approach to this problem is Exponential Weighted Average(EWA) [4], which is fundamentally very similar to RWM.",
"Instead of using only zero-one loss function as is used in RWM, it exploits a convex loss function, and instead of finding a mistake bound, EWA obtains a regret bound.",
"All the above mentioned methods are based on the definition of the best expert.",
"The best expert is the expert with the minimum average error during the experiment.",
"Therefore, intuitively, the best expert does not necessarily have both the highest true negative rate and the highest true positive rate in the experiment.",
"Our experiments reveal the fact that finding the best classifier for positive and negative output regions separately and predicting based on them, leads to a significant improvement in the performance of the classification.",
"In this paper, we propose a simple and effective method to find the experts that have the lowest false positive and the lowest false negative rates besides the overall best expert, simultaneously.",
"The proposed method is called cascading randomized weighted majority (CRWM), and presents a cascade version of RWMs to find these best experts.",
"Theoretically, we show that CRWM converges to the best experts with a mistake bound tighter than that of RWMs in exposure to sufficient number of data points.",
"Practically, our experiments on a wide range of well-known datasets support our contribution and show outstanding performance in comparison to not only RWM methods, but also some other well-known online ensemble methods.",
"While we introduce the cascading method based on RWM, considering the similarities between RWM and EWA, CRWM can be similarly applied to EWA as a framework.",
"The rest of this paper is organized as follows: In the next section, online ensemble learning as an approach to the problem of predicting with expert advice is discussed.",
"In section 3, the proposed algorithm and its mistake bound is presented.",
"Section 4 evaluates the proposed algorithm and compares the experimental results to several other online ensemble methods.",
"Finally, the conclusion is presented in section 5."
],
[
"Related Work",
"In nearly all online learning algorithms for classification problem, there is a common scenario which consist of these phases: First of all, the learner is given with an instance, then the learner assigns a label to the given instance, and at the end the correct label of that instance is given to the learner; moreover, the learner learns this new labeled data to increase its performance.",
"In the following we define predicting with expert advice problem which exploits a similar scenario, and Randomized Weighted Majority algorithm as one of its well-known solution."
],
[
"predicting with expert advice",
"Let us consider a simple intuitive problem from [30]; a learning algorithm is given the task of predicting the weather each day that if it will rain today or not.",
"In order to make this prediction, the thought of several experts is given to the algorithm.",
"Each day, every expert says yes or no to this question and the learning algorithm should exploit this information to predict its opinion about the weather.",
"After making the prediction, the algorithm is told how the weather is today.",
"It will be decent if the learner can determine who the best expert is and predict the best expert's opinion as its output.",
"Since we do not know who the best expert is, our goal instead would be performing nearly as the best expert's opinion so far.",
"It means a good learner should guarantee that at any time, it has not performed much worse than none of the experts.",
"An algorithm that solves this problem is consist of the following stages.",
"First, it receives the predictions of the experts.",
"Second, Makes its own perdition and third, Finally, it is told the correct answer."
],
[
"Randomized weighted majority",
"Weighted Majority algorithm and its randomized version are the most famous solutions of predicting with expert advice problem.",
"The Randomized Weighted Majority, which is the fundamental part of the proposed algorithm, has several base classifiers(expert) and each classifier has a weight factor.",
"Every time a new instance is received, each of these base classifiers predicts a label for that instance, and the algorithm decides a label based on these predictions and the weight factors of the classifiers.",
"Whenever the true label of that instance arrives, the weight factors of the classifiers should be updated in a way that each classifier that predicts a wrong label would be penalized by a constant factor.",
"This algorithm is proven that converges to the best classifier among all the base classifiers.",
"Algorithm 1 describes the pseudo code of the randomized weighted majority algorithm.",
"Randomized weighted majority [27] $w_i \\leftarrow 1$ $\\vee $ $x_{ij}$ $\\leftarrow $ prediction of $i^{th}$ expert on $j^{th}$ data $\\vee $ $n \\leftarrow $ number of experts $\\vee $ $C_j \\leftarrow $ correct label of $j^{th}$ data $\\vee $ $N \\leftarrow $ number of data $\\vee $ $ \\beta \\leftarrow $ the penalty parameter $y_j$ = output label of $j^{th}$ data $j = 1 \\rightarrow N$ W = $\\sum _{i} w_i$ $y_j = x_{ij}$ , with probability $w_i/W$ $i = 1 \\rightarrow n$ $x_{ij} \\ne C_j$ $w_i \\leftarrow w_i \\times \\beta $ It has been shown that, on any trial sequence, the expected number of Mistakes (M) made by randomized weighted majority Algorithm satisfies the following inequality [27]: $M \\le \\frac{m \\ln (1/\\beta ) + \\ln n}{(1-\\beta )},$ where m is the number of mistakes made by the best expert so far and $\\beta $ is a constant penalizing value."
],
[
" The cascading randomized weighted majority algorithm ",
"When the data size becomes too large, the randomized weighted majority (RWM) algorithm tends to decide according to the best expert's opinion.",
"It takes time for the algorithm to converge to the best expert, and the learner may make more mistakes compared to the best one.",
"However, when the best expert is discovered by the learner, one should be sure that the algorithm did not make many mistakes more than the best one, and we can say that it predicts whatever the best expert says from now on.",
"In this section, we propose an online learning algorithm which its main idea is to define more than one best expert, each for a number of data instances.",
"In fact, the algorithm tries to find the best experts, and for every new data instance, decides which expert is the most suitable one, and predicts according to the opinion of that expert.",
"Each expert is actually a classifier.",
"As we studied numerous classifiers, we observed that they often do not have low false positive (FP) rates at the same time as having low false negative (FN) rates.",
"Figure REF shows a two-class dataset and three different linear classifiers.",
"Although, $C_o$ has lowest error rate its FP rate and also FN rate are not the best among these three classifiers.",
"As it is shown in the figure, $C_p$ has lower FP rate than $C_o$ and $C_n$ has lower FN rate than $C_o$ , either.",
"As a result, instead of looking for the expert with the lowest error rate, we look for experts with lowest FP and FN rates leading us to define three best classifiers.",
"Figure: The different classifiers for the given data setFinding the overall best expert in an ensemble of online classifiers is the goal of Weighted Majority algorithm; However, finding the best positive expert and the best negative expert simultaneously is still a problem and we propose the following algorithm to solve this problem.",
"Figure: The structure of the CRWM algorithmAs it is shown in the Figure REF , the proposed algorithm has $n$ base classifiers, and three online learners with RWM mechanism that exploit these classifiers.",
"In other words, there are $3 \\times n$ weight factors, $n$ weight factors for each learner.",
"For every new data instance, each of these $n$ base classifiers predicts a label.",
"Using the corresponding weight factors, these predictions are given to the learners in order for them to make their predictions.",
"Since the first algorithm is responsible for predicting negative labels, If this algorithm predicts the label as negative, the label is set to negative, otherwise, the second algorithm will be applied.",
"If the second algorithm predicts the label as positive, the output label will be positive, otherwise the third algorithm will be applied and the output of this algorithm would be the output label for the instance.",
"Although the described scenario states the algorithm for two class datasets, the scenario is so similar for multi-class datasets.",
"For a multi-class dataset, the algorithms would still use $n$ base classifiers, but it needs $L+1$ online learners, where $L$ is the number of different classes.",
"One for each output class besides a final learner.",
"So, in an $L$ class dataset the algorithm would have $(L+1) \\times n$ weight factors.",
"The rest of algorithm is same as the algorithm for two class datasets.",
"For simplicity; in what follows, we explain about two class datasets.",
"However, all the following; including the proof of the algorithm, still stands for multi-class datasets.",
"Whenever the correct label of the instance arrived, this correct label will be given only to the learner that produced the output label.",
"Therefore, on every data instance, the weight factors of only one learner can be updated.",
"As the learners are using RWM mechanism; in the specified learner, the weight factor of the classifiers that made the wrong prediction would be penalized by a constant factor $\\beta $ .",
"Since the CRWM algorithm is using three RWM learners in three levels, and due to its cascading structure, we called it cascading randomized weighted majority.",
"Algorithm 2 is the pseudo code of CRWM algorithm.",
"Cascading randomized weighted majority algorithm $w_ij \\leftarrow 1$ $\\vee $ $x_{ijk}$ $\\leftarrow $ prediction of $i^{th}$ base classifier of $j^{th}$ learner on $k^{th}$ data $\\vee $ $n \\leftarrow $ number of experts $\\vee $ $C_j \\leftarrow $ correct label of $j^{th}$ data $\\vee $ $N \\leftarrow $ number of data $\\vee $ $ \\beta \\leftarrow $ the penalty parameter $y_j$ = output label of $j^{th}$ data $j = 1 \\rightarrow N$ $W_1 = \\sum _{i} w_{i1}$ $y1 = x_{i1k}$ , with probability $w_{i1}/W_1$ $W_2 = \\sum _{i} w_{i2}$ $y2 = x_{i2k}$ , with probability $w_{i2}/W_2$ $W_3 = \\sum _{i} w_{i3}$ $y3 = x_{i3k}$ , with probability $w_{i3}/W_3$ y1 = 0 $i = 1 \\rightarrow n$ $x_{ij1} \\ne C_j$ $w_{i1} \\leftarrow w_{i1} \\times \\beta $ y2 = 1 $i = 1 \\rightarrow n$ $x_{ij2} \\ne C_j$ $w_{i2} \\leftarrow w_{i2} \\times \\beta $ $i = 1 \\rightarrow n$ $x_{ij3} \\ne C_j$ $w_{i3} \\leftarrow w_{i3} \\times \\beta $"
],
[
"The Mistake Bound",
"In this section, we intend to find the bound on the number of mistakes which probably occurred by this algorithm in a series of predictions.",
"We know that there is a mistake bound for RWM algorithm.",
"First, we use the method thereby RWM mistake bound was calculated to find a mistake bound for CRWM and subsequently, show that this mistake bound is better than RWM's mistake bound, when the data size is large enough.",
"On any sequence of trials, the expected number of mistakes ($M_j$ ) made by $j^{th}$ learner ($OL_1$ , $OL_2$ , $OL_3$ ) in cascading randomized weighted majority algorithm satisfies the following condition.",
"$M_j \\le \\frac{m_{kj} \\times \\ln (1/ \\beta ) + \\ln n}{(1- \\beta )}, \\hspace{5.69054pt}\\forall (j, k),$ where $m_{kj}$ is the number of mistakes made by the $k^{th}$ expert in $j^{th}$ learner so far and $n$ is the number of base classifiers(experts).",
"Define $F_{ij}$ as the fraction of the total weight on the wrong answer at the $i^{th}$ trial on the $j^{th}$ learner($OL_1$ , $OL_2$ , $OL_3$ ), and let $M_j$ be the expected number of mistakes of the $j^{th}$ learner so far.",
"So after the $t^{th}$ trial, we would have $ M_j = \\sum _{i=1}^{t}{F_{ij}}$ .",
"On the $i^{th}$ instance that classified by the $j^{th}$ learner, the total weight of the $j^{th}$ learner where is defined by $W_j$ changes according to: $W_j \\leftarrow W_{j} (1-(1-\\beta )F_{ij}).$ Since when the data is classified by $j^{th}$ learner, we multiply the weights of experts in $j^{th}$ learner that made a mistake by $\\beta $ and there is an $F_{ij}$ fraction of the weight on these experts.",
"Regarding to the initial value of weight factor for each base classifier which is set to 1, and considering $n$ as the number of base classifiers, the final total weight for the $j^{th}$ learner is: $W_j = n \\prod _{i=1}^t {(1-(1-\\beta )F_{ij})}.$ Let $m_{kj}$ be the number of total mistakes of the $k^{th}$ base classifier in the $j^{th}$ learner so far, therefore its weight factor would be $\\beta ^{m_{kj}}$ at this time.",
"Using the fact that the total weight must be at least as large as the weight of the $k^{th}$ classifier; for each value of j and k, we have: $n \\prod _{i=1}^t {(1-(1-\\beta )F_{ij})} \\ge \\beta ^{m_{kj}}.$ Taking the natural log of both side we get: $\\ln n + \\sum _{i=1}^t \\ln {(1-(1-\\beta )F_{ij})} \\ge m_{kj} \\ln \\beta $ $- \\ln n - \\sum _{i=1}^t \\ln {(1-(1-\\beta )F_{ij})} \\le m_{kj} \\ln (1/\\beta ).$ Since, $\\forall \\hspace{2.84526pt} 0<x<1 , - \\ln (1-x) > x $ , following equation will be obtained: $- \\ln n + (1-\\beta ) \\sum _{i=1}^t {F_{ij}} \\le m_{kj} \\ln (1/\\beta ).$ Using $ M_j = \\sum _{i=1}^t {F_ij}$ , we conclude: $M_j \\le \\frac{m_{kj} \\times \\ln (1/\\beta ) + \\ln n}{(1-B)}, \\hspace{5.69054pt}\\forall (j, k).$ Which completes the proof of theorem.",
"Now we have a bound for the expected number of mistakes that every learner will do in a sequence of trials.",
"Since for each instance only one of these three learners respondes and predicts the output label, in the following theorem we intend to find the total expected number of mistakes for the algorithm by aggregating the expected number of mistakes of these learners.",
"On any sequence of trials, the expected number of mistakes made by cascading randomized weighted majority algorithm$(M_{CRWM})$ satisfies the following condition.",
"$M_{CRWM} \\le \\frac{ \\sum _{i=1}^3 m_i \\times \\ln (1/ \\beta ) + 3 \\ln n}{(1- \\beta )},$ where $m_i$ is the number of mistakes made by the best expert of $i^{th}$ learner so far, and $n$ is the number of experts in each learner.",
"Since, each instance is classified by exactly one of the three learners, the expected number of mistakes made by cascading randomized weighted majority algorithm can be obtained with the following equation: $M_{CRWM} = \\sum _{j=1}^3 M_j$ By using Theorem 1 and defining $k, k^{\\prime }$ and $k\"$ as the index of the best expert of $OL_1, OL_2$ and $OL_3$ , respectively, we have: $M_{CRWM} \\le \\frac{(m_{k1} + m_{k^{\\prime }2} + m_{k\"3}) \\times \\ln (1/ \\beta ) + 3 \\ln n}{(1- \\beta )}.$ For convenience we define $m_i$ as the number of mistakes made by the best expert of $i^{th}$ learner, So we would have: $M_{CRWM} \\le \\frac{ \\sum _{i=1}^3 m_i \\times \\ln (1/ \\beta ) + 3 \\ln n}{(1- \\beta )}.$ Which completes the proof of the theorem.",
"Now we need to know how suitable is this bound.",
"For this purpose, we will compare the bound of CRWM algorithm with the bound of RWM algorithm in the following theorem.",
"The mistake bound for CRWM algorithm is better than RWM mistake bound, when the data size is going to be large.",
"We know that the expected number of mistake for RWM Algorithm is bounded by the following inequality: $M_{RWM} \\le \\frac{m \\ln (1/B) + \\ln n}{(1-B)} = Bound_{RWM}.$ We have already defined $Bound_{RWM}$ in the above equation In addition; using eq.",
"(12), we define $Bound_{CRWM}$ as follows: $Bound_{CRWM} = \\frac{(m_{k1} + m_{k^{\\prime }2} + m_{k\"3}) \\times \\ln (1/ \\beta ) + 3 \\ln n}{(1- \\beta )}.$ Now, it is needed to show that by increasing the number of incoming data, $Bound_{CRWM}$ is less than $Bound_{RWM}$ , so we should have: $Bound_{CRWM} - Bound{RWM} < 0.$ Using the eq.",
"(14) and the eq.",
"(15) we derive the following inequality: $\\frac{(m_{k1} + m_{k^{\\prime }2} + m_{k\"3}) \\times \\ln (1/ \\beta ) + 3 \\ln n}{(1- \\beta )} - \\frac{m \\ln (1/B) + \\ln n}{(1-B)} < 0,$ where $m$ is the number of overall errors of the best expert.",
"Lets suppose the index of the overall best expert is $p$ .",
"So, the number of mistakes made by the overall best expert in $OL_1$ is defined by $m_{p1}$ and similarly $m_{p2}$ and $m_{p3}$ are the number of mistakes of the overall best expert in $OL_2$ and $OL_3$ , respectively.",
"By using these definitions and considering the fact that $ m = m_{p1}+m_{p2}+m_{p3}$ , we can rewrite the above inequality as follows: $\\frac{(m_{k1}-m_{p1} + m_{k^{\\prime }2}-m_{p2} + m_{k\"3}-m_{p3}) \\times \\ln (1/ \\beta ) + 2 \\ln n}{(1- \\beta )} <0.$ Since $k$ is the index of best expert in $OL_1$ , obviously $m_{k1} \\le m_{p1}$ .",
"This also stands for $m_{k^{\\prime }2}$ and $m_{k\"3}$ , so we have $m_{k^{\\prime }2} \\le m_{p2}$ and $m_{k\"3} \\le m_{p3}$ .",
"By considering the region of instances that has been classified by $j^{th}$ learner, we define $K_j$ as the number of instances in this region and $X_{ij}$ as the error rate of the $i^{th}$ expert in $j^{th}$ learner in the specified region of instances.",
"So we would have: $m_{ij} = K_j * X_{ij}, \\hspace{5.69054pt} \\forall (i,j).$ As we mentioned earlier as our main hypothesis, the overall best expert does not have the best error rate in both positive and negative regions.",
"So, following inequality holds.",
"$X_{k1} < X_{p1} \\hspace{5.69054pt} or \\hspace{5.69054pt} X_{k^{\\prime }2} < X_{p2}.$ Without loss of generality, suppose $X_{k1} < X_{p1}$ .",
"So, we have $X_{k^{\\prime }2} = X_{p2}$ and $X_{k\"3} = X_{p3}$ .",
"Using these facts and also eq.",
"(19) we can rewrite the eq.",
"(18) as follows: $\\frac{K_1\\times (X_{k1}-X_{p1}) \\times \\ln (1/ \\beta ) + 2 \\ln n}{(1- \\beta )} <0.$ By some algebraic simplification, we obtain: $K_1 \\times (X_{p1}-X_{k1}) > \\frac{2\\ln n}{ln(1/B)}.$ In the above inequality, the only parameters that will be raised with increasing the size of dataset is $K_1$ and the other parameters, which are given bellow, are constants and limited.",
"$X_{p1}-X_{k1} = C_0 > 0$ $\\frac{2\\ln n}{ln(1/B)}= C_1 \\ge 0.$ Using the above definitions we can rewrite eq.",
"(22) as follows: $K_1 \\times C_0 > C_1.$ Since $K_1$ is increasing with increase of the number of data instances, hence by increasing the number of data instances, the above equation would be true which means the mistake bound of CRWM would be better than the mistake bound of RWM, which completes the proof of the theorem."
],
[
"Experimental Results",
"In this section, we compared the classification performance of the proposed method with randomized weighted majority, Online bagging and Online boosting on 14 datasets.",
"These datasets are from the UCI Machine Learning repository [31] to evaluate different aspects of the algorithms.",
"The different characteristics of these datasets are shown in table 1.",
"The number of instances in these datasets vary from 208 to 490000 and the number of attributes vary from 4 to 64.",
"In order to show the effectiveness of the proposed method, some multi-class datasets are chosen as well as two-class datasets.",
"All the four algorithms are implemented in JAVA using MOA framework.",
"In all the implementations, we exploit naive Bayes as the base classifier algorithm, due to the fact that it is highly fast and can easily updated algorithm as well.",
"In [19] the best performance of online bagging and online boosting achieved using 100 number of base classifiers.",
"To have fair comparison we have used the same number of naive Bayes base classifiers in all of these methods.",
"Table: Datasets used for evaluation of algorithmsThere is only one parameters in the proposed algorithms, which is the penalty parameter ($\\beta $ ) in both RWM and CRWM.",
"We use $\\beta = 0.5$ in all the experiments.",
"Fig.",
"3 illustrates how the accuracies of RWM and CRWM algorithms depend on $\\beta $ .",
"In this figure, the results of an experiment on only 4 datasets are shown.",
"These results show that using this value for $\\beta $ the algorithms often do near the best performance.",
"Figure: Sensitivity of Accuracy to parameter β\\beta in RWM and CRWMThe idea of looking for the experts with lowest FP rate seems problematic.",
"For instance, if the expert predicts everything as negative, FP rate of it would be zero and similarly, concern is there for lowest FN rate.",
"In order to avoid such problems, we can exploit a biased estimation of FP rate instead of FP rate itself.",
"$FP\\hspace{2.84526pt}Rate\\hspace{2.84526pt}Biased\\hspace{2.84526pt}Estimation = \\frac{FP + n_p}{ FP + TP + n},$ where, $n_p$ and $n$ are constant.",
"For simplicity we can set $n_p$ to 1 and $n$ to 2.",
"We also define the biased estimation of FN rate in the same way.",
"To determine the performance of the proposed algorithm, we compare the accuracy of CRWM algorithm with RWM, online bagging and online boosting.",
"Table 2 shows the results of each of the algorithms on all the datasets.",
"The performance measure used for comparison is the accuracy of the algorithms, which is the average accuracy of algorithms in 50 independent runs.",
"The scenario which is used for evaluating the accuracy is the exact scenario of online learning; in which, every new instance is given to the algorithm and the algorithm makes its prediction.",
"Then, the correct label would be given and will be compared to the output prediction of the algorithm.",
"Table: Accuracy of different algorithmsAs it has shown in table 2, the results of CRWM are dominant in most of the datasets.",
"Another fact is that in the remaining datasets it has the second best performance.",
"These stable results can be considered as a great power of the proposed method.",
"Another result that comes from table 2 is that the accuracy of CRWM is higher compared to RWM in all the datasets, which means our theoretical bound is fully supported by our experimental results.",
"Although we have shown that the CRWM obtains more reasonable results when the number of data is increased, we can see even in datasets with a few number of instances that the CRWM algorithm has better accuracy compared to the RWM, which shows another power of the CRWM algorithm.",
"While in some data sets there is significant difference between accuracy of CRWM and other algorithms, in some others the difference is not very clear.",
"So, we have arranged student's t-test to clarify this ambiguity.",
"Table 3 shows the result of paired student's t-test between CRWM and the other three algorithms.",
"The output of this test is P-value.",
"A P-value below 0.05 is generally considered statistically significant, So the one who has better average is considered to have better results than the other.",
"while one of 0.05 or greater indicates no difference between the groups.",
"In this table, a blue cell shows there is no difference.",
"Using the output P-value of this test we have arranged table3.",
"In which, a red cell means CRWM is worse than the specified algorithm on specified dataset, while black cells; which cover most of the table (36/42 number of cells), show that CRWM is better in compared to that algorithm on that dataset.",
"As it is clear, only 3 cells of the table are red and just 6 cells are blue, which means the great performance of CRWM algorithm in compared to other algorithms and it confirms the results of table 2.",
"Table: Paired student's t-test beetween CRWM and other algorithmsConsidering the cascading structure of CRWM and also its outstanding results, it seems that this excellence is obtained with the cost of more tunning time.",
"However, the results of table 4 refuse this view.",
"Table 4 shows the running time required for each algorithm on every dataset.",
"The results of this table show that there is no significant difference in running time between CRWM, RWM and Online Bagging, Even though the number of weight factors in CRWM depend on the number of classes, it still exploits the same number of base classifiers as other algorithms, which is the main factor in running time of the algorithms.",
"In addition, after receiving the correct label, CRWM gives it to only one of the learners to learn and justify its weights.",
"These points of view indicate why there is no great difference between the running times of this algorithms.",
"The only reason that makes a little difference in running times is the overhead of creating and using more weight factors, Table: Running times (sec.",
")Table 5 aims to confirm our main hypothesis about comparison of the values of best FP rate, best FN rate and best error rate among the base classifiers.",
"A green cell in the best FP rate and best FN rate columns means that the value is better than the corresponding error rate value, and existence of such cell in every row is exactly what what we have assumed to be true in every groups of base classifiers.",
"Table: best FP, FN and error rate among the base classifiersTable 6 aims to compare the mistake bound of CRWM and RWM with each other for the two-class datasets used in our experiments.",
"It also compares these theoretical mistake bounds with the experimental results.",
"In this experiment, the value of theoretical mistake bound is calculated using the results of table 5 and some other parameters that exist in corresponding formulas.",
"As it is shown in table 6, the theoretical mistake bound is always greater than the corresponding experimental result, which confirms the accuracy of calculated mistake bound.",
"In addition, when the size of datasets is small, the mistake bound of RWM is lower than the one of CRWM.",
"However, in larger datasets the mistake bound of CRWM excels its rival, which is exactly the point that we have mentioned in theorem 3.",
"Table: Number of Mistakes in Theoretical Mistake Bound vs.",
"Experimental Result"
],
[
"Conclusion and Future Works",
"In this paper, we proposed a new online ensemble learning algorithm, called CRWM.",
"It is shown that CRWM's mistake bound is better than that of RWM's when the size of the input is increased.",
"In addition, the experimental results reveal that CRWM obtains a better accuracy compared to RWM with a wide range of input sizes.",
"By carrying out several experiments, we have shown that this new algorithm outperforms other online ensemble learning algorithms.",
"It usually acquires the best performance or the second best performance among these algorithms, indicating its superiority among them.",
"In this study, we did not address imbalanced datasets.",
"However, the structure of CRWM that focuses on each class separately, provides us a powerful facility in defining different cost functions on each class.",
"Besides, it provides us with a means to change the order of online learners.",
"For instance, for classes that are needed to have more true positive we can move their corresponding learners to the top of the structure.",
"Inversely, whenever the false positive of a specified class is significant we can move the corresponding learner to the bottom of CRWM structure.",
"This dynamic structure is a powerful feature that distinguishes CRWM from other similar algorithms.",
"Clearly, using different base classifiers for each learner may lead to a better accuracy.",
"In addition, utilizing one-class classifiers as base classifiers would cause great effects on accuracy.",
"However it is needless to say that, using more base classifiers lead to an increase in running time of the algorithm.",
"While, using just different weight factors, as we did in CRWM, does not affect it so much.",
"This means that, whenever the running time is not an important factor for the algorithm we can use different base classifiers for each learner or even use one-class classifiers to get better results."
]
] | 1403.0388 |
[
[
"Intersection of paraboloids and application to Minkowski-type problems"
],
[
"Abstract In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution.",
"This study is motivated by a Minkowski-type problem arising in geometric optics.",
"We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere.",
"We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids.",
"We provide an algorithm to compute these intersections using the exact geometric computation paradigm.",
"This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem."
],
[
"Introduction",
"The computation of intersection of half-spaces is a well-studied problem in computational geometry, which by duality is equivalent to the computation of a convex hull.",
"Similarly, the computation of intersections or unions of spheres is also well studied and can be done by using power diagrams [2].",
"In this article, we study the computation and the complexity of the intersection of the convex hull of confocal paraboloids of revolution, showing that it is equivalent to intersecting a certain power diagram with the unit sphere.",
"Union of convex hull of confocal paraboloids of revolution, and intersection or union of convex hull of confocal ellipsoids of revolution can be studied using the same tools.",
"These studies are motivated by inverse problems similar to Minkowski problem that arise in geometric optics.",
"We show how the algorithm we developed to compute the intersection of paraboloids are used to solve large instances of one of these problems.",
"A theorem of Minkowski asserts that given a family of directions $(y_i)_{1\\le i\\le N}$ and a family of non-negative numbers $(\\alpha _i)_{1\\le i\\le N}$ , one can construct a convex polytope with exactly $N$ facets, such that the $i$ th facet has exterior normal $y_i$ and area $\\alpha _i$ .",
"Aurenhammer, Hoffman and Aronov [3] studied a variant of this problem involving power diagrams and showed its equivalence with the so-called constrained least-square matching problem.",
"This article is motivated by yet another problem of Minkowski-type that arises in geometric optics, which is called the far-field reflector problem in the literature [9], [8].",
"Recall that a paraboloid of revolution is defined by three parameters: its focal point, its focal distance $\\lambda $ and its direction $y$ .",
"We assume that all paraboloids are focused at the origin, and we denote $P(y,\\lambda )$ the convex hull of a paraboloid of revolution with direction $y$ and focal distance $\\lambda $ .",
"We will say in the following that $P(y,\\lambda )$ is a solid paraboloid.",
"Paraboloids of revolution have the well-known optical property that any ray of light emanating from the origin is reflected by the surface $\\partial P(y,\\lambda )$ in the direction $y$ .",
"Assume first that one wants to send the light emited from the origin in $N$ prescribed directions $y_1,\\hdots ,y_N$ .",
"From the property of a paraboloid of revolution, this can be done by considering a surface made of pieces of paraboloids of revolution whose directions are among the $(y_i)$ .",
"In the far-field reflector problem, one would also like to prescribe the amount of light $\\alpha _i$ that is reflected in the direction $y_i$ .",
"A theorem of Oliker-Caffarelli [9] ensures the existence of a solution to this problem: there exist unique (up to a common multiplicative constant) focal distances $\\lambda _1,\\hdots ,\\lambda _N$ such that the surface $\\partial ( \\cap _{1\\le i \\le N} P(y_i,\\lambda _i)) $ reflects exactly the amount $\\alpha _i$ in each direction $y_i$ .",
"Other types of inverse problems in geometric optics can be formulated as Minkowski-type problems involving the union of confocal solid paraboloids, and the union or intersection of confocal ellipsoids [17], [14].",
"Motivated by these Minkowski-type problems, our goal is to compute the union and intersection of solid confocal paraboloids and ellipsoids of revolution.",
"Using a radial parameterization, each of these computations is equivalent to the computation of a decomposition of the unit sphere into cells, that are not necessarily connected.",
"Our contributions are the following: We show that each of the four types of cells can be computed by intersecting a certain power diagram with the unit sphere (Propositions REF , REF and REF ).",
"The approach is similar to the computation of union and intersection of balls using power diagrams in [2], or to the computation of multiplicatively weighted power diagrams in $^{d-1}$ using power diagrams in $^d$ [5].",
"We show that the complexity bounds of these four diagram types are different.",
"In the case of intersection of solid confocal paraboloids in $^3$ , the complexity of the intersection diagram is $O(N)$ (Theorem REF ).",
"This is in contrast with the $\\Omega (N^2)$ complexity of the intersection of a power diagram with a paraboloid in $^3$ [5].",
"In the case of the union and intersection of solid confocal ellipsoids, we recover this $\\Omega (N^2)$ complexity (Theorem REF ).",
"Finally, the case of the union of paraboloids is very different from the case of the intersection of paraboloids.",
"Indeed, in the latter case, the corresponding cells on the sphere are connected, while in the former case the number of connected component of a single cell can be $\\Omega (N)$ (Proposition REF ).",
"The complexity of the diagram in this case is unknown.",
"In Section , we describe an algorithm for computing the intersection of a power diagram with the unit sphere.",
"This algorithm uses the exact geometric computation paradigm and can be applied to the four types of unions and intersections.",
"It is optimal for the union and intersection of ellipsoids, but its optimality for the case of intersection of paraboloids is open.",
"This algorithm is then used for the numerical resolution of the far-field reflector problem.",
"Using a known optimal transport formulation [21], [11] and similar techniques to [3], we cast this problem into a concave maximization problem in Theorem REF .",
"This allows us to solve instances with up to 15k paraboloids, improving by several order of magnitudes upon existing numerical implementations [8]."
],
[
"Intersection of confocal paraboloids of revolution",
"Because of their optical properties, finite intersections of solid paraboloids of revolutions with the same focal point play a crucial role in an inverse problem called the far-field reflector problem.",
"This inverse problem is explained in more detail in Section .",
"Here we study the computation and complexity of such an intersection when the focal point lies at the origin.",
"We call this type of intersection a paraboloid intersection diagram."
],
[
"Paraboloid intersection diagram",
"A paraboloid of revolution in $^d$ with focal point at the origin is uniquely defined by two parameters: its focal distance $\\lambda $ and its direction, described by a unit vector $y$ .",
"We denote the convex hull of such a paraboloid by $P(y,\\lambda )$ .",
"The boundary surface $\\partial P(y, \\lambda )$ can be parameterized in spherical coordinates by the radial map $u\\in ^{d-1} \\mapsto \\rho _{y,\\lambda }(u)\\ u$ , where the function $\\rho _{y,\\lambda }$ is defined by: $\\rho _{y,\\lambda }: u \\in ^{d-1} \\mapsto \\frac{\\lambda }{1-{y}{u}}.$ Given a family $Y=(y_i)_{1\\le i\\le N}$ of unit vectors and a family $\\lambda = (\\lambda _i)_{1\\le i\\le N}$ of positive focal distances, the boundary of the intersection of the solid paraboloids $(P(y_i,\\lambda _i))_{1\\le i \\le N}$ is parameterized in spherical coordinates by the function: $\\rho _{Y,\\lambda }(u) := \\min _{1\\le i \\le N} \\rho _{y_i,\\lambda _i}(u) = \\min _{y \\in Y} \\frac{\\lambda _i}{1-{y_i}{u}}.$ The paraboloid intersection diagram associated to a family of solid paraboloids $(P(y_i,\\lambda _i))_{1\\le i\\le N}$ is a decomposition of the unit sphere into $N$ cells defined by: $_{Y}^\\lambda (y_i) := \\lbrace u \\in ^{d-1};~ \\forall j \\in \\lbrace 1,\\hdots ,N\\rbrace , \\rho _{y_i,\\lambda _i}(u) \\le \\rho _{y_j,\\lambda _j}(u)\\rbrace .$ The paraboloid intersection diagram corresponds to the decomposition of the unit sphere given by the lower envelope of the functions $(\\rho _{y_i,\\lambda _i})_{1\\le i \\le N}$ ."
],
[
"Power diagram formulation",
"We show in this section that each cell of the paraboloid intersection diagram is the intersection of a cell of a certain power diagram with the unit sphere.",
"We first recall the definition of a power diagram.",
"Let $P = (p_i)_{1\\le i\\le N}$ be a family of points in $^d$ and $(\\omega _i)_{1\\le i\\le N}$ a family of weights.",
"The power diagram is a decomposition of $^d$ into $N$ convex cells, called power cells, defined by $_{P}^{\\omega }(p_i) :=\\Big \\lbrace x\\in ^d,\\ \\forall j\\in \\lbrace 1,\\hdots ,N\\rbrace \\ {x-p_i}^2 + \\omega _i \\le {x-p_j}^2+\\omega _j\\Big \\rbrace .$ Let $(P(y_i,\\lambda _i))_{1\\le i \\le N}$ be a family of confocal paraboloids.",
"One has $\\forall i\\in \\lbrace 1,\\hdots ,N\\rbrace \\quad _{Y}^{\\lambda }(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i),$ where $P=(p_i)_{1\\le i \\le N}$ and $(\\omega _i)_{1\\le i\\le N}$ are defined by $p_i = - (\\lambda _i^{-1}/2) y_i$ and $\\omega _i = - \\lambda _i^{-1} -\\lambda _i^{-2}/4$ .",
"For any point $u\\in ^{d-1}$ , we have the following equivalence : $u \\hbox{ belongs to } _{Y}^\\lambda (y_k)\\Longleftrightarrow k = \\arg \\min _{1\\le i\\le N} \\frac{\\lambda _i}{1-{y_i}{u}}\\Longleftrightarrow k = \\arg \\max _{1 \\le i \\le N}\\lambda _i^{-1}-{u}{\\lambda _i^{-1}y_i}.$ An easy computation gives : $\\begin{array}{rl}\\max _{1 \\le i \\le N}\\lambda _i^{-1}-{u}{\\lambda _i^{-1}y_i} &=\\max _{1 \\le i \\le N} \\lambda _i^{-1} -{u + \\frac{1}{2}\\lambda _i^{-1} y_i}^2 +{u}^2 + \\frac{1}{4} {\\lambda _i^{-1} y_i}^2 \\\\&= {u}^2 -\\min _{1\\le i \\le N}\\left({u + \\frac{1}{2}\\lambda _i^{-1} y_i}^2 - \\lambda _i^{-1}- \\frac{1}{4}\\lambda _i^{-2}\\right).\\end{array}$ This implies that a unit vector $u$ belongs to the paraboloid intersection cell $_{Y}^{\\lambda }(y_i)$ if and only if it lies in the power cell $_P^{\\omega }(p_i)$ .",
"One can remark that the paraboloid intersection diagram does not change if all the focal distances $\\lambda _i$ are multiplied by the same positive constant.",
"This implies that the intersection with the sphere of the power cells defined in the above proposition does not change under a uniform scaling by $\\lambda $ (even though the whole power cells change)."
],
[
"Complexity of the paraboloid intersection diagram in $^3$",
"In this section, we show that in dimension three, the complexity of the paraboloids intersection diagram is linear in the number of paraboloids.",
"Let $(P(y_i,\\lambda _i))_{1\\le i \\le N}$ be a family of solid paraboloids of $^3$ .",
"Then the number of edges, vertices and faces of its paraboloid intersection diagram is in $O(N)$ .",
"The proof of this theorem strongly relies on the following proposition, which shows that each cell $_{Y}^\\lambda (y_i)$ can be transformed into a finite intersection of discs, and is thus connected.",
"Note that while it is stated only in dimension 3, this proposition holds in any ambient dimension.",
"For any two solid paraboloids $P(y,\\lambda )$ and $P(z,\\mu )$ , the projection of the set $\\mathcal {L} := (\\partial P(y,\\lambda )) \\cap P(z,\\mu )$ onto the plane $\\lbrace y\\rbrace ^$ orthogonal to $y$ is a disc with center and radius $ c[(y,\\lambda ),(z,\\mu )] = 2\\lambda \\pi _y(z) / \\Vert y - z\\Vert ^2 \\qquad r[(y,\\lambda ),(z,\\mu )] = 2\\sqrt{\\lambda \\mu }/\\Vert y - z\\Vert , $ where $\\pi _y$ denotes the orthogonal projection on $\\lbrace y\\rbrace ^$ .",
"Moreover, given a solid paraboloid $P(y,\\lambda )$ , the following map is one-to-one: $\\begin{array}{rl}F_{(y,\\lambda )}: (^{d-1}\\setminus \\lbrace y \\rbrace ) \\times ^+ &\\rightarrow \\lbrace y\\rbrace ^{} \\times ^+\\\\(z,\\mu ) &\\mapsto \\left(c[(y,\\lambda ),(z,\\mu )], r[(y,\\lambda ),(z,\\mu )]\\right).\\end{array}$ The proof of the first half of this proposition can be found in [7], but we include it here for the sake of completeness.",
"We first show that the orthogonal projection onto the plane $\\lbrace y\\rbrace ^$ of the intersection $\\mathcal {L}^{\\prime }:= \\partial P(y,\\lambda ) \\cap \\partial P(z,\\nu )$ is a circle.",
"Without loss of generality, we assume that $y$ is the last basis vector $(0,\\hdots ,0,1)$ .",
"Recall that a paraboloid of revolution $\\partial P(y,\\lambda )$ is defined implicitly by the relation ${x} = {x}{y} + \\lambda $ .",
"Hence, any point $x$ in $\\mathcal {L}^{\\prime }$ belongs to the hyperplane defined by ${x}{z-y} =\\lambda - \\mu $ .",
"If we denote by $z^{\\prime }=\\pi _y(z)$ , $z_d={z}{y}$ , $x^{\\prime }=\\pi _y(x)$ and $x_d={x}{y}$ , one has $x_d = \\frac{{z^{\\prime }}{x^{\\prime }}}{1-z_d} + \\frac{\\mu - \\lambda }{1-z_d}.$ The surface $\\partial P(y,\\lambda )$ can be parameterized over the plane $\\lbrace y\\rbrace ^$ by the equation $x_d ={x^{\\prime }}^2 / 2 \\lambda - \\lambda / 2$ .",
"Combining this with the relations ${z^{\\prime }}^2 + z_d^2=1$ and ${y-z}^2= 2(1-z_d)$ , we get ${ x^{\\prime } - \\frac{2 \\lambda }{{y-z}^2} z^{\\prime }}^2 = \\frac{4 \\lambda \\mu }{{y-z}^2}.$ We deduce that the projection of $\\mathcal {L}^{\\prime }$ onto the plane $\\lbrace y\\rbrace ^{\\prime }$ is a circle of center $c=2\\lambda z^{\\prime } / \\Vert y - z\\Vert ^2$ and of radius $r=2\\sqrt{\\lambda \\mu } / \\Vert y - z\\Vert $ .",
"Therefore, the projection $\\pi _y(\\mathcal {L})$ is either the disc enclosed by this circle or its complementary.",
"In order to exclude the latter case, we remark that the intersection $P(y,\\lambda )\\cap P(z,\\mu )$ is a compact set, because it is convex and does not include a ray (assuming $y\\ne z$ ).",
"Hence, the projection $\\pi _y(\\mathcal {L}) \\subseteq \\pi _y(P(y,\\lambda )\\cap P(z,\\mu ))$ is also compact, and therefore it is the disc of center $c$ and radius $r$ .",
"Let us now show that the map $F_{(y,\\lambda )}$ is one-to-one.",
"For a fixed positive $\\mu $ , let $c(z) = c([y,\\lambda ],[z,\\mu ])$ .",
"For every point $z$ in $^{2} \\setminus \\lbrace \\pm y\\rbrace $ , denote $\\pi _y^1(z) =\\pi _y(z) / {\\pi _y(z)}$ .",
"This point belongs to the unit circle in $\\lbrace y\\rbrace ^$ , which coincides with the equator $E_y$ of the sphere $^{2}$ which is equidistant to the points $\\lbrace \\pm y\\rbrace $ .",
"Then, given any constant-speed geodesic $z(.",
")$ such that $z(\\pm 1) = \\pm y$ and such that $z(0) = z_0 \\in E_y$ , i.e., $z(t) = \\sin (t \\pi /2) y +\\cos (t\\pi /2) z_0$ , the following formula holds $ c(z(t)) = 2\\lambda \\frac{{\\pi _y(z(t))}}{{y-z(t)}^2} z_0 =\\lambda \\frac{\\cos (t\\pi /2)}{1 - \\sin (t\\pi /2)} z_0.$ One easily checks that the function $t \\in [-1,1) \\mapsto \\frac{\\cos (t\\pi /2)}{1 -\\sin (t\\pi /2)}$ is increasing and maps $[-1,1)$ to $[0,+\\infty )$ .",
"The mapping $z \\in ^2\\setminus \\lbrace y\\rbrace \\mapsto c(z) \\in \\lbrace y\\rbrace ^$ thus transforms bijectively every geodesic arc joining the points $-y$ and $y$ into a ray joining the origin to the infinity on the plane $\\lbrace y\\rbrace ^$ , and is therefore bijective.",
"From the bijectivity of $c$ and the formula defining the radius, one deduces that the map $F_{(y,\\lambda )}$ is one-to-one.",
"Proposition REF implies that the projection of the set ${\\mathcal {L}}_i = \\lbrace \\rho _{Y,\\lambda }(u) u, u \\in _{Y}^\\lambda (y_i)\\rbrace = (\\partial P(y_i, \\lambda _i)) \\cap \\Big (\\bigcap _{j\\ne i} P(y_j,\\lambda _j)\\Big )$ onto the plane orthogonal to $y_i$ is a finite intersection of discs, and therefore convex.",
"Since the surface $\\partial P(y_i,\\lambda _i)$ is a graph over the plane $\\lbrace y_i\\rbrace ^$ , we deduce that $\\mathcal {L}_i$ is connected.",
"This implies that its radial projection on the sphere, namely the paraboloid intersection cell $_{Y}^\\lambda (y_i)$ , is also connected.",
"We denote by $V$ (resp.",
"$E$ , $F$ ) the number of vertices (resp.",
"edges, faces).",
"Since, each cell $_{Y}^\\lambda (y_i)$ is connected, the number of faces $F$ is bounded by $N$ .",
"Moreover, since there are at least three incident edges for each vertex, we have that $3V \\le 2E$ .",
"Then, by Euler's formula, we have that $V \\le 2F-4$ and $E \\le 3F-6$ .",
"Even though the complexity of the paraboloid intersection diagram is $O(N)$ , it can not be computed faster than $\\Omega (N \\log N)$ , as stated in the proposition below.",
"We first define the genericity condition used in the statement of this proposition.",
"A family of solid paraboloids $(P(y_i, \\lambda _i))_{1\\le i \\le N}$ in $^3$ is called in generic position if for any subset $(i_k)_{1\\le k\\le 4}$ of $\\lbrace 1,\\hdots , N\\rbrace $ , the intersection $\\bigcap _{1\\le k\\le 4} \\partial P(y_{i_k},\\lambda _{i_k})$ is empty.",
"Remark that the intersection of four paraboloids $(\\partial P(y_{i_1},\\lambda _{i_k}))_{1\\le k\\le 4}$ contains a point $x$ if and only if the projection $u=x/{x}$ of this point on the unit sphere satisfies the equations ${u-p_{i_1}}^2+\\omega _{i_1}=\\hdots ={u-p_{i_4}}^2+\\omega _{i_4}$ , where the points $(p_i)$ and the weights $(\\omega _i)$ are defined by Proposition REF .",
"The genericity condition is then equivalent to the condition that for any quadruple of weighted points $(p_{i_k},\\omega _{i_k})_{1\\le k \\le 4}$ , the weighted circumcenter does not lie on $^2$ .",
"The complexity of the computation of the paraboloid intersection diagram is $\\Omega (N\\log (N))$ under the algebraic tree model, and even under an assumption of genericity.",
"We take a family of $N$ real numbers $(t_i)_{1\\le i \\le N}$ .",
"For every $i\\in \\lbrace 1,\\hdots ,N\\rbrace $ , we put $\\lambda _i=1$ and $y_i=\\varphi (t_i)$ , where the map $\\varphi :\\rightarrow ^3$ defined by $\\varphi (t)=(\\frac{t^2-1}{1+t^2},\\frac{2t}{1+t^2},0)$ is a parameterization of the equator $^2 \\cap \\lbrace z=0\\rbrace $ from which we removed the point $(1,0,0)$ .",
"The family of paraboloids $(P(y_i,\\lambda ))_{1\\le i \\le N}$ is such that every cell $_{Y}^\\lambda (y_i)$ is delimited by two half great circles between the two poles, each of these half circles being shared by two cells.",
"We add the points $y_{N+1}=(1,0,0)$ , $y_{N+2}=(0,1,0)$ , $y_{N+3}=(-1,0,0)$ , $y_{N+4}=(0,0-1,0)$ , $y_{N+5}=(0,0,1)$ and $y_{N+6}=(0,0,-1)$ , so that $(P(y_i,\\lambda ))_{1\\le i \\le N+6}$ is in general position.",
"More precisely, the four points $y_{N+1},\\hdots , y_{N+4}$ are added to ensure that every points of the equator is at a distance strictly less than $\\sqrt{2}/2$ from $\\lbrace y_1,\\hdots ,y_{N+4}\\rbrace $ .",
"The cells of the two poles $y_{N+5}$ and $y_{N+6}$ then do not intersect the equator and we keep the property that there exists a cycle with the $N+4$ vertices of $\\lbrace y_1,\\hdots ,y_{N+4}\\rbrace $ in the dual of the paraboloid intersection diagram.",
"Finding this cycle then amounts to sorting the values $(t_i)_{1\\le i \\le N+4}$ .",
"The conclusion holds from the fact that a sorting algorithm has a complexity $\\Omega (N\\log (N))$ under the algebraic tree model."
],
[
"Other types of union and intersections",
"Other quadrics, such as the ellipsoid of revolution, or one sheet of a two-sheeted hyperboloid of revolution can also be parametrized over a unit sphere by the inverse of an affine map [19].",
"In this section, we study the combinatorics of the intersection of solid ellipsoids, one of whose focal points lie at the origin.",
"Note that this intersection naturally appears in the near-field reflector problem, where one wants to illuminate points in the space instead of directions (as in the far-field reflector problem) [14].",
"Furthermore, for both the ellipsoid and the paraboloid cases, we also study the union of the convex hulls.",
"We show that in these three cases, the combinatorics is still given by the intersection of a power diagram with the unit sphere.",
"However, the complexity might be higher.",
"We show that there exists configuration of $N$ ellipsoids whose intersection and union have complexity $\\Omega (N^2)$ .",
"An algorithm that matches this lower bound is provided in Section ."
],
[
"Union of confocal paraboloids of revolution",
"The union of a family of solid paraboloids $(P(y_i,\\lambda _i))_{1\\le i \\le N}$ is star-shaped with respect to the origin.",
"Moreover, its boundary can be parameterized by the radial function $u \\in ^{d-1}\\mapsto \\left(\\max _{1\\le i\\le N} \\rho _{y_i,\\lambda _i}(u)\\right)\\cdot u.$ The paraboloid union diagram is a decomposition of the sphere into cells associated to the upper envelope of the functions $(\\rho _{y_i,\\lambda _i})_{1\\le i \\le N}$ : $_{Y}^\\lambda (y_i) := \\lbrace u \\in ^{d-1};~ \\forall j \\in \\lbrace 1,\\hdots ,N\\rbrace , \\rho _{y_i,\\lambda _i}(u) \\ge \\rho _{y_j,\\lambda _j}(u)\\rbrace .$ As before, these cells can be seen as the intersection of certain power cells with the unit sphere.",
"Given a family $(P(y_i,\\lambda _i))_{1\\le i \\le N}$ of solid paraboloids, one has for all $i$ , $_{Y}^{\\lambda }(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i),$ where the points and weights are given by $p_i = \\frac{1}{2} \\lambda _i^{-1} y_i$ and $\\omega _i =\\lambda _i^{-1} - \\frac{1}{4}\\lambda _i^{-2}.$ Proposition REF implies that for every $i$ , the projection of ${\\mathcal {L}}_i = (\\partial P(y_i, \\lambda _i)) \\cap \\partial \\left(\\bigcup _{1\\le j \\le N} P(y_j,\\lambda _j)\\right)$ onto the plane orthogonal to $y_i$ is a finite intersection of complements of discs.",
"In particular, this set does not need to be connected in general, and neither does the corresponding cell.",
"This situation can happen in practice (see Proposition REF ).",
"Consequently, one cannot use the connectedness argument as in the proof of Theorem REF to show that the paraboloid union diagram has complexity $O(N)$ in dimension three.",
"Actually, the following proposition shows that a unique cell may have $\\Omega (N)$ distinct connected component.",
"One can construct a family of paraboloids $(P(y_i,\\lambda _i))_{0\\le i\\le N}$ such that the paraboloid union cell $_Y^\\lambda (y_0)$ has $\\Omega (N)$ connected components.",
"Let $y_0$ be an arbitrary point on the sphere, and let $\\lambda _0=1$ .",
"Now, consider a family of disks $D_i$ in the plane $H = \\lbrace y_0\\rbrace ^$ with centers and radii $(c_i,r_i)_{1\\le i \\le N}$ , and such that the set $ U= \\bigcup _{i=1}^N (H\\setminus D_i) = H \\setminus \\bigcup _{i=1}^N D_i$ has $\\Omega (N)$ connected components.",
"This is possible by setting up a flower shape (see Figure REF ), i.e., $D_1$ is the unit ball and $D_2,\\hdots ,D_N$ are set up in a flower shape around $D_1$ .",
"By the second part of Proposition REF , one can construct paraboloids $(P(y_i,\\lambda _i))_{1\\le i\\le N}$ such that $F_{(y_0,\\lambda _0)}(y_i,\\lambda _i) = (c_i,r_i)$ .",
"Then, the first part of Proposition REF shows that the paraboloid union cell $_{Y}^\\lambda (y_0)$ is homeomorphic to $U$ , and has therefore $\\Omega (N)$ connected components.",
"Figure: A flowerOne can also underline that the complexity of each cell is $O(N)$ .",
"This is a direct consequence of Proposition REF and the fact that the complexity of the union of $N$ planar discs is $O(N)$ [2]."
],
[
"Intersection and union of confocal ellipsoids of revolution",
"An ellipsoid of revolution whose one focal point lies at the origin is characterized by two other parameters: its second focal point $y$ and its eccentricity $e$ in $(0,1)$ .",
"We denote the convex hull of such an ellipsoid of revolution $E(y,e)$ .",
"The surface $\\partial E(y,e)$ of this set is parameterized in spherical coordinates by the function $\\sigma _{y,e}(m) := \\frac{d}{1-e {m}{\\frac{y}{\\Vert y\\Vert }}} \\quad \\mbox{where } d=\\frac{\\Vert y\\Vert (1-e^2)}{2 e}.$ Note that the value $d$ is fully determined by $e$ and $y$ and is introduced only to simplify the computations.",
"Let $Y=(y_i)_{1\\le i \\le N}$ be a family of distinct points in $^d$ and $e=(e_i)_{1\\le i \\le N}$ be a family of real numbers in the interval $(0,1)$ .",
"The boundary of the intersection of solid ellipsoids $\\bigcap _{1\\le i\\le N} E(y_i, e_i)$ is parameterized in spherical coordinates by the lower envelope of the functions $(\\sigma _{y_i,e_i})_{1\\le i \\le N}$ .",
"The ellipsoid intersection diagram of this family of ellipsoids is the decomposition of the unit sphere into cells associated to the lower envelope of the functions $(\\sigma _{y_i,e_i})_{1\\le i \\le N}$ : $_{Y}^e(y_i) := \\lbrace u \\in ^{d-1};~ \\forall j \\in \\lbrace 1,\\hdots ,N\\rbrace , \\sigma _{y_i,e_i}(u) \\le \\sigma _{y_j,e_j}(u)\\rbrace .$ Similarly, the cells of the ellipsoid union diagram are associated to the upper envelope of the functions $(\\sigma _{y_i,e_i})_{1\\le i \\le N}$ as follows: $_{Y}^e(y_i) := \\lbrace u \\in ^{d-1};~ \\forall j \\in \\lbrace 1,\\hdots ,N\\rbrace , \\sigma _{y_i,e_i}(u) \\ge \\sigma _{y_j,e_j}(u)\\rbrace .$ As in the case of paraboloids, the computation of each diagram amounts to compute the intersection of a power diagram with the unit sphere.",
"Let $(E(y_i,e_i))_{1\\le i \\le N}$ be a family of solid confocal ellipsoids.",
"Then, (i) The cells of the ellipsoid intersection diagram are given by $_{Y}^{e}(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i),$ where $p_i = -\\frac{e_i}{2d_i} \\frac{y_i}{\\Vert y_i\\Vert }$ and $\\omega _i = -\\frac{1}{d_i} -\\frac{e_i^2}{4d_i^2}.$ (ii) The cells of the ellipsoid union diagram are given by : $_{Y}^{e}(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i),$ where $p_i = \\frac{e_i}{2d_i} \\frac{y_i}{\\Vert y_i\\Vert }$ and $\\omega _i =\\frac{1}{d_i} -\\frac{e_i^2}{4d_i^2}.$ The theorem below shows that in dimension three the complexity of these diagrams can be quadratic in the number of ellipsoids.",
"This is in sharp contrast with the case of the paraboloid intersection diagram, where the complexity is linear in the number of paraboloids.",
"In $^3$ , there exists a configuration of confocal ellipsoids of revolution such that the number of vertices and edges in the ellipsoid intersection diagram (resp.",
"the ellipsoids union diagram) is $\\Omega (N^2)$ .",
"The proof of this theorem strongly relies on the following lemma, that transfers the problem to a complexity problem of power diagrams intersected by the unit sphere.",
"Let $(p_i,\\omega _i)_{1\\le i \\le N}$ be a family of weighted points such that no point $p_i$ lie at the origin.",
"Then, there exists: (i) a family of ellipsoids $(E(y_i,e_i))_{1\\le i \\le N}$ , such that for all $i$ , $_{Y}^{e}(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i).$ (ii) a family of ellipsoids $(E(y_i,e_i))_{1\\le i \\le N}$ , such that for all $i$ , $_{Y}^{e}(y_i) = ^{d-1} \\cap _P^{\\omega }(p_i).$ We prove only the first assertion, the second one being similar.",
"Let $i\\in \\lbrace 1,\\hdots , N\\rbrace $ .",
"By inverting the equations $p_i = -\\frac{e_i}{2d_i} \\frac{y_i}{\\Vert y_i\\Vert }$ and $\\omega _i =\\frac{1}{d_i} +\\frac{e_i^2}{4d_i^2}$ , we get $y_i=\\frac{-4}{(\\omega _i +{p_i}^2)^2 - 4 {p_i}^2}\\ p_i\\quad \\mbox{and} \\quad e_i=\\frac{-2{p_i}}{\\omega _i + {p_i}^2}.$ The condition $e_i \\in (0,1)$ is equivalent to $\\omega _i < -{p_i}^2 \\mbox{ and } \\omega _i < -{p_i}^2 - {p_i} $ .",
"These inequalities can always be satisfied by substracting a large constant to all the weights $\\omega _i$ , an operation that does not change the power cells.",
"We recall that the Voronoi diagram of a point cloud $(p_i)_{1\\le i \\le N}$ of $^3$ is the decomposition of the space in $N$ convex cells defined by $\\mathrm {Vor}_P(p_i):=\\lbrace x \\in ^3,\\ {x-p_i} \\le {x-p_j} \\rbrace .$ The next proof is illustrated by Figure REF .",
"Thanks to Lemma REF , it is sufficient to build an example of Voronoi diagram whose intersection with $^2$ has a quadratic number of edges and vertices.",
"We can consider without restriction that $N=2k$ is even.",
"Let $\\varepsilon >0$ be a small number.",
"We let $p_1,\\hdots ,p_k$ be $k$ points uniformly distributed on the circle centered at the origin in the plane $\\lbrace z=0\\rbrace $ , and with radius $2-\\varepsilon $ .",
"We also consider $k$ evenly distributed points $q_1,\\hdots ,q_k$ on $(A,B)\\setminus \\lbrace O\\rbrace $ , where $A=(0,0,-\\varepsilon /4)$ , $B=(0,0,\\varepsilon /4)$ and $O=(0,0,0)$ .",
"Our goal is now to show that for any $i$ and $j$ in between 1 and $k$ , the intersection $_P(p_i) \\cap _P(q_j) \\cap ^2$ is non-empty, where $P =\\lbrace p_i\\rbrace \\cup \\lbrace q_j\\rbrace $ .",
"We denote by $m_{i,j}$ the unique point which is equidistant from $p_i$ and $q_j$ , and which lies in the horizontal plane containing $q_j$ and in the (vertical) plane passing through $p_i$ , $q_j$ and the origin.",
"A simple computation shows that the distance $\\delta _{i,j}$ between $m_{i,j}$ and $q_{j}$ satisfies $\\delta _{i,j} \\le 1-\\varepsilon / 2 +\\varepsilon ^2/(64-32\\varepsilon )$ .",
"Taking $\\varepsilon $ small enough, this implies that the point $m_{i,j}$ belongs to the unit ball.",
"The line passing through $m_{i,j}$ and orthogonal to the plane passing through $p_i$ , $q_j$ and the origin cuts the unit sphere at two points $r^\\pm _{i,j}$ .",
"These two points belong to the same horizontal plane as $q_j$ , and they both belong to $_P(q_j)$ .",
"Moreover, the distance between $m_{i,j}$ and each of these two points is less than $(1-\\delta _{i,j}^2)^{1/2}$ .",
"We can take $\\varepsilon $ small enough so that this distance is less than $\\sin (\\pi /k)$ .",
"This implies that the points $r_{i,j}^\\pm $ also belong to the Voronoi cell $\\mathrm {Vor}(p_i)$ and therefore to the intersection $_P(p_i) \\cap _P(q_j) \\cap ^2$ .",
"Thus, the intersection between the Voronoi diagram and the sphere has at least $k^2=\\frac{1}{4}N^2$ spherical edges.",
"Figure: An Ω(N 2 )\\Omega (N^2) construction.",
"From a total of NN sites, half of them in a segment inside the sphere, and half of them on a circle outside the sphere, we obtain a total of Ω(N 2 )\\Omega (N^2) edges embedded on the sphere."
],
[
"Computing the intersection of a power diagram with a sphere",
"In this section, we describe a robust and efficient algorithm to compute the intersection between the power diagram of weighted points $(P,\\omega )$ and the unit sphere.",
"We call this intersection diagram.",
"Given the set of weighted points $(p_i,\\omega _i)$ , we define the $i$ th intersection cell as $_P^\\omega (p_i) = ^{2} \\cap _P^\\omega (p_i)$ .",
"This algorithm is implemented using the Computational Geometry Algorithms Library (CGAL) [1], and bears some similarity to [5], [10].",
"We concentrate our efforts in obtaining the exact combinatorial structure of the diagram on the sphere for weighted points whose coordinates and weight are rational.",
"Exact constructions can be obtained, but are not our focus, since in our applications, the results of this section are used as input of a numerical optimization procedure; see Section .",
"Note that the intersection diagram has complex combinatorics.",
"If one assumes that the weighted points are given by a paraboloid intersection diagram, Proposition REF shows that the intersection cells are connected.",
"But even in this case, the intersection between two intersection cells $_P^\\omega (p_i)$ and $_P^\\omega (p_j)$ can have multiple connected components.",
"Consequently, one cannot hope to be able to reconstruct these cells from the adjacency graph (i.e., the graph that contains the points in $P$ as vertices, and where two vertices are connected by an arc if the two cells intersect in a non-trivial circular arc), even in this simple case.",
"In general, the cells of the intersection diagram can be disconnected and they can have holes, as shown in Figure REF .",
"In the next paragraph, we propose an algorithm to compute a boundary representation of these cells.",
"Recall the following definitions concerning power diagrams: When two power cells $_P^\\omega (p_{i})\\cap _P^\\omega (p_{j})$ intersect in a 2-dimensional set, that is a (possibly unbounded) polygon in $^3$ , they determine a facet of the power diagram.",
"Similarly, a ridge is a 1-dimensional intersection $_P^\\omega (p_{i})\\cap _P^\\omega (p_{j}) \\cap _P^\\omega (p_{k})$ between three power cells.",
"A ridge can be either a ray or a segment.",
"The boundary of each facet can be written as a finite union of ridges.",
"Figure: Examples of diagram on the sphere.",
"Left: an intersection diagram containing faces with no vertices, faces with holes and faces with two vertices or more.",
"Right: a diagram corresponding to the intersection of a cube and a sphere; there are seven faces, six faces with no vertices, and a face with no vertices and six holes.",
"In the remainder of this section, we explain how to compute a single intersection cell $_P^\\omega (p_i)$ , for a given $p_i$ , without any assumption on the points and weights.",
"This intersection cell can be quite intricate (multiple connected components, holes), and will therefore be represented by its oriented boundary.",
"More precisely, the boundary $_P^\\omega (p_i)$ is a finite union of closed curves called cycles.",
"These cycles are oriented so that for someone walking on the sphere following a cycle, the intersection cell $_P^\\omega (p_i)$ lies to the right.",
"The cell is uniquely determined by its oriented boundary.",
"The following theorem is the main result of this section.",
"There is an $O(N \\log N + C)$ algorithm for obtaining the intersection diagram of a set $P$ of $N$ weighted points in $\\mathbb {R}^3$ , where $C$ is the complexity of the power diagram of $P$ .",
"From Theorem REF , the size of the output in the worst case for the intersection (or union) of confocal ellipsoids is $\\Omega (N^2)$ .",
"This implies that the algorithm described below is optimal for computing an ellipsoid intersection (or union) diagram.",
"The optimality of the algorithm for the paraboloid intersection diagram is an open problem, which can be phrased as follows: Is the complexity of the power diagram given by Proposition REF bounded by $O(N)$ ?"
],
[
"Remark.",
"Even if the question above has a negative answer, at least for the particular case of the intersection diagram of confocal paraboloids, there exists an easy randomized incremental algorithm attaining the lower bound in Proposition REF .",
"If the input is $N$ confocal paraboloids, from Theorem REF , we know that the complexity of the paraboloid intersection diagram is $O(N)$ .",
"Moreover the resulting cells are connected.",
"Adopting the randomized incremental construction paradigm (see [20], [16] for a comprehensive study), this means that the expected complexity of the cell of the $i$ th inserted paraboloid is $O(N)/i$ .",
"Besides, thanks to the connectedness of the cells, we can detect the conflicts with a breadth-first search in linear time as well, without any additional structure.",
"Therefore, the total expected cost is $\\sum _{i=1}^n O(N)/i$ , which is in $O(N \\log {N})$ .",
"Notice that this analysis relies on the fact that the cells are connected, which is not necessarily true in the case of union of paraboloids and in the ellipsoids cases.",
"In our application, we always use the generic and robust algorithm based on the intersections of power cells, using CGAL."
],
[
"Predicates",
"Geometric algorithms often rely on predicates, i.e., estimation of finite-valued geometric quantities such as the orientation of a quadruple of points in 3D, or the number of intersections between a straight line and a surface.",
"These predicates need to be evaluated exactly: if this is not the case, some geometric algorithm may even not terminate [12].",
"We adopt the dynamic filtering technique for the computation of our predicates in CGAL.",
"This means that for every arithmetic operation, we also compute an error bound for the result.",
"This can be automatized using interval arithmetic [6].",
"If the error bound is too large to determine the result of the predicate, the arithmetic operation is performed again with exact rational arithmetic, allowing us to return an exact answer for the predicate.",
"Our algorithm requires the following three predicates: (i) has_on(Point p) returns $+1$ , 0, or $-1$ , if the point is outside, on, or inside the unit sphere respectively.",
"(ii) number_of_intersections(Ridge r) returns the number of intersections between $r$ and the unit sphere ($0,1$ or 2), where $r$ can be a segment or a ray.",
"By convention, segment or ray touching the sphere tangentially has two intersection points, with same coordinates.",
"Remark that the result of the first predicate can be obtained directly from the intermediate computations of this predicate.",
"As we use them intertwined, this shortcut is adopted in our implementation.",
"(iii) plane_crossing_sphere(Facet f) returns the number of intersections between the supporting plane of $f$ and the sphere ($0,1$ or $+\\infty $ ).",
"One may notice that this predicate is just the dual of the first one.",
"Even though the algorithm below is presented as working directly on the cells of the power diagram, our actual implementation uses CGAL, and we work with the regular triangulation of the set of points (that is, the dual of the power diagram).",
"This triangulation is a simplicial complex with 0-, 1-, 2- and 3-simplices.",
"In CGAL, only the 0- and 3-simplices are actually stored in memory, but it remains possible to traverse the whole structure in linear time in the number of simplices.",
"In order to handle the boundary simplices, CGAL takes the usual approach of adding a point at infinity to the initial set of points.",
"In practice, this means the following.",
"A ridge of the power digram can be either a segment or a ray, and is dual of a 2-simplex in the triangulation.",
"In CGAL, such a ridge corresponds to two adjacent tetrahedra, one of whom may contain the point at infinity.",
"The predicates above need to be adapted in order to handle $k$ -simplices that are incident to the point at infinity.",
"The general predicates have been implemented, but the details are omitted here."
],
[
"Computation of the oriented boundary",
"The output of our algorithm is the oriented boundary of the intersection cell $_P^\\omega (p_i)$ .",
"Note that this is a purely combinatorial object, and no geometric construction are performed during its computation.",
"It is described as a sequence of vertices on the sphere, oriented arcs, and oriented cycles.",
"Figure: Obtaining the arcs of P ω (p i )_P^\\omega (p_i).",
"Three distinct cases: (a) inside the circle, (b) intersecting the circle, and (c) enclosing the circle.",
"There is still a case not depict above, ff can be outside the circle as well.",
"The arcs with zero length, such as the one in (b) can be removed from the diagram depending on its use.",
"$\\bullet $ A vertex on the sphere is the result of an intersection between a ridge of the power diagram and the unit sphere.",
"A vertex on the sphere is not constructed explicitly, but stored as an ordered pair of extremities of the corresponding ridge.",
"Since a ridge $[x_1,x_2]$ can intersect the sphere twice, the order determines which intersection point to consider, i.e., the pair $(x_1,x_2)$ describes the intersection point that is closest to $x_1$ .",
"This convention allows us to handle infinite ridges, and to uniquely identify vertices on the sphere.",
"In degenerate cases, vertices may correspond to different ordered pairs, but having the same coordinates.",
"$\\bullet $ An oriented arc corresponds to the (oriented) intersection between $_P^\\omega (p_i)$ , another cell $_P^\\omega (p_j)$ and the unit sphere.",
"Two situations arise.",
"Either the arc is a complete circle, in which case it is also a cycle.",
"Or the two extreme points are vertices as defined above, that is, they are the intersection between ridges and the unit sphere.",
"An arc is oriented so that, when walking on the sphere from the first point to the second, the cell lies on the right.",
"$\\bullet $ An oriented cycle is a connected component of the oriented boundary of $_P^\\omega (p_i)$ .",
"A cycle can have two or more vertices on the sphere, in this case, they are represented by a cyclic sequence of arcs.",
"However, a cycle can also have no vertices, and in such a case, it is represented by a single full arc.",
"The orientation of a cycle is given by the orientation of its arcs.",
"Notice that the boundary of a power cell is composed of several convex facets, that can possibly be unbounded.",
"The intersection of such facets with the unit sphere gives the arcs of the intersection diagram.",
"Our algorithm constructs the oriented boundary of the intersection cell $_P^\\omega (p_i)$ by iterating over each facet of the power cell $_P^\\omega (p_i)$ and obtaining implicitly both the set of vertices on the sphere $V$ and the set of oriented arcs $E$ .",
"The oriented graph $G=(V,E)$ is called the boundary graph.",
"Once $G$ is constructed, the oriented cycles in the boundary $\\partial _p^\\omega (p_i)$ coincide with the (oriented) connected components of $G$ , and can be obtained by a simple traversal.",
"In the next paragraph, we explain the construction of $G=(V,E)$ ."
],
[
"Computation of the boundary graph",
"We start by discarding the facets that do not contribute to the intersection diagram.",
"We iterate over all facets $f$ of $_P^\\omega (p_i)$ , and keep $f$ only if the predicate plane_crossing_sphere$(f)$ returns $\\infty $ .",
"(The case where it returns 1 corresponds to a trivial intersection between the facet and the sphere and can be safely ignored.)",
"Assuming that $f$ is not discarded, the intersection of its supporting plane with the unit sphere is a circle in 3D.",
"If all vertices of $f$ are on or inside the sphere (i.e., has_on$\\le 0$ ), then $f$ is inside or only touching the circle, and is discarded.",
"See Figure fig:obtainingarcs:inside.",
"Given a facet $f = _P^\\omega (p_i) \\cap _P^\\omega (p_j)$ that was not discarded previously, we build a sequence of vertices on the sphere by traversing $f$ in a clockwise sequence of ridges, oriented with respect to the vector $p_j - p_i$ We do not need orientation predicates at this point, since, in CGAL, 3-simplices of a triangulation are positively oriented.. We distinguish two different kinds of vertices on the sphere, the source vertices and the target vertices, as shown on Figure fig:obtainingarcs:intersecting.",
"Given an oriented ridge $r=[x_1,x_2]$ , we compute the value of number_of_intersections(r).",
"If the ridge has zero intersections with the sphere, we skip it.",
"If there is only one intersection between $r$ and the unit sphere, the corresponding vertex is a source if $x_1$ is inside the sphere (i.e., has_on(x_1) $\\le 0$ ) and it is a target vertex in the other case.",
"Finally, if the ridge intersects the sphere twice, the oriented pair $(x_1,x_2)$ describes the target vertex, and the pair $(x_2,x_1)$ is the source vertex.",
"The arcs of the boundary graph $G$ corresponding to this facet are obtained by matching each source vertex with a target vertex using the same cyclic sequence.",
"They are represented by an ordered pair of vertex indices.",
"Finally, we need to consider the case where none of the ridges of $f$ intersect the sphere.",
"There are two possibilities: either the facet $f$ encloses the circle $\\pi \\cap ^2$ , where $\\pi $ is the affine plane spanned by $f$ , or the circle is completely outside from $f$ .",
"In the former case, we obtain an arc without vertex in the boundary of $_P^\\omega (p_i)$ , as shown in Figure fig:obtainingarcs:enclosing.",
"Its orientation is clockwise with respect to the vector $p_j - p_i$ .",
"To detect whether this happens, we simply need to determine whether the center of the circle lies inside or outside the convex polygon $f$ .",
"This is a classical routine that can be performed using signed volumes.",
"For this test, we only need to use rational numbers because the center of the circle is the projection of the origin on a plane described with rational coordinates.",
"First we build the regular triangulation of the set of $N$ weighted points in $O(N \\log N + C)$ time.",
"Running the algorithm described in this section for every $p_i$ makes us visiting each 1-simplex, 2-simplex, and 3-simplex of the primal no more than two, three, and four times respectively; and for each visit, it takes $O(1)$ time, since we compute constant-degree predicates.",
"Then we visit each generated cycle at most twice.",
"Therefore, the complexity of our algorithm is in $O(N \\log N + C + D)$ , where $D$ is the complexity of the diagram on the sphere.",
"The fact that $D = O(C)$ completes the proof."
],
[
"Application to Minkowski-type problems involving paraboloids",
"This section deals with the far-field reflector problem [9], which is a Minkowski-type problem involving intersection of confocal paraboloids of revolution.",
"Consider a point source light located at the origin $O$ of $^d$ that emits lights in all directions.",
"The intensity of the light emitted in the direction $x \\in ^{d-1}$ is denoted by $\\rho (x)$ .",
"Now, consider a hypersurface $R$ of $^d$ .",
"By Snell's law, every ray $x$ emitted by the source light that intersects $R$ at a point whose normal vector is $n$ is reflected in the direction $y=x-2{x}{n}n$ .",
"This means that after reflection on $R$ , the distribution of light at the origin given by $\\rho $ is transformed into a distribution $\\nu _R$ on the set of directions at infinity, a set that can also be described by the unit sphere.",
"The far-field reflector problem consists in the following inverse problem: given a density $\\rho $ on the source sphere $^2$ and a distribution $\\mu $ on the sphere at infinity, the problem is to find an hypersurface $R$ such that $\\mu _R$ coincides with $\\mu $ .",
"For computational purposes, we assume that the target measure $\\nu $ is supported on a finite set of directions $Y:=(y_i)_{1\\le i \\le N}$ , and it is thus natural to consider a reflector made of pieces of paraboloids with directions $Y$ .",
"This problem is numerically solved using an optimal transport formulation due to [21], [11].",
"Figure: Numerical computation.",
"Calculations were done with N=1000N=1000 paraboloids for thefirst row and N=15000N=15000 paraboloids for the second row.",
"(a)Paraboloid intersection diagram for an initial(λ i ) 1≤i≤N (\\lambda _i)_{1\\le i \\le N}.",
"(b,d) Final intersection diagramafter optimization.",
"(e) Reflector surface defined by theintersection of paraboloids.",
"(c,f) Simulation of theillumination at infinity from a punctual light sourcelighting uniformly - 2 _-^2, using LuxRender, aphysically accurate raytracer engine."
],
[
"Far-field reflector problem.",
"We consider a finite family $Y =(y_1,\\hdots ,y_N)$ of unit vectors that describe directions at infinity, non-negative numbers $(\\alpha _i)_{1\\le i\\le N}$ such that $\\sum _{1\\le i\\le N} \\alpha _i = 1$ and a probability density $\\rho $ on the unit sphere.",
"The far-field reflector problem consists in finding a vector of non-negative focal distances $(\\lambda _i)_{1\\le i\\le N}$ , such that $\\forall i \\in \\lbrace 1,\\hdots ,N\\rbrace ,~~ \\rho (_Y^\\lambda (y_i)) = \\alpha _i,\\qquad \\mathrm {(FF)}$ where for a subset $X$ of the sphere, $\\rho (X):=\\int _{X} \\rho (u) u$ is the weighted area of $X$ .",
"The far-field reflector problem can be transformed into the maximization of a concave functional, combining ideas from [21], [11], [3].",
"A vector of focal distances $(\\lambda _i)_{1\\le i\\le N}$ solves the far-field reflector problem (REF ) if and only if the vector $(\\gamma _i)_{1\\le i\\le N}$ defined by $\\gamma _i = \\log (\\lambda _i)$ is a global maximizer of the following $^1$ concave function: $\\Phi (\\gamma ) := \\left[\\sum _{i=1}^N \\int _{_Y^{\\exp \\gamma }(y_i)} (c(u,y_i) + \\gamma _i) \\rho (u)u \\right] - \\sum _{i=1}^N \\gamma _i \\alpha _i$ where $c(u,v) := - \\log (1-{u}{v})$ and with the convention $\\log (0)= -\\infty $ .",
"Moreover, the gradient of the function $\\Phi $ is given by $\\nabla \\Phi (\\gamma ) :=(\\rho (_Y^{\\exp (\\gamma )}(y_i)) - \\alpha _i)_{1\\le i\\le N}.$ There is a similar formulation for a reflector defined by the union of solid confocal paraboloids [11].",
"However, there is no known variational formulation for the reflector problem involving intersection of ellipsoids.",
"A numerical approach has been proposed in [18].",
"We now turn to the proof of Theorem REF .",
"This proof combines the results of Section 5 of [3] with the optimal transport formulation of the far-field reflector problem [21], [11].",
"Let us first recall some properties of supdifferential of concave functions.",
"Given a function $\\Phi $ and $\\lambda $ in $^N$ , the supdifferential of $\\Phi $ at $\\lambda $ , denoted $\\partial ^+ \\Phi (\\lambda )$ is the set of vectors $v$ such that $ \\forall \\kappa \\in ^N,~ \\Phi (\\kappa ) \\le \\Phi (\\lambda ) +{\\kappa -\\lambda }{v}.$ A function $\\Phi $ is concave if and only if for every $\\lambda $ , the supdifferential $\\partial ^+ \\Phi (\\lambda )$ is nonempty.",
"If this is the case, $\\Phi $ is differentiable almost everywhere, and at points of differentiability the supdifferential $\\partial ^+ \\Phi (\\lambda )$ coincides with the singleton $\\lbrace \\nabla \\Phi (\\lambda )\\rbrace $ .",
"Finally, $\\lambda $ is a global maximum of $\\Phi $ if and only if $\\partial ^+ \\Phi (\\lambda )$ contains the zero vector.",
"We consider a vector $\\gamma $ in $^N$ .",
"First remark that $c(u,y_i) + \\gamma _i =\\log (\\exp (\\gamma _i)/(1-{u}{y_i})$ .",
"Therefore, $u \\in _Y^{\\exp (\\gamma )}(y_i)&\\Longleftrightarrow \\forall j\\in \\lbrace 1,\\hdots ,N\\rbrace ,\\frac{\\exp (\\gamma _i)}{1 - {u}{y_i}} \\le \\frac{\\exp (\\gamma _j)}{1 - {u}{y_j}} \\\\&\\Longleftrightarrow \\forall j\\in \\lbrace 1,\\hdots ,N\\rbrace ,c(u,y_i) + \\gamma _i \\le c(u,y_j) + \\gamma _j$ Therefore, the function $\\Phi $ can be reformulated as follows: $\\Phi (\\gamma ) = \\int _{^{d-1}} \\left(\\min _{1\\le i\\le N}c(u,y_i) + \\gamma _i\\right)\\rho (u) u - \\sum _{i=1}^N \\gamma _i\\alpha _i.$ We now define $T_\\gamma $ as the function that maps a point $u$ on the unit sphere to the point $y_i$ such that $u$ belongs to $_Y^{\\exp (\\gamma )}(y_i)$ .",
"Then, $\\Phi (\\gamma ) = \\int _{^{d-1}} (c(u,T_\\gamma (u)) +\\gamma _{T_\\gamma (u)})\\rho (u) u - \\sum _{i=1}^N \\gamma _i\\alpha _i.$ Moreover, for any $\\kappa $ in $^d$ , one has $\\min _{1\\le i\\le N} c(u,y_i) + \\kappa _i \\le c(u,T_\\gamma (u)) +\\kappa _{T_\\gamma (u)}$ .",
"Integrating this inequality, and substracting (REF ), we get $\\Phi (\\kappa ) - \\Phi (\\gamma )&\\le \\int _{^{d-1}} (\\kappa _{T_\\gamma (u)} - \\gamma _{T_\\gamma (u)})\\rho (u)u - \\sum _{1\\le i\\le N} (\\kappa _i - \\gamma _i) \\alpha _i \\\\&\\le \\sum _{1\\le i\\le N} \\left(\\int _{_Y^{\\exp \\gamma }(y_i)} \\rho (u)u - \\alpha _i\\right) (\\kappa _i - \\gamma _i) = {D\\Phi (\\gamma )}{\\kappa - \\gamma } \\\\\\hbox{where }D\\Phi (\\gamma ) &:=(\\rho (_Y^{\\exp (\\gamma )}(y_i)) - \\alpha _i)_{1\\le i\\le N}.$ The above inequality shows that $D\\Phi (\\gamma )$ lies in $\\partial ^+\\Phi (\\gamma )$ , i.e., this set is never empty and the function $\\Phi $ is concave.",
"Since the vector $D\\Phi (\\gamma )$ depends continuously on $\\gamma $ , we deduce that the function $\\Phi $ is $^1$ smooth, and that $\\nabla \\Phi (\\gamma ) = D\\Phi (\\gamma )$ everywhere.",
"By Equation (REF ), a vector $\\lambda :=\\exp (\\gamma )$ solves the far-field reflector problem (REF ) if and only if $D\\Phi (\\gamma )=0$ , i.e., if and only if $\\gamma $ is a global maximizer of $\\Phi $ ."
],
[
"Implementation details.",
"The implementation of the maximization of the functional $\\Phi $ follows closely [15].",
"We rely on a quasi-Newton method, which only requires being able to evaluate the value of $\\Phi $ and the value of its gradient at any point $\\gamma $ , as given by Equations (REF )–(REF ).",
"The computations of these values are performed in two steps.",
"First, we compute the boundary of the paraboloid intersection cells $_Y^{\\exp (\\gamma )}(y_i)$ , using the algorithm described in Section .",
"These cells are then tessellated, and the integrals in Equations (REF )–(REF ) are evaluated numerically using a simple Gaussian quadrature.",
"In the experiments illustrated in Figure REF , we constructed the measure $\\sum _{i} \\alpha _i \\delta _{y_i}$ so as to approximate a picture of Gaspard Monge (projected on a part of the half-sphere $^2_+ := ^2\\cap \\lbrace z\\ge 0\\rbrace $ ).",
"The density $\\rho $ is constant in the half-sphere $^2_-$ and vanishes in the other half.",
"To the best of our knowledge, the only other numerical implementation of this formulation of the far-field reflector problem has been proposed in [8].",
"The authors develop an algorithm, called Supporting paraboloids which bears resemblance to Bertsekas' auction algorithm for the assignment problem [4].",
"They use it to solve the far-field reflector problem with 19 paraboloids.",
"Using the quasi-Newton approach presented above, and the algorithm developed in Section , we are able to solve this problem for 15,000 paraboloids in less than 10 minutes on a desktop computer.",
"Note that the algorithm of Section would probably also allow a faster and robust implementation of the Supporting paraboloids algorithm [8]."
],
[
"Conclusion",
"In addition to the open problems mentioned earlier, let us mention a perspective.",
"In a recent article [13], the algorithm of supporting paraboloids was extended to optimal transport problems involving a cost function $c$ that satisfies the so-called Ma-Trudinger-Wang regularity condition.",
"For this algorithm to be practical, one needs to compute the generalized Voronoi cells efficiently, defined for any function $\\psi :Y\\rightarrow $ by $_c^\\psi (y) = \\lbrace x \\in X;~\\forall z \\in Y,~c(x,y) + \\psi (y) \\le c(x,z)+\\psi (z)\\rbrace .$ For general costs, and even in 2D, one cannot hope to do this in time below $\\Omega (N^2)$ .",
"However, the MTW regularity condition ensures an analog of Proposition REF and in particular, it implies that these generalized Voronoi cells are connected.",
"One might wonder whether a randomized iterative construction could be used in this setting to yield a construction in expected time $O(N\\log N)$ in 2D.",
"This would open the way to practical algorithms for the resolution of optimal transport problems that are intractable to PDE-based methods."
],
[
"Acknowledgements.",
"The authors would like to thank Dominique Attali, Olivier Devillers and Francis Lazarus for interesting discussions.",
"Olivier Devillers suggested the approach used in the proof of the lower complexity bound for intersection of ellipsoids.",
"The first author is supported by grant FACEPE/INRIA,APQ-0055-1.03/12.",
"The second and third author would like to acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-11-BS01-014-01 (TOMMI) and ANR-13-BS01-0008-03 (TOPDATA) respectively."
]
] | 1403.0062 |
[
[
"Finite group actions on homology spheres and manifolds with nonzero\n Euler characteristic"
],
[
"Abstract Let $X$ be a smooth manifold belonging to one of these three collections: acyclic manifolds (compact or not, possibly with boundary), compact connected manifolds (possibly with boundary) with nonzero Euler characteristic, integral homology spheres.",
"We prove that $Diff(X)$ is Jordan.",
"This means that there exists a constant $C$ such that any finite subgroup $G$ of $Diff(X)$ has an abelian subgroup whose index in $G$ is at most $C$.",
"Using a result of Randall and Petrie we deduce that the automorphism groups of connected, non necessarily compact, smooth real affine varieties with nonzero Euler characteristic are Jordan."
],
[
"Introduction",
"In this paper we are interested on smooth finite group actions on manifolds.",
"Two general and natural questions in this subject are to characterize which finite groups admit effective actions on a given manifold, and to study geometric properties such as existence of fixed points for general actions.",
"These questions have attracted attention from the first works on finite transformation groups and have been a continuous source of inspiration ever since (see [2], [3], [9] for general introductions).",
"Despite all the progress, a general classification of which groups admit effective actions on a given manifold seems to be completely out of reach at present, except in very particular examples (either low dimensional manifolds, or manifolds which have none or very few finite order automorphisms, see e.g.",
"[7]).",
"Our aim is to study weak versions of the preceding questions.",
"More precisely, given a manifold $X$ , we want to find properties ${{P}}$ admitting a constant $C$ (depending on $X$ and ${{P}}$ ) such that any finite group $G$ acting effectively on $X$ has some subgroup $G_0$ satisfying ${{P}}$ and $[G:G_0]\\le C$ .",
"The properties ${{P}}$ which we are going to consider are commutativity and existence of fixed points."
],
[
"This is the first main result of this paper.",
"Theorem 1.1 Let $X$ be a smooth manifold belonging to one of these three collections: (1) acyclic manifolds (compact or not, possibly with boundary), (2) compact manifolds (possibly with boundary) with nonzero Euler characteristic, and (3) homology spheres.",
"There exist constants $C,d$ such that any finite group $G$ acting effectively and smoothly on $X$ has an abelian subgroup $A$ which can be generated by $d$ elements and has index $[G:A]\\le C$ .",
"Replacing $C$ by a bigger constant, we can assume that $A$ is normal in $G$ (indeed, for any inclusion of finite groups $\\Gamma ^{\\prime }\\subseteq \\Gamma $ there is a subgroup $\\Gamma _0\\subseteq \\Gamma ^{\\prime }$ which is normal in $\\Gamma $ and such that $[\\Gamma :\\Gamma _0]\\le [\\Gamma :\\Gamma ^{\\prime }]!$ ).",
"So if $X$ is any of the manifolds included in Theorem REF then there is an integer $\\delta $ and a finite collection of finite groups $G_1,\\dots ,G_r$ such that any finite group $G$ acting effectively and smoothly on $X$ sits in an exact sequence $1\\rightarrow {Z}_{n_1}\\oplus \\dots \\oplus {Z}_{n_{\\delta }}\\rightarrow G\\rightarrow G_j \\rightarrow 1,$ where ${Z}_n={Z}/n{Z}$ and $\\delta \\le d$ .",
"Hence, at least theoretically, it should be possible to classify all finite groups admitting effective actions on such manifolds in terms of finitely many parameters: namely, the group $G_j$ , the numbers $n_1,\\dots ,n_{\\delta }$ , and the extension class of (REF ); the latter is classified by a $G_j$ -module structure on $M:={Z}_{n_1}\\oplus \\dots \\oplus {Z}_{n_{\\delta }}$ and an element of $H^2(G_j;M)$ (see e.g.",
"[4]).",
"The proof of Theorem REF uses the classification of finite simple groups through a theorem proved jointly by Alexandre Turull and the author in [31] (see Theorem REF below).",
"Theorem REF gives a positive partial answer to a conjecture of Étienne Ghys, according to which for any compact manifold $X$ there is a constant $C$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ of index at most $C$ (see Question 13.1 in [12]This conjecture was discussed in several talks by Ghys [14] (I thank É. Ghys for this information), but apparently it was in [12] when it appeared in print for the first time.).",
"The particular case in which $X$ is a sphere was also independently asked in several talks by Walter FeitI thank I. Hambleton for this information.",
"and later by Bruno Zimmermann [39].",
"The bound on the number of generators of the abelian group in the statement of Theorem REF is not part of Ghys's conjecture, and follows immediately from a theorem of Mann and Su [22].",
"Ghys's conjecture was inspired by the following classic result of Jordan [20].",
"Theorem 1.2 (Jordan) For any $n\\in {N}$ there exists a constant $C_n$ such that if $G$ is a finite subgroup of $\\operatorname{GL}(n,{R})$ then $G$ has an abelian subgroup $A\\subseteq G$ of index at most $C_n$ .",
"Jordan's theorem has also inspired the following terminology, introduced by Popov [34]: a group $\\Gamma $ is said to be Jordan if any finite subgroup $G\\subseteq \\Gamma $ has an abelian subgroup whose index in $G$ is bounded above by a constant depending only on $\\Gamma $ .",
"Hence, Jordan's theorem can be restated by saying that $\\operatorname{GL}(n,{C})$ is Jordan for every $n$ , and Ghys's conjecture states that the diffeomorphism group of any compact manifold is Jordan.",
"The question of whether automorphism groups of a given geometric structure is Jordan has been asked in other contexts besides smooth manifolds.",
"Serre asked whether high dimensional Cremona groups are Jordan [38], and Popov extended the question to birational groups and automorphism groups of general algebraic varieties (see [36] for a survey).",
"Note that Theorem REF implies in particular that the group of automorphisms of ${C}^n$ as an affine algebraic manifold is Jordan.",
"The question is also interesting in symplectic geometry (see [30] and below).",
"Other partial cases of Ghys's conjecture were previously known to be true: it is an easy exercise to prove it for compact manifolds of dimension at most 2 (see [25] for the case of surfaces); for compact 3-manifolds it was proved by Zimmermann in [40]; for compact 4-manifolds with nonzero Euler characteristic it was proved by the author in [27]; finally, the author proved in [25] Ghys's conjecture for closed manifolds admitting a nonzero top dimensional cohomology class expressible as a product of one dimensional classes (e.g.",
"tori).",
"For open manifolds the analogous statement of Ghys conjecture was known to be true in some cases.",
"For ${R}$ and ${R}^2$ it is an easy exercise; for ${R}^3$ it follows from the work of Meeks and Yau on minimal surfaces, see Theorem 4 in [23]; Guazzi, Mecchia and Zimmermann proved in [15] that the diffeomorphism groups of ${R}^3$ and ${R}^4$ are Jordan using much more elementary methods than those of [23]; finally, Zimmermann proved in [40] that the diffeomorphism groups of ${R}^5$ and ${R}^6$ are Jordan using [31].",
"An example of connected open 4-manifold whose diffeomorphism group is not Jordan was given by Popov in [35].",
"Recently, Csikós, Pyber, and Szabó [8] found a counterexample to Ghys's conjecture, by showing that the diffeomorphism group of $T^2\\times S^2$ is not Jordan (in contrast, the symplectomorphism group of any symplectic form on $T^2\\times S^2$ is Jordan, see [30]).",
"Many other counterexamples to Ghys's conjecture can be obtained using the ideas in [8]: in particular, for any manifold $M$ supporting an effective action of $\\operatorname{SU}(2)$ or $\\operatorname{SO}(3,{R})$ the diffeomorphism group of $T^2\\times M$ is not Jordan (see [29]).",
"In view of this, it is an intriguing question to characterize for which compact smooth manifolds Ghys's conjecture is true."
],
[
"Our second main result concerns existence of fixed points.",
"We say that a compact manifold $X$ has no odd cohomology if its integral cohomology is torsion free and supported in even degrees.",
"This implies that the Euler characteristic of any connected component of $X$ is nonzero.",
"Note that if $X$ is orientable and closed then the assumption that $H^*(X;{Z})$ is supported in even degrees implies, by Poincaré duality and the universal coefficient theorem, that the cohomology is torsion free.",
"Theorem 1.3 Let $X$ be a compact smooth manifold (possibly with boundary) without odd cohomology.",
"There exists a constant $C$ such that any finite abelian group $A$ acting smoothly on $X$ contains a subgroup $A_0$ satisfying $[A:A_0]\\le C$ and $\\chi (Y^{A_0})=\\chi (Y)$ for every connected component $Y\\subseteq X$ .",
"Theorem REF implies that $A_0$ preserves the connected components of $X$ and for any connected component $Y\\subseteq X$ the fixed point set $Y^{A_0}$ is nonempty.",
"We remark that Theorem REF is not a formal consequence of the particular case in which $X$ is connected.",
"To prove Theorem REF we introduce a condition on finite abelian smooth group actions on the manifold $X$ called $\\lambda $ -stability, where $\\lambda $ is an integer.",
"We prove that if $\\lambda $ is big enough (depending on $X$ ) then for any $\\lambda $ -stable smooth action of a finite abelian group $\\Gamma $ on $X$ there exists some $\\gamma \\in \\Gamma $ satisfying $X^{\\Gamma }=X^{\\gamma }$ ; then the Euler characteristic of $X^{\\gamma }$ can be computed using Lefschetz' formula [9].",
"Furthermore, for any $\\lambda $ there exists a constant $C_{\\lambda }$ (depending on $\\lambda $ and $X$ ) such that for any abelian group $\\Gamma $ acting smoothly on $X$ has a subgroup $\\Gamma _0$ whose action on $X$ is $\\lambda $ -stable and $[\\Gamma :\\Gamma _0]\\le C_{\\lambda }$ .",
"Theorem REF can be combined with Theorem REF to yield the following.",
"Corollary 1.4 Let $X$ be a smooth compact manifold (possibly with boundary) without odd cohomology.",
"There exists a constant $C$ such that any finite group $G$ acting smoothly on $X$ has a subgroup $G_0$ satisfying $[G:G_0]\\le C$ and $|X^{G_0}|\\ge \\chi (X)$ .",
"This corollary is nontrivial even for high dimensional disks, since disks of high enough dimension support finite group actions without fixed points.",
"The first example of such actions was found by Floyd and Richardson (see [13] and also [3]), and a complete characterization of finite groups admitting smooth actions on disks without fixed points was given by Oliver (see Theorem 7 of [32]).",
"In particular, Oliver proves that a finite abelian group has a smooth fixed point free action on a disk if and only if it has three or more noncyclic Sylow subgroups.",
"There is a similar story for finite group actions on spheres with a unique fixed point, see [33].",
"Neither the methods of [32] nor those of [33] give a precise control on the dimension of the disks or spheres supporting the finite group action.",
"Some partial results, restricted either to low dimensions or to actions of some concrete finite groups, on the dimensions of disks (resp.",
"spheres) supporting group actions with none (resp.",
"one) fixed points, are proved in [1], [5].",
"Compactness is an essential condition in Theorem REF .",
"Indeed, the analogous statement for smooth finite group actions on ${R}^n$ is false for $n\\ge 7$ : by a theorem of Haynes, Kwasik, Mast and Schultz [17], if $n,r$ are natural numbers, $n\\ge 7$ and $r$ is not a prime power, then there exists a smooth diffeomorphism of ${R}^n$ of order $r$ without fixed points; taking $r=pq$ with $p,q$ different primes, it follows that there is a smooth action of $G={Z}_r$ on ${R}^n$ such that for every $x\\in {R}^n$ the isotropy group $G_x$ is trivial or isomorphic to ${Z}_p$ or ${Z}_q$ ; in particular $[G:G_x]\\ge \\min \\lbrace p,q\\rbrace $ .",
"Since $p,q$ can both be chosen to be arbitrarily big, Theorem REF can not be true for actions on ${R}^n$ .",
"(I thank I. Hambleton for this observation.)"
],
[
"The constants in Theorems ",
"A natural question which we do not address here is to find the optimal values of $C$ in Theorem REF .",
"We also do not estimate the constants that arise from our arguments; doing so would require in particular estimating the constants in [31], which plays a crucial role in the proof of Theorem REF .",
"We believe that neither the proofs in this paper nor the ones in [31] are close to giving sharp estimates.",
"Our method of proof implies that for an acyclic manifold $X$ the constant $C$ in Theorem REF can be bounded in terms of the dimension of $X$ , and the same happens for homology spheres; if $X$ is a compact manifold with nonzero Euler characteristic then $C$ can be bounded in terms of the dimension of $X$ and the Betti numbers.",
"Similar comments apply to Theorem REF , although in this case it is easier to give concrete (albeit probably far from sharp) bounds.",
"Instead of giving a general bound, we state two theorems giving bounds on actions on disks and even dimensional spheres.",
"The proofs use the same ideas as the proof of Theorem REF , but restricting to actions on disks and spheres allows us to get stronger bounds than in the general case.",
"Define a map $f:{Z}\\rightarrow {N}$ as follows.",
"For any nonnegative integer $k$ let $f(k)=2^k\\prod _{p\\ge 3}p^{[k/p]},$ where the product is over the set of odd primes, and set $f(k)=1$ for every negative integer $k$ .",
"Note that if $k$ is nonnegative then $f(k)$ divides $2^{k-[k/2]}k!$ .",
"Theorem 1.5 Let $n$ be a natural number and let $X$ be the $n$ -dimensional disk.",
"Let $k=[(n-3)/2]$ .",
"Any finite abelian group $A$ acting smoothly on $X$ has a subgroup $A^{\\prime }\\subseteq A$ such that $[A:A^{\\prime }]$ divides $f(k)$ , and $\\chi (X^{A^{\\prime }})=1$ .",
"For any finite abelian group $A$ acting smoothly on $X$ such that all prime divisors $p$ of $|A|$ satisfy $p>\\max \\lbrace 2,k\\rbrace $ we have $\\chi (X^{A})=1$ .",
"Theorem 1.6 Let $m$ be a natural number and let $X$ be a smooth $2m$ -dimensional homology sphere.",
"Any finite abelian group $A$ acting smoothly on $X$ has a subgroup $A^{\\prime }\\subseteq A$ such that $[A:A^{\\prime }]$ divides $2^{m+1}f(m-1)$ , and $|X^{A^{\\prime }}|\\ge 2$ .",
"For any finite abelian group $A$ acting smoothly on $X$ such that all prime divisors $p$ of $|A|$ satisfy $p>\\max \\lbrace 2,m-1\\rbrace $ we have $|X^{A}|\\ge 2$ ."
],
[
"Notation and conventions",
"We denote inclusion of sets with the symbol $\\subseteq $ ; the symbol $\\subset $ is reserved for strict inclusion.",
"As usual in the theory of finite transformation groups in this paper ${Z}_n$ denotes ${Z}/n{Z}$ , not to be mistaken, when $n$ is a prime $p$ , with the $p$ -adic integers.",
"If $p$ is a prime we denote by ${F}_p$ the field of $p$ elements.",
"When we say that a group $G$ can be generated by $d$ elements we mean that there are elements $g_1,\\dots ,g_d\\in G$ , not necessarily distinct, which generate $G$ .",
"All manifolds appearing in the text may, unless we say the contrary, be open and have boundary.",
"If a group $G$ acts on a set $X$ we denote the stabiliser of $x\\in X$ by $G_x$ , and for any subset $S\\subset G$ we denote $X^S=\\lbrace x\\in X\\mid S\\subseteq G_x\\rbrace $ ."
],
[
"Remark",
"This paper is the result of combining substantial revisions of [26] and [28].",
"We have left out an entire section of [26] which gives a different proof of Theorem REF for compact manifolds without odd cohomology using Theorem REF .",
"This proof leads to stronger results than the geometric arguments in the present paper, but apart from requiring more restrictive hypothesis it is somewhat involved and it still uses the classification of finite simple groups.",
"(However, in dimension 4 the arguments of [26] allow to prove Ghys's conjecture for manifolds with nonzero Euler characteristic without using the classification of finite simple groups, see [27].)"
],
[
"Contents of the paper",
"Section contains some preliminary results.",
"In Section we recall the main result in [31] and prove a slight strengthening of it.",
"In Section we consider the situation of an action of a finite group on a manifold preserving a submanifold on which the induced action is abelian, and prove a lemma which plays a crucial role in the proof of Theorem REF .",
"In Section we treat the cases of open acyclic manifolds (Theorem REF ) and compact manifolds with nonzero Euler characteristic (Theorem REF ), and in Section we treat the case of homology spheres (Theorem REF ).",
"In Section we introduce the notion of $\\lambda $ -stable action and its basic properties.",
"These are used in Section to prove Theorem REF .",
"Finally, in Section we prove Theorems REF and REF ."
],
[
"Acknowledgement",
"I am very pleased to acknowledge my indebtedness to Alexandre Turull.",
"It's thanks to him that Theorem REF , which in earlier versions of this paper referred only to finite solvable groups, has become a theorem on arbitrary finite groups."
],
[
"Local linearization of smooth finite group actions",
"The following result is well known.",
"We recall it because of its crucial role in some of the arguments of this paper.",
"Statement (1) implies that the fixed point set of a finite group action on a manifold with boundary is a neat submanifold in the sense of §1.4 in [18].",
"Lemma 2.1 Let a finite group $\\Gamma $ act smoothly on a manifold $X$ , and let $x\\in X^{\\Gamma }$ .",
"The tangent space $T_xX$ carries a linear action of $\\Gamma $ , defined as the derivative at $x$ of the action on $X$ , satisfying the following properties.",
"There exist neighborhoods $U\\subset T_xX$ and $V\\subset X$ , of $0\\in T_xX$ and $x\\in X$ respectively, such that: if $x\\notin \\partial X$ then there is a $\\Gamma $ -equivariant diffeomorphism $\\phi \\colon U\\rightarrow V$ ; if $x\\in \\partial X$ then there is $\\Gamma $ -equivariant diffeomorphism $\\phi \\colon U\\cap \\lbrace \\xi \\ge 0\\rbrace \\rightarrow V$ , where $\\xi $ is a nonzero $\\Gamma $ -invariant element of $(T_xX)^*$ such that $\\operatorname{Ker}\\xi =T_x\\partial X$ .",
"If the action of $\\Gamma $ is effective and $X$ is connected then the action of $\\Gamma $ on $T_xX$ is effective, so it induces an inclusion $\\Gamma \\hookrightarrow \\operatorname{GL}(T_xX)$ .",
"If $\\Gamma ^{\\prime }\\vartriangleleft \\Gamma $ and $\\dim _xX^{\\Gamma }<\\dim _xX^{\\Gamma ^{\\prime }}$ then there exists an irreducible $\\Gamma $ -submodule $V\\subset T_xX$ on which the action of $\\Gamma $ is nontrivial but the action of $\\Gamma ^{\\prime }$ is trivial.",
"We first construct a $\\Gamma $ -invariant Riemannian metric $g$ on $X$ with respect to which $\\partial X\\subset X$ is totally geodesic.",
"Take any tangent vector field on a neighborhood of $\\partial X$ whose restriction to $\\partial X$ points inward; averaging over the action of $\\Gamma $ , we get a $\\Gamma $ -invariant vector field which still points inward, and its flow at short time defines an embedding $\\psi :\\partial X\\times [0,\\epsilon )\\rightarrow X$ for some small $\\epsilon >0$ such that $\\psi (x,0)=x$ and $\\psi (\\gamma \\cdot x,t)=\\gamma \\cdot \\psi (x,t)$ for any $x\\in \\partial X$ and $t\\in [0,\\epsilon )$ .",
"Let $h$ be a $\\Gamma $ -invariant Riemannian metric on $\\partial X$ and consider any Riemannian metric on $X$ whose restriction to $\\psi (\\partial X\\times [0,\\epsilon /2])$ is equal to $h+dt^2$ .",
"Averaging this metric over the action of $\\Gamma $ we obtain a metric $g$ with the desired property.",
"The exponential map with respect to $g$ gives the local diffeomorphism in (1).",
"To prove (2), assume that the action of $\\Gamma $ on $X$ is effective.",
"(1) implies that for any subgroup $\\Gamma ^{\\prime }\\subseteq \\Gamma $ the fixed point set $X^{\\Gamma ^{\\prime }}$ is a submanifold of $X$ and that $\\dim _xX=\\dim (T_xX)^{\\Gamma ^{\\prime }}$ for any $x\\in X^{\\Gamma ^{\\prime }}$ ; furthermore, $X^{\\Gamma ^{\\prime }}$ is closed by the continuity of the action.",
"So if some element $\\gamma \\in \\Gamma $ acts trivially on $T_xX$ , then $X^{\\gamma }$ is a closed submanifold of $X$ satisfying $\\dim _xX^{\\gamma }=\\dim X$ .",
"Since $X$ is connected this implies $X^{\\gamma }=X$ , so $\\gamma =1$ , because the action of $\\Gamma $ on $X$ is effective.",
"Finally, (3) follows from (1) ($V$ can be defined as any of the irreducible factors in the $\\Gamma $ -module given by the perpendicular of $T_xX^{\\Gamma }$ in $T_xX^{\\Gamma ^{\\prime }}$ )."
],
[
"Points with big stabilizer for actions of $p$ -groups on compact manifolds",
"Let $G$ be a group and let ${{C}}$ be a simplicial complex endowed with an action of $G$ .",
"We say that this action is good if for any $g\\in G$ and any $\\sigma \\in {{C}}$ such that $g(\\sigma )=\\sigma $ we have $g(\\sigma ^{\\prime })=\\sigma ^{\\prime }$ for any subsimplex $\\sigma ^{\\prime }\\subseteq \\sigma $ (equivalently, the restriction of the action of $g$ to $|\\sigma |\\subset |{{C}}|$ is the identity).",
"This property is called condition (A) in [3].",
"If ${{C}}$ is a simplicial complex and $G$ acts on ${{C}}$ , then the induced action of $G$ on the barycentric subdivision $\\operatorname{sd}{{C}}$ is good (see [3]).",
"Suppose that $G$ acts on a compact manifold $Y$ , possibly with boundary.",
"A $G$ -good triangulation of $Y$ is a pair $({{C}},\\phi )$ , where ${{C}}$ is a finite simplicial complex endowed with a good action of $G$ and $\\phi \\colon Y\\rightarrow |{{C}}|$ is a $G$ -equivariant homeomorphism.",
"For any smooth action of a finite group $G$ on a manifold $X$ there exist $G$ -good triangulations of $X$ (by the previous comments it suffices to prove the existence of a $G$ -equivariant triangulation; this can be easily obtained adapting the construction of triangulations of smooth manifolds given in [6] to the finitely equivariant setting; for much more detailed results, see [19]).",
"Lemma 2.2 Let $Y$ be a compact smooth manifold, possibly with boundary, satisfying $\\chi (Y)\\ne 0$ .",
"Let $p$ be a prime, and let $G$ be a finite $p$ -group acting smoothly on $Y$ .",
"Let $r$ be the biggest nonnegative integer such that $p^r$ divides $\\chi (X)$ .",
"There exists some $y\\in Y$ whose stabilizer $G_y$ satisfies $[G:G_y]\\le p^r$ .",
"Let $({{C}},\\phi )$ be a $\\Gamma $ -good triangulation of $Y$ .",
"The cardinal of each of the orbits of $G$ acting on ${{C}}$ is a power of $p$ .",
"If the cardinal of all orbits were divisible by $p^{r+1}$ , then for each $d$ the cardinal of the set of $d$ -dimensional simplices in ${{C}}$ would be divisible by $p^{r+1}$ , and consequently $\\chi (Y)=\\chi ({{C}})$ would also be divisible by $p^{r+1}$ , contradicting the definition of $r$ .",
"Hence, there must be at least one simplex $\\sigma \\in {{C}}$ whose orbit has at most $p^r$ elements.",
"This means that the stabilizer $G_{\\sigma }$ of $\\sigma $ has index at most $p^r$ .",
"If $y\\in Y$ is a point such that $\\phi (y)\\in |\\sigma |\\subseteq |{{C}}|$ , then $y$ is fixed by $G_{\\sigma }$ , because the triangulation is good."
],
[
"Fixed point loci of actions of abelian $p$ -groups",
"Lemma 2.3 Let $X$ be a manifold, let $p$ be a prime, and let $G$ be a finite $p$ -group acting continuously on $X$ .",
"We have $\\sum _j b_j(X^G;{F}_p)\\le \\sum _j b_j(X;{F}_p).$ If $|G|=p$ then the statement follows from [2].",
"For general $G$ use ascending induction on $|G|$ .",
"In the induction step, choose a central subgroup $G_0\\subset G$ of order $p$ and apply the inductive hypothesis to the action of $G/G_0$ on $X^{G_0}$ ."
],
[
"Testing Jordan's property on $\\lbrace p,q\\rbrace $ -groups",
"Suppose that ${{C}}$ is a set of finite groups.",
"We denote by ${{T}}({{C}})$ the set of all $T \\in {{C}}$ such that there exist primes $p$ and $q$ , a Sylow $p$ -subgroup $P$ of $T$ (which might be trivial), and a normal Sylow $q$ -subgroup $Q$ of $T$ , such that $T = PQ$ .",
"(In particular, if $T\\in {{T}}({{C}})$ then $|T|=p^{\\alpha }q^{\\beta }$ for some primes $p$ and $q$ and nonnegative integers $\\alpha ,\\beta $ .)",
"Let $C$ and $d$ be positive integers.",
"We say that a set of groups ${{C}}$ satisfies (the Jordan property) ${{J}}(C,d)$ if each $G\\in {{C}}$ has an abelian subgroup $A$ such that $[G:A]\\le C$ and $A$ can be generated by $d$ elements.",
"For convenience, we will say that ${{C}}$ satisfies the Jordan property, without specifying any constants, whenever there exist some $C$ and $d$ such that ${{C}}$ satisfies ${{J}}(C,d)$ .",
"The following is the main result in [31]: Theorem 3.1 Let $d$ and $M$ be positive integers.",
"Let ${{C}}$ be a set of finite groups which is closed under taking subgroups and such that ${{T}}({{C}})$ satisfies ${{J}}(M,d)$ .",
"Then there exists a positive integer $C_0$ such that $\\mathcal {C}$ satisfies ${{J}}(C_0,d)$ .",
"We next prove a refinement of this theorem.",
"Given a set of finite groups ${{C}}$ , we define ${{T}_A}({{C}})$ exactly like ${{T}}({{C}})$ but imposing additionally that the Sylow subgroups are abelian (in particular, ${{T}_A}({{C}})\\subset {{T}}({{C}})$ ).",
"In concrete terms, a group $G\\in {{C}}$ belongs to ${{T}_A}({{C}})$ if and only if there exist primes $p$ and $q$ , an abelian Sylow $p$ -subgroup $P\\subseteq G$ and a normal abelian Sylow $q$ -subgroup $Q\\subseteq G$ , such that $G = PQ$ .",
"Denote by ${{P}}({{C}})$ the set of groups $G\\in {{C}}$ with the property that there exists a prime $p$ such that $G$ is a $p$ -group.",
"If $G$ is a (possibly infinite) group we denote by ${{C}}(G)$ the set of finite subgroups of $G$ and we let ${{T}_A}(G):={{T}_A}({{C}}(G))$ .",
"Corollary 3.2 Let $d$ and $M$ be positive integers.",
"Let ${{C}}$ be a set of finite groups which is closed under taking subgroups and such that ${{P}}({{C}})\\cup {{T}_A}({{C}})$ satisfies ${{J}}(M,d)$ .",
"Then there exists a positive integer $C_0$ such that $\\mathcal {C}$ satisfies ${{J}}(C_0,d)$ .",
"Let $d$ and $M$ be positive integers, suppose that ${{C}}$ is a set of finite groups which closed under taking subgroups, and assume that ${{P}}({{C}})\\cup {{T}_A}({{C}})$ satisfies ${{J}}(M,d)$ .",
"Let $C:=M^2(M!",
")^d$ .",
"We claim that ${{T}}({{C}})$ satisfies ${{J}}(C,d)$ .",
"This immediately implies our result, in view of Theorem REF .",
"Since ${{P}}({{C}})$ satisfies ${{J}}(M,d)$ , to justify the claim it suffices to prove the following fact.",
"Let $G$ be a finite group, let $p,q$ be distinct prime numbers, let $P\\subseteq G$ be a $p$ -Sylow subgroup, let $Q\\subseteq G$ be a normal $q$ -Sylow subgroup, and assume that $G=PQ$ ; if there exist abelian subgroups $P_0\\subseteq P$ and $Q_0\\subseteq Q$ such that $[P:P_0]\\le M$ , $[Q:Q_0]\\le M$ , and $Q_0$ can be generated by $d$ elements, then there exists some $G^{\\prime }\\in {{T}_A}(G)$ such that $[G:G^{\\prime }]\\le C$ .",
"To prove this, define $Q^{\\prime }:=\\bigcap _{\\phi \\in \\operatorname{Aut}(Q)}\\phi (Q_0)$ , where $\\operatorname{Aut}(Q)$ denotes the group of automorphisms of $Q$ .",
"Clearly $Q^{\\prime }$ is an abelian characteristic subgroup of $Q$ , so it is normal in $G$ (because $Q$ is normal in $G$ ).",
"Define $G^{\\prime }:=P_0Q^{\\prime }$ .",
"Then $G^{\\prime }\\in {{T}_A}(G)$ , so we only need to prove that $[G:G^{\\prime }]\\le C$ .",
"Suppose that $\\lbrace g_1,\\dots ,g_\\delta \\rbrace $ is a generating set of $Q_0$ such that $\\delta \\le d$ and $Q_0\\simeq \\prod _j\\langle g_j\\rangle $ , where $\\langle g_j\\rangle \\subseteq Q_0$ is the subgroup generated by $g_j$ (such generating set exists because $Q_0$ is abelian and can be generated by $d$ elements).",
"If $\\Gamma \\subseteq Q_0$ is any subgroup of index at most $M$ , then $g_j^{M!",
"}$ belongs to $\\Gamma $ for each $j$ .",
"Consequently, the subgroup $Q^{\\prime \\prime }\\subseteq Q_0$ generated by $\\lbrace g_1^{M!",
"},\\dots ,g_{\\delta }^{M!",
"}\\rbrace $ is contained in any subgroup $\\Gamma \\subseteq Q_0$ of index at most $M$ .",
"In particular $Q^{\\prime \\prime }\\subseteq Q^{\\prime }$ , because for any $\\phi \\in \\operatorname{Aut}(Q)$ we have $[Q_0:Q_0\\cap \\phi (Q_0)]\\le M$ .",
"On the other hand, $[Q_0:Q^{\\prime \\prime }]\\le (M!",
")^{\\delta }\\le (M!",
")^d$ , so a fortiori $[Q_0:Q^{\\prime }]\\le (M!",
")^d$ .",
"Since $[G:G^{\\prime }]=[P:P_0][Q:Q^{\\prime }]=[P:P_0][Q:Q_0][Q_0:Q^{\\prime }]$ , the result follows."
],
[
"Actions of finite groups on real vector bundles",
"Define, for any smooth manifold $Y$ , ${{T}_A}(Y):={{T}_A}(\\operatorname{Diff}(Y)).$ Suppose that $E\\rightarrow Y$ be a smooth real vector bundle.",
"Denote by $\\operatorname{Diff}(E\\rightarrow Y)$ the group of smooth bundle automorphisms lifting arbitrary diffeomorphisms of the base $Y$ (equivalently, diffeomorphisms of $E$ which map fibers to fibers and whose restriction to each fiber is a linear map).",
"Denote by $\\pi :\\operatorname{Diff}(E\\rightarrow Y)\\rightarrow \\operatorname{Diff}(Y)$ the map which assigns to each $\\phi \\in \\operatorname{Diff}(E\\rightarrow Y)$ the diffeomorphism of $\\psi \\in \\operatorname{Diff}(Y)$ such that $\\phi (E_y)=\\phi (E_{\\psi (y)})$ for every $y$ , where $E_y$ is the fiber of $E$ over $y$ .",
"Define also ${{T}_A}(E\\rightarrow Y):={{T}_A}(\\operatorname{Diff}(E\\rightarrow Y)).$ Since the properties defining the groups in ${{T}_A}({{C}})$ in Section are preserved by passing to quotients, for any $G\\in {{T}_A}(E\\rightarrow Y$ ) we have $\\pi (G)\\in {{T}_A}(Y)$ .",
"Lemma 4.1 Assume that $Y$ is connected and let $r$ be the rank of $E$ .",
"Suppose that $G\\in {{T}_A}(E\\rightarrow Y)$ and that $\\pi (G)\\in {{T}_A}(Y)$ is abelian.",
"Then $G$ has an abelian subgroup $A\\subseteq G$ satisfying $[G:A]\\le r!$ .",
"Take a group $G\\in {{T}_A}(E\\rightarrow Y)$ such that $\\pi (G)$ is abelian.",
"There exist two distinct primes $p$ and $q$ , a $p$ -Sylow subgroup $P\\subseteq G$ , and a normal $q$ -Sylow subgroup $Q\\subset G$ , such that $G=PQ$ .",
"Furthermore, both $P$ and $Q$ are abelian.",
"Let $Q_0=Q\\cap \\operatorname{Ker}\\pi \\subseteq Q$ .",
"Then $Q_0$ is normal in $G$ , since $Q_0=Q\\cap (G\\cap \\operatorname{Ker}\\pi )$ and both $Q$ and $G\\cap \\operatorname{Ker}\\pi $ are normal in $G$ .",
"So the action of $P$ on $G$ given by conjugation preserves both $Q$ and $Q_0$ .",
"Furthermore, since $\\pi (G)$ is abelian we have, for any $\\gamma \\in P$ and $\\eta \\in Q$ , $\\pi (\\gamma \\eta \\gamma ^{-1}\\eta ^{-1})=\\pi (\\gamma )\\pi (\\eta )\\pi (\\gamma )^{-1}\\pi (\\eta )^{-1}=1$ , which is equivalent to $\\gamma \\eta \\gamma ^{-1}\\eta ^{-1}\\in Q_0$ .",
"The complexified vector bundle $E\\otimes {C}$ splits as a direct sum of subbundles indexed by the characters of $Q_0$ , $E\\otimes {C}=\\bigoplus _{\\rho \\in \\operatorname{Mor}(Q_0,{C}^*)}E_{\\rho },$ where $v\\in E_{\\rho }$ if and only if $\\eta \\cdot v=\\rho (\\eta )v$ for any $\\eta \\in Q_0$ .",
"The action of $P$ on $E\\otimes {C}$ permutes the summands $\\lbrace E_{\\rho }\\rbrace $ .",
"In concrete terms, if we define for any $\\rho \\in \\operatorname{Mor}(Q_0,{C}^*)$ and $\\gamma \\in P$ the character $\\rho _{\\gamma }\\in \\operatorname{Mor}(Q_0,{C}^*)$ by $\\rho _{\\gamma }(\\eta )=\\rho (\\gamma ^{-1}\\eta \\gamma )$ for any $\\eta \\in Q_0$ , then we have $\\gamma \\cdot E_{\\rho }=E_{\\rho _{\\gamma }}$ .",
"Since there are at most $r$ nonzero summands in (REF ) (because $Y$ is connected and the rank of $E$ is $r$ ), the subgroup $P^{\\prime }\\subseteq P$ consisting of those elements which preserve each nonzero subbundle $E_{\\rho }$ satisfies $[P:P^{\\prime }]\\le r!$ .",
"Furthermore, each element of $P^{\\prime }$ commutes with all the elements in $Q_0$ , because $P^{\\prime }$ acts linearly on $E\\otimes {C}$ preserving the summands in (REF ) and the action of $Q_0$ on each summand is given by homothecies (in particular, the action of each element of $Q_0$ lifts the identity on $Y$ ).",
"Hence, the action of $P^{\\prime }$ on $Q$ given by conjugation gives a morphism $P^{\\prime }\\rightarrow B:=\\lbrace \\phi \\in \\operatorname{Aut}(Q)\\mid \\phi (\\eta )=\\eta \\text{ for each $\\eta \\in Q_0$},\\quad \\phi (\\eta )\\eta ^{-1}\\in Q_0\\text{ for each $\\eta \\in Q$}\\rbrace .$ We now prove that $B$ is $q$ -group.",
"Let $\\phi \\in B$ be any element, and define a map $f:Q\\rightarrow Q_0$ by the condition that $f(\\eta )=\\phi (\\eta )\\eta ^{-1}$ for every $\\eta $ .",
"Since $Q$ is abelian and $\\phi $ is a morphism of groups, $f$ is also a morphism of groups.",
"Furthermore, $f(f(\\eta ))=1$ for every $\\eta \\in Q$ , because $Q_0\\subseteq \\operatorname{Ker}f$ .",
"Using induction it follows that $\\phi ^k(\\eta )=f(\\eta )^k\\eta $ for every $k\\in {N}$ .",
"Hence the order of $\\phi \\in B$ divides $|Q_0|$ , which is a power of $q$ .",
"Since $P^{\\prime }$ is a $p$ -group and $p\\ne q$ , any morphism $P^{\\prime }\\rightarrow B$ is trivial.",
"This implies that $P^{\\prime }$ commutes with $Q$ .",
"Setting $A:=P^{\\prime }Q$ , the result follows.",
"Lemma 4.2 Suppose that $X$ is a smooth connected manifold and that $G\\in {{T}_A}(X)$ .",
"Assume that there is a $G$ -invariant connected submanifold $Y\\subseteq X$ .",
"Let $G_Y\\subseteq \\operatorname{Diff}(Y)$ be the group consisting of all diffeomorphisms of $Y$ which are induced by restricting to $Y$ the action of the elements of $G$ .",
"Let $r:=\\dim X-\\dim Y$ .",
"If $G_Y$ is abelian, then there is an abelian subgroup $A\\subseteq G$ satisfying $[G:A]\\le r!$ .",
"There is an inclusion of vector bundles $TY\\hookrightarrow TX|_Y$ .",
"Consider the quotient bundle $E:=TX|_Y/TY\\rightarrow Y$ , which is the normal bundle of the inclusion $Y\\hookrightarrow X$ .",
"Let $\\operatorname{Diff}(X,Y)$ be the group of diffeomorphisms of $X$ which preserve $Y$ .",
"There is a natural restriction map $\\rho :\\operatorname{Diff}(X,Y)\\rightarrow \\operatorname{Diff}(E\\rightarrow Y)$ given by restricting the diffeomorphisms in $\\operatorname{Diff}(X,Y)$ to the first jet of the inclusion $Y\\hookrightarrow X$ , which gives a bundle automorphism $TX|_Y\\rightarrow TX|_Y$ preserving $TY$ , and then projecting to an automorphism of $E$ .",
"Furthermore, if $\\Gamma \\subseteq \\operatorname{Diff}(X,Y)$ is a finite group then by (2) in Lemma REF the restriction $\\rho |_\\Gamma :\\Gamma \\rightarrow \\rho (\\Gamma )$ is injective.",
"Applying this to the group $G\\in {{T}_A}(X)$ in the statement of the lemma we obtain a group $G_E:=\\rho (G)\\in {{T}_A}(E\\rightarrow Y)$ which is isomorphic to $G$ .",
"Furthermore, if $\\pi :\\operatorname{Diff}(E\\rightarrow Y)\\rightarrow \\operatorname{Diff}(Y)$ is the map (REF ), then $\\pi (G_E)\\in {{T}_A}(Y)$ coincides with $G_Y$ , which by hypothesis is abelian.",
"We are thus in the setting of Lemma REF , so we deduce that $G_E$ (and hence $G$ ) has an abelian subgroup of index at most $r!$ ."
],
[
"Actions of finite groups on acyclic manifolds and on compact manifolds\nwith $\\chi \\ne 0$",
"Theorem 5.1 For any $n$ there is a constant $C$ such that any finite group acting smoothly and effectively on an acyclic smooth $n$ -dimensional manifold $X$ has an abelian subgroup of index at most $C$ .",
"Fix $n$ and let $X$ be an acyclic smooth $n$ -dimensional manifold.",
"Let ${{C}}$ be the set of all finite subgroups of $\\operatorname{Diff}(X)$ .",
"By Corollary REF it suffices to prove that ${{P}}({{C}})\\cup {{T}_A}({{C}})$ satisfies the Jordan property.",
"We prove it first for ${{P}}({{C}})$ and then for ${{T}_A}({{C}})$ .",
"Let $G\\in {{P}}({{C}})$ be a finite $p$ -group, where $p$ is a prime.",
"By Smith theory the fixed point set $X^G$ is ${F}_p$ -acyclic, hence nonempty (see [2] for the case $G={Z}_p$ and use induction on $|G|$ for the general case, as in the proof of Lemma REF ).",
"Let $x\\in X^G$ .",
"By (2) in Lemma REF , linearizing the action of $G$ at $x$ we get an injective morphism $G\\hookrightarrow \\operatorname{GL}(T_xX)\\simeq \\operatorname{GL}(n,{R})$ .",
"It follows from Jordan's Theorem REF that there is an abelian subgroup $A\\subseteq G$ such that $[G:A]\\le C_n$ , where $C_n$ depends only on $n$ .",
"Furthermore, since $A$ can be identified with a subgroup of $\\operatorname{GL}(n,{R})$ , it can be generated by at most $n$ elements.",
"We have thus proved that ${{P}}({{C}})$ satisfies ${{J}}(C_n,n)$ .",
"The same argument also proves that any elementary $p$ -group acting effectively on $X$ has rank at most $n$ .",
"Now let $G\\in {{T}_A}({{C}})$ .",
"By definition, there are two distinct primes $p$ and $q$ , an abelian $p$ -Sylow subgroup $P\\subseteq G$ , and a normal abelian $q$ -Sylow subgroup $Q\\subseteq G$ , such that $G=PQ$ .",
"Let $Y:=X^Q$ .",
"By Smith theory, $Y$ is a ${F}_q$ -acyclic manifold (combine again [2] with induction on $|Q|$ as before); in particular, $Y$ is nonempty and connected.",
"Since $Q$ is normal in $G$ , the action of $G$ on $X$ preserves $Y$ .",
"Finally, since the elements of $Q$ act trivially on $Y$ , the action of $G$ on $Y$ given by restriction defines an abelian subgroup of $\\operatorname{Diff}(Y)$ .",
"This means that we are in the setting of Lemma REF , and we can deduce that $G$ has an abelian subgroup $A\\subseteq G$ of index at most $n!$ .",
"Since, as explained in the previous paragraph, any elementary $p$ -group acting on $X$ has rank at most $n$ , $A$ can be generated by at most $n$ elements.",
"We have thus proved that ${{T}_A}({{C}})$ satisfies ${{J}}(n!,n)$ , and the proof of the theorem is now complete.",
"Theorem 5.2 Let $X$ be a compact connected smooth manifold, possibly with boundary, and satisfying $\\chi (X)\\ne 0$ .",
"There exists a constant $C$ such that any finite group acting smoothly and effectively on $X$ has an abelian subgroup of index at most $C$ .",
"Let ${{C}}$ be the set of all finite subgroups of $\\operatorname{Diff}(X)$ .",
"We will again deduce the theorem from Corollary REF , so we only need to prove that ${{P}}({{C}})\\cup {{T}_A}({{C}})$ satisfies the Jordan property.",
"To prove that ${{P}}({{C}})$ satisfies the Jordan property, let $s:=q^e$ be the biggest prime power dividing $\\chi (X)$ .",
"Let $p$ be any prime, and consider a $p$ -group $G\\in {{P}}(G)$ .",
"By Lemma REF , there is a point $x\\in X$ whose stabiliser $G_x$ satisfies $[G:G_x]\\le p^r$ , where $p^r$ divides $\\chi (X)$ .",
"In particular, $[G:G_x]\\le s$ .",
"By Lemma REF there is an inclusion $G_x\\hookrightarrow \\operatorname{GL}(T_xX)\\simeq \\operatorname{GL}(n,{R})$ , where $n=\\dim X$ .",
"By the same argument as in the proof of Theorem REF , there is an abelian subgroup $A\\subseteq G_x$ of index $[G_x:A]\\le C_n$ (with $C_n$ depending only on $n$ ) and which can be generated by $n$ elements.",
"Hence, ${{P}}({{C}})$ satisfies ${{J}}(sC_n,n)$ .",
"We now prove that ${{T}_A}({{C}})$ satisfies the Jordan property.",
"Let $G\\in {{T}_A}({{C}})$ .",
"There exist two distinct primes $p$ and $q$ , an abelian $p$ -Sylow subgroup $P\\subseteq G$ , and a normal abelian $q$ -Sylow subgroup $Q\\subseteq G$ , such that $G=PQ$ .",
"By the arguments used before (involving Lemma REF ) there is a point $x\\in X$ whose stabiliser $Q_x$ satisfies $[Q:Q_x]\\le s$ for some positive integer $s$ depending only on $\\chi (X)$ .",
"Since $Q_x$ is abelian and we have an inclusion $Q_x\\hookrightarrow \\operatorname{GL}(T_xX)$ , we know that $Q_x$ can be generated by $n$ elements.",
"Define $Q^{\\prime }:=\\bigcap _{\\phi \\in \\operatorname{Aut}(Q)}\\phi (Q_x).$ By the arguments at the end of the proof of Corollary REF , we have $[Q_x:Q^{\\prime }]\\le (s!",
")^n$ .",
"Since $Q^{\\prime }\\subset Q_x$ , we have $x\\in X^{Q^{\\prime }}$ , so $X^{Q^{\\prime }}$ is nonempty.",
"Also, $Q^{\\prime }$ does not contain any elementary $q$ -group of rank greater than $n$ (because it is a subgroup of $G_x$ ), so it can be generated by $n$ elements.",
"By Lemma REF we have $\\sum _j b_j(X^{Q^{\\prime }};{F}_q)\\le \\sum _j b_j(X;{F}_q)\\le K:=\\sum _j\\max \\lbrace b_j(X;{F}_p)\\mid p\\text{ prime}\\rbrace ,$ where $K$ is finite because $X$ is compact.",
"In particular, $X^{Q^{\\prime }}$ has at most $K$ connected components.",
"On the other hand, $Q^{\\prime }$ is a characteristic subgroup of $Q$ , and since $Q$ is normal in $G$ it follows that $Q^{\\prime }$ is also normal in $G$ .",
"Hence the action of $P$ on $X$ preserves $X^{Q^{\\prime }}$ .",
"Since the latter has at most $K$ connected components, there exists a subgroup $P_0\\subseteq P$ of index $[P:P_0]\\le K$ and a connected component $Y\\subseteq X^{Q^{\\prime }}$ such that $P_0$ preserves $Y$ .",
"By Lemma REF there is also a subgroup $P^{\\prime }\\subseteq P_0$ which fixes some point in $X$ and such that $[P_0:P^{\\prime }]\\le s$ , which implies as before that $P^{\\prime }$ can be generated by at most $n$ elements.",
"Let $G^{\\prime }:=P^{\\prime }Q^{\\prime }$ .",
"We can bound $[G:G^{\\prime }]=[P:P_0][P_0:P^{\\prime }][Q:Q_x][Q_x:Q^{\\prime }]\\le sKs(s!",
")^n.$ On the other hand, $G^{\\prime }$ preserves $Y$ , and its induced action on $Y$ is abelian (because $Q^{\\prime }$ acts trivially on $Y$ and $P^{\\prime }$ is abelian).",
"By Lemma REF , there exists an abelian subgroup $G^{\\prime \\prime }\\subseteq G^{\\prime }$ satisfying $[G^{\\prime }:G^{\\prime \\prime }]\\le n!$ .",
"It follows that $[G:G^{\\prime \\prime }]=[G:G^{\\prime }][G^{\\prime }:G^{\\prime \\prime }]\\le M:=sKs(s!",
")^nn!.$ Since both $P^{\\prime }$ and $Q^{\\prime }$ can be generated by at most $n$ elements, $G^{\\prime }$ does not contain any elementary $p$ -group or $q$ -group of rank greater than $n$ , which implies that $G^{\\prime \\prime }$ can be generated by $n$ elements.",
"We have thus proved that ${{T}_A}({{C}})$ satisfies ${{J}}(M,n)$ , so the proof of the theorem is complete."
],
[
"Actions on homology spheres",
"Recall some standard terminology: given a ring $R$ and an integer $n\\ge 0$ , an $R$ -homology $n$ -sphere is a topological $n$ -manifoldThis definition of homology sphere is more restrictive than that in [2] or [21], where homology spheres are not required to be topological manifolds.",
"$M$ satisfying $H_*(M;R)\\simeq H_*(S^n;R)$ .",
"A homology $n$ -sphere is a ${Z}$ -homology $n$ -sphere.",
"By the universal coefficient theorem any homology $n$ -sphere is a ${Z}_p$ -homology $n$ -sphere for any prime $p$ .",
"Standard properties of topological manifolds imply that for any prime $p$ and any integer $n$ any ${Z}_p$ -homology $n$ -sphere is compact and orientable.",
"Theorem 6.1 For any $n$ there is a constant $C$ such that any finite group acting smoothly and effectively on a smooth homology $n$ -sphere has an abelian subgroup of index at most $C$ .",
"Before proving the theorem we collect a few facts which will be used in our arguments.",
"In what follows $p$ denotes an arbitrary prime.",
"Let $p$ be any prime, and let $G$ be a finite $p$ -group acting on a ${Z}_p$ -homology $n$ -sphere $S$ .",
"Then the fixed point set $S^{G}$ is a ${Z}_p$ -homology $n(G)$ -sphere for some integer $-1\\le n(G)\\le n$ , where a $(-1)$ -sphere is by convention the empty set (see [2] for the case $G={Z}_p$ ; the general case follows by induction, see [2]).",
"Furthermore, if $S$ is smooth and the action of $G$ is smooth and nontrivial, then $S^{G}$ is a smooth proper submanifold of $S$ and $n(G)<n$ .",
"Suppose that $G\\simeq ({Z}_p)^r$ is an elementary abelian $p$ -group acting on a ${Z}_p$ -homology $n$ -sphere $S$ .",
"The following formula ([2]) was proved by Borel: $n-n(G)=\\sum _{H\\subseteq G \\text{ subgroup}\\atop [G:H]=p}(n(H)-n(G)).$ Now suppose that $G$ is a finite $p$ -group acting on a smooth ${Z}_p$ -homology $n$ -sphere $S$ .",
"Dotzel and Hamrick proved in [10] that there exists a real representation of $\\rho :G\\rightarrow \\operatorname{GL}(V)$ such that $\\dim _{{R}}V^H=n(H)+1$ for each subgroup $H\\subseteq G$ .",
"This implies that $\\dim _{{R}}V=n+1$ (take $H=\\lbrace 1\\rbrace $ ) and that, if the action is effective, $\\rho $ is injective (because for any nontrivial $H\\subseteq G$ we have $n(H)<n$ ).",
"Consequently, if $G$ acts effectively on $S$ then we can identify $G$ with a subgroup of $\\operatorname{GL}(n+1,{R})$ (which, when $S=S^n$ , does not mean that the original action of $G$ on $S^n$ is necessarily linear!).",
"In particular, if $G\\simeq ({Z}_p)^r$ acts effectively on $S$ , then $r\\le n+1$ .",
"Alternatively, $r$ can be bounded using a general theorem of Mann and Su [22].",
"Let $p$ be now an odd prime, let $d$ be a positive integer, and consider a morphism $\\psi :{Z}_d\\rightarrow \\operatorname{Aut}({Z}_p)$ .",
"Suppose that the image under $\\psi $ of a generator $g\\in {Z}_d$ is multiplication by some $y\\in {Z}_p^*$ (here we use additive notation on ${Z}_p$ ).",
"The following is a very slight modification of a result of Guazzi and Zimmermann [16]: Lemma 6.2 If ${Z}_p {\\rtimes _{\\psi }}{Z}_d$ acts effectively on a smooth ${Z}_p$ -homology $n$ -sphere $S$ and the restriction of the action to ${Z}_p$ is free then $y^{n+1}=1\\in {Z}_p^*$ The original result of Guazzi and Zimmermann does not require the restriction of the action to ${Z}_p$ to be free, but it requires the action of ${Z}_p {\\rtimes _{\\psi }}{Z}_d$ to be orientation preserving.",
"The proof we give of Lemma REF is essentially the same as [16]; we give it to justify that the result is valid without assuming that ${Z}_p {\\rtimes _{\\psi }}{Z}_d$ acts orientation-preservingly.",
"Since $S$ is smooth and compact, $p$ is odd, and the action of ${Z}_p$ is free, $n$ must be odd, say $n=2\\nu +1$ .",
"Let $\\pi :S_{{Z}_p}\\rightarrow B{Z}_p$ be the Borel construction and let $\\zeta :S/{Z}_p\\rightarrow B{Z}_p$ be the composition of a homotopy equivalence $S/{Z}_p\\simeq S_{{Z}_p}$ (which exists because ${Z}_p$ acts freely on $S$ ) with $\\pi $ .",
"A simple argument using the Serre spectral sequence for $\\zeta $ proves that the map $H^k(\\zeta ):H^k(B{Z}_p;{Z}_p)\\rightarrow H^k(S/{Z}_p;{Z}_p)$ is an isomorphism for $0\\le k\\le n$ (note that, since $p$ is odd, the action of ${Z}_p$ on $S$ is orientation preserving, so the second page of the Serre spectral sequence for $\\zeta $ has entries $H^u(B{Z}_p;{Z}_p)\\otimes H^v(S;{Z}_p)$ ).",
"We have $H^*(B{Z}_p;{Z}_p)\\simeq \\Lambda (\\alpha )\\otimes {Z}_p[\\beta ]$ where $\\deg \\alpha =1$ , $\\deg \\beta =2$ , and $\\beta =\\text{b}(\\alpha )$ , where $\\text{b}$ denotes the Bockstein.",
"The action of ${Z}_d$ on ${Z}_p$ induces an action $\\phi ^*:{Z}_d\\rightarrow \\operatorname{Aut}(H^*(B{Z}_p;{Z}_p))$ satisfying $\\phi ^*(g)(\\alpha )=y\\alpha $ , and by naturality and linearity of the Bockstein we also have $\\phi ^*(g)(\\beta )=y\\beta $ .",
"This implies that the action of $g$ on $H^n(B{Z}_p;{Z}_p)={Z}_p\\langle \\alpha \\beta ^{\\nu }\\rangle $ is multiplication by $y^{1+\\nu }$ .",
"Since the map $\\zeta $ is ${Z}_d$ -equivariant, the action of $g$ on $H^n(S/{Z}_p;{Z}_p)$ is also multiplication by $y^{1+\\nu }$ .",
"This is the reduction mod $p$ of the action of $g$ on $H^n(S/{Z}_p;{Z})\\simeq {Z}$ (the isomorphism follows from the fact that $S$ is compact and orientable, and that the action of ${Z}_p$ is orientation preserving).",
"Since the action of $g$ is by a diffeomorphism, it follows that $y^{1+\\nu }$ is the reduction mod $p$ of $\\pm 1$ , so $y^{2(1+\\nu )}=y^{n+1}=1\\in {Z}_p^*$ ."
],
[
"Proof of Theorem ",
"Fix some $n\\ge 1$ , let $S$ be a smooth homology $n$ -sphere, and let ${{C}}$ be the set of finite subgroups of $\\operatorname{Diff}(S)$ .",
"We are going to use Corollary REF , so we need to prove that ${{P}}({{C}})\\cup {{T}_A}({{C}})$ satisfies the Jordan property.",
"As in the proofs of the other two theorems of this paper, we treat separately ${{P}}({{C}})$ and ${{T}_A}({{C}})$ .",
"If $G\\in {{P}}({{C}})$ then by the theorem of Dotzel and Hamrick [10] we may identify $G$ with a subgroup of $\\operatorname{GL}(n+1,{R})$ .",
"By Theorem REF there is an abelian subgroup $A\\subseteq G$ of index at most $C_{n+1}$ .",
"Furthermore, $A$ can be generated by $n+1$ elements.",
"Consequently, ${{P}}({{C}})$ satisfies ${{J}}(C_{n+1},n+1)$ .",
"The fact that ${{T}_A}({{C}})$ satisfies the Jordan property is a consequence of the following lemma, combined with the existence of a uniform upper bound, for any prime $p$ , on the rank of elementary $p$ -groups acting effectively on $S^n$ (such bound follows, as we said, either from the theorem of Dotzel and Hamrick [10] or from that of Mann and Su [22]).",
"Lemma 6.3 Given two integers $m\\ge 0,r\\ge 1$ there exists an integer $K_{m,r}\\ge 1$ such that for any two distinct primes $p$ and $q$ , any abelian $p$ -group $P$ of rank at most $r$ , any abelian $q$ -group $Q$ , any morphism $\\phi :P\\rightarrow \\operatorname{Aut}(Q)$ , any smooth ${Z}_q$ -homology $m$ -sphere $S$ , and any smooth and effective action of $G:=Q\\rtimes _{\\phi } P$ on $S$ , there is an abelian subgroup $A\\subseteq G$ satisfying $[G:A]\\le K_{m,r}$ .",
"Fix some integer $r\\ge 1$ .",
"We prove the lemma, for this fixed value of $r$ , using induction on $m$ .",
"The case $m=0$ being obvious, we may suppose that $m>0$ and assume that Lemma REF is true for smaller values of $m$ .",
"Let $p,q,P,Q,\\phi ,G,S$ be as in the statement of the lemma, and take a smooth effective action of $G$ on $S$ .",
"Suppose first that $S^Q\\ne \\emptyset $ .",
"Then $S^Q$ is a smooth ${Z}_q$ -homology sphere of smaller dimension than $S$ .",
"Furthermore, since $Q$ is normal in $G$ , the action of $G$ on $S$ preserves $S^Q$ .",
"Let $G_0\\subseteq \\operatorname{Diff}(S^Q)$ be the diffeomorphisms of $S^Q$ induced by restricting the action of the elements of $G$ on $S$ to $S^Q$ .",
"Then $G_0$ is a quotient of $G$ , and this implies that $G_0\\simeq Q_0\\rtimes P_0$ , where $P_0$ (resp.",
"$Q_0$ ) is a quotient of $P$ (resp.",
"$Q$ ).",
"Hence we may apply the inductive hypothesis to the action of $G_0$ on $S^Q$ and deduce that there exists an abelian subgroup $A_0\\subseteq G_0$ satisfying $[G_0:A_0]\\le K_{m-1,r}$ .",
"Let $\\pi :G\\rightarrow G_0$ be the quotient map (i.e., the restriction of the action to $S^Q$ ), and let $G^{\\prime }:=\\pi ^{-1}(A_0)\\subseteq G$ .",
"Then $[G:G^{\\prime }]\\le K_{m-1,r}$ and the action of $G^{\\prime }$ satisfies the hypothesis of Lemma REF with $X=S$ and $Y=S^Q$ .",
"Hence, there is an abelian subgroup $A\\subseteq G^{\\prime }$ satisfying $[G^{\\prime }:A]\\le m!$ .",
"By the previous estimates $A$ is an abelian subgroup of $G$ of index $[G:A]\\le m!K_{m-1,r}$ .",
"Now assume that $S^Q=\\emptyset $ .",
"Let $Q^{\\prime }:=\\lbrace \\eta \\in Q\\mid \\eta ^q=1\\rbrace \\subseteq Q$ .",
"Then $Q^{\\prime }\\simeq ({Z}_q)^l$ .",
"Since $Q^{\\prime }$ is a characteristic subgroup of $Q$ and $Q$ is normal in $G$ , $Q^{\\prime }$ is normal in $G$ .",
"We distinguish two cases.",
"Suppose first that $l\\ge 2$ .",
"The Borel formula (REF ) applied to the action of $Q^{\\prime }$ on $S$ gives $m+1=\\sum _{H\\subseteq Q^{\\prime } \\text{ subgroup}\\atop [Q^{\\prime }:H]=p}(n(H)+1).$ All summands on the RHS are nonnegative integers.",
"So at least one summand is strictly positive, and there are at most $m+1$ strictly positive summands.",
"Hence, the set ${{H}}:=\\lbrace H\\text{ subgroup of }Q^{\\prime }\\mid S^H\\ne \\emptyset ,\\,[Q^{\\prime }:H]=q\\rbrace $ is nonempty and has at most $m+1$ elements.",
"The action of $G$ on $Q^{\\prime }$ by conjugation permutes the elements of ${{H}}$ , so there is a subgroup $G^{\\prime }\\subset G$ fixing some element $H\\in {{H}}$ and such that $[G:G^{\\prime }]\\le m+1$ .",
"The fact that $G^{\\prime }$ fixes $H$ as an element of ${{H}}$ means that $H$ is a normal subgroup of $G^{\\prime }$ , so the action of $G^{\\prime }$ on $S$ preserves $S^H\\ne \\emptyset $ .",
"We now proceed along similar lines to the previous case.",
"Let $G^{\\prime \\prime }\\subseteq \\operatorname{Diff}(S^H)$ be the diffeomorphisms of $S^H$ induced by restricting the action of the elements of $G^{\\prime }$ on $S$ to $S^H$ .",
"Then $G^{\\prime \\prime }$ is a quotient of $G^{\\prime }$ and $S^H$ is a smooth ${Z}_q$ -homology sphere of dimension strictly smaller than $m$ ; we may thus apply the inductive hypothesis and deduce the existence of an abelian subgroup $A^{\\prime }\\subseteq G^{\\prime \\prime }$ of index at most $K_{m-1,r}$ .",
"Letting $G_a\\subseteq G^{\\prime }$ be the preimage of $A^{\\prime }$ under the projection map $G^{\\prime }\\rightarrow G^{\\prime \\prime }$ we apply Lemma REF to the action of $G_a$ near the submanifold $S^H\\subseteq S$ and conclude that $G_a$ has an abelian subgroup $A$ of index at most $m!$ .",
"Then $[G:A]\\le (m+1)m!K_{m-1,r}$ .",
"Finally, suppose that $l=1$ .",
"In this case $Q^{\\prime }$ acts freely on $S$ and $Q\\simeq ({Z}_{q^s})$ .",
"If $q=2$ then $|\\operatorname{Aut}({Z}_{q^s})|=(q-1)q^{s-1}$ is equal to $2^{s-1}$ and since $p\\ne 2$ the morphism $\\phi :P\\rightarrow \\operatorname{Aut}(Q)$ is trivial, which means that $G$ is abelian and there is nothing to prove.",
"Assume then that $q$ is odd.",
"Taking an isomorphism $P\\simeq {Z}_{p^{e_1}}\\times \\dots \\times {Z}_{p^{e_c}}$ (with $c\\le r$ , by our assumption on the $p$ -rank of $P$ ) we may apply Lemma REF to the restriction of $\\phi $ to each summand, $\\phi |_{{Z}_{p^{e_i}}}:{Z}_{p^{e_i}}\\rightarrow \\operatorname{Aut}({Z}_q)$ and conclude that there is a subgroup $\\Gamma _i\\subset {Z}_{p^{e_i}}$ of index at most $m+1$ such that $\\phi (\\Gamma _i)$ contains only the trivial automorphism of ${Z}_q$ , i.e., $\\Gamma _i$ commutes with $Q^{\\prime }\\simeq {Z}_q$ .",
"Let $P_0:=\\Gamma _1\\times \\dots \\times \\Gamma _c$ .",
"Then $[P:P_0]\\le (m+1)^c$ .",
"Finally, since, if we identify ${Z}_q$ with the $q$ -torsion of ${Z}_{q^s}$ , the group $\\lbrace \\alpha \\in \\operatorname{Aut}({Z}_{q^s})\\mid \\alpha (t)=t\\text{ for every $t\\in {Z}_q$}\\rbrace $ is a $q$ -group, it follows that $P_0$ not only commutes with $Q^{\\prime }$ , but also with all the elements of $Q$ .",
"Consequently $G_0:=P_0Q$ is an abelian group, and we have $[G:G_0]\\le (m+1)^c\\le (m+1)^r.$ This completes the proof of the induction step, and with it that of Lemma REF ."
],
[
"$\\lambda $ -stable actions of abelian groups",
"In this section all manifolds will be compact, possibly with boundary, and non necessarily connected.",
"If $X$ is a manifold we call the dimension of $X$ (denoted by $\\dim X$ ) the maximum of the dimensions of the connected components of $X$ ."
],
[
"Preliminaries",
"Lemma 7.1 Suppose that $m,k$ are non negative integers.",
"If $X$ is a smooth manifold of dimension $m$ , $X_1\\subset X_2\\subset \\dots \\subset X_r\\subseteq X$ are strict inclusions of neatSee §1.4 in [18].",
"submanifolds, and each $X_i$ has at most $k$ connected components, then $r\\le \\left(m+k+1\\atop m+1\\right).$ Let $X_1\\subset X_2\\subset \\dots \\subset X_r\\subseteq X$ be as in the statement of the lemma.",
"For any $i$ let $d(X_i)=(d_m(X_i),\\dots ,d_{0}(X_i))$ , where $d_{j}(X_i)$ denotes the number of connected components of $X_i$ of dimension $j$ .",
"Each $X_i$ has at most $k$ connected components, so $d(X_i)$ belongs to ${{D}}=\\lbrace (d_m,\\dots ,d_1,d_0)\\in {Z}_{\\ge 0}^{m+1}\\mid \\sum d_j\\le k\\rbrace .$ Consider the lexicographic order on ${{D}}$ .",
"We claim that for each $i$ we have $d(X_i)>d(X_{i-1})$ .",
"To prove the claim, let us denote by $X_{i-1,1},\\dots ,X_{i-1,r}$ (resp.",
"$X_{i,1},\\dots ,X_{i,s}$ ) the connected components of $X_{i-1}$ (resp.",
"$X_i$ ), labelled in such a way that $\\dim X_{i-1,j-1}\\ge \\dim X_{i-1,j}$ and $\\dim X_{i,j-1}\\ge \\dim X_{i,j}$ for each $j$ .",
"Since $X_{i-1}\\subset X_i$ , there exists a map $f\\colon \\lbrace 1,\\dots ,r\\rbrace \\rightarrow \\lbrace 1,\\dots ,s\\rbrace $ such that $X_{i-1,j}\\subseteq X_{i,f(j)}$ , which implies that $\\dim X_{i-1,j}\\le \\dim X_{i,f(j)}$ .",
"Let $J$ be the set of indices $j$ such that $\\dim X_{i-1,j}<\\dim X_{i,f(j)}$ .",
"We distinguish two cases.",
"If $J=\\emptyset $ , so that $\\dim X_{i-1,j}=\\dim X_{i,f(j)}$ for each $j$ , then $X_{i-1}\\ne X_i$ implies that $X_i=X_{i-1}\\sqcup X_i^{\\prime }$ for some nonempty and possibly disconnected $X_i^{\\prime }\\subset X$ , because by assumption $X_{i-1}$ is a neat submanifold of $X$ .",
"This implies that $d_{\\delta }(X_i)\\ge d_{\\delta }(X_{i-1})$ for each $\\delta $ , and the inequality is strict for at least one $\\delta $ .",
"Hence $d(X_i)>d(X_{i-1})$ .",
"Now suppose that $J\\ne \\emptyset $ .",
"Let $l=\\dim X_{i,\\min f(J)}$ .",
"If $l+1\\le \\delta \\le m$ then any $\\delta $ -dimensional connected component of $X_{i-1}$ is also a connected component of $X_i$ , so $d_{\\delta }(X_i)\\ge d_{\\delta }(X_{i-1})$ (this is not necessarily an equality, since there might be some $\\delta $ -dimensional connected component of $X_i$ which does not contain any connected component of $X_{i-1}$ ), whereas $d_l(X_i)>d_l(X_{i-1})$ .",
"This implies again that $d(X_i)>d(X_{i-1})$ , so the proof of the claim is complete.",
"The claim implies that $r\\le |{{D}}|$ , and an easy computation gives $|{{D}}|=\\left( m \\atop m\\right)+\\left( m+1 \\atop m\\right)+\\dots +\\left( m+k \\atop m\\right)=\\left( m+k+1 \\atop m+1\\right),$ so the proof of the lemma is complete.",
"The following result of Mann and Su (see [22]) has already been mentioned in an earlier section: for any compact manifold $X$ there exists some integer $\\mu (X)\\in {Z}$ with the property that for any prime $p$ and any elementary $p$ -group ${Z}_p^r$ admitting an effective action on $X$ we have $r\\le \\mu (X)$ .",
"This implies that any finite abelian $p$ -group acting effectively on $X$ is isomorphic to ${Z}_{p^{e_1}}\\oplus \\dots \\oplus {Z}_{p^{e_r}}$ , where $r\\le \\mu (X)$ and $e_1,\\dots ,e_r$ are natural numbers.",
"Lemma 7.2 Let $X$ be a compact manifold and let $n$ be a natural number.",
"Any finite abelian $p$ -group $\\Gamma $ acting effectively on $X$ has a subgroup $G\\subset \\Gamma $ of index $[\\Gamma :G]\\le p^{n\\mu (X)}$ which is contained in each subgroup $\\Gamma ^{\\prime }\\subset \\Gamma $ of index $[\\Gamma :\\Gamma ^{\\prime }]\\le p^n$ .",
"We may assume that $\\Gamma \\simeq \\prod _{j=1}^r\\langle \\gamma _j\\rangle $ , where $r\\le \\mu (X)$ and $\\gamma _1,\\dots ,\\gamma _r\\in \\Gamma $ .",
"We claim that $G:=\\langle \\gamma _1^{p^n},\\dots ,\\gamma _r^{p^n}\\rangle $ has the desired property.",
"Indeed, if $\\Gamma ^{\\prime }\\subset \\Gamma $ is a subgroup satisfying $[\\Gamma :\\Gamma ^{\\prime }]\\le p^n$ then the exponent of $\\Gamma /\\Gamma ^{\\prime }$ divides $p^n$ , and this implies that $\\gamma _j^{p^n}\\in \\Gamma ^{\\prime }$ for each $j$ , so $G\\subseteq \\Gamma ^{\\prime }$ .",
"Clearly, $[\\Gamma :G]\\le p^{nr}\\le p^{n\\mu (X)}$ .",
"Lemma 7.3 Let $X$ be a compact smooth manifold and let $p$ be any prime number.",
"There exists a natural number $C_{p,\\chi }$ with the following property.",
"For any finite $p$ -group $\\Gamma $ acting smoothly on $X$ there exists a subgroup $\\Gamma _{\\chi }\\subseteq \\Gamma $ of index at most $C_{p,\\chi }$ satisfying $[\\Gamma :\\Gamma _{\\chi }]\\le C_{p,\\chi }$ ; for any subgroup $\\Gamma _0\\subseteq \\Gamma _{\\chi }$ we have $\\chi (X^{\\Gamma _0})=\\chi (X).$ Furthermore, there exists some $P_{\\chi }$ such that if $p\\ge P_{\\chi }$ then $C_{p,\\chi }$ can be taken to be 1.",
"Let $n$ be the smallest integer such that $p^{n+1}>2\\sum _jb_j(X;{F}_p)$ .",
"We have $|\\chi (X)+ap^{n+1}|>\\sum _jb_j(X;{F}_p)\\qquad \\text{for any nonzero integer $a$.",
"}$ We are going to prove that $C_{p,\\chi }:=p^{n\\mu (X)}$ does the job.",
"Let $\\Gamma $ be a $p$ -group acting on $X$ .",
"Let $\\Gamma _{\\operatorname{tr}}\\subseteq \\Gamma $ be the kernel of the morphism $\\Gamma \\rightarrow \\operatorname{Diff}(X)$ given by the action.",
"For the purposes of proving the lemma we may replace $\\Gamma $ by $\\Gamma /\\Gamma _{\\operatorname{tr}}$ and hence assume that $\\Gamma $ acts effectively on $X$ .",
"Take, using Lemma REF , a subgroup $\\Gamma _{\\chi }\\subseteq \\Gamma $ of index $[\\Gamma :\\Gamma _{\\chi }]\\le p^{n\\mu (X)}$ such that for any subgroup $\\Gamma ^{\\prime }\\subseteq \\Gamma $ of index $[\\Gamma :\\Gamma ^{\\prime }]\\le p^n$ we have $\\Gamma _{\\chi }\\subseteq \\Gamma ^{\\prime }$ .",
"We now prove that any subgroup $\\Gamma _0\\subseteq \\Gamma _{\\chi }$ satisfies $\\chi (X^{\\Gamma _0})=\\chi (X)$ .",
"Consider a $\\Gamma $ -good triangulation $({{C}},\\phi )$ of $X$ .",
"We have $|{{C}}|^{\\Gamma _0}=|{{C}}^{\\Gamma _0}|$ , so $\\chi (X)-\\chi (X^{\\Gamma _0})=\\chi ({{C}})-\\chi ({{C}}^{\\Gamma _0})=\\sum _{j\\ge 0}(-1)^j\\sharp \\lbrace \\sigma \\in {{C}}\\setminus {{C}}^{\\Gamma _0}\\mid \\dim \\sigma =j\\rbrace .$ If $\\sigma \\in {{C}}\\setminus {{C}}^{\\Gamma _0}$ then the stabilizer $\\Gamma _{\\sigma }=\\lbrace \\gamma \\in \\Gamma \\mid \\gamma \\cdot \\sigma =\\sigma \\rbrace $ does not contain $\\Gamma _0$ .",
"This implies that $[\\Gamma :\\Gamma _{\\sigma }]\\ge p^{n+1}$ , for otherwise $\\Gamma _{\\sigma }$ would contain $\\Gamma _{\\chi }$ and hence also $\\Gamma _0$ .",
"Consequently, the cardinal of the orbit $\\Gamma \\cdot \\sigma $ is divisible by $p^{n+1}$ .",
"Repeating this argument for all $\\sigma \\in {{C}}\\setminus {{C}}^{\\Gamma _0}$ and using (REF ), we conclude that $\\chi (X)-\\chi (X^{\\Gamma _0})$ is divisible by $p^{n+1}$ .",
"Now, we have $|\\chi (X^{\\Gamma _0})|\\le \\sum _j b_j(X^{\\Gamma _0};{F}_p)\\le \\sum _j b_j(X;{F}_p)$ (the first inequality is obvious, and the second one follows from Lemma REF ).",
"By our choice of $n$ , the congruence $\\chi (X^{\\Gamma _0})\\equiv \\chi (X)\\mod {p}^{n+1}$ and the inequality $|\\chi (X^{\\Gamma _0})|\\le \\sum _j b_j(X;{F}_p)$ imply that $\\chi (X^{\\Gamma _0})=\\chi (X)$ .",
"We now prove the last statement.",
"Since $X$ is compact, its cohomology is finitely generated, so in particular the torsion of its integral cohomology is bounded.",
"Hence there exists some $p_0$ such that if $p\\ge p_0$ then $b_j(X;{F}_p)=b_j(X)$ for every $j$ .",
"Define $P_{\\chi }=\\max \\lbrace p_0,2\\sum _j b_j(X)+1\\rbrace $ .",
"If $p\\ge P_{\\chi }$ then the number $n$ defined at the beginning of the proof is equal to 0, so $C_{p,\\chi }$ can be taken to be 1."
],
[
"$\\lambda $ -stable actions: abelian {{formula:9f2f5ad7-c5c0-4151-9781-a37a451243f5}} -groups",
"Let $p$ be a prime and let $\\Gamma $ be a finite abelian $p$ -group acting smoothly on a smooth compact manifold $X$ .",
"Recall that for any $x\\in X^{\\Gamma }$ the space $T_xX/T_x^{\\Gamma }X$ (which is the fiber over $x$ of the normal bundle of the inclusion of $X^{\\Gamma }$ in $X$ ) carries a linear action of $\\Gamma $ , induced by the derivative at $x$ of the action on $X$ , and depending up to isomorphism only on the connected component of $X^{\\Gamma }$ to which $x$ belongs.",
"Let $\\lambda $ be a natural number.",
"We say that the action of $\\Gamma $ on $X$ is $\\lambda $ -stable if: $\\chi (X^{\\Gamma _0})=\\chi (X)$ for any subgroup $\\Gamma _0\\subseteq \\Gamma $ ; for any $x\\in X^{\\Gamma }$ and any character $\\theta \\colon \\Gamma \\rightarrow {C}^*$ occurring in the $\\Gamma $ -module $T_xX/T_xX^{\\Gamma }$ we have $[\\Gamma :\\operatorname{Ker}\\theta ]> \\lambda .$ Note that if $\\Gamma $ acts trivially on $X$ then the action is $\\lambda $ -stable for any $\\lambda $ .",
"When the manifold $X$ and the action of $\\Gamma $ on $X$ are clear from the context, we will sometimes abusively say that $\\Gamma $ is $\\lambda $ -stable.",
"For example, if a group $G$ acts on $X$ we will say that an abelian $p$ -subgroup $\\Gamma \\subseteq G$ is $\\lambda $ -stable if the restriction of the action of $G$ to $\\Gamma $ is $\\lambda $ -stable.",
"Lemma 7.4 Let $\\Gamma $ be a finite abelian $p$ -group acting smoothly on $X$ so that for any subgroup $\\Gamma ^{\\prime }\\subseteq \\Gamma $ we have $\\chi (X^{\\Gamma ^{\\prime }})=\\chi (X)$ .",
"If $p>\\lambda $ then $\\Gamma $ is $\\lambda $ -stable.",
"If $p\\le \\lambda $ then there exists a $\\lambda $ -stable subgroup $\\Gamma _{\\operatorname{st}}\\subseteq \\Gamma $ satisfying $[\\Gamma :\\Gamma _{\\operatorname{st}}]\\le \\lambda ^e,\\qquad e=\\left(m+k+1\\atop m+1\\right),$ where $m=\\dim X$ and $k=\\sum _jb_j(X;{F}_p)$ .",
"Suppose that $p$ is a prime number satisfying $p>\\lambda $ , and that $\\Gamma $ is a finite abelian $p$ -group satisfying the properties in the statement of the lemma.",
"Then $\\Gamma $ is $\\lambda $ -stable, because for any $x\\in X^{\\Gamma }$ and any character $\\theta :\\Gamma \\rightarrow {C}^*$ occurring in $T_xX/T_xX^{\\Gamma }$ the subgroup $\\operatorname{Ker}\\theta \\subset \\Gamma $ , being a strict subgroup (by (1) in Lemma REF ), satisfies $[\\Gamma :\\operatorname{Ker}\\theta ]\\ge p>\\lambda $ .",
"Now suppose that $p$ is a prime satisfying $p\\le \\lambda $ and that $\\Gamma $ is a finite abelian $p$ -group satisfying the properties in the statement of the lemma.",
"Let also $m,k,e$ be as in the statement.",
"We are going to prove that there exists some $\\lambda $ -stable subgroup $\\Gamma _{\\operatorname{st}}\\subseteq \\Gamma $ satisfying $[\\Gamma :\\Gamma _{\\operatorname{st}}]\\le \\lambda ^e$ .",
"Let $\\Gamma _0=\\Gamma $ .",
"If $\\Gamma _0$ is $\\lambda $ -stable, we define $\\Gamma _{\\operatorname{st}}:=\\Gamma _0$ and we are done.",
"If $\\Gamma _0$ is not $\\lambda $ -stable, then there exists some $x\\in X^{\\Gamma _0}$ and a character $\\theta \\colon \\Gamma _0\\rightarrow {C}^*$ occurring in the $\\Gamma _0$ -module $T_xX/T_xX^{\\Gamma _0}$ such that $[\\Gamma _0:\\operatorname{Ker}\\theta ]\\le \\lambda $ .",
"Choose one such $x$ and $\\theta $ and define $\\Gamma _1:=\\operatorname{Ker}\\theta $ .",
"Then clearly $[\\Gamma _0:\\Gamma _1]\\le \\lambda $ and, by (1) in Lemma REF , $X^{\\Gamma _0}\\subset X^{\\Gamma _1}$ .",
"If $\\Gamma _1$ is $\\lambda $ -stable, then we define $\\Gamma _{\\operatorname{st}}:=\\Gamma _1$ and we stop, otherwise we repeat the same procedure with $\\Gamma _0$ replaced by $\\Gamma _1$ and define a subgroup $\\Gamma _2\\subset \\Gamma _1$ satisfying $[\\Gamma _1:\\Gamma _2]\\le \\lambda $ and $X^{\\Gamma _1}\\subset X^{\\Gamma _2}$ .",
"And so on.",
"Each time we repeat this procedure, we go from one group $\\Gamma _i$ to a subgroup $\\Gamma _{i+1}$ satisfying $[\\Gamma _i:\\Gamma _{i+1}]\\le \\lambda $ and $X^{\\Gamma _i}\\subset X^{\\Gamma _{i+1}}$ .",
"Suppose that we have been able to repeat the previous procedure $e$ steps, so that we have a decreasing sequence of subgroups $\\Gamma =\\Gamma _0\\supset \\Gamma _1\\supset \\dots \\supset \\Gamma _{e}$ giving strict inclusions $X^{\\Gamma _0}\\subset X^{\\Gamma _1}\\subset \\dots X^{\\Gamma _{e}}\\subseteq X.$ For each $j$ the manifold $X^{\\Gamma _j}$ is a neat submanifold of $X$ (by (1) in Lemma REF ) and the number of connected components of $X^{\\Gamma _j}$ satisfies (by Lemma REF ) $|\\pi _0(X^{\\Gamma _j})|=b_0(X^{\\Gamma _j};{F}_p)\\le \\sum _j b_j(X^{\\Gamma _j};{F}_p)\\le \\sum _j b_j(X;{F}_p)=k.$ So our assumption leads to a contradiction with Lemma REF .",
"It follows that the previous procedure must stop before reaching the $e$ -th step, so its outcome is a sequence of subgroups $\\Gamma =\\Gamma _0\\supset \\dots \\supset \\Gamma _{f}$ satisfying $[\\Gamma _i:\\Gamma _{i+1}]\\le \\lambda $ , $f<e$ , and $\\Gamma _{\\operatorname{st}}:=\\Gamma _f$ is $\\lambda $ -stable.",
"We also have $[\\Gamma :\\Gamma _{\\operatorname{st}}]\\le \\lambda ^f\\le \\lambda ^e$ , so the proof of the lemma is complete."
],
[
"Fixed point sets and inclusions of groups",
"Let $X$ be a compact manifold.",
"If $A,B$ are submanifolds of $X$ , we will write $A\\preccurlyeq B$ whenever $A\\subseteq B$ and each connected component of $A$ is a connected component of $B$ .",
"Let $p$ be a prime.",
"Lemma 7.5 Let $\\lambda $ be a natural number.",
"Let $\\Gamma $ be a finite abelian $p$ -group acting smoothly on a compact manifold $X$ in a $\\lambda $ -stable way.",
"If a subgroup $\\Gamma _0\\subseteq \\Gamma $ satisfies $[\\Gamma :\\Gamma _0]\\le \\lambda $ then $X^{\\Gamma }\\preccurlyeq X^{\\Gamma _0}$ .",
"We clearly have $X^{\\Gamma }\\subseteq X^{\\Gamma _0}$ , so it suffices to prove that for each $x\\in X^{\\Gamma }$ we have $\\dim _xX^{\\Gamma }=\\dim _xX^{\\Gamma _0}$ .",
"If this is not the case for some $x\\in X^{\\Gamma }$ then, by (REF ) in Lemma REF , there exist an irreducible $\\Gamma $ -submodule of $T_xX/T_xX^{\\Gamma }$ on which the action of $\\Gamma _0$ is trivial.",
"Let $\\theta \\colon \\Gamma \\rightarrow {C}^*$ be the character associated to this submodule.",
"Then $\\Gamma _0\\subseteq \\operatorname{Ker}\\theta $ , which implies that $[\\Gamma :\\operatorname{Ker}\\theta ]\\le \\lambda $ , contradicting the hypothesis that $\\Gamma $ is $\\lambda $ -stable.",
"Lemma 7.6 Suppose that $\\lambda \\ge (\\dim X)(\\sum _jb_j(X;{F}_p))$ , and let $\\Gamma $ be a finite abelian $p$ -group acting on $X$ in a $\\lambda $ -stable way.",
"There exists an element $\\gamma \\in \\Gamma $ such that $X^{\\Gamma }\\preccurlyeq X^{\\gamma }$ .",
"Let $\\nu (\\Gamma )$ be the collection of subgroups of $\\Gamma $ of the form $\\operatorname{Ker}\\theta $ , where $\\theta \\colon \\Gamma \\rightarrow {C}^*$ runs over the set of characters appearing in the action of $\\Gamma $ on the fibers of the normal bundle of the inclusion of $X^{\\Gamma }$ in $X$ .",
"Since $\\Gamma $ is finite, its representations are rigid, so the irreducible representations in the action of $\\Gamma $ on the normal fibers of the inclusion $X^{\\Gamma }\\hookrightarrow X$ are locally constant on $X^{\\Gamma }$ .",
"For each $x\\in X^{\\Gamma }$ the representation of $\\Gamma $ on $T_xX/T_x^{\\Gamma }$ splits as the sum of at most $\\dim X$ different irreducible representations.",
"Consequently, $\\nu (\\Gamma )$ has at most $\\dim X |\\pi _0(X^{\\Gamma })|$ elements.",
"By Lemma REF , $|\\pi _0(X^{\\Gamma })|\\le \\sum _jb_j(X;{F}_p)$ .",
"Since $\\Gamma $ is $\\lambda $ -stable, we have $|\\Gamma ^{\\prime }|<\\lambda ^{-1}|\\Gamma |$ for each $\\Gamma ^{\\prime }\\in \\nu (\\Gamma )$ , so $\\left|\\bigcup _{\\Gamma ^{\\prime }\\in \\nu (\\Gamma )}\\Gamma ^{\\prime }\\right|\\le \\lambda ^{-1}|\\Gamma ||\\nu (\\Gamma )|\\le \\lambda ^{-1}|\\Gamma |\\dim X\\left(\\sum _jb_j(X;{F}_p)\\right)<|\\Gamma |.$ Consequently, there exists at least one element $\\gamma \\in \\Gamma $ not contained in $\\bigcup _{\\Gamma ^{\\prime }\\in \\nu (\\Gamma )}\\Gamma ^{\\prime }$ .",
"By Lemma REF we have $X^{\\Gamma }\\preccurlyeq X^{\\gamma }$ ."
],
[
"$\\lambda $ -stable actions: arbitrary abelian groups",
"Let $X$ be a compact manifold and let $\\Gamma $ be a finite abelian group acting on $X$ .",
"For any prime $p$ we denote by $\\Gamma _{p}$ the $p$ -part of $\\Gamma $ .",
"We say that the action of $\\Gamma $ on $X$ is $\\lambda $ -stable if and only if for any prime $p$ the restriction of the action to $\\Gamma _{p}$ is $\\lambda $ -stable (recall that any action of the trivial group is $\\lambda $ -stable).",
"As for $p$ -groups, when the manifold $X$ and the action are clear from the context, we will sometimes say that $\\Gamma $ is $\\lambda $ -stable (this will be often the case when talking about subgroups of a group acting on $X$ ).",
"Theorem 7.7 Let $\\lambda $ be a natural number.",
"There exists a constant $C_{\\lambda }$ , depending only on $X$ and $\\lambda $ , such that any finite abelian group $\\Gamma $ acting on $X$ has a $\\lambda $ -stable subgroup of index at most $C_{\\lambda }$ .",
"Let $P_{\\chi }$ be the number defined in Lemma REF .",
"Define $C_{\\lambda }:=\\left(\\prod _{p\\le P_{\\chi }}C_{p,\\chi }\\right)\\left(\\prod _{p\\le \\lambda }\\lambda ^e\\right),$ where in both products $p$ runs over the set of primes satisfying the inequality and $e=\\left( m+K+1\\atop m+1\\right),\\qquad m=\\dim X,\\qquad K=\\sum _j\\max \\lbrace b_j(X;{F}_p)\\mid p\\text{ prime}\\rbrace .$ The theorem follows from combining Lemmas REF and Lemma REF applied to each of the factors of $\\Gamma \\simeq \\prod _{p|d}\\Gamma _p$ , where $d=|\\Gamma |$ ."
],
[
"$\\lambda $ -stable actions on manifolds without odd cohomology",
"In this section $X$ denotes a manifold without odd cohomology.",
"Let $p$ be any prime number.",
"Applying cohomology to the exact sequence of locally constant sheaves on $X$ $0\\rightarrow \\underline{{Z}}\\stackrel{\\cdot p}{\\longrightarrow }\\underline{{Z}}\\rightarrow \\underline{{F}_p}\\rightarrow 0$ and using the fact that $X$ has no odd cohomology we obtain $b_j(X;{F}_p)=b_j(X)\\quad \\text{for any $j$}\\qquad \\Longrightarrow \\qquad \\chi (X)=\\sum _jb_j(X)=\\sum _jb_j(X;{F}_p).$ Lemma 7.8 Let $p$ be any prime number.",
"Suppose that a finite abelian $p$ -group $\\Gamma $ acts on $X$ and that there is a subgroup $\\Gamma ^{\\prime }\\subseteq \\Gamma $ such that $X^{\\Gamma }\\preccurlyeq X^{\\Gamma ^{\\prime }}$ and $\\chi (X^{\\Gamma })=\\chi (X^{\\Gamma ^{\\prime }})=\\chi (X)$ .",
"Then $X^{\\Gamma }=X^{\\Gamma ^{\\prime }}$ .",
"By Lemma REF we have $\\sum _jb_j(X^{\\Gamma };{F}_p)\\le \\sum _jb_j(X;{F}_p)$ so, using (REF ), $\\chi (X^{\\Gamma })\\le \\sum _jb_j(X^{\\Gamma };{F}_p)\\le \\sum _jb_j(X;{F}_p)=\\chi (X).$ Since $\\chi (X^{\\Gamma })=\\chi (X)$ we have $\\sum _jb_j(X^{\\Gamma };{F}_p)=\\sum _jb_j(X;{F}_p).$ Applying the same arguments to $\\Gamma ^{\\prime }$ we conclude that $\\sum _jb_j(X^{\\Gamma };{F}_p)=\\sum _jb_j(X^{\\Gamma ^{\\prime }};{F}_p)$ .",
"Combining this with $X^{\\Gamma }\\preccurlyeq X^{\\Gamma ^{\\prime }}$ we deduce $X^{\\Gamma }=X^{\\Gamma ^{\\prime }}$ .",
"Lemma 7.9 Let $\\lambda _\\chi =\\chi (X)\\dim X$ .",
"If $\\Gamma $ is a finite abelian $p$ -group acting on $X$ in a $\\lambda _\\chi $ -stable way, then there exists some $\\gamma \\in X$ such that $X^{\\Gamma }=X^{\\gamma }$ .",
"This follows from combining Lemma REF , equality (REF ), and Lemma REF ."
],
[
"Proof of Theorem ",
"Let $X$ be a compact manifold without odd cohomology.",
"Let $A$ be a finite abelian group acting smoothly on $X$ .",
"Let $b_j=b_j(X)$ and let $b=\\sum _jb_j^2$ .",
"We claim that there exists a subgroup $G\\subseteq A$ whose action on the cohomology $H^*(X;{Z})$ is trivial and which satisfies $[A:G]\\le 3^{b}.$ To prove the claim, recall that a well known lemma of Minkowski states that for any $n$ and any finite group $H\\subseteq \\operatorname{GL}(n,{Z})$ the restriction of the quotient map $q_n:\\operatorname{GL}(n,{Z})\\rightarrow \\operatorname{GL}(n,{F}_3)$ to $H$ is injective (see e.g.",
"[24], [37]; the proof is easy: it suffices to check that for any nonzero $M\\in \\operatorname{Mat}_{n\\times n}({Z})$ and nonzero integer $k$ the matrix $(\\operatorname{Id}_n+3M)^k$ is different from the identity, see e.g.",
"[11]).",
"Choosing a homogeneous basis of $H^*(X;{Z})$ the action of $A$ on the cohomology can be encoded in a morphism of groups $\\phi :A\\rightarrow \\prod _j\\operatorname{GL}(b_j,{Z}).$ Then $G:=\\operatorname{Ker}(q\\circ \\phi ),\\qquad q=(q_{b_0},\\dots ,q_{b_n}),\\qquad n=\\dim X$ has the required property, because $|\\prod _j\\operatorname{GL}(b_j,{F}_3)|\\le 3^{b}$ .",
"Let $\\lambda _{\\chi }=\\chi (X)\\dim X$ .",
"By Theorem REF there exists a subgroup $\\Gamma \\subseteq G$ whose action on $X$ is $\\lambda _{\\chi }$ -stable and which satisfies $[G:\\Gamma ]\\le C_{\\lambda _{\\chi }}$ , where $C_{\\lambda _{\\chi }}$ depends on $\\lambda _{\\chi }$ and $X$ , but not on the group $G$ .",
"There is an isomorphism $\\Gamma \\simeq \\Gamma _{p_1}\\times \\dots \\times \\Gamma _{p_k}$ , where $p_1,\\dots ,p_k$ are the prime divisors of $|\\Gamma |$ .",
"Since the action of $\\Gamma $ is $\\lambda _\\chi $ -stable so is, by definition, its restriction to each $\\Gamma _{p_i}$ , so by Lemma REF there exists, for each $i$ , an element $\\gamma _i\\in \\Gamma _{p_i}$ such that $X^{\\gamma _i}=X^{\\Gamma _{p_i}}$ .",
"Let $\\gamma =\\gamma _1\\dots \\gamma _k$ .",
"Then $X^{\\Gamma }=\\bigcap _i X^{\\Gamma _i}\\subseteq X^{\\gamma }.$ By the Chinese remainder theorem and the fact that the elements $\\gamma _1,\\dots ,\\gamma _k$ commute, for each $i$ there exists some $e$ such that $\\gamma ^e=\\gamma _i$ .",
"Hence $X^{\\gamma }\\subseteq X^{\\gamma ^e}=X^{\\gamma _i}=X^{\\Gamma _{p_i}}.$ Taking the intersection over all $i$ we get $X^{\\gamma }\\subset \\bigcap _i X^{\\Gamma _i}=X^{\\Gamma }.$ Combining the two inclusions we have $X^{\\gamma }=X^{\\Gamma }$ .",
"Since $\\gamma \\in G$ , the action of $\\gamma $ on $X$ induces the trivial action on $H^*(X;{Z})$ , so in particular it preserves the connected components of $X$ .",
"Let $Y\\subseteq X$ be any connected component.",
"Applying Lefschetz's formula [9] to the action of $\\gamma $ on $Y$ we conclude that $\\chi (Y^{\\gamma })=\\chi (Y)$ .",
"Since $Y^{\\gamma }=Y^{\\Gamma }$ , it follows that $A_0:=\\Gamma $ has the desired properties.",
"Finally, $[A:A_0]\\le 3^{b}C_{\\lambda _{\\chi }}.$"
],
[
"Proof of Theorem ",
"The following well known fact immediately proves the theorem in the cases $n=1,2$ .",
"Lemma 9.1 Let $n$ be either 1 or 2, and let $X$ be the $n$ -dimensional disk.",
"The fixed point locus of any smooth action of a finite group on $X$ is contractible.",
"Let now $n\\ge 3$ be a natural number and let $X$ be the $n$ -dimensional disk.",
"Let $p$ be a prime and suppose that a finite abelian $p$ -group $A$ acts on $X$ .",
"Smith theory implies that the fixed point set $X^{A}$ is ${F}_p$ -acyclic (see the proof of Theorem REF ).",
"In particular, $X^{A}$ is nonempty and connected.",
"Lemma 9.2 Let $x\\in X^{A}$ be any point, and let $\\theta _1,\\dots ,\\theta _r$ ($\\theta _j:A\\rightarrow \\operatorname{GL}(W_j)$ ) be the different (real) irreducible representations of $A$ appearing in $T_xX/T_xX^{A}$ .",
"Let $l=\\dim X^{A}$ .",
"If $p=2$ then $r\\le n-l$ .",
"If $p$ is odd then $n-l$ is even and $r\\le (n-l)/2$ .",
"There exists some $\\gamma \\in A$ and a subgroup $A^{\\prime }\\subseteq A$ such that $X^{\\gamma }=X^{A^{\\prime }}$ and $[A:A^{\\prime }]$ divides $p^{[r/p]}$ .",
"(1) is clear.",
"To prove (2), define $A_j:=\\operatorname{Ker}\\theta _j$ , let $e_j=\\log _p[A:A_j]$ , and consider the function $I:A\\rightarrow {Z}$ defined as $I(\\gamma )=\\sum _{j\\mid \\gamma \\in A_j}e_j.$ Since each $\\theta _j$ is nontrivial (e.g.",
"by (2) in Lemma REF ) we have $e_j\\ge 1$ for every $j$ .",
"Now we estimate $\\sum _{\\gamma \\in A}I(\\gamma )=\\sum _{\\gamma \\in A}\\sum _{j\\mid \\gamma \\in A_j}e_j=\\sum _{j=1}^r|A_j|e_j=\\sum _{j=1}^r\\frac{|A|}{p^{e_j}}e_j\\le \\sum _{j=1}^r\\frac{|A|}{p}=\\frac{r}{p}|A|,$ where the inequality follows from the fact that the function ${N}\\ni n\\mapsto n/q^n$ is non increasing for any integer $q\\ge 2$ .",
"Hence the average value of $I$ is not bigger than $r/p$ , so there exists some $\\gamma \\in A$ , which we fix for the rest of the argument, such that $I(\\gamma )\\le [r/p]$ .",
"Let $A^{\\prime }:=\\bigcap _{j\\mid \\gamma \\in A_j}A_j$ .",
"We claim that $X^{\\gamma }=X^{A^{\\prime }}$ .",
"Since $\\gamma \\in A^{\\prime }$ , the inclusion $X^{A^{\\prime }}\\subseteq X^{\\gamma }$ is clear.",
"To prove the reverse inclusion observe that, by Smith theory, both $X^{A^{\\prime }}$ and $X^{\\gamma }$ are acyclic, hence connected, so it suffices to prove that $T_xX^{\\gamma }\\subseteq T_xX^{A}$ .",
"Let $T_xX/T_xX^{A}=V_1\\oplus \\dots \\oplus V_r$ be the decomposition in isotypical real representations of $A$ , where $V_j$ is isomorphic to the direct sum of a number of copies of $W_j$ .",
"Since we clearly have $T_xX^{A}\\oplus \\bigoplus _{j\\mid \\gamma \\in A_j}V_j\\subseteq T_xX^{A^{\\prime }},$ it suffices to prove that $T_xX^{\\gamma }\\subseteq T_xX^{A}\\oplus \\bigoplus _{j\\mid \\gamma \\in A_j}V_j.$ The latter is equivalent to proving that $T_xX^{\\gamma }\\cap V_i=\\operatorname{Ker}(\\theta _i^V(\\gamma )-\\operatorname{Id})=\\lbrace 0\\rbrace $ for every $i$ such that $\\gamma \\notin A_i$ (here $\\theta _i^V:A\\rightarrow \\operatorname{GL}(V_j)$ is given by restricting the action of $A$ on $T_xX/T_x^{A}$ ).",
"Since $V_i$ is isotypical and $\\gamma $ is central in $A$ , $\\operatorname{Ker}(\\theta _i^V(\\gamma )-\\operatorname{Id})$ is either $\\lbrace 0\\rbrace $ or $V_i$ .",
"But $\\operatorname{Ker}(\\theta _i^V(\\gamma )-\\operatorname{Id})=V_i$ would imply $\\gamma \\in \\operatorname{Ker}\\theta _i$ , contradicting the choice of $i$ .",
"Hence $\\operatorname{Ker}(\\theta _i^V(\\gamma )-\\operatorname{Id})=\\lbrace 0\\rbrace $ and the proof that $X^{\\gamma }=X^{A^{\\prime }}$ is complete.",
"To finish the proof of the lemma we observe that $[A:A^{\\prime }]$ divides $\\prod _{j\\mid \\gamma \\in A_j}[A:A_j]=p^{I(\\gamma )}$ .",
"Since $I(\\gamma )\\le [r/p]$ , we deduce that $[A:A^{\\prime }]$ divides $p^{[r/p]}$ .",
"Now let $A$ be a finite abelian group acting on $X$ .",
"For each prime $p$ let $A_p\\subseteq A$ denote the $p$ -part of $A$ , so that $A=\\prod _pA_p$ .",
"If for some $p$ we have $\\dim X^{A_p}\\le 2$ , then the classification of manifolds (with boundary) of dimension at most 2 implies that $\\dim X^{A_p}$ is a disk, because $\\dim X^{A_p}$ is ${F}_p$ -acyclic; so applying Lemma REF to the action of $A$ on $X^{A_p}$ we deduce that $\\chi (X^{A})=\\chi ((X^{A_p})^{A})=1$ and the proof of the theorem is complete.",
"Hence it suffices to consider the case when $\\dim X^{A_p}\\ge 3$ for each $p$ .",
"Let $k=[(n-3)/2]$ .",
"Applying Lemma REF for each prime $p$ we deduce that there exists some subgroup $A_2^{\\prime }\\subseteq A_2$ satisfying $[A_2:A_2^{\\prime }]\\le 2^k$ and an element $\\gamma _2\\in A_2^{\\prime }$ such that $X^{\\gamma _2}=X^{A_2^{\\prime }}$ and, for each odd prime $p$ , there exists some subgroup $A_p^{\\prime }\\subseteq A_p$ satisfying $[A_p:A_p^{\\prime }]\\le p^{[k/p]}$ and an element $\\gamma _p\\in A_p^{\\prime }$ such that $X^{\\gamma _p}=X^{A_p^{\\prime }}$ .",
"Let $A^{\\prime }=\\prod A_p^{\\prime }$ and let $\\gamma =\\prod \\gamma _p$ .",
"Arguing as in Section we prove that $X^{\\gamma }=X^{A^{\\prime }}$ .",
"Clearly $[A:A^{\\prime }]$ divides $f(k)$ , so statement (1) of Theorem REF is proved.",
"Statement (2) follows immediately from statement (1), because none of the odd prime divisors of $f(k)$ is bigger than $k$ ."
],
[
"Proof of Theorem ",
"We follow a scheme similar to the proof of Theorem REF .",
"Let $m$ be a natural number, let $p$ be a prime and let $Y$ be a ${F}_p$ -homology $2m$ -sphere.",
"For any smooth action of ${Z}_p$ on $X$ the fixed point set is a ${F}_p$ -homology $s$ -sphere [2].",
"Furthermore, if $p$ is odd, the difference $2m-s$ is even [2].",
"Hence we may apply the same inductive scheme as in Lemma REF (or [2]) and deduce the following.",
"Lemma 9.3 For any odd prime $p$ , any finite $p$ -group $A$ , and any action of $A$ on a ${F}_p$ -homology even dimensional sphere the fixed point set is a ${F}_p$ -homology even dimensional sphere (in particular, the fixed point set is nonempty).",
"The case $p=2$ works differently.",
"Suppose that $Y$ is a smooth $n$ -dimensional manifold and that $Y$ is a ${F}_2$ -homology $n$ -sphere.",
"Suppose that ${Z}_2$ acts smoothly on $Y$ .",
"Then $Y^{{Z}_2}$ is a ${F}_2$ -homology $s$ -sphere [2].",
"The fixed point set $Y^{{Z}_2}$ is also a smooth submanifold of $Y$ and $s$ coincides with the dimension of $Y^{{Z}_2}$ as a manifold.",
"The condition that $Y$ is a ${F}_2$ -homology sphere implies that $Y$ is compact and orientable.",
"One checks, using Lefschetz' formula [9] and arguing in terms of volume forms, that $n-s$ is even if and only if the action of the nontrivial element of ${Z}_2$ on $Y$ is orientation preserving (this works more generally for continuous actions on finite Hausdorff spaces whose integral homology is finitely generated and whose ${F}_2$ -homology is isomorphic to $H_*(S^n;{F}_2)$ , by a theorem of Liao [21], see also [2]).",
"Lemma 9.4 For any finite 2-group $A$ and any smooth action of $A$ on a smooth ${F}_2$ -homology $2m$ -sphere ($m\\in {Z}_{\\ge 0}$ ) there exists a subgroup $A_0\\subseteq A$ whose index $[A:A_0]$ divides $2^{m+1}$ and whose fixed point set $X^{A_0}$ is a smooth ${F}_2$ -homology even dimensional sphere (in particular, $X^{A_0}$ is nonempty).",
"We use ascending induction on $|A|$ .",
"The case $|A|=2$ being obvious, suppose that $|A|>2$ and that the lemma is true for 2-groups with less elements than $A$ .",
"Suppose that $A$ acts smoothly on a compact smooth ${F}_2$ -homology $2r$ -sphere $Y$ .",
"If the action of $A$ is not effective, then it factors through a quotient of $A$ , and applying the inductive hypothesis the lemma follows.",
"So assume that the action of $A$ on $Y$ is effective.",
"Let $A^{\\prime }\\subseteq A$ be the subgroup consisting of those elements whose action is orientation preserving.",
"Then $[A:A^{\\prime }]$ divides 2.",
"Let $A^{\\prime \\prime }\\subseteq A^{\\prime }$ be a central subgroup isomorphic to ${Z}_2$ .",
"Then $Y^{A^{\\prime \\prime }}$ is a compact smooth ${F}_2$ -homology even dimensional sphere satisfying $\\dim Y^{A^{\\prime \\prime }}\\le 2r-2$ .",
"Furthermore, $A^{\\prime }/A^{\\prime \\prime }$ acts smoothly on $Y^{A^{\\prime \\prime }}$ .",
"To finish the proof, apply the inductive hypothesis to this action.",
"Let now $X$ be a smooth homology $2r$ -sphere and suppose that a finite abelian group $A$ acts smoothly on $X$ .",
"For any prime $p$ let $A_p\\subseteq A$ denote the $p$ -part.",
"By Lemma REF , for any odd prime $p$ the fixed point set $X^{A_p}$ is an even dimensional ${F}_p$ -homology sphere.",
"By Lemma REF , $A_2$ has a subgroup $A_{2,0}$ whose index divides $2^{r+1}$ and whose fixed point set is an even dimensional ${F}_2$ -homology sphere (in particular, it is nonempty).",
"Replace $A_2$ by $A_{2,0}$ and define $A_0:=\\prod _pA_p$ , so that $[A:A_0]$ divides $2^{r+1}$ .",
"Suppose that for some prime $p$ the fixed point set $X^{A_p}$ is 0-dimensional.",
"Then $X^{A_p}$ consists of two points, $A$ acts on $X^{A_p}$ , and there is a subgroup $A^{\\prime }\\subseteq A$ whose index divides 2 and whose action on $X^{A_p}$ is trivial.",
"It follows that $|X^{A^{\\prime }}|\\ge 2$ and we are done.",
"Suppose now that for each prime $p$ the fixed point set $X^{A_p}$ is an even dimensional ${F}_p$ -homology sphere of dimension at least 2.",
"In particular, $X^{A_p}$ is nonempty and connected for every $p$ , and so is $X^{A_p^{\\prime }}$ for every subgroup $A_p^{\\prime }\\subseteq A_p$ (since $X^{A_p^{\\prime }}$ is a ${F}_p$ -homology sphere and, given the inclusion $X^{A_p}\\subseteq X^{A_p^{\\prime }}$ , the dimension of $X^{A_p^{\\prime }}$ is at least 2).",
"This property allows to use the same arguments in the proof of Lemma REF to prove the following.",
"Lemma 9.5 Let $p$ be any prime, let $x\\in X^{A_p}$ be any point, and let $\\theta _1,\\dots ,\\theta _r$ be the different real irreducible representations of $A_p$ appearing in $T_xX/T_xX^{A_p}$ .",
"Let $2l=\\dim X^{A_p}$ .",
"If $p=2$ then $r\\le 2m-2l$ .",
"If $p$ is odd then $r\\le m-l$ .",
"There exists some $\\gamma \\in A_p$ and a subgroup $A_p^{\\prime }\\subseteq A_p$ such that $X^{\\gamma }=X^{A_p^{\\prime }}$ and $[A_p:A_p^{\\prime }]$ divides $p^{[r/p]}$ .",
"By Lemma REF there exists some subgroup $A_2^{\\prime }\\subseteq A_2$ satisfying $[A_2:A_2^{\\prime }]\\le 2^{2m-2}$ and an element $\\gamma _2\\in A_2^{\\prime }$ such that $X^{\\gamma _2}=X^{A_2^{\\prime }}$ and, for each odd prime $p$ , there exists some subgroup $A_p^{\\prime }\\subseteq A_p$ satisfying $[A_p:A_p^{\\prime }]\\le p^{[m-1/p]}$ and an element $\\gamma _p\\in A_p^{\\prime }$ such that $X^{\\gamma _p}=X^{A_p^{\\prime }}$ .",
"Let $A^{\\prime }=\\prod A_p^{\\prime }$ and let $\\gamma =\\prod \\gamma _p$ .",
"As in Section we have $X^{\\gamma }=X^{A^{\\prime }}$ .",
"The index $[A:A^{\\prime }]$ divides $2^{m+1}f(m-1)$ , so statement (1) of Theorem REF is proved.",
"Statement (2) follows immediately from statement (1), because none of the odd prime divisors of $f(m-1)$ is bigger than $m-1$ ."
]
] | 1403.0383 |
[
[
"Vortex and half-vortex dynamics in a spinor quantum fluid of interacting\n polaritons"
],
[
"Abstract Spinorial or multi-component Bose-Einstein condensates may sustain fractional quanta of circulation, vorticant topological excitations with half integer windings of phase and polarization.",
"Matter-light quantum fluids, such as microcavity polaritons, represent a unique test bed for realising strongly interacting and out-of-equilibrium condensates.",
"The direct access to the phase of their wavefunction enables us to pursue the quest of whether half vortices ---rather than full integer vortices--- are the fundamental topological excitations of a spinor polariton fluid.",
"Here, we are able to directly generate by resonant pulsed excitations, a polariton fluid carrying either the half or full vortex states as initial condition, and to follow their coherent evolution using ultrafast holography.",
"Surprisingly we observe a rich phenomenology that shows a stable evolution of a phase singularity in a single component as well as in the full vortex state, spiraling, splitting and branching of the initial cores under different regimes and the proliferation of many vortex anti-vortex pairs in self generated circular ripples.",
"This allows us to devise the interplay of nonlinearity and sample disorder in shaping the fluid and driving the phase singularities dynamics"
],
[
"3 super Vortex and half-vortex dynamics in a spinor quantum fluid of interacting polaritons Lorenzo Dominici lorenzo.dominici@gmail.com NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Istituto Italiano di Tecnologia, IIT–Lecce, Via Barsanti, 73010 Lecce, Italy Galbadrakh Dagvadorj Department of Physics, University of Warwick, Coventry CV47AL, UK Jonathan M. Fellows j.fellows@warwick.ac.uk Department of Physics, University of Warwick, Coventry CV47AL, UK Stefano Donati NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Istituto Italiano di Tecnologia, IIT–Lecce, Via Barsanti, 73010 Lecce, Italy Dario Ballarini NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Milena De Giorgi NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Francesca M. Marchetti Departamento de Física Teórica de la Materia Condensada, UAM, Madrid 28049, Spain Bruno Piccirillo Dipartimento di Fisica, Università Federico II di Napoli, 80126 Napoli, Italy Lorenzo Marrucci Dipartimento di Fisica, Università Federico II di Napoli, 80126 Napoli, Italy Alberto Bramati Laboratoire Kastler Brossel, UPMC-Paris 6, ENS et CNRS, 75005 Paris, France Giuseppe Gigli NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Marzena H. Szymańska Department of Physics and Astronomy, UCL, London WC1E6BT, UK Daniele Sanvitto NANOTEC, Istituto di Nanotecnologia–CNR, Via Arnesano, 73100 Lecce, Italy Spinorial or multi-component Bose-Einstein condensates may sustain fractional quanta of circulation, vorticant topological excitations with half integer windings of phase and polarization.",
"Matter-light quantum fluids, such as microcavity polaritons, represent a unique test bed for realising strongly interacting and out-of-equilibrium condensates.",
"The direct access to the phase of their wavefunction enables us to pursue the quest of whether half vortices —rather than full integer vortices— are the fundamental topological excitations of a spinor polariton fluid.",
"Here, we are able to directly generate by resonant pulsed excitations, a polariton fluid carrying either the half or full vortex states as initial condition, and to follow their coherent evolution using ultrafast holography.",
"Surprisingly we observe a rich phenomenology that shows a stable evolution of a phase singularity in a single component as well as in the full vortex state, spiraling, splitting and branching of the initial cores under different regimes and the proliferation of many vortex anti-vortex pairs in self generated circular ripples.",
"This allows us to devise the interplay of nonlinearity and sample disorder in shaping the fluid and driving the phase singularities dynamics.",
"Vortices and topological excitations play a crucial role in our understanding of the universe, recurring in the fields of subatomic particles, quantum fluids, condensed matter and nonlinear optics, being involved in fluid dynamics and phase transitions ranging up to the cosmologic scale [1].",
"Spacetime could be analogue to a superfluid [2] and elementary particles the excitations of a medium called the quantum vacuum [3]; whetever this modern view will take strength or not, phase singularities (i.e., vortices) of a quantum fluid (e.g., of a superfluid) are point-like and quantized quasi-particles by excellence.",
"Here we make use of a specific experimental “quantum interface”: polariton condensates [4], which are bosonic hybrid light-matter particles consisting of strongly coupled excitons and photons.",
"The $\\pm 1$ spin components of the excitons couple to different polarisation states of light making the Bose-degenerate polariton gas a spinor condensate.",
"The realisation of exciton polariton condensates in semiconductor microcavities [5], [6] has paved the way for a prolific series of studies into quantum hydrodynamics in two-dimensional systems [7], [8], [9], [10], [11], [12], [13].",
"Microcavity polaritons are particularly advantageous systems for the study of topological excitations in interacting superfluids, thanks to the stronger nonlinearities and peculiar dispersive and dissipative features, with respect to both atomic condensates and nonlinear optics.",
"Figure: Generation of optical and polariton FVsand HVs.",
"(a,b) Experimental scheme for creation of optical full-(a) and half-vortex (b) state via a q-plate.",
"The disks and helicsrepresent the isophase surfaces for Gaussian and vortex beams,respectively, in the radial regions of larger intensity.",
"Red andyellow colours refer to the σ + \\sigma _{+} (σ - \\sigma _{-}) circularpolarizations.",
"(c,d) Emission density of the polariton fluid atthe time of initial generation and the corresponding phase maps.For the equilibrium spinor polariton fluid, in which the drive and decay processes are ignored, the lowest energy topological excitations have been predicted to be “half vortices” (HV) [14], [15].",
"These carry a phase singularity in only one circular polarisation, such that in the linear polarisation basis they have a half-integer winding number for both the phase and field-direction [16].",
"Such an excitation is complementary to a “full vortex” (FV), which instead has a singularity in each circular polarisation.",
"Even in this simplified equilibrium scenario the question of either HVs or FVs are dynamically stable has led to some debate [17], [18], [19] due to the presence of an inherent TE-TM splitting, which often arises in semiconductor microcavities and couples HVs with opposite spin.",
"[14], [17].",
"The issue is even more complicated in a real polariton system, which is always subject to drive and dissipation, and is intrinsically out of equilibrium [20].",
"Indeed, in the case of an incoherently pumped polariton superfluid, in contrast to the equilibrium predictions, it has been theoretically demonstrated [21] that both FV and HV are dynamically stable in the absence of a symmetry breaking between the linear polarisation states, while in its presence only full vortex states are seen to be stable.",
"On the experimental side, the recent work of Manni et al [22] shows the splitting of a spontaneously formed linear polarised vortex state (FV) into two circularly polarised vortices (HVs) under non-resonant pulsed excitation.",
"However, in this case, formation and motion/pinning of these vortices are caused by strong inhomogeneities and disorder in specific locations of the sample rather than by any fundamental process intrinsic to the fluid.",
"In general, the stability of vortex states in polariton condensates remains an open issue of fundamental importance, given that the nature of the elementary excitations is likely to affect the macroscopic properties of the system such as, for example, the conditions for the Berezinsky-Kosterlitz-Thouless (BKT) transitions to the superfluid state.",
"On the application side, polariton vortices have been proposed also for ultra-sensitive gyroscopes [23] or information processing [24].",
"Experimental system In this work we have been able for the first time to study the dynamics of half and full vortices created into a polariton condensate in a variety of initial conditions and in a controlled manner, taking advantage of the versatility of the resonant pumping scheme.",
"We take care to generate the polariton vortex in a specific position on the sample with sufficiently weak disorder that the biasing effects of sample inhomogeneities can be screened out for a wide range of fluid densities.",
"In order to shape the phase profile of the incoming laser beam, we use a q-plate (Fig.",
"REF ), a patterned liquid crystal retarder recently developed to study laser windings and optical vorticity [25], [26], [27].",
"The q-plate allows us, through appropriate optical and electrical tuning, to transform a Gaussian pulse into either a FV or a HV, according to the simplified schemes shown in Fig.",
"REF a,b.",
"One advantage of a q-plate over using a typical SLM (space light modulator) is evident in the fact that the latter device works for a given linear polarization, and two SLM are needed to create a HV.",
"The exciting pulse is sent resonant on the microcavity sample to directly create a polariton fluid carrying either a full or half vortex, as shown from the emission maps in Fig.",
"REF c,d.",
"Using a time-resolved digital holography [28], [29] technique for the detection, we measure both the instantaneous amplitude and phase of the polariton condensate [30] in all its polarization components.",
"Each phase singularity can be digitally tracked so as to record the evolution of the resonantly created vortices after the initial pulse has gone but before the population has decayed away.",
"The lifetime of the 2D polariton fluid in our microcavity sample [31], [32] kept at 10K is 10 ps, and we excite it by means of a 80 MHz train of 4 ps laser pulses resonant with the lower polariton energy at 836 nm.",
"Dynamics of half and full vortex The creation of a HV is shown in Fig.",
"REF at different pulse powers.",
"In panels (a,b) the trajectory of the primary vortex ($\\Delta $ t:5-15 ps, $\\delta $ t=0.5 ps) is superimposed to the amplitude map of the opposite spin (taken at t=15 ps).",
"For both powers, the singularity of the primary HV is seen moving along a circular trajectory around the density maximum of the opposing Gaussian state, keeping itself orbiting during few tenths of ps.",
"Such curves are better depicted in Fig.",
"REF (c,d), which are the $(x(t),y(t),t)$ trajectories ($\\Delta $ t:5-40 ps, $\\delta $ t=0.5 ps) relative to cases (a,b), respectively, and in panel (e) reporting the angle $\\theta $ and distance $d$ between the primary HV core and the Gaussian center of mass (see also Movie SM1).",
"Figure: Evolution of the main singularity upon HV injection.The Gaussian map is showntogether with the core trajectory in the opposite σ\\sigma ,both 15 ps into the evolution,at the power of P 1 P_{1}=0.77 mW and P 2 P_{2}=1.8 mW in (a,b), respectively(see also Supporting Movie SM1 for power P 1 P_{1}).The complete (x,y,t)(x,y,t) vortex trajectories (time range Δt=5-40ps\\Delta t=5-40\\text{ ps},step δt=0.5ps\\delta t=0.5\\text{ ps}) are shown in (c) and (d),with the blue spheres representing the Gaussian centroidand the red ones the phase singularity.The angle θ\\theta and distance ddbetween the HV core and the opposite spin centroidare represented in (e) for 3 different powers.Panel (f) is the phase-intensity plot along a vertical cutfor P 2 P_{2} and t=22 ps (arrows follow y),higlighting a π\\pi -jump in the phase between adiacent maxima (i.e., when crossing the dark ring).The orbital-like trajectories suggests the presence of interactions between the vortex of $\\sigma _+$ polaritons and the opposite $\\sigma _-$ density.",
"Such dynamical configuration resembles the metastable rotating vortex state, predicted in [33], [34], supported by a harmonic trap, although this effective potential is dynamically modified by the intra-spin repulsive forces, e.g., by the deformation of the initial Gaussian.",
"Indeed the nonlinearities induce a breaking of radial symmetry, with the formation of circular ripples in the density.",
"The dark ripple shown in Fig.",
"REF (b) relative to the $\\sigma _-$ Gaussian component, presents a $\\pi $ -jump in the phase $\\phi $ , panel (f), which is a possible signature of a self-induced ring dark soliton (RDS), considered its nonlinear drive.",
"It is known that RDS are possible solutions of a 2D fluid with repulsive interactions [35], [31], [36].",
"Yet, the displacement of the singularity (density minimum) with respect to the centroid (opposite spin maximum), is consistent with attractive inter-spin forces.",
"This is the first time that the manifestation of opposite spin interactions in polariton condensates is directly observed through their fluid dynamic effects.",
"In Fig.",
"REF we show the generation of a vortex with winding number $n=1$ for each circular polarisation —i.e., a FV— that can then be detected separately.",
"Panels (a-c) represent the amplitude maps of one population ($\\sigma _{+}$ ) at t= 20 ps with superposition of the vortices positions (trajectories for (a,b), instant positions for (c)) for three increasing pulse powers.",
"The evolution of the primary singularities has been shown using 3D plots, i.e., $(x(t),y(t),t)$ curves, in the panels (d-f) corresponding to (a-c), respectively.",
"In the linear regime, at which the polariton density is low (a,d, and Supporting Movie SM2), the opposite polarisation vortices evolve jointly for the first few picoseconds once the pulse has gone.",
"As the density starts to drop, the vortex cores show an increasing separation in space, panel (g, orange), adopting independent trajectories.",
"This suggests that the FV state is not intrinsically unstable, even though it may undergo splitting supposedly driven by the sample disorder; this is triggered when the density decreases below some critical value.",
"Figure: Density maps and phase singularities uponresonant injection of FV states at different power regimes.",
"(a-c)are the σ + \\sigma _{+} density at t=20pst=20\\text{ ps} with superimposedphase singularities for both polarization, marked by symbols(circle for V, star for AV, colour for spin) (see MoviesSM2–SM4).",
"The trajectories of the primary vortices appear in (d-f)as 3D curves (x,y,t)(x,y,t) (time range Δt=5-26ps\\Delta t=5-26\\text{ ps}, step δt=0.5ps\\delta t=0.5\\text{ ps})(see Movie SM5 for P 1 P_{1}),and the evolution of the inter-core distance is resumed in(g).",
"The final panel (h) shows the proliferation of secondarypairs (at t=30pst=30\\text{ ps}) upon increasing pump power.The used laser powers are P 1-5 =0.17,0.77,1.8,3.1P_{1-5}=0.17,~0.77,~1.8,~3.1~and4.4mW~4.4~\\text{mW},which correspond to an initial excitation of0.2,1.0,1.8,2.20.2, 1.0, 1.8, 2.2 and 2.6·10 6 2.6\\cdot 10^6 total polaritons, respectively.At larger polariton densities, Fig.",
"REF (b,e) and Movie SM3, at which the disorder is expected to be screened out, the twin singularities of the injected FV move together while the fluid is reshaped under the drive of the nonlinear interactions and the increase of radial flow.",
"Here, they also undergo a spiraling similar to the HV case.",
"Interestingly, the twin cores appear to follow the same initial path, see also panel (g, violet), hence indicating the lack of any intrinsic tendency of the FV state to split.",
"This is confirmed by increasing the polariton density further, Fig.",
"REF (c,f) and (g, cyan), where the twin cores remain together for even longer times.",
"Any potential instability of a FV, and the consequent tendency to split into two HV, is not observed here, differently from what observed in [22], where the splitting after non-resonant pumping was due to marked sample inhomogeneities.",
"On the contrary our results show that at high densities, for which the internal currents should prevail, there is a strong inclination for the system to keep the full vortex state together.",
"However, note that the increased density eventually causes circular density ripples, which appear due to nonlinear radial currents and lead to the proliferation of vortex-antivortex (V-AV) pairs in both polarisations.",
"In particular, secondary vortices nucleate in the low density regions of those circular ripples (Fig.",
"REF c,h), which additionally disrupt the original vortex core (see also Movie SM4).",
"Theoretical Modeling In order to get a better understanding of the experimental vortex dynamics and interactions between the fundamental excitations, we have performed numerical simulations.",
"The theoretical analysis performed by Rubo and collaborators in Ref.",
"[14] is based on the minimisation of the total energy for an equilibrium polariton condensate of infinite dimensions, i.e., where the density profile far from the vortex core is homogeneous.",
"This analysis allows to establish a phase diagram for the stability/instability of different vortex excitations.",
"In contrast, here, we study the dynamics of finite size FV and HV states and their stability during the dissipative and nonlinear evolution of interacting spinorial components, by dynamical simulations.",
"We consider a generalised dissipative Gross-Pitaevskii equations for coupled two-component excitons $\\phi _{\\pm }(x,y,t)$ and microcavity photon $\\psi _{\\pm }(x,y,t)$ fields: $i\\hbar \\frac{\\partial \\phi _{\\pm }}{\\partial t} &=\\left(-\\frac{\\hbar ^{2}}{2m_{\\phi }}\\nabla ^{2}-i\\frac{\\hbar }{\\tau _{\\phi }}\\right)\\phi _{\\pm }+\\frac{\\hbar \\Omega _{R}}{2}\\psi _{\\pm }\\nonumber \\\\& + g|\\phi _{\\pm }|^{2}\\phi _{\\pm } + \\alpha |\\phi _{\\mp }|^{2}\\phi _{\\pm }\\\\i\\hbar \\frac{\\partial \\psi _{\\pm }}{\\partial t} &=\\left(-\\frac{\\hbar ^{2}}{2m_{\\psi }}\\nabla ^{2}-i\\frac{\\hbar }{\\tau _{\\psi }}\\right)\\psi _{\\pm }+\\frac{\\hbar \\Omega _{R}}{2}\\phi _{\\pm }\\nonumber \\\\& + D(x,y)\\phi _{\\pm } + \\beta \\left(\\frac{\\partial }{\\partial x}\\pm i\\frac{\\partial }{\\partial y}\\right)^{2}\\psi _{\\mp }+F_{\\pm }\\nonumber $ In order to reproduce the experimental conditions, we introduce a disorder term $D(x,y)$ for the photon field to match the inhomogeneities of the cavity mirror.",
"The potential $D(x,y)$ is a Gaussian correlated potential with an amplitude of strength $50~\\mu eV$ and a $1~\\mu m$ correlation length.",
"Since the effective mass of the excitons, $m_{\\phi }$ , is 4-5 orders of magnitude greater than that of the microcavity photons, $m_{\\psi }$ , we may safely neglect the kinetic energy of the excitons.",
"The parameters in Eq.",
"(1) are fixed so that to reproduce the experimental conditions, with an exciton and photon lifetimes of $\\tau _{\\phi }=1000~\\text{ps}$ and $\\tau _{\\psi }=5~\\text{ps}$ , respectively, a Rabi splitting $\\Omega _{R}=5.4~\\text{meV}$ and the exciton-exciton interactions strength $g=2~\\mu \\text{eV}\\cdot \\mu \\text{m}^{2}$ .",
"We take the strength of the inter-spin exciton interaction to be an order of magnitude weaker than the intra-spin interaction [37], so that $\\alpha =-0.1g$ .",
"The coupling between different polarisations is given by the inter-spin interaction $\\alpha $ and by the TE-TM splitting term $\\beta $ .",
"Following Hivet et al [38], we fix the ratio between the two effective masses $m_{\\psi }^{\\text{TE}}/m_{\\psi }^{\\text{TM}}$ to $0.95$ in order to have an intermediate TE-TM splitting $\\beta =\\frac{\\hbar ^{2}}{4}(\\frac{1}{m_{\\psi }^{\\text{TE}}}-\\frac{1}{m_{\\psi }^{\\text{TM}}})=0.026\\times \\frac{\\hbar ^{2}}{2m_{\\psi }}$ .",
"The initial laser pulse is modelled as a pulsed Laguerre-Gauss $F_{\\pm }$ : $F_{\\pm }(\\mathbf {r})=f_{\\pm }r^{|n_{\\pm }|}e^{-\\frac{1}{2}\\frac{r^{2}}{\\sigma _{\\text{r}}^{2}}}e^{in_{\\pm }\\theta }e^{-\\frac{1}{2}\\frac{\\left(t-t_{\\text{0}}\\right)^{2}}{\\sigma _{t}^{2}}}e^{i(\\mathbf {k}_{\\text{p}}\\cdot \\mathbf {r}-\\omega _{\\text{p}}t)}$ where the winding number of the vortex component in the $\\pm $ polarisation is $n_{\\pm }$ .",
"The strength, $f$ , has been selected so as to replicate the observed total photon output.",
"The $\\sigma _{\\text{r}}$ and $\\sigma _{t}$ parameters were chosen in order to have space width and time duration (FWHM) of the pump $20~\\mu \\text{m}$ and $4~\\text{ps}$ , respectively, in line with the experimental settings.",
"The pump is slowly switched on into the simulation, reaching its maximum at $t_{\\text{0}}=5.5~\\text{ps}$ and cut out completely after $5\\sigma _{t}$ so as to avoid any unintended phase-locking.",
"We follow the dynamics of both full and half vortices shined resonantly with the lower polariton dispersion at $\\mathbf {k}_{\\text{p}}=0$ and $\\omega _{\\text{p}}=-1$ .",
"Figure: Theoretical trajectories of primary singularities forFV state simulated at 3 increasing powers.",
"(a-c) are the 3D(x,y,t)(x,y,t) curves with δt=0.4ps\\delta t=0.4\\text{ ps} stepin a Δt=0-60ps\\Delta t=0-60\\text{ ps} span and theevolution of the inter-core distance is resumed in (d).Our simulations show that only in the presence of the disorder term the imprinted vortex excitations undergo an erratic movement, both in the half and full vortex configurations.",
"In agreement with the experiments the splitting of the FV is observed in the simulations only in the presence of disorder.",
"In Fig.",
"REF (a,b,c) we plot the trajectories for different increasing powers $P_{1-3}$ .",
"The dissociation is seen at earlier times at low initial density, when the sample disorder potential is expected to play a pivotal role.",
"At larger power the disorder and splitting are partially screened out, the main charges move jointly for a longer time.",
"These results are resumed in the panel (d) and are in a good qualitative agreement with the experimental ones of Fig.",
"REF .",
"Simulations without disorder show that charges are dynamically stable, immune to any internal splitting.",
"This holds in our simulations even with artificially enlarged $\\alpha $ , confirming that any dissociation is an external rather than an intrinsic effect, at least during the polariton lifetime.",
"In other terms, even though the thermodynamics would prefer HVs, based upon energy minimization [14], the kinetics are too slow to observe such effect in a real system.",
"Figure: Branching dynamics of a HV polaritoncondensate created at an intermediate power regime (1.8mW\\text{mW}).",
"The four rows (a-d) show frames, taken at t = 8,16, 24 and 32 ps, with densities and vortices in the first columnand associated phase maps for σ - \\sigma _{-}in the second column.",
"The initialcondensate (a, orange due to overlap of red and yellowσ ± \\sigma _{\\pm } intensity scale) undergoes the formation ofconcentric ripples (b-d, see also Movie SM6).",
"Spontaneous fullV-AV formation with quadrupole symmetry for the initially Gaussianpopulation is tracked and represented as (x,y,t)(x,y,t) vortex stringswith 0.5ps0.5\\text{ ps} time stepin a 5-35ps5-35\\text{ ps} time span (e, see Movie SM7).Branching and secondary vortices In the experiments, as already stated, at large densities both the HV and FV develop concentric ripples, and this is causing generation of secondary vortices.",
"For the HV, this effect is firstly seen by increasing the power in the initially vortex-free Gaussian component, where the same amount of total particles are concentrated in a smaller area than in the vortex counterpart.",
"An exemplificative case of this regime is shown in Fig.",
"REF , which reports in the first column the overlapped density maps of the two populations (red and yellow intensity scales) together with the vortices, and in the second column the $\\sigma _{-}$ phase maps.",
"The condensate evolves from the initial time (a), where only the primary core of the HV is present, with the Gaussian developing more marked ripples, generating a first V-AV couple (b) and then a second one (c), which take positions in a 4-fold symmetric structure (see Movie SM6).",
"This effect is not driven by disorder.",
"It is intrinsic and observed in a very large number of realizations and in different polarizations.",
"Generation of secondary V-AV pairs is also seen in simulations, where the disorder term is removed (see Fig REF ), confirming that this effect is not caused by the sample disorder.",
"The branching dynamics and its symmetry can be clearly seen also in the 3D (xyt) trajectories of panel (e) (see also Movie SM7).",
"Only at later time (d), when the density decreases substantially, also the $\\sigma _{+}$ component develops secondary pairs but in an external region where the density drops locally.",
"It is worth noting that at this later stage (d), the primary core of the HV, which was moving around, is seen to merge with a secondary vortex of the opposite polarization (but same winding), thus giving rise to the formation of a FV.",
"Figure: Branching dynamics of a FV polaritoncondensate created at an intermediate power regime (1.8mW\\text{mW}).",
"(a-c) are density frames and vortices taken at t =8, 12 and 24 ps, respectively, while (d-f) are the correspondingphase maps for just one polarisation (σ - \\sigma _{-}).",
"The initialcondensate (a, orange due to overlap of red and yellowσ ± \\sigma _{\\pm }) develops concentric ripples (b-c, see also MovieSM4).",
"Spontaneous full V-AV formation is tracked as (x,y,t)(x,y,t)vortex branches with time step ofδt=0.5ps\\delta t=0.5\\text{ ps}and Δt=6-24ps\\Delta t=6-24\\text{ ps} rangein (g, see MovieSM8), for both the populations.",
"Each secondary HV stay close toits spin counterpart until quite late into the dynamics.The generation of secondary vortices is seen also in case of the FV, as shown in Fig.",
"REF , at $P=1.8~\\text{mW}$ .",
"The panels (a-c) represent the joint population and vortices at different time frames, while the corresponding phase maps (d-f) are reported only for one polarization.",
"We observe that while the primary FV (a) rotates, it undergoes a displacement a moment before the creation of the first V-AV pair, which is followed by a second one, (b) and (c), respectively.",
"The two secondary V-AV pairs are created in succession, and jointly between the two $\\sigma $ states: in other terms, the secondary topological charges are created as full vortex and anti-vortex.",
"The $\\sigma _{+}$ and $\\sigma _{-}$ cores of the primary and the first secondary FVs move together in a FV configuration for quite a long time.",
"The branching and its partial symmetry, can be seen also in the phase maps (d-f), and in the branch structure of Fig.",
"REF (g) (see Movie SM8), with the $xyt$ trajectories of the vortices.",
"At later times the central region, initially dark, is partially filled with fluid and some degree of asymmetry is present between the two polariton distributions.",
"We found that at different densities, localized transient structures with 3, 4 or 6-fold symmetries may arise too (see also [20]).",
"Figure: Theoretical density maps and phasesingularities in the case of FV without the disorderpotential.",
"(a-c) Each row corresponds to a different time, (a)t=8pst=8\\text{ ps}, (b) 24ps24\\text{ ps} and (c) 40ps40\\text{ ps}.",
"Leftand right columns represent the σ + \\sigma _{+} and σ - \\sigma _{-}density, respectively, with superimposed their phasesingularities, marked by symbols (circle for V, star for AV,colour for spin, see Movie SM9).In the simulations we see the emergence of density ripples (radial symmetry breaking), as observed in the experiment, above certain density (pump power) threshold with or without the disorder.",
"It is in the very bottom of these ripples, where the density is almost zero, that spontaneous V-AV pairs nucleate.",
"Figure REF shows the theoretical evolution of the density maps for the two components of a FV, on each column, respectively (see also Movie SM9).",
"The main difference compared with experiments is that here the secondary couples are generated in different positions for the two polarisations.",
"Yet, they keep rotating along a direction depending on their winding, and not on their spins.",
"We have reasons to believe that the direction of circulation could be associated to the winding sign and the direction of the fluid reshaping (i.e., contracting or expanding), but the study of such aspect is well beyond the scope of the present work.",
"Conclusions To conclude, we have investigated the dynamics and branching of half and full vortices resonantly injected in an out-of-equilibrium polariton quantum fluid.",
"The dynamics of these topological defects is ruled by the interplay between the non-linearity and the disorder landscape.",
"Our main conclusion is that, surprisingly, both FV and HV states are intrinsically dynamically stable, i.e., the topological charges in the two spin components do not split because of intrinsic energy considerations during the lifetime of the polaritons, nor the singularity of a half-vortex is seen to attract an opposite spin counterpart.",
"The splitting effects, we observe, can be attributed to the fact that at low density (long time) the fluid streamlines are affected more by the sample landscape, with disorder guiding the displacement of the vortices, and eventually separating the cores when a symmetry breaking term such as anisotropic or TE-TM splitting is at action.",
"At intermediate density regimes, when sample inhomogeneities are screened out and nonlinear turbulence is moderate, the charges stay together for longer times.",
"It is at even larger densities, when the main charges stay together up to tens of ps, that they are also seen to move in a marked precessing trajectory, both for the HV and FV states.",
"Here, the nonlinearities drive radial flows with the reshaping of the fluid into circular ripples of alternating high and low density regions, where secondary vortices nucleate.",
"This nucleation is systematic and distinct from the proliferation of vortices at very low densities, which are pinned by disorder, as demonstrated by the theoretical simulations performed in an homogeneous landscape —the secondary charges nucleate in pairs of opposite winding in each of the two spin populations, and their evolution is seen as quasi-ordered branching of 3D (2D+t) singularity trees.",
"Our observations suggest that quantum phase-singularities might be seen as an analogue of fundamental particles, whose features can span from quantized events such as pair creation and recombination to vortex strings.",
"Moreover, with both topological states seemingly stable during the typical polariton lifetimes, an interesting question left to be addressed is which excitations are relevant for the Kosterlitz-Thouless-type transition in these systems.",
"Acknowledgments We acknowledge Giovanni Lerario for fruitful discussions, R. Houdré for the growth of the microcavity sample and the project ERC POLAFLOW for financial support.",
"MHS acknowledges support from EPSRC (EP/I028900/2 and EP/K003623/2).",
"FMM acknowledges financial support from the Ministerio de Economía y Competitividad (MINECO), projects No.",
"MAT2011-22997 and No. MAT2014-53119-C2-1-R.",
"Supplemental Information Supporting Movies are available online at this URL ."
]
] | 1403.0487 |
[
[
"Massive neutron stars with hyperonic core : a case study with the IUFSU\n model"
],
[
"Abstract The recent discoveries of massive neutron stars, such as PSR J$0348+0432$ and PSR J$1614-2230$, have raised questions about the existence of exotic matter such as hyperons in the neutron star core.",
"The validity of many established equations of states (EoS's) like the GM1 and FSUGold are also questioned.",
"We investigate the existence of hyperonic matter in the central regions of massive neutron stars using Relativistic Mean Field (RMF) theory with the recently proposed IUFSU model.",
"The IUFSU model is extended by including hyperons to study the neutron star in $\\beta$ equilibrium.",
"The effect of different hyperonic potentials, namely $\\Sigma$ and $\\Xi$ potentials, on the EoS and hence the maximum mass of neutron stars has been studied.",
"We have also considered the effect of stellar rotation since the observed massive stars are pulsars.",
"It has been found that a maximum mass of $1.93M_{\\odot}$, which is within the 3$\\sigma$ limit of the observed mass of PSR J$0348+0432$, can be obtained for rotating stars, with certain choices of the hyperonic potentials.",
"The said star contains a fair amount of hyperons near the core."
],
[
"Introduction",
"The recent discoveries of the massive neutron stars PSR J$0348+0432$ [1] and PSR J$1614-2230$ [2] have brought new challenges for theories of dense matter beyond the nuclear saturation density.",
"Recently the radio timing measurements of the pulsar PSR J$0348+0432$ and its white dwarf companion have confirmed the mass of the pulsar to be in the range of $1.97 - 2.05$ M$_{\\odot }$ at $68.27\\%$ or $1.90 - 2.18$ M$_{\\odot }$ at $99.73\\%$ confidence [1].",
"This is only the second neutron star(NS) with a precisely determined mass around 2M$_{\\odot }$ , after PSR J$1614-2230$ and has a 3$\\sigma $ lower mass limit $0.05$ M$_{\\odot }$ higher than the latter.",
"It therefore provides the tightest reliable lower bound on the maximum mass of neutron stars.",
"Compact stars provide the perfect astrophysical environment for testing theories of cold and dense matter.",
"Densities at the core of neutron stars can reach values of several times of $10^{15} gm \\,\\, cm^{-3}$ .",
"At such high densities, the energies of the particles are high enough to favour the appearance of exotic particles in the core.",
"Since the lifetime of neutron stars are much greater than those associated with the weak interaction, strangeness conservation can be violated in the core due to the weak interaction.",
"This would result in the appearance of strange particles such as hyperons.",
"The appearance of such particles produces new degrees of freedom, which results in a softer equation of state (EoS) in the neutron star interior.",
"The observable properties of compact stars depend crucially on the EoS.",
"According to the existing models of dense matter the presence of strangeness in the neutron star interior leads to a considerable softening of the EoS, resulting in a reduction of the maximum mass of the neutron star [3], [4], [5], [6].",
"Therefore many existing theories involving hyperons cannot explain the large pulsar masses [7].",
"Most relativistic models obtain maximum neutron star masses in the range $1.4-1.8M_{\\odot }$ [8], [9], [10], [11], [12], [13], [14], [15], when hyperons are included.",
"Some authors have tackled this problem by including a strong vector repulsion in the strange sector or by pushing the threshold for the appearance of hyperons to higher densities [15], [16], [17], [18], [19], [20], [21], [22].",
"In several studies the maximum neutron star masses were generally found to be lower than $1.6M_\\odot $ [4], [5], [6], [23], [24], [25], [26], [27] which is in contradiction with observed pulsar masses.",
"However, neutron stars with maximum mass larger than $2M_{\\odot }$ have been obtained theoretically.",
"Bednarek et al.",
"[28] achieved a stiffening of the EoS by using a non-linear relativistic mean field (RMF) model with quartic terms involving the strange vector meson.",
"Lastowiecki et al.",
"[29] obtained massive stars including a quark matter core.",
"Taurines et al.",
"[30] achieved large neutron star masses including hyperons by considering a model with density dependent coupling constants.",
"The coupling constants were varied nonlinearly with the scalar field.",
"Bonanno and Sedrakian [31] also modeled massive neutron stars including hyperons and quark core using a fairly stiff EoS and vector repulsion among quarks.",
"Authors in ref.",
"[32] incorporated higher order couplings in the RMF theory in addition to kaonic interactions to obtain the maximum neutron star mass.",
"Agrawal et al.",
"[33] have optimized the parameters of the extended RMF model using a selected set of global observables which includes binding energies and charge radii for nuclei along several isotopic and isotonic chains and the iso-scalar giant monopole resonance energies for the $^{90}$ Zr and $^{208}$ Pb nuclei.",
"Weissenborn et al.",
"[34] investigated the vector meson-hyperon coupling, going from SU(6) quark model to a broader SU(3), and concluded that the maximum mass of a neutron star decreases linearly with the strangeness content of the neutron star core independent of the nuclear EoS.",
"On the other hand, H. Dapo et al.",
"[6] found that for several different bare hyperon-nucleon potentials and a wide range of nuclear matter parameters the hyperons in neutron stars are always present.",
"The parameters of the RMF model are fitted to the saturation properties of the infinite nuclear matter and/or the properties of finite nuclei.",
"As a result extrapolation to higher densities and asymmetry involve uncertainties.",
"Three of these properties of the infinite nuclear matter are more precisely known: (a) the saturation density, (b) the binding energy and (c) the asymmetry energy, compared to the remaining ones - the effective nucleon mass and the compression modulus of the nuclear matter.",
"The uncertainty in the dense matter EoS is basically related to the uncertainty in these two saturation properties.",
"It has been seen that to reproduce the giant monopole resonance (GMR) in $^{208}$ Pb, accurately fitted non-relativistic and relativistic models predict compression modulus in the symmetric nuclear matter ($K$ ) that differ by about $25\\%$ .",
"The reason for this discrepancy being the density dependence of the symmetry energy.",
"Moreover, the alluded correlation between $K$ and the density dependence of the symmetry energy results in an underestimation of the frequency of oscillations of neutrons against protons, the so-called isovector giant dipole resonance (IVGDR) in $^{208}$ Pb.",
"FSUGold is a recently proposed accurately calibrated relativistic parameterization.",
"It simultaneously describes the GMR in $^{90}$ Zr and $^{208}$ Pb and the IVGDR in $^{208}$ Pb without compromising the success in reproducing the ground-state observables [35].",
"The main virtue of this parameterization is the softening of both the EoS of symmetric nuclear matter and the symmetry energy.",
"This softening appears to be required for an accurate description of different collective modes having different neutron-to-proton ratios.",
"As a result, the FSUGold effective interaction predicts neutron star radii that are too large and a maximum stellar mass that is too small [36].",
"The Indiana University-Florida State University (IUFSU) interaction, is a new relativistic parameter set, derived from FSUGold.",
"It is simultaneously constrained by the properties of finite nuclei, their collective excitations and the neutron star properties by adjusting two of the parameters of the theory - the neutron skin thickness of $^{208}Pb$ and the maximum neutron star mass [37].",
"As a result the new effective interaction softens the EoS at intermediate densities and stiffens the EoS at high density.",
"As it stands now, the new IUFSU interaction reproduces the binding energies and charge radii of closed-shell nuclei, various nuclear giant (monopole and dipole) resonances, the low-density behavior of pure neutron matter, the high-density behavior of the symmetric nuclear matter and the mass-radius relationship of neutron stars.",
"Whether this new EoS can accommodate the hyperons inside the compact stars, with the severe constraints imposed by the recent observations of $\\sim 2M_\\odot $ pulsars, needs to be explored.",
"In this work we plan to make a detailed study of such a possibility.",
"For this purpose we have extended the IUFSU interaction by including the full baryon octet.",
"A new EoS is constructed to investigate the neutron star properties with hyperons.",
"Table: Parameter sets for the two models discussed in the text.The nucleon mass and the meson masses are kept fixed at m n m_n = 939 MeV, m σ m_\\sigma = 491.5 MeV,m ω m_\\omega = 782.5 MeV, m ρ m_\\rho = 763 MeV and m φ m_\\phi = 1020 MeV in both the models.The paper is organized as follows.",
"In section 2, we briefly discuss the model used and the resulting EoS.",
"In the next section we use this EoS to look at static and rotating star properties.",
"We give a brief summary in section 4."
],
[
" IUFSU with hyperons",
"One of the possible approaches to describe neutron star matter is to adopt an RMF model subject to $\\beta $ equilibrium and charge neutrality.",
"For our investigation of nucleons and hyperons in the compact star matter we choose the full standard baryon octet as well as electrons and muons.",
"Contribution from neutrinos are not taken into account assuming that they can escape freely from the system.",
"In this model, baryon-baryon interaction is mediated by the exchange of scalar ($\\sigma $ ), vector ($\\omega $ ), isovector ($\\rho $ ) and the strange vector ($\\phi $ ) mesons.",
"The Lagrangian density we consider is given by [37] $\\mathcal {L} &=& \\sum _{B}\\bar{\\psi }_{B}[i\\gamma ^{\\mu }\\partial _{\\mu }- m_{B}+g_{\\sigma B}\\sigma - g_{\\omega B}\\gamma ^{\\mu }\\omega _{\\mu } - g_{\\phi B}\\gamma ^{\\mu }\\phi _{\\mu }- \\frac{g_{\\rho B}}{2}\\gamma ^{\\mu }\\vec{\\tau }\\cdot \\vec{\\rho }^{\\mu }]{\\psi }_{B} +\\frac{1}{2}\\partial _{\\mu }\\sigma \\partial ^{\\mu }\\sigma - \\frac{1}{2} m_\\sigma ^2\\sigma ^2 \\nonumber \\\\&& - \\frac{\\kappa }{3!",
"}(g_{\\sigma N}\\sigma )^3 -\\frac{\\lambda }{4!",
"}(g_{\\sigma N}\\sigma )^4 - \\frac{1}{4}F_{\\mu \\nu }F^{\\mu \\nu } +\\frac{1}{2}m_\\omega ^2\\omega _\\mu \\omega ^\\mu + \\frac{\\zeta }{4!",
"}(g^{2}_{\\omega N}\\omega _\\mu \\omega ^\\mu )^2+\\frac{1}{2}m_\\rho ^2\\vec{\\rho }_{\\mu }\\cdot \\vec{\\rho }^{\\mu } - \\frac{1}{4}\\vec{G}_{\\mu \\nu }\\vec{G}^{\\mu \\nu }\\nonumber \\\\&&+ \\Lambda _{v}(g^{2}_{\\rho N}\\vec{\\rho }_{\\mu }\\cdot \\vec{\\rho }^{\\mu })(g^{2}_{\\omega N}\\omega _\\mu \\omega ^\\mu )+ \\frac{1}{2}m_\\phi ^2\\phi _\\mu \\phi ^\\mu -\\frac{1}{4}H_{\\mu \\nu }H^{\\mu \\nu }+\\sum _{l}\\bar{\\psi }_{l}[i\\gamma ^{\\mu }\\partial _{\\mu } - m_{l}]{\\psi }_{l}$ where the symbol B stands for the baryon octet ($p$ , $n$ , $\\Lambda $ , $\\Sigma ^{+}$ , $\\Sigma ^{0}$ , $\\Sigma ^{-}$ , $\\Xi ^{-}$ , $\\Xi ^{0}$ ) and $l$ represents $e^{-}$ and $\\mu ^{-}$ .",
"The masses $m_B$ , $m_\\sigma $ , $m_\\omega $ , $m_\\rho $ and $m_\\phi $ are respectively for baryon, $\\sigma $ , $\\omega $ , $\\rho $ and $\\phi $ mesons.",
"The antisymmetric tensors of vector mesons take the forms ${F}_{\\mu \\nu }$ = $\\partial _{\\mu }\\omega _{\\nu } - \\partial _{\\nu }\\omega _{\\mu }$ , ${G}_{\\mu \\nu }$ = $\\partial _{\\mu }\\vec{\\rho }_{\\nu } - \\partial _{\\nu }\\vec{\\rho }_{\\mu } + g[\\vec{\\rho }_{\\mu },{\\vec{\\rho }}_{\\nu }]$ and ${H}_{\\mu \\nu }$ = $\\partial _{\\mu }{\\phi }_{\\nu } - \\partial _{\\nu }{\\phi }_{\\mu }$ .",
"The isoscalar meson self-interactions (via $\\kappa $ , $\\lambda $ and $\\zeta $ terms) are necessary for the appropriate EoS of the symmetric nuclear matter [38].",
"The new additional isoscalar-isovector coupling ($\\Lambda _v$ ) term is used to modify the density dependence of the symmetry energy and the neutron-skin thickness of heavy nuclei [36], [37].",
"The meson-baryon coupling constants are given by $g_{\\sigma B}$ , $g_{\\omega B}$ , $g_{\\rho B}$ and $g_{\\phi B}$ .",
"All the nucleon-meson parameters used in this work are shown in Table REF .",
"The saturation properties of the symmetric nuclear matter produced by IUFSU are: saturation density $n_0=0.155$ $fm^{-3}$ , binding energy per nucleon $\\varepsilon _0= -16.40$ MeV and compression modulus $K = 231.2$ MeV.",
"The hyperon-meson couplings are taken from the SU(6) quark model [39], [40] as, $g_{\\rho \\Lambda }$ = 0, $g_{\\rho \\Sigma }$ = $2g_{\\rho \\Xi }$ = $2g_{\\rho N}$ $g_{\\omega \\Lambda }$ = $g_{\\omega \\Sigma }$ = $2g_{\\omega \\Xi }$ = $\\frac{2}{3}g_{\\omega N}$ $2g_{\\phi \\Lambda }$ = $2g_{\\phi \\Sigma }$ = $g_{\\phi \\Xi }$ = $\\frac{-2\\sqrt{2}}{3} g_{\\omega N}$ The scalar couplings are determined by fitting the hyperonic potential, $U^{(N)}_Y = g_{\\omega Y}\\omega _0 + g_{\\sigma Y}\\sigma _0$ where Y stands for the hyperon and $\\sigma _0$ , $\\omega _0$ are the values of the scalar and vector meson fields at saturation density [9].",
"The values of $U^{(N)}_Y$ are taken from the available hypernuclear data.",
"The best known hyperonic potential is that of $\\Lambda $ , having a value of about $U^{(N)}_\\Lambda $ = -30 MeV [41].",
"In case of $\\Sigma $ and $\\Xi $ hyperons, the potential depths are not as clearly known as in the case of $\\Lambda $ .",
"However, analyses of laboratory experiments indicate that at nuclear densities the $\\Lambda $ -nucleon potential is attractive but the $\\Sigma ^-$ -nucleon potential is repulsive [42].",
"Therefore, we have varied both $U^{(N)}_\\Sigma $ and $U^{(N)}_\\Xi $ in the range of -40 MeV to +40 MeV to investigate the properties of neutron star matter.",
"For neutron star matter, with baryons and charged leptons, the $\\beta $ -equilibrium conditions are guaranteed with the following relations between chemical potentials for different particles: $\\mu _p &=& \\mu _{\\Sigma ^{+}} = \\mu _n - \\mu _e \\nonumber \\\\\\mu _{\\Lambda } &=& \\mu _{\\Sigma ^{0}} = \\mu _{\\Xi ^{0}} = \\mu _n \\nonumber \\\\\\mu _{\\Sigma ^{-}} &=& \\mu _{\\Xi ^{-}} = \\mu _n+\\mu _e \\nonumber \\\\\\mu _{\\mu } &=& \\mu _e$ and the charge neutrality condition is fulfilled by $n_p + n_{\\Sigma ^{+}} = n_e+n_{\\mu ^{-}}+n_{\\Sigma ^{-}}+n_{\\Xi ^{-}}$ where $n_i$ is the number density of the i'th particle.",
"The effective chemical potentials of baryons and leptons can be given by $\\mu _B = \\sqrt{{k_{F}^{B}}^2+{m_{B}^{\\ast ^2}}}+g_{\\omega B}\\omega +g_{\\rho B}\\tau _{3B}\\rho $ $\\mu _l = \\sqrt{{K_F^{l}}^2+m_l^2}$ where $m_{B}^{\\ast } = m_B-g_{\\sigma B}\\sigma $ is the baryon effective mass and $K_F^l$ is the Fermi momentum of the lepton (e, $\\mu $ ).",
"The EoS of neutron star matter can be given by, Figure: (color online) a) EoS obtained with varying U Σ (N) U^{(N)}_\\Sigma at fixed U Ξ (N) U^{(N)}_\\Xi .The upper branch shows the EoS for a system containing nucleons, leptons and all the nonstrange mesons.",
"The middle branch shows the EoS for a system containing the whole baryon octet, the leptonsand σ\\sigma , ω\\omega , ρ\\rho and φ\\phi mesons.",
"The lower branch shows the EoSfor the particles contained in the middle branch except φ\\phi .b) EoS obtained with varying U Ξ (N) U^{(N)}_\\Xi at fixed U Σ (N) U^{(N)}_\\Sigma .",
"The compositionsof the upper, middle and lower branches are same as those of a) respectively.${\\varepsilon } &=& \\frac{1}{2}m_\\sigma ^2 \\sigma ^2+ \\frac{\\kappa }{6} g_{\\sigma N}^3 \\sigma ^3 + \\frac{\\lambda }{24} g_{\\sigma N}^4 \\sigma ^4 + \\frac{1}{2} m_\\omega ^2 \\omega ^2+ \\frac{\\zeta }{8}g_{\\omega N}^4 \\omega ^4+ \\frac{1}{2} m_\\rho ^2 \\rho ^2 + 3\\Lambda _v g_{\\rho N}^2 g_{\\omega N}^2{\\omega }^2{\\rho }^2 \\nonumber \\\\&& + \\frac{1}{2}m_\\phi ^2 \\phi ^2 + \\sum _B\\frac{\\gamma _B}{(2\\pi )^3}\\int _0^{k_{F}^B} \\sqrt{k^2+m^{* 2}_B} \\ d^3 k+ \\frac{1}{\\pi ^2}\\sum _l\\int _0^{K_F^l} \\sqrt{k^2+m^2_l} \\ k^2 dk$ $P &=& - \\frac{1}{2}m_\\sigma ^2 \\sigma ^2- \\frac{\\kappa }{6} g_{\\sigma N}^3 \\sigma ^3 - \\frac{\\lambda }{24} g_{\\sigma N}^4 \\sigma ^4 + \\frac{1}{2} m_\\omega ^2 \\omega ^2 + \\frac{\\zeta }{24}g_{\\omega N}^4 \\omega ^4 + \\Lambda _v g_{\\rho N}^2 g_{\\omega N}^2{\\omega }^2{\\rho }^2+ \\frac{1}{2} m_\\rho ^2 \\rho ^2 \\nonumber \\\\&& + \\frac{1}{2}m_\\phi ^2 \\phi ^2 + \\frac{1}{3}\\sum _B \\frac{\\gamma _B}{(2\\pi )^3}\\int _0^{k_F^B}\\frac{k^2 \\ d^3 k}{(k^2+m^{* 2}_B)^{1/2}}+ \\frac{1}{3} \\sum _{l} \\frac{1}{\\pi ^2}\\int _0^{K_F^l}\\frac{k^4 \\ dk}{(k^2+m^2_l)^{1/2}}~\\nonumber \\\\$ where $\\varepsilon $ and $P$ stand for energy density and pressure respectively and $\\gamma _B$ is the baryon spin-isospin degeneracy factor.",
"Figure: (color online) Particle fractions for different Σ\\Sigma potential depths: a)for “σωρ\\sigma \\omega \\rho ”with U Σ (N) =-30U^{(N)}_\\Sigma = -30 MeV, b) for “σωρ\\sigma \\omega \\rho ”with U Σ (N) =+30U^{(N)}_\\Sigma = +30 MeV, c) for “σωρφ\\sigma \\omega \\rho \\phi ”with U Σ (N) =-30U^{(N)}_\\Sigma = -30 MeV, d) for “σωρφ\\sigma \\omega \\rho \\phi ”with U Σ (N) =+30U^{(N)}_\\Sigma = +30 MeV.",
"U Ξ (N) U^{(N)}_\\Xi is fixed at -18 MeV in each case.In fig.",
"REF we plot the EoS for different values of the hyperonic potentials.",
"The upper branch is for the usual nuclear matter which does not contain any strange particle.",
"The middle and lower branches are for full baryon octet, leptons and $\\sigma $ , $\\omega $ , $\\rho $ mesons.",
"In addition, the middle branch contains the $\\phi $ meson.",
"In the left panel, i.e.",
"in fig.",
"REF a, we keep $U^{(N)}_\\Xi $ fixed at -18 MeV, this value is generally adopted from hypernuclear experimental data [43].",
"For the middle and lower branches we vary the $\\Sigma $ potential from -40 MeV to +40 MeV in steps of 20 MeV.",
"The lower branch shows that for an attractive $\\Sigma $ potential the EoS gets stiffer as $U_{\\Sigma }^{(N)}$ increases.",
"However as $U_{\\Sigma }^{(N)}$ becomes positive the EoS seems to become independent of $U_{\\Sigma }^{(N)}$ .",
"We see from fig.",
"REF a that for $U_{\\Sigma }^{(N)} > 0$ MeV the EoS remains identical to that for $U_{\\Sigma }^{(N)} = 0$ MeV.",
"However, once we add $\\phi $ meson to the system, the EoS continues to get stiffer as $U_{\\Sigma }^{(N)}$ moves to more positive side (middle branch of fig.",
"REF a).",
"Figure: (color online) Particle fractions for different Ξ\\Xi potential depths: a) for “σωρ\\sigma \\omega \\rho ”with U Ξ (N) =-30U^{(N)}_\\Xi = -30 MeV, b) for “σωρ\\sigma \\omega \\rho ”with U Ξ (N) =+30U^{(N)}_\\Xi = +30 MeV, c) for “σωρφ\\sigma \\omega \\rho \\phi ”with U Ξ (N) =-30U^{(N)}_\\Xi = -30 MeV, d) for “σωρφ\\sigma \\omega \\rho \\phi ”with U Ξ (N) =+30U^{(N)}_\\Xi = +30 MeV.",
"U Σ (N) U^{(N)}_\\Sigma is fixed at +30 MeV in each case.We then fix $U^{(N)}_\\Sigma $ and vary $U^{(N)}_\\Xi $ .",
"This is represented in fig.",
"REF b, where we have fixed the value of $U^{(N)}_\\Sigma = +30$ MeV (adopted from hypernuclear experimental data [43]).",
"We vary $U^{(N)}_\\Xi $ from -40 MeV to +40 MeV.",
"We see that for the lower branch, i.e the case without the $\\phi $ meson, the EoS gets stiffer with the increase in $\\Xi $ potential up to $U^{(N)}_\\Xi = 0$ MeV.",
"However, for positive values of $U^{(N)}_\\Xi $ the EoS remains unchanged.",
"Adding an extra repulsion to the system by including the $\\phi $ meson changes the scenario altogether.",
"The EoS becomes totally independent of the $\\Xi $ potential (middle branch of fig.",
"REF b).",
"From figures 1a and 1b one can generally conclude that the inclusion of $\\phi $ meson makes the EoS stiffer, however, hyperonic EoS is much softer than the usual nuclear matter EoS.",
"Figure: (color online) Mass-radius curves for static star fixing the a) Ξ\\Xi potential depthat U Ξ (N) =+40U^{(N)}_\\Xi = +40 MeV and varying the U Σ (N) U^{(N)}_\\Sigma .b) Σ\\Sigma potential depthat U Σ (N) =+40U^{(N)}_\\Sigma = +40 MeV and varying the U Ξ (N) U^{(N)}_\\Xi .",
"The uppermost curve in each case corresponds to the pure nuclear matter.In fig.",
"REF we have plotted the particle fractions for an attractive $\\Sigma $ potential $U^{(N)}_\\Sigma = -30$ MeV and a repulsive potential $U^{(N)}_\\Sigma = +30$ MeV keeping $U^{(N)}_\\Xi $ fixed at -18 MeV, with and without $\\phi $ in each case.",
"From fig.",
"REF a, when $\\phi $ is not present, we see that all the hyperons contribute to the particle fractions for an attractive $\\Sigma $ potential whereas for repulsive $U^{(N)}_\\Sigma $ there is no $\\Sigma $ present in the matter (fig.",
"REF b).",
"The appearance of $\\Lambda $ is also pushed to higher density compared to the case of an attractive potential.",
"When $\\phi $ is included in the system $\\Sigma ^0$ and $\\Sigma ^-$ appear with $\\Lambda $ for $U^{(N)}_\\Sigma = -30$ MeV (fig.",
"REF c).",
"However, for $U^{(N)}_\\Sigma = +30$ MeV (fig.",
"REF d), the threshold of $\\Sigma ^-$ is pushed to higher density compared to the case of $U^{(N)}_\\Sigma = -30$ MeV, $\\Sigma ^0$ disappears and $\\Xi ^-$ appears in the system.",
"We also note that in the case of attractive $\\Sigma $ potential, $\\Sigma ^-$ is always the first hyperon to appear in the system.",
"For repulsive $U^{(N)}_\\Sigma $ , $\\Xi ^-$ appears before others in the “$\\sigma \\omega \\rho $ ” case and $\\Lambda $ is the the first hyperon to appear in case of “$\\sigma \\omega \\rho \\phi $ ”.",
"From fig.",
"REF we see that for negative values of $U^{(N)}_\\Sigma $ , the $\\Sigma $ 's are bound in matter and the effective mesonic interaction would be more attractive as the potential gets deeper.",
"As a result, the EoS gets softer with more attractive $U^{(N)}_\\Sigma $ (see fig.",
"REF a).",
"For $U^{(N)}_\\Sigma \\ge 0$ , $\\Sigma $ 's are no longer bound to matter and the effective mesonic interaction becomes more and more repulsive with increasing $U^{(N)}_\\Sigma $ .",
"This should, in principle, stiffen the EoS.",
"However, for the “$\\sigma \\omega \\rho $ ” case, up to neutron star densities, i.e about $n_B \\lesssim (4-7)n_0$ , $\\Sigma $ 's are not present in the matter when the potential is repulsive and hence the EoS up to these densities becomes insensitive to $U^{(N)}_\\Sigma $ .",
"Figure: (color online) Mass-radius curves for rotating stars for two cases: a) U Ξ (N) =+40U^{(N)}_\\Xi = +40 MeVand -40MeV≤U Σ (N) ≤+40-40 MeV \\le U^{(N)}_\\Sigma \\le +40 MeV and b) U Σ (N) =+40U^{(N)}_\\Sigma = +40 MeVand -40MeV≤U Ξ (N) ≤+40-40 MeV \\le U^{(N)}_\\Xi \\le +40 MeV.",
"The uppermost curve in each case corresponds to the pure nuclear matter.In fig.",
"REF the particle fractions are plotted for an attractive $\\Xi $ potential $U^{(N)}_\\Xi = -30$ MeV and a repulsive potential $U^{(N)}_\\Xi = +30$ MeV keeping $U^{(N)}_\\Sigma $ fixed at +30 MeV.",
"We see that in the first case i.e.",
"when $\\phi $ is not present and the potential is attractive (fig.",
"REF a), all the hyperons except $\\Sigma $ 's are present in the system and the $\\Lambda $ hyperon dominates.",
"When the $\\Xi $ potential becomes positive (fig.",
"REF b) $\\Xi ^0$ disappears and the threshold for appearance of $\\Xi ^-$ shifts to much higher density.",
"However $\\Sigma ^-$ is present in matter in this potential and it appears before $\\Xi ^-$ .",
"When $\\phi $ is introduced in the system, for an attractive $\\Xi $ potential (fig.",
"REF c), again $\\Sigma ^-$ and $\\Xi ^-$ are present along with $\\Lambda $ .",
"However, the difference from fig.",
"REF b i.e “$\\sigma \\omega \\rho $ ” case and $U^{(N)}_\\Xi \\ge $ 0 is that, here $\\Xi ^-$ appears much before $\\Sigma ^-$ .",
"In the last case (fig.",
"REF d), we see that as a result of the combined effects of inclusion of $\\phi $ and repulsive potentials, only the $\\Lambda $ and $\\Sigma ^-$ are present in the system.",
"From both figures REF and REF , we see that, inclusion of $\\phi $ meson decreases the density of hyperons.",
"Since $\\phi $ is a strange particle, further strangeness is suppressed and as a result the hyperon densities are reduced compared to the “$\\sigma \\omega \\rho $ ” case."
],
[
"static and rotating stars",
"In this section we are going to discuss the properties of static and rotating axisymmetric stars using the EoS which we have studied in the last section.",
"The EoS without $\\phi $ meson is softer compared to that with $\\phi $ meson.",
"So we do not discuss the EoS without $\\phi $ as it results in less maximum mass.",
"The stationary, axisymmetric space-time used to model the compact stars are defined through the metric $ds^2 = -e^{\\gamma +\\rho } dt^2 + e^ {2\\alpha }(dr^2+r^2d\\theta ^2)\\nonumber \\\\+ e^{\\gamma -\\rho }r^2 sin^2{\\theta }(d\\phi -\\omega dt)^2$ where $\\alpha $ , $\\gamma $ , $\\rho $ and $\\omega $ are the gravitational potentials which depend on r and $\\theta $ only.",
"In this work we adopt the procedure of Komatsu et al.",
"[44] to look into the observable properties of static and rotating stars.",
"Einstein's equations for the three gravitational potentials $\\gamma $ , $\\rho $ and $\\omega $ can be solved using Green's function technique.",
"The fourth potential $\\alpha $ can be determined using these three potentials.",
"Once these potentials are determined one can calculate all the observable quantities using those.",
"The solution of the potentials and hence the determination of physical quantities is numerically quite an involved process.",
"For this purpose the “rns” code [45] is used in this work.",
"This code, developed by Stergoilas, is very efficient in calculating the rotating star observables.",
"We discuss the properties of static stars first.",
"In fig.",
"REF we have plotted the mass-radius curves of static stars using the EoS with “$\\sigma \\omega \\rho \\phi $ ”.",
"A plot for the pure nuclear matter case is also given for comparison (uppermost curve of both the panels).",
"The maximum mass of pure nuclear matter star in the static case is $1.92 M_\\odot $ with a radius of $11.24$ km.",
"We have found that the mass of hyperonic star becomes maximum for $U_\\Sigma ^{N} = +40$ MeV and $U_\\Xi ^{N} \\ge 0$ MeV.",
"Hence in fig.",
"REF and fig.",
"REF we have shown the effect of these potentials on the maximum mass of neutron stars by fixing one of the potentials at +40 MeV and varying the other.",
"The left panel, i.e.",
"fig.",
"REF a, corresponds to $U_\\Xi = +40$ MeV and $U_{\\Sigma }$ varying from -40 MeV to +40 MeV.",
"In the right panel, i.e.",
"in fig.",
"REF b, it is the other way round.",
"From fig.",
"REF a one can see that the maximum mass of the star increases with $U_\\Sigma ^{(N)}$ .",
"For $U_\\Sigma ^{(N)} = +40$ MeV the maximum mass is $1.62 M_\\odot $ with a radius of $10.82$ km.",
"The central energy density of such a star is $\\epsilon _c = 2.46 \\times 10^{15} gm \\,\\, cm^{-3}$ .",
"This is a reflection of the EoS shown in fig.",
"REF a, which shows that the EoS becomes stiffer with increase in $U_\\Sigma ^{(N)}$ .",
"However, as seen from fig.",
"REF b, the maximum mass of static stars is insensitive to $U_\\Xi ^{(N)}$ , which should be obvious from fig.",
"REF b as the EoS is independent of the cascade potential.",
"Furthermore, from fig.",
"REF d one can see that there is no cascade present in the medium.",
"So the insensitivity of the EoS and hence the maximum mass, towards the cascade potential is expected.",
"One should note that the maximum mass we obtain for the static stars is less than the observed mass of PSR J$0348+0432$ .",
"So the static stars with hyperons in the IUFSU parameter set can not incorporate a maximum mass $\\sim 2M_\\odot $ .",
"This result is consistent with the findings in Ref. [46].",
"However, since both of the observed $\\sim 2M_\\odot $ stars are pulsars, it would be a better idea to compare the observations with results from the rotating stars, which we do in the next part.",
"Figure: (color online) Particle densities varying with radius along the equator.The potential depths for which particle densities are plotted are U Ξ (N) =0U^{(N)}_\\Xi = 0 and U Σ (N) =+40U^{(N)}_\\Sigma = +40 MeV.In fig.",
"REF we plot the mass-radius curves for stars rotating with Keplerian velocities, for two cases.",
"In fig.",
"REF a we fix the cascade potential at $U^{(N)}_\\Xi = +40$ MeV and vary $U^{(N)}_\\Sigma $ from $-40$ MeV to $+40$ MeV.",
"In fig.",
"REF b it is the other way round.",
"The pure nuclear matter case is also shown in the uppermost curve.",
"The maximum mass for the pure nucleonic star is $2.29 M_\\odot $ with a radius of $15.31$ km.",
"We see that the maximum mass obtained for a rotating star with hyperonic core is $1.93M_{\\odot }$ with a radius of $14.7$ km in the Keplerian limit with angular velocity $\\Omega = 0.86\\times 10^4 s^{-1}$ , for $U^{(N)}_\\Sigma = +40$ MeV and $U^{(N)}_\\Xi \\ge 0$ .",
"As in the case of static sequence, we see that the maximum mass for the rotating case also increases with $U^{(N)}_\\Sigma $ as we go towards more positive values of this potential.",
"At $U^{(N)}_\\Sigma = -40$ MeV we get a maximum mass of $1.79 M_{\\odot }$ whereas for $U^{(N)}_\\Sigma = +40$ MeV the maximum mass is $1.93 M_{\\odot }$ .",
"The effect of $U^{(N)}_\\Xi $ is much less significant on the maximum mass.",
"From $U^{(N)}_\\Xi = -40$ MeV to $U^{(N)}_\\Xi = +40$ MeV mass is changed only by $\\bigtriangleup M = 0.03 M_{\\odot }$ .",
"In order to have a look at the composition of the maximum mass star, we have plotted the particle densities as a function of radius along the equator in fig.",
"REF .",
"For $U^{(N)}_\\Xi = 0$ and $U^{(N)}_\\Sigma = +40$ MeV, we see that a fair amount of hyperons are present in the core.",
"There are $\\Lambda $ , $\\Sigma ^-$ and $\\Xi ^-$ present.",
"Another interesting observation is that near the core, the density of $\\Lambda $ is much more compared to that of protons and it continues up to a distance of about 5 km from the center."
],
[
"Summary and conclusions",
"To summarize, we have studied the static and rotating axisymmetric stars with hyperons using IUFSU model.",
"The original FSUGold parameter set has been very successful in describing the properties of finite nuclei.",
"With the discovery of highly massive neutron stars the reliability of this model was questioned.",
"It was then revised in the form of IUFSU to accommodate such highly massive stars leaving the low density finite nuclear properties unchanged.",
"In this work we have studied this new parameter set in the context of the possibility of having a hyperonic core in such massive stars.",
"We have included the full octet of baryons in IUFSU.",
"The EoS gets softened due to the inclusion of hyperons whereas the inclusion of the $\\phi $ meson makes the EoS stiffer.",
"We have also investigated the influence of $\\Sigma $ and $\\Xi $ potentials on the EoS.",
"For static stars with hyperonic core we get a maximum mass of $1.62 M_\\odot $ .",
"So IUFSU with hyperons cannot reproduce the observed mass of static stars.",
"However, as the observed $\\sim 2M_\\odot $ neutron stars are both pulsars, we compare the results in the rotating limit.",
"In the Keplerian limit we get a maximum mass of $1.93 M_{\\odot }$ , which is within the 3$\\sigma $ limit of the mass of PSR J$0348+0432$ and 1$\\sigma $ limit of the earlier observation of PSR J$1614-2230$ .",
"We have looked at the particle densities inside the star having the maximum mass and found that a considerable amount of hyperons are present near the core.",
"Therefore, our results are consistent with the recent observations of highly massive pulsars confirming the presence of hyperons in the core of such massive neutron stars.",
"To conclude, IUFSU model, which reproduces the properties of finite nuclei quite successfully also reproduces the recent observations of $\\sim 2M_{\\odot }$ stars, in case of stars having exotic core and rotating in the Keplerian limit.",
"It will be interesting to see whether such a star can hold a quark core.",
"Related work is in progress."
],
[
"Acknowledgement",
"This work is funded by the University Grants Commission (RFSMS, DSKPDF and DRS) and Department of Science and Technology, Government of India."
]
] | 1403.0341 |
[
[
"On a discrete-to-continuum convergence result for a two dimensional\n brittle material in the small displacement regime"
],
[
"Abstract We consider a two-dimensional atomic mass spring system and show that in the small displacement regime the corresponding discrete energies can be related to a continuum Griffith energy functional in the sense of Gamma-convergence.",
"We also analyze the continuum problem for a rectangular bar under tensile boundary conditions and find that depending on the boundary loading the minimizers are either homogeneous elastic deformations or configurations that are completely cracked generically along a crystallographic line.",
"As applications we discuss cleavage properties of strained crystals and an effective continuum fracture energy for magnets."
],
[
"Introduction",
"A fundamental problem in static fracture mechanics is to determine the behavior of a brittle material which is subject to certain displacements imposed at its boundary.",
"Of particular interest is the identification of critical loads at which failure occurs.",
"A natural framework to treat such free discontinuity problems with variational methods is given by Griffith energy functionals introduced by Francfort and Marigo [21] comprising elastic bulk contributions and surface terms comparable to the size of the crack (see also [17]).",
"Often these models contain anisotropic surface terms (see e.g.",
"[2], [19], [26]) modeling the fact that due to the crystalline structure of the materials certain directions for the formation of cracks are energetically favored.",
"Indeed, fracture typically occurs in the form of cleavage along crystallographic planes.",
"Ultimately, such a continuum model should be identified as an effective theory derived from atomistic interactions.",
"Specifying the set-up even further, a basic experiment to infer material properties of brittle materials is to probe the specimen by applying a uniaxial tensile strain which allows to determine its Poisson ratio in the elastic regime and a critical load beyond which the body fails due to fracture.",
"From a theoretical point of view this problem has been studied recently by Mora-Corral in [25], where he investigates a rectangular bar of brittle, incompressible, homogeneous and isotropic material subject to uniaxial extension and shows that, depending on the loading, the minimizers are either given by purely elastic configurations or deformations with horizontal fracture.",
"An atomistic model problem with surface contributions sensitive to the crack geometry has been studied by the authors in [22] leading to a complete analysis of the asymptotically optimal configurations under uniaxial extension in the discrete-to-continuum limit: The body shows pure elastic behavior in the subcritical case and for supercritical boundary values generically cleavage occurs along a specific crystallographic line.",
"However, for a certain symmetric orientation of the lattice cleavage may fail more complicated crack geometries are possible.",
"The goal of this work is to show that in the small displacement regime the energies associated to such a discrete system can be related to a continuum Griffith energy functional with anisotropic surface contributions in the sense of $\\Gamma $ -convergence.",
"Moreover, we analyze the continuum problem under tensile boundary conditions.",
"In this way we (1) obtain convergence scheme which in certain applications to be discussed below allows to identify effective continuum fracture energies, (2) extend the results of [25] to anisotropic and compressible materials and (3) re-derive in part the aforementioned convergence results of [22].",
"In the theory of fracture mechanics the passage from discrete systems to continuum models via $\\Gamma $ -convergence is by now well understood for one-dimensional chains, see e.g.",
"[7], [8], [9].",
"In the higher dimensional setting there are results for scalar valued models (see [10]) and approximations of vector valued free discontinuity problems where the elastic bulk part of the energy is characterized by linearized terms (see [2]) or by a quasiconvex stored energy density (see [19]).",
"However, in more than one dimension the energy density of discrete systems such as well-known mass spring models is in general not given in terms of a discretized continuum quasiconvex function.",
"For large strains these lattices typically become even unstable, see e.g.",
"the basic model discussed in [24].",
"Consequently, in the regime of finite elasticity it is a subtle question if minimizers for given boundary data exist at all.",
"On the other hand, for sufficiently small strains one may expect the Cauchy-Born rule to apply so that individual atoms do in fact follow a macroscopic deformation gradient, see [24], [14].",
"In particular this applies to the regime of infinitesimal elastic strains.",
"For purely elastic interactions this relation has also been obtained in the sense of $\\Gamma $ -convergence for a simultaneous passage from discrete to continuum and linearization process in [12], [27].",
"The model considered in [22] as well as the one-dimesional seminal paper [11] suggest that the most interesting regime for the elastic strains is given by $\\sqrt{\\varepsilon }$ ($\\varepsilon $ denotes the typical interatomic distance) as in this particular regime the elastic and the crack energy are of the same order.",
"This is in accordance to the observation that brittle materials develop cracks already at moderately large strains.",
"Moreover, it shows that a discrete-to-continuum $\\Gamma $ -limit for the discrete energies under consideration naturally involves a linearization process.",
"Identifying all possible limiting continuum configurations and energies is a challenging task as necessary smallness assumptions on the discrete gradient can not be inferred from suitable energy bounds and deriving rigidity estimates being essential in the passage from nonlinear to linearized theory (see [12], [27]) is a subtle problem.",
"Partial results have been obtain in [22] for almost minimizers of a boundary value problem describing uniaxial extension.",
"A general analysis in two dimensions is deferred to a subsequent work.",
"In the present context we make the simplifying assumption that we consider deformations lying $\\sqrt{\\varepsilon }$ -close to the identity mapping.",
"However, we will also see that there are physically interesting applications e.g.",
"to magnetic materials where such an assumption can be justified rigorously.",
"It then turns out that the derivation of the continuum limit is an issue similar to those considered in [2], [10], [19].",
"Nevertheless, we believe that the present $\\Gamma $ -convergence result is interesting as (1) it gives rise to a limiting Griffith functional in the realm of linearized elasticity which can be explicitly investigated for cleavage, (2) there are applications to systems with small displacements for small energies and (3) to the best of our knowledge our approach to the problem differs from techniques which are predominantly used when treating discrete systems in the framework of fracture mechanics.",
"The reduction to one-dimensional sections using slicing properties for (special) functions of bounded variation turned out to be a useful tool not only to derive general properties of these function spaces but also to study discrete systems and variational approximation of free discontinuity problems.",
"E.g., the original proofs of the main compactness and closure theorems in $SBV$ (see [3]) as well as the $\\Gamma $ -convergence results in [10], [19] make use of this integralgeometric approach.",
"Similar to the fact that there are simplified proofs of these compactness theorems being derived without the slicing technique (see [1]), we show that in our framework the lower bound of the $\\Gamma $ -limit can be achieved in a different way.",
"In fact, we carefully construct the crack shapes of discrete configurations in an explicit way which allows us to directly appeal to lower semicontinuity results for $SBV$ functions.",
"The paper is organized as follows.",
"We first introduce our discrete model and state our main results in Section .",
"Here we also briefly discuss how these results shed new light on our findings in [22] on crystal cleavage and study an application to fractured magnets in an external field.",
"Section is devoted to the derivation of the continuum energy functional via $\\Gamma $ -convergence.",
"The main idea for the lower bound relies on a separation of the energy into elastic and surface contributions by introducing an interpolation with discontinuities on triangles where large expansion occurs.",
"By constructing the set of discontinuity points in a suitable way the surface energy can be estimated using lower semicontinuity results for $SBV$ functions.",
"The elastic part can be treated similarly as in [23], [27].",
"Finally, in Section we analyze the continuum problem under tensile boundary values and extend the results obtained in [25] to anisotropic and compressible materials.",
"A careful analysis of the anisotropic surface contribution shows that in the generic case there is a unique optimal direction for the formation of fracture, while in a symmetrically degenerate case cleavage fails and all energetically optimal crack geometries can be characterized by specific Lipschitz curves.",
"As in [25] the proof makes use of a qualitative rigidity result for $SBV$ functions (see [13]) and of the structure theorem on the boundary of sets of finite perimeter by Federer [18]."
],
[
"The discrete model",
"Let ${\\cal L}$ denote the rotated triangular lattice $ {\\cal L}= R_{\\cal L} \\begin{pmatrix} 1 & \\frac{1}{2} \\\\ 0 & \\frac{\\sqrt{3}}{2} \\end{pmatrix} Z^2= \\lbrace \\lambda _1 \\mathbf {v}_1 + \\lambda _2 \\mathbf {v}_2 : \\lambda _1, \\lambda _2 \\in Z\\rbrace , $ where $R_{\\cal L} = \\begin{footnotesize} \\begin{pmatrix} \\cos \\phi & - \\sin \\phi \\\\ \\sin \\phi & \\cos \\phi \\end{pmatrix} \\in SO(2) \\end{footnotesize}$ is some rotation and $\\mathbf {v}_1$ , $\\mathbf {v}_2$ are the lattice vectors $\\mathbf {v}_1 = R_{\\cal L} \\mathbf {e}_1$ and $\\mathbf {v}_2 = R_{\\cal L}(\\frac{1}{2} \\mathbf {e}_1 + \\frac{\\sqrt{3}}{2} \\mathbf {e}_2)$ , respectively.",
"Without loss of generality we can assume $\\phi \\in [0, \\frac{\\pi }{3})$ .",
"We collect the basic lattice vectors in the set ${\\cal V} = \\left\\lbrace \\mathbf {v}_1,\\mathbf {v}_2,\\mathbf {v}_2 - \\mathbf {v}_1\\right\\rbrace $ .",
"The macroscopic region $\\Omega \\subset R^2$ occupied by the body is supposed to be a bounded domain with Lipschitz boundary.",
"In the reference configuration the positions of the specimen's atoms are given by the points of the scaled lattice $\\varepsilon \\mathcal {L}$ that lie within $\\Omega $ .",
"Here $\\varepsilon $ is a small parameter defining the length scale of the typical interatomic distances.",
"The deformations of our system are mappings $y : \\varepsilon \\mathcal {L}\\cap \\Omega \\rightarrow R^2$ .",
"The energy associated to such a deformation $y$ is assumed to be given by nearest neighbor interactions as $E_{\\varepsilon }(y)= \\frac{1}{2} \\sum _{x,x^{\\prime } \\in \\varepsilon {\\cal L} \\cap \\Omega \\atop |x-x^{\\prime }| = \\varepsilon } W \\left( \\frac{|y(x) - y(x^{\\prime })|}{\\varepsilon } \\right).$ Note that the scaling factor $\\frac{1}{\\varepsilon }$ in the argument of $W$ takes account of the scaling of the interatomic distances with $\\varepsilon $ .",
"The pair interaction potential $W:[0,\\infty ) \\rightarrow [0, \\infty ]$ is supposed to be of `Lennard-Jones-type': (i) $W \\ge 0$ and $W(r) = 0$ if and only if $r = 1$ .",
"(ii) $W$ is continuous on $[0, \\infty )$ and $C^2$ in a neighborhood of 1 with $\\alpha := W^{\\prime \\prime }(1) > 0$ .",
"(iii) $\\lim _{r \\rightarrow \\infty } W(r) = \\beta > 0$ .",
"In order to analyze the passage to the limit as $\\varepsilon \\rightarrow 0$ it will be useful to interpolate and rewrite the energy as an integral functional.",
"Let ${\\cal C}_{\\varepsilon }$ be the set of equilateral triangles $\\triangle \\subset \\Omega $ of sidelength $\\varepsilon $ with vertices in $\\varepsilon {\\cal L}$ and define $\\Omega _{\\varepsilon } = \\bigcup _{\\triangle \\in {\\cal C}_{\\varepsilon }} \\triangle $ .",
"By $\\tilde{y} : \\Omega _{\\varepsilon } \\rightarrow R^2$ we denote the interpolation of $y$ , which is affine on each $\\triangle \\in {\\cal C}_{\\varepsilon }$ .",
"The derivative of $\\tilde{y}$ is denoted by $\\nabla \\tilde{y}$ , whereas we write $(y)_{\\triangle }$ for the (constant) value of the derivative on a triangle $\\triangle \\in {\\cal C}_{\\varepsilon }$ .",
"Then (REF ) can be rewritten as $\\begin{split}E_{\\varepsilon }(y)&= \\sum _{\\triangle \\in {\\cal C}_{\\varepsilon }} W_{\\triangle } ((\\tilde{y})_{\\triangle })+ E_{\\varepsilon }^{\\rm boundary}(y) \\\\&= \\frac{4}{\\sqrt{3}\\varepsilon ^2} \\int _{\\Omega _{\\varepsilon }} W_{\\triangle } (\\nabla \\tilde{y}) \\, dx+ E_{\\varepsilon }^{\\rm boundary}(y),\\end{split} $ where $W_{\\triangle }(F)&= \\frac{1}{2} \\Big ( W(|F \\mathbf {v}_1|) + W(|F \\mathbf {v}_2|) + W(|F (\\mathbf {v}_2 - \\mathbf {v}_1)|) \\Big ).$ Here we used that $|\\triangle | = \\sqrt{3}\\varepsilon ^2/4$ .",
"The boundary term is the sum of pair interaction energies $\\frac{1}{4} W(\\frac{|y(x) - y(x^{\\prime })|}{\\varepsilon })$ or $\\frac{1}{2} W(\\frac{|y(x) - y(x^{\\prime })|}{\\varepsilon })$ over nearest neighbor pairs which form the side of only one or no triangle in ${\\cal C}_{\\varepsilon }$ , respectively.",
"Due to the discreteness of the underlying atomic lattice, Dirichlet boundary conditions have to be imposed in a small neighborhood of the boundary as otherwise cracks near the boundary may become energetically more favorable.",
"Assume that $\\tilde{\\Omega } \\supset \\Omega $ is a bounded, open domain in $R^2$ with Lipschitz boundary defining the Dirichlet boundary $\\partial _D\\Omega = \\partial \\Omega \\cap \\tilde{\\Omega }$ of $\\Omega $ .",
"For $g \\in W^{1, \\infty }(\\tilde{\\Omega })$ we define the class of discrete displacements assuming the boundary value $g$ on $\\partial _D\\Omega $ as ${\\cal A}_g= \\big \\lbrace u : \\varepsilon {\\cal L} \\cap \\tilde{\\Omega } \\rightarrow R^2 :u(x) = g(x) \\text{ for } x \\in \\varepsilon {\\cal L} \\cap \\Omega _{D,\\varepsilon } \\big \\rbrace ,$ where $\\Omega _{D,\\varepsilon } := \\lbrace x\\in \\tilde{\\Omega }: \\operatorname{dist}(x, \\tilde{\\Omega }\\setminus \\Omega ) \\le \\varepsilon \\rbrace $ .",
"For the corresponding deformations $y = \\mathbf {id}+ u$ this amounts to requiring $y(x) = x + g(x)$ for $x \\in \\varepsilon {\\cal L} \\cap \\Omega _{D,\\varepsilon }$ .",
"Similar as before, we let $\\tilde{\\cal C}_{\\varepsilon }$ be the set of equilateral triangles $\\triangle \\subset \\tilde{\\Omega }$ with vertices in $\\varepsilon {\\cal L}$ and define $\\tilde{\\Omega }_{\\varepsilon } = \\bigcup _{\\triangle \\in \\tilde{\\cal C}_{\\varepsilon }} \\triangle $ .",
"By $\\tilde{y} : \\tilde{\\Omega }_{\\varepsilon } \\rightarrow R^2$ we again denote the piecewise affine interpolation of $y$ .",
"It is easy to see that the formation of a crack of finite length resulting from a number of largely deformed triangles scaling with $\\frac{1}{\\varepsilon }$ leads to an energy contribution to $E_{\\varepsilon }$ scaling with $\\varepsilon $ .",
"The most interesting regime is when the elastic energy contributions to $E_{\\varepsilon }$ and the energy cost of a cracked configurations are of the same order.",
"We are thus particularly interested in boundary displacements $g_{\\varepsilon }$ scaling with $\\sqrt{\\varepsilon }$ .",
"For then there are also completely elastic deformations for which $E_\\varepsilon $ scales with $\\varepsilon $ , e.g.",
"$E_{\\varepsilon }(\\mathbf {id}+ g_{\\varepsilon }) = O(\\varepsilon )$ .",
"In order to obtain finite energies and displacements in the limit $\\varepsilon \\rightarrow 0$ , we accordingly rescale the displacement field to $u = \\frac{1}{\\sqrt{\\varepsilon }}(y -\\mathbf {id})$ and the energy $E_{\\varepsilon }$ to $ {\\cal E}_{\\varepsilon }(u) := \\varepsilon E_{\\varepsilon }(y) = \\varepsilon E_{\\varepsilon }(\\mathbf {id}+ \\sqrt{\\varepsilon } u).$ Moreover, we will assume $u \\in {\\cal A}_{g_\\varepsilon }$ for some $g_\\varepsilon \\in W^{1,\\infty }(\\tilde{\\Omega })$ .",
"We also introduce the functionals ${\\cal E}^\\chi _\\varepsilon $ which arise from ${\\cal E}_\\varepsilon $ by replacing $W_\\Delta $ by $W_{\\Delta , \\chi } = W_\\Delta + \\chi $ , where $\\chi : R^{2 \\times 2} \\rightarrow [0, \\infty ]$ is a frame indifferent penalty term with $\\chi \\ge c_\\chi >0$ in a neighborhood of $O(2)\\setminus SO(2)$ and $ \\chi \\equiv 0$ in a neighborhood of $SO(2) \\cup \\lbrace \\infty \\rbrace $ .",
"This term is a mild extra assumption to assure that the orientation of the triangles is preserved in the elastic regime and unphysical effects are avoided."
],
[
"Convergence of the variational problems",
"Our convergence analysis applies to discrete deformations which may elongate a number scaling with $\\frac{1}{\\varepsilon }$ of springs very largely, leading to cracks of finite length in the continuum limit.",
"On triangles not adjacent to such essentially broken springs, the defomations are $\\sqrt{\\varepsilon }$ -close to the identity mapping, so that the accordingly rescaled displacements are of bounded $L^2$ -norm.",
"Note that the first of these assumptions can be inferred from suitable energy bounds.",
"By way of example, however, we will see that this cannot be true for the displacement estimates in the bulk: The sequence of functionals $({\\cal E}_{\\varepsilon })_{\\varepsilon }$ is not equicoercive.",
"Nevertheless, it is interesting to investigate this regime in order to identify a corresponding continuum functional which describes the system in the realm of Griffith models with linearized elasticity.",
"In fact, below we will discuss two specific models where external fields or boundary conditions break the rotational symmetry whence the sequence $({\\cal E}^{\\chi }_{\\varepsilon })_{ \\varepsilon }$ satisfies suitable equicoercivity conditions.",
"Recall that the space $SBV(\\Omega ; R^2)$ , abbreviated as $SBV(\\Omega )$ hereafter, of special functions of bounded variation consists of functions $u \\in L^1(\\Omega ; R^2)$ whose distributional derivative $Du$ is a finite Radon measure, which splits into an absolutely continuous part with density $\\nabla u$ with respect to Lebesgue measure and a singular part $D^j u$ whose Cantor part vanishes and thus is of the form $ D^j u = [u] \\otimes \\nu _u {\\cal H}^1 \\lfloor J_u, $ where ${\\cal H}^1$ denotes the one-dimensional Hausdorff measure, $J_u$ (the `crack path') is an ${\\cal H}^1$ -rectifiable set in $\\Omega $ , $\\nu _u$ is a normal of $J_u$ and $[u] = u^+ - u^-$ (the `crack opening') with $u^{\\pm }$ being the one-sided limits of $u$ at $J_u$ .",
"If in addition $\\nabla u \\in L^2(\\Omega )$ and ${\\cal H}^1(J_u) < \\infty $ , we write $u \\in SBV^2(\\Omega )$ .",
"See [5] for the basic properties of these function spaces.",
"The sense in which discrete displacements are considered convergent to a limiting displacement in SBV is made precise in the following definition.",
"Definition 2.1 Suppose $ u_{\\varepsilon } : \\varepsilon {\\cal L} \\cap \\tilde{\\Omega } \\rightarrow R^2$ is a sequence of discrete displacements.",
"We say that $u_{\\varepsilon }$ converges to some $u \\in SBV^2(\\tilde{\\Omega })$ : $u_{\\varepsilon } \\rightarrow u$ , if (i) $\\chi _{\\tilde{\\Omega }_{\\varepsilon }} \\tilde{u}_{\\varepsilon } \\rightarrow u$ in $L^{1}(\\tilde{\\Omega })$ and there exists a sequence ${\\cal C}^*_{\\varepsilon } \\subset \\tilde{\\cal C}_{\\varepsilon }$ with $\\# {\\cal C}^*_{\\varepsilon } \\le \\frac{C}{\\varepsilon }$ for a constant $C$ independent of $\\varepsilon $ such that (ii) $\\Vert \\nabla \\tilde{u}_{\\varepsilon } \\Vert _{L^2(\\tilde{\\Omega } \\setminus \\cup _{\\triangle \\in {\\cal C}^*_{\\varepsilon }} \\triangle )} \\le C$ .",
"The main idea will be to separate the energy into elastic and crack surface contributions by introducing a threshold such that triangles $\\triangle $ with $(y)_{\\triangle }$ beyond that threshold are considered as cracked and $\\tilde{y}$ is modified there to a discontinuous function.",
"The treatment of the elastic part draws ideas from [27] and [23].",
"To derive the crack energy, one could use a slicing technique, see, e.g., [10].",
"Although also possible in our framework, we follow a different approach here: We carefully construct crack shapes of discrete configurations in an explicit way which allows us to directly appeal to lower semicontinuity results for $SBV$ functions in order to derive the main energy estimates.",
"Consider the limiting functional ${\\cal E}(u) =\\frac{4}{\\sqrt{3}} \\int _{\\Omega } \\frac{1}{2} Q(e(u)) \\, dx + \\int _{J_{u}} \\sum _{\\mathbf {v}\\in {\\cal V}} \\frac{2\\beta }{\\sqrt{3}} |\\mathbf {v}\\cdot \\nu _{u}|\\, d{\\cal H}^1$ for $u \\in SBV^2(\\tilde{\\Omega })$ , where $e(u)=\\frac{1}{2}\\left(\\nabla u^{T} + \\nabla u\\right)$ denotes the symmetric part of the gradient.",
"$Q$ is the linearization of $W_{\\triangle }$ about the identity matrix $\\mathbf {Id}$ (see Lemma REF for its explicit form).",
"Note that for a displacement field $u$ , which is the limit of a sequence $(u_{\\varepsilon }) \\subset {\\cal A}_{g_{\\varepsilon }}$ converging in the sense of Definition REF , we get $u = g$ on $\\tilde{\\Omega } \\setminus \\Omega $ , where $g = L^1\\text{-}\\lim _{\\varepsilon \\rightarrow 0} g_\\varepsilon $ .",
"Therefore, if $u|_\\Omega $ does not attain the boundary condition $g$ on the Dirichlet boundary $\\partial _D \\Omega $ (in the sense of traces), this will be penalized in the energy ${\\cal E}(u)$ .",
"In Section we prove the following $\\Gamma $ -convergence result (see [16] for an exhaustive treatment of $\\Gamma $ -convergence): Theorem 2.2 (i) Let $(g_\\varepsilon )_\\varepsilon \\subset W^{1,\\infty }(\\tilde{\\Omega })$ with $\\sup _\\varepsilon \\Vert g_\\varepsilon \\Vert _{W^{1,\\infty }(\\tilde{\\Omega })} < + \\infty $ .",
"If $(u_{\\varepsilon })_\\varepsilon $ is a sequence of discrete displacements with $u_{\\varepsilon } \\in {\\cal A}_{g_\\varepsilon }$ and $u_\\varepsilon \\rightarrow u \\in SBV^2(\\tilde{\\Omega })$ , then $ \\liminf _{\\varepsilon \\rightarrow 0} {\\cal E}_{\\varepsilon }(u_{\\varepsilon })\\ge {\\cal E}(u).",
"$ (ii) For every $u \\in SBV^2(\\tilde{\\Omega })$ and $g \\in W^{1,\\infty }(\\tilde{\\Omega })$ with $u=g$ on $\\tilde{\\Omega } \\setminus \\Omega $ there is a sequence $(u_{\\varepsilon })_\\varepsilon $ of discrete displacements such that $u_{\\varepsilon } \\in {\\cal A}_{g}$ , $u_{\\varepsilon } \\rightarrow u \\in SBV^2(\\tilde{\\Omega })$ and $ \\lim _{\\varepsilon \\rightarrow 0} {\\cal E}^{ \\chi }_{\\varepsilon }(u_{\\varepsilon })= {\\cal E}(u).",
"$ Note that the recovery sequence is obtained for the energy ${\\cal E}^\\chi _\\varepsilon $ which includes the frame indifferent penalty term.",
"Due to the frame indifference of $W$ , $({\\cal E}_{\\varepsilon })$ and $({\\cal E}^\\chi _{\\varepsilon })$ are not equicoercive as the following example shows.",
"Example.",
"Assume that the specimen satisfying the boundary conditions is broken into three parts by two even cracks where the middle part is subject to a rotation $R \\ne \\mathbf {Id}$ so that $ \\nabla \\tilde{y}_{\\varepsilon } = R \\text{ for } p \\le x_1 \\le q, \\ 0 < p < q < l. $ In particular, the energy of the configuration is of order 1.",
"But for $p \\le x_1 \\le q$ $ |\\nabla \\tilde{u}_{\\varepsilon }(x)|= \\left|\\frac{1}{\\sqrt{\\varepsilon }} \\left(R - \\mathbf {Id}\\right)\\right|\\rightarrow \\infty \\text{ for } \\varepsilon \\rightarrow 0.$ Thus, $\\nabla \\tilde{u}_{\\varepsilon }$ is not bounded in $L^1$ and so $u_{\\varepsilon }$ does not converge.",
"We now add a term to ${\\cal E}_{\\varepsilon }$ such that the sequence becomes equicoercive.",
"Let $\\hat{m}: R^{2 \\times 2} \\rightarrow S^1$ be a function satisfying $\\hat{m}(RF) = R\\hat{m}(F) \\ \\text{ for all } F \\in R^{2 \\times 2}, R \\in SO(2), \\ \\ \\ \\ \\hat{m}(\\mathbf {Id}) = \\mathbf {e}_1 .$ Moreover, assume that $\\hat{m}$ is $C^2$ in a neighborhood of $SO(2)$ and $R^{2 \\times 2}_{\\rm sym} \\subset {\\rm ker}(D\\hat{m}(\\mathbf {Id}))$ .",
"Let ${\\cal F}_\\varepsilon (u) = {\\cal E}_\\varepsilon (u) + \\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon } f_{\\kappa }(\\nabla \\tilde{y})$ with $f_{\\kappa }(F) = {\\left\\lbrace \\begin{array}{ll}\\kappa (1- \\mathbf {e}_1 \\cdot \\hat{m}(F)), & |F| \\le T, \\\\ 0 & \\text{else},\\end{array}\\right.",
"}$ for $F \\in R^{2 \\times 2}$ , where $T,\\kappa >0$ .",
"Likewise, we define ${\\cal F}^\\chi _\\varepsilon $ .",
"In Lemma REF below we show that $W_{\\Delta ,\\chi }(F) + f_{\\kappa }(F) \\ge C|F - \\mathbf {Id}|^2$ for all $F \\in R^{2 \\times 2}$ with $|F| \\le T$ .",
"This implies that the sequence $({\\cal F}^\\chi _\\varepsilon )_\\varepsilon $ is equicoercive: Given a sequence of displacement fields $(u_\\varepsilon )_\\varepsilon $ with ${\\cal F}^\\chi _\\varepsilon (u_\\varepsilon ) + \\Vert u_\\varepsilon \\Vert _\\infty \\le C$ we find a subsequence converging in the sense of Definition REF .",
"Indeed, we get that $\\# {\\cal C}^*_\\varepsilon \\le \\frac{C}{\\varepsilon }$ , where ${\\cal C}^*_\\varepsilon := \\lbrace \\Delta \\in \\tilde{\\cal C}_\\varepsilon : |(\\mathbf {Id}+ \\sqrt{\\varepsilon } \\tilde{u}_\\varepsilon )_\\Delta | > T \\rbrace $ .",
"By Lemma REF we then get $\\Vert \\nabla \\tilde{u}_{\\varepsilon } \\Vert _{L^2(\\tilde{\\Omega } \\setminus \\cup _{\\triangle \\in {\\cal C}^*_{\\varepsilon }} \\triangle )} \\le C$ and therefore condition (ii) in Definition REF is satisfied.",
"By an $SBV$ compactness theorem (see [5]) we then find a (not relabeled) subsequence such that $\\tilde{u}_\\varepsilon \\chi _{\\tilde{\\Omega }_\\varepsilon \\setminus \\cup _{\\triangle \\in {\\cal C}^*_{\\varepsilon }} \\triangle } \\rightarrow u$ in $L^1$ for some $u\\in SBV^2(\\tilde{\\Omega })$ .",
"This together with $\\Vert u_\\varepsilon \\Vert _\\infty \\le C$ and $|\\bigcup _{\\triangle \\in {\\cal C}^*_{\\varepsilon }} \\triangle | \\le C\\varepsilon $ implies that also condition (i) in Definition REF (i) holds with this function $u$ .",
"Define $\\hat{m}_1: R^{2 \\times 2} \\rightarrow [-1,1]$ by $\\hat{m}_1 = \\mathbf {e}_1 \\cdot \\hat{m}$ and let $\\hat{Q} = D^2\\hat{m}_1(\\mathbf {Id})$ be the Hessian at the identity.",
"We introduce the limiting functional ${\\cal F}: SBV^2(\\tilde{\\Omega }) \\rightarrow [0,\\infty )$ given by ${\\cal F}(u) = {\\cal E}(u) - \\frac{\\kappa }{2}\\int _{\\Omega } \\hat{Q}(\\nabla u).$ We then obtain a $\\Gamma $ -convergence result similar to Theorem REF .",
"Theorem 2.3 The assertions of Theorem REF remain true when ${\\cal E}_\\varepsilon $ , ${\\cal E}^\\chi _\\varepsilon $ and ${\\cal E}$ are replaced by ${\\cal F}_\\varepsilon $ , ${\\cal F}^\\chi _\\varepsilon $ and ${\\cal F}$ , respectively."
],
[
"Analysis of a limiting cleavage problem",
"We now analyze the limiting functional ${\\cal E}$ for a rectangular slab $\\Omega = (0, l) \\times (0, 1)$ with $l \\ge \\frac{1}{\\sqrt{3}}$ under uniaxial extension in $\\mathbf {e}_1$ direction.",
"We determine the minimizers and prove uniqueness up to translation of the specimen and the crack line for the boundary conditions $u_1 = 0 \\text{ for } x_1 = 0 \\quad \\text{ and } \\quad u_1= a l \\text{ for } x_1 = l.$ (More precisely: $u \\in SBV^2((-\\eta , l+ \\eta ) \\times (0, 1))$ with $u_1(x) = 0$ for $x \\le 0$ and $u_1(x) = al$ for $x \\ge l$ .)",
"Note that we can investigate the limiting problem without any assumption on the second component of the boundary displacement.",
"Let $\\gamma = \\max \\lbrace |\\mathbf {v}_1 \\cdot \\mathbf {e}_2|, |\\mathbf {v}_2 \\cdot \\mathbf {e}_2|, |(\\mathbf {v}_2 - \\mathbf {v}_1) \\cdot \\mathbf {e}_2|\\rbrace $ and $\\mathbf {v}_{\\gamma } \\in {\\cal V}$ such that $\\gamma = |\\mathbf {v}_{\\gamma } \\cdot \\mathbf {e}_2|$ .",
"We note that $\\gamma $ takes values in $[\\frac{\\sqrt{3}}{2},1]$ and $\\mathbf {v}_{\\gamma }$ is unique if and only if $\\phi \\ne 0$ .",
"It turns out that the specimen shows perfect elastic behavior up to the critical boundary displacement $ a_{\\rm crit} = \\sqrt{\\frac{2\\sqrt{3}\\beta }{\\alpha \\gamma l}}.",
"$ Beyond critical loading the body fails by breaking into two pieces.",
"Theorem 2.4 Let $a \\ne a_{\\rm crit}$ .",
"Then $ \\min \\big \\lbrace {\\cal E}(u) : u \\text{ satisfies (\\ref {eq:BC})} \\big \\rbrace = \\min \\left\\lbrace \\frac{ \\alpha l}{\\sqrt{3}} a^2 , \\frac{2 \\beta }{\\gamma } \\right\\rbrace .",
"$ All minimizers of ${\\cal E}$ subject to (REF ) are of the following form: (i) If $a < a_{\\rm crit}$ , then $ u^{\\rm el}(x)= (0,s) + \\begin{pmatrix}a & 0 \\\\ 0 & -\\frac{a}{3}\\end{pmatrix} x $ for some $s \\in R$ .",
"(ii) If $a > a_{\\rm crit}$ and $\\phi \\ne 0$ then $ u^{\\rm cr}(x)= {\\left\\lbrace \\begin{array}{ll}(0,s) & \\mbox{\\rm for } x \\mbox{\\rm \\ to the left of } (p,0) + R\\mathbf {v}_{\\gamma }, \\\\(a l, t) & \\mbox{\\rm for } x \\mbox{\\rm \\ to the right of } (p,0) + R\\mathbf {v}_{\\gamma },\\end{array}\\right.}",
"$ for some $s,t \\in R$ and $p \\in (0,l)$ such that $(p,0) + R\\mathbf {v}_{\\gamma }$ intersects both the segments $(0, l) \\times \\lbrace 0\\rbrace $ and $(0, l) \\times \\lbrace 1\\rbrace $ .",
"(iii) If $a > a_{\\rm crit}$ and $\\phi = 0$ then $ u^{\\rm cr}(x)= {\\left\\lbrace \\begin{array}{ll}(0,s) & \\mbox{\\rm if } 0 < x_1 < h(x_2), \\\\(a l, t) & \\mbox{\\rm if } h(x_2) < x_1 < l,\\end{array}\\right.}",
"$ for a Lipschitz function $h:(0,1) \\rightarrow [0,l]$ with $|h^{\\prime }| \\le \\frac{1}{\\sqrt{3}}$ a.e.",
"and constants $s,t \\in R$ .",
"This theorem will be addressed in Section .",
"An analogous result for isotropic, incompressible materials has been obtained recently by Mora-Corral [25].",
"Theorem REF is an extension of this result to anisotropic, compressible brittle materials in the framework of linearized elasticity.",
"In particular, as mentioned above we see that all the optimal configurations show purely elastic behavior in the subcritical case and complete fracture in the supercritical regime.",
"The crack minimizer in (ii) for $\\phi \\ne 0$ is broken parallel to $R\\mathbf {v}_{\\gamma }$ which proves that cleavage occurs along crystallographic lines, while in the symmetric case $\\phi = 0$ cleavage in general fails."
],
[
"Applications: Cleaved crystals and fractured magnets",
"As applications of the converging results for the energy functionals ${\\cal E}^\\chi _\\varepsilon $ and ${\\cal F}^\\chi _\\varepsilon $ we consider cleaved crystals and fractured magnets, respectively.",
"In the first model a mild equicoercivity of the sequence $({\\cal E}^\\chi _\\varepsilon )_\\varepsilon $ is guaranteed by investigating a specific boundary value problem, in the latter model an external field provides an even stronger equicoercivity condition."
],
[
"Uniaxially strained crystals",
"Theorem REF in combination with Theorem REF gives a new perspective to the results on crystal cleavage of [22].",
"Let $\\Omega = (0, l) \\times (0, 1)$ with $l \\ge \\frac{1}{\\sqrt{3}}$ .",
"For $\\tilde{\\Omega } = (-\\eta , l + \\eta ) \\times (0,1)$ and $a \\ge 0$ set $\\begin{split}{\\cal A}( a)= \\big \\lbrace u = &(u_1, u_2) : \\varepsilon {\\cal L} \\cap \\tilde{\\Omega } \\rightarrow R^2 : \\\\ & u(x) = g(x) \\text{ for } x_1 \\le \\varepsilon \\text{ and }x_1 \\ge l - \\varepsilon \\text{ for some } g \\in {\\cal G}(a)\\big \\rbrace ,\\end{split}$ where ${\\cal G}(a) := \\lbrace g \\in W^{1,\\infty }(\\tilde{\\Omega }): g_1(x) = 0 \\text{ for } x_1 \\le \\varepsilon , ~ g_1(x) = al \\text{ for } x_1 \\ge l - \\varepsilon \\rbrace $ .",
"In [22] we proved that the limiting minimal energy leads to a universal cleavage law of the form $\\lim _{\\varepsilon \\rightarrow 0} \\inf \\left\\lbrace {\\cal E}_{\\varepsilon }( u) : u \\in {\\cal A}(a) \\right\\rbrace = \\min \\left\\lbrace \\frac{ \\alpha l}{\\sqrt{3}} a^2, \\frac{2 \\beta }{\\gamma } \\right\\rbrace ,$ independent of the particular shape of the interatomic potential $W$ .",
"Optimal configurations are given by the constant sequences $u_\\varepsilon = u^{\\rm el}$ in the subcritical case $a \\le a_{\\rm crit}$ and $u_\\varepsilon = u^{\\rm cr}$ in the supercritical case $a \\ge a_{\\rm crit}$ , respectively, with $u^{\\rm el}$ and $u^{\\rm cr}$ as in Theorem REF .",
"In fact, the above given configurations provide a characterization of all minimizing sequences in the sense that, all low energy sequences $(u_\\varepsilon )_\\varepsilon $ satisfying ${\\cal E}^\\chi _\\varepsilon (u_\\varepsilon ) = \\inf \\lbrace {\\cal E}^\\chi _\\varepsilon (u): u \\in {\\cal A}(a) \\rbrace + O(\\varepsilon )$ and $\\sup _{\\varepsilon } \\Vert u_{\\varepsilon } \\Vert _{\\infty }< \\infty $ converge–up to subsequences–in the sense of Definition REF to $u^{\\rm el}$ if $a < a_{\\rm crit}$ or $u^{\\rm cr}$ if $a > a_{\\rm crit}$ for suitable $s, t, p$ and $g$ , respectively.",
"This is a direct consequence of [22].",
"(The convergence obtained in [22] is even stronger.)",
"One implication of [22] is that, under the tensile boundary conditions $ u_{\\varepsilon } \\in {\\cal A}(a)$ , the requirement that $ u_{\\varepsilon }$ be an almost energy minimizer satisfying (REF ), guarantees the existence of a subsequence converging in the sense of Definition REF .",
"In particular, the sequence $({\\cal E}^\\chi _{\\varepsilon })$ is mildly equicoercive.",
"A fundamental theorem of $\\Gamma $ -convergence (see, e.g., [6]) implies that such low energy sequences converge to limiting configurations $u^{\\rm el}$ , respectively, $u^{\\rm cr}$ , in the sense of Definition REF .",
"Consequently, in this way we have re-derived the convergence result [22] (in the sense of Definition REF )."
],
[
"Permanent magnets in an external field",
"Assume that the material is a permanent magnet and let $\\mathbf {e}_1$ be the magnetization direction.",
"We suppose that there is a constitutive relation between $\\nabla \\tilde{y}(x)$ and the local magnetization direction $\\hat{m}(\\tilde{y},x) \\in S^1$ of the deformed configuration $\\tilde{y}$ at some point $x \\in \\Omega $ , which is of the form $\\hat{m}(\\tilde{y},x) = \\hat{m} (\\nabla \\tilde{y}(x))$ with $\\hat{m}$ as defined in Section REF .",
"Let $H_{\\rm ext}: R^2 \\rightarrow R^2$ be an external magnetic field.",
"The magnetic energy corresponding to the deformation $y = \\mathbf {id}+ \\sqrt{\\varepsilon } u$ is then given by ${\\cal E}^{\\rm mag}_\\varepsilon (u) = -\\frac{1}{\\varepsilon } \\int _{\\Omega _\\varepsilon } H_{\\rm ext} \\cdot \\hat{m}(\\nabla \\tilde{y}),$ i.e.",
"alignment of the magnetization direction with the external field is energetically favored.",
"The total energy of the system is given by ${\\cal E}^{\\rm tot}_\\varepsilon = {\\cal E}^\\chi _\\varepsilon + {\\cal E}^{\\rm mag}_\\varepsilon .$ We now suppose that the external field is homogeneous and satisfies without restriction $H_{\\rm ext} = \\kappa \\mathbf {e}_1$ for $\\kappa >0$ .",
"We then see that ${\\cal F}_\\varepsilon = {\\cal E}^{\\rm tot}_\\varepsilon - \\frac{\\kappa }{\\varepsilon }|\\Omega _\\varepsilon | $ with $f_{\\kappa }$ as in (REF ) and corresponding ${\\cal F}_\\varepsilon $ .",
"By Theorem REF we get that ${\\cal E}^{\\rm tot}_\\varepsilon $ $\\Gamma $ -converges to ${\\cal E}^{\\rm tot} = {\\cal F}$ after renormalization with the sequence of constants $(\\frac{\\kappa }{\\varepsilon }|\\Omega _\\varepsilon |)_{\\varepsilon }$ .",
"(Obviously, a configuration minimizes ${\\cal E}^{\\rm tot}_\\varepsilon $ if and only if it minimizes ${\\cal F}_{\\varepsilon }$ .)",
"We consider a boundary value problem $\\min _{u \\in {\\cal A}_g} {\\cal E}^{\\rm tot}(u)$ for $g \\in W^{1,\\infty }(\\tilde{\\Omega })$ .",
"Since the sequence $({\\cal E}^{\\rm tot}_\\varepsilon )_\\varepsilon $ is equicoercive as discussed in Section REF , the theory of $\\Gamma $ -convergence implies $\\lim _{\\varepsilon \\rightarrow 0 }\\min _{u \\in {\\cal A}_g} {\\cal F}_\\varepsilon (u) = \\min _{u \\in {\\cal A}_g} {\\cal F}^{\\rm tot}(u)$ and also convergence of the corresponding (almost) minimizers of ${\\cal E}^{\\rm tot}_{\\varepsilon }$ in the sense of Definition REF is guaranteed.",
"In this context, note that by a truncation argument taking $g\\in W^{1,\\infty }(\\tilde{\\Omega })$ into account, we may indeed assume that a low energy sequence satisfies $\\sup _\\varepsilon \\Vert u_\\varepsilon \\Vert _\\varepsilon < + \\infty $ ."
],
[
"Preparations",
"The goal of this section is the derivation of the $\\Gamma $ -convergence result for ${\\cal E}_{\\varepsilon }$ .",
"We first collect some properties of the cell energy $W_{\\triangle }$ proven in [22] provided that $W$ satisfies the assumptions (i), (ii) and (iii).",
"Lemma 3.1 $W_{\\triangle }$ is (i) frame indifferent: $W_{\\triangle }(QF) = W_{\\triangle }(F)$ for all $F \\in R^{2 \\times 2}$ , $Q \\in O(2)$ , (ii) non-negative and satisfies $W_{\\triangle }(F) = 0$ if and only if $F \\in O(2)$ and (iii) $\\liminf _{|F| \\rightarrow \\infty } W_{\\triangle }(F) = \\liminf _{|F| \\rightarrow \\infty } W_{\\triangle ,\\chi }(F) = \\beta $ .",
"Lemma 3.2 Let $F = \\mathbf {Id}+ G$ for $G \\in R^{2 \\times 2}$ .",
"Then for $|G|$ small $ W_{\\triangle }(F) = \\frac{1}{2} Q(G) + o(|G|^2),$ where $Q(G) = \\frac{3 \\alpha }{16} \\left( 3 g_{11}^2 + 3 g_{22}^2 + 2 g_{11} g_{22} + 4 \\left(\\frac{g_{12}+g_{21}}{2} \\right)^2 \\right)$ .",
"In particular, $Q(G)$ only depends on the symmetric part $\\left(G^{T} + G\\right)/2$ of $G$ .",
"$Q$ is positive semidefinite and thus convex on $R^{2 \\times 2}$ and positive definite and strictly convex on the subspace $R^{2 \\times 2}_{\\rm sym}$ of symmetric matrices.",
"The following lemma provides useful lower bounds for the energy $W_{\\triangle }$ and the pair interaction potential $W$ .",
"Lemma 3.3 For all $T>1$ one has: (i) There exists some $c > 0$ such that $c \\operatorname{dist}^2(F,O(2)) \\le W_{\\triangle } (F)$ for all $F \\in R^{2 \\times 2}$ satisfying $|F|\\le T$ .",
"(ii) For $\\rho > 0$ there is an increasing, subadditive function $\\psi ^{\\rho }:[0,\\infty ) \\rightarrow (0, \\infty )$ which satisfies $\\psi ^{\\rho }(r) - \\rho \\le W(r+1)$ for all $r\\ge 0$ and $\\psi (r) = \\beta $ for all $r \\ge c_{\\rho }$ for some constant $c_{\\rho }$ only depending on $\\rho $ .",
"Proof.",
"(i) This essentially follows from the expansion given in Lemma REF .",
"For details we refer to [22].",
"(ii) We define $ \\bar{\\psi }(r)= {\\left\\lbrace \\begin{array}{ll}\\eta r& \\text{for } 0 \\le r \\le \\frac{\\beta }{\\eta } , \\\\\\beta & \\text{for } r \\ge \\frac{\\beta }{\\eta },\\end{array}\\right.}",
"$ for some $\\eta > 0$ (depending on $\\rho $ ) such that $\\bar{\\psi } - \\rho \\le W$ .",
"Then we set $\\psi ^{\\rho }(r) = \\bar{\\psi }(r+1)$ .",
"As $\\psi ^{\\rho }$ is a concave function with $\\psi ^{\\rho }(0) > 0$ , it is subadditive.",
"$\\Box $ Moreover, we provide a lower bound for $W_{\\Delta ,\\chi }(F) + f_{\\kappa }(F)$ which implies the equicoercivity of $({\\cal F}^\\chi _\\varepsilon )_\\varepsilon $ .",
"Lemma 3.4 Let $T > \\sqrt{2}$ .",
"Then there are constants $C_1,C_2>0$ such that for all $F \\in R^{2 \\times 2}$ with $|F| \\le T$ we obtain i) $|\\hat{m}(F) - \\hat{m}(R(F))| \\le C_1|F - R(F)|^2$ , where $R(F) \\in SO(2)$ is a solution of $ |F - R(F)| = \\min _{R \\in SO(2)}|F - R|$ , (ii) $W_{\\Delta ,\\chi }(F) + f_{\\kappa }(F) \\ge C_2|F - \\mathbf {Id}|^2$ .",
"Proof.",
"(i) Without restriction we may assume that $|F - R(F)|$ is small as otherwise the assertion is clear.",
"So in particular, $R(F)$ is uniquely determined.",
"Moreover, it suffices to consider $F \\in R^{2 \\times 2}_{\\rm sym}$ and $R(F) = \\mathbf {Id}$ .",
"Indeed, once this is proved, we find $|\\hat{m}(F) - \\hat{m}(R(F))| = |R(F)\\hat{m}(R(F)^T F) - R(F)\\hat{m}(\\mathbf {Id}))| \\le C|R(F)^T F - \\mathbf {Id}|^2$ , as desired.",
"Let $F \\in R^{2 \\times 2}_{\\rm sym}$ , $R(F) = \\mathbf {Id}$ and set $G = F - \\mathbf {Id}$ with $G \\in R^{2 \\times 2}_{\\rm sym}$ small.",
"As $\\hat{m}$ is $C^2$ in a neighborhood of $SO(2)$ we derive $|\\hat{m}(F)- \\hat{m}(\\mathbf {Id})| \\le |D\\hat{m}(\\mathbf {Id}) \\, G| + C|G|^2 = C|G|^2$ as $R^{2 \\times 2}_{\\rm sym} \\subset {\\rm ker}(D\\hat{m}(\\mathbf {Id}))$ .",
"(ii) By Lemma REF (i) the assertion is clear for all $|F| \\le T$ with $c_0 \\le \\operatorname{dist}(F,O(2))$ for $c_0 >0$ and $C=C(c_0,T)$ sufficiently small.",
"Otherwise, we again apply Lemma REF (i) to obtain for $c_0$ small enough $W_{\\Delta ,\\chi }(F) \\ge C\\operatorname{dist}^2(F,O(2)) + \\chi (F) \\ge C\\operatorname{dist}^2(F,SO(2)) = C|F - R(F)|^2.$ For convenience we write $r_{ij} = \\mathbf {e}^T_i R(F) \\mathbf {e}_j$ for $i,j = 1,2$ .",
"As $r_{12}^2 = r_{21}^2 = 1- r_{11}^2$ we find $1- r_{11} = 1- r^2_{11} + r_{11}(r_{11}-1) = r_{12}^2 + (1- r_{11})^2 -(1- r_{11})$ .",
"Thus, recalling $\\hat{m}(R) = R\\mathbf {e}_1$ for all $R \\in SO(2)$ and applying (i) we get for $0 < c \\le \\kappa $ small enough $\\begin{split}W_{\\Delta ,\\chi }&(F) + f_{\\kappa }(F) \\\\& \\ge C|F - R(F)|^2 + c(1 - \\mathbf {e}_1 \\cdot \\hat{m}(R(F))) + c\\mathbf {e}_1\\cdot (\\hat{m}(R(F)) - \\hat{m}(F)) \\\\ &\\ge C|F - R(F)|^2 + c(1 - \\mathbf {e}_1^T R(F) \\mathbf {e}_1) - c C_{1}|F - R(F)|^2 \\\\& \\ge \\frac{C}{2} |F - R(F)|^2 + \\frac{c}{2} (1 - r_{11})^2 + \\frac{c}{2} r_{12}^2\\ge C_{2} |F - \\mathbf {Id}|^2,\\end{split}$ as desired.",
"$\\Box $ As a further preparation we modify the interpolation $\\tilde{y}$ on triangles with large deformation: We fix a threshold explicitly as $R = 7$ and let $\\bar{{\\cal C}}_{\\varepsilon } \\subset \\tilde{{\\cal C}}_{\\varepsilon }$ be the set of those triangles where $|(\\tilde{y})_{\\triangle }| > R$ .",
"By definition of the boundary values in (REF ) we find $\\bar{{\\cal C}}_{\\varepsilon } \\subset {{\\cal C}}_{\\varepsilon }$ for $\\varepsilon $ small enough.",
"We introduce another interpolation $y^{\\prime }$ which leaves $\\tilde{y}$ unchanged on $\\triangle \\in \\tilde{\\cal C}_{\\varepsilon } \\setminus \\bar{{\\cal C}}_{\\varepsilon }$ and replaces $\\tilde{y}$ on $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ by a discontinuous function with constant derivative satisfying $|(y^{\\prime })_{\\triangle }| \\le R$ .",
"In fact, by introducing jumps we achieve a release of the elastic energy.",
"Note that $y^{\\prime } \\in SBV (\\tilde{\\Omega }_\\varepsilon )$ .",
"More precisely, note that on $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ we have $|(\\tilde{y})_{\\triangle } \\, \\mathbf {v}| \\ge 2$ for at least two springs $\\mathbf {v}\\in {\\cal V}$ .",
"Indeed, using the elementary identity $\\sum _{\\mathbf {v}\\in {\\cal V}} \\langle \\mathbf {v}, H \\mathbf {v}\\rangle ^2= \\frac{3}{8} \\left( 2\\operatorname{trace}(H^2) + (\\operatorname{trace}H)^2 \\right)\\ge \\frac{3}{8} (\\operatorname{trace}H)^2$ for any $H \\in R^{2 \\times 2}_{\\rm sym}$ , we find that $|F| > 7$ implies $\\sum _{\\mathbf {v}\\in {\\cal V}} |F \\mathbf {v}|^4= \\sum _{\\mathbf {v}\\in {\\cal V}} \\langle \\mathbf {v}, F^T F \\mathbf {v}\\rangle ^{2}\\ge \\frac{3}{8} (\\operatorname{trace}(F^T F))^2= \\frac{3}{8} |F|^4$ and so $\\max _{\\mathbf {v}\\in {\\cal V}} |F \\mathbf {v}|^{4} > \\frac{7^4}{8} > 4^4$ .",
"Hence, $|F \\mathbf {v}| > 4$ for at least one $\\mathbf {v}\\in {\\cal V}$ and at least two springs are elongated by a factor larger than 2.",
"For $m=2,3$ let $\\bar{{\\cal C}}_{\\varepsilon ,m} \\subset \\bar{{\\cal C}}_{\\varepsilon }$ be the set of triangles where $|(\\tilde{y})_{\\triangle } \\, \\mathbf {v}| \\ge 2$ holds for exactly $m$ springs $\\mathbf {v}\\in {\\cal V}$ .",
"For $i,j,k=1,2,3$ pairwise distinct let $h_{i}$ denote the segment between the centers of the sides in $\\mathbf {v}_{j}$ and $\\mathbf {v}_{k}$ direction and define the set $V_{i}= h_{j} \\cup h_{k}$ .",
"We now construct $y^{\\prime } \\in SBV^2(\\tilde{\\Omega }_{\\varepsilon })$ .",
"On $\\triangle \\in \\tilde{\\cal C}_{\\varepsilon } \\setminus \\bar{{\\cal C}}_{\\varepsilon }$ we simply set $y^{\\prime } = \\tilde{y}$ .",
"On $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,2}$ , assuming $|(\\tilde{y})_{\\triangle } \\, \\mathbf {v}_i|\\le 2$ , we choose $y^{\\prime }$ such that $\\nabla y^{\\prime }$ assumes the constant value $(y^{\\prime })_{\\triangle }$ on $\\triangle $ with $(y^{\\prime })_{\\triangle } \\, \\mathbf {v}_i = (\\tilde{y})_{\\triangle } \\, \\mathbf {v}_i$ and $|(y^{\\prime })_{\\triangle } \\, \\mathbf {v}| = 1$ for $\\mathbf {v}\\in {\\cal V}\\setminus \\left\\lbrace \\mathbf {v}_i\\right\\rbrace $ .",
"Moreover, we ask that $y^{\\prime } = \\tilde{y}$ at the three vertices and on the side orientated in $\\mathbf {v}_i$ direction.",
"This can and will be done in such a way that $y^{\\prime }$ is continuous on $\\text{int}(\\triangle )\\setminus h_{i}$ .",
"We note that the definition of $(y^{\\prime })_{\\triangle }$ is unique up to a reflection, unless $(\\tilde{y})_{\\triangle } \\mathbf {v}_{i} = 0$ .",
"We may and will assume that $\\operatorname{dist}\\left((y^{\\prime })_{\\triangle }, SO(2)\\right) \\le \\operatorname{dist}\\left((y^{\\prime })_{\\triangle }, O(2) \\setminus SO(2)\\right).$ For $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ we set $(y^{\\prime })_{\\triangle } = \\mathbf {Id}$ and $y^{\\prime } = \\tilde{y}$ at the three vertices such that $y^{\\prime }$ is continuous on $\\text{int}(\\triangle )\\setminus V_{i}$ for some $i \\in \\lbrace 1,2,3\\rbrace $ .",
"Here, the index $i$ can be taken arbitrarily at first.",
"However, in what follows it will also be necessary to use the following unambiguously defined `variants' of $y^{\\prime }$ : If on every $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ the set $V_{i}$ is chosen as the jump set of $y^{\\prime }$ we denote this interplation explicitly as $y^{\\prime }_{V_{i}}$ .",
"We define the interpolation $u^{\\prime }$ for the rescaled displacement field by $u^{\\prime } = \\frac{1}{\\sqrt{\\varepsilon }} (y^{\\prime } - \\mathbf {id})$ .",
"We note that by construction also on an edge $[p,q] \\subset \\partial \\triangle $ for $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ jumps may occur.",
"There, however, the jump height $|[u_{\\varepsilon }]|$ can be bounded by $|[u^{\\prime }_{\\varepsilon }](x)|\\le \\varepsilon \\left\\Vert \\nabla u^{\\prime }_{\\varepsilon }\\right\\Vert _{\\infty } \\le \\varepsilon \\cdot c\\varepsilon ^{-\\frac{1}{2}} = c\\sqrt{\\varepsilon }$ for a constant $c > 0$ independent of $\\varepsilon $ and $x \\in [p, q]$ .",
"This holds since the interpolations are continuous at the vertices.",
"The following lemma shows that we may pass from $\\tilde{u}_{\\varepsilon }$ to $u^{\\prime }_{\\varepsilon }$ without changing the limit.",
"Lemma 3.5 If $u_{\\varepsilon } \\rightarrow u$ in the sense of Definition REF and ${\\cal E}(u_{\\varepsilon })$ is uniformly bounded, then $\\chi _{\\tilde{\\Omega }_{\\varepsilon }} u^{\\prime }_{\\varepsilon } \\rightarrow u$ in $L^{1}(\\tilde{\\Omega })$ , $\\chi _{\\tilde{\\Omega }_{\\varepsilon }}\\nabla u_{\\varepsilon }^{\\prime } \\rightharpoonup \\nabla u$ in $L^2(\\tilde{\\Omega })$ and ${\\cal H}^1(J_{u^{\\prime }_{\\varepsilon }})$ is uniformly bounded.",
"Proof.",
"We first note that there is some $M > 0$ such that $\\#\\bar{{\\cal C}}_{\\varepsilon } \\le \\frac{M}{\\varepsilon }$ for all $\\varepsilon >0$ .",
"To see this, we just recall that every triangle $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ provides at least the energy $\\varepsilon \\inf \\left\\lbrace W(r): r \\ge 2\\right\\rbrace $ .",
"In fact we may assume that ${\\cal C}^*_{\\varepsilon } = \\bar{\\cal C}_{\\varepsilon }$ in Definition REF as for $\\Delta \\in {\\cal C}^*_{\\varepsilon } \\setminus \\bar{\\cal C}_{\\varepsilon }$ we have $|(\\tilde{u}_{\\varepsilon })_{\\triangle }| \\le \\frac{C}{\\sqrt{\\varepsilon }} |(\\tilde{y}_{\\varepsilon })_{\\triangle } - \\mathbf {Id}| \\le \\frac{C}{\\sqrt{\\varepsilon }}$ and so $\\Vert \\nabla \\tilde{u}_{\\varepsilon } \\Vert _{L^2(\\tilde{\\Omega }_{\\varepsilon } \\setminus \\cup _{\\triangle \\in \\bar{\\cal C}_{\\varepsilon }} \\triangle )}&\\le \\Vert \\nabla \\tilde{u}_{\\varepsilon } \\Vert _{L^2(\\tilde{\\Omega }_{\\varepsilon } \\setminus \\cup _{\\triangle \\in {\\cal C}^*_{\\varepsilon }} \\triangle )}+ \\Vert \\nabla \\tilde{u}_{\\varepsilon } \\Vert _{L^2(\\cup _{\\triangle \\in {\\cal C}^*_{\\varepsilon } \\setminus \\bar{\\cal C}_{\\varepsilon }} \\triangle )} \\\\& \\le C + \\left( \\# ( {\\cal C}^*_{\\varepsilon } \\setminus \\bar{\\cal C}_{\\varepsilon } ) \\frac{\\sqrt{3}\\varepsilon ^2}{4} \\cdot \\frac{C}{\\varepsilon } \\right)^{\\frac{1}{2}}\\le C.$ It follows that $\\chi _{\\tilde{\\Omega }_{\\varepsilon }}\\nabla u_{\\varepsilon }^{\\prime }$ is bounded uniformly in $L^2$ and, in particular, equiintegrable.",
"Finally, the jump lengths ${\\cal H}^1(J_{u^{\\prime }_{\\varepsilon }})$ are readlily seen to be bounded by $C \\varepsilon \\#\\bar{{\\cal C}}_{\\varepsilon } \\le C$ .",
"But then Ambrosio's compactness Theorem for GSBV [4] shows that indeed $\\chi _{\\tilde{\\Omega }_{\\varepsilon }}\\nabla u_{\\varepsilon }^{\\prime } \\rightharpoonup \\nabla u$ in $L^2(\\tilde{\\Omega })$ .",
"$\\Box $"
],
[
"The $\\Gamma $ -{{formula:d930f20f-3c3d-4767-8730-fe3a6053e731}} -inequality",
"With the above preparations at hand, we may now prove the $\\Gamma $ -$\\liminf $ -inequality in Theorem REF .",
"Proof of Theorem REF (i).",
"Let $(g_\\varepsilon )_\\varepsilon \\in W^{1,\\infty }(\\tilde{\\Omega })$ with $ \\sup _{\\varepsilon } \\Vert g_\\varepsilon \\Vert _{W^{1,\\infty }(\\tilde{\\Omega })} < + \\infty $ be given.",
"Let $u \\in SBV^2(\\tilde{\\Omega })$ and consider a sequence $u_{\\varepsilon } \\subset SBV^2(\\tilde{\\Omega }_{\\varepsilon })$ with $u_{\\varepsilon } \\in {\\cal A}_{g_{\\varepsilon }}$ converging to $u$ in $SBV^2$ in the sense of Definition REF .",
"We split up the energy into bulk and crack parts neglecting the contribution $\\varepsilon E^{\\rm boundary}_{\\varepsilon }$ from the boundary layers: $\\begin{split}{\\cal E}_{\\varepsilon }(u_{\\varepsilon })& \\ge \\varepsilon \\sum _{\\triangle \\in {\\cal C}_{\\varepsilon } \\setminus \\bar{{\\cal C}}_{\\varepsilon }} W_{\\triangle }((\\tilde{y}_{\\varepsilon })_{\\triangle }) + \\varepsilon \\sum _{\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }} W_{\\triangle }((\\tilde{y}_{\\varepsilon })_{\\triangle })\\\\& = \\frac{4}{\\sqrt{3}\\varepsilon } \\int _{\\Omega _{\\varepsilon }} W_{\\triangle }\\left(\\mathbf {Id}+ \\sqrt{\\varepsilon } \\nabla u^{\\prime }_{\\varepsilon }\\right) + \\varepsilon \\sum _{\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }} \\ \\sum _{\\mathbf {v}\\in {\\cal V}, \\atop |(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}|>2} \\frac{1}{2} W\\left(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}|\\right)\\\\& =: {\\cal E}^{\\rm elastic}_{\\varepsilon } (u_{\\varepsilon }) + {\\cal E}^{\\rm crack}_{\\varepsilon } (u_{\\varepsilon }).\\end{split}$ We note that by contruction of the interpolation $u^{\\prime }_{\\varepsilon }$ we may take the integral over $\\Omega _{\\varepsilon }$ .",
"As both parts separate completely in the limit, we discuss them individually.",
"Elastic energy.",
"We first concern ourselves with the elastic part of the energy.",
"We recall $W_{\\triangle }(\\mathbf {Id}+ G) = \\frac{1}{2}Q(G) + \\omega (G)$ with $\\sup \\left\\lbrace \\frac{\\omega (F)}{|F|^2} : |F|\\le \\rho \\right\\rbrace \\rightarrow 0$ as $\\rho \\rightarrow 0$ .",
"Let $\\chi _{\\varepsilon }(x) := \\chi _{[0,\\varepsilon ^{-1/4})}(|\\nabla u^{\\prime }_{\\varepsilon }(x)|)$ .",
"Note that for $F \\in R^{2 \\times 2}$ , $r>0$ one has $Q(rF) = r^2 Q(F)$ .",
"We compute ${\\cal E}^{\\rm elastic}_{\\varepsilon } (u_{\\varepsilon })& \\ge \\frac{4}{\\sqrt{3}} \\int _{\\Omega _{\\varepsilon }} \\chi _{\\varepsilon }(x)\\left( \\frac{1}{2} Q( \\nabla u^{\\prime }_{\\varepsilon }) + \\frac{1}{\\varepsilon } \\omega \\left(\\sqrt{\\varepsilon }\\nabla u^{\\prime }_{\\varepsilon }(x)\\right) \\right) \\, dx.$ The second term of the integral can be bounded by $\\chi _{\\varepsilon } |\\nabla u^{\\prime }_{\\varepsilon }|^2\\frac{\\omega \\left(\\sqrt{\\varepsilon } \\nabla u^{\\prime }_{\\varepsilon } \\right)}{|\\sqrt{\\varepsilon }\\nabla u^{\\prime }_{\\varepsilon } |^2}.$ Since $\\nabla u^{\\prime }_{\\varepsilon }$ is bounded in $L^2$ and $\\chi _{\\varepsilon } \\frac{\\omega \\left(\\sqrt{\\varepsilon } \\nabla u^{\\prime }_{\\varepsilon } \\right)}{|\\sqrt{\\varepsilon } \\nabla u^{\\prime }_{\\varepsilon } |^2}$ converges uniformly to 0 as $\\varepsilon \\rightarrow 0$ it follows that $\\liminf _{\\varepsilon \\rightarrow 0}{\\cal E}^{\\rm elastic}_{\\varepsilon } (u_{\\varepsilon }) & \\ge \\liminf _{\\varepsilon \\rightarrow 0} \\frac{4}{\\sqrt{3}} \\int _{\\Omega _{\\varepsilon }} \\chi _{\\varepsilon }(x) \\frac{1}{2} Q( \\nabla u^{\\prime }_{\\varepsilon }(x)) \\, dx\\\\& \\ge \\liminf _{\\varepsilon \\rightarrow 0} \\frac{4}{\\sqrt{3}} \\int _{\\Omega } \\frac{1}{2} Q(\\chi _{\\Omega _{\\varepsilon }} \\chi _{\\varepsilon }(x) \\nabla u^{\\prime }_{\\varepsilon }(x)) \\, dx.$ By assumption $\\chi _{\\Omega _{\\varepsilon }} \\nabla u^{\\prime }_{\\varepsilon } \\rightharpoonup \\nabla u$ weakly in $L^2$ .",
"As $\\chi _{\\varepsilon } \\rightarrow 1$ boundedly in measure on $\\Omega $ , it follows $\\chi _{\\Omega _{\\varepsilon }}\\chi _{\\varepsilon } \\nabla u^{\\prime }_{\\varepsilon } \\rightharpoonup u$ weakly in $L^2(\\Omega )$ .",
"By lower semicontinuity (Q is convex by Lemma REF ) we conclude recalling that $Q$ only depends on the symmetric part of the gradient: $\\liminf _{\\varepsilon \\rightarrow 0}{\\cal E}^{\\rm elastic}_{\\varepsilon } (u_{\\varepsilon })\\ge \\frac{4}{\\sqrt{3}} \\int _{\\Omega } \\frac{1}{2} Q(e(u(x))) \\, dx.$ Crack energy.",
"By construction the functions $u^{\\prime }_{\\varepsilon }$ have jumps on destroyed triangles $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ .",
"We now write the energy of such a triangle in terms of the jump height $\\left[u\\right] = u^{+} - u^{-}$ .",
"We first concern ourselves with a triangle $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ .",
"For the variant $u^{\\prime }_{\\varepsilon ,V_{i}}$ , $i=1,2,3$ we consider the springs in $\\mathbf {v}_{j},\\mathbf {v}_{k}$ direction for $j,k \\ne i$ .",
"Thus, we compute $\\varepsilon (\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j} = \\varepsilon (y^{\\prime }_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j} + [y^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}} = \\varepsilon \\mathbf {v}_{j} + \\sqrt{\\varepsilon } [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}},$ where $[u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}$ denotes the jump height on the set $h_{\\mathbf {v}_{k}}$ .",
"Here and in the following equations, the same holds true if we interchange the roles of $j$ and $k$ .",
"We claim that $|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}|\\ge \\varepsilon ^{\\frac{1}{4}} \\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right| + 1.$ Indeed, for $|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}| \\le \\varepsilon ^{-\\frac{1}{4}}$ this is clear since $|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}| \\ge 2$ .",
"Otherwise, applying (REF ) we compute for $\\varepsilon $ small enough: $|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}|&= \\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}} + \\mathbf {v}_{j}\\right|\\ge \\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right| - 1 \\\\&\\ge \\varepsilon ^{\\frac{1}{4}}\\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right| + \\left(1 - \\varepsilon ^{\\frac{1}{4}}\\right)\\varepsilon ^{-\\frac{1}{4}} - 1 \\\\&= \\varepsilon ^{\\frac{1}{4}}\\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right| - 2 + \\varepsilon ^{-\\frac{1}{4}}\\ge \\varepsilon ^{\\frac{1}{4}}\\left|\\frac{1}{\\sqrt{\\varepsilon }} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right| + 1.$ Let $\\rho > 0$ sufficiently small.",
"Applying Lemma REF (ii) there is an increasing subadditive function $\\psi ^{\\rho } 0$ with $\\psi ^{\\rho }(r-1) - \\rho \\le W(r)$ for $r \\ge 1$ .",
"We define $\\tilde{\\psi }^{\\rho } = \\psi ^{\\rho } - \\rho $ .",
"The monotonicity of $\\psi ^{\\rho }$ and (REF ) yield $W(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}|)\\ge \\tilde{\\psi }^{\\rho }(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}|-1)\\ge \\tilde{\\psi }^{\\rho }\\left(\\left|\\varepsilon ^{-\\frac{1}{4}} [u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}\\right|\\right).$ Now for $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ we may estimate the energy as follows: $W_{\\triangle } \\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right)&= \\frac{1}{2} \\sum ^{3}_{l=1} W(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{l}|) \\\\&\\ge \\frac{1}{4} \\sum ^{3}_{i=1} \\left\\lbrace \\tilde{\\psi }^{\\rho }\\left(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{k}}}|\\right) + \\tilde{\\psi }^{\\rho }\\left(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon , V_{i}}]_{h_{\\mathbf {v}_{j}}}|\\right) \\right\\rbrace =: W_{\\triangle ,3}\\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right),$ where $i,j,k =1,2,3$ are pairwise distinct.",
"With $\\nu ^{(i)}_u = \\nu _{u^{\\prime }_{\\varepsilon , V_{i}}}$ we can also write $ W_{\\triangle ,3}\\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right)= \\frac{1}{4} \\cdot \\frac{2}{\\varepsilon } \\cdot \\frac{2}{\\sqrt{3}} \\sum ^{3}_{i=1} \\int _{h_{\\mathbf {v}_{j}} \\cup h_{\\mathbf {v}_{k}}} \\tilde{\\psi }^{\\rho }\\left(\\varepsilon ^{-\\frac{1}{4}} |[u^{\\prime }_{\\varepsilon , V_{i}}]|\\right) \\left(|\\mathbf {v}_{j}\\cdot \\nu ^{(i)}_{u}| + |\\mathbf {v}_{k}\\cdot \\nu ^{(i)}_{u}|\\right) d{\\cal H}^1.",
"$ The factors in front occur since ${\\cal H}^1(h_{\\mathbf {v}_{j}}) = \\frac{\\varepsilon }{2}$ and, letting $\\nu _{j}$ be a normal of $h_{\\mathbf {v}_{j}}$ , one has $|\\nu _{j} \\cdot \\mathbf {v}_{j}| = 0$ and $|\\nu _{j} \\cdot \\mathbf {v}_{k}| = \\frac{\\sqrt{3}}{2}$ .",
"Consequently, defining $\\phi ^{\\rho }_{i}(r,\\nu ) = \\psi ^{\\rho }(r) \\left(|\\mathbf {v}_{j}\\cdot \\nu | + |\\mathbf {v}_{k}\\cdot \\nu |\\right)$ and $\\tilde{\\phi }^{\\rho }_{i}(r,\\nu ) = \\tilde{\\psi }^{\\rho }(r) \\left(|\\mathbf {v}_{j}\\cdot \\nu | + |\\mathbf {v}_{k}\\cdot \\nu |\\right)$ , respectively, we get $W_{\\triangle ,3}\\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right)= \\frac{1}{\\sqrt{3}{\\varepsilon }} \\sum ^{3}_{i=1} \\int _{J_{u^{\\prime }_{\\varepsilon , V_{i}}} \\cap {\\text{int}}(\\triangle )} \\tilde{\\phi }^{\\rho }_{i}(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon , V_{i}}]|,\\nu ^{(i)}_{u}) \\, d{\\cal H}^1$ on every $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ .",
"For $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,2}$ we proceed analogously.",
"Assuming $|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{i}| \\le 2$ we compute for the springs in $\\mathbf {v}_{j}, \\mathbf {v}_{k}$ direction (abbreviated by $\\mathbf {v}_{j,k}$ ) as in (REF ) $\\varepsilon (\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j,k} = \\varepsilon (y^{\\prime }_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j,k} + \\sqrt{\\varepsilon } [u^{\\prime }_{\\varepsilon }]_{h_{\\mathbf {v}_{i}}}.$ Note that in this case we do not have to take a special variant of $u^{\\prime }_{\\varepsilon }$ into account.",
"Repeating the steps (REF ) and (REF ) we find $\\frac{1}{2} \\left(W(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{j}|) + W(|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}_{k}|) \\right)\\ge \\tilde{\\psi }^{\\rho }\\left(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon }]_{h_{\\mathbf {v}_{i}}}|\\right)=: W_{\\triangle ,2}\\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right).$ Noting that $|\\mathbf {v}_j \\cdot \\nu _i| = |\\mathbf {v}_k \\cdot \\nu _i| = \\frac{\\sqrt{3}}{2}$ , $|\\mathbf {v}_i \\cdot \\nu _i| = 0$ and that every of these terms occurs twice in the sum of the right hand side of the following formula, it is not hard to see that this energy satisfies the same integral representation formula as $W_{\\triangle ,3}$ : $W_{\\triangle ,2}\\left( (\\tilde{y}_{\\varepsilon })_{\\triangle }\\right)= \\frac{1}{\\sqrt{3}{\\varepsilon }} \\sum ^{3}_{i=1} \\int _{J_{u^{\\prime }_{\\varepsilon , V_{i}}} \\cap {\\text{int}}(\\triangle )} \\tilde{\\phi }^{\\rho }_{i}(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon , V_{i}}]|,\\nu ^{(i)}_{u}) \\, d{\\cal H}^1.$ (Recall that the interpolation variant $u^{\\prime }_{\\varepsilon , V_{i}}$ and its crack normal $\\nu ^{(i)}_{u}$ do not depend on $i$ on $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,2}$ .)",
"Let $\\sigma > 0$ .",
"Note that $\\bar{{\\cal C}}_{\\varepsilon } \\subset {{\\cal C}}_{\\varepsilon }$ for $\\varepsilon $ sufficiently small as $\\sup _{\\varepsilon } \\Vert g_\\varepsilon \\Vert _{W^{1,\\infty }(\\tilde{\\Omega })} < + \\infty $ .",
"Thus, the crack energy can be estimated by ${\\cal E}^{\\rm crack}_{\\varepsilon } (u_{\\varepsilon })& \\ge \\frac{1}{\\sqrt{3}} \\sum _{i} \\int _{J_{u^{\\prime }_{\\varepsilon , V_{i}}} \\cap \\tilde{\\Omega }_{\\varepsilon }} \\tilde{\\phi }^{\\rho }_{i}(\\varepsilon ^{-\\frac{1}{4}}|[u^{\\prime }_{\\varepsilon , V_{i}}]|,\\nu ^{(i)}_{u}) \\, d{\\cal H}^1 - E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right) \\\\& \\ge \\frac{1}{\\sqrt{3}} \\sum _{i} \\int _{J_{u^{\\prime }_{\\varepsilon , V_{i}}} \\cap \\tilde{\\Omega }_{\\varepsilon }} \\left(\\phi ^{\\rho }_{i}(\\sigma ^{-1}|[u^{\\prime }_{\\varepsilon , V_{i}}]|,\\nu ^{(i)}_{u}) - 2 \\rho \\right) \\, d{\\cal H}^1 - E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right),$ where $E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right)$ compensates for the extra contribution provided by jumps lying on the boundary of some $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ .",
"We will show that this term vanishes in the limit.",
"Now by construction the $\\phi ^{\\rho }_{i}(r, \\nu )$ , $i = 1,2,3$ , are products of a positive, increasing and concave function in $r$ and a norm in $\\nu $ .",
"Moreover, $u^{\\prime }_{\\varepsilon }$ and its variants converge to $u$ in $L^1$ with $\\nabla u^{\\prime }_{\\varepsilon }$ bounded in $L^2$ and thus equiintegrable.",
"By Ambrosio's lower semicontinuity Theorem [4] we obtain $ \\liminf _{\\varepsilon \\rightarrow 0} {\\cal E}^{\\rm crack}_{\\varepsilon } (u_{\\varepsilon }) \\ge \\frac{1}{\\sqrt{3}} \\int _{J_{u}} \\sum _{i} \\phi ^{\\rho }_{i}(\\sigma ^{-1}|[u]|,\\nu _{u})\\, d{\\cal H}^1 - C M \\rho - \\limsup _{\\varepsilon \\rightarrow 0} E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right),$ where we used that $\\sup _{\\varepsilon } {\\cal H}^1(J_{u^{\\prime }_{\\varepsilon }}) \\le CM$ for a constant $C > 0$ by (REF ).",
"We recall that $\\psi ^{\\rho }(r) \\rightarrow \\beta $ for $r \\rightarrow \\infty $ .",
"In the limit $\\sigma \\rightarrow 0$ this yields $\\liminf _{\\varepsilon \\rightarrow 0} {\\cal E}^{\\rm crack}_{\\varepsilon } (u_{\\varepsilon }) \\ge \\frac{1}{\\sqrt{3}} \\int _{J_{u}} 2\\beta \\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu _{u}|\\, d{\\cal H}^1 - CM\\rho - \\limsup _{\\varepsilon \\rightarrow 0} E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right).$ Taking (REF ) and (REF ) into account we compute $\\limsup _{\\varepsilon \\rightarrow 0} \\sum _{\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }} \\int _{\\partial \\triangle }|\\tilde{\\psi }^{\\rho }\\left(\\varepsilon ^{-\\frac{1}{4}} |[u^{\\prime }_{\\varepsilon }]|\\right)|&\\le \\lim _{\\varepsilon \\rightarrow 0} CM \\sup \\left\\lbrace |\\psi ^{\\rho }\\left(r\\right) - \\rho |: \\, r\\le \\varepsilon ^{-\\frac{1}{4}} \\cdot c\\varepsilon ^{\\frac{1}{2}}\\right\\rbrace \\\\&= CM \\rho .$ This proves $\\limsup _{\\varepsilon } |E^{\\rho }_{\\varepsilon , \\cup \\partial \\triangle }\\left(\\tilde{y}_{\\varepsilon }\\right)| \\le \\tilde{C}M \\rho $ for some $\\tilde{C} > 0$ .",
"We finally let $\\rho \\rightarrow 0$ in (REF ).",
"This finishes the proof of (i).",
"$\\Box $ We now prove the $\\Gamma $ -$\\liminf $ -inequality in Theorem REF .",
"Proof of Theorem REF , first part.",
"Following the proof of Theorem REF (i) it suffices to show $\\liminf _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon }\\chi _\\varepsilon f_{\\kappa }(\\nabla y^{\\prime }_\\varepsilon ) \\ge - \\frac{\\kappa }{2}\\int _\\Omega \\hat{Q}(\\nabla u),$ where $\\hat{Q} = D^2 \\hat{m}_1(\\mathbf {Id})$ .",
"Let $u^{\\prime }_\\varepsilon = \\frac{1}{\\sqrt{\\varepsilon }} (y^{\\prime }_\\varepsilon - \\mathbf {id})$ .",
"With a slight abuse of notation we set $e(F) = \\frac{1}{2}(F^T + F)$ and $a(F) = F - e(F)$ for matrices $F \\in R^{2 \\times 2}$ .",
"Let $F = \\mathbf {Id}+ \\sqrt{\\varepsilon }G$ for $G \\in R^{2 \\times 2}$ .",
"Linearization around the identity matrix yields $\\operatorname{dist}(F,SO(2)) = \\sqrt{\\varepsilon }|e(G)| + \\varepsilon O(|G|^2)$ .",
"It is not hard to see that this implies $R(F) = \\mathbf {Id}+ \\sqrt{\\varepsilon } a(G) + \\varepsilon O(|G|^2),$ where $R(F) \\in SO(2)$ is as defined in Lemma REF .",
"As $\\hat{m}(\\mathbf {Id}) =\\mathbf {e}_1$ and $e(G) \\in {\\rm ker}(D\\hat{m}(\\mathbf {Id}))$ , we find by expanding $\\hat{m}_1$ $\\hat{m}_1(F) = 1 + \\sqrt{\\varepsilon }D \\hat{m}_{1}(\\mathbf {Id}) a(G) + \\frac{\\varepsilon }{2} \\hat{Q}( G) + \\omega (\\sqrt{\\varepsilon } G)$ with $\\sup \\left\\lbrace \\frac{\\omega (H)}{|H|^2} : |H|\\le \\rho \\right\\rbrace \\rightarrow 0$ as $\\rho \\rightarrow 0$ .",
"We concern ourselves with the term $D \\hat{m}_{1}( \\mathbf {Id}) a(G)$ .",
"Recall that $|\\hat{m}(R(F)) - \\hat{m}(F)| \\le C |R(F) - F|^2$ by Lemma REF (i).",
"For $F = \\mathbf {Id}+ \\sqrt{\\varepsilon }G $ this implies by (REF ) $D \\hat{m}_{1}( \\mathbf {Id}) a(G)&= \\mathbf {e}_1 \\cdot D\\hat{m}(\\mathbf {Id}) G = \\lim _{\\varepsilon \\rightarrow 0} \\mathbf {e}_1 \\cdot \\frac{\\hat{m}(F) - \\hat{m}(\\mathbf {Id})}{\\sqrt{\\varepsilon }} \\\\& = \\lim _{\\varepsilon \\rightarrow 0} \\mathbf {e}_1 \\cdot \\frac{\\hat{m}(R(F)) - \\mathbf {e}_1}{\\sqrt{\\varepsilon }} + O(\\sqrt{\\varepsilon }) = \\lim _{\\varepsilon \\rightarrow 0} \\mathbf {e}_1 \\cdot a(G)\\mathbf {e}_1 + O(\\sqrt{\\varepsilon }) = 0.$ In particular, (REF ) then implies $0 \\le \\frac{1}{\\varepsilon } f_\\kappa (F) = -\\frac{\\kappa }{2} \\hat{Q}(G) - \\frac{1}{\\varepsilon }\\omega (\\sqrt{\\varepsilon }G)$ and thus $-\\hat{Q}$ is positive semidefinite.",
"We proceed exactly as in the proof of Theorem REF (i) and conclude $\\liminf _{\\varepsilon \\rightarrow 0}\\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon } \\chi _\\varepsilon f_{\\kappa }(\\nabla y^{\\prime }_\\varepsilon ) &\\ge \\liminf _{\\varepsilon \\rightarrow 0} -\\int _{\\Omega _\\varepsilon } \\chi _\\varepsilon \\Big ( \\frac{\\kappa }{2} \\hat{Q}( \\nabla u^{\\prime }_\\varepsilon ) + \\frac{\\kappa }{\\varepsilon } \\omega (\\sqrt{\\varepsilon } \\nabla u^{\\prime }_\\varepsilon )\\Big )\\\\ &\\ge -\\frac{\\kappa }{2}\\int _{\\Omega } \\hat{Q}(\\nabla u).$ $\\Box $"
],
[
"Recovery sequences",
"It remains to construct recovery sequences in order to complete the proof of Theorem REF .",
"Proof of Theorem REF (ii).",
"The basic tool for the proof of the $\\Gamma $ -limsup-inequality is a density result for $SBV$ functions due to Cortesani and Toader [15].",
"Moreover, a proof very similar to that of Proposition 2.5 in [20] shows that we may also impose suitable boundary conditions on the approximating sequence.",
"We suppose ${\\cal W}(\\Omega , R^2)$ is the space of all $SBV$ functions $u \\in SBV(\\Omega , R^2)$ such that $J_{u}$ is a finite union of (disjoint) segments and $u \\in W^{k,\\infty }(\\Omega \\setminus J_{u}, R^2)$ for all $k$ .",
"Then ${\\cal W}(\\Omega , R^2)$ is dense in $SBV^2(\\Omega , R^2) \\cap L^{\\infty }(\\Omega , R^2)$ in the following way: For every $u \\in SBV^2(\\tilde{\\Omega }, R^2) \\cap L^{\\infty }(\\tilde{\\Omega }, R^2)$ with $u=g$ on $\\tilde{\\Omega } \\setminus \\Omega $ , there exists a sequence $u_n$ and a sequence of neighborhoods $U_n \\subset \\tilde{\\Omega }$ of $\\tilde{\\Omega }\\setminus \\Omega $ such that $u_n = g$ on $\\Omega _{D,\\frac{1}{n}}$ (recall (REF )), $u_n \\in W^{1,\\infty }(U_n)$ and $u_n|_{V_n} \\in {\\cal W}(V_n, R^2)$ , where $V_n \\subset \\Omega $ is some neighborhood of $\\Omega \\setminus U_n$ , such that $\\left\\Vert u_n\\right\\Vert _{\\infty } \\le \\left\\Vert u\\right\\Vert _{\\infty }$ and (i) $u_{ n} \\rightarrow u$ strongly in $L^1(\\Omega ,R^2)$ , $\\nabla u_n \\rightarrow \\nabla u$ strongly in $L^2(\\Omega , R^2)$ , (ii) $\\limsup _{n \\rightarrow \\infty } \\int _{J_{u_{n}}} \\phi (\\nu _{u_{n}})d{\\cal H}^1 \\le \\int _{J_{u} }\\phi (\\nu _{u})d{\\cal H}^1$ for every upper semicontinuous function $\\phi :S^1 \\rightarrow [0,\\infty )$ satisfying $\\phi (\\nu ) = \\phi (-\\nu )$ for every $\\nu \\in S^1$ .",
"Let $u \\in SBV^2(\\tilde{\\Omega }, R^2)$ with $u=g$ on $\\tilde{\\Omega } \\setminus \\Omega $ .",
"Without restriction we can assume $u \\in \\cap L^{\\infty }(\\tilde{\\Omega }, R^2)$ as this hypothesis my be dropped by applying a truncation argument and taking $Q(F) \\le C |F|^2$ into account.",
"In fact, it suffices to provide a recovery sequence for an approximation $u_n$ defined above.",
"Although our notion of convergence in Definition REF is not given in terms of a specific metric, similarly to a general density result in the theory of $\\Gamma $ -convergence this can be seen by a diagonal sequence argument.",
"The crucial point is that due to (REF ) below we may assume that for $\\varepsilon $ sufficiently small (depending on $n$ ) $ \\# {\\cal C}^*_{\\varepsilon } = \\# {\\cal D}_{\\varepsilon } \\le \\frac{C {\\cal H}^1(J_{u_n})}{\\varepsilon } \\le \\frac{C {\\cal H}^1(J_{u})}{\\varepsilon }, $ where $C$ is independent of $n$ and $\\varepsilon $ .",
"If $(u_{n,\\varepsilon })_{\\varepsilon }$ is a recovery for $u_n$ , one may therefore pass to a diagonal sequence which is a recovery sequence for $u$ , in particular converging to $u$ the sense of Definition REF .",
"For simplicity write $u$ instead of $u_n$ in what follows.",
"Let $\\delta >0$ and define $J^{\\delta }_{u} = \\left\\lbrace x \\in J_{u}, |[u](x)| \\ge \\delta \\right\\rbrace $ .",
"Since $|[u]|$ is Lipschitz continuous on $J_u$ , it cannot oscillate infinitely often between values $\\le \\delta $ and values $\\ge 2\\delta $ on a single segment.",
"Consequently, there is a finite number $N_u^{\\delta }$ of disjoint subsegments $S_1, \\ldots , S_{N_u^{\\delta }}$ in $J_u$ such that $|[u]| < 2 \\delta $ on every $S_j$ and $|[u]| > \\delta $ on $J_u \\setminus (S_1 \\cup \\ldots \\cup S_{N_u^{\\delta }})$ .",
"Note that ${\\cal H}^1( \\bigcup ^{N_u^{\\delta }}_{i=1} S_i) \\le {\\cal H}^1(J_u \\setminus J^{2\\delta }_u)=: \\rho (\\delta ) \\rightarrow 0$ for $\\delta \\rightarrow 0$ .",
"We cover $S_1, \\ldots , S_{N_u^{\\delta }}$ by pairwise disjoint rectangles $Q_1, \\ldots Q_{N_u^{\\delta }}$ which satisfy $\\sum _j {\\cal H}^1(\\partial Q_i) + |Q_i| \\le C\\rho (\\delta )$ .",
"It is not hard to see that $|u({ x}) - u(y)| \\le C{\\cal H}^1(\\partial Q_i) + 2\\delta $ for $x,y \\in Q_j$ as $\\nabla u \\in L^\\infty (\\tilde{\\Omega })$ .",
"We modify $u$ on the rectangles $Q_i$ : Let $u_\\delta = u$ on $\\tilde{\\Omega } \\setminus \\bigcup ^{N_u^{\\delta }}_{i=1} Q_j$ and define $u_\\delta = c_j$ on $Q_j$ for $c_j \\in R^2$ in such a way that $J_{u_\\delta } = J^\\delta _{u_\\delta }$ up to an ${\\cal H}^1$ -negligible set.",
"As $u \\in L^\\infty (\\tilde{\\Omega })$ , $\\nabla u \\in L^\\infty (\\tilde{\\Omega })$ we find $u_\\delta \\rightarrow u$ in $L^1(\\tilde{\\Omega })$ and $\\nabla u_\\delta \\rightarrow \\nabla u$ in $L^2(\\tilde{\\Omega })$ .",
"Moreover, we have ${\\cal H}^1(J_u \\Delta J_{u_\\delta }) \\le C\\rho (\\delta ) \\rightarrow 0$ for $\\delta \\rightarrow 0$ .",
"Consequently, it suffices to establish a recovery sequence for a function $u \\in {\\cal W}(\\Omega )$ with $u = g$ in a neighborhood of $\\tilde{\\Omega } \\setminus \\Omega $ and $J_u = J^\\delta _u$ for some $\\delta >0$ .",
"Note after the above modification the segments of $J_u$ might not be pairwise disjoint.",
"We define $u_{\\varepsilon }(x) = u(x)$ for $x \\in {\\cal L}_{\\varepsilon } \\cap \\tilde{\\Omega }$ and let $y_{\\varepsilon }(x) = \\mathbf {id}+ \\sqrt{\\varepsilon } u_{\\varepsilon }(x)$ .",
"Clearly we have $u_{\\varepsilon } \\in {\\cal A}_{g_\\varepsilon }$ for all $\\varepsilon $ .",
"By $\\tilde{u}_{\\varepsilon }, u^{\\prime }_{\\varepsilon }$ we again denote the interpolations on $\\tilde{\\Omega }_{\\varepsilon }$ .",
"Up to considering a translation of $u$ of order $\\varepsilon $ , we may assume that $J_{u} \\cap {\\cal L}_{\\varepsilon } = \\emptyset $ .",
"Let ${\\cal D}_{\\varepsilon }$ be the sets of triangles where $J_{u}$ crosses at least one side of the triangle.",
"Then $\\# {\\cal D}_{\\varepsilon } \\le \\frac{C {\\cal H}^1(J_{u})}{\\varepsilon } + C N_u$ for a constant $C > 0$ independent of $u \\in {\\cal W}(\\tilde{\\Omega }, R^2)$ and $\\varepsilon $ , where $N_u$ denotes the (smallest) number of segments whose union gives $J_{u}$ .",
"From now on for the local nature of the arguments we may assume that $J_{u}$ consists of one segment only.",
"Indeed, if $J_u$ consists of segments $S_1,\\ldots , S_{N_u}$ , which are possibly not disjoint, the number of triangles $\\Delta \\in \\tilde{\\cal C}_\\varepsilon $ with $\\triangle \\cap S_{i_1} \\cap S_{i_2} \\ne \\emptyset $ for $1\\le i_i < i_2 \\le N_u$ scales like $N_u$ and therefore their energy contribution is negligible in the limit.",
"We show $\\bar{{\\cal C}}_{\\varepsilon } = {\\cal D}_{\\varepsilon }$ for $\\varepsilon $ small enough.",
"Let $\\triangle \\in {\\cal D}_\\varepsilon $ .",
"We see that, if $J_{u} = J^{\\delta }_{u}$ crosses a spring $\\mathbf {v}$ at point $x_{*}$ , say, then a computation similar as in (REF ) together with $\\nabla u \\in L^{\\infty }$ shows $\\left| (\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}\\right|= \\left| \\frac{1}{\\sqrt{\\varepsilon }} [u(x_{*})] + O(1) \\right|\\ge \\frac{\\delta }{\\sqrt{\\varepsilon }} + O(1).$ Thus, $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon }$ for $\\varepsilon $ small enough.",
"On the other hand, if we assume $\\triangle \\notin {\\cal D}_{\\varepsilon }$ , then for at least two springs $\\mathbf {v}\\in {\\cal V}$ we have $|(\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}| \\le 1 + \\sqrt{\\varepsilon } \\left\\Vert \\nabla u \\right\\Vert _{\\infty } < 2$ for $\\varepsilon $ small enough leading to $\\triangle \\notin \\bar{{\\cal C}}_{\\varepsilon }$ .",
"We claim that $\\Vert \\nabla u^{\\prime }_{\\varepsilon }\\Vert _{L^\\infty (\\tilde{\\Omega })} \\le C.$ This is clear for $\\triangle \\notin {\\cal D}_{\\varepsilon } = \\bar{\\cal C}_{\\varepsilon }$ as $\\nabla u \\in L^{\\infty }$ .",
"For $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,3}$ it follows by construction.",
"For $\\triangle \\in \\bar{{\\cal C}}_{\\varepsilon ,2}$ there is a $\\mathbf {v}\\in {\\cal V}$ such that $(y^{\\prime }_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}= (\\tilde{y}_{\\varepsilon })_{\\triangle } \\, \\mathbf {v}= \\mathbf {v}+ O(\\sqrt{\\varepsilon })$ .",
"By Lemma REF (i) and (REF ) we get a rotation $R_{\\varepsilon } \\in SO(2)$ such that $ |R_{\\varepsilon } - (y^{\\prime }_{\\varepsilon })_{\\triangle }|^2= \\operatorname{dist}^2((y^{\\prime }_{\\varepsilon })_{\\triangle }, SO(2))= \\operatorname{dist}^2((y^{\\prime }_{\\varepsilon })_{\\triangle }, O(2))\\le C W_{\\triangle }((y^{\\prime }_{\\varepsilon })_{\\triangle })= O(\\varepsilon ).$ This yields $|(y^{\\prime }_{\\varepsilon })_{\\triangle } - \\mathbf {Id}| = O(\\sqrt{\\varepsilon })$ and thus $|(u^{\\prime }_{\\varepsilon })_{\\triangle }| = O(1)$ .",
"We note that $\\chi _{\\tilde{\\Omega }_{\\varepsilon }} \\tilde{u}_{\\varepsilon } \\rightarrow u$ in $L^1$ as $u$ and thus every $\\tilde{u}_{\\varepsilon }$ is bounded uniformly in $L^{\\infty }$ and, $u$ being Lipschitz away from $J_u$ , $\\tilde{u}_{\\varepsilon } \\rightarrow u$ uniformly on $\\tilde{\\Omega }_{\\varepsilon } \\setminus \\bigcup _{\\triangle \\in {\\cal D}_{\\varepsilon }} \\triangle $ , where $|\\bigcup _{\\triangle \\in {\\cal D}_{\\varepsilon }} \\triangle | \\le C \\varepsilon $ .",
"Letting ${\\cal C}^*_{\\varepsilon } = {\\cal D}_{\\varepsilon }$ this shows that $u_{\\varepsilon } \\rightarrow u$ in the sense of Definition REF recalling (REF ) and the fact that $|(\\tilde{u}_{\\varepsilon })_{\\triangle }| = O(1)$ for $\\triangle \\notin {\\cal D}_{\\varepsilon }$ .",
"We next establish an even stronger convergence of the derivatives.",
"Consider $\\nabla \\tilde{u}_{\\varepsilon }$ on triangles in ${\\cal C}_{\\varepsilon } \\setminus {\\cal D}_{\\varepsilon }$ .",
"As $u$ is Lipschitz there, the oscillation on such a triangle, $\\text{osc}^{\\triangle }_{\\varepsilon } (\\nabla u) := \\sup \\left\\lbrace \\left\\Vert \\nabla u(x) - \\nabla u(x^{\\prime })\\right\\Vert _\\infty , x,x^{\\prime } \\in \\triangle \\right\\rbrace $ , tends to zero uniformly (i.e., not depending on the choice of the triangle).",
"We thus obtain $ \\int _{\\tilde{\\Omega }_{\\varepsilon } \\setminus \\cup _{\\triangle \\in {\\cal D}_{\\varepsilon }} \\triangle }\\Vert \\nabla \\tilde{u}_{\\varepsilon } - \\nabla u\\Vert _{\\infty }^2\\le \\int _{\\tilde{\\Omega }_{\\varepsilon } \\setminus \\cup _{\\triangle \\in {\\cal D}_{\\varepsilon }} \\triangle }(\\text{osc}^{\\triangle }_{\\varepsilon } (\\nabla u))^2 \\rightarrow 0$ for $\\varepsilon \\rightarrow 0$ , so that even $\\chi _{\\tilde{\\Omega }_{\\varepsilon } \\setminus \\cup _{\\triangle \\in {\\cal D}_{\\varepsilon }} \\triangle }\\nabla \\tilde{u}_{\\varepsilon } \\rightarrow \\nabla u$ strongly in $L^2(\\tilde{\\Omega })$ .",
"Note that in fact $\\chi _{\\tilde{\\Omega }_{\\varepsilon }} \\nabla u^{\\prime }_{\\varepsilon } \\rightarrow \\nabla u$ in $L^2(\\Omega )$ .",
"Indeed, recall $\\# {\\cal D}_{\\varepsilon } \\le C\\varepsilon ^{-1}$ by (REF ).",
"Using (REF ) on the set of broken triangles we then get $\\int _{\\bigcup _{\\triangle \\in {{\\cal D}}_{\\varepsilon }} \\triangle } |\\nabla u^{\\prime }_{\\varepsilon } - \\nabla u|^2 \\le C \\#\\bar{{\\cal D}}_{\\varepsilon } \\varepsilon ^2 \\rightarrow 0$ for $\\varepsilon \\rightarrow 0$ .",
"We now split up the energy in bulk and surface parts ${\\cal E}^{\\chi }_{\\varepsilon }(u_{\\varepsilon })& = {\\cal E}^{\\rm elastic}_{\\varepsilon }(u_{\\varepsilon }) + {\\cal E}^{\\rm crack}_{\\varepsilon }(u_{\\varepsilon }) + O(\\varepsilon ) + \\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon } \\chi (\\nabla \\tilde{y}_\\varepsilon )$ as defined in (REF ).",
"Note that indeed the contribution $\\varepsilon E^{\\rm boundary}_{\\varepsilon }$ is of order $O(\\varepsilon )$ as $\\nabla u \\in L^\\infty (\\tilde{\\Omega })$ and $J_u \\subset \\Omega $ since $u = g$ in a neighborhood of $\\tilde{\\Omega }\\setminus \\Omega $ .",
"We first observe that $\\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon } \\chi (\\nabla \\tilde{y}_\\varepsilon ) = 0$ for $\\varepsilon $ small enough.",
"Indeed, for $\\Delta \\in \\bar{\\cal C}_\\varepsilon $ this follows from (REF ).",
"For $\\Delta \\notin {\\cal D}_\\varepsilon $ it suffices to recall $|(\\tilde{u}_{\\varepsilon })_{\\triangle }| = O(1)$ which implies that $(\\tilde{u}_{\\varepsilon })_{\\triangle }$ is near $SO(2)$ .",
"Repeating the steps in the elastic energy estimate in (i), applying $\\chi _{\\Omega _{\\varepsilon }} \\nabla u^{\\prime }_{\\varepsilon } \\rightarrow \\nabla u$ strongly in $L^2(\\Omega )$ , (REF ) and $Q(F) \\le C|F|^2$ for a constant $C > 0$ we conclude that $\\limsup _{\\varepsilon \\rightarrow 0}{\\cal E}^{\\rm elastic}_{\\varepsilon }(u_{\\varepsilon })= \\frac{4}{\\sqrt{3}} \\int _{\\Omega } \\frac{1}{2} Q(e(u(x))) \\, dx.$ It is elementary to see that $J_{u}$ crosses ${\\cal H}^1(J_{u}) \\frac{2 |\\nu _{u} \\cdot \\mathbf {v}|}{\\sqrt{3} \\varepsilon } + O(1)$ springs in $\\mathbf {v}$ -direction for $\\mathbf {v}\\in {\\cal V}$ , where $\\nu _u$ is a normal to the segment $J_u$ .",
"Recalling (REF ), the crack energy may be estimated by $&\\limsup _{\\varepsilon \\rightarrow 0}{\\cal E}^{\\rm crack}_{\\varepsilon }(u_{\\varepsilon }) \\\\&\\le \\limsup _{\\varepsilon \\rightarrow 0} {\\cal H}^1(J_{u}) \\ \\sup \\left\\lbrace W(r): r \\ge \\delta \\varepsilon ^{-\\frac{1}{2}} + O(1) \\right\\rbrace \\frac{2}{\\sqrt{3}} \\sum _{\\mathbf {v}\\in {\\cal V}} |\\nu _{u} \\cdot \\mathbf {v}| +O(\\varepsilon ) \\\\& = {\\cal H}^1(J_{u}) \\ \\beta \\ \\frac{2}{\\sqrt{3}} \\sum _{\\mathbf {v}\\in {\\cal V}} |\\nu _{u} \\cdot \\mathbf {v}|.$ This together with (REF ) and (REF ) shows that $u_{\\varepsilon }$ is a recovery sequence for $u$ .",
"$\\Box $ Finally, we construct recovery sequences for the functionals ${\\cal F}^\\chi _\\varepsilon $ to conclude the proof of Theorem REF .",
"Proof of Theorem REF , second part.",
"Following the proof of Theorem REF (ii) it suffices to show $\\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon }f_{\\kappa }(\\nabla \\tilde{y}_\\varepsilon ) = -\\frac{\\kappa }{2}\\int _\\Omega \\hat{Q}(\\nabla u).$ First, by (REF ) and the definition of $f_{\\kappa }$ we get $\\int _{\\bigcup _{\\Delta \\in {\\cal D}_\\varepsilon } \\Delta } f_{\\kappa }(\\nabla \\tilde{y}_\\varepsilon ) = 0$ for $\\varepsilon $ small enough.",
"For $\\Delta \\notin {\\cal D}_\\varepsilon $ we have $(\\nabla \\tilde{y}_\\varepsilon )_\\Delta = (\\nabla y^{\\prime }_\\varepsilon )_\\Delta $ and thus we find $f_{\\kappa }((\\nabla \\tilde{y}_\\varepsilon )_\\Delta ) = - \\varepsilon \\frac{\\kappa }{2} \\hat{Q}( ( \\nabla u^{\\prime }_\\varepsilon )_\\Delta ) - \\kappa \\omega (\\sqrt{\\varepsilon } \\nabla (u^{\\prime }_\\varepsilon )_\\Delta )$ by (REF ).",
"We obtain $\\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon }f_{\\kappa }(\\nabla \\tilde{y}_\\varepsilon ) &=\\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon \\setminus \\bigcup _{\\Delta \\in {\\cal D}_\\varepsilon } \\Delta } f_{\\kappa }(\\nabla y^{\\prime }_\\varepsilon ) \\\\& \\le -\\frac{\\kappa }{2}\\int _{\\Omega _\\varepsilon \\setminus \\bigcup _{\\Delta \\in {\\cal D}_\\varepsilon } \\Delta } \\hat{Q}(\\nabla u^{\\prime }_\\varepsilon ) +\\frac{C}{\\varepsilon }\\int _{\\Omega _\\varepsilon } \\omega (\\sqrt{\\varepsilon } \\nabla u^{\\prime }_\\varepsilon ).$ Using (REF ) and the definition of $\\omega $ we observe $\\frac{1}{\\varepsilon }\\Vert \\omega (\\sqrt{\\varepsilon } \\nabla u^{\\prime }_\\varepsilon )\\Vert _\\infty \\rightarrow 0$ for $\\varepsilon \\rightarrow 0$ .",
"This together with strong convergence $\\chi _{\\Omega _\\varepsilon } \\nabla u^{\\prime }_\\varepsilon \\rightarrow \\nabla u$ in $L^2(\\Omega )$ shows $\\limsup _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon }\\int _{\\Omega _\\varepsilon }f_{\\kappa }(\\nabla \\tilde{y}_\\varepsilon ) \\le -\\frac{\\kappa }{2}\\int _\\Omega \\hat{Q}(\\nabla u).$ $\\Box $"
],
[
"Analysis of the limiting variational problem",
"We finally give the proof of Theorem REF determining the minimizers of the limiting functional ${\\cal E}$ .",
"An analogous result for isotropic energy functionals has been obtained in [25].",
"We thus do not repeat all the steps of the proof provided in [25] but rather concentrate on the additional arguments necessary to handle anisotropic surface contributions.",
"Proof of Theorem REF .",
"We first establish a lower bound for the energy ${\\cal E}$ .",
"To this end, we begin to estimate $\\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu |$ for $\\nu \\in S^1$ .",
"We recall that $\\gamma \\in [\\frac{\\sqrt{3}}{2},1]$ and define $P:[\\frac{\\sqrt{3}}{2},1] \\times S^1 \\rightarrow [0,\\infty )$ by $P(\\gamma ,\\nu )= {\\left\\lbrace \\begin{array}{ll} \\left(1- \\sqrt{3} \\frac{\\sqrt{1 - \\gamma ^2}}{\\gamma }\\right) |\\mathbf {v}_\\gamma \\cdot \\nu |, & \\gamma >\\frac{\\sqrt{3}}{2}, \\\\ \\max \\big \\lbrace \\sqrt{3} |\\mathbf {e}_2 \\cdot \\nu | - |\\mathbf {e}_1 \\cdot \\nu |,0\\big \\rbrace , & \\gamma = \\frac{\\sqrt{3}}{2}.",
"\\end{array}\\right.",
"}$ As $\\mathbf {v}_\\gamma $ is unique for $\\gamma > \\frac{\\sqrt{3}}{2}$ , the function $P$ is well defined.",
"In the generic case, i.e.",
"for $\\gamma > \\frac{\\sqrt{3}}{2}$ , an elementary computation yields $\\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu |&\\ge |\\mathbf {v}_{\\gamma } \\cdot \\nu | + \\sqrt{3} |\\mathbf {v}^{\\bot }_{\\gamma } \\cdot \\nu |= |\\mathbf {v}_{\\gamma } \\cdot \\nu |+ \\sqrt{3} \\left| \\pm \\frac{1}{\\gamma } \\mathbf {e}_1 \\cdot \\nu \\pm \\frac{\\sqrt{1 - \\gamma ^2}}{\\gamma } \\mathbf {v}_{\\gamma } \\cdot \\nu \\right| \\\\&\\ge \\frac{\\sqrt{3}}{\\gamma }|\\mathbf {e}_1 \\cdot \\nu | + P(\\gamma , \\nu )$ for $\\nu \\in S^1$ .",
"In the first step we used that $\\sum _{\\mathbf {v}\\in {\\cal V}\\setminus \\left\\lbrace \\mathbf {v}_{\\gamma }\\right\\rbrace } \\mathbf {v}= \\pm \\sqrt{3}\\mathbf {v}^{\\bot }_{\\gamma }$ .",
"In the special case $\\phi = 0 \\Leftrightarrow \\gamma = \\frac{\\sqrt{3}}{2}$ , i.e.",
"$\\mathbf {v}_1 = \\mathbf {e}_1$ , $\\mathbf {v}_{2,3} = \\pm \\frac{1}{2} \\mathbf {e}_1 + \\frac{\\sqrt{3}}{2} \\mathbf {e}_2$ we obtain $\\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu | = |\\mathbf {e}_1\\cdot \\nu | + \\sqrt{3} |\\mathbf {e}_2 \\cdot \\nu |$ for $|\\nu _2|>\\frac{1}{2}$ and $\\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu | = 2|\\mathbf {e}_1\\cdot \\nu |$ for $|\\nu _2|\\le \\frac{1}{2}$ , $\\nu \\in S^1$ .",
"Consequently, it is not hard to see that $\\sum _{\\mathbf {v}\\in {\\cal V}} |\\mathbf {v}\\cdot \\nu |\\ge \\frac{\\sqrt{3}}{\\gamma } |\\mathbf {e}_1 \\cdot \\nu | + P(\\gamma ,\\nu )$ also holds for $\\gamma = \\frac{\\sqrt{3}}{2}$ .",
"Thus, we get $ {\\cal E}(u)\\ge \\frac{4}{\\sqrt{3}} \\int _{\\Omega } \\frac{1}{2} Q(e(u(x))) \\, dx+ \\int _{J_{u}} \\frac{2\\beta }{\\gamma } |\\mathbf {e}_1 \\cdot \\nu _{u}|+ \\frac{2\\beta }{\\sqrt{3}} P(\\gamma ,\\nu _u) \\, d{\\cal H}^1.",
"$ By Lemma REF we obtain $\\min \\lbrace Q(F): \\mathbf {e}_1^T F\\mathbf {e}_1 =r\\rbrace = \\frac{\\alpha }{2} r^2 $ .",
"Then using the slicing method (see, e.g., [5]) we get ${\\cal E}(u) \\ge \\int ^{1}_{0}\\left( \\int ^{l}_{0}\\frac{\\alpha }{\\sqrt{3}}\\left(\\mathbf {e}_1^{T}\\nabla u(x_1,x_2)\\mathbf {e}_1\\right)^2 \\, dx_1+ \\frac{2\\beta }{\\gamma } \\#S^{x_2}(u)\\right) \\,dx_2 + {\\cal E}^{\\gamma }(u),$ where $\\# S^{x_2}$ denotes the number of jumps on a slice $(0,l) \\times \\left\\lbrace x_2\\right\\rbrace $ and $ {\\cal E}^{\\gamma }(u)= \\int _{J_{u}} \\frac{2\\beta }{\\sqrt{3}P(\\gamma , \\nu _u)} \\, d{\\cal H}^1.$ In case $\\#S^{x_2}(u) \\ge 1$ , the inner integral in (REF ) is obviously bounded from below by $\\frac{2 \\beta }{\\gamma }$ .",
"If $\\#S^{x_2}(u) =0$ , by applyig Jensen's inequality we find that this term is bounded from below by $\\alpha l a^2$ due to the boundary conditions.",
"We thus obtain $\\inf {\\cal E} \\ge \\min \\big \\lbrace \\frac{\\alpha l a^2}{\\sqrt{3}}, \\frac{2 \\beta }{\\gamma }\\big \\rbrace $ .",
"On the other hand, it is straighforward to check that ${\\cal E}(u^{\\rm el}) = \\alpha l a^2$ and ${\\cal E}(u^{\\rm cr}) = \\frac{2 \\beta }{\\gamma }$ , which shows that $u^{\\rm el}$ is a minimizer for $a < a_{\\rm crit}$ and $u^{\\rm cr}$ is a minimizer for $a > a_{\\rm crit}$ .",
"It remains to prove uniqueness: (i) Let $a < a_{\\rm crit}$ and $u$ be a minimizer of ${\\cal E}$ .",
"Since ${\\cal E}(u) = {\\cal E}(u^{\\rm el})$ we infer from (REF ) that $u$ has no jump on a.e.",
"slice $(0,l) \\times \\left\\lbrace x_2\\right\\rbrace $ and satisfies $\\mathbf {e}_1^{T} \\nabla u \\, \\mathbf {e}_1 = a$ a.e.",
"by the imposed boundary values and strict convexity of the mapping $t \\mapsto t^2$ on $[0, \\infty )$ .",
"Thus, if $J_{u} \\ne \\emptyset $ , a crack normal must satisfy $\\nu _{u} = \\pm \\mathbf {e}_2$ ${\\cal H}^1$ -a.e.",
"Taking ${\\cal E}^{\\gamma }(u)$ and the fact that $P(\\gamma ,\\mathbf {e}_2)>0$ for $\\gamma \\in [\\frac{\\sqrt{3}}{2},1]$ into account, we then may assume $J_{u} =\\emptyset $ up to an ${\\cal H}^1$ negligible set, i.e., $u \\in H^1(\\Omega )$ .",
"We find $u_1(x_1,x_2) = a x_1 + f(x_2)$ a.e.",
"for a suitable function $f$ , and the boundary condition $u_1(0,x_2)=0$ yields $f = 0$ a.e.",
"In particular, $\\mathbf {e}_1^{T}\\nabla u \\, \\mathbf {e}_2 = 0$ a.e.",
"Applying strict convexity of $Q$ on symmetric matrices (Lemma REF ) we now observe $\\mathbf {e}_2^{T} \\nabla u \\, \\mathbf {e}_2 = -\\frac{a}{3}$ and $\\mathbf {e}_1^{T} \\nabla u \\, \\mathbf {e}_2+ \\mathbf {e}_2^{T} \\nabla u \\, \\mathbf {e}_1 = 0$ a.e.",
"So the derivative has the form $ \\nabla u(x) = \\begin{footnotesize} \\begin{pmatrix} a & 0 \\\\ 0 &-\\frac{a}{3} \\end{pmatrix} \\end{footnotesize} \\text{ for a.e.\\ $x$}.",
"$ Since $\\Omega $ is connected, we conclude $u(x)= (0,s) + F^{a} x = u^{\\rm el}(x)$ a.e.",
"(ii) Let $a > a_{\\rm crit}$ , $\\phi \\ne 0$ and $u$ be a minimizer of ${\\cal E}$ .",
"We again consider the lower bound (REF ) for the energy ${\\cal E}$ and now obtain that on a.e.",
"slice $(0,l) \\times \\left\\lbrace x_2\\right\\rbrace $ a minimizer $u$ has precisely one jump and that $\\mathbf {e}_1^{T} \\nabla u \\, \\mathbf {e}_1 = 0$ a.e.",
"Now Lemma REF shows that $\\nabla u$ is antisymmetric a.e.",
"As a consequence, the linearized rigidity estimate for SBD functions of Chambolle, Giacomini and Ponsiglione [13] yields that there is a Caccioppoli partition $(E_i)$ of $\\Omega $ such that $ u(x) = \\sum _{i} (A_i x + b_i) \\chi _{E_i} \\quad \\text{and} \\quad J_u = \\bigcup _{i} \\partial ^* E_i, $ where $A_i^T = -A_i \\in R^{2 \\times 2}$ and $b_i \\in R^2$ .",
"(See [5] for the definition and basic properties of Caccioppoli partitions.)",
"As ${\\cal E}^{\\gamma }(u) = 0$ , we also note that $\\nu _u \\perp \\mathbf {v}_{\\gamma }$ a.e.",
"on $J_u$ .",
"Following the arguments in [25], in particular using regularity results for boundary curves of sets of finite perimeter and exhausting the sets $\\partial ^* E_i$ with Jordan curves, we find that $ J_u = \\bigcup _{i} \\partial ^* E_i \\subset (p, 0) + R\\mathbf {v}_{\\gamma } $ for some $p$ such that $(p, 0) + R\\mathbf {v}_{\\gamma }$ intersects both segments $(0, l) \\times \\lbrace 0\\rbrace $ and $(0, l) \\times \\lbrace 1\\rbrace $ .",
"We thus obtain that $(E_i)$ consists of only two sets: $E_1$ to the left and $E_2$ to the right of $(p, 0) + R\\mathbf {v}_{\\gamma }$ , say.",
"Due to the boundary conditions we conclude that $A_1 = A_2 = 0$ and $b_1 = (0,s)$ , $b_2 = (al, t)$ for suitable $s, t \\in R$ .",
"(iii) Let $a > a_{\\rm crit}$ , $\\phi = 0$ and $u$ be a minimizer of ${\\cal E}$ .",
"We follow the lines of the proof in (ii).",
"The only difference is that ${\\cal E}^\\gamma (u) = 0$ now implies that $|\\nu _u \\cdot \\mathbf {e}_1| \\ge \\frac{\\sqrt{3}}{2}$ a.e.",
"and then arguing similarly as before we obtain $J_u \\subset h((0,1))$ up to an ${\\cal H}^1$ -negligible set, where $h:(0,1) \\rightarrow [0,l]$ is a Lipschitz function with $|h^{\\prime }| \\le \\frac{1}{\\sqrt{3}}$ a.e.",
"We now conclude as in (ii).",
"$\\Box $"
]
] | 1403.0443 |
[
[
"Scaling instability of the buckling load in axially compressed circular\n cylindrical shells"
],
[
"Abstract In this paper we initiate a program of rigorous analytical investigation of the paradoxical buckling behavior of circular cylindrical shells under axial compression.",
"This is done by the development and systematic application of general theory of \"near-flip\" buckling of 3D slender bodies to cylindrical shells.",
"The theory predicts scaling instability of the buckling load due to imperfections of load.",
"It also suggests a more dramatic scaling instability caused by shape imperfections.",
"The experimentally determined scaling exponent 1.5 of the critical stress as a function of shell thickness appears in our analysis as the scaling of the lower bound on safe loads given by the Korn constant.",
"While the results of this paper fall short of a definitive explanation of the buckling behavior of cylindrical shells, we believe that our approach is capable of providing reliable estimates of the buckling loads of axially compressed cylindrical shells."
],
[
"Introduction",
"A circular cylindrical shell loaded by an axial compressive stress will buckle producing a variety of buckling patterns[4], [20], [6], including the single-dimple buckle [32], [15], shown in Figure REF .",
"Figure: Single-dimple buckling pattern in buckled soda cans .In the soda can experiments [14] this dimple consistently appeared with an audible click, corresponding to the drop in load in Figure REF and disappears (also with a click) upon unloading.",
"This suggests that the local material response is still linearly elastic, while the global non-linearity is purely geometric.",
"The abrupt nature of the observed buckling suggests that the trivial branch, whose stress and strain are well-approximated by linear elasticity, becomes unstable with respect to the observed buckling variation.",
"The classical shell theory supplies the following formula for the critical stress [24], [27] (see also [28]): $\\sigma _{\\rm cr}=\\frac{Eh}{\\sqrt{3(1-\\nu ^{2})}},$ where $E$ and $\\nu $ are the Young modulus and the Poisson ratio, respectively, and $h=t/R$ is the ratio of the wall thickness to the radius of the cylinder.",
"A large body of experimental results summarized in [20], [32] show that not only the theoretical value of the buckling load is about 4 to 5 times higher than the one observed in experiments, but the critical stress $\\sigma _{\\rm cr}$ scales like $h^{3/2}$ with $h$ , in stark contradiction to (REF ).",
"Such paradoxical behavior is generally attributed to the sensitivity of the buckling load to imperfections of load and shape [1], [26], [29], [10], [30], due to the subcritical nature of the bifurcation [18], [19], [23], [16] in the von-Kármán-Donnell equations.",
"Yet, such an interpretation of the experimental results does not give a quantification of sensitivity to imperfections, and does little to explain the paradoxical $h^{1.5}$ scaling of the critical stress.",
"These questions have been raised in [5], [32], [15], where a combination of heuristic arguments and numerical simulations were used to address the problem.",
"In situations where the classical shell theory gives predictions inconsistent with experiment, one can question whether “sensitivity to imperfections” is the true source of the inconsistency, or whether the failure of the heuristic models to adequately describe stability of slender bodies is at play.",
"In a companion paper [12] we give a rigorous proof of the asymptotical correctness of (REF ), showing that the second variation of the energy of the compressed shell, regarded as a 3D hyperelastic body, becomes negative when the load exceeds the critical value (REF ).",
"Recent years have seen significant progress in the rigorous analysis of dimensionally reduced theories of plates and shells based on $\\Gamma $ -convergence [8], [25], [9], [21], [22].",
"In this approach, one must postulate the scaling of energy and the forces a priori, whereby different scaling assumptions lead to different dimensionally reduced plate and shell equations.",
"These analyses show that the tacit assumptions of validity of specific shell theories must be justified before conclusions about the elastic behavior of such shells can be regarded as rigorous.",
"By contrast, the theory in [13], based on the study of second variation, has no need for such a priori assumptions, since it pursues a more modest goal of identifying a critical load at which the “trivial branch”, or “fundamental state”, of equilibria becomes weakly unstable.",
"This exclusive targeting of the instability without any attempt to compute $\\Gamma $ -limiting models or capture a global bifurcation picture of post-buckling behavior leads to a significant simplification in the rigorous analysis of stability of slender structures.",
"Most notably, our approach does not require compactness of arbitrary low energy sequences as in [8].",
"In particular, our method is applicable in situations where compactness fails, as is the case for the axially compressed cylindrical shells.",
"In this paper we prove that a constitutively reduced characterization of the buckling load, derived in [13], captures the buckling mode as well.",
"A more convenient criterion for the validity of this characterization of buckling, derived in this paper, makes the theory applicable to a broader, compared to [13], class of slender structures, including axially compressed cylindrical shells.",
"While our approach is capable of providing a rigorous proof of the classical formula (REF ) [12], it also reveals a possible mechanism of imperfection sensitivity that may explain the experimental results and their discrepancy with the classical theory.",
"Specifically, we show that generic imperfections in loading will change the scaling law of $\\sigma _{\\rm cr}$ to $h^{5/4}$ .",
"Shape imperfections may lead to an even bigger jump in the scaling exponent of $\\sigma _{\\rm cr}$ from $h$ to $h^{3/2}$ .",
"The power law $h^{3/2}$ arises as the scaling of the Korn constant [11], shown to describe the universal lower bound on safe loads in [13].",
"This explanation of the experimentally observed scaling of the critical stress could be viewed as an improvement of the ingenuous but somewhat intuitive arguments in [5], [32].",
"Generically, shape imperfections eliminate sharp bifurcation transitions [2].",
"However, the abrupt appearance of the dimple-shaped buckle accompanied by an audible click in our experiments suggests that, in the case of a cylindrical shell, shape imperfections do not eliminate bifurcation instability.",
"If this is indeed the case, our methods would be able to accurately predict the critical load and the corresponding buckling mode.",
"However, the rigorous analysis of an imperfect cylindrical shell is beyond the scope of this paper, since all the estimates proved here and in our companion paper [11] are for the specific circular cylindrical shell geometry.",
"This paper is organized as follows.",
"In Section we extend the theory of buckling of general slender bodies based on the asymptotic analysis of second variation [13].",
"We define “compression tensor” and further develop the method of buckling equivalence [13].",
"Section applies the theory in Section and the asymptotics of the Korn and Korn-type constants proved in [11] to the computation of the scaling law of the buckling load.",
"We next demonstrate scaling instabilities due to generic imperfections of load.",
"We conclude the paper with a less rigorous discussion of imperfections of shape."
],
[
"Buckling of slender structures",
"In this section we revisit the general theory of buckling developed in [13] in order to extend and apply it to buckling of axially compressed cylindrical shells.",
"The theory provides a recipe for computing the asymptotics of buckling loads of slender structures, as the slenderness parameter goes to zero.",
"We follow the established tradition and use the energy criterion of stability.",
"Namely, we say that the deformation $ y= y( x)$ , $ x\\in \\Omega \\subset \\mathbb { R}^{3}$ is stable if it is a weak local minimizer of the energy ${\\mathcal {E}}( y)=\\int _{\\Omega }W(\\nabla y)d x-\\int _{\\partial \\Omega } y\\cdot t( x)dS( x),$ where $W( F)$ is the energy density function of the body and $ t( x)$ is the vector of dead load tractions.",
"The energy density $W( F)$ satisfies the four fundamental properties: (P1) Absence of prestress: $W_{ F}( I)= 0$ ; (P2) Frame indifference: $W( F R)=W( F)$ for every $ R\\in SO(3)$ ; (P3) Local stability of the trivial deformation $ y( x)= x$ : $(\\mathsf {L}_{0}\\xi ,\\xi )\\ge 0$ for any $\\xi \\in \\mathbb { R}^{3\\times 3}$ , where $\\mathsf {L}_{0}=W_{ F F}( I)$ is the linearly elastic tensor of material properties; (P4) Non-degeneracy: $(\\mathsf {L}_{0}\\xi ,\\xi )=0$ if and only if $\\xi ^{T}=-\\xi $ .",
"Here, and elsewhere in this paper we use the notation $( A, B)=\\mathrm {Tr}\\,( A B^{T})$ for the Frobenius inner product on the space of $3\\times 3$ matrices.",
"In [13] we attribute the universal nature of buckling to the two universal properties (P1) and (P2) of the energy density function because they guarantee non-convexity of $W( F)$ in any neighborhood of the identity $ I$ .",
"We also remark that properties (P3) and (P4) of $\\mathsf {L}_{0}$ imply a uniform lower bound $(\\mathsf {L}_{0}\\xi ,\\xi )\\ge \\alpha _{\\mathsf {L}_{0}}|\\xi _{\\rm sym}|^{2},\\qquad \\xi _{\\rm sym}=\\displaystyle \\frac{1}{2}(\\xi +\\xi ^{T})$ for some $\\alpha _{\\mathsf {L}_{0}}>0$ ."
],
[
"Trivial branch",
"Consider a sequence of progressively slenderThe appropriate notion of slenderness, introduced in [13] is made precise in Defintion REF .",
"domains $\\Omega _{h}$ parametrized by a dimensionless parameter $h$ .",
"For example, for circular cylindrical shells, $h$ is the ratio of cylinder wall thickness to the cylinder radius (we keep the ratio of cylinder height to its radius constant).",
"We consider a loading program parametrized by the loading parameter $\\lambda $ describing the magnitude of the applied tractions $ t( x;h,\\lambda )=\\lambda t^{h}( x)+O(\\lambda ^{2})$ , as $\\lambda \\rightarrow 0$ , or as a measure of the prescribed strain.",
"Here and below $O(\\lambda ^{\\alpha })$ is understood uniformly in $ x\\in \\Omega _{h}$ and $h\\in [0,h_{0}]$ .",
"Let $ y( x;h,\\lambda )$ be a family of Lipschitz equilibria of ${\\mathcal {E}}( y;h,\\lambda )=\\int _{\\Omega _{h}}W(\\nabla y)d x-\\int _{\\partial \\Omega _{h}} y( x)\\cdot t( x;h,\\lambda )dS( x).$ defined on $\\overline{\\Omega }\\times [0,h]\\times [0,\\lambda _{0}]$ for some $h_{0}>0$ and $\\lambda _{0}>0$ .",
"The general theory can treat a wide range of boundary conditions.",
"To describe one, we restrict $ y$ to an affine subspace of $W^{1,\\infty }(\\Omega _{h};\\mathbb { R}^{3})$ , given by $ y\\in \\overline{ y}( x;h,\\lambda )+V_{h}^{\\circ },$ where $V_{h}^{\\circ }$ is a linear subspace of $W^{1,\\infty }(\\Omega _{h};\\mathbb { R}^{3})$ that contains $W_{0}^{1,\\infty }(\\Omega _{h};\\mathbb { R}^{3})$ and does not depend on the loading parameter $\\lambda $ .",
"The given function $\\overline{ y}( x;h,\\lambda )\\in W^{1,\\infty }(\\Omega _{h};\\mathbb { R}^{3})$ describes the “Dirichlet part” of the boundary conditions, while the traction vector $ t( x;h,\\lambda )$ describes the Neumann-partThe use of a general subspace $V_{h}^{\\circ }$ permits one to describe loadings in which desired linear combinations of the displacement and traction components are prescribed on the boundary.. An example of such a description of the boundary conditions for the cylindrical shell will be given in Section REF .",
"Definition 2.1 We call the family of Lipschitz equilibria $ y( x;h,\\lambda )$ of ${\\mathcal {E}}( y;h,\\lambda )$ a linearly elastic trivial branch if there exist $h_{0}>0$ and $\\lambda _{0}>0$ , so that for every $h\\in [0,h_{0}]$ and $\\lambda \\in [0,\\lambda _{0}]$ (i) $ y( x;h,0)= x$ (ii) There exist a family of Lipschitz functions $ u^{h}( x)$ , independent of $\\lambda $ , such that $\\Vert \\nabla y( x;h,\\lambda )- I-\\lambda \\nabla u^{h}( x)\\Vert _{L^{\\infty }(\\Omega _{h})}\\le C\\lambda ^{2},$ (iii) $\\Vert \\displaystyle \\frac{\\partial (\\nabla y)}{\\partial \\lambda }( x;h,\\lambda )-\\nabla u^{h}( x)\\Vert _{L^{\\infty }(\\Omega _{h})}\\le C\\lambda $ where the constant $C$ is independent of $h$ and $\\lambda $ .",
"We remark that neither uniqueness nor stability of the trivial branch are assumed.",
"The equilibrium equations and the boundary conditions satisfied by the trivial branch $ y( x;h,\\lambda )$ can be written explicitly in the weak form: $\\int _{\\Omega _{h}}(W_{ F}(\\nabla y( x;h,\\lambda )),\\nabla \\phi )d x-\\int _{\\partial \\Omega _{h}}\\phi \\cdot t( x;h,\\lambda )dS=0,\\qquad \\phi \\in V_{h}^{\\circ },$ Differentiating (REF ) in $\\lambda $ at $\\lambda =0$ , which is allowed due to (REF ), we obtain $\\int _{\\Omega _{h}}(\\mathsf {L}_{0}\\nabla u^{h}( x),\\nabla \\phi )d x-\\int _{\\partial \\Omega _{h}}\\phi \\cdot t^{h}( x)dS=0,\\qquad \\phi \\in V_{h}^{\\circ },$ In [13] the notion of the near-flip buckling is defined when for any $h\\in [0,h_{0}]$ the trivial branch $ y( x;h,\\lambda )$ becomes unstable for $\\lambda >\\lambda (h)$ , where $\\lambda (h)\\rightarrow 0$ , as $h\\rightarrow 0$ .",
"This happens because it becomes energetically more advantageous to activate bending modes rather than store more compressive stress.",
"This exchange of stability is detected by the change in sign of the second variation of energy $\\delta ^{2}{\\mathcal {E}}(\\phi ;h,\\lambda )=\\int _{\\Omega _{h}}(W_{ F F}(\\nabla y( x;h,\\lambda ))\\nabla \\phi ,\\nabla \\phi )d x,\\qquad \\phi \\in V_{h},$ where $V_{h}=\\overline{V_{h}^{\\circ }}$ is the closure of $V_{h}^{\\circ }$ in $W^{1,2}(\\Omega _{h};\\mathbb { R}^{3})$ .",
"The second variation is always non negative, when $0<\\lambda <\\lambda (h)$ and can become negative for some choice of the admissible variation $\\phi \\in V_{h}$ , when $\\lambda >\\lambda (h)$ .",
"It was understood in [13] that this failure of weak stability is due to properties (P1)–(P4) of $W( F)$ and is intimately related to flip instability in soft device."
],
[
"Buckling load and buckling mode",
"Using the second variation criterion for stability we define the buckling load as $\\lambda ^{*}(h)=\\inf \\lbrace \\lambda >0:\\delta ^{2}{\\mathcal {E}}(\\phi ;h,\\lambda )<0\\text{ for some }\\phi \\in V_{h} \\rbrace .$ Definition 2.2 We say that the body undergoes a near-flip buckling if $\\lambda ^{*}(h)>0$ for all $h\\in (0,h_{0})$ , for some $h_{0}>0$ , and $\\lambda ^{*}(h)\\rightarrow 0$ , as $h\\rightarrow 0$ .",
"We refer to [13] for a discussion of why this terminology is appropriate.",
"The buckling mode is generally understood as the variation $\\phi ^{*}_{h}\\in V_{h}\\setminus \\lbrace 0\\rbrace $ , such that $\\delta ^{2}{\\mathcal {E}}(\\phi ^{*}_{h};h,\\lambda ^{*}(h))=0$The question of existence of the buckling mode $\\phi ^{*}_{h}$ is irrelevant here, since the goal of this discussion is to explain the intuitive meaning of the formal definition of a buckling mode, made in Definition REF ..",
"However, if we are only interested in the asymptotics of the critical load, as $h\\rightarrow 0$ , then we would not distinguish between $\\lambda ^{*}(h)$ and $\\lambda (h)$ , as long as $\\lambda (h)/\\lambda ^{*}(h)\\rightarrow 1$ , as $h\\rightarrow 0$ .",
"If we replace $\\lambda ^{*}(h)$ with $\\lambda _{\\epsilon }(h)=\\lambda ^{*}(h)(1+\\epsilon )$ , then we estimate $\\delta ^{2}{\\mathcal {E}}(\\phi ^{*}_{h};h,\\lambda ^{*}(h)(1+\\epsilon ))\\approx \\lambda ^{*}(h)\\epsilon \\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi ^{*}_{h};h,\\lambda ^{*}(h)).$ This means that for the purposes of asymptotics we should not distinguish differences in values of second variation that are infinitesimal, compared to $\\lambda ^{*}(h)\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi ^{*}_{h};h,\\lambda ^{*}(h)).$ In keeping with these observations, we redefine the notion of the buckling load and buckling mode, under the assumption that the body undergoes a near-flip buckling in the sense of Definition REF .",
"Definition 2.3 We say that $\\lambda (h)\\rightarrow 0$ , as $h\\rightarrow 0$ is a buckling load if $\\lim _{h\\rightarrow 0}\\frac{\\lambda (h)}{\\lambda ^{*}(h)}=1.$ A buckling mode is a family of variations $\\phi _{h}\\in V_{h}\\setminus \\lbrace 0\\rbrace $ , such that $\\lim _{h\\rightarrow 0}\\frac{\\delta ^{2}{\\mathcal {E}}(\\phi _{h};h,\\lambda ^{*}(h))}{\\lambda ^{*}(h)\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi _{h};h,\\lambda ^{*}(h))}=0.$ The most important insight in [13] is that at the critical load $\\lambda ^{*}(h)\\rightarrow 0$ , as $h\\rightarrow 0$ , the local material response is well inside the linearly elastic regime and the instability can be detected by a simpler constitutively linearized second variation: $\\delta ^{2}{\\mathcal {E}}_{cl}(\\phi ;h,\\lambda )=\\int _{\\Omega _{h}}\\lbrace (\\mathsf {L}_{0}e(\\phi ),e(\\phi ))+\\lambda (\\sigma _{h},\\nabla \\phi ^{T}\\nabla \\phi )\\rbrace d x, \\qquad \\phi \\in V_{h}$ where $e(\\phi )=\\displaystyle \\frac{1}{2}(\\nabla \\phi +(\\nabla \\phi )^{T})$ and $\\sigma _{h}( x)=\\mathsf {L}_{0}e( u^{h}( x))$ is the linear elastic stress.",
"We define $\\mathfrak {C}_{h}(\\phi )=\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}}_{cl})}{\\partial \\lambda }(\\phi ;h,\\lambda )=\\int _{\\Omega _{h}}(\\sigma _{h},\\nabla \\phi ^{T}\\nabla \\phi )d x.$ Observe that ${\\mathcal {A}}_{h}=\\left\\lbrace \\phi \\in V_{h}:\\mathfrak {C}_{h}(\\phi )<0\\right\\rbrace $ can be regarded as the set of all destabilizing variations for (REF ).",
"We assume that the applied loading has a compressive nature.",
"In particular, we assume that the sets ${\\mathcal {A}}_{h}$ are non-empty for all $h\\in (0,h_{0})$ for some $h_{0}>0$ .",
"In parallel with our discussion of the asymptotics of the critical load and buckling mode we define the functional $\\mathfrak {R}(h,\\phi )=-\\frac{\\int _{\\Omega _{h}}(\\mathsf {L}_{0}e(\\phi ),e(\\phi ))d x}{\\int _{\\Omega _{h}}(\\sigma _{h},\\nabla \\phi ^{T}\\nabla \\phi )d x}=-\\frac{\\mathfrak {S}_{h}(\\phi )}{\\mathfrak {C}_{h}(\\phi )},$ where $\\mathfrak {S}_{h}(\\phi )=\\int _{\\Omega _{h}}(\\mathsf {L}_{0}e(\\phi ),e(\\phi ))d x.$ is the measure of stability of the trivial branch.",
"The constitutively linearized buckling load and buckling mode are then defined by analogy with the original second variation.",
"Definition 2.4 The constitutively linearized buckling load $\\widehat{\\lambda }(h)$ is defined by $\\widehat{\\lambda }(h)=\\inf _{\\phi \\in {\\mathcal {A}}_{h}}\\mathfrak {R}(h,\\phi ).$ We say that the family of variations $\\lbrace \\phi _{h}\\in {\\mathcal {A}}_{h}:h\\in (0,h_{0})\\rbrace $ is a constitutively linearized buckling mode if $\\lim _{h\\rightarrow 0}\\frac{\\mathfrak {R}(h,\\phi _{h})}{\\widehat{\\lambda }(h)}=1.$ We now need to prove that under some reasonable assumptions the constitutively linearized buckling load and buckling mode are buckling mode and buckling mode, respectively, in the sense of Definition REF .",
"Recall the definition of the Korn constant $K(V_{h})=\\inf _{\\phi \\in V_{h}}\\frac{\\Vert e(\\phi )\\Vert ^{2}}{\\Vert \\nabla \\phi \\Vert ^{2}}.$ Here and elsewhere in this paper $\\Vert \\cdot \\Vert $ always denotes the $L^{2}$ -norm on $\\Omega _{h}$ .",
"Definition 2.5 We say that the body $\\Omega _{h}$ is slender if $\\lim _{h\\rightarrow 0}K(V_{h})=0.$ We remark that this notion of slenderness, introduced in [13], is not purely geometric, but depends on the type of loading described by the subspace $V_{h}$ .",
"On the one hand, a thin rod or a plate in the hard device will not be regarded as slender, since their Korn constant is 1/2, regardless of their geometric slenderness.",
"On the other hand, a geometrically non-slender body, such as a ball or a cube will not be slender under our definition, for any boundary conditions that exclude all rigid body motions.",
"Theorem 2.6 (Asymptotics of the critical load) Suppose that the body is slender in the sense of Definition REF .",
"Assume that the constitutively linearized critical load $\\widehat{\\lambda }(h)$ , defined in (REF ) satisfies $\\widehat{\\lambda }(h)>0$ for all sufficiently small $h$ and $\\lim _{h\\rightarrow 0}\\frac{\\widehat{\\lambda }(h)^{2}}{K(V_{h})}=0.$ Then $\\widehat{\\lambda }(h)$ is the buckling load and any constitutively linearized buckling mode $\\phi _{h}$ is a buckling mode in the sense of Definition REF .",
"The theorem is proved by means of the basic estimate, which is a simple modification of the estimates in [13] used in the derivation of the formula for $\\delta ^{2}{\\mathcal {E}}_{cl}(\\phi ;h,\\lambda )$ : Lemma 2.7 Suppose $ y( x;h,\\lambda )$ satisfies (REF ) and $W( F)$ has the properties (P1)–(P4).",
"Then $\\left|\\delta ^{2}{\\mathcal {E}}(\\phi ;h,\\lambda )-\\delta ^{2}{\\mathcal {E}}_{cl}(\\phi ;h,\\lambda )\\right|\\le C\\left(\\frac{\\lambda }{\\sqrt{K(V_{h})}}+\\frac{\\lambda ^{2}}{K(V_{h})}\\right)\\mathfrak {S}_{h}(\\phi ).$ and $\\left|\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi ;h,\\lambda )-\\mathfrak {C}_{h}(\\phi )\\right|\\le C\\left(\\frac{1}{\\sqrt{K(V_{h})}}+\\frac{\\lambda }{K(V_{h})}\\right)\\mathfrak {S}_{h}(\\phi ).$ where the constant $C$ is independent of $h$ , $\\lambda $ and $\\phi $ .",
"According to the frame indifference property (P2), $W( F)=\\widehat{W}( F^{T} F)$ .",
"Differentiating this formula twice we obtain $(W_{ F F}( F)\\xi ,\\xi )=4(\\widehat{W}_{ C C}( C)( F^{T}\\xi ), F^{T}\\xi )+2(\\widehat{W}_{ C}( C),\\xi ^{T}\\xi ),\\qquad C= F^{T} F.$ We can estimate $|(\\widehat{W}_{ C C}( C)( F^{T}\\xi ), F^{T}\\xi )-(\\widehat{W}_{ C C}( I)\\xi ,\\xi )|\\le |(\\widehat{W}_{ C C}( C)( F^{T}- I)\\xi ,( F^{T}- I)\\xi )|+$ $|((\\widehat{W}_{ C C}( C)-\\widehat{W}_{ C C}( I))\\xi ,\\xi )|+2|(\\widehat{W}_{ C C}( C)\\xi ,( F^{T}- I)\\xi )|$ When $ F$ is uniformly bounded we obtain $|(\\widehat{W}_{ C C}( C)( F^{T}\\xi ), F^{T}\\xi )-(\\widehat{W}_{ C C}( I)\\xi ,\\xi )|\\le C\\left(| F- I|^{2}|\\xi |^{2}+| C- I||\\xi _{\\rm sym}|^{2}+| F- I||\\xi _{\\rm sym}||\\xi |\\right).$ Similarly, $|(\\widehat{W}_{ C}( C)-\\widehat{W}_{ C C}( I)( C- I),\\xi ^{T}\\xi )|\\le C| C- I|^{2}|\\xi |^{2}$ When $ F=\\nabla y( x;h,\\lambda )$ and $\\xi =\\nabla \\phi $ we obtain, taking into account (REF ), that $| F- I|\\le C\\lambda ,\\qquad | C- I|\\le C\\lambda .$ Observing that $4\\widehat{W}_{ C C}( I)=W_{ F F}( I)=\\mathsf {L}_{0},\\qquad | C- I-2\\lambda e( u^{h})|\\le C\\lambda ^{2}.$ we obtain the estimate $|(W_{ F F}( F)\\xi ,\\xi )-(\\mathsf {L}_{0}\\xi _{\\rm sym},\\xi _{\\rm sym})-\\lambda (\\sigma _{h},\\xi ^{T}\\xi )|\\le C(\\lambda |\\xi _{\\rm sym}||\\xi |+\\lambda ^{2}|\\xi |^{2}).$ Integrating over $\\Omega _{h}$ as using the coercivity (REF ) of $\\mathsf {L}_{0}$ we obtain the estimate (REF ).",
"In order to prove the estimate (REF ) we substitute $ F=\\nabla y( x;h,\\lambda )$ and $\\xi =\\nabla \\phi $ into (REF ) and differentiate in $\\lambda $ , obtaining $\\displaystyle \\frac{\\partial (W_{ F F}( F)\\xi ,\\xi )}{\\partial \\lambda }=4((\\widehat{W}_{ C C C}( C)\\dot{ C})( F^{T}\\xi ), F^{T}\\xi )+8(\\widehat{W}_{ C C}( C)( F^{T}\\xi ),\\dot{ F}^{T}\\xi )+2(\\widehat{W}_{ C C}( C)\\dot{ C},\\xi ^{T}\\xi ),$ where $\\dot{ C}$ and $\\dot{ F}$ denote differentiation with respect to $\\lambda $ .",
"Using the uniform boundedness of $\\dot{ C}$ , which is a corollary of (REF ), as well as (REF ) we estimate $|((\\widehat{W}_{ C C C}( C)\\dot{ C})( F^{T}\\xi ), F^{T}\\xi )|\\le C(|\\xi _{\\rm sym}|^{2}+\\lambda |\\xi ||\\xi _{\\rm sym}|).$ and $|(\\widehat{W}_{ C C}( C)( F^{T}\\xi ),\\dot{ F}^{T}\\xi )|\\le C(|\\xi ||\\xi _{\\rm sym}|+\\lambda |\\xi |^{2}).$ We also estimate, using $| C- I|\\le C\\lambda $ and $|\\dot{ C}-2e( u^{h})|\\le C\\lambda $ , that are consequences of (REF ) and (REF ): $|2(\\widehat{W}_{ C C}( C)\\dot{ C},\\xi ^{T}\\xi )-(\\sigma _{h},\\xi ^{T}\\xi )|\\le C\\lambda |\\xi |^{2}.$ [Proof of Theorem REF ] By definition of $\\widehat{\\lambda }(h)$ , for any $\\epsilon >0$ and any $h\\in (0,h_{0})$ there exists $\\phi _h\\in {\\mathcal {A}}_h$ such that $\\mathfrak {S}_{h}(\\phi _{h})+\\widehat{\\lambda }(h)(1+\\epsilon )\\mathfrak {C}_{h}(\\phi _{h})<0,$ thus, $\\delta ^{2}{\\mathcal {E}}_{cl}(\\phi _{h};h,\\widehat{\\lambda }(h)(1+2\\epsilon ))\\le -\\frac{\\epsilon \\mathfrak {S}_{h}(\\phi _{h})}{1+\\epsilon }.$ The estimate (REF ) gives the upper bound on the second variation: $\\delta ^{2}{\\mathcal {E}}(\\phi _h;h,\\widehat{\\lambda }(h)(1+2\\epsilon ))\\le \\left(-\\frac{\\epsilon }{(1+\\epsilon )}+C\\left(\\frac{\\widehat{\\lambda }(h)}{\\sqrt{K(V_{h})}}+\\frac{\\widehat{\\lambda }(h)^{2}}{K(V_{h})}\\right)\\right)\\mathfrak {S}_{h}(\\phi _{h}),$ Thus, due to (REF ), for sufficiently small $h$ , we have $\\delta ^{2}{\\mathcal {E}}(\\phi _h;h,\\widehat{\\lambda }(h)(1+2\\epsilon ))<0$ , and hence $\\lambda ^{*}(h)\\le \\widehat{\\lambda }(h)(1+2\\epsilon )$ .",
"We conclude that $\\mathop {\\overline{\\lim }}_{h\\rightarrow 0}\\frac{\\lambda ^{*}(h)}{\\widehat{\\lambda }(h)}\\le 1.$ To prove the opposite inequality we observe that by definition of $\\widehat{\\lambda }(h)$ we have $\\mathfrak {S}_{h}(\\phi )+\\widehat{\\lambda }(h)\\mathfrak {C}_{h}(\\phi )\\ge 0$ for any $\\phi \\in V_{h}$ .",
"Therefore, for any $\\epsilon >0$ and any $0<\\lambda \\le \\widehat{\\lambda }(h)(1-\\epsilon ))$ we have $\\delta ^{2}{\\mathcal {E}}_{cl}(\\phi ;h,\\lambda )\\ge \\epsilon \\mathfrak {S}_{h}(\\phi ).$ The estimate (REF ) now gives the lower bound on the second variation: $\\delta ^{2}{\\mathcal {E}}(\\phi ;h,\\lambda )\\ge \\left(\\epsilon -C\\left(\\frac{\\widehat{\\lambda }(h)}{\\sqrt{K(V_{h})}}+\\frac{\\widehat{\\lambda }(h)^{2}}{K(V_{h})}\\right)\\right)\\mathfrak {S}_{h}(\\phi ),$ Thus for all sufficiently small $h$ and all $\\phi \\in V_{h}\\setminus \\lbrace 0\\rbrace $ we have $\\delta ^{2}{\\mathcal {E}}(\\phi ;h,\\lambda )>0$ for all $0<\\lambda \\le \\widehat{\\lambda }(h)(1-\\epsilon )$ , which means that $\\lambda ^{*}(h)\\ge \\widehat{\\lambda }(h)(1-\\epsilon )$ .",
"This implies $\\mathop {\\underline{\\lim }}_{h\\rightarrow 0}\\frac{\\lambda ^{*}(h)}{\\widehat{\\lambda }(h)}\\ge 1.$ Combining (REF ) and (REF ) we conclude that $\\widehat{\\lambda }(h)$ is the buckling load.",
"Assume now that $\\phi _h$ is a constitutively linearized buckling mode, i.e.",
"(REF ) holds.",
"Set $\\lambda =\\lambda ^{*}(h)$ and $\\phi =\\phi _h$ in the inequality (REF ).",
"Then, dividing both sides of the inequality by $-\\lambda ^{*}(h)\\mathfrak {C}_{h}(\\phi _{h})>0$ we obtain $\\left|\\frac{\\delta ^{2}{\\mathcal {E}}(\\phi _{h};h,\\lambda ^{*}(h))}{-\\lambda ^{*}(h)\\mathfrak {C}_{h}(\\phi _{h})}-\\left(\\frac{\\mathfrak {R}(h,\\phi _{h})}{\\lambda ^{*}(h)}-1\\right)\\right|\\le C\\left(\\frac{\\lambda ^{*}(h)}{\\sqrt{K(V_{h})}}+\\frac{(\\lambda ^{*}(h))^{2}}{K(V_{h})}\\right)\\frac{\\mathfrak {R}(h,\\phi _{h})}{\\lambda ^{*}(h)}.$ Since we have proved that $\\widehat{\\lambda }(h)$ is the buckling load we conclude that $\\lim _{h\\rightarrow 0}\\frac{\\delta ^{2}{\\mathcal {E}}(\\phi _{h};h,\\lambda ^{*}(h))}{\\lambda ^{*}(h)\\mathfrak {C}_{h}(\\phi _{h})}=0.$ Similarly, setting $\\lambda =\\lambda ^{*}(h)$ and $\\phi =\\phi _h$ in the inequality (REF ) and dividing both sides of the inequality by $-\\mathfrak {C}_{h}(\\phi _{h})>0$ we obtain $\\left|\\frac{\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi _{h};h,\\lambda ^{*}(h))}{-\\mathfrak {C}_{h}(\\phi _{h})}+1\\right|\\le C\\left(\\frac{\\lambda ^{*}(h)}{\\sqrt{K(V_{h})}}+\\frac{(\\lambda ^{*}(h))^{2}}{K(V_{h})}\\right)\\frac{\\mathfrak {R}(h,\\phi _{h})}{\\lambda ^{*}(h)}.$ We conclude that $\\lim _{h\\rightarrow 0}\\frac{\\displaystyle \\frac{\\partial (\\delta ^{2}{\\mathcal {E}})}{\\partial \\lambda }(\\phi _{h};h,\\lambda ^{*}(h))}{\\mathfrak {C}_{h}(\\phi _{h})}=1.$ It follows now that $\\phi _{h}$ satisfies (REF ), and the theorem is proved.",
"An immediate consequence of our Rayleigh quotient characterization of buckling load (REF ) is the safe load estimate: $\\widehat{\\lambda }(h)=\\inf _{\\phi \\in {\\mathcal {A}}_{h}}\\mathfrak {R}(h,\\phi )\\ge \\inf _{\\phi \\in V_{h}}\\frac{\\mathfrak {S}_{h}(\\phi )}{\\Vert \\sigma _{h}\\Vert _{\\infty }\\Vert \\nabla \\phi \\Vert ^{2}}=\\frac{K_{\\mathsf {L}_{0}}(V_{h})}{\\Vert \\sigma _{h}\\Vert _{\\infty }},$ where $K_{\\mathsf {L}_{0}}(V_{h})=\\inf _{\\phi \\in V_{h}}\\frac{\\int _{\\Omega _{h}}(\\mathsf {L}_{0}e(\\phi ),e(\\phi ))d x}{\\Vert \\nabla \\phi \\Vert ^{2}}.$ We remark that rods and plates have buckling loads proportional to $K_{\\mathsf {L}_{0}}(V_{h})$ , while the theoretical buckling load for axially compressed circular cylindrical shells is much higher.",
"Based on this, we conjecture that buckling loads that scale with $K(V_{h})$ should not exhibit sensitivity to imperfections."
],
[
"Buckling equivalence",
"In the previous subsection we showed that the asymptotics of the critical load and buckling mode can be captured by a constitutively linearized functional $\\mathfrak {R}(h,\\phi )$ .",
"Even though such a characterization of buckling represents a significant simplification, compared to the characterization based on the second variation of a fully non-linear energy functional, further simplifications may be necessary to obtain explicit analytic expressions in specific problems.",
"We envision two ways in which the analysis of buckling can be simplified.",
"One is the simplification of the functional $\\mathfrak {R}(h,\\phi )$ .",
"The other is replacing the space of all admissible functions ${\\mathcal {A}}_{h}$ with a smaller space ${\\mathcal {B}}_{h}$ .",
"For example, we may want to use a specific ansatz, like the Kirchhoff ansatz in buckling of rods and plates.",
"In order to formalize our simplification procedure we make the following definitions.",
"Definition 2.8 Assume that $J(h,\\phi )$ is a variational functional defined on ${\\mathcal {B}}_h\\subset {\\mathcal {A}}_{h}$ .",
"We say that the pair $({\\mathcal {B}}_h, J(h,\\phi ))$ characterizes buckling if the following three conditions are satisfied Characterization of the buckling load: If $\\lambda (h)=\\inf _{\\phi \\in {\\mathcal {B}}_{h}}J(h,\\phi ),$ then $\\lambda (h)$ is a buckling load in the sense of Definition REF .",
"Characterization of the buckling mode: If $\\phi _{h}\\in {\\mathcal {B}}_{h}$ is a buckling mode in the sense of Definition REF , then $\\lim _{h\\rightarrow 0}\\frac{J(h,\\phi _{h})}{\\lambda (h)}=1.$ Faithful representation of the buckling mode: If $\\phi _{h}\\in {\\mathcal {B}}_{h}$ satisfies (REF ) then it is a buckling mode.",
"Definition 2.9 Two pairs $({\\mathcal {B}}_h, J_{1}(h,\\phi ))$ and $({\\mathcal {C}}_h, J_{2}(h,\\phi ))$ are called buckling equivalent if the pair $({\\mathcal {B}}_h,J_{1}(h,\\phi ))$ characterizes buckling if and only if $({\\mathcal {C}}_h,J_2(h,\\phi ))$ does.",
"The notion of buckling equivalence of functionals $({\\mathcal {B}}_h, J(h,\\phi ))$ is an extension of B-equivalence, introduced in [13], in that it also captures buckling modes in addition to buckling loads.",
"Let us first address a simple question of restricting the space of functions ${\\mathcal {B}}_{h}$ to an “ansatz” ${\\mathcal {C}}_{h}$ .",
"Lemma 2.10 Suppose the pair $({\\mathcal {B}}_{h},J(h,\\phi ))$ characterizes buckling.",
"Let ${\\mathcal {C}}_{h}\\subset {\\mathcal {B}}_{h}$ be such that it contains a buckling mode.",
"Then the pair $({\\mathcal {C}}_{h},J(h,\\phi ))$ characterizes buckling.",
"Let $\\lambda (h)=\\inf _{\\phi \\in {\\mathcal {B}}_{h}}J(h,\\phi ),\\qquad \\widetilde{\\lambda }(h)=\\inf _{\\phi \\in {\\mathcal {C}}_{h}}J(h,\\phi ).$ Then, clearly, $\\widetilde{\\lambda }(h)\\ge \\lambda (h)$ .",
"By assumption there exists a buckling mode $\\phi _{h}\\in {\\mathcal {C}}_{h}\\subset {\\mathcal {B}}_{h}$ .",
"Therefore, $\\mathop {\\overline{\\lim }}_{h\\rightarrow 0}\\frac{\\widetilde{\\lambda }(h)}{\\lambda (h)}\\le \\lim _{h\\rightarrow 0}\\frac{J(h,\\phi _{h})}{\\lambda (h)}=1,$ since the pair $({\\mathcal {B}}_{h},J(h,\\phi ))$ characterizes buckling.",
"Hence $\\lim _{h\\rightarrow 0}\\frac{\\widetilde{\\lambda }(h)}{\\lambda (h)}=1,$ and part (a) of Definition REF is established.",
"If $\\phi _{h}\\in {\\mathcal {C}}_{h}\\subset {\\mathcal {B}}_{h}$ is a buckling mode then $\\lim _{h\\rightarrow 0}\\frac{J(h,\\phi _{h})}{\\lambda (h)}=1,$ since the pair $({\\mathcal {B}}_{h},J(h,\\phi ))$ characterizes buckling.",
"Part (b) now follows from (REF ).",
"Finally, if $\\phi _{h}\\in {\\mathcal {C}}_{h}$ satisfies $\\lim _{h\\rightarrow 0}\\frac{J(h,\\phi _{h})}{\\widetilde{\\lambda }(h)}=1,$ then, $\\phi _{h}\\in {\\mathcal {B}}_{h}$ and by (REF ) we also have $\\lim _{h\\rightarrow 0}\\frac{J(h,\\phi _{h})}{\\lambda (h)}=1.$ Therefore, $\\phi _{h}$ is a buckling mode.",
"The Lemma is proved now.",
"Our key tool for simplification of the functionals $J(h,\\phi )$ characterizing buckling is the following theorem.",
"Theorem 2.11 (Buckling equivalence) Suppose that $\\lambda (h)$ is a buckling load in the sense of Definition REF .",
"If either $\\lim _{h\\rightarrow 0}\\lambda (h)\\sup _{\\phi \\in {\\mathcal {B}}_{h}}\\left|\\frac{1}{J_1(h,\\phi )}-\\frac{1}{J_2(h,\\phi )}\\right|=0,$ or $\\lim _{h\\rightarrow 0}\\displaystyle \\frac{1}{\\lambda (h)}\\sup _{\\phi \\in {\\mathcal {B}}_{h}}|J_1(h,\\phi )-J_2(h,\\phi )|=0,$ then the pairs $({\\mathcal {B}}_h, J_1(h,\\phi ))$ and $({\\mathcal {B}}_h, J_2(h,\\phi ))$ are buckling equivalent in the sense of Definition REF .",
"Let us introduce the following notation: $\\lambda _i(h)=\\inf _{\\phi \\in {\\mathcal {B}}_h}J_i(h,\\phi ),\\quad i=1,2.$ $\\delta _{1}(h)=\\lambda (h)\\sup _{\\phi \\in {\\mathcal {B}}_{h}}\\left|\\frac{1}{J_1(h,\\phi )}-\\frac{1}{J_2(h,\\phi )}\\right|.$ $\\delta _{2}(h)=\\displaystyle \\frac{1}{\\lambda (h)}\\sup _{\\phi \\in {\\mathcal {B}}_{h}}|J_1(h,\\phi )-J_2(h,\\phi )|.$ Then $\\left|\\frac{\\lambda (h)}{\\lambda _1(h)}-\\frac{\\lambda (h)}{\\lambda _2(h)}\\right|=\\lambda (h)\\left|\\sup _{\\phi \\in {\\mathcal {B}}_{h}}\\frac{1}{J_1(h,\\phi )}-\\sup _{\\phi \\in {\\mathcal {B}}_{h}}\\frac{1}{J_2(h,\\phi )}\\right|\\le \\delta _{1}(h)$ and $\\frac{|\\lambda _1(h)-\\lambda _{2}(h)|}{\\lambda (h)}=\\displaystyle \\frac{1}{\\lambda (h)}\\left|\\inf _{\\phi \\in {\\mathcal {B}}_{h}}J_1(h,\\phi )-\\inf _{\\phi \\in {\\mathcal {B}}_{h}}J_2(h,\\phi )\\right|\\le \\delta _{2}(h)$ Assume that $({\\mathcal {B}}_h, J_1(h,\\phi ))$ characterizes buckling.",
"Then we have just proved that if either $\\delta _{1}(h)\\rightarrow 0$ or $\\delta _{2}(h)\\rightarrow 0$ , as $h\\rightarrow 0$ , then $\\lambda _2(h)/\\lambda (h)\\rightarrow 1$ , as $h\\rightarrow 0$ , and condition (a) in Definition REF is proved for $J_{2}(h,\\phi )$ .",
"Observe that by parts (b) and (c) of Definition REF $\\phi _{h}\\in {\\mathcal {B}}_{h}$ is the buckling mode if and only if $\\lim _{h\\rightarrow 0}\\frac{J_{1}(h,\\phi _{h})}{\\lambda _{1}(h)}=1.$ This is equivalent to $\\lim _{h\\rightarrow 0}\\frac{\\lambda (h)}{J_{1}(h,\\phi _{h})}=1.$ Therefore, $\\lim _{h\\rightarrow 0}\\frac{J_{2}(h,\\phi _{h})}{\\lambda (h)}=1,$ since either $\\left|\\frac{\\lambda (h)}{J_{1}(h,\\phi _{h})}-\\frac{\\lambda (h)}{J_{2}(h,\\phi _{h})}\\right|\\le \\delta _{1}(h)$ or $\\frac{|J_{1}(h,\\phi _{h})-J_{2}(h,\\phi _{h})|}{\\lambda (h)}\\le \\delta _{2}(h)$ Thus, in view of part (a), $\\phi _{h}$ is a buckling mode if and only if $\\lim _{h\\rightarrow 0}\\frac{J_{2}(h,\\phi _{h})}{\\lambda _{2}(h)}=1.$ As an application of Theorem REF we show that we can simplify the Rayleigh quotient $\\mathfrak {R}(h,\\phi )$ further.",
"Theorem 2.12 Suppose that the critical load $\\widehat{\\lambda }(h)$ satisfies (REF ).",
"Let $\\mathfrak {R}_{0}(h,\\phi )=-\\frac{\\int _{\\Omega _{h}}(\\mathsf {L}_{0}e(\\phi ),e(\\phi ))d x}{\\displaystyle \\frac{1}{4}\\int _{\\Omega _{h}}(\\widetilde{\\sigma }_{h}\\nabla \\times \\phi ,\\nabla \\times \\phi )d x}=-\\frac{\\mathfrak {S}_{h}(\\phi )}{\\mathfrak {C}_{h}^{0}(\\phi )},$ where $\\widetilde{\\sigma }_{h}=(\\mathrm {Tr}\\,\\sigma _{h}) I-\\sigma _{h}$ is the compression tensor.",
"Then $({\\mathcal {A}}_{h},\\mathfrak {R}(h,\\phi ))$ and $({\\mathcal {A}}_{h},\\mathfrak {R}_{0}(h,\\phi ))$ are buckling equivalent.",
"For $ a\\in \\mathbb { R}^{3}$ , let $\\pi ( a)$ denote a $3\\times 3$ antisymmetric matrix defined by the cross-product map: $\\pi ( a) u= a\\times u.$ Then $\\nabla \\phi -(\\nabla \\phi )^{T}=\\pi (\\nabla \\times \\phi )$ .",
"We observe that replacing $\\nabla \\phi $ with $e(\\phi )-\\pi (\\nabla \\times \\phi )/2$ in $\\mathfrak {C}_{h}(\\phi )$ we obtain $\\mathfrak {C}_{h}(\\phi )=\\int _{\\Omega _{h}}\\left(\\sigma _{h},e(\\phi )^{2}+e(\\phi )\\pi (\\nabla \\times \\phi )\\right)d x+\\mathfrak {C}^{0}_{h}(\\phi ).$ It follows that for every $\\phi \\in V_{h}$ $|\\mathfrak {C}_{h}(\\phi )-\\mathfrak {C}^{0}_{h}(\\phi )|\\le \\Vert \\sigma _{h}\\Vert _{\\infty }(\\Vert e(\\phi )\\Vert ^{2}+2\\Vert e(\\phi )\\Vert \\Vert \\nabla \\phi \\Vert )\\le \\Vert \\sigma _{h}\\Vert _{\\infty }\\Vert e(\\phi )\\Vert ^{2}\\left(1+\\frac{2}{\\sqrt{K(V_{h})}}\\right).$ Recalling that, due to (REF ), $\\mathfrak {S}_{h}(\\phi )\\ge \\alpha _{\\mathsf {L}_{0}}\\Vert e(\\phi )\\Vert ^{2}$ we obtain $\\widehat{\\lambda }(h)\\left|\\displaystyle \\frac{1}{\\mathfrak {R}(h,\\phi )}-\\displaystyle \\frac{1}{\\mathfrak {R}_{0}(h,\\phi ))}\\right|\\le \\frac{\\Vert \\sigma _{h}\\Vert _{\\infty }}{\\alpha _{\\mathsf {L}_{0}}}\\left(\\widehat{\\lambda }(h)+\\frac{2\\widehat{\\lambda }(h)}{\\sqrt{K(V_{h})}}\\right).$ Thus (REF ) implies that the sufficient condition (REF ) for buckling equivalence is satisfied.",
"The theorem is proved.",
"We remark that $\\nabla \\times \\phi $ is a scalar in 2D, and similar calculations show that the functional $\\mathfrak {R}(h,\\phi )$ can be replaced in 2D by $\\mathfrak {R}_{0}^{2D}(h,\\phi )=-\\frac{\\int _{\\Omega _{h}}(\\mathsf {L}_{0}e(\\phi ),e(\\phi ))d x}{\\displaystyle \\frac{1}{2}\\int _{\\Omega _{h}}\\mathrm {Tr}\\,\\sigma _{h}|\\nabla \\phi |^{2}d x}$ Therefore, in the case of a homogeneous compressive trivial branch $\\lim _{h\\rightarrow 0}\\mathrm {Tr}\\,\\sigma _{h}=\\mathfrak {c}<0$ we have a general formula for the critical load [13]: $\\widehat{\\lambda }(h)=\\frac{2K_{\\mathsf {L}_{0}}(V_{h})}{\\mathfrak {c}}.$ By contrast, the situation in 3D is much more nuanced.",
"Even in the case of a homogeneous trivial branch, the critical load formula demands further study."
],
[
"Buckling of circular cylindrical shells",
"In this section we apply the theory of near-flip buckling developed in Section to the buckling of circular cylindrical shells under axial compression."
],
[
"Trivial branch ",
"Consider the circular cylindrical shell given in cylindrical coordinates $(r,\\theta ,z)$ as follows: ${\\mathcal {C}}_{h}=I_{h}\\times \\mathbb { T}\\times [0,L],\\qquad I_{h}=[1-h/2,1+h/2],$ where $\\mathbb { T}$ is a 1-dimensional torus (circle) describing $2\\pi $ -periodicity in $\\theta $ .",
"In this paper we consider the axial compression of the shell where the deformation $ y:{\\mathcal {C}}_{h}\\rightarrow \\mathbb {R}^3$ satisfies the following boundary conditions: $y_{\\theta }(r,\\theta ,0)=y_{z}(r,\\theta ,0)=y_{\\theta }(r,\\theta ,L)=0,\\quad y_{z}(r,\\theta ,L)=(1-\\lambda )L,\\quad t( x;h,\\lambda )= 0,$ where $ t$ is the vector of tractions in (REF ).",
"The loading is parametrized by the compressive strain $\\lambda $ in the axial direction.",
"To apply our theory of buckling we need to describe the boundary conditions in the form (REF ).",
"This is done by defining $\\overline{ y}( x;h,\\lambda )=(1-\\lambda )z e_{z},$ and $V_{h}^{\\circ }=\\lbrace \\phi \\in W^{1,\\infty }({\\mathcal {C}}_{h};\\mathbb {R}^3):\\phi _{\\theta }(r,\\theta ,0)=\\phi _{z}(r,\\theta ,0)=\\phi _{\\theta }(r,\\theta ,L)=\\phi _{z}(r,\\theta ,L)=0\\rbrace ,$ which gives $V_{h}=\\lbrace \\phi \\in W^{1,2}({\\mathcal {C}}_{h};\\mathbb {R}^3):\\phi _{\\theta }(r,\\theta ,0)=\\phi _{z}(r,\\theta ,0)=\\phi _{\\theta }(r,\\theta ,L)=\\phi _{z}(r,\\theta ,L)=0\\rbrace .$ These boundary conditions allow the shell to “breathe”, since the radial displacements are not prescribed at either end.",
"In our notation the dependence on $L$ will be consistently suppressed, while the essential dependence on $h$ will be emphasized.",
"We observe that during buckling the Cauchy-Green strain tensor $ C= F^{T} F$ is close to the identity.",
"Therefore, considering the energy which is quadratic in $ E=( C- I)/2$ should capture all the effects associated with buckling.",
"Hence, we assume, for the purposes of exhibiting the explicit form of the trivial branch, that in the vicinity of the identity matrix the energy density has the Saint Venant-Kirchhoff form: $W( F)=\\displaystyle \\frac{1}{2}(\\mathsf {L}_{0} E, E),\\qquad E=\\displaystyle \\frac{1}{2}( F^{T} F- I).$ where the elastic tensor $\\mathsf {L}_{0}$ is isotropic.",
"We study stability of the homogeneous trivial branch $ y( x;h,\\lambda )$ given in cylindrical coordinates by $y_{r}=(a(\\lambda )+1)r,\\qquad y_{\\theta }=0,\\qquad y_{z}=(1-\\lambda )z.$ We compute, using the formula $\\nabla \\phi =\\begin{bmatrix}\\phi _{r,r} & \\dfrac{\\phi _{r,\\theta }-\\phi _\\theta }{r} & \\phi _{r,z}\\\\\\phi _{\\theta ,r} & \\dfrac{\\phi _{\\theta ,\\theta }+\\phi _r}{r} & \\phi _{\\theta ,z}\\\\\\phi _{z,r} & \\dfrac{\\phi _{z,\\theta }}{r} & \\phi _{z,z}\\\\\\end{bmatrix}.$ for the gradient of the vector field $\\phi =\\phi _r e_r+\\phi _\\theta e_\\theta +\\phi _z e_z$ in cylindrical coordinates $ F=\\nabla y=\\left[\\begin{array}{ccc}1+a & 0 &0\\\\0 & 1+a& 0\\\\0 & 0& 1-\\lambda \\end{array}\\right],\\quad E=\\left[\\begin{array}{ccc}a+\\frac{a^{2}}{2} & 0 &0\\\\0 & a+\\frac{a^{2}}{2}& 0\\\\0 & 0& \\frac{\\lambda ^{2}}{2}-\\lambda \\end{array}\\right]$ Then we compute $ P= F(\\mathsf {L}_{0} E)$ , and the traction-free condition $ P e_{r}= 0$ on the lateral boundary leads to the expression for $a(\\lambda )$ : $a(\\lambda )=\\sqrt{1+2\\nu \\lambda -\\nu \\lambda ^{2}}-1,$ where $\\nu $ is the Poisson's ratio for $\\mathsf {L}_{0}$ .",
"We now see that the fundamental assumptions (REF ) and (REF ) are satisfied, since the trivial branch does not depend on $h$ explicitly.",
"We compute $\\sigma _{h}=-E e_{z}\\otimes e_{z},$ where $E$ is the Young's modulus.",
"The compression tensor $\\widetilde{\\sigma }_{h}$ defined in (REF ) is given by $\\widetilde{\\sigma }_{h}=-E\\left[\\begin{array}{ccc}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 0\\end{array}\\right].$ We see that the compression tensor is degenerate.",
"This degeneracy in the compression tensor is one of the factors contributing to the sensitivity of the critical load to imperfections."
],
[
"Scaling of the critical load",
"Theorem 3.1 Suppose that $\\sigma _{h}$ is given by (REF ).",
"Then there exist constants $c>0$ and $C>0$ depending only on $L$ and the elastic moduli, such that $ch\\le \\widehat{\\lambda }(h)\\le Ch.$ Observe that $\\mathfrak {C}_{h}(\\phi )=\\int _{{\\mathcal {C}}_{h}}(\\sigma _{h},\\nabla \\phi ^{T}\\nabla \\phi )d x=-E(\\Vert \\phi _{r,z}\\Vert ^2+\\Vert \\phi _{\\theta ,z}\\Vert ^2+\\Vert \\phi _{z,z}\\Vert ^2),$ and there exist constants $\\alpha >0$ and $\\beta >0$ (depending only on the elastic moduli) such that $\\alpha \\Vert e(\\phi )\\Vert ^{2}\\le \\mathfrak {S}_{h}(\\phi )\\le \\beta \\Vert e(\\phi )\\Vert ^{2}.$ Thus, in order to compute the scaling of $\\widehat{\\lambda }(h)$ , given by (REF ) and verify conditions of Theorem REF we need to estimate the Korn constant $K(V_{h})$ , as well as the norms of gradient components $\\Vert \\phi _{r,z}\\Vert ^2$ , $\\Vert \\phi _{\\theta ,z}\\Vert ^2$ and $\\Vert \\phi _{z,z}\\Vert ^2$ in terms of $\\Vert e(\\phi )\\Vert $ .",
"This was accomplished in our companion paper [11].",
"The desired estimates are stated in the following lemma.",
"Lemma 3.2 (Korn-type inequalities) There exist constants $C(L),c(L)>0$ depending only on $L$ such that $c(L)h^{3/2}\\le K(V_{h})\\le C(L)h^{3/2}.$ $\\Vert \\phi _{\\theta ,z}\\Vert ^{2}\\le \\frac{C(L)}{\\sqrt{h}}\\Vert e(\\phi )\\Vert ^{2},$ $\\Vert \\phi _{r,z}\\Vert ^{2}\\le \\frac{C(L)}{h}\\Vert e(\\phi )\\Vert ^{2}.$ Moreover, the powers of $h$ in the inequalities (REF )–(REF ) are optimal, achieved simultaneously by the ansatz ${\\left\\lbrace \\begin{array}{ll}\\phi ^{h}_{r}(r,\\theta ,z)=&-W_{,\\eta \\eta }\\left(\\frac{\\theta }{\\@root 4 \\of {h}},z\\right)\\\\[2ex]\\phi ^{h}_{\\theta }(r,\\theta ,z)=&r\\@root 4 \\of {h}W_{,\\eta }\\left(\\frac{\\theta }{\\@root 4 \\of {h}},z\\right)+\\frac{r-1}{\\@root 4 \\of {h}}W_{,\\eta \\eta \\eta }\\left(\\frac{\\theta }{\\@root 4 \\of {h}},z\\right),\\\\[2ex]\\phi ^{h}_{z}(r,\\theta ,z)=&(r-1)W_{,\\eta \\eta z}\\left(\\frac{\\theta }{\\@root 4 \\of {h}},z\\right)-\\sqrt{h}W_{,z}\\left(\\frac{\\theta }{\\@root 4 \\of {h}},z\\right),\\end{array}\\right.",
"}$ for any smooth compactly supported function $W(\\eta ,z)$ on $(-1,1)\\times (0,L)$ , with the understanding that the function $\\phi ^{h}(\\theta ,z)$ is extended $2\\pi $ -periodically in $\\theta \\in \\mathbb { R}$ .",
"Adding inequalities (REF ), (REF ) and an obvious inequality $\\Vert \\phi _{z,z}\\Vert ^{2}\\le \\Vert e(\\phi )\\Vert ^{2}$ we obtain $-\\mathfrak {C}_{h}(\\phi )\\le Ch\\mathfrak {S}_{h}(\\phi ).$ The power of $h$ in (REF ) is optimal, achieved by the ansatz (REF ).",
"Hence, the estimates (REF ) are proved.",
"We remark that the upper bound in (REF ) implies that condition (REF ) in Theorem REF is satisfied.",
"But then $\\widehat{\\lambda }(h)$ is the buckling load in the sense of Definition REF .",
"Figure: Yoshimura buckling pattern on the umbrella cover at theMathematisches Forschungsinstitut, Oberwolfach, Germany.",
"Photo by AntonioDeSimone.Remark 3.3 We remark that the scaling of the critical strain $\\lambda ^{*}(h)\\sim h$ implies that the elastic energy stored in the critically strained cylinder is of order $h^{3}$ , since the stress remains proportional to the strain at the onset of buckling.",
"Thus, the $\\Gamma $ -limit theorem from [7] applies.",
"However, that theorem misses the structure of low energy sequences, since the set of $W^{2,2}$ isometries of the cylindrical surface consists of rigid motions, and the limiting energy is zero.",
"The non-trivial isometries are non-smooth (Lipschitz), given by the Yoshimura buckling pattern [31] (see Figure REF ), which seems to be captured by some of the theoretical buckling modes in [12]."
],
[
"Scaling instability",
"In this section we exhibit scaling instability of the critical load under imperfections of load and shape.",
"The discussion of shape imperfections here is not rigorous, since the key Korn and Korn-type inequalities from [11] are rigorously proved for perfectly circular cylindrical shells.",
"However, once the necessary technical inequalities are established for an imperfect shell, its critical load can be estimated in a definitive and rigorous way, following the same strategy, as for a perfect shell."
],
[
"Imperfections of load",
"Consider a perfect isotropic circular cylindrical shell undergoing a compressive deformation satisfying the boundary conditions (REF ), which are perturbed arbitrarily, but only at $z=L$ .",
"We also assume that the modified boundary conditions do not violate the trivial branch regularity assumptions in Definition REF .",
"Let us make an additional assumption that the family of Lipschitz functions $ u^{h}$ from Definition REF depends regularly on $r$ and $h$ .",
"This means that $ u^{h}(r,\\theta ,z)\\approx \\widetilde{ u}^{h}(r,\\theta ,z)= u^{0}(\\theta ,z)+(r-1) u^{1}(\\theta ,z)+\\frac{(r-1)^{2}}{2} u^{2}(\\theta ,z),$ understood in the following sense: $\\lim _{h\\rightarrow 0} u^{h}=\\lim _{h\\rightarrow 0}\\widetilde{ u}^{h}= u^{0},\\qquad \\lim _{h\\rightarrow 0}\\nabla u^{h}=\\lim _{h\\rightarrow 0}\\nabla \\widetilde{ u}^{h}=\\lim _{h\\rightarrow 0}\\nabla ( u^{0}+(r-1) u^{1}),$ $\\lim _{h\\rightarrow 0}\\nabla \\nabla u^{h}=\\lim _{h\\rightarrow 0}\\nabla \\nabla \\widetilde{ u}^{h}$ where the first two limits are understood in the a.e.",
"sense, while the last limit is understood in the sense of distributions.",
"According to the formula (REF ), the buckling load depends only on $\\sigma ^{0}(\\theta ,z)=\\lim _{h\\rightarrow 0}\\sigma _{h}(r,\\theta ,z)=\\lim _{h\\rightarrow 0}\\mathsf {L}_{0}e( u^{h}).$ Under these assumptions generic load imperfections at $z=L$ may involve several functions of $\\theta $ .",
"We will show now that somewhat surprisingly, the regularity assumptions (REF ) guarantee that the set of possible limits $\\sigma ^{0}(\\theta ,z)$ depends only on 2 scalar parameters.",
"Theorem 3.4 Suppose that $ u^{h}(r,\\theta ,z)$ depends on $r$ and $h$ regularly, in the sense of (REF ).",
"Suppose further that (i) $\\nabla \\cdot (\\mathsf {L}_{0}e( u^{h}))= 0$ , (ii) $\\sigma _{h} e_{r}= 0$ at $r=1\\pm h/2$ , where $\\sigma _{h}=\\mathsf {L}_{0}e( u^{h})$ , (iii) $u^{h}_{z}(r,\\theta ,0)=u^{h}_{\\theta }(r,\\theta ,0)=0$ .",
"Then there exist two constants $s$ and $t$ , such that $u^{0}_{r}+(r-1)u_{r}^{1}=-\\frac{t\\nu }{E}r,\\quad u^{0}_{\\theta }+(r-1)u_{\\theta }^{1}=\\frac{2(1+\\nu )s}{E}rz,\\quad u^{0}_{z}+(r-1)u_{z}^{1}=\\frac{t}{E}z,$ and consequently $\\sigma ^{0}=\\begin{bmatrix}0 & 0 & 0\\\\0 & 0 & s\\\\0 & s & t\\end{bmatrix}$ By the assumptions of regularity (REF ) and by condition (i) we have $\\lim _{h\\rightarrow 0}\\nabla \\cdot \\sigma _{h}=\\lim _{h\\rightarrow 0}\\nabla \\cdot (\\sigma ^{0}(\\theta ,z)+(r-1)\\sigma ^{1}(\\theta ,z))=0,$ where $\\sigma ^{0}=\\lim _{h\\rightarrow 0}\\mathsf {L}_{0}e( u^{0}+(r-1) u^{1}),\\qquad \\sigma ^{1}=\\lim _{h\\rightarrow 0}\\mathsf {L}_{0}e\\left( u^{1}+\\frac{r-1}{2} u^{2}\\right).$ Passing to the limit as $h\\rightarrow 0$ in (REF ), we obtain ${\\left\\lbrace \\begin{array}{ll}\\sigma ^{1}_{rr}+\\sigma ^{0}_{r\\theta ,\\theta }+\\sigma ^{0}_{rr}-\\sigma ^{0}_{\\theta \\theta }+\\sigma ^{0}_{rz,z}=0,\\\\\\sigma ^{1}_{r\\theta }+\\sigma ^{0}_{\\theta \\theta ,\\theta }+2\\sigma ^{0}_{r\\theta }+\\sigma ^{0}_{\\theta z,z}=0,\\\\\\sigma ^{1}_{rz}+\\sigma ^{0}_{\\theta z,\\theta }+\\sigma ^{0}_{rz}+\\sigma ^{0}_{zz,z}=0.\\end{array}\\right.",
"}$ The traction-free boundary conditions $\\sigma _{h} e_{r}= 0$ at $r=1\\pm h/2$ imply that $\\sigma ^{0}(\\theta ,z) e_{r}=\\sigma ^{1}(\\theta ,z) e_{r}= 0$ for all $(\\theta ,z)\\in \\mathbb { T}\\times (0,L)$ .",
"Substituting these equations into (REF ) we obtain $\\sigma ^{0}_{\\theta \\theta }=0,\\quad \\sigma ^{0}_{\\theta z,z}=0,\\qquad \\sigma ^{0}_{\\theta z,\\theta }+\\sigma ^{0}_{zz,z}=0.$ Solving these equations we obtain $\\sigma ^{0}(\\theta ,z)=\\begin{bmatrix}0 & 0 & 0\\\\0 & 0 & s(\\theta )\\\\0 & s(\\theta ) & t(\\theta )-zs^{\\prime }(\\theta )\\end{bmatrix}.$ for some functions $s(\\theta )$ and $t(\\theta )$ .",
"The first equation in (REF ) can now be written as ${\\left\\lbrace \\begin{array}{ll}u^{1}_{\\theta }=u^{0}_{\\theta }-u^{0}_{r,\\theta },\\quad u^{1}_{z}=-u^{0}_{r,z},\\quad u^{0}_{\\theta ,z}+u^{0}_{z,\\theta }=\\frac{2(1+\\nu )}{E}s(\\theta ),\\\\[2ex]u^{1}_{r}=-\\frac{\\nu }{1-\\nu }(u^{0}_{r}+u^{0}_{\\theta ,\\theta }+u^{0}_{z,z}),\\quad u^{0}_{r}+u^{0}_{\\theta ,\\theta }+\\frac{\\nu }{1-\\nu }(u^{1}_{r}+u^{0}_{z,z})=0,\\\\[2ex]u^{0}_{z,z}+\\frac{\\nu }{1-\\nu }(u^{1}_{r}+u^{0}_{r}+u^{0}_{\\theta ,\\theta })=\\frac{(1+\\nu )(1-2\\nu )}{E(1-\\nu )}(t(\\theta )-zs^{\\prime }(\\theta )).\\end{array}\\right.",
"}$ Solving these equations subject to the conditions $u^{0}_{z}(\\theta ,0)=u^{0}_{\\theta }(\\theta ,0)=u^{1}_{z}(\\theta ,0)=u^{1}_{\\theta }(\\theta ,0)=0$ we conclude that the functions $s(\\theta )$ and $t(\\theta )$ have to be constantWe note that $s=$ constant is a consequence of $u^{1}_{z}(\\theta ,0)=0$ , while $t=$ constant is a consequence of $u^{1}_{\\theta }(\\theta ,0)=0$ .",
"and that the formulas (REF ) hold.",
"The formula (REF ) follows from (REF ).",
"Thus, the effect of generic imperfections of load on $\\sigma ^{0}$ may manifest themselves only through a small perturbation of $(z,z)$ -component, and the appearance of a small constant $(\\theta z)$ -component.",
"In order to prove rigorously that imperfections of shape can indeed result in $\\sigma ^{0}$ of the form (REF ) with $s\\ne 0$ we need to exhibit a fully non-linear trivial branch satisfying all our assumptions and leading to (REF ).",
"It is clear that the non-linear trivial branch satisfying the perturbed boundary conditions may no longer be homogeneous.",
"This prevents us from exhibiting it explicitly for the Saint Venant-Kirchhoff energy, as in (REF ).",
"However, for an incompressible Mooney-Rivlin material the desired non-linear trivial branch can be computed explicitly (see Appendix ).",
"It is an important feature of our approach, that in order to compute the asymptotics of the buckling load and the buckling mode we do not need to know the non-linear trivial branch explicitly.",
"(We only need to know that the linearly elastic trivial branch, in the sense of Definition REF , exists.)",
"The desired asymptotics is given by Theorem REF in terms of the solution $ u^{h}$ of the equations of linear elasticity.",
"In order to obtain $\\sigma ^{0}$ of the form (REF ) we observe that $u^{h}_{r}=\\nu r,\\qquad u^{h}_{\\theta }=\\epsilon rz,\\qquad u^{h}_{z}=-z,$ solves ${\\left\\lbrace \\begin{array}{ll}\\nabla \\cdot (\\mathsf {L}_{0}e( u^{h}))= 0,&\\text{ in }{\\mathcal {C}}_{h},\\\\\\sigma _{h} e_{r}= 0,&r=1\\pm \\frac{h}{2},\\\\u^{h}_{z}=u^{h}_{\\theta }=0,&z=0,\\end{array}\\right.",
"}$ resulting in $\\sigma _{h}=\\left[\\begin{array}{ccc}0 & 0 & 0\\\\0 & 0 & \\dfrac{\\epsilon Er}{2(\\nu +1)}\\\\0 & \\dfrac{\\epsilon Er}{2(\\nu +1)} & -E\\end{array}\\right],\\qquad \\sigma ^{0}=\\left[\\begin{array}{ccc}0 & 0 & 0\\\\0 & 0 & \\dfrac{\\epsilon E}{2(\\nu +1)}\\\\0 & \\dfrac{\\epsilon E}{2(\\nu +1)} & -E\\end{array}\\right].$ In this explicit solution the imperfections of load are described by a single small, in absolute value, parameter $\\epsilon $ .",
"This specific representation of $\\sigma ^{0}$ is, nevertheless, generic for arbitrary imperfections of load at $z=L$ , according to Theorem REF .",
"Similarly to Theorem REF , the formulas (REF ) determine the scaling of the critical load with $h$ , which, for every fixed $\\epsilon $ , is different from (REF ).",
"Theorem 3.5 Suppose that $\\sigma ^{0}$ is given by (REF ).",
"Then there are positive constants $c$ and $C$ , depending only on $L$ and the elastic moduli, such that $\\frac{ch^{5/4}}{\\epsilon +h^{1/4}}\\le \\widehat{\\lambda }(h)\\le \\frac{Ch^{5/4}}{\\epsilon +h^{1/4}},$ when $h$ and $\\epsilon $ are sufficiently small.",
"Let $\\mathfrak {C}_{h}^{0}(\\phi )=\\int _{{\\mathcal {C}}_{h}}(\\sigma ^{0},\\nabla \\phi ^{T}\\nabla \\phi )d x.$ We first prove the lower bound on $\\widehat{\\lambda }(h)$ by observing that $-\\mathfrak {C}_{h}^{0}(\\phi )=E(\\Vert (\\nabla \\phi )_{rz}\\Vert ^{2}+\\Vert (\\nabla \\phi )_{\\theta z}\\Vert ^{2})-\\dfrac{\\epsilon E}{2(\\nu +1)}((\\nabla \\phi )_{rz},(\\nabla \\phi )_{r\\theta })+R_{h}(\\phi ),$ where $(f,g)$ denotes the inner product in $L^{2}({\\mathcal {C}}_{h})$ and $R_{h}(\\phi )=E\\Vert (\\nabla \\phi )_{zz}\\Vert ^{2}-\\dfrac{\\epsilon E}{2(\\nu +1)}\\lbrace ((\\nabla \\phi )_{\\theta z},(\\nabla \\phi )_{\\theta \\theta })+(\\nabla \\phi )_{z\\theta },(\\nabla \\phi )_{zz})\\rbrace .$ Then, for every $\\phi \\in V_{h}$ $|R_{h}|\\le C(\\Vert e(\\phi )\\Vert ^{2}+\\Vert e(\\phi )\\Vert \\Vert \\nabla \\phi \\Vert )\\le \\frac{C\\Vert e(\\phi )\\Vert ^{2}}{\\sqrt{K(V_{h})}}.$ Let $\\widetilde{\\mathfrak {R}}(h,\\phi )=\\frac{\\mathfrak {S}_{h}(\\phi )}{E(\\Vert (\\nabla \\phi )_{rz}\\Vert ^{2}+\\Vert (\\nabla \\phi )_{\\theta z}\\Vert ^{2})-\\dfrac{\\epsilon E}{2(\\nu +1)}((\\nabla \\phi )_{rz},(\\nabla \\phi )_{\\theta r})}.$ Then $\\left|\\displaystyle \\frac{1}{\\widetilde{\\mathfrak {R}}(h,\\phi )}-\\displaystyle \\frac{1}{\\mathfrak {R}(h,\\phi )}\\right|\\le \\frac{C}{\\sqrt{K(V_{h})}},$ and hence, by Theorem REF , the pair $(\\widetilde{\\mathfrak {R}}(h,\\phi ),V_{h})$ is buckling equivalent to the pair $(\\mathfrak {R}(h,\\phi ),V_{h})$ .",
"By Lemma REF we obtain, applying the Cauchy-Schwarz inequality, $|((\\nabla \\phi )_{rz},(\\nabla \\phi )_{\\theta r})|\\le \\Vert (\\nabla \\phi )_{rz}\\Vert \\Vert \\nabla \\phi \\Vert \\le \\frac{C\\Vert e(\\phi )\\Vert }{\\sqrt{h}}K(V_{h})\\Vert e(\\phi )\\Vert \\le \\frac{C\\Vert e(\\phi )\\Vert ^{2}}{h^{5/4}}.$ Applying Lemma REF and (REF ) we obtain $\\widetilde{\\mathfrak {R}}(h,\\phi )\\ge \\frac{\\alpha _{\\mathsf {L}_{0}}\\Vert e(\\phi )\\Vert ^{2}}{C\\Vert e(\\phi )\\Vert ^{2}(h^{-1}+h^{-1/2}+\\epsilon h^{-5/4})}\\ge \\frac{Ch^{5/4}}{\\epsilon +h^{1/4}},$ To obtain an upper bound on the critical load, we use test functions $\\phi ^{h}$ given by (REF ) in the estimate $\\widehat{\\lambda }(h)\\le C\\widetilde{\\mathfrak {R}}(h,\\phi ^{h}).$ Using the explicit formulas (REF ) for $\\phi ^{h}$ we compute $\\lim _{h\\rightarrow 0}\\displaystyle \\frac{1}{h}((\\nabla \\phi ^{h})_{rz},(\\nabla \\phi ^{h})_{\\theta r})=\\int _{0}^{2\\pi }\\int _{0}^{L}W_{,\\eta \\eta \\eta }(\\eta ,z)W_{,\\eta \\eta z}(\\eta ,z)d\\eta dz.$ By Lemma REF , in order to prove the upper bound in (REF ) we only need to exhibit a fixed compactly supported function $W(\\eta ,z)$ , such that the right-hand side in (REF ) is non-zero.",
"This is done by choosing two arbitrary non-zero compactly supported functions $\\phi (\\eta )$ and $\\psi (z)$ and setting $W(\\eta ,z)=\\phi (\\eta )\\psi ^{\\prime }(z)+\\phi ^{\\prime }(\\eta )\\psi (z).$ Then $W_{,\\eta \\eta \\eta }W_{,\\eta \\eta z}=\\displaystyle \\frac{1}{4}(\\psi ^{\\prime }(z)^{2})^{\\prime }(\\phi ^{\\prime \\prime }(\\eta )^{2})^{\\prime }+(\\phi ^{\\prime \\prime \\prime }(\\eta )^{2})^{\\prime }(\\psi (z)^{2})^{\\prime }+(\\phi ^{\\prime \\prime \\prime }(\\eta )\\phi ^{\\prime \\prime }(\\eta ))^{\\prime }\\psi (z)\\psi ^{\\prime \\prime }(z)\\\\-\\phi ^{\\prime \\prime \\prime }(\\eta )^{2}(\\psi (z)\\psi ^{\\prime }(z))^{\\prime }+2\\phi ^{\\prime \\prime \\prime }(\\eta )^{2}\\psi ^{\\prime }(z)^{2}.$ This shows that $\\int _{0}^{2\\pi }\\int _{0}^{L}W_{,\\eta \\eta \\eta }W_{,\\eta \\eta z}d\\eta dz=2\\int _{0}^{2\\pi }\\int _{0}^{L}\\phi ^{\\prime \\prime \\prime }(\\eta )^{2}\\psi ^{\\prime }(z)^{2}d\\eta dz>0.$ From the estimates (REF ) we see that in order for the scaling $h^{5/4}$ to be experimentally significant, $|\\epsilon |$ must be much larger than $h^{1/4}$ .",
"This is unlikely for the typical values of $h\\approx 10^{-4}$ .",
"Nevertheless, Theorem REF (together with Appendix ) demonstrates rigorously that axially compressed cylindrical shells exhibit scaling instability under imperfections of load.",
"We can also view this result as a strong indication that it is the imperfections of shape that are largely responsible for the discrepancy between the theory and experiment."
],
[
"Imperfection of shape",
"In the case of shape imperfections our Korn inequalities for gradient and gradient components, strictly speaking, cannot be applied, since the domain is no longer ${\\mathcal {C}}_{h}$ .",
"In this case we conjecture that for some shape imperfections, such as small localized dents the trivial branch would still exist and satisfy out assumptions (REF ), while the Korn constant retains its $h^{3/2}$ asymptotics.",
"While the arguments below are not exactly rigorous, we believe that they do shed new light on the question of rigorous estimation of the critical load for an imperfect cylindrical shell.",
"The key insight achieved in the foregoing analysis is that the reason for the difference in scaling laws of the critical strain and the Korn constant is the structure of the stress in the trivial branch (which in a perfect axially compressed cylinder has only $zz$ -component that is non-zero).",
"The failure of the imperfections of load to modify this structure in a significant way (see Theorem REF ) is due to the traction-free boundary conditions on the lateral surfaces of the shell.",
"This observation leads to the idea that if the shell is “dented”, the normal to the lateral surface may undergo a non-negligible change in a small region.",
"To model this mathematically we assume that the dented cylindrical shell is given by $\\widetilde{{\\mathcal {C}}}_{h}=\\left\\lbrace (r,\\theta ,z):\\theta \\in \\mathbb { T},\\ z\\in [0,L],1+\\epsilon ^{2}\\rho \\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right)-\\frac{h}{2}\\le r\\le 1+\\epsilon ^{2}\\rho \\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right)+\\frac{h}{2}\\right\\rbrace ,$ where the function $\\rho (\\eta ,\\zeta )$ is compactly supported on a unit ball in $\\mathbb { R}^{2}$ , where $\\rho (\\theta /\\epsilon ,(z-z_{0})/\\epsilon )$ is meant for $\\theta \\in [-\\pi ,\\pi ]$ and is understood as a $2\\pi $ -periodic function.",
"We assume that $\\epsilon =\\epsilon (h)\\rightarrow 0$ , as $h\\rightarrow 0$ and $h/\\epsilon (h)\\rightarrow 0$ , as $h\\rightarrow 0$ .",
"For the “proof-of-concept” demonstration we assume, without proof, that the linear stress in the trivial branch can be written as $\\sigma ^{h}(r,\\theta ,z)=\\sigma ^{h}_{p}+\\widetilde{\\sigma }^{h}(\\theta ,z)+(r-1)\\tau ^{h}(\\theta ,z)+o(h),$ where $\\sigma ^{h}_{p}$ is the stress in the perfect shell, given by (REF ).",
"We assume that $\\widetilde{\\sigma }^{h}=O(1)$ and $\\tau ^{h}=O(1)$ , as $h\\rightarrow 0$ , while $\\nabla \\sigma ^{h}(r,\\theta ,z)=\\nabla (\\sigma ^{h}_{p}+\\widetilde{\\sigma }^{h}(\\theta ,z)+(r-1)\\tau ^{h}(\\theta ,z))+o(1).$ The normal to the traction-free surface of the imperfect cylinder $\\widetilde{{\\mathcal {C}}}_{h}$ is now $ N_{h}= e_{r}-\\epsilon (\\rho _{,\\eta } e_{\\theta }+\\rho _{,\\zeta } e_{z}).$ According to (REF ) we must have $\\widetilde{\\sigma }^{h} N_{h}+\\sigma ^{h}_{p} N_{h}=o(h),\\qquad \\tau ^{h} N_{h}=o(1).$ In components this implies ${\\left\\lbrace \\begin{array}{ll}\\widetilde{\\sigma }^{h}_{rr}=\\epsilon (\\rho _{,\\eta }\\widetilde{\\sigma }^{h}_{r\\theta }+\\rho _{,\\zeta }\\widetilde{\\sigma }^{h}_{rz})+o(h),\\\\\\widetilde{\\sigma }^{h}_{r\\theta }=\\epsilon (\\rho _{,\\eta }\\widetilde{\\sigma }^{h}_{\\theta \\theta }+\\rho _{,\\zeta }\\widetilde{\\sigma }^{h}_{\\theta z})+o(h),\\\\\\widetilde{\\sigma }^{h}_{rz}=\\epsilon (-E\\rho _{,\\zeta }+\\rho _{,\\eta }\\widetilde{\\sigma }^{h}_{\\theta z}+\\rho _{,\\zeta }\\widetilde{\\sigma }^{h}_{zz})+o(h),\\end{array}\\right.",
"}\\qquad {\\left\\lbrace \\begin{array}{ll}\\tau ^{h}_{rr}=o(1),\\\\\\tau ^{h}_{r\\theta }=o(1),\\\\\\tau ^{h}_{rz}=o(1).\\end{array}\\right.",
"}$ The balance equations $\\nabla \\cdot \\sigma ^{h}= 0$ then become ${\\left\\lbrace \\begin{array}{ll}\\epsilon \\displaystyle \\frac{\\partial }{\\partial \\theta }(\\rho _{,\\eta }\\widetilde{\\sigma }^{h}_{\\theta \\theta }+\\rho _{,\\zeta }\\widetilde{\\sigma }^{h}_{\\theta z})-\\widetilde{\\sigma }^{h}_{\\theta \\theta }+\\epsilon \\displaystyle \\frac{\\partial }{\\partial z}(-E\\rho _{,\\zeta }+\\rho _{,\\eta }\\widetilde{\\sigma }^{h}_{\\theta z}+\\rho _{,\\zeta }\\widetilde{\\sigma }^{h}_{zz})=o(1),\\\\\\widetilde{\\sigma }^{h}_{\\theta \\theta ,\\theta }+\\widetilde{\\sigma }^{h}_{\\theta z,z}=o(1),\\\\\\widetilde{\\sigma }^{h}_{\\theta z,\\theta }+\\widetilde{\\sigma }^{h}_{zz,z}=o(1).\\end{array}\\right.",
"}$ At this point we abandon any semblance of rigor and set the right-hand sides in (REF ) to zero and assume that $\\widetilde{\\sigma }^{h}=\\widehat{\\sigma }\\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right).$ The last two equations in (REF ) then implies that $\\widetilde{\\sigma }^{h}_{\\theta \\theta }=s_{,\\zeta \\zeta }\\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right),\\qquad \\widetilde{\\sigma }^{h}_{\\theta z}=-s_{,\\eta \\zeta }\\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right),\\qquad \\widetilde{\\sigma }^{h}_{zz}=s_{,\\eta \\eta }\\left(\\frac{\\theta }{\\epsilon },\\frac{z-z_{0}}{\\epsilon }\\right).$ The first equation in (REF ) becomes $\\rho _{,\\eta \\eta }s_{,\\zeta \\zeta }+s_{,\\eta \\eta }\\rho _{,\\zeta \\zeta }-2\\rho _{,\\eta \\zeta }s_{,\\eta \\zeta }=s_{,\\zeta \\zeta }+E\\rho _{,\\zeta \\zeta }.$ If we assume that $\\rho _{,\\eta \\eta }(\\eta ,\\zeta )$ and $\\rho _{,\\eta \\zeta }$ are uniformly small (i.e.",
"the dent is localized significantly more in the $z$ direction than in $\\theta $ ), then $s\\approx -E\\rho $ .",
"In order to trigger the mode of instability with the critical strain scaling like the Korn constant $\\lambda (h)\\sim h^{3/2}$ we require $\\sigma ^{h}_{\\theta \\theta }<-\\alpha <0$ in a neighborhood of a point $(0,z_{0})$ , i.e.",
"$\\rho _{,\\zeta \\zeta }(0,0)>0$ .",
"This can be achieved only on “inward dents”.",
"In general, we assume that there exists a decaying at infinity solution $s(\\eta ,\\zeta )$ of (REF ), such that $s_{,\\zeta \\zeta }(0,0)<0$ .",
"In conclusion we note that the exponents $5/4=1.25$ , associated with load imperfections and $3/2=1.5$ , associated with imperfections of shape are close to the upper and lower limits of experimentally determined behavior of the buckling load, respectively, [5], [17].",
"We also note that the observed buckling load of the real imperfect structure may be further affected by the subcritical nature of the respective bifurcations (see [3] for a lucid explanation why).",
"Acknowledgments.",
"We are grateful to Eric Clement, Stefan Luckhaus, Mark Peletier and Lev Truskinovsky for insightful comments and questions.",
"This material is based upon work supported by the National Science Foundation under Grants No.",
"1008092."
],
[
"Non-linear trivial branch for an incompressible Mooney-Rivlin material.",
"Consider an incompressible Mooney-Rivlin type material with strain energy function $W( F)=\\frac{E}{6}(| F|^{2}-3),\\qquad \\det F=1.$ We are looking for a trivial branch in a cylindrical shell, given in cylindrical coordinates by $y_{r}=\\psi (r)\\cos (\\alpha z),\\qquad y_{\\theta }=\\psi (r)\\sin (\\alpha z),\\qquad y_{z}=(1-\\lambda )z.$ It is expected that $\\psi (r)$ also depends on $\\alpha $ , $\\lambda $ and $h$ .",
"When $\\alpha =0$ we expect that $\\psi (r)$ will reduce to $(a(\\lambda )+1)r$ , as in (REF ).",
"We remark that, in principle, the ansatz (REF ) should also work for compressible materials, except the resulting non-linear second order ODE for $\\psi (r)$ cannot be solved explicitly.",
"We compute $\\det (\\nabla y)=(1-\\lambda )\\psi ^{\\prime }(r)\\frac{\\psi (r)}{r}.$ For an incompressible material we must have $\\det (\\nabla y)=1$ , and hence $\\psi (r)=\\sqrt{\\frac{r^{2}}{1-\\lambda }+\\beta }$ for some $\\beta >-1$ .",
"The Piola-Kirchhoff stress function is $ P( F)=\\frac{E}{3}\\left( F-\\frac{3\\hat{p}}{E}\\mathrm {cof}( F)\\right),$ where the Lagrange multiplier $\\hat{p}$ plays the role of pressure.",
"For $ y$ , given by (REF ) and $ F=\\nabla y$ we compute $ F^{T} F=\\begin{bmatrix}(\\psi ^{\\prime }(r))^{2} & 0 & 0\\\\0 & \\frac{\\psi (r)^{2}}{r^{2}} & \\frac{\\alpha \\psi (r)^{2}}{r}\\\\0 & \\frac{\\alpha \\psi (r)^{2}}{r} & \\alpha ^{2}\\psi (r)^{2} + (1-\\lambda )^{2}\\end{bmatrix}.$ The traction-free condition $ P e_{r}= 0$ on $r=1\\pm h/2$ can be written as $ F^{T} F e_{r}=p e_{r},\\quad r=1\\pm \\frac{h}{2},\\qquad p=3\\hat{p}/E.$ The formula for $ F^{T} F$ , together with $\\det F=1$ , implies that $p(r,\\theta ,z)=(\\psi ^{\\prime }(r))^{2},\\quad r=1\\pm \\frac{h}{2}.$ This suggests that it is reasonable to look for the trivial branch for which the function $p(r,\\theta ,z)$ depends only on $r$ .",
"Under this assumption we compute $\\frac{3}{E} P=\\begin{bmatrix}s_{1}(r)\\cos (\\alpha z) & -s_{2}(r)\\sin (\\alpha z) & -s_{3}(r)\\sin (\\alpha z)\\\\s_{1}(r)\\sin (\\alpha z) & s_{2}(r)\\cos (\\alpha z) & s_{3}(r)\\cos (\\alpha z)\\\\0 & q_{1}(r) & q_{2}(r)\\end{bmatrix},$ where $s_{1}=\\psi ^{\\prime }-\\frac{p}{\\psi ^{\\prime }},\\quad s_{2}=\\frac{\\psi }{r}-\\frac{rp}{\\psi },\\quad s_{3}=\\alpha \\psi ,\\quad q_{1}=\\frac{\\alpha rp}{1-\\lambda },\\quad q_{2}=1-\\lambda -\\frac{p}{1-\\lambda }.$ It follows that $\\nabla \\cdot P= 0$ results in a single ODE for $p(r)$ : $(rs_{1})^{\\prime }=s_{2}+\\alpha rs_{3}.$ Substituting (REF ) for $\\psi (r)$ into (REF ) and solving for $p(r)$ we obtain $p(r)=\\displaystyle \\frac{1}{2(1-\\lambda )}\\left(\\ln \\left(\\frac{1}{1-\\lambda }+\\frac{\\beta }{r^{2}}\\right)-r^{2}\\alpha ^{2}-\\frac{\\beta (1-\\lambda )}{r^{2}+\\beta (1-\\lambda )}+\\gamma \\right).$ The traction-free boundary conditions (REF ) become $\\frac{r^{2}}{r^{2}+\\beta (1-\\lambda )}=\\ln \\left(\\frac{1}{1-\\lambda }+\\frac{\\beta }{r^{2}}\\right)-r^{2}\\alpha ^{2}+\\gamma -1,\\quad r=1\\pm \\frac{h}{2}.$ Let $\\Phi (r;\\lambda ,\\beta )=\\ln \\left(\\frac{1}{1-\\lambda }+\\frac{\\beta }{r^{2}}\\right)-\\frac{r^{2}}{r^{2}+\\beta (1-\\lambda )}.$ Then, ${\\left\\lbrace \\begin{array}{ll}\\alpha ^{2}\\left(1+\\frac{h}{2}\\right)^{2}=\\Phi \\left(1+\\frac{h}{2};\\lambda ,\\beta \\right)+\\gamma -1,\\\\\\alpha ^{2}\\left(1-\\frac{h}{2}\\right)^{2}=\\Phi \\left(1-\\frac{h}{2};\\lambda ,\\beta \\right)+\\gamma -1\\end{array}\\right.",
"}$ Eliminating $\\gamma $ from (REF ) we obtain $\\alpha ^{2}=\\displaystyle \\frac{1}{2h}\\left(\\Phi \\left(1+\\frac{h}{2};\\lambda ,\\beta \\right)-\\Phi \\left(1-\\frac{h}{2};\\lambda ,\\beta \\right)\\right).$ when $h$ is small $\\alpha ^{2}\\approx \\displaystyle \\frac{1}{2}\\Phi ^{\\prime }(1;\\lambda ,\\beta )=-\\frac{\\beta (1-\\lambda )(2+\\beta (1-\\lambda ))}{(1+\\beta (1-\\lambda ))^{2}}.$ Thus, when $(h,\\lambda )\\rightarrow (0,0)$ , $\\beta \\approx -\\alpha ^{2}/2$ .",
"We conclude that $\\alpha $ , and, therefore, $\\beta $ must go to zero, as $\\lambda \\rightarrow 0$ , since otherwise, the trivial branch $ y( x;h,\\lambda )$ , given by (REF ), (REF ) will not emanate from the undeformed state.",
"The regularity of the trivial branch in $\\lambda $ demands that $\\alpha (h,\\lambda )\\sim \\alpha _{0}(h)\\lambda $ , as $\\lambda \\rightarrow 0$ .",
"Thus, for an arbitrary fixed parameter $\\beta _{0}>0$ we set $\\beta =-\\beta _{0}^{2}\\lambda ^{2}/2$ , resulting in the explicit expression for the parameter $\\alpha $ : $\\alpha (\\lambda ,h)=\\sqrt{\\frac{\\Phi (1+h/2;\\lambda ,-\\beta _{0}^{2}\\lambda ^{2}/2)-\\Phi (1-h/2;\\lambda ,-\\beta _{0}^{2}\\lambda ^{2}/2)}{2h}}.$ We compute $\\left.\\displaystyle \\frac{\\partial \\alpha }{\\partial \\lambda }\\right|_{\\lambda =0}=\\frac{4\\beta _{0}}{4-h^{2}},\\qquad \\left.\\displaystyle \\frac{\\partial \\psi }{\\partial \\lambda }\\right|_{\\lambda =0}=\\frac{r}{2}.$ Therefore, the linearized trivial branch displacement $ u^{h}$ is given by $u^{h}_{r}=\\left.\\displaystyle \\frac{\\partial y_{r}}{\\partial \\lambda }\\right|_{\\lambda =0}=\\frac{r}{2},\\qquad u^{h}_{\\theta }=\\left.\\displaystyle \\frac{\\partial y_{\\theta }}{\\partial \\lambda }\\right|_{\\lambda =0}=\\frac{4\\beta _{0}rz}{4-h^{2}},\\qquad u^{h}_{z}=\\left.\\displaystyle \\frac{\\partial y_{z}}{\\partial \\lambda }\\right|_{\\lambda =0}=-z.$ The corresponding linear stress and its $h\\rightarrow 0$ limit are $\\sigma _{h}=E\\begin{bmatrix}0 & 0 & 0\\\\0 & 0 & \\frac{4\\beta _{0}r}{3(4-h^{2})}\\\\0 & \\frac{4\\beta _{0}r}{3(4-h^{2})} & -1\\end{bmatrix},\\qquad \\sigma ^{0}=E\\begin{bmatrix}0 & 0 & 0\\\\0 & 0 & \\frac{\\beta _{0}}{3}\\\\0 & \\frac{\\beta _{0}}{3} & -1\\end{bmatrix}.$ These agree with formulas (REF ), (REF ) for $\\nu =1/2$ ."
]
] | 1403.0287 |
[
[
"Fast, low-power manipulation of spin ensembles in superconducting\n microresonators"
],
[
"Abstract We demonstrate the use of high-Q superconducting coplanar waveguide (CPW) microresonators to perform rapid manipulations on a randomly distributed spin ensemble using very low microwave power (400 nW).",
"This power is compatible with dilution refrigerators, making microwave manipulation of spin ensembles feasible for quantum computing applications.",
"We also describe the use of adiabatic microwave pulses to overcome microwave magnetic field ($B_{1}$) inhomogeneities inherent to CPW resonators.",
"This allows for uniform control over a randomly distributed spin ensemble.",
"Sensitivity data are reported showing a single shot (no signal averaging) sensitivity to $10^{7}$ spins or $3 \\times 10^{4}$ spins/$\\sqrt{Hz}$ with averaging."
],
[
"Fast, low-power manipulation of spin ensembles in superconducting microresonators A. J. Sigillito []asigilli@princeton.edu Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA H. Malissa Current address: Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA A. M. Tyryshkin Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA H. Riemann, N. V. Abrosimov Institut für Kristallzüchtung, D-12489 Berlin, Germany P. Becker Physikalisch-Technische Bundesanstalt, D-38116 Braunschweig, Germany H.-J.",
"Pohl VITCON Projectconsult GMBH, D-07745 Jena, Germany M. L. W. Thewalt Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A 1S6, Canada K. M. Itoh School of Fundamental Science and Technology, Keio University, Yokohama, Kanagawa 2238522, Japan J. J. L. Morton London Centre for Nanotechnology, University College London, London WC1H 0AH, UK A.",
"A. Houck Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA D. I. Schuster Department of Physics and James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA S. A. Lyon Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA We demonstrate the use of high-Q superconducting coplanar waveguide (CPW) microresonators to perform rapid manipulations on a randomly distributed spin ensemble using very low microwave power (400 nW).",
"This power is compatible with dilution refrigerators, making microwave manipulation of spin ensembles feasible for quantum computing applications.",
"We also describe the use of adiabatic microwave pulses to overcome microwave magnetic field ($B_{1}$ ) inhomogeneities inherent to CPW resonators.",
"This allows for uniform control over a randomly distributed spin ensemble.",
"Sensitivity data are reported showing a single shot (no signal averaging) sensitivity to $10^{7}$ spins or $3 \\times 10^{4}$ spins/$\\sqrt{Hz}$ with averaging.",
"Superconducting coplanar waveguide (CPW) resonators are a good alternative to conventional volume resonators for many applications because of their high sensitivity, low power requirements, and small size[1], [2], [3].",
"They are of particular interest in the construction of hybrid quantum systems utilizing the long coherence times of spin-based qubits and the strong coupling of superconducting qubits[4], [5], [6], [7].",
"Hybrid quantum systems have been realized for nitrogen-vacancy (N-V) centers coupled to transmon qubits [4].",
"These systems are limited by dephasing of the spin ensemble ($T_{2}^{*}$ ) which is often hundreds of nanoseconds in solids.",
"This can be improved by employing refocusing techniques common in pulsed electron spin resonance (ESR), enabling spin memories effective over the full coherence time of the electron spin ($T_{2}$ ), which can be 10 s in Si[8].",
"However, refocusing pulses have been difficult to implement because the microwave magnetic field ($B_{1}$ ) in a CPW is inhomogeneous.",
"Furthermore, it has been suggested that driving ensemble rotations with microwave pulses is too slow for quantum computing and can lead to excessive microwave heating of the system[9].",
"In this letter we address these issues.",
"We report CPW resonators capable of performing $\\pi $ -rotations on a spin ensemble in 40 ns while using powers compatible with dilution refrigerators (40 $\\mu W$ peak).",
"We also present data showing the use of adiabatic microwave pulses to overcome $B_{1}$ inhomogeneities, enabling accurate spin manipulations over a randomly distributed ensemble.",
"The spin ensembles discussed in this letter are donor electron spins in Si.",
"Donors in Si have the advantage over N-Vs of long coherence times exceeding seconds[8], [10] and a wealth of experience in fabrication techniques.",
"However, only donors with large zero-field splittings are compatible with these hybrid systems because superconducting qubits have a low critical field.",
"Bismuth donors in Si have been proposed as a good candidate for coupling to superconducting qubits due to their long coherence times, existence of clock transitions, and large zero-field splitting[11].",
"While the data presented in this letter focuses on P donors, the methodology and results are not unique to P and easily extended to other spin ensembles.",
"One major challenge to performing rotations on a spin ensemble at low temperature is microwave heating.",
"A typical X-band pulsed spectrometer performing 40 ns $\\pi $ -rotations can require an input power of tens of watts for a high quality factor (Q) volume resonator or up to a kilowatt of power for the lower-Q resonators used in studying systems with short coherence times.",
"Most of that energy is reflected from the resonator, but heating can be significant.",
"However, the resonators we report have a substantially smaller mode volume, so less power is required to drive ensemble rotations.",
"The small mode volume is possible because CPW resonators require only one dimension to be on the order of the resonant frequency wavelength.",
"The other two dimensions can be made arbitrarily small.",
"Another challenge to manipulating an ensemble of randomly distributed spins is $B_{1}$ inhomogeneity, which is intrinsic to CPW designs.",
"Field inhomogeneities lead to non-uniform control over a macroscopic ensemble, because the tipping angle of a given spin is proportional to the driving field strength.",
"Spins in regions of large $B_{1}$ will be rotated more than spins in regions of small $B_{1}$ .",
"There are essentially two approaches to overcoming these inhomogeneities.",
"The first is to tailor the device geometry such that spins are located in regions of homogeneous $B_{1}$ .",
"This is accomplished by either changing the resonator structure [12] or by confining the ensemble to a small region where $B_{1}$ is uniform.",
"These methods typically require that the volume of the spin ensemble be smaller than the mode volume of the resonator, leading to weaker coupling.",
"The second approach is to construct microwave pulses that compensate for $B_{1}$ inhomogeneities.",
"This allows for uniform control over an ensemble filling nearly the entire mode volume of the resonator.",
"We have chosen the latter method and utilize adiabatic microwave pulses, which produce $B_{1}$ -insensitive spin rotations.",
"Such adiabatic pulses are known in the nuclear magnetic resonance community, and we have tested several varieties[13], [14], [15].",
"The best results were obtained by combining a WURST-20 (Wideband, Uniform Rate, Smooth Truncation 20) envelope shape[15] with a BIR-4 ($B_{1}$ Insensitive Rotation 4) phase compensation[14].",
"The WURST-20 envelope shapes the pulse as $sin^{20}(\\pi t /t_{p})$ where $t$ is time and $t_{p}$ is the pulse length.",
"The BIR-4 technique breaks the WURST-20 pulse in half and combines four of these waveforms in a time-reversed order.",
"The BIR-4 technique is robust against off-resonance effects, and specifically compensate for geometrical phase errors.",
"Individually, BIR-4 and WURST-20 have been discussed in the literature[13], [14], [15].",
"The spin ensemble consisted of a 25 $\\mu $ m epitaxial layer of $^{28}$ Si grown on high resistivity p-type Si.",
"The epi-layer was doped to a concentration of $8 \\times 10^{14}$ P donors/cm$^{3}$ .",
"This layer had 50 nm of Al$_{2}$ O$_{3}$ grown on the surface to protect against a SF$_{6}$ plasma used in the device fabrication.",
"Six CPW resonators were patterned in 50 nm thick Nb films directly on the Al$_{2}$ O$_{3}$ surface and they are shown in Fig.",
"REF a.",
"The fabrication techniques have been described previously[1].",
"The CPW center conductor width was 30 $\\mu $ m, with a gap width of 17.4 $\\mu $ m defining an impedance of 50 $\\Omega $ .",
"Each resonator had a unique frequency, spanning a range from 7 GHz - 8 GHz, and all were nearly critically coupled to a common transmission line.",
"Most of the results reported in this letter were obtained using one resonator with a 7.17 GHz center frequency, Q of $\\sim $ 2000, and coupling coefficient of 1.15.",
"Resonators were wire bonded to copper printed circuit boards equipped with microwave connectors and cooled to 1.7 K. The output of the resonator transmission line was attached to a low-noise cryogenic preamplifier (Caltech LNA 1-12).",
"We applied a direct current (DC) magnetic field to the sample, taking care that the plane of the Nb film remained parallel to the field.",
"Careful alignment prevented the trapping of magnetic flux vortices in the superconducting film, which are lossy and serve as a decoherence mechanism[1].",
"Two-pulse Hahn echo experiments (nominally $\\pi $ /2(+x) – $\\tau $ – $\\pi $ (+y) – $\\tau $ – echo) were conducted with a delay time ($\\tau $ ) of 15 $\\mu $ s and the results are shown in Fig.",
"REF b.",
"Because the relaxation time (T$_{1}$ ) of P donors at 1.7 K is on the order of minutes, the back side of the sample was illuminated with a 1050 nm light emitting diode for 50 ms prior to each two pulse experiment.",
"The light relaxes the spins allowing fast repetition rates.",
"For 400 ns rectangular $\\pi $ -pulses, the optimal microwave power was $-34$ dBm (400 nW).",
"The experiment was repeated using BIR-4-WURST-20 adiabatic pulses.",
"For our devices, the optimal adiabatic pulse chirped from 2 MHz below to 2 MHz above the resonant frequency in 11 $\\mu $ s. This chirp bandwidth was chosen to be wider than the ESR linewidth of 0.3 MHz in order to excite the entire spin ensemble.",
"The corresponding peak microwave power was $-30$ dBm (1 $\\mu $ W).",
"The integrated signal-to-noise (S/N) ratio for the single shot (no signal averaging) rectangular and adiabatic pulse experiments are 84 and 146, respectively.",
"Thus, by using adiabatic pulses the signal increased by a factor of 1.74.",
"To demonstrate that microwave manipulation of spin ensembles can occur on timescales compatible with quantum computing, shorter rectangular pulses were tested on a second device.",
"The device had a Q of 3200, center frequency of 7.14 GHz, and coupling coefficient of 1.15.",
"The optimal power for 400 ns and 40 ns rectangular $\\pi $ -pulses was $-33$ dBm and $-13$ dBm, respectively.",
"We expect the 40 ns $\\pi $ -pulse to be distorted by the high-Q resonator.",
"However, a 20 dB increase in power still led to order of magnitude shorter excitation pulses.",
"Figure: (a) Optical micrograph of the device.",
"Six resonators (serpentine structures) are capacitively coupled to a common transmission line.",
"(b) Single shot spin echoes acquired using adiabatic (top black) and rectangular (bottom red) pulses.",
"The adiabatic pulse echo has been shifted by 0.8 for clarity.",
"Data were taken at 1.7 K in a DC magnetic field of 0.26 T.Two-pulse experiments (nominally $\\pi $ /2(+x) – $\\tau $ – $\\theta $ (+y) – $\\tau $ – echo) were also performed where the tipping angle of the second pulse was varied.",
"For rectangular pulses, a microwave power of $-27$ dBm was used, with a 200 ns first pulse.",
"The second pulse was varied from 0 ns to 1400 ns.",
"The echo intensity as a function of pulse length is plotted in Fig.",
"REF a.",
"This is compared to a similar experiment conducted using adiabatic pulses, shown in Fig.",
"REF b.",
"When using adiabatic pulses, tipping angles were well defined such that the first pulse performed a $\\pi $ /2 rotation and the second pulse tipping angle varied from 0 to 4$\\pi $ .",
"An optimal peak microwave power of $-30$ dBm was used for these experiments.",
"It is clear from the data that the $B_{1}$ inhomogeneity greatly affects the rectangular pulse experiment, which shows no Rabi oscillations, while the adiabatic pulses produce Rabi oscillations as expected.",
"To understand these experiments, a model was developed to simulate the results.",
"The normalized $B_{1}$ distribution in the CPW resonator was computed using a conformal mapping technique, and the echo intensity was determined by summing over the contribution of each individual spin, as previously described[1].",
"The contribution of a single spin to the echo is given by $signal(r) = g_{s}(r) sin(\\theta _{1}(r)) sin^{2}(\\theta _{2}(r)/2)$ where $g_{s}(r)$ is the coupling of a spin at position $r$ to the resonator, $\\theta _{1}(r)$ is the tipping angle of the first pulse (the first pulse is nominally $\\pi $ /2, but the actual tipping angle varies with spin position), and $\\theta _{2}(r)$ is the tipping angle of the second pulse.",
"For rectangular pulses, the tipping angle is $g\\mu _{B} B_{1}(r) t_{p} / \\hbar $ , where $g$ is the electron g-factor, $\\mu _{B}$ is the Bohr magneton, and $\\hbar $ is the reduced Planck constant.",
"The spin-resonator coupling is linearly proportional to $B_{1}(r)$ , the microwave magnetic field.",
"We can write $B_{1}(r) = C B_{1n}(r)$ where $B_{1n}$ is the normalized $B_{1}$ , and $C$ depends on the microwave power, cavity coupling, and Q.",
"Thus, by writing $g_{s}(r) = A B_{1n}(r)$ , the total signal becomes $ signal = \\int dr A C B_{1n}(r)sin\\biggl (\\frac{g \\mu C B_{1n}(r) t_{1}}{\\hbar }\\biggr ) \\\\ sin^{2}\\biggl (\\frac{g \\mu C B_{1n}(r) t_{2}}{2 \\hbar }\\biggr )$ where $t_{1}$ is the first pulse duration and $t_{2}$ is the second pulse duration.",
"The constant, $A$ , simply normalizes the vertical scale whereas $C$ determines the shape of the curve shown in Fig.",
"REF a.",
"By varying $C$ for a given microwave power, we obtain a good fit to the data.",
"From $C$ , we compute $B_{1}(r)$ , which includes resonator Q, losses, and coupling.",
"Figure: Echo intensity as a function of tipping angle for (a) rectangular pulses and (b) adiabatic pulses.",
"Experimental data is represented by solid squares.",
"The curve in (a) is the best fit curve from the model (Eq.",
"2) and the curve in (b) assumes ideal spin rotations.",
"Data were taken at 1.7 K in a DC magnetic field of 0.26 T.To evaluate the performance of adiabatic pulses quantitatively, we performed echo experiments using a high-homogeneity commercial dielectric resonator (Bruker MD5).",
"We used a bulk doped $^{28}$ Si crystal[16] with $3.3 \\times 10^{15}$ P donors/cm$^{3}$ .",
"The sample volume was 1 mm $\\times $ 2 mm $\\times $ 4 mm, and $B_{1}$ homogeneity varied by no more than 5% over the sample volume.",
"In these experiments, adiabatic pulse tipping angles were defined as $\\pi $ /2 and $\\pi $ , while the microwave power and thus $B_{1}$ was varied.",
"The integrated echo intensity as a function of $B_{1}$ is shown in Fig.",
"REF .",
"As a comparison, the experiment was repeated using rectangular pulses with a $\\pi $ -pulse width of 400 ns ($B_{1}\\sim $ 10 times the ESR linewidth), and these data are also shown.",
"At least half of the maximum echo intensity is observed for $B_{1}$ in the range of 6 $\\mu $ T to 83 $\\mu $ T for adiabatic pulses, while the range is 24 $\\mu $ T to 61 $\\mu $ T for rectangular pulses.",
"This comparison shows that adiabatic pulses correct $B_{1}$ inhomogeneity over an order of magnitude (two orders of magnitude in microwave power).",
"Figure: Plot of echo amplitude as a function of B 1 _{1} for a sample in a high-homogeneity resonator with adiabatic pulses (black) and rectangular pulses (red).",
"Data were taken at 4.6 K in a DC magnetic field of 0.34 T. The solid line is a spline fit to the data and is used later in a simulation.Combining the simulated $B_{1}$ distribution with our measurements of echo intensity as a function of $B_{1}$ (Fig.",
"REF ), we identified the regions of the sample contributing most to the echo signal.",
"Fig.",
"REF a is a cross section of the CPW resonator at an antinode in the magnetic field (near the shorted end of the resonator).",
"The CPW is depicted at the top of the plot, and the magnitude of $B_{1}$ in the sample is shown with contours (the 8 $\\mu $ T contour is labeled, and $B_{1}$ for each subsequent contour increases by a factor of two).",
"The hatched regions in the figure denote where 2/3 of the signal originates for the rectangular and adiabatic pulses.",
"The Si sample used for these experiments had a 25 $\\mu $ m thick P-doped $^{28}$ Si epi-layer, and thus the ensemble volume only extends down to the green cross-hatched region.",
"Using the $B_{1}$ distribution, the contribution of all spins, at each value of $B_{1}$ , to the echo was computed and is plotted in Fig.",
"REF b for both adiabatic and rectangular pulses.",
"By integrating over these curves we obtained the total signal intensity and found that adiabatic pulses produce a signal that is 1.73 times larger than the signal produced by rectangular pulses.",
"This is in excellent agreement with the value of 1.74 observed in experiment.",
"Note that because adiabatic pulses are sensitive to a volume larger than the epi-layer, the adiabatic pulse signal would increase when using a bulk doped sample.",
"We also note that rectangular pulses are sensitive to a thin region of spins which could allow for high resolution tomography experiments.",
"Figure: (a) Cross section of resonator at an antinode in B 1 B_{1} with contour lines indicating B 1 B_{1} magnitude for a 400 nW input power.",
"The hatched regions denote the location of spins contributing to 2/3 of the echo intensity for rectangular (red) pulses and adiabatic (blue) pulses (violet where they overlap).",
"The green-hatched region below 25 μ\\mu m denotes the undoped portion of the sample.",
"(b) Plot of the echo intensity contribution of all spins at particular values of B 1 B_{1}.From these simulations, we estimate the sample volume coupled to our resonators to be $3.9 \\times 10^{-6}$ cm$^{3}$ .",
"The doping density of the sample is $8 \\times 10^{14}$ donors/cm$^{3}$ , so there are $1.6 \\times 10^{9}$ spins coupled to the resonator per hyperfine line.",
"We measured these spins in a single shot (using no signal averaging and utilizing a single two-pulse sequence) and found a S/N of 84 using rectangular pulses and 146 with adiabatic pulses.",
"Scaling to S/N = 1, we have sensitivity to $2 \\times 10^{7}$ spins when using rectangular pulses and $1 \\times 10^{7}$ spins using adiabatic pulses.",
"ESR sensitivity is often reported in units of spins/$\\sqrt{Hz}$ .",
"These units are appropriate for continuous wave experiments.",
"However, in pulsed electron spin resonance, this sensitivity is limited by the shot repetition rate.",
"Our typical shot repetition rate is 100 Hz (determined by optical spin relaxation) giving a sensitivity of $10^{6}$ spins/$\\sqrt{Hz}$ .",
"By employing refocusing pulses in a CPMG (Carr-Purcell-Meiboom-Gill) sequence, much faster repetition rates have been achieved[17].",
"Our projected sensitivity when using rectangular pulses in a CPMG sequence is $3 \\times 10^{4}$ spins/$\\sqrt{Hz}$ .",
"This represents an order of magnitude improvement over recently reported values for spin resonance detected by a superconducting qubit[2] and is on par with sensitivities reported using surface loop-gap microresonators[17].",
"This value should improve by another factor of 5 by measuring at lower temperatures where spins are fully polarized.",
"In summary, we have demonstrated the use of superconducting CPW resonators to perform pulsed electron spin resonance using an ultra low power of -34 dBm (400 nW) with a $\\pi $ -pulse length of 400 ns.",
"We also verify that 40 ns $\\pi $ -rotations can be achieved using peak powers of about 50 $\\mu $ W, making CPW resonators compatible with dilution refrigerators.",
"We report a single-shot sensitivity to $10^{7}$ spins or $3 \\times 10^{4}$ spins/$\\sqrt{Hz}$ .",
"This is comparable to the best results reported thus far.",
"Finally, we have shown that BIR-4-WURST-20 pulses can be used to compensate for $B_{1}$ inhomogeneities spanning an order of magnitude.",
"These adiabatic pulses improved our S/N by a factor of 1.74 and substantially improve the uniformity of microwave manipulations of a randomly distributed spin ensemble.",
"Work at Princeton and UCL was supported in part by the NSF and EPSRC through the Materials World Network Program (DMR-1107606 and EP/I035536/1).",
"Work at Princeton was also supported by the ARO (W911NF-13-1-0179) and Princeton MRSEC (DMR-0819860), at Keio by the Grant-in-Aid for Scientific Research and Project for Developing Innovation Systems by the Ministry of Education, Culture, Sports, Science and Technology, the FIRST Program by the Japan Society for the Promotion of Science, and the Japan Science and Technology Agency/UK EPSRC (EP/H025952/1), at LBNL by the US Department of Energy (DE-AC02-05CH11231) and the NSA (100000080295), at The University of Chicago by DARPA (N66001-11-1-4123) and NSF through the Chicago MRSEC (DMR-0820054), and at Simon Fraser University by the Natural Sciences and Engineering Research Council of Canada.",
"J.J.L.M.",
"is supported by the Royal Society."
]
] | 1403.0018 |
[
[
"Connections of Zero Curvature and Applications to Nonlinear Partial\n Differential Equations"
],
[
"Abstract A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed.",
"It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well.",
"It is also shown that the connection coefficients can be defined so that the partial differential equation to be studied appears as the curvature term in the structure equations.",
"It is discussed how Lax pairs and Backlund transformations can be formulated for such equations.",
"It is discussed how Lax pairs and Backlund transformations can be formulated for such equations that occur as zero curvature terms."
],
[
"Introduction",
"Connections which determine representations of zero curvature have turned out to be a very useful and innovative approach for studying nonlinear partial differential equations.",
"These connection forms have the capacity to produce results which can be used to obtain Lax pairs as well as Bäcklund transformations in a very direct way provided information concerning the structural differential forms of special fiber bundles can be specified.",
"These types of connection have a special property in that the curvature tensor of such a connection contains a subtensor which is directly proportional to a partial differential equation which is of interest.",
"For the case in which the connection tensor with these components vanishes, as on the corresponding lifts of solutions of a given nonlinear equation, it is said the connection determines a representation of zero curvature.",
"The main ideas which have led to these developments began several decades ago and can be traced to the work of people such as Estabrook and Wahlquist [1-4] and by R. Hermann [5] as well.",
"Hermann first introduced at one point a particular connection of basically this type.",
"He proposed early on to interpret the Bäcklund transformation as a connection similar in a certain sense to the connection which defines a representation of zero curvature.",
"He first introduced the concept of a Bäcklund connection which is defined by the way the connection form is specified.",
"Hermann then formulates Bäcklund's problem as that of finding a section in a bundle space on whose pull-back the Bäcklund connection is plane.",
"He has presented the basic idea in [6], and an introductory outline can be given based on that.",
"Let $M$ be a manifold and consider two sorts of object on $M$ .",
"First $I$ will be a differential ideal of differential forms on $M$ , and $R$ a Pfaffian system or submodule of the set of differential one-forms on $M$ .",
"Thus, $F^* (M)$ denotes the exterior algebra of differential forms on $M$ , and $R$ is called a prolongation of $I$ if the following condition is satisfied $d R \\subset F^{*} (M) \\wedge R + I.$ In the initial approach taken by Estabrook and Wahlquist, they primarily start off with $I$ and then search for $R$ .",
"If $I=0$ , then (REF ) expresses the fact that $R$ is completely integrable.",
"The Frobenius complete integrability theorem [7] then asserts that there are, locally, one-forms $\\omega _1, \\cdots , \\omega _n \\in R$ forming a basis and such that $d \\omega _1 = \\cdots = d \\omega _n =0$ .",
"Second, if $R$ is generated by a single element, $\\omega $ , such that $d \\omega \\in I$ , then $\\omega $ is a conservation law for $I$ .",
"Studying the relation (REF ) in more advanced ways and further generalizations has led to an entire geometric approach to the classic AKNS program [8-9], and the study of the geometric properties of non-linear partial differential equations and their associated solutions.",
"There has been much interest in this approach [10-13], and has led to many insights between integrable evolution equations and pseudo-spherical surfaces as well [14-16].",
"The objective of this work is to go beyond this more primitive formulation which has just been described by starting with a jet-bundle $J^r E$ of $r$ -jets over a lower dimensional bundle $E$ [17].",
"For purposes here, $r$ is usually two or three when second or third order equations are involved, however, a formulation which doesn't specify $r$ at first will be given.",
"Structure equations are established for the systems of forms on these bundles.",
"A very novel approach to the formulation of zero curvature connections is presented in detail.",
"Several theorems and different proofs of these are presented as well which establish a general theory of the subject from a specific abstract viewpoint.",
"It is shown how the choice of particular connection coefficients can lead to an expression for the curvature, and an expression for the curvature tensor under the assumed form of the coefficients is found and satisfies a particular relation.",
"It is also shown how prolongations of the connections can be generated, and the resulting connections remain zero curvature.",
"Out of this comes a method for writing Lax pairs and Bäcklund transformations [18] for the equations involved.",
"In fact, one of the remarkable features of these differential systems is that once they have been specified, they can be used to yield Lax pairs very easily as well as Bäcklund transformations for the equations which appear as the zero curvature terms in the structure equations.",
"It is explained in detail how these can be constructed.",
"The difficult part as far as applications are concerned is to be able to write down the specific system of connection one-forms to initialize the process.",
"These same forms contain the relevant information for producing these additional structures.",
"Finally, it will be shown how the formalism can be applied in practice to obtain Bäcklund transformations between the Liouville equation and the wave equation.",
"Differential systems which are the zero curvature representations for these two different nonlinear equations will be written down.",
"They will be shown to have the right zero curvature structure and moreover how information from these differential forms needed to write down Lax pairs and Bäcklund transformations can be extracted."
],
[
"Framework",
"The main purpose in formulating connections which define representations of zero curvature is to study nonlinear partial differential equations in a systematic way.",
"By this it is intended that useful structures relevant to the study of these equations, such as Lax pairs and Bäcklund transformations, can be produced.",
"For definiteness, a general third order equation is of the form $F ( x^i, u, u_j, u_{jk}, u_{jkl}) =0.$ By enlarging the manifold which supports (REF ), equations of this type can be written in a more general form as $F ( x^i, u, \\lambda _j, \\lambda _{jk}, \\lambda _{jkl})=0,$ This notation is common and can be found in [19-20].",
"The $\\lbrace x^i, u\\rbrace $ are adapted local coordinates in the $(n+1)$ -dimensional bundle $E$ over the $n$ -dimensional base $M$ , whose local coordinates are given by $\\lbrace x^i \\rbrace $ where $i,j,k= 1, \\cdots , n$ .",
"This larger manifold called $J^r E$ over which (REF ) is defined is called the space of holonomic $r$ -jets of the local sections of the manifold $E$ .",
"It carries the system of coordinates $\\lbrace x^i$ , $u$ , $\\lambda _{j_1, \\cdots , j_k} \\rbrace $ with $k=1, \\cdots , r$ .",
"Thus, there exist the following inclusions, $M \\subset E \\subset J^r E$ .",
"Let $\\omega ^i$ , $\\omega ^{n+1}$ , $\\omega _j^i$ , $\\omega ^{n+1}_j$ , $\\omega ^{n+1}_{n+1}$ , $\\omega ^i_{jk},\\cdots $ be a sequence of structural forms of the holonomic frames of the manifold $E$ , symmetric in the subscripts.",
"The forms $\\omega ^i$ , $\\omega ^{n+1}$ , $\\omega ^{n+1}_{i_1, \\cdots , i_k}$ , for $k=1, \\cdots , r$ , are referred to as principal forms in the bundle of holonomic $r$ -jets, $J^r E$ [21].",
"These forms will satisfy systems of structural equations which have the form, $\\begin{array}{c}d \\omega ^i = \\omega ^k \\wedge \\omega _j^i, \\\\\\\\d \\omega ^{n+1} = \\omega ^j \\wedge \\omega ^{n+1}_j+ \\omega ^{n+1} \\wedge \\omega _{n+1}^{n+1}, \\\\\\end{array}$ as well as equations which arise in the process of regular prolongation of these by means of Cartan's lemma.",
"That is to say, taking the exterior derivative of the first equation in (REF ) gives $0 = d^2 \\omega ^i = d \\omega ^k \\wedge \\omega _k^i- \\omega ^k \\wedge d \\omega _k^i= \\omega ^s \\wedge ( \\omega _s^k \\wedge \\omega _k^i- d \\omega _s^i).$ By the generalized Cartan lemma, the coefficients in the brackets can be expanded in terms of the forms $\\omega ^i$ $d \\omega _s^i - \\omega _s^k \\wedge \\omega _k^i= \\omega ^k \\wedge \\omega ^i_{sk}.$ This can be differentiated in turn and when the process is repeated, a tower of forms can be constructed [22].",
"It is important in the course of this work to be able to evaluate appropriate sections in these bundles, and it is carried out in the following way.",
"For any section $\\Sigma \\subset E$ which is defined by the equation $u = u (x^1, \\cdots , x^n)$ , sections in $\\Sigma ^r \\subset J^r E$ are defined by the equations $u= u(x^1, \\cdots , x^n),\\qquad \\lambda _{i_1, \\cdots , i_k} = u_{i_1, \\cdots , i_k},\\qquad k=1, \\cdots , r.$ The subscripts $i+1, \\cdots , i_k$ on the function $u$ now denote partial derivatives.",
"Consequently, under this process, the equation (REF ) is mapped onto (REF ), the equation of interest.",
"If contact forms are chosen as principal forms on the manifold $J^r E$ , then the pull-backs are integral manifolds of the system of Pfaffian equations $\\omega ^{n+1} = \\omega ^{n+1}_i = \\cdots = \\omega ^{n+1}_{i_1 \\cdots i_k}=0.$"
],
[
"Principle Bundle",
"To begin with, based on this sequence of manifolds, consider the principle bundle $P (J^r E, G)$ over $J^r E$ along with the $g$ parameter structure group $G$ .",
"Let $P (J^r E, G)$ have structural forms $\\omega ^A$ , ($A, B =1, \\cdots , g$ ) which satisfy structure equations of the form $d \\omega ^A = \\frac{1}{2} C_{BC}^A \\, \\omega ^B \\wedge \\omega ^C+ \\omega ^{\\delta } \\wedge \\omega _{\\delta }^A.$ In (REF ), the $C_{BC}^A$ are the structure constants pertaining to the Lie group $G$ .",
"They are skew-symmetric with respect to the lower indices and satisfy the Jacobi identity $C^A_{BK} C_{LM}^B + C_{BL}^A C_{MK}^B + C_{BM}^A C_{KL}^B =0.$ The forms $\\omega ^{\\delta }$ will be principle forms of the base $J^r E$ , and will be completely integrable.",
"Thus, their differentials satisfy structure equations of the form $d \\omega ^{\\delta } = \\omega ^{\\mu } \\wedge \\omega _{\\mu }^{\\delta }.$"
],
[
"General Zero-Curvature Formulation",
"To show exactly how zero curvature representations can be developed from a rigorous point of view, a connection in the principle bundle $P (J^r E, G)$ has to be defined [19-20].",
"One way of doing this is to specify the object of connection.",
"This is made precise in the following theorem.",
"Theorem 3.1 A connection in the principle bundle $P (J^r E, G)$ can be given by the field of a connection object on $J^r E$ which has components $\\Gamma ^A_{\\epsilon }$ that satisfy the system of differential equations $d \\Gamma ^A_{\\epsilon } + C_{BC}^A \\Gamma ^B_{\\epsilon } \\omega ^C- \\Gamma ^A_{\\delta } \\omega ^{\\delta }_{\\epsilon } - \\omega _{\\epsilon }^A= \\Gamma _{\\epsilon \\delta }^A \\omega ^{\\delta },$ The forms $\\omega _{\\epsilon }^{\\delta }$ are determined from (REF ).",
"The associated connection forms $\\tilde{\\omega }^A = \\omega ^A + \\Gamma ^A_{\\epsilon } \\, \\omega ^{\\epsilon }$ satisfy the structure equations $d \\tilde{\\omega }^A = \\frac{1}{2} C_{BC}^A \\tilde{\\omega }^B \\wedge \\tilde{\\omega }^C + \\Omega ^A.$ The $\\Omega ^A$ in (REF ) are curvature forms given by $\\Omega ^A = R^A_{\\epsilon \\delta } \\omega ^{\\epsilon } \\wedge \\omega ^{\\delta }.$ Proof: Differentiating the connection forms in (REF ) and requiring the exterior derivative be consistent with (REF ), yields $d \\omega ^A + d ( \\Gamma ^A_{\\delta } \\omega ^{\\delta })= \\frac{1}{2} C_{BC}^A ( \\omega ^B + \\Gamma ^B_{\\epsilon } \\omega ^{\\epsilon })\\wedge ( \\omega ^C + \\Gamma ^C_{\\delta } \\omega ^{\\delta }) + \\Omega ^A.$ Expanding this out, the following expression results, $d \\omega ^A + d \\Gamma ^A_{\\delta } \\wedge \\omega ^{\\delta } + \\Gamma _{\\delta }^Ad \\omega ^{\\delta } = \\frac{1}{2} C_{BC}^A \\omega ^B \\wedge \\omega ^C +\\frac{1}{2} C_{BC}^A \\omega ^B \\wedge \\Gamma ^C_{\\delta } \\omega ^{\\delta }+ \\frac{1}{2} C_{BC}^A \\Gamma ^B_{\\epsilon } \\omega ^{\\epsilon } \\wedge \\omega ^C + \\frac{1}{2} C_{BC}^A \\Gamma _{\\epsilon }^B \\Gamma ^C_{\\delta }\\omega ^{\\epsilon } \\wedge \\omega ^{\\delta } + \\Omega ^A.$ Substituting (REF ) and (REF ) into this, we obtain, $d \\omega ^A - \\frac{1}{2} C_{BC}^A \\omega ^B \\wedge \\omega ^C - \\omega ^{\\delta }\\wedge \\omega ^A_{\\delta } + ( - C_{BC}^A \\Gamma ^B_{\\delta } \\omega ^C+ \\Gamma ^A_{\\sigma } \\omega ^{\\sigma }_{\\delta } + \\omega ^A_{\\delta }+ \\Gamma _{\\delta \\sigma }^A \\omega ^{\\sigma }) \\wedge \\omega ^{\\delta }+ \\Gamma ^A_{\\delta } \\omega ^{\\epsilon } \\wedge \\omega ^{\\delta }_{\\epsilon }$ $= - \\omega ^{\\delta } \\wedge \\omega ^A_{\\delta } + \\frac{1}{2} C_{BC}^A\\Gamma ^C_{\\delta } \\omega ^B \\wedge \\omega ^{\\delta } + \\frac{1}{2}C_{BC}^A \\Gamma ^C_{\\delta } \\omega ^{\\delta } \\wedge \\omega ^B+ \\frac{1}{2} C_{BC}^A \\Gamma ^B_{\\epsilon } \\Gamma ^C_{\\delta }\\omega ^{\\epsilon } \\wedge \\omega ^{\\delta } + \\Omega ^A.$ Now replace $d \\omega ^A$ using (REF ) to obtain $- C_{BC}^A \\Gamma ^B_{\\delta } \\omega ^C \\wedge \\omega ^{\\delta }+ \\Gamma ^A_{\\sigma } \\omega ^{\\sigma }_{\\delta } \\wedge \\omega ^{\\delta } + \\omega ^A_{\\delta }\\wedge \\omega ^{\\delta } + \\Gamma ^A_{\\delta \\sigma } \\omega ^{\\sigma }\\wedge \\omega ^{\\delta } + \\Gamma ^A_{\\delta } \\omega ^{\\epsilon }\\wedge \\omega _{\\epsilon }^{\\delta }$ $=- \\omega ^{\\delta } \\wedge \\omega ^A_{\\delta } + C_{BC}^A\\Gamma ^C_{\\delta } \\omega ^B \\wedge \\omega ^{\\delta } + \\frac{1}{2}C_{BC}^A \\Gamma ^B_{\\epsilon } \\Gamma ^C_{\\delta } \\omega ^{\\epsilon }\\wedge \\omega ^{\\delta } + \\Omega ^A.$ The fact that the $C_{BC}^A$ are antisymmetric in the lower indices simplifies this result to the form, $\\Omega ^A = \\Gamma ^A_{\\delta \\sigma } \\omega ^{\\sigma } \\wedge \\omega ^{\\delta } - \\frac{1}{2} C_{BC}^A \\Gamma ^B_{\\epsilon }\\Gamma ^C_{\\delta } \\, \\omega ^{\\epsilon } \\wedge \\omega ^{\\delta }.$ Factoring the one-forms in the first part of $\\Omega ^A$ , it is found that $\\Omega ^A =- \\frac{1}{2} ( \\Gamma _{\\epsilon \\delta }^A- \\Gamma _{\\delta \\epsilon }^A + C_{BC}^A \\Gamma ^B_{\\epsilon }\\Gamma _{\\delta }^C) \\, \\omega ^{\\epsilon } \\wedge \\omega ^{\\delta }.$ This gives $\\Omega ^A$ explicitly and finishes the proof.",
"The coefficients of $\\Omega ^A$ in (REF ) give the components of $R^A_{\\epsilon \\delta }$ and the theorem allows us to identify the components of the curvature tensor as $R^A_{\\epsilon \\delta } =- \\frac{1}{2} ( \\Gamma ^A_{\\epsilon \\delta }- \\Gamma ^A_{\\delta \\epsilon } + C_{BC}^A \\Gamma ^B_{\\epsilon }\\Gamma ^C_{\\delta } ).$ Theorem 3.2 The curvature tensor satisfies the following relation $d R_{\\lambda \\mu }^A + R_{\\lambda \\mu }^B C_{BC}^A \\omega ^C- R_{\\sigma \\mu }^A \\omega _{\\lambda }^{\\sigma }- R_{\\lambda \\sigma }^A \\omega _{\\mu }^{\\sigma } =0,\\qquad \\mod {\\,} \\omega ^{\\Delta },$ where $\\omega ^{\\Delta }$ are principle forms of the jet manifold.",
"Proof: Differentiating both sides of (REF ) exteriorly, it is found that $0= \\frac{1}{2} C_{BC}^A \\, d \\tilde{\\omega }^B \\wedge \\tilde{\\omega }^C- \\frac{1}{2} C_{BC}^A \\tilde{\\omega }^B \\wedge d \\tilde{\\omega }^C+ d R_{\\lambda \\mu }^A \\wedge \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }+ R_{\\lambda \\mu }^A d \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }- R_{\\lambda \\mu }^A \\, \\omega ^{\\lambda } \\wedge d \\omega ^{\\mu }$ $= C_{BC}^A ( \\frac{1}{2} C_{DQ}^B \\tilde{\\omega }^D \\wedge \\tilde{\\omega }^Q+ R_{\\lambda \\mu }^B \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }) \\wedge \\tilde{\\omega }^C+ d R_{\\lambda \\mu }^A \\wedge \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }+ R_{\\lambda \\mu }^A \\omega ^{\\sigma } \\wedge \\omega _{\\sigma }^{\\lambda }\\wedge \\omega ^{\\mu } - R_{\\lambda \\mu }^A \\, \\omega ^{\\lambda } \\wedge \\omega ^{\\sigma } \\wedge \\omega _{\\sigma }^{\\mu }$ $= \\frac{1}{2} C_{TC}^A C_{DB}^T \\tilde{\\omega }^D \\wedge \\tilde{\\omega }^B\\wedge \\tilde{\\omega }^C + C_{BC}^A R_{\\lambda \\mu }^B \\tilde{\\omega }^C\\wedge \\omega ^{\\lambda } \\wedge \\omega ^{\\mu } + d R_{\\lambda \\mu }^A \\wedge \\omega ^{\\lambda }\\wedge \\omega ^{\\mu } - R_{\\lambda \\mu }^A \\omega _{\\sigma }^{\\lambda } \\wedge \\omega ^{\\sigma } \\wedge \\omega ^{\\mu } - R_{\\lambda \\mu }^A \\, \\omega _{\\sigma }^{\\mu }\\wedge \\omega ^{\\lambda } \\wedge \\omega ^{\\sigma }.$ Invoking the Jacobi identity (REF ), this result reduces to the following form $( d R_{\\lambda \\mu }^A + R_{\\lambda \\mu }^B C_{BC}^A \\tilde{\\omega }^C- R_{\\sigma \\mu }^A \\omega _{\\lambda }^{ \\sigma } - R_{\\lambda \\sigma }^A\\omega _{\\mu }^{\\sigma } ) \\wedge \\omega ^{\\lambda } \\wedge \\omega ^{\\mu } =0.$ This implies that the coefficient of $\\omega ^{\\lambda } \\wedge \\omega ^{\\mu }$ is zero $\\mod {\\omega }^{\\Delta }$ , the principle forms of the jet manifold, so that $\\tilde{\\omega }^C = \\omega ^C$ .",
"The result in (REF ) then follows.",
"Thus, the curvature tensor components include, in particular, the components $R_{kl}^A$ .",
"As a consequence of these theorems, the following result is very important as far as the application of the zero-curvature idea to specific nonlinear differential equations is concerned.",
"Theorem 3.3 For the connection given in the principle bundle $P (J^r E, G)$ to define the representation of zero curvature which corresponds to an equation $F (x^i, u, \\lambda _j, \\lambda _{jk}, \\cdots )=0$ , it is necessary and sufficient that the components $R_{kl}^{A}$ of the curvature vanish on the pull-backs of the solutions to the equation.",
"Proof: Since the vanishing of the forms of curvature $\\Omega ^A= R_{\\lambda \\mu }^A \\, \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }$ on the pull-backs of solutions is invariant, it suffices to show the statement for some special choice of the principle forms.",
"The statement then becomes obvious if contact forms are taken as principle forms since, in this case, the relations $\\Omega ^A= R^A_{kl} \\omega ^k \\wedge \\omega ^l$ hold on the pull-back of any section $\\Sigma \\subset E$ .",
"In practical terms, the curvature tensor will be, or will have a subtensor, which is proportional to the equation under consideration, and will clearly vanish identically on solutions of that equation.",
"Thus, a connection is called a connection determining a representation of zero curvature for a differential equation if the curvature form vanishes on the solutions, or on the corresponding lifts of solutions, and only on solutions."
],
[
"Prolongations on These Spaces",
"An additional bundle associated with the principle bundle $P (J^r E, E)$ , which is called $F (P (J^r E, G))$ , can now be constructed.",
"A larger space is now being associated with $P$ .",
"The typical fiber of this new bundle is a space $F$ which is an $N$ -dimensional space of the representation of the Lie group $G$ .",
"The representation of the group $G$ as a group of transformations of the space $F$ can be defined by the specification of the system of Pfaffian equations $d X^I - \\xi _A^I (X) w^a=0.$ In (REF ), the $w^A$ are invariant forms of the group $G$ which satisfy the structural equations $d w^A = \\frac{1}{2} \\, C_{BC}^A \\, w^B \\wedge w^C.$ Indeed, it is worth recalling that if $G$ is connected, any diffeomorphism $f : G \\rightarrow G$ which preserves left-invariant forms, $\\theta ^{\\alpha }$ , so that $f^* \\theta ^{\\alpha } = \\theta ^{\\alpha }$ is left translation.",
"If $N$ is a smooth manifold and $w^{\\alpha }$ linearly independent forms on $N$ satisfying (REF ), then for any point in $N$ , there exists a neighborhood $U$ and a diffeomorphism $f: U \\rightarrow G$ such that $\\theta ^{\\alpha } =f^* (w^{\\alpha })$ .",
"The following theorem will produce a condition that, when satisfied, will guarantee that system (REF ) is completely integrable.",
"Theorem 4.1 Pfaffian system (REF ) is completely integrable provided the set of $\\xi ^I_A (X)$ satisfy the following constraint, $\\xi _B^K \\frac{\\partial \\xi _C^I}{\\partial X^K}- \\xi ^K_C \\frac{\\partial \\xi ^I_B}{\\partial X^K}+ \\xi _A^I C_{BC}^A =0.$ Proof: Differentiate both sides of system (REF ) to obtain, $\\frac{\\partial \\xi _A^I}{\\partial X^K} \\xi _C^K (X) \\, w^C \\wedge w^A+ \\frac{1}{2} \\xi _B^I (X) C_{BC}^A \\, w^B \\wedge w^C =0.$ The first term in this equation can be put in the form $\\frac{1}{2} \\lbrace \\xi _B^K (X) \\frac{\\partial \\xi _C^I}{\\partial X^K} \\,w^B \\wedge w^C + \\xi _C^K (X) \\frac{\\partial \\xi _B^I}{\\partial X^K}\\, w^C \\wedge w^B \\rbrace + \\frac{1}{2} \\xi _A^I (X) C_{BC}^A \\, w^B \\wedge w^C =0.$ Equating the coefficient of $w^B \\wedge w^C$ to zero, the condition (REF ) for complete integrability is obtained.",
"These conditions are often referred to as the Lie identities.",
"If there exists a connection in $P (J^r E, G)$ which determines a representation of zero curvature, it is remarkable that the same property holds in the associated bundle $F(P(J^r E,G))$ .",
"The $N$ -dimensional space $F$ is coordinatized by means of coordinates $ \\lbrace X^i \\rbrace _1^N$ and carries a representation of the group.",
"Moreover, the curvature forms of $F ( P( J^r E, G))$ are defined by $\\theta ^I = d X^I - \\xi ^I_A (X^1, \\cdots , X^N) \\omega ^A,\\quad I,J,K =1, \\cdots , N.$ In (REF ), the $\\omega ^A$ are structural forms of the principle bundle.",
"If a connection with the connection forms $\\tilde{\\omega }^A = \\omega ^A + \\Gamma ^A_{\\lambda } \\omega ^{\\lambda },$ is defined in the principle bundle, then along with this connection in the principle bundle, a connection is induced in the associated bundle $F (P (J^r E, G))$ and it has connection forms $\\tilde{\\theta }^I = d X^I - \\xi ^I_A (X) \\tilde{\\omega }^A.$ Proposition 4.1 The Pfaffian system $\\tilde{\\theta }^I$ satisfies the system of structural equations $d \\tilde{\\theta }^I = \\tilde{\\theta }^K \\wedge \\tilde{\\theta }^I_K- \\xi ^I_A (X) \\, R^A_{\\lambda \\mu } \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }.$ The $\\xi _A^I (X)$ satisfy the Lie identities (REF ) and the $\\tilde{\\theta }_K^I$ are given by $\\tilde{\\theta }_K^I =- \\frac{\\partial \\xi _A^I}{\\partial X^K}\\tilde{\\omega }^A.$ The $R^A_{\\lambda \\mu }$ are the components of the curvature tensor defined in $P (J^r E, G)$ .",
"Proof: Differentiating the set of forms in (REF ), it is found that $d \\tilde{\\theta }^I =- \\frac{\\partial \\xi _A^I}{\\partial X^K} d X^K\\wedge \\tilde{\\omega }^A - \\xi _A^I (X) d \\tilde{\\omega }^A$ $=- \\frac{\\partial \\xi _A^I}{\\partial X^K} ( \\tilde{\\theta }^K+ \\xi _C^K (X) \\tilde{\\omega }^C ) \\wedge \\tilde{\\omega }^A- \\xi _A^I (X) \\, d \\tilde{\\omega }^A$ $=- \\frac{\\partial \\xi ^I_A}{\\partial X^K} \\tilde{\\theta }^K \\wedge \\tilde{\\omega }^A - \\xi ^K_C (X) \\frac{\\partial \\xi _A^I}{\\partial X^K}\\tilde{\\omega }^C \\wedge \\tilde{\\omega }^A - \\xi _A^I (X) \\, d \\tilde{\\omega }^A$ $= \\tilde{\\theta }^K \\wedge (- \\frac{\\partial \\xi ^I_A}{\\partial X^K}) \\tilde{\\omega }^A- \\xi ^K_B (X) \\frac{\\partial \\xi ^I_C}{\\partial X^K} \\tilde{\\omega }^B \\wedge \\tilde{\\omega }^C - \\frac{1}{2} C_{BC}^A \\xi ^I_A (X) \\tilde{\\omega }^B \\wedge \\tilde{\\omega }^C - \\xi _A^I (X) \\, R^A_{\\lambda \\mu }\\omega ^{\\lambda } \\wedge \\omega ^{\\mu }.$ Assuming that the Lie identities (REF ) hold and $\\tilde{\\theta }^I_K$ are defined by (REF ), the desired result (REF ) appears directly, $d \\tilde{\\theta }^I = \\tilde{\\theta }^K \\wedge \\tilde{\\theta }^I_K - \\xi ^I_A (X)\\, R^A_{\\lambda \\mu } \\omega ^{\\lambda } \\wedge \\omega ^{\\mu }.$ Therefore, if the connection defined in the principle bundle specifies a representation of zero curvature for an equation, then the related connection just defined in the associated bundle generated by it will define a representation of zero curvature as well.",
"Its curvature tensor $\\xi ^I_A R^A_{\\lambda \\mu }$ vanishes on sections $\\Sigma \\subset E$ if and only if the sections are solutions of the equations.",
"This has established the following.",
"Corollary 4.1 The system of forms $\\tilde{\\theta }^I$ defined by (REF ) is completely integrable on the pull-backs of solutions to the associated equation and only on these solutions.",
"The theoretical advantage then in introducing the general formalism is that the $R^A_{\\lambda \\mu }$ can be interpreted as curvature forms with respect to this larger manifold.",
"This also suggests an application for these results.",
"It is possible that a system of forms $\\tilde{\\theta }^K$ can be found such that a set of equations of the form (REF ) obtain.",
"The curvature terms may automatically vanish or be proportional to some nonlinear partial differential equation of interest which vanishes on some transverse integral manifold of solutions.",
"Along with Bäcklund connections on bundles having one-dimensional fibers, Bäcklund connections on bundles with two-dimensional fibers can be studied; for example, on a bundle associated to a two-dimensional vector space of the representation of the group $Sl (2)$ .",
"This connection is often referred to as a Lax connection as it can be made to lead directly to formulation of Lax pairs for the equation.",
"In this event, the specific forms can then be used to generate both Lax pairs and Bäcklund transformations.",
"This will be illustrated clearly in the following general theorem below [23].",
"Hermann used a one-form with the structure (REF ) for the KdV equation and realized that it could be written in a particular way [5].",
"He inferred that the Wahlquist-Estabrook prolongation structure could be interpreted as a type of connection.",
"As for the form $\\tilde{\\theta }$ , it is a form of connection in a bundle with a one-dimensional typical fiber associated with the principal bundle $P (J^r E, Sl (2))$ .",
"This connection is also a connection defining a representation of zero curvature.",
"Note that a one-form is a connection form in a bundle with a one-dimensional typical fiber associated with the principal bundle $P ( J^r E, Sl (2))$ if and only if it takes the form $dy - \\xi (y) \\tilde{\\theta }_0 - \\xi _1^2 (y) \\tilde{\\theta }_1- \\xi _2^1 (y) \\tilde{\\theta }_2.$ The Lie identities satisfied by these coefficients are obtained from the system $\\frac{\\partial \\xi _B^I}{\\partial y^K} \\xi _C^K- \\frac{\\partial \\xi _C^I}{\\partial y^K} \\xi _B^K= \\xi _A^I C^A_{BC}.$ Consider a Bäcklund mapping in the one-dimensional case.",
"In this case the system of Pfaff equations that define the Bäcklund mapping consist of a single equation $dy - \\xi (y) \\tilde{\\omega } - \\xi _1^2 (y) \\tilde{\\omega }^1_2- \\xi ^1_2 (y) \\tilde{\\omega }^2_1 =0.$ The Lie identities satisfied by the coefficients in (REF ) are of the following form $\\xi \\frac{\\partial \\xi _1^2}{\\partial y} - \\xi ^2_1\\frac{\\partial \\xi }{\\partial y} = \\xi _1^2,$ $\\xi \\frac{\\partial \\xi ^1_2}{\\partial y} - \\xi _2^1\\frac{\\partial \\xi }{\\partial y} =- \\xi _2^1,$ $\\xi _1^2 \\frac{\\partial \\xi ^1_2}{\\partial y}- \\xi _2^1 \\frac{\\partial \\xi ^2_1}{\\partial y}= 2 \\xi .$ Theorem 4.2.",
"The Pfaff equation (REF ) which defines the Bäcklund mapping with the associated space of the structure group $G$ of dimension one can be represented in either of the two forms, $\\begin{array}{c}d \\varphi - \\tilde{\\omega }^2_1 - \\varphi \\tilde{\\omega }+ \\varphi ^2 \\tilde{\\omega }^1_2 =0, \\\\\\\\d \\psi - \\tilde{\\omega }^1_2 - \\psi \\tilde{\\omega }- \\psi ^2 \\tilde{\\omega }^2_1 =0.\\end{array}$ Proof: Take the second equation in (REF ) and divide it by $(\\xi _2^1)^2$ to obtain $- \\frac{\\xi }{( \\xi ^1_2)^2} \\, d \\xi ^1_2 + \\frac{d \\xi }{\\xi _2^1} = \\frac{dy}{\\xi ^1_2}.$ This is equivalent to $d ( \\frac{\\xi }{\\xi ^1_2}) = \\frac{dy}{\\xi _2^1}.$ Define the variable $\\varphi = \\xi / \\xi ^1_2$ and use it in this result to give, $d \\varphi = \\frac{dy}{\\xi ^1_2}.$ Dividing by $( \\xi ^1_2)^2$ , the third equation becomes $- \\frac{\\xi _1^2}{( \\xi _2^1)^2} \\, d \\xi ^1_2 + \\frac{d \\xi _1^2}{\\xi _2^1}= -2 \\frac{\\xi }{(\\xi _2^1)^2} \\, dy.$ Consequently, using (REF ), $d ( \\frac{\\xi ^2_1}{\\xi ^1_2}) =- 2 \\frac{\\xi }{\\xi ^1_2} \\frac{dy}{\\xi ^1_2}=- d \\varphi ^2.$ Thus, we can identify $- \\varphi ^2 = \\xi ^2_1 / \\xi _2^1$ .",
"Since the form (REF ) can be written in the following way, $\\frac{dy}{\\xi _2^1} - \\tilde{\\omega }^2_1 - \\frac{\\xi (y)}{\\xi _2^1 (y)}\\tilde{\\omega } - \\frac{\\xi ^2_1 (y)}{\\xi ^1_2 (y)} \\tilde{\\omega }^1_2 =0,$ the required first equation in (REF ) follows by substituting these results for $\\varphi $ and $\\varphi ^2$ into (REF ).",
"The second equation in (REF ) follows in a similar fashion.",
"An example which shows how the results in these last two sections can be combined and made into something useful will be presented.",
"Here $M$ will be the two-dimensional base manifold which is coordinatized by the coordinates $(x^1, x^2 ) = (x, t)$ .",
"Now consider the following application which starts with Theorem 3.1.",
"A system of structural forms $\\tilde{\\omega }^A$ is required to satisfy the structure equations (REF ) expressed as $d \\tilde{\\omega }^1 =2 \\tilde{\\omega }^2 \\wedge \\tilde{\\omega }^3+ R_{12} \\, d x^1 \\wedge d x^2,\\quad d \\tilde{\\omega }^2 = \\tilde{\\omega }^1 \\wedge \\tilde{\\omega }^2+ R^2_{112} \\, dx^1 \\wedge d x^2,\\quad d \\tilde{\\omega }^3 = \\tilde{\\omega }^3 \\wedge \\tilde{\\omega }^1+ R^1_{212} \\, dx^1 \\wedge dx^2.$ The last terms in these are the curvature terms which are required to vanish when they are considered on the lifting of a section.",
"This will result in producing a particular equation in the end.",
"In the notation of (REF ), take for the forms $\\tilde{\\omega }^A$ $\\tilde{\\omega }^1 = 2 \\lambda _1 \\, d x^2,\\quad \\tilde{\\omega }^2 = \\frac{1}{2} \\lambda _1 \\, d x^1 +( u \\lambda _1 - \\lambda _{11} ) \\, d x^2,\\quad \\tilde{\\omega }^3 = d x^1 + 2 u \\, d x^2.$ It is easily verified that these forms satisfy system (REF ).",
"The curvature term in the first and third is zero.",
"The second is satisfied provided that considered on the lifting of a section in which the notation reverts to that of (REF ), $u$ satisfies the following Burgers-type equation $-\\frac{1}{2} u_{12}+ \\frac{1}{2} (u^2)_{11} - u_{111} + (u_1)^2 =0$ .",
"Replacing $(x^1, x^2)=(x,t)$ in this, the following form for the equation is obtained, $u_{xt} = (u^2)_{xx} - 2 u_{xxx} + 2 (u_x)^2.$ Following along the lines of Theorem 4.2, there should be a Bäcklund transformation of the form $dy + \\tilde{\\omega }^2 - y \\tilde{\\omega }^1- y^2 \\tilde{\\omega }^3 =0$ .",
"Substituting the forms (REF ) into this, the following differential system is obtained $y_x = - \\frac{1}{2} u_x + y^2,\\qquad y_t = u_{xx} - u u_x + 2 u_x y + 2 u y^2.$ Evaluating the derivatives $y_{xt}$ and $y_{tx}$ , and subtracting, all higher order terms in the expression above $y^0$ are found to cancel.",
"Only the $y^0$ term remains and it is precisely the equation (REF ).",
"Another approach to Lax and Bäcklund systems will be presented in the next section."
],
[
"Lax and Bäcklund Systems",
"Perhaps the most interesting aspect of the theoretical development presented so far is that there exists a clear relationship between connections which define a representation of zero curvature and specific Lax and Bäcklund systems for the equation.",
"Let the group be $G = Gl (2)$ , so that $r$ is selected to suit the system under consideration.",
"In fact, for the example given here, we take $r=2$ , and the following theorem holds.",
"Theorem 5.1 Given a connection in $P (J^r E, G)$ , where $G= Gl(2)$ or a subgroup, which defines a representation of zero curvature corresponding to an equation of the form (REF ), a Lax system exists which can be defined in terms of the connection coefficients.",
"Proof: Let $\\tilde{\\omega }^i_j = \\omega _j^i + \\Gamma ^i_{j \\lambda } \\omega ^{\\lambda }$ be connection forms in the principle bundle $P (J^r E, Gl (2))$ which define the representation of zero curvature for the $\\tilde{\\omega }^A$ .",
"This connection which is defined in the principle bundle generates a connection in the associated bundle whose typical fiber is a two-dimensional linear space.",
"The connection forms in the associated bundle corresponding to the connection in $P$ can be written in the form (REF ) $\\tilde{\\theta }^i = d X^i + X^j \\tilde{\\omega }^i_j.$ As for the connection in $P$ , the connection in the associated bundle is also a connection which defines a representation of zero curvature for the equation.",
"Consequently, the restriction of the $\\tilde{\\theta }^i$ to the corresponding pull-back of the section $\\Sigma \\subset E$ defined by $u = u(x,y)$ is completely integrable if and only if the section $\\Sigma \\subset E$ is a solution of the equation.",
"$\\clubsuit $ In practical terms, if contact forms are taken as principle forms then $\\omega _j^i$ will be equal to zero and the forms $\\tilde{\\theta }^i$ take the form $\\tilde{\\theta }^i = d X^i + X^j \\Gamma ^i_{j \\lambda } \\omega ^{\\lambda }.$ In this case, with $(x^1, x^2)= (x,y)$ , the system of equations $\\tilde{\\theta }^i |_{\\Sigma } =0$ have the form $d X^i + X^j \\Gamma ^i_{j1} (x,y,u,u_k,u_{kl}) \\, dx+ X^j \\Gamma _{j2}(x,y,u,u_k,u_{kl}) \\, dy =0.$ Of course, this is equivalent to the following system of partial differential equations $X^i_x =- \\Gamma ^i_{j1} (x,y, u,u_k,u_{kl}) X^j,\\qquad X^i_y =- \\Gamma ^i_{j2} (x,y,u,u_k,u_{kl}) X^j.$ In matrix form for a two-dimensional representation of $G$ , (REF ) can be written as $\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_x =\\begin{pmatrix}- \\Gamma ^1_{11} & - \\Gamma ^1_{21} \\\\- \\Gamma ^2_{11} & - \\Gamma ^2_{21} \\\\\\end{pmatrix}\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix},\\qquad \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_y =\\begin{pmatrix}- \\Gamma ^1_{12} & -\\Gamma ^1_{22} \\\\- \\Gamma ^2_{12} & - \\Gamma ^2_{22} \\\\\\end{pmatrix}\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}.$ This system is completely integrable and has solutions satisfying any initial conditions if and only if $u=u (x,y)$ is a solution of the associated nonlinear equation.",
"There are relationships between Bäcklund transformations and the connections defining representations of zero curvature, as Hermann pointed out [3].",
"Consider restricting the problem to investigate how to write Bäcklund transformations between two second order equations.",
"Suppose $x,y,u$ and $x,y,v$ are adapted local coordinates in bundles $E_1$ and $E_2$ respectively which share a common base manifold $M$ with local coordinates $x$ , $y$ .",
"The variables $x,y,u, \\lambda _i, \\lambda _{jk}$ and $x,y,v, \\mu _i,\\mu _{jk}$ are local coordinates in the bundles of second order jets $J^2 E_1$ and $J^2 E_2$ .",
"In this case, $x,y,u, \\lambda _i$ and $x,y, v, \\mu _i$ are local coordinates in the corresponding bundles of first order jets $J^1 E_1$ and $J^1 E_2$ .",
"In this event, the equations then take the form $F_1 ( x,y, u, \\lambda _i, \\lambda _{jk} )=0,$ and, $F_2 (x, y, v, \\mu _i, \\mu _{jk}) =0.",
"\\qquad $ A Bäcklund transformation between these two equations can be defined as a system of equations $\\Phi ( x,y, u,v, u_i, v_j ) =0.$ Equation (REF ) will be integrable over $u$ if and only if $v = v(x,y)$ is a solution of (REF ) and integrable over $v$ if and only if $u = u(x,y)$ is a solution of (REF ).",
"For any specified solution $u$ of (REF ), or $v$ of (REF ), (REF ) makes it possible to determine a certain solution $v$ of (REF ), or of (REF ), respectively.",
"It is said that a Bäcklund transformation is established between (REF ) and (REF ) if connections have been defined in the two principle bundles $P ( J^1 E_1, G_1)$ and $P (J^1 E_2, G_2)$ which define representations of zero curvature for each equation.",
"In each of the manifolds $E_1$ and $E_2$ a structure of the bundle is defined with a one-dimensional fiber associated.",
"In the case of $E_2$ , it is with the principle bundle $P (J^1 E_1, G_1)$ and in the case of $E_1$ with $P ( J^1 E_2, G_2)$ .",
"Therefore, the connections which are defined in the principle bundles and specify representations of zero curvature generate corresponding representations of zero curvature in the associated bundles.",
"The forms for these two connection forms are written ${\\theta }$ and ${\\vartheta }$ .",
"For the case in which $G_1=G_2 = Gl (2)$ , the forms $\\theta $ and $\\vartheta $ take the form $\\theta = dv - \\xi _j^i (v) \\tilde{\\omega }_i^j,$ and, $\\vartheta = du - \\eta _j^i (u) \\tilde{\\pi }_i^j.$ The structure forms on the right of (REF ) and (REF ) are given by $\\tilde{\\omega }_j^i = \\omega _j^i + \\Gamma ^i_{jk} (x,y,u,\\lambda _l)\\omega ^k,\\quad \\tilde{\\pi }^i_j = \\pi _j^i + \\Phi ^i_{jk} (x,y,v, \\mu _l) \\omega ^k,\\qquad i,j=1,2.$ These will be connection forms in $P (J^1 E_1, Gl (2))$ and $P ( J^1 E_2, Gl(2))$ , respectively.",
"If contact forms are selected as principle forms in the bundle of jets, then $\\omega _j^i =0$ and $\\pi _j^i =0$ hold.",
"The forms in (REF ) simplify to $\\tilde{\\omega }_j^i = \\Gamma ^i_{jk} (x,y,u, \\lambda _l) \\, \\omega ^k,\\quad \\tilde{\\pi }_j^i = \\Phi ^i_{jk} (x,y,v, \\mu _l) \\omega ^k.$ In this case, the equations $\\theta =0$ and $\\vartheta =0$ considered on pull-backs of solutions of the equations (REF ) and (REF ), respectively, are written as $\\begin{array}{c}dv - \\xi ^i_j (v) \\Gamma ^j_{i1} (x,y,u, u_k) \\, dx- \\xi ^i_j (v) \\Gamma ^j_{i2} (x,y,u,u_k) \\, dy =0, \\\\\\\\du - \\eta _j^i (u) \\Phi _{i1}^j (x,y,v,v_k) \\, dx- \\eta ^i_j (u) \\Phi ^j_{i2} (x,y,v,v_k) \\, dy =0.",
"\\\\\\end{array}$ Of course, (REF ) are equivalent to the following systems of partial differential equations $v_x = \\xi ^i_j (v) \\Gamma ^j_{i1} (x,y,u,u_k),\\qquad v_y = \\xi ^i_j (v) \\Gamma ^j_{i2} (x,y,u,u_k),$ and $u_x = \\eta ^i_j (u) \\Phi _{i1}^j (x,y,v, v_k),\\qquad u_y = \\eta ^i_j (u) \\Phi ^j_{i2} (x,y,v,v_k).$"
],
[
"An Application of the Theory",
"This formalism is now applied to obtain Bäcklund transformations between the Liouville equation $u_{xy} =e^u$ and the wave equation $v_{xy}=0$ .",
"These can now be defined by specifying the connections in two principle bundles which define representations of zero curvature, and the corresponding connections in the associated bundles.",
"In this case, the connection forms in the principle bundles are defined as in (REF ).",
"A system of forms which will accomplish the task can be specified as follows $\\begin{array}{cccc}\\tilde{\\omega }^1_1 =- \\displaystyle \\frac{\\lambda _1}{4} dx + \\displaystyle \\frac{\\lambda _2}{4} dy, &\\tilde{\\omega }^2_2 = \\displaystyle \\frac{\\lambda _1}{4} dx - \\displaystyle \\frac{\\lambda _2}{4} dy, &\\tilde{\\omega }^2_1 = \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2} dx, &\\tilde{\\omega }^1_2 = \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2} \\, dy \\\\& & & \\\\\\tilde{\\pi }^1_1 = \\displaystyle \\frac{\\mu _1}{4} dx - \\displaystyle \\frac{\\mu _2}{4} dy, &\\tilde{\\pi }_2^2 =- \\displaystyle \\frac{\\mu _1}{4} dx + \\displaystyle \\frac{\\mu _2}{4} dy, &\\tilde{\\pi }^2_1 = \\sqrt{2} (e^{-v/2} dx + e^{v/2} dy), &\\tilde{\\pi }^1_2 =0.",
"\\\\\\end{array}$ Based on this collection of definitions, the required coefficients $\\Gamma ^j_{ik}$ and $\\Phi ^j_{ik}$ can be read off $\\begin{array}{cccc}\\Gamma ^1_{11} =- \\displaystyle \\frac{\\lambda _1}{4}, & \\Gamma _{21}^2 = \\displaystyle \\frac{\\lambda _1}{4}, &\\Gamma _{11}^2 = \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2}, & \\Gamma ^1_{21} =0, \\\\& & & \\\\\\Gamma ^1_{12} = \\displaystyle \\frac{\\lambda _2}{4}, & \\Gamma _{22}^2 =- \\displaystyle \\frac{\\lambda _2}{4}, &\\Gamma _{12}^2 =0, & \\Gamma ^1_{22} = \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2}.",
"\\\\\\end{array}$ and as well, $\\begin{array}{cccc}\\Phi _{11}^1 = \\displaystyle \\frac{\\mu _1}{4}, & \\Phi _{21}^2 =- \\displaystyle \\frac{\\mu _1}{4}, &\\Phi _{11}^2 = \\sqrt{2} e^{-v/2}, & \\Phi _{21}^1 =0, \\\\& & & \\\\\\Phi _{12}^1 =- \\displaystyle \\frac{\\mu _2}{4}, & \\Phi _{22}^2 = \\displaystyle \\frac{\\mu _2}{4}, &\\Phi _{12}^2 = \\sqrt{2} e^{v/2}, & \\Phi _{22}^1 = 0.",
"\\\\\\end{array}$ Now the corresponding sets of forms $\\theta ^i$ and $\\vartheta ^i$ are defined in terms of the structural forms (REF ), $\\begin{array}{ccc}\\theta ^1 = 2 ( \\tilde{\\omega }^1_1 - \\tilde{\\omega }_2^2), &\\theta ^2 = 2 \\tilde{\\omega }^2_1, & \\theta ^3 =-2 \\tilde{\\omega }^1_2, \\\\& & \\\\\\vartheta ^1 = 2 ( \\tilde{\\pi }^1_1 - \\tilde{\\pi }^2_2 ), &\\vartheta ^2 = \\tilde{\\pi }_1^2, & \\vartheta ^3 =0.",
"\\\\\\end{array}$ It will be shown explicitly that these forms define representations of zero curvature for the two equations above.",
"The first three structure equations in the $\\theta ^i$ are given by $d \\theta ^1 + \\theta ^2 \\wedge \\theta ^3 =- d \\lambda _1 \\wedge dx + d \\lambda _2 \\wedge dy- 2 e^u \\, dx \\wedge dy,$ $d \\theta ^2 - \\frac{1}{2} \\theta ^1 \\wedge \\theta ^2 = \\frac{1}{\\sqrt{2}} e^{u/2}(du - \\lambda _2 dy) \\wedge dx,$ $d \\theta ^3 + \\frac{1}{2} \\theta ^1 \\wedge \\theta ^3 = \\frac{1}{\\sqrt{2}} e^{u/2}(-du + \\lambda _1 \\, dx) \\wedge dy.$ On a section $\\Sigma _1 \\subset E_1$ , using (REF ) it follows that $\\lambda _1 =u_x$ , $\\lambda _2 = u_y$ and all three of these equations vanish provided that $u$ satisfies $u_{xy}= e^u$ .",
"Similarly, for the forms $\\vartheta ^i$ , it is found that $\\begin{array}{c}d \\vartheta ^1 + \\vartheta ^2 \\wedge \\vartheta ^3 = d \\mu _1 \\wedge dx - d \\mu _2 \\wedge dy, \\\\\\\\d \\vartheta ^2 - \\frac{1}{2} \\vartheta ^1 \\wedge \\vartheta ^2 =\\displaystyle \\frac{1}{\\sqrt{2}} [- e^{v/2} (dv \\wedge dx + \\mu _2 dx \\wedge dy)+ e^{v/2} (dv \\wedge dy - \\mu _1 dx \\wedge dy) ].\\end{array}$ The third vanishes identically since $\\vartheta ^3 =0$ .",
"On a section $\\Sigma _2 \\subset E_2$ , by applying (REF ), it follows that $\\mu _1 = v_x$ , $\\mu _2 = v_y$ and these equations vanish provided that $v$ satisfies the equation $v_{xy}=0$ .",
"Using (REF ) and the definitions in (REF ), then under the assignment $\\xi _1^1 =2 , \\quad \\xi _2^2 =- 2,\\quad \\xi _2^1 = 2 e^{-v/2}, \\quad \\xi _1^2 =- 2 e^{v/2}.$ the equation for $dv$ in (REF ) is written as $dv - \\theta ^1 - e^{-v/2} \\theta ^2 - e^{v/2} \\theta ^3 =0.$ Similarly, using (REF ) and identifying $\\eta _1^1 =-2, \\qquad \\eta _2^2 =2, \\qquad \\eta _2^1 = e^{u/2},$ the equation for $d u$ in (REF ) becomes $du + \\vartheta ^1 - e^{u/2} \\vartheta ^2 =0.$ It can be observed that the one-form of (REF ) is a closed form, whereas (REF ) is not closed, but leads to a consistent result.",
"Now all of the required information is at hand to write down Bäcklund transformations between these two equations.",
"Substituting (REF ) and (REF ) into (REF ), there results the system $u_x + v_x = \\sqrt{2} e^{(u-v)/2}, \\qquad u_y -v_y = \\sqrt{2} e^{(u+v)/2}.$ Substituting (REF ) and (REF ) into (REF ), it is found that the same pair appears, $u_x + v_x = \\sqrt{2} e^{(u-v)/2}, \\qquad u_y - v_y = \\sqrt{2} e^{(u+v)/2}.$ Theorem 6.1.",
"The exterior derivatives of the one-forms in (REF ) and (REF ) vanish modulo the sets of forms $\\lbrace dv, d \\theta ^i \\rbrace $ and $\\lbrace du, d \\vartheta ^i \\rbrace $ , respectively.",
"Proof: Let $\\tau $ denote the one-form on the laft-hand side of (REF ).",
"Differentiate $\\tau $ exteriorly and there results, $d \\tau =- d \\theta ^1 + \\frac{1}{2} e^{-v/2} dv \\wedge \\theta ^2 -e^{-v/2} d \\theta ^2 - \\frac{1}{2} e^{v/2} dv \\wedge \\theta ^3- e^{v/2} d \\theta ^3.$ Replacing the known forms $d \\theta ^i$ from (REF ) and $dv$ (REF ), we obtain that $d \\tau = \\theta ^2 \\wedge \\theta ^3 + \\frac{1}{2} e^{-v/2} \\theta ^1 \\wedge \\theta ^2+ \\frac{1}{2} \\theta ^3 \\wedge \\theta ^2 - \\frac{1}{2} e^{-v/2} \\theta ^1 \\wedge \\theta ^2 - \\frac{1}{2} e^{v/2} \\theta ^1 \\wedge \\theta ^3 - \\frac{1}{2}\\theta ^2 \\wedge \\theta ^3 + \\frac{1}{2} e^{v/2} \\theta ^1 \\wedge \\theta ^3 =0,$ as required.",
"In the same way, differentiate (REF ) and substitute $d \\vartheta ^i$ from (REF ) and $du$ from (REF ).",
"This provides another way to get the $\\xi ^i_j$ and $\\eta ^i_j$ which appear in (REF ) and (REF ).",
"Theorem 6.2.",
"Equations (REF ) and (REF ) form a system of Bäcklund transformations which connect the equations $u_{xy} = e^u$ and $v_{xy}=0$ respectively.",
"Proof: Differentiating the pair of equations in (REF ) and replacing the first derivatives on the right-hand side, it is found that $(u + v)_{xy} = \\frac{1}{\\sqrt{2}} (u-v)_y e^{(u-v)/2} = e^u,\\qquad (u- v)_{yx} = \\frac{1}{\\sqrt{2}} (u+v)_x e^{(u+v)/2} = e^u.$ Adding these two second derivatives, the Liouville equation $u_{xy} = e^u$ results.",
"Upon subtracting this pair, the wave equation $v_{xy} =0$ is obtained."
],
[
"Outlook and Summary",
"A very general and useful formalism has been examined which makes use of connections of zero curvature.",
"The first few sections present one way of giving an abstrct formulation to this subject, and the latter part transfers this to the more concrete aspect of actually calculating some differential systems for a pair of specific equations.",
"If the forms are selected in the right way, it should be possible to create auto-Bäcklund transformations, that is transformations between solutions of the same equation.",
"It has been shown that these types of connection have the potential to produce Lax pairs and Bäcklund transformations for nonlinear partial differential equations.",
"In fact the results of the previous section can be used to write Lax pairs for the respective equations.",
"Using coefficients (REF ) for the equation $u_{xy}=e^u$ , the following Lax pair is obtained $\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_x =\\begin{pmatrix}\\displaystyle \\frac{u_x}{4} & 0 \\\\- \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2} & - \\displaystyle \\frac{u_x}{4} \\\\\\end{pmatrix} \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix},\\qquad \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_y= \\begin{pmatrix}- \\displaystyle \\frac{u_y}{4} & - \\displaystyle \\frac{1}{\\sqrt{2}} e^{u/2} \\\\0 & \\displaystyle \\frac{u_y}{4} \\\\\\end{pmatrix} \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}.$ The compatibility condition for this pair can be calculated by differentiating the first matrix equation with respect to $y$ and the second with respect to $x$ .",
"It is seen to hold provided that $u$ satisfies the equation $u_{xy}=e^u$ .",
"Similarly, using the results in (REF ) for the equation $v_{xy}=0$ , the following Lax pair results $\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_x =\\begin{pmatrix}- \\displaystyle \\frac{v_x}{4} & 0 \\\\- \\displaystyle \\sqrt{2} e^{-v/2} & \\displaystyle \\frac{v_x}{4} \\\\\\end{pmatrix} \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix},\\qquad \\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}_y = \\begin{pmatrix}\\displaystyle \\frac{v_y}{4} & 0 \\\\- \\displaystyle \\sqrt{2} e^{-v/2} & - \\displaystyle \\frac{v_y}{4} \\\\\\end{pmatrix}\\begin{pmatrix}X^1 \\\\X^2 \\\\\\end{pmatrix}.$ The compatibility condition is again found to hold provided that $v$ satisfies $v_{xy}=0$ .",
"It might be conjectured as a further application of this work that if Lax pairs of the form (REF ) can be produced by some means, their matrix elements might be used to generate connections of zero curvature as discussed here.",
"If they are found to have zero curvature structure, the results obtained here would be of use in generating Bäcklund transformations for the equations involved."
],
[
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]
] | 1403.0028 |
[
[
"Coronal Mass Ejections and Angular Momentum Loss in Young Stars"
],
[
"Abstract In our own solar system, the necessity of understanding space weather is readily evident.",
"Fortunately for Earth, our nearest stellar neighbor is relatively quiet, exhibiting activity levels several orders of magnitude lower than young, solar-type stars.",
"In protoplanetary systems, stellar magnetic phenomena observed are analogous to the solar case, but dramatically enhanced on all physical scales: bigger, more energetic, more frequent.",
"While coronal mass ejections (CMEs) could play a significant role in the evolution of protoplanets, they could also affect the evolution of the central star itself.",
"To assess the consequences of prominence eruption/CMEs, we have invoked the solar-stellar connection to estimate, for young, solar-type stars, how frequently stellar CMEs may occur and their attendant mass and angular momentum loss rates.",
"We will demonstrate the necessary conditions under which CMEs could slow stellar rotation."
],
[
"Introduction",
"On young stars, we observe flares hundreds to ten thousand times more energetic and frequent than solar flares.",
"Along with energy scales greater by orders of magnitude, we also observe physical scales far greater than in the solar case: while solar prominences soar around 1 R$_{\\odot }$ above the solar surface and CMEs launch from similar radii in the Sun's atmosphere, magnetic structures on T Tauri Stars (TTS, young solar analogs)–post-flare loops and prominences–can extend tens of stellar radii from the star's surface.",
"The discovery of such large magnetic structures arose from solar-stellar analogy, applying solar flare models to the X-ray light curve data from young stars [17].",
"In characterizing the solar-stellar connection, overwhelming evidence has been found in support of the idea that the fundamental physics of magnetic reconnection is the same, despite differences in stellar parameters (e.g., mass, radius, $B$ , age).",
"As such, we approach analysis of young stars' flares and CMEs under this supposition, and aim to assess how the physical properties of these events–and their frequency–may scale accordingly with stellar parameters.",
"Ultimately, we seek to understand the consequences of exoplanetary space weather on protostellar systems and their forming planets."
],
[
"Estimating Angular Momentum Loss via Stellar CMEs",
"The inference of magnetic loops many stellar radii in extent [11] inspired three questions: one, are these loops interacting with circumstellar disks?",
"In [4], we did not find evidence for this.",
"Second, if there is not a star-disk link, how do the loops remain stable for the multiple rotation periods over which the X-ray flares are observed to decay?",
"We showed in [2] that when modeled as hot prominences, the addition of a scaled-up wind consistent with TTS observations provided sufficient support for the loops to be stable.",
"Finally, here [1] we address the third question of what happens when stability is lost: at many stellar radii, is the specific angular momentum shed significant enough to slow stellar rotation?",
"In order to estimate the effects of eruptive prominences and stellar CMEs on the rotation of young stars, we must procure two ingredients: the mass lost via these events, and their frequency of occurrence.",
"Despite ongoing and historical efforts to observe stellar CMEs, we lack definitive detections and thus frequency distributions.",
"It is known that at times, magnetic reconnection on the Sun will produce both a flare and an associated CME; for stars, the flare is the observable quantity, and so we characterize stellar CME frequency by using stellar flare frequency as a proxy.",
"In Fig.",
"REF , we show our solar flare energy/CME mass relationship [3] extrapolated up to the energies of young stellar flares.",
"We calculate loop masses for the 32 “superflaring” stars from the Chandra Orion Ultradeep Project [12] from the parameters reported by F05.",
"Interestingly, these loop masses are close in parameter space to the extrapolated solar relationship.",
"This is perhaps unsurprising, as the plasma confined in a post-flare loop has properties which relate to the energy of the flare.",
"In AA11, we found that for associated flares and CMEs, that is to say, flares and CMEs which likely originated from a shared magnetic reconnection event, the CME mass and flare energy were related.",
"As such, the post-flare loop mass and mass of an associated CME should then also be related.",
"From [18], we can estimate the total energy released by a flare is related to the total magnetic energy in a flare loop: $E_{mag} = \\frac{B^2 L^3}{8 \\pi }.$ If we substitute the loop volume expressed in terms of mass (i.e., $L^3 \\sim V=m_{\\rm loop}/\\rho $ ), we find a relationship between the total flare energy and the mass confined in the magnetic loop (dashed, parallel lines in Fig.",
"REF ).",
"Figure: CME mass/flare energy relationship of AA11 shown with TTS post-flare loop masses (black diamonds) derivedin AA12 plotted as a function of the flare's energy.",
"Point size denotes the mass of the star on which the flare wasobserved.Black dotted lines show predicted post-flare stellar loop masses (Eqn.",
"and discussion in text)for a range of observed densities from 10 10 ^{10}-10 12 ^{12} cm -3 ^{-3} and assuming a confining field strength of 50G.Gray, solid-lined boxes denote the range of observed X-ray flare energies(Maggio et al.",
"2000, Collier Cameron et al.",
"1988), and coolHα\\alpha prominence mass estimates (Collier Cameron & Robinson, 1989a,b), for AB Dor and Speedy Mic, 20-50Myr K dwarfs.Here, to be consistent with the analysis of F05, we assume an Euclidian loop filling but do note that recent solar X-ray flare imaging has indicated that a fractal scaling of V(L)$\\propto $ L$^{2.4}$ is likely a more accurate characterization [6].",
"It is remarkable that the post-flare loop masses even lie near the extrapolated solar CME mass/flare energy relationship, several orders of magnitude away in parameter space.",
"In Fig.",
"REF , we have also shown representative ranges of X-ray flare energy and prominence mass for two K dwarfs intermediate in age to the TTS sample and the Sun; with an eruptive prominence thought to be the core of a CME, these mass ranges likely represent lower limits on the range of CME masses on these stars.",
"In the following calculations, to represent a fiducial TTS case, we will simply extrapolate the solar relationship to generate a stellar CME mass distribution.",
"In Fig.",
"REF , we show the frequency distributions for flares observed on the Sun, TTS, M dwarfs, and active, main sequence G stars.",
"For this work, we use the TTS frequency distribution.",
"Clearly, not all CMEs are flare-associated, nor are all flares CME-associated; AA11 found, however, that the association fraction increases with increasing flare energy, so for young stars for which we observe flares several orders of magnitude more energetic than in the solar case, we simply assume this association fraction to be of order unity.",
"Figure: Flare frequencies for the Sun, M dwarfs (Hilton et al.",
"2011),TTS in the ONC (Albacete Colombo et al.",
"2007), and active, main sequence G starsfrom Kepler (Maehara et al.",
"2012).",
"Interestingly, the active G stars are almost indistinguishablefrom the TTS.",
"Note the solar and TTS frequencies are derived from X-ray flare data, while the M dwarf and G stars areoptical flare frequencies."
],
[
"Angular momentum loss",
"In AA12, we extrapolate the solar CME mass/flare energy relationship (Fig.",
"REF ) to TTS flare energies and frequencies (Fig.",
"REF ) and construct a CME frequency distribution as a function of CME mass.",
"Given the observational completeness limits on the flare distributions that went into the CME distribution, we derive lower and upper limits on the mass loss rate by empirical (integrating the distribution) and analytical (integrating a fit to the distribution) means.",
"The range of mass loss rates we estimate for the TTS case is 10$^{-12}$ -10$^{-9}$ M$_{\\odot }$ yr$^{-1}$ .",
"To assess the torque applied against stellar rotation by these CMEs, we apply stellar wind models with mass loss rates set as determined above.",
"Given the episodic nature of CMEs, we included an efficiency parameter to account for the fact that steady-state winds are more efficient at removing angular momentum than “clumpy” winds (cf.",
"AA12 and references therein).",
"We adopt a dipolar field with strength 600G, consistent with observations of TTS fields, and allow the stellar radius to contract as stellar evolution models predict.",
"In a protostellar system, multiple torques act simultaneously to spin up and spin down the star.",
"In this analysis, we compared spin up due to contraction and spin down due to mass loss from stellar CMEs to see if, at any point in the pre-main sequence our fiducial TTS could have its rotation slowed due to CMEs.",
"Comparing parameters of efficiency and the range of mass loss rates we calculated, it became clear that only towards the end of the pre-main sequence phase (ages $\\gtrsim $ 6 Myr) could a very efficient, high CME mass loss rate begin to counteract spin up from contraction.",
"We have left out factors such as spin up from accretion and mass loss via stellar wind; [16] explore these two torques in depth and the necessary conditions for an accretion powered stellar wind to slow stellar rotation."
],
[
"Discussion",
"We have shown that for young, solar-type stars, spin down due to CMEs might play a significant role in stellar rotation evolution after the star has ceased accreting.",
"Our Figs.",
"REF and REF illustrate a critical selection effect in performing this kind of calculation: we only have data for the most active young stars, or the star conveniently located at 1 AU.",
"There is a dearth of data for older, less active stars, and we suggest that filling in the gaps in flare X-ray energy could trace age evolution in these parameter spaces.",
"The addition of data from the $\\sim $ 20-50 Myr old K dwarfs in Fig.",
"REF hints at this, but more data are needed to conclusively show age dependence.",
"In both figures, we have taken care to specify the masses of the stars involved: how would evolution with stellar age look in these parameter spaces as a function of stellar mass?",
"While the fundamental physics are the same, the scaling could change, and the ramifications certainly would.",
"For low-mass stars in particular, high activity levels are observed for longer fractions of the stars' lives; this could have grave implications for exoplanets as these stars' habitable zones could be within range of extreme exo-weather.",
"A. N. A. thanks K. Shibata for helpful discussion regarding stellar post-flare loops."
]
] | 1403.0006 |
[
[
"Bethe ansatz solution of the topological Kondo model"
],
[
"Abstract Conduction electrons coupled to a mesoscopic superconducting island hosting Majorana bound states have been shown to display a topological Kondo effect with robust non-Fermi liquid correlations.",
"With $M$ bound states coupled to $M$ leads, this is an SO($M$) Kondo problem, with the asymptotic high and low energy theories known from bosonization and conformal field theory studies.",
"Here we complement these approaches by analyzing the Bethe ansatz equations describing the exact solution of these models at all energy scales.",
"We apply our findings to obtain nonperturbative results on the thermodynamics of $M\\rightarrow M-2$ crossovers induced by tunnel couplings between adjacent Majorana bound states."
],
[
"Introduction",
"Majorana fermion states are presently among the most intensively studied objects in condensed matter physics.",
"This is mainly due to the non-Abelian anyon statistics of defects binding Majorana fermions, with promising applications to topological quantum information processing [1], [2], [3], [4], [5], [6].",
"Majorana fermions are zero energy bound states, pairs of which form “topological qubits\" encoding ordinary fermion degrees of freedom in a non-local manner, see Refs.",
"[7], [8], [9] for recent reviews.",
"These exotic objects are predicted to arise in heterostructures combining simple $s$ -wave superconductors and materials with strong spin-orbit coupling, and first experimental results on potential realizations based on semiconductor nanowires [10], [11], [12], [13], [14] or topological insulators [15], [16] are under current investigation.",
"Much of the work aimed at detecting [17], [18], [19], [20], [21], [22] and manipulating [23], [24], [25], [26], [27], [28] Majorana fermions has so far been concerned with effectively noninteracting physics.",
"However, as we have shown in a series of recent papers [29], [30], [31], [32], [33], [34], Majorana fermions can also be a source of rich strongly correlated physics.",
"Such a setup is realized by coupling a “floating” mesoscopic superconducting island — which has a finite charging energy $E_c$ — to normal-conducting lead electrodes via $M>2$ Majorana modes.",
"The device setup is sketched in Fig.",
"REF and, in the parameter regime discussed below, implies the “topological Kondo effect” [29].",
"The Majorana fermions residing on the island thereby represent a quantum “impurity” which, at low energy scales, becomes massively entangled with the conduction electrons in the attached leads through exchange-type processes.",
"At low temperatures, this system is predicted to display exotic non-Fermi liquid correlations, similar to but different from the well-known overscreened multi-channel Kondo effect [35], [36], [37], [38].",
"While achieving such non-Fermi liquid correlations in conventional Kondo devices is usually hindered by the competition of various couplings and, in particular, by the fact that channel anisotropy is a relevant perturbation destroying the non-Fermi liquid fixed point [39], the couplings acting against the topological Kondo effect can be eliminated to exponential accuracy simply by ensuring that adjacent Majorana fermions are sufficiently far apart.",
"We note in passing that related physics has also been predicted in junctions of transverse Ising spin chains [40], [41], [42].",
"However, despite of superficial similarities, the topological Kondo effect found in the Majorana device in Fig.",
"REF is substantially different and always characterized by non-Fermi liquid behavior.",
"Figure: Schematic device setup for the topological Kondo effect with M=6M=6 Majorana fermions γ j \\gamma _j coupled to normal-conducting leads.",
"The Majorana bound states are realized as end states of spin-orbit coupled nanowires on a floating superconducting island with charging energy E c E_c.",
"The tunnel couplings h j<k h_{j<k} between pairs of Majoranas act like a Zeeman field on the “Majorana spin” with components iγ j γ k i\\gamma _j\\gamma _k.The source of the topological Kondo effect is the combination of the island's charging energy, $E_c$ , and the presence of the topological qubit degrees of freedom associated to the Majoranas.",
"The former ensures that the island has a definite number of electrons in its ground state, while the latter transforms the island into an effective non-local “Majorana spin” at energies much below both $E_c$ and the gap to non-Majorana (quasi-particle) excitations.",
"In this work we only discuss the physics on those low energy scales.",
"When conduction electrons in a given lead electrode (labeled by the index $j$ ) have a finite tunnel coupling, $t_j$ , to this non-local Majorana spin via the Majorana fermion $\\gamma _j$ , the screening correlations ultimately responsible for non-Fermi liquid behavior arise.",
"As the $M$ Majorana fermions are described by operators $\\gamma _j=\\gamma _j^\\dagger $ subject to the Clifford algebra $\\lbrace \\gamma _j,\\gamma _k\\rbrace = 2\\delta _{jk}$ [7], [8], [9], the relevant symmetry group for this Kondo problem is SO($M$ ).",
"Indeed, the $\\lbrace \\gamma _j \\rbrace $ compose a spinor representation of the SO($M$ ) group, where the $M(M-1)/2$ different products $i\\gamma _j\\gamma _k$ represent the different components of the “bare” (i.e., uncoupled to leads) Majorana spin [33].",
"The Kondo limit is realized when the charging energy $E_c \\gg {\\rm max}(t_j^2/v)$ ; in what follows we set the Fermi velocity $v=1$ and use units with $\\hbar =k_B=1$ .",
"The effective Hamiltonian describing the system at low energy scales, i.e., below $E_c$ and the energy of non-Majorana excitations, is [29], [30], [31], [32] $ H &=& -i\\sum _{j=1}^M \\int _{-\\infty }^\\infty dx \\ \\psi _j^\\dagger (x)\\partial _x \\psi _j^{}(x) \\\\ &+&\\nonumber \\sum _{j\\ne k} \\lambda _{jk}\\gamma _{j} \\gamma _{k} \\psi _{k}^{\\dagger }(0) \\psi ^{}_{j}(0)+ i\\sum _{j\\ne k} h_{jk}\\gamma _j\\gamma _k.$ Here $\\psi _{j}(x)$ is an effectively spinless right-moving fermion field describing the $j$ th lead, where we unfolded from the physical lead coordinates $x<0$ to the full line; $x=0$ is the coordinate of the tunnel contact.",
"The symmetric matrix $\\lambda _{jk}\\approx t_j t_k/E_c >0$ is the analogue of the exchange coupling in the usual Kondo problem.",
"The non-local couplings $h_{jk}=-h_{kj}$ correspond to overlaps between Majorana bound states.",
"While they can in principle be suppressed to exponential accuracy by separating the Majoranas from each other, we include them here for two reasons.",
"First, instead of suppressing these couplings, they can be deliberately switched on (by applying gate voltages) and used as handles to probe the physics of the Kondo screened non-local SO($M$ ) spin [33].",
"Second, their inclusion also allows one to study the eventual fate of the non-Fermi liquid physics at the lowest energy scales [33], [34].",
"Note that in the Kondo language, the couplings $h_{jk}$ act like Zeeman fields.",
"We mention in passing that the $M$ Majoranas appearing in Eq.",
"(REF ) might only be a subset of all Majorana bound states on the island.",
"Regarding the complementary Majoranas which are not coupled to any lead electrode, we assume that these have no direct tunnel couplings with the $\\gamma _j$ .",
"In the absence of the Zeeman field, the weak-coupling renormalization group (RG) analysis for Eq.",
"(REF ) indicates a flow of the exchange couplings towards a strong-coupling isotropic limit [29], [30], [31], [32].",
"Taking all exchange couplings equal, $\\lambda _{j\\ne k} = \\bar{\\lambda }$ , the perturbative RG approach turns out to break down on energy scales below the Kondo temperature $T_K\\simeq E_c \\exp \\left(- \\frac{\\pi }{(M-2) \\bar{\\lambda }}\\right).$ For temperatures $T\\ll T_K$ , one enters the non-Fermi liquid regime of the topological Kondo effect corresponding to an SO$_2(M)$ Wess-Zumino-Novikov-Witten boundary conformal field theory (BCFT) [38], [43].",
"Crucially, anisotropy in the $\\lambda _{jk}$ is an irrelevant perturbation around this fixed point, in contrast to conventional overscreened multi-channel Kondo fixed points.",
"However, anisotropy is likely to break integrability for $M>4$ .",
"The asymptotic low and high energy physics of the topological Kondo effect has been studied in Refs.",
"[29], [30], [31], [32] for $h_{jk}=0$ , while the effects of the Zeeman field components $h_{jk}$ have been addressed in Ref. [33].",
"For the $M=3$ case, Ref.",
"[34] has explored the full crossover from high to low energies, with and without Zeeman fields, using the numerical renormalization group technique.",
"Here we complement those previous studies by providing analytical results obtained from exact Bethe ansatz (BA) calculations.",
"Note that BA results for related spin chain junctions can be found in Ref. [42].",
"Before turning to the detailed BA solution of the topological Kondo model (REF ), we first summarize the main results of this paper."
],
[
"Summary of results",
"In principle, the BA solution allows one to obtain very detailed information about the system, including the entire energy spectrum (for finite-length leads).",
"This can provide access to the full thermodynamic information for arbitrary temperature and Zeeman field.",
"The BA solution certainly exists for $M=3$ , where Eq.",
"(REF ) is equivalent to the four-channel Kondo model with spin $S=1/2$ [44].",
"For general $M$ , we make a plausible conjecture for the BA and justify it by running various checks.",
"In particular, we verify that the BA reproduces the results obtained by means of BCFT, including the ground-state impurity entropy.",
"As a concrete application, we will discuss the thermodynamic response of the topological Kondo system to a Zeeman field by computing the “magnetization”, i.e., the expectation value of the Majorana spin, $ {\\cal M}_{jk}=\\langle i\\gamma _j\\gamma _k\\rangle ,$ for which a measurement scheme has been proposed in Ref. [33].",
"In particular, we show that in the presence of just one Zeeman field component, say $h_{12}> 0$ , a crossover between a Kondo problem with SO($M)$ symmetry to another one with the symmetry group SO($M-2$ ) is induced upon lowering the temperature.",
"For $M\\le 4$ , instead of another Kondo fixed point, the $M\\rightarrow M-2$ crossover terminates at a Fermi liquid state [45].",
"The BA solution discussed below allows one to address this crossover in a nonperturbative fashion.",
"This is an attractive feature since the Zeeman field is an RG-relevant perturbation destabilizing the topological Kondo fixed point on temperatures below the temperature scale $T_h \\simeq T_K ({h_{12}}/T_K)^{M/2} , $ which follows from simple dimensional scaling arguments [33].",
"Effectively, for $T\\ll T_h$ , the Majorana modes $\\gamma _1$ and $\\gamma _2$ coupled by the Zeeman field move to finite energy and thus disappear from the low-energy theory.",
"The remaining $M-2$ Majoranas then realize a non-Fermi liquid SO$_2(M-2)$ Kondo fixed point (assuming $M>4$ ).",
"In this flow, the Kondo temperature $T_K$ represents the high-energy cutoff, while the “effective\" Kondo temperature for the emergent low-temperature fixed point of SO$_2(M-2$ ) symmetry is set by $T_K^{(M-2)}=T_h$ .",
"Our BA solution, explicitly constructed for $M=3$ up to $M=6$ below, nicely confirms this intuitive picture.",
"For each of the considered $M$ , we also compute the ground-state impurity entropy $S_{\\rm imp}$ .",
"In the absence of the Zeeman field, we find that a BCFT calculation yields $S_{\\rm imp}=\\ln d_M,\\quad d_M=\\left\\lbrace \\begin{array}{ll} \\sqrt{M},& M\\ \\textrm {odd},\\\\ \\sqrt{M/2}, & M\\ \\textrm {even}\\end{array} \\right.$ The quantum dimension $d_M$ thus strongly responds to the parity of $M$ .",
"This BCFT result follows from arguments very similar to those in Ref.",
"[46], and the perfect agreement with our BA results provides a consistency check for the latter.",
"Let us start with the case $M=3$ , where the model (REF ) maps [44] to the four-channel SU(2) Kondo problem with impurity spin $S=1/2$ .",
"This correspondence allows us to use BA results obtained for the latter model [35], [37].",
"In particular, the ground-state magnetization ${\\cal M}_{12}$ can be expressed as scaling function of the variable $Y=h_{12}/\\kappa T_K$ , where $\\kappa =\\frac{64\\pi }{e^2}\\simeq 27.21$ and $T_K$ is given by Eq.",
"(REF ).",
"For $Y\\ll 1$ and $T=0$ , we find $ {\\cal M}_{12} \\simeq 0.868 \\sqrt{Y} + 0.034 Y \\ln Y + O\\left( Y^3 \\right).$ The case $M=3$ is the one where the singularities in the thermodynamic quantities and correlation functions are the strongest.",
"Indeed, the susceptibility, $\\chi _{12}=\\partial {\\cal M}_{12}/\\partial h_{12}$ , has a $Y^{-1/2}$ singularity for $Y\\rightarrow 0$ .",
"The $M\\rightarrow M-2$ crossover picture here turns out to be consistent with a $T=0$ fixed point describing a conventional Coulomb blockade regime, where the island ultimately decouples from the environment.",
"For $M > 4$ , the $M\\rightarrow M-2$ crossover induced by lowering the temperature instead terminates in another Kondo fixed point with symmetry group SO$(M-2$ ).",
"In this case, the breakdown of perturbation theory at low temperatures $T\\ll T_h$ signals in a weak singularity.",
"For instance, for $M=6$ , we obtain a logarithmically divergent second derivative of ${\\cal M}_{12}$ , $\\frac{\\partial ^2{\\cal M}_{12}}{\\partial h^2_{12}} \\sim T_K^{-2}\\ln (T_K/h_{12}).",
"$ To observe the flow $M \\rightarrow M-2$ we turn on an additional weak field component $h_{34}\\ll h_{12}$ .",
"Then at $h_{34} \\ll T_{h}$ , already the first derivative of the corresponding magnetization component diverges, $\\frac{\\partial {\\cal M}_{34}}{\\partial h_{34}} \\sim T_h^{-1}\\ln (T_h/h_{34}),$ in a manner well known for the two-channel SU$(2)$ Kondo model [35], [37].",
"This behavior can be understood from the equivalence of the SO(4) and two-channel SU$(2)$ Kondo fixed points [30], which is also recovered from the structure of the BA equations.",
"The rest of the paper is organized as follows.",
"We begin by discussing the problem for $M=3$ and $M=6$ , including a study of the $M\\rightarrow M-2$ crossover in the latter case.",
"In these cases, as well as for $M=4$ , the BA analysis is aided by the equivalence relations ${\\rm SO}(3)\\sim {\\rm SU}(2), \\quad {\\rm SO}(6)\\sim {\\rm SU}(4), \\quad {\\rm SO}(4)\\sim {\\rm SU}(2)\\times {\\rm SU}(2),$ which establish links to previously studied SU($N$ ) Kondo models [35], [36], [37], [47].",
"We will then combine our $M=6$ equations with general results, linking the group algebra to the structure of the BA equations, to suggest a generalization for arbitrary even $M$ .",
"We also propose the corresponding equations for odd $M>3$ , and discuss the $M=5\\rightarrow 3$ crossover.",
"This case (unlike $M=3,4,6$ ) does not correspond to impurity models previously studied in the SU($N$ ) Kondo context."
],
[
"Bethe ansatz solution",
"The strategy pursued below is as follows.",
"The general classification scheme of possible BA equations for models with Lie group symmetry is known.",
"This allows us to address the problem with equal coupling constants $\\lambda _{jk} = \\bar{\\lambda }$ ; we thus assume isotropic exchange couplings unless stated otherwise.",
"We consider finite length $L$ of the leads with periodic boundary conditions, and take the thermodynamic limit afterwards.",
"According to Refs.",
"[48] and [49], the general form of the BA equations is then dictated by the Dynkin diagram of the corresponding group.",
"The next step is to determine the position of the so-called driving terms and their precise form.",
"These are determined by representations of the bulk Hamiltonian and the impurity spin.",
"We also check our BA equations against previously known results."
],
[
"Case $M=3$",
"For $M=3$ , the model (REF ) is equivalent to the four-channel SU(2) Kondo problem with impurity spin $S=1/2$ [44].",
"To see this equivalence, let us introduce the operators $J_j = \\frac{i}{4}\\varepsilon _{jkl}\\gamma _k\\gamma _l$ (with $j=1,2,3=x,y,z$ and summation convention), which are equivalent to the components of a spin $S=1/2$ operator [50].",
"Then the exchange interaction in Eq.",
"(REF ) reads $H_K &=& 4i\\lambda _{12} J_x\\psi ^\\dagger _1(0)\\psi _2(0) +4i \\lambda _{23} J_y\\psi ^\\dagger _2(0)\\psi _3(0) \\nonumber \\\\&+& 4i \\lambda _{13} J_z \\psi ^\\dagger _3(0)\\psi _1(0) + \\textrm {h.c.}$ Expanding the bulk fermions in their real and imaginary Majorana components, $\\psi _j(x) = \\chi _j(x) + i\\xi _j(x)$ , we get $i\\varepsilon _{jkl}(\\psi ^\\dagger _k\\psi _l - \\psi ^\\dagger _l\\psi _k) =i\\varepsilon _{jkl}(\\chi _k\\chi _l + \\xi _l\\xi _k)$ , i.e., the sum of two SO$_1(3)$ currents equals the SU$_4$ (2) current.",
"The result is the anisotropic spin $S=1/2$ four-channel Kondo model, which is integrable.",
"The BA solution for this model has been thoroughly studied [35], [37], and thereby applies also to the SO(3) topological Kondo effect.",
"In particular one finds Eq.",
"(REF ) for the magnetization, and $S_{\\rm imp}=\\ln \\sqrt{3}$ in the absence of the Zeeman field."
],
[
"Case $M=6$",
"Next we discuss the case $M=6$ .",
"We demonstrate that a Zeeman field coupling just one pair of Majoranas drives the system from $M=6$ to $M=4$ , which in turn is equivalent to the SU$_2$ (2) Kondo effect [30].",
"We restrict our analysis to Zeeman fields $h_{jk}$ that couple to commuting pairs of Majoranas forming a Cartan subalgebra.",
"These pairs could be chosen arbitrarily but should not overlap.",
"For $M$ wires, we then have $[M/2]$ pairs ($[x]$ is the integer part of $x$ ).",
"For $M=6$ , the non-vanishing Zeeman couplings are taken as $H_1=h_{12},\\quad H_2=h_{34}, \\quad H_3=h_{56}.´$ For $M=6$ , the BA equations have the same general form as for the SU$_2$ (4) Kondo model because the corresponding Dynkin diagrams coincide.",
"Since creation and annihilation operators of the bulk fermions transform according to the vector representation of the O(6) group, this representation is isomorphic to the representation of the SU(4) group, where the Young tableaux have one column containing two boxes.",
"The suggested BA equations are, with rapidities $x_a^{(j)}$ , given by $&& e_1(x_a^{(1)} -1/\\bar{\\lambda })\\prod _{b=1}^{M_2}e_{1}(x_a^{(1)} -x_b^{(2)}) =\\prod _{b=1}^{M_1}e_2(x_a^{(1)} -x_b^{(1)}),\\\\ \\nonumber && \\left[e_2(x_a^{(2)})\\right]^N \\prod _{b=1}^{M_2}e_{1}(x_a^{(2)} -x_b^{(1)})\\prod _{b=1}^{M_3} e_{1} (x_a^{(2)} -x_b^{(3)} ) =\\prod _{b=1}^{M_2}e_2(x_a^{(2)} -x_b^{(2)}), \\\\ \\nonumber && \\prod _{b=1}^{M_2}e_{1}(x_a^{(3)} -x_b^{(2)}) =\\prod _{b=1}^{M_3}e_2 (x_a^{(3)} -x_b^{(3)}),$ where the energy follows as $E = \\sum _{a=1}^{M_2} \\frac{1}{2i}\\ln e_2(x_a^{(2)}),\\quad e_n(x) = \\frac{x- in/2}{x+in/2}.$ Here $N$ is the number of particles in the Fermi sea, and $M_{1,2,3}$ are integer numbers equal to linear combinations of eigenvalues of the Cartan operators of the group [49].",
"The driving term for the bulk is located in the second equation, as it fits the vector representation of the O(6) group.",
"Since the impurity spin is in the spinor representation, its driving term is in the first equation.",
"As a first step in the derivation of the thermodynamic Bethe ansatz (TBA) equations, we then classify solutions of the bare BA equations (REF ).",
"These solutions are complex, but in the thermodynamic limit, $L \\rightarrow \\infty $ with $N/L$ and $M_{1,2,3}/L$ finite, their imaginary parts are simple: They group into clusters with common real part $X_{\\alpha }^{(j,n)}$ , the so-called 'strings', where the rapidities are given by $x_{n,p;\\alpha }^{(j)} = X_{\\alpha }^{(j,n)} +\\frac{i}{2}(n+1-2p) + O\\left(e^{-c_0L}\\right),$ with $n=1,2,\\ldots $ , $p = 1,\\ldots ,n$ , and $c_0>0$ .",
"As next step, we introduce distribution functions for rapidities of string centers, $\\rho _n^{(j)}(x)$ , and unoccupied spaces, $\\tilde{\\rho }_n^{(j)}(x)$ .",
"The discrete equations (REF ) are thereby transformed into integral equations relating $\\tilde{\\rho }$ and $\\rho $ , $&& \\tilde{\\rho }_n^{(j)} + A_{nm}*C^{jl}*\\rho _m^{(l)} = A_{n,2}*s(x) \\delta ^{j,2} +\\frac{1}{N}a_n(x-1/\\bar{\\lambda })\\delta ^{j,1}, $ where $n = 1,2,\\ldots $ , $j=1,2,3$ , we use the summation convention, convolutions are denoted by a star, i.e., $f*g(x) = \\int dy f(x-y)g(y)$ , and $A_{nm}(\\omega ) &=& \\coth (|\\omega |/2) \\left( e^{-|n-m||\\omega |/2} -e^{-(n+m)|\\omega |/2}\\right), \\nonumber \\\\C_{nm}(\\omega ) &=& \\delta _{nm} - s(\\omega ) \\left(\\delta _{n,m-1} +\\delta _{n,m+1}\\right), \\nonumber \\\\s(\\omega ) &=& \\frac{1}{2\\cosh (\\omega /2)}, \\quad a_n(\\omega ) = e^{-n|\\omega |/2}.",
"$ The TBA equations now follow by minimization of the generalized free energy, $F=E-TS$ , subject to the constraints imposed by Eq.",
"(REF ) and with the entropy $\\nonumber S &=& N\\sum _{n=0}^{\\infty }\\sum _j\\int d x\\Big [ (\\rho _n^{(j)} + \\tilde{\\rho }_n^{(j)})\\ln (\\rho _n^{(j)} +\\tilde{\\rho }_n^{(j)}) \\\\&& - \\rho _n^{(j)}\\ln \\rho _n^{(j)} - \\tilde{\\rho }_n^{(j)}\\ln \\tilde{\\rho }_n^{(j)}\\Big ].$ The TBA equations determine ratios of the distribution functions, which are collected in $\\phi _n^{(j)}$ functions according to $\\tilde{\\rho }_n^{(j)}(x)/\\rho _n^{(j)}(x) = e^{-\\phi _n^{(j)}(x)}.$ For $M=6$ , the TBA equations in the scaling limit coincide with those of the SU$_2(4)$ Kondo (or Coqblin-Schrieffer) model, with the impurity in the fundamental (single box) representation [47], $&& F_{\\rm imp} = - T\\sum _{j=1}^3\\int dx f_j \\left[x + \\frac{\\ln (T_K/T)}{\\pi }\\right]\\ln \\left(1 +e^{\\phi _1^{(j)}(x)}\\right), \\nonumber \\\\&& \\ln \\left(1 + e^{-\\phi _n^{(j)}}\\right) - {\\cal A}_{jl}*C_{nm}*\\ln \\left(1 +e^{\\phi _m^{(l)}}\\right)= \\\\ \\nonumber &&= \\delta _{n,2}\\sin (\\pi j/4) e^{-\\pi x/2} ,$ where $j,l=1,2,3$ and $n,m=1,2,\\ldots $ .",
"The Zeeman fields $H_j$ in Eq.",
"(REF ) enter through the constraint $\\lim _{n\\rightarrow \\infty } \\frac{\\phi _n^{(j)}}{n} = H_j/T,$ and the Fourier transforms of the above kernels are given by $&& f_j(\\omega ) = \\frac{\\sinh [(2-j/2)\\omega ]}{\\sinh (2\\omega )}, \\quad {\\cal A}_{jl} = [C^{-1}]_{jl} = \\nonumber \\\\&& =2\\coth (\\omega /2)\\frac{\\sinh [(2 - \\mbox{max}(j,l)/2)\\omega ]\\sinh [\\mbox{min}(j,l)\\omega /2]}{\\sinh (2\\omega )}.$ The ground-state impurity entropy is then determined by the asymptotics of $\\phi _1^{(j)}(-\\infty )$ .",
"In the absence of the Zeeman field, the solution for the general SU$_k(N)$ case is $1+e^{\\phi _n^{(j)}(-\\infty )} = \\frac{\\sin \\left[\\frac{\\pi (n+ N-j)}{k+N}\\right]\\sin \\left[\\frac{\\pi (n+j)}{k+N}\\right]}{\\sin \\left[\\frac{\\pi (N-j)}{k+N}\\right]\\sin \\left[\\frac{\\pi j}{k+N}\\right]}.$ Substituting this into the above equations and putting $n=1,$ $N=4,$ and $k=2$ , as is appropriate for our SO$_2(6$ ) problem [see Eq.",
"(REF )], we obtain $S_{\\rm imp} = \\ln \\sqrt{3}$ in accordance with Eq.",
"(REF ).",
"Below we consider the thermodynamics at $T=0$ , such that the equations for the ground-state root densities suffice.",
"All roots of the BA equations are in $n=2$ strings of different 'colors', $j =1,2,3$ .",
"In that case, Eqs.",
"(REF ) are reduced to a set of Wiener-Hopf equations, $&& s(x)\\delta ^{j,2} +\\frac{\\delta ^{j,1}}{N}[s*s](x-1/\\bar{\\lambda })= [A_{2,2}]^{-1}\\tilde{\\rho }^{(2)}_{2}(x) +C_{jl}*\\rho ^{(l)}_{2}(x).",
"$ Let us then isolate the terms proportional to $1/N$ , which are associated with the impurity, $\\rho = \\rho _{b} + \\frac{1}{N}\\rho _{\\rm imp},$ where $\\rho =\\rho _2$ or $\\tilde{\\rho }_2$ .",
"In the ground state, the $j$ th-order roots fill the interval $(-\\infty , B_j)$ , where the limits $B_j$ are determined by the Zeeman fields $H_j$ in Eq.",
"(REF ), $\\chi H_j = \\int _{B_j}^{\\infty } d x [\\tilde{\\rho }^{(j)}(x)]_b,$ with the bulk susceptibility $\\chi =1/(2\\pi )$ (we use $v=1$ ).",
"To proceed from this point on, we need to specify the precise Zeeman field configuration."
],
[
"All $B_j$ equal.",
"With Eq.",
"(REF ), we consider the Zeeman field configuration with $H_j = h_0\\sin (\\pi j/4),\\quad j=1,2,3,$ where all $B_j$ are equal.",
"In this case, Eqs.",
"(REF ) describe the vicinity of the SU$_2$ (4) fixed point for small $h_0$ .",
"The system of Wiener-Hopf equations (REF ) with equal limits of integration is then solved by $&& \\rho ^{(j),+}(\\omega =0) =\\frac{1}{16\\pi i}\\sum _{l=1}^3\\sin (\\pi jl/4) \\\\ \\nonumber &&\\times \\int \\frac{d\\omega }{\\omega -i 0^+}\\frac{f_l^{(-)}(\\omega ) \\exp \\left[\\frac{2i\\omega }{\\pi }\\ln \\left(h_0/ \\left[f_1^{(-)}(-i\\pi /2) T_K\\right]\\right)\\right]}{2\\cosh (\\omega /2)[\\cosh (\\omega /2)-\\cos (\\pi l/4)]},\\\\&& f_l^{(-)}(\\omega ) =\\frac{ \\left( \\nonumber \\frac{i\\omega +0^+}{\\pi e}\\right)^{i\\omega /\\pi }\\left[\\left(\\frac{\\omega -i0^+}{4\\pi }\\right)^2 +(l/8)^2\\right]^{1/2}}{\\Gamma (\\frac{1}{2} +i \\frac{\\omega }{2\\pi })\\Gamma (1 - \\frac{l}{8 }+ i\\frac{\\omega }{4\\pi })\\Gamma (1 +\\frac{l}{8} + i\\frac{\\omega }{4\\pi })},$ with the Gamma function $\\Gamma $ .",
"For small Zeeman fields, $h_0 \\ll T_K$ , the result is dominated by the linear term but acquires a non-Fermi liquid correction, $\\rho ^{(j),+}(\\omega =0) = \\sin (\\pi j/4)\\frac{h_0}{2\\pi T_K} + b_j \\frac{h_0^2}{T_K^2} \\ln (T_K/h_0) + \\cdots ,$ where $b_j$ is a numerical coefficient.",
"The second (non-Fermi liquid) term originates from the double pole in the integrand at $\\omega = -i\\pi $ .",
"From here on it is straightforward to obtain Eq.",
"(REF ) for the magnetization, which has been quoted in Sec.",
"."
],
[
"Case $H_1\\gg H_{2,3}$ and {{formula:8d10645f-918f-457f-9ad0-01ccddbe8a26}} flow.",
"Consider next a Zeeman field where the amount of holes (unoccupied spaces) in the $j=2$ equation, see Eq.",
"(REF ), strongly exceeds their amount in the $j=1,3$ equations, such that $B_2 \\ll B_{1,3}$ .",
"This situation is realized when one of the Zeeman field components by far exceeds the others, for instance, $H_{1} \\gg H_{2}, H_{3}$ in Eq.",
"(REF ).",
"The $H_1$ field then generates the temperature scale $T_h$ in Eq.",
"(REF ), below which the physics is expected to be determined by the SO(4) $\\sim $ SU$_2$ (2) Kondo effect.",
"In this limit, we can neglect $\\tilde{\\rho }^{(1,3)}$ and rewrite Eq.",
"(REF ) as $&& K*\\tilde{\\rho }^{(1)} + \\rho ^{(1)} = s*\\rho ^{(2)} + \\frac{1}{N}[s*s](x-1/\\bar{\\lambda }),\\\\&& K*\\tilde{\\rho }^{(3)} + \\rho ^{(3)} = s*\\rho ^{(2)}, \\\\&& \\rho ^{(2)} +K*{\\cal A}_{2,2}*\\tilde{\\rho }^{(2)} ={\\cal A}_{2,2}*s + \\frac{1}{N}{\\cal A}_{2,2}*[s*s](x-1/\\bar{\\lambda }), \\\\ \\nonumber && K(\\omega ) = [A_{2,2}]^{-1}=\\frac{1}{1+ e^{-2|\\omega |}}.$ The densities $\\tilde{\\rho }^{(3)}$ and $\\rho ^{(3)}$ do not contribute to the impurity thermodynamics.",
"In the scaling limit, we have to keep only the asymptotics of the bulk driving term, and the explicit form of Eq.",
"() is $&& \\rho ^{(2)}(x) + \\int _{B_2}^{\\infty } d y K(x-y)\\tilde{\\rho }^{(2)}(y) =\\nonumber \\\\ && =\\frac{1}{\\sqrt{2}}e^{-\\pi x/2} + \\frac{1}{N} \\frac{1}{2\\cosh [\\pi (x- 1/\\bar{\\lambda })/2]}.",
"$ Equations (REF ) and () determine the magnetization components ${\\cal M}_{jk}$ with $(j,k)\\ne (1,2)$ , which are not directly affected by the large Zeeman field $H_1=h_{12}$ .",
"Since $B_2 \\ll B_{1,3}$ , one can approximate $s*\\rho ^{(2)}$ by the asymptotic expression $\\nonumber && s*\\rho ^{(2)} \\approx (A + \\frac{1}{N}A^{\\prime }) e^{-\\pi x}, \\\\&& A + \\frac{1}{N}A^{\\prime }= \\int _{-\\infty }^{B_2} d y e^{\\pi y}\\rho ^{(2)}(y).$ Then Eq.",
"(REF ) coincides with the equation for the ground-state root density of the SU$_2$ (2) Kondo model, and Eq.",
"() coincides with the equation for the SU$_1$ (2) Kondo model in the Fermi liquid limit.",
"Indeed, the impurity part $\\sim 1/N$ of Eq.",
"(REF ) is also $\\sim e^{- \\pi x}$ , and we have $&& K*\\tilde{\\rho }^{(1)} + \\rho ^{(1)} = Ae^{-\\pi x} + \\frac{1}{N}[s*s](x-1/\\bar{\\lambda }),\\\\&& K*\\tilde{\\rho }^{(3)} + \\rho ^{(3)} = A e^{-\\pi x} +\\frac{1}{N}A^{\\prime } e^{-\\pi x}.",
"$ Eliminating the prefactor $A$ in Eq.",
"(REF ) by a shift of $x$ , we bring Eq.",
"(REF ) to the canonical form for the SU$_2$ (2) model, with the renormalized Kondo temperature for the emergent $M=4$ Kondo model, $T_K^{(4)} = A^{-1}e^{-\\pi /\\bar{\\lambda }}.",
"$ The factor $A$ can now be extracted from the solution of the Wiener-Hopf equation (REF ), $&& \\rho ^{(+)}_2(\\omega ) = \\frac{2}{\\pi G^{(-)}(-i\\pi /2)G^{(+)}(\\omega )}e^{-\\pi B_2/2} + \\\\&& \\frac{1}{N}\\frac{i}{2\\pi G^{(+)}(\\omega )}\\int \\frac{d \\omega ^{\\prime }}{\\omega ^{\\prime }-\\omega +i 0^+}\\frac{e^{-i\\omega ^{\\prime }(B_2-1/\\bar{\\lambda })}}{2\\cosh (\\omega ^{\\prime }) G^{(-)}(\\omega ^{\\prime })}, \\nonumber \\\\ \\nonumber && G^{(+)}(-\\omega ) = G^{(-)}(\\omega ) =\\frac{\\Gamma (1/2 + i\\omega /\\pi )}{\\sqrt{\\pi }}\\left(\\frac{i\\omega - 0^+}{\\pi e}\\right)^{-i\\omega /\\pi }.$ The bulk part of this expression yields the bulk magnetic moment $\\chi H_1$ , and thus determines the value of $B_2$ .",
"In fact, we find $\\chi H_1 =\\frac{ \\sqrt{8} e^{-\\pi B_2/2}}{\\pi G^{(-)}(-i\\pi /2)}.$ Substituting this into Eq.",
"(REF ), we get $A = (\\chi H_1)^3/\\pi $ .",
"We now plug this result back into Eq.",
"(REF ) and take into account that the $M=6$ Kondo temperature $T_K^{(6)}$ (defined for $h_{jk}=0$ ) is given by Eq.",
"(REF ).",
"With an extra factor two in the exponent because of the different normalization of SO(6) and SU(4) generators, the $M=6$ Kondo temperature here reads $T_K^{(6)} = \\chi ^{-1}\\exp \\left(-\\frac{\\pi }{2\\bar{\\lambda }}\\right),$ and therefore we finally get the Kondo temperature of the effective low-energy SO(4) model, realized at $T\\ll T_h$ , in the form $T_K^{(4)} = \\frac{1}{\\pi } T_K^{(6)}\\left(H_1/T_K^{(6)}\\right)^3.$ Up to a prefactor of order unity, this scale coincides with the crossover scale $T_h$ in Eq.",
"(REF ).",
"We have thus shown that $T_h$ acts as the Kondo temperature for the emergent SO($M-2)$ topological Kondo effect.",
"The well-known result for the magnetization of the two-channel Kondo model [35], $\\langle S^z\\rangle \\sim \\frac{H_1}{2\\pi T_K}\\ln (T_K/H_1),$ then dictates the $T=0$ magnetization behavior announced above in Eq.",
"(REF ).",
"Moreover, the results of Ref.",
"[37] yield the ground-state impurity entropy for the SU$_2$ (2) case corresponding to $M=4$ , $S_{\\rm imp} = \\ln \\sqrt{2}$ , which is again in agreement with Eq.",
"(REF ).",
"Now that we have successfully run the checks for $M=6$ , we can write down the BA equations for arbitrary even $M$ .",
"Putting $M=2K$ , they read $&& e_1(x_a^{(K-1)} -1/\\bar{\\lambda })\\prod _{b=1}^{M_{K-2}}e_{1}(x_a^{(K-1)} -x_b^{(K-2)}) \\nonumber =\\\\ &&\\prod _{b=1}^{M_{K-1}}e_2(x_a^{(K-1)} -x_b^{(K-1)}),\\nonumber \\\\&& \\prod _{b=1}^{M_{K-1}}e_{1}(x_a^{(K-2)} -x_b^{(K-1)})\\prod _{b=1}^{M_K}e_{1}(x_a^{(K-2)} -x_b^{(K)})\\\\\\nonumber && \\prod _{b=1}^{M_{K-3}}e_1(x_a^{(K-2)} -x_b^{(K-3)}) = \\prod _{b=1}^{M_{K-2}}e_2(x_a^{(K-2)} -x_b^{(K-2)}), \\\\ \\nonumber && \\prod _{b=1}^{M_{p-1}}e_1(x_a^{(p)} -x_b^{(p-1)})\\prod _{b=1}^{M_{p+1}}e_1(x_a^{(p)} -x_b^{(p+1)})\\\\ \\nonumber &&= \\prod _{b=1}^{M_p}e_2(x_a^{(p)} - x_b^{(p)}), \\quad p=2,\\ldots ,K-1,\\\\&& [e_2(x_a^{(1)})]^N\\prod _{b=1}^{M_{2}}e_1(x_a^{(1)} -x_b^{(2)}) =\\prod _{b=1}^{M_{1}}e_2(x_a^{(1)} -x_b^{(1)}), \\nonumber \\\\&& \\prod _{b=1}^{M_{K-1}}e_{1}(x_a^{(K)} -x_b^{(K-1)}) =\\prod _{b=1}^{M_K}e_2(x_a^{(K)} -x_b^{(K)}), \\nonumber \\\\&& E = \\sum _{a=1}^{M_1} \\frac{1}{2i}\\ln e_2(x_a^{(1)}).$ With these equations, see also Ref.",
"[42], one can obtain thermodynamic observables in an exact manner for arbitrary even $M$ ."
],
[
"Odd $M$ . Detailed description of {{formula:1e71ca11-4d9c-434b-adb3-e8639d402595}}",
"For the SO($M=2K+1$ ) group, the BA equations (up to the driving terms) can be extracted, for instance, from Ref. [49].",
"The positions of the bulk and the impurity driving terms are determined by the same logic as before, that is by representation theory considerations and the $M \\rightarrow M-2$ flow.",
"For the SO$_2$ ($2K+1$ ) model, we suggest the following bare BA equations, see also Ref.",
"[42], $&& e_{1/2}(x_a^{(K)} -1/\\bar{\\lambda })\\prod _{b=1}^{M_{K-1}}e_1(x_a^{(K)} - x_b^{(K-1)}) = \\prod _{b=1}^{M_K}e_1(x_a^{(K)} - x_b^{(K)}),\\nonumber \\\\\\nonumber &&\\prod _{b=1}^{M_{p-1}}e_1(x_a^{(p)} -x_b^{(p-1)})\\prod _{b=1}^{M_{p+1}}e_1(x_a^{(p)} -x_b^{(p+1)})\\\\&&= \\prod _{b=1}^{M_p}e_2(x_a^{(p)} - x_b^{(p)}),\\quad p=2,\\ldots ,K-1,\\nonumber \\\\&& [e_2(x_a^{(1)})]^{N}\\prod _{b=1}^{M_2}e_1(x_a^{(1)} -x_b^{(2)}) = \\prod _{b=1}^{M_1}e_2(x_a^{(1)} - x_b^{(1)}),\\nonumber \\\\&& E = \\frac{1}{2i}\\sum _a \\ln [e_k(x_a^{(1)})].",
"$ The impurity is in the spinor representation, and its driving term is in the first equation.",
"For $M=5$ , we shall see that this is consistent with the flow $M=5 \\rightarrow M=3$ driven by a single-component Zeeman field.",
"To illustrate the case of odd $M$ , we now analyze Eqs.",
"(REF ) for $K=2$ , i.e., for the group SO(5).",
"The corresponding equations for the densities are $&& s*A_{n,2} = \\tilde{\\sigma }_n + A_{nm}*\\sigma _m - [s(\\omega /2)*A_{2n,m}(\\omega /2)]*\\rho _m,\\\\&& \\frac{1}{N}[a_n(\\omega /2)]e^{i\\omega /\\bar{\\lambda }} =\\\\&& -[s(\\omega /2)*A_{n,2m}(\\omega /2)]*\\sigma _m +\\tilde{\\rho }_n + [A_{nm}(\\omega /2)]*\\rho _m.",
"\\nonumber $ From Eq.",
"(), we can then derive the TBA equations.",
"There are two types of energies, where $\\phi _n$ $(\\xi _n$ ) is related to $\\rho _n$ ($\\sigma _n$ ).",
"With $n=0,1,2,\\ldots $ , we obtain $&& \\phi _{2n+1} = s_{1/2}*\\ln \\left[(1+e^{\\phi _{2n}})(1+e^{\\phi _{2n+2}})\\right],\\nonumber \\\\&& \\phi _{2n} = - \\delta _{n,2}e^{-2\\pi x/3} + s_{1/2}*\\ln \\left[(1+e^{\\phi _{2n-1}})(1+e^{\\phi _{2n+1}})\\right] - \\nonumber \\\\&& \\frac{s_{1/2}}{1-s}*\\Big [s_{1/2}C_{2n,m}*\\ln (1+e^{\\phi _m}) +C_{2n,m}*\\ln (1+e^{\\xi _m})\\Big ],\\nonumber \\\\&&-\\ln (1+e^{-\\xi _n}) = -\\frac{C_{nm}}{1-s}*\\ln (1+e^{\\xi _{m}}) -\\nonumber \\\\&& \\frac{s_{1/2}*C_{nm}}{1-s}*\\ln (1+ e^{\\phi _{2m}}) -\\delta _{n,2}e^{-2\\pi x/3}.$ The impurity free energy reads $F_{\\rm imp} &=& - T\\int d x \\Big \\lbrace s_{3/2}[x + \\frac{3}{2\\pi }\\ln (T_K/T)]\\ln (1 + e^{\\phi _2(x)}) \\nonumber \\\\&+& s_{1/2}[x + \\frac{3}{2\\pi }\\ln (T_K/T)]\\ln (1 + e^{\\phi _1(x)}) \\nonumber \\\\&+& s_{1/2}*s_{3/2}[x + \\frac{3}{2\\pi }\\ln (T_K/T)]\\ln (1+ e^{\\xi _1(x)})\\Big \\rbrace ,$ where $s_{n/2} = s(n\\omega /2)$ .",
"We now address the flow $M=5 \\rightarrow M=3$ with just one non-zero Zeeman field component $h_{12}=h_0$ , putting $T=0$ for simplicity.",
"The nonvanishing densities are $\\sigma _{2}$ , $\\tilde{\\sigma }_2$ , $\\rho _{4}$ , and $\\tilde{\\rho }_4$ , with the corresponding equations $&& \\frac{1}{2\\cosh (\\omega /2)} = [A_{2,2}]^{-1}*\\tilde{\\sigma }_2 +\\sigma _2 - \\left[\\frac{\\cosh (\\omega /4)}{\\cosh (\\omega /2)}\\right]*\\rho _{4},\\\\&& \\frac{e^{i\\omega /\\bar{\\lambda }}}{N}\\frac{\\tanh (\\omega /4)}{\\sinh (\\omega )}= \\left[A_{4,4}(\\omega /2)\\right]^{-1}*\\tilde{\\rho }_{4} + \\nonumber \\\\ && + \\rho _{4} - \\left[\\frac{1}{2\\cosh (\\omega /4)}\\right]*\\sigma _{2}.",
"$ In the absence of the Zeeman field, $\\sigma _2,\\rho _{4} \\ne 0$ on the entire real axis, and $\\tilde{\\rho }_{4} = \\tilde{\\sigma }_2 =0$ .",
"In the opposite case of large $h_0$ , however, $\\tilde{\\sigma }_{2}$ is significant and corresponds to the progressive emptiness of $\\sigma _2$ .",
"Now $\\sigma _2(x) \\ne 0$ only at $x<B$ , where $B$ is determined by the Zeeman field.",
"As a result, the asymptotics of $\\rho _{4}(x)$ at $+\\infty $ , which implies the low-energy behavior of the free energy, is determined by Eq. ().",
"Here we can approximate $\\left[\\frac{1}{2\\cosh (\\omega /2)}\\right]*\\sigma _{2} \\approx A e^{-2\\pi x} ,\\quad A =\\int _{B}^{-\\infty }d y e^{2\\pi y}\\sigma _2(y).$ In the end, we find that Eq.",
"() coincides with the equation for the SU$_{4}$ (2) Kondo model with impurity spin $S=1/2$ , corresponding to the SO(3) topological Kondo effect.",
"This once more illustrates the flow $M \\rightarrow M-2$ induced by lowering temperature below $T_h$ .",
"Equation (REF ) also allows one to calculate the ground-state impurity entropy.",
"Solving Eqs.",
"(REF ) for $x\\rightarrow -\\infty $ , we find that $\\xi _2, \\phi _4 \\rightarrow -\\infty $ .",
"As a consequence, the equations for $\\phi _{1,2,3}$ and $\\xi _1$ decouple from the rest.",
"Their solution is given by $e^{\\phi _1} =e^{\\phi _3} = 3/2,\\quad 1+e^{\\phi _2} = e^{2\\phi _1},\\quad e^{\\xi _1} = 1/3.$ Substituting this into Eq.",
"(REF ), we find $S_{\\rm imp} = \\ln \\sqrt{5}$ , in accordance with the result quoted in Sec.",
"."
],
[
"Conclusions",
"To conclude, we have formulated a Bethe ansatz solution for the SO$(M)$ topological Kondo problem realized by a mesoscopic superconducting island coupled to external leads via $M>2$ Majorana fermions.",
"In our previous paper [33], we reported that in this model, the Majorana spin non-locally encoded by the Majorana fermions exhibits rich and observable dynamics characterized by nonvanishing multi-point correlations and nonperturbative crossovers between different non-Fermi liquid Kondo fixed points.",
"The Bethe ansatz results provided in the present work support these conclusions and provide a nonperturbative approach to the model spectrum and its thermodynamics.",
"We thank A.A. Nersesyan and V. Kravtsov for valuable discussions, and acknowledge financial support by the SFB TR12 and the SPP 1666 of the DFG, a Royal Society URF, and the DOE under Contract No.",
"DE-AC02-98CH10886."
]
] | 1403.0113 |
[
[
"Optical and Ultraviolet Observations of the Narrow-Lined Type Ia SN\n 2012fr in NGC 1365"
],
[
"Abstract Extensive optical and ultraviolet (UV) observations of the type Ia supernova (SN Ia) 2012fr are presented in this paper.",
"It has a relatively high luminosity, with an absolute $B$-band peak magnitude of about $-19.5$ mag and a smaller post-maximum decline rate than normal SNe Ia [e.g., $\\Delta m _{15}$($B$) $= 0.85 \\pm 0.05$ mag].",
"Based on the UV and optical light curves, we derived that a $^{56}$Ni mass of about 0.88 solar masses was synthesized in the explosion.",
"The earlier spectra are characterized by noticeable high-velocity features of \\ion{Si}{2} $\\lambda$6355 and \\ion{Ca}{2} with velocities in the range of $\\sim22,000$--$25,000$ km s$^{-1}$.",
"At around the maximum light, these spectral features are dominated by the photospheric components which are noticeably narrower than normal SNe Ia.",
"The post-maximum velocity of the photosphere remains almost constant at $\\sim$12,000 km s$^{-1}$ for about one month, reminiscent of the behavior of some luminous SNe Ia like SN 1991T.",
"We propose that SN 2012fr may represent a subset of the SN 1991T-like SNe Ia viewed in a direction with a clumpy or shell-like structure of ejecta, in terms of a significant level of polarization reported in Maund et al.",
"(2013)."
],
[
"Introduction",
"Type Ia supernovae (SNe Ia) are widely accepted as the results of thermonuclear explosion of accreting carbon-oxygen white dwarf (WD) with a mass close to the Chandrasekhar limit ($\\sim $ 1.4 M$_{\\odot }$ ) in a binary system [29], [63], [39].",
"They play important roles in many aspects of astrophysics, especially in observational cosmology because of their use as distance indicators probing the expansion history of the universe [49], [54], [43].",
"Observationally, most of the SNe Ia show strikingly similar photometric and spectroscopic behavior (i.e., [61], [19]), and the remained scatter can be better understood in terms of an empirical relation between light-curve width and luminosity (i.e., [45]).",
"Nevertheless, there is increasing evidence for the observed diversity that cannot be explained with such a relation.",
"The representative subclasses include: (1) overluminous group like SN 1991T characterized by weak Si2 absorption and prominent iron in the near-maximum light spectra [17], [44]; (2) underluminous events like SN 1991bg that exhibit strong Si2 $\\lambda $ 5972 and $\\sim $ 4000Å Ti features (i.e.,[18], [2]); (3) peculiar objects like SN 2002cx which have extremely low ejecta velocities and luminosities [34]; (4) super-Chandrasekhar mass SNe Ia like SN 2007if [52] and SN 2009dc[56], which are characteristic of extremely high luminosity but relatively low expansion velocity.",
"In addition, there are also reports of circumstellar interaction in some SNe Ia such as SN 2002ic [27] and PTF 11kx [16], though their classifications are still controversial because of bearing similarities to type IIn supernovae.",
"The existence of above peculiar subtypes indicate that there are possibly multiple channels leading to SN Ia explosions.",
"Besides peculiar SN Ia events, specific classification schemes have recently been proposed to highlight the diversity of relatively normal SNe Ia.",
"For example, [2] found that normal SNe Ia can be further subclassified by the temporal velocity gradient of the Si2 $\\lambda $ 6355 line, i.e., the group with a high-velocity gradient (HVG) and the group with a low-velocity gradient (LVG).",
"Based on the equivalent width (EW) of the absorption features of Si2 $\\lambda $ 5972 and Si2 $\\lambda $ 6355, [6], [7] suggested dividing the SN Ia sample into four groups: cool (CL), shallow silicon (SS), core normal (CN), and broad line (BL); the CL and SS groups mainly consist of peculiar objects like SN 1991bg and SN 1991T, respectively.",
"[67] proposed using the expansion velocity of the Si2 $\\lambda $ 6355 line to distinguish the subclass with a higher Si2 velocity (HV) from that with a normal velocity (NV).",
"In spite of different criteria adopted in these classifications, the respective subsamples show some overlap with each other.",
"For example, the FAINT subclass from [2] tend to match the SN 1991bg/CL subclass.",
"The HV subclass overlap with the BL and the HVG ones, as suggested by the Berkeley and CfA spectral datasets of supernova [4], [57].",
"In particular, the HV SNe Ia are found to have redder $$ colors [67] and different locations within host galaxies in comparison to the NV ones [69], suggesting that the properties of their progenitors may be different.",
"SN 2012fr is a type Ia supernova discovered at a relatively young phase in nearby galaxy NGC 1365 [33], [11].",
"Owing to its brightness, extensive follow-up observations were performed in multi-wavebands for this object immediately after the discovery.",
"Childress et al.",
"(2013, hereafter C13) have presented observations of earlier optical spectra for SN 2012fr, showing clear signatures of high-velocity features (HVFs) that are detached from the photospheric components.",
"After comparing with subclasses of SNe Ia defined in different classification schemes, they suggested that SN 2012fr may represent a transitional event between nominal spectroscopic subclasses of SN Ia, with important dissimilarities with the overluminous SN 1991T-like subclass of SNe Ia.",
"[36] presented the spectropolarimetric observations of this object, spanning from $-$ 11 days to +24 days with respect to $B$ -band maximum light.",
"They found that the high-velocity components of the spectral features are highly polarized in the earlier phases but the polarization decreases as these features become weaker.",
"However, the continuum polarization for the SN is always low $<$ 0.1%, suggestive of an overall symmetry of the photosphere for SN 2012fr.",
"In this paper, we present extensive optical and ultraviolet (UV) observations of SN 2012fr.",
"The large dataset of SN 2012fr can help us further understand diversity of SNe Ia, its physical origin, and impact on cosmological applications.",
"The paper is organized as follows.",
"Observations and data reductions are described in Section .",
"Section presents the UV and optical light and color curves, while Section presents the spectral evolution.",
"In Section we constructed the bolometric luminosity of SN 2012fr and discussed its classification.",
"The conclusions are given in Section ."
],
[
"Observations and Data Reduction",
"SN 2012fr was discovered by [33] on 2012 Oct. 27.05 (UT is used throughout this paper) using the 0.25 m robotic telescope TAROT La Silla observatory, Chile.",
"Its coordinates are R.A. = 03h33m35s.99, Decl.",
"= $-36^\\circ $ 0737.7(J2000), located at 3 west and 52 north of the center of the nearby galaxy NGC 1365 (see Figure REF ).",
"The host is a faced-on, Sb-type galaxy with a giant bar in the Fornax cluster, and has been extensively studied because of harboring a Seyfert 1.8 nucleus (see a review by Lindblad 1999).",
"The supernova is located in a faint region that is not far from the bar of the host galaxy (see Figure 1).",
"Figure: SN 2012fr in NGC 1365.",
"Composite color image obtained with the Li-Jiang 2.4-m telescope.The supernova and eight local reference stars are marked.An optical spectrum taken one day after the discovery shows that SN 2012fr was a young SN Ia (Childress et al.",
"2012), with extremely HVFs of Si2 and Ca2.",
"Our observations of this supernova started from 2012 Nov. 04 with the Yunnan Faint Object Spectrograph and Camera (YFOSC) mounted on the Li-Jiang 2.4-m telescope (hereafter LJT) of Yunnan Astronomical Observatories (YNAO), China.",
"The YFOSC observation system is equipped with a 2.1K$\\times $ 4.5K back-illuminated, blue sensitive CCD, which works in both the imaging and long-slit spectroscopic modes (see Zhang et al.",
"2012 for detailed descriptions of the YFOSC).",
"In the imaging mode, the CCD has a field of view of $9.6\\times 9.6$ (corresponding to an angular resolution of $0.28$ per pixel).",
"With this system, we collected a total of 42 photometric datapoints and 20 spectra in the optical.",
"Extensive UV and optical photometry were also obtained with the Ultraviolet/Optical Telescope (UVOT) on the space-based $Swift$ telescope."
],
[
"Optical Observations from Li-Jiang 2.4-m Telescope",
"Our optical photometry for SN 2012fr was obtained in the $BVRI$ bands with the LJT and YFOSC system, covering the period from t = $\\sim $ +11 days to t =$\\sim $ +71 days since the $B$ -band maximum light.",
"All of the CCD images were reduced using the IRAF IRAF, the Image Reduction and Analysis Facility, is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy(AURA), Inc. under cooperative agreement with the National Science Foundation(NSF).",
"standard procedure, including the corrections for bias, flat field, and removal of cosmic rays.",
"As SN 2012fr was very bright and in a faint region of the host galaxy (see Figure 1), we skipped the step of subtracting the galaxy template in the photometry.",
"The standard point-spread function (PSF) fitting method from the IRAF DAOPHOT package [58] was used to measure the instrumental magnitudes for both the SN and the local standard stars.",
"These magnitudes were then converted to the standard Johnson $UBV$ [31] and Kron-Cousins $RI$ [15] systems through transformations established by observing [37] standards during photometric nights.",
"The average value of the photometric zeropoints, determined on four photometric nights, was used to calibrate the local standard stars in the field of SN 2012fr.",
"Table REF lists the standard $BVRI$ magnitudes and the corresponding uncertainties of 8 local standard stars labeled in Figure REF .",
"The magnitudes of these stars are then used to transform the instrumental magnitudes of SN 2012fr to those of the standard $BVRI$ system, and the final results of the photometry from the LJT are listed in Table REF .",
"ccccc[!th] Magnitudes of the Photometric Standards in the Field of SN 2012fr 0pt Star $B$ (mag) $V$ (mag) $R$ (mag) $I$ (mag) 1 15.59(04) 15.04(03) 14.61(02) 14.24(03) 2 14.71(02) 14.39(04) 12.54(02) 11.86(04) 3 17.31(05) 16.66(05) 16.21(03) 15.76(02) 4 14.22(04) 13.56(04) 13.07(01) 13.55(01) 5 14.89(03) 14.57(03) 14.26(02) 13.97(05) 6 15.52(04) 15.12(03) 14.74(03) 14.39(06) 7 14.06(04) 13.59(02) 13.16(03) 12.76(04) 8 16.42(02) 15.32(03) 14.46(03) 13.77(04) See Figure 1 for the finder chart of SN 2012fr and the comparison stars.",
"Uncertainties, in units of 0.01 mag, are 1 $\\sigma $ .",
"cccccccccc 0pt The $BVRI$ Photometry of SN 2012fr from Li-Jiang 2.4-m telescope and YFOSC.",
"MJD Daya $B$ (mag) $V$ (mag) $R$ (mag) $I$ (mag) 56254.18 10.68 12.52(03) 12.21(01) 12.21(03) 12.65(02) 56256.20 12.70 12.69(03) 12.34(02) 12.35(02) 12.81(03) 56260.17 16.67 13.06(02) 12.57(02) 12.52(03) 12.87(03) 56266.17 22.67 13.53(02) 12.83(01) 12.70(01) 56273.14 29.64 14.12(03) 13.08(01) 12.81(02) 12.57(04) 56278.11 34.61 14.47(04) 13.37(01) 13.05(02) 12.77(03) 56280.13 36.63 14.64(03) 13.54(01) 13.16(01) 12.91(04) 56290.10 46.60 14.04(02) 13.76(02) 13.50(03) 56298.10 54.60 14.99(04) 14.26(01) 14.00(02) 13.91(04) 56306.05 62.55 15.09(03) 14.41(01) 14.28(02) 14.27(03) 56314.04 70.54 15.19(04) 14.71(02) 14.52(02) 14.58(03) Uncertainties, in units of 0.01 mag, are 1$\\sigma $ ; MJD = JD-2400000.",
"aRelative to the epoch of $B$ -band maximum (MJD =56243.50)"
],
[
"Optical/UV Observations from $Swfit$ UVOT",
"SN 2012fr was also intensively observed with the UVOT [50] on board the $Swift$ satellite [24], spanning from $t \\approx -14$ days to $t \\approx +126$ days relative to the $B$ -band maximum light.",
"The photometric observations were performed in three UV filters ($uvw2$ , $uvm2$ , and $uvw1$ ) and three broadband optical filters ($U$ ,$B$ and $V$ ).",
"All of the $Swift$ images were reduced using HEASoft (the High Energy Astrophysics Software)http://www.swift.ac.uk/analysis/software.php– a “uvotsource\" program with the latest $Swift$ Calibration Databasehttp://heasarc.gsfc.nasa.gov/docs/heasarc/caldb/swift/.",
"As the PSF profile is slightly dependent on the count rate of the source, we adopted the aperture photometry in the reduction.",
"The photometric aperture and the background value were set according to [47].",
"Since the UVOT is a photon-counting detector and suffers from coincidence loss (C-loss) for bright sources, the observed counts need corrections for such losses.",
"This has now been automatically done by the “uvotsource\" program based on an empirical relation given by Poole et al.",
"(2008).",
"As the instrumental response curves of the UVOT optical filters do not follow exactly those of the Johnson $UBV$ system, color-term corrections have been further applied to the magnitudes.",
"Table REF lists the final UVOT UV/optical magnitudes of SN 2012fr.",
"A total of 20 low-resolution spectra of SN 2012fr were obtained with the YFOSC on the LJT, with the wavelength covering from $\\sim $ 3400Å to $\\sim $ 9100Å.",
"A few very early-time spectra obtained through Astronomical Ring for Access to Spectroscopy (ARAShttp://www.astrosurf.com/aras) were also included in our analysis.",
"A journal of our spectroscopic observations is given in Table REF .",
"All spectra were reduced using standard IRAF routines.",
"The spectra were flux-calibrated with the spectrophotometric flux standard stars (e.g., LTT-1020) observed at a similar air mass on the same night.",
"The spectra were further corrected for the continuum atmospheric extinction at the Li-Jiang Observatory; moreover, telluric lines were also removed from the data.",
"Figure: A spectrum of SN 2012fr taken on Jan. 13 2013, with the flux calibrated by the spectrophotometric star LTT-1020 (black) andanother two standard stars of GD71 and G193-74 (red).cccccccc[!th] 0pt $Swift$ UVOT Photometry of SN 2012fr MJD Daya uvw2(mag) uvm2(mag) uvw1(mag) $U$ (mag) $B$ (mag) $V$ (mag) 56229.28 -14.22 19.19(21) 20.49(34) 17.50(12) 15.93(05) 15.21(03) 14.88(04) 56229.29 -14.21 19.04(20) 20.32(22) 17.83(14) 15.83(05) 15.15(03) 14.91(04) 56231.56 -11.94 17.14(09) 19.36(22) 15.62(08) 13.76(03) 13.62(03) 13.62(03) 56232.08 -11.42 16.83(08) 18.81(15) 15.21(08) 13.43(03) 13.38(03) 13.42(03) 56232.35 -11.15 16.73(08) 18.44(12) 15.07(07) 13.29(03) 13.30(03) 13.34(03) 56232.42 -11.08 16.73(08) 18.58(13) 14.97(07) 13.25(03) 13.26(03) 13.31(03) 56233.90 -9.60 15.97(07) 17.63(14) 14.18(07) 12.59(03) 12.85(03) 12.88(03) 56233.96 -9.54 15.96(07) 17.64(10) 14.18(07) 12.59(03) 12.80(03) 12.88(03) 56235.58 -7.92 15.45(04) 17.14(05) 13.74(03) 12.15(03) 12.65(03) 12.57(01) 56235.64 -7.86 15.54(08) 16.88(10) 13.87(08) 12.24(04) 12.58(04) 56235.71 -7.79 15.52(08) 17.01(11) 13.61(08) 12.13(04) 12.55(04) 56235.78 -7.72 15.48(10) 16.95(11) 13.67(08) 12.16(04) 12.53(04) 56235.91 -7.59 15.53(08) 17.80(16) 13.57(03) 12.18(05) 12.60(04) 56237.70 -5.80 15.07(04) 16.56(05) 13.27(03) 11.97(03) 12.40(03) 12.34(01) 56237.90 -5.60 15.21(07) 16.65(09) 13.41(07) 12.32(04) 56239.57 -3.93 14.92(04) 16.21(05) 13.11(03) 11.79(03) 12.14(03) 12.15(01) 56241.64 -1.86 14.88(03) 15.96(05) 13.12(03) 11.74(03) 12.09(03) 12.06(01) 56244.18 +0.68 14.94(03) 15.82(05) 13.29(03) 11.85(01) 12.03(03) 12.01(01) 56246.32 +2.82 12.02(02) 12.08(02) 56248.06 +4.56 15.17(03) 15.87(04) 13.70(03) 12.12(01) 12.15(03) 12.05(01) 56248.38 +4.88 12.15(06) 12.18(03) 56249.52 +6.02 12.22(02) 12.24(02) 12.13(05) 56250.13 +6.63 15.21(03) 15.89(04) 13.72(07) 12.25(03) 12.10(03) 56251.54 +8.04 12.38(02) 12.36(02) 12.11(05) 56251.66 +8.16 15.33(05) 15.98(06) 13.93(08) 12.36(04) 12.12(04) 56251.73 +8.23 15.38(04) 15.94(05) 13.91(07) 12.44(04) 12.13(03) 56254.00 +10.50 15.54(04) 16.17(05) 14.13(07) 12.68(03) 12.58(04) 12.26(03) 56256.35 +12.85 12.94(03) 12.84(06) 12.43(05) 56256.41 +12.91 15.80(04) 16.48(05) 14.50(07) 12.90(03) 12.79(03) 12.41(03) 56257.88 +14.38 16.03(04) 16.63(06) 14.66(07) 13.07(03) 12.89(03) 12.49(03) 56257.95 +14.45 13.07(05) 13.02(05) 12.46(05) 56260.21 +16.71 16.27(05) 16.86(07) 14.90(08) 13.38(03) 13.12(03) 12.57(03) 56260.27 +16.77 13.36(03) 13.12(05) 12.61(05) 56262.35 +18.85 16.55(05) 17.12(07) 15.14(08) 13.60(03) 13.28(03) 12.75(03) 56262.42 +18.92 13.61(03) 13.32(05) 12.70(02) 56264.08 +20.58 13.70(06) 13.43(02) 12.73(03) 56264.15 +20.65 16.69(07) 17.20(09) 15.37(08) 13.73(03) 13.46(03) 12.78(03) 56266.22 +22.72 13.99(03) 13.61(02) 12.83(05) 56266.29 +22.79 16.77(06) 17.30(08) 15.61(08) 14.01(03) 13.60(03) 12.84(03) 56267.51 +24.01 14.16(03) 13.86(05) 12.96(06) 56268.23 +24.73 17.12(07) 17.54(09) 15.73(08) 14.17(04) 13.79(02) 12.91(03) 56269.63 +26.13 14.30(03) 13.91(05) 12.94(05) 56270.09 +26.59 17.03(07) 17.60(10) 15.88(09) 14.28(04) 13.93(03) 12.94(03) 56275.63 +32.13 17.36(08) 17.74(10) 16.23(09) 14.66(04) 14.33(03) 13.24(03) 56280.18 +36.68 17.52(08) 18.03(10) 16.89(10) 14.96(04) 14.63(03) 13.53(03) 56283.51 +40.01 17.60(11) 18.21(16) 16.85(15) 15.18(05) 14.75(04) 13.68(04) 56287.92 +44.42 17.87(11) 18.22(14) 16.97(14) 15.24(05) 14.84(03) 13.88(04) 56291.67 +48.17 18.02(11) 18.31(12) 17.13(13) 15.27(04) 14.93(03) 14.03(03) 56296.01 +52.51 18.11(11) 18.53(15) 17.18(14) 15.34(04) 14.97(03) 14.21(04) 56300.22 +57.22 18.36(13) 18.48(13) 17.24(13) 15.48(04) 15.01(03) 14.33(04) 56303.69 +60.69 18.12(17) 18.51(18) 17.32(15) 15.52(07) 15.05(04) 14.39(05) 56314.65 +71.65 18.61(17) 18.53(15) 17.47(17) 15.69(06) 15.20(04) 14.65(05) 56320.98 +77.98 18.62(20) 18.99(31) 17.65(20) 15.81(05) 15.37(03) 14.81(04) 56326.66 +83.66 18.71(20) 18.96(26) 17.64(19) 15.85(07) 15.43(05) 14.98(06) 56332.75 +89.75 18.68(16) 19.08(29) 17.62(23) 15.91(06) 15.48(06) 15.02(05) 56339.02 +96.02 18.83(18) 19.21(22) 17.63(16) 16.08(06) 15.61(05) 15.34(06) 56344.50 +101.50 18.98(20) 19.52(28) 17.91(19) 16.26(07) 15.73(04) 15.43(06) 56350.64 +107.64 19.00(20) 19.50(26) 18.00(26) 16.30(07) 15.81(07) 15.62(06) 56356.65 +113.65 19.03(18) 19.41(21) 18.19(24) 16.37(07) 15.86(04) 15.73(06) 56362.53 +119.53 19.08(18) 19.72(32) 18.21(21) 16.54(09) 15.95(05) 15.83(07) 56368.95 +125.95 19.12(21) 19.94(34) 18.41(21) 16.66(09) 16.09(05) 15.85(07) Uncertainties, in units of 0.01 mag, are 1$\\sigma $ ; MJD = JD-2400000.",
"aRelative to the epoch of $B$ -band Maximum"
],
[
"Spectrophotometry",
"As it can be seen that the spectra obtained with the LJT for SN 2012fr have a better phase coverage compared with the photometry.",
"In particular, these spectra have better flux calibrations and cover well the wavelength range of the $BVR$ filters, as shown in Figure REF .",
"This enables us to fill in the gaps of the light curves through spectrophotometry.",
"The synthetic magnitudes were derived by convolving the observed spectra with the transmission curves of the filters, and are listed in Table REF .",
"The spectrophotometry is overall consistent with the photometry to within 0.10 mag in the $BVR$ bands, but larger differences emerge in the $I$ band because of limited wavelength coverage of our spectra.",
"The errors listed in Table REF include uncertainties in slit loss, sky transparency loss, and flux calibration of the spectra.",
"A slit width of $1.8$ was used in the observations, which is larger than the typical seeing at Li-Jiang observatory (i.e., $\\sim $ 1$$ ).",
"The slit-loss is therefore negligible for our spectra.",
"The uncertainty due to the sky transparency loss should be small as the flux standard star LTT-1020 is very close to SN 2012fr.",
"In the $I$ band, the synthetic magnitudes still suffer from the uncertainty in the adopted spectral shape of the wavelength region from 9100Å to 11000Å.",
"A spectral template with a similar $\\Delta m_{15}(B)$ was taken from Hsiao et al.",
"(2007), and an uncertainty of 10% was assumed in the calculation.",
"Note that the uncertainties on the derived magnitudes may be underestimated as compared to those listed in Table REF because of large airmasses through which SN 2012fr and LTT-1020 were observed at Li-Jiang Observatory.",
"cccccc 0pt Magnitudes of the Spectrophotometry of SN 2012fr MJD Daya $B$ (mag) $V$ (mag) $R$ (mag) $I$ (mag) 236.25 -7.25 12.44(10) 12.66(08) 12.30(08) 12.62(10) 237.21 -6.29 12.30(10) 12.52(08) 12.23(08) 12.48(10) 239.22 -4.28 12.11(10) 12.24(08) 12.05(08) 12.37(10) 241.22 -2.28 12.02(20) 12.14(08) 11.93(08) 12.28(10) 243.23 -0.27 11.95(10) 12.04(08) 11.89(08) 12.29(10) 247.22 +3.72 12.08(10) 11.95(08) 11.97(08) 12.30(10) 250.24 +6.74 12.18(10) 12.05(08) 12.12(08) 12.37(10) 252.22 +8.72 12.42(10) 12.21(08) 12.21(08) 12.54(10) 254.18 +10.68 12.51(10) 12.29(08) 12.22(08) 12.64(10) 256.20 +12.70 12.65(10) 12.40(08) 12.38(08) 12.82(10) 260.17 +16.67 12.99(10) 12.60(08) 12.59(08) 12.99(10) 266.17 +22.67 13.59(10) 12.71(08) 12.63(08) 12.85(10) 273.14 +29.64 14.17(10) 13.04(08) 12.79(08) 12.62(10) 277.11 +33.61 14.32(20) 13.16(08) 12.97(08) 12.78(10) 280.13 +36.63 14.44(10) 13.48(08) 13.12(08) 12.83(10) 290.10 +46.60 14.81(10) 14.00(08) 13.67(08) 13.49(10) 298.10 +54.60 14.96(10) 14.19(08) 13.89(08) 13.86(10) 306.05 +62.55 15.11(10) 14.49(08) 14.21(08) 14.20(10) 314.04 +70.54 15.25(10) 14.63(08) 14.44(08) 14.56(10) Uncertainties, in units of 0.01 mag, are 1$\\sigma $ ; MJD = JD-2456000.",
"The spectra have been corrected for reddening and redshift before being used for spectrophotometry.",
"aRelative to the epoch of $B$ -band Maximum cccccccccc[!th] 0pt Journal of Spectroscopic Observations of SN 2012fr UT DateMJD Epoch Res Range Telescope Flux Airmass Exp.time Observer (-240000) (days) (Å/pix) (Å) (+Instrument) (Standard) (sec) 2012 Oct. 29 56230.07 -13.43 4.65 4000-6500 C11+Lhires 1.05 7$\\times $ 600 BHa 2012 Oct. 30 56230.99 -12.51 6.67 4000-7000 C11+LISA HD 27376 1.22 3$\\times $ 600 TBb 2012 Oct. 31 56231.97 -11.53 6.67 4000-7000 C11+LISA HD 27376 1.33 8$\\times $ 600 TB 2012 Nov. 4 56236.25 -7.25 2.85 3400-9100 LJT+YFOSC LTT1020 2.25 600.00 ZJc 2012 Nov. 5 56237.21 -6.29 2.85 3400-9100 LJT+YFOSC LTT1020 2.17 600.00 ZJ 2012 Nov. 7 56239.22 -4.28 2.85 3400-9100 LJT+YFOSC LTT1020 2.18 600.00 ZJ 2012 Nov. 9 56241.22 -2.28 2.85 3400-9100 LJT+YFOSC LTT1020 2.19 600.00 ZJ 2012 Nov. 11 56243.23 -0.27 2.85 3400-9100 LJT+YFOSC LTT1020 2.06 600.00 ZJ 2012 Nov. 14 56246.24 +2.74 2.85 3400-9100 LJT+YFOSC d 2.31 600.00 ZJ 2012 Nov. 15 56247.22 +3.72 2.85 3400-9100 LJT+YFOSC LTT1020 2.23 600.00 ZJ 2012 Nov. 18 56250.24 +6.74 2.85 3400-9100 LJT+YFOSC LTT1020 2.42 600.00 ZJ 2012 Nov.20 56252.22 +8.72 2.85 3400-9100 LJT+YFOSC LTT1020 2.30 600.00 ZJ 2012 Nov.22 56254.18 +10.68 2.85 3400-9100 LJT+YFOSC LTT1020 2.17 600.00 ZJ 2012 Nov.24 56256.20 +12.70 2.85 3400-9100 LJT+YFOSC LTT1020 2.27 600.00 ZJ 2012 Nov. 28 56260.17 +16.67 2.85 3400-9100 LJT+YFOSC LTT1020 2.19 600.00 ZJ 2012 Dec. 4 56266.17 +22.67 2.85 3400-9100 LJT+YFOSC LTT1020 2.26 600.00 ZJ 2012 Dec. 11 56273.14 +29.64 2.85 3400-9100 LJT+YFOSC LTT1020 2.21 600.00 ZJ 2012 Dec. 15 56277.11 +33.61 2.85 3400-9100 LJT+YFOSC LTT1020 2.17 600.00 ZJ 2012 Dec. 18 56280.13 +36.63 2.85 3400-9100 LJT+YFOSC LTT1020 2.24 600.00 ZJ 2012 Dec. 28 56290.10 +46.60 2.85 3400-9100 LJT+YFOSC LTT1020 2.24 600.00 ZJ 2013 Jan. 5 56298.10 +54.60 2.85 3400-9100 LJT+YFOSC LTT1020 2.33 900.00 ZJ 2013 Jan. 13 56306.05 +62.55 2.85 3400-9100 LJT+YFOSC LTT1020 2.19 900.00 ZJ 2013 Jan. 21 56314.04 +70.54 2.85 3400-9100 LJT+YFOSC LTT1020 2.25 900.00 ZJ aBernard Heathcote.",
"Facility: Celestron11 + LhiresIII150 + ATIK314L.",
"Observatory: Melbourne (Australia) bTerry Bohlsen.",
"Facility: Celestron11 + LISA ST8XME.",
"Observatory: Armidale (Australia) cZhang Jujia and staff of the Li-Jiang Observatory.",
"Facility: LJT+YFOSC.",
"Observatory: YNAO (China) dflux calibration was obtained through the observations of LTT1020 on Nov. 13 and Nov. 15 2012.",
"Figure REF shows the optical and UV light curves of SN 2012fr.",
"Besides the color-term corrections, additional magnitude corrections such as S-corrections (Stritzinger et al.",
"2002) are also applied to the light curves to account for the differences between the instrumental responses and those defined by Bessell (1992).",
"No S-corrections were applied to the UV data.",
"In the optical bands, the observations of the LJT and YFOSC system agree well with the $Swift$ observations.",
"Detailed analysis of the light and color curves are presented in the following subsections.",
"Figure: Ultraviolet and optical light curves of SN 2012fr.",
"The light curves are shifted vertically for better display."
],
[
"Optical and Ultraviolet Light Curves",
"With the LJT and $Swift$ light curves, we derived the parameters of peak magnitudes, maximum dates, and light-curve decline rates $\\Delta m_{15}$ (i.e., Phillips 1993), as listed in Table REF .",
"It is found that SN 2012fr reached a $B$ -band maximum brightness of $12.01\\pm 0.01$ mag on JD 2456243.50$\\pm $ 0.30 (2012 Nov. 12.00 UT), and it reached at the $V$ -band maximum of 11.99$\\pm $ 0.01 mag at about 1.5 days later.",
"The observed light-curve decline rate $\\Delta m_{15}$ (B) and the maximum-light color $B_{max} - V_{max}$ are estimated as 0.85$\\pm $ 0.05 mag and $0.02\\pm $ 0.03 mag, respectively.",
"Figure: Light curves of SN 2012fr compared with some representative sample of SNe Ia.",
"Left: Comparison of SN 2012fr with SN 2005cf, SN 2007S, SN 2009ig, and SN 2011fe in SwiftSwift UV bands (see text for references); Right: Comparison in the UBVRIUBVRI bands (see text for references).Our estimates of these parameters are consistent with those reported by C13 who derived that $B$ -band light curve has a peak brightness of $\\approx $ 12.0 mag on 2012 Nov. 12.04 and a decline rate $\\Delta m _{15}$ (B) = 0.80 mag.",
"A smaller decline rate suggests that SN 2012fr should be intrinsically luminous if it follows the light curve width-luminosity relation (i.e., Phillips 1993; Riess et al.",
"1996; Goldhabor et al.",
"2001; Guy et al.",
"2005).",
"After corrections for the Galactic reddening of E$()$ = 0.02 mag (Schlegel et al.",
"1998), the corresponding color index of $B_{max}-V_{max}$ becomes 0$\\pm $ 0.03 mag.",
"This suggests insignificant host-galaxy reddening for SN 2012fr according to the intrinsic colors of SNe Ia (e.g., Phillips et al.",
"1999; Wang et al.",
"2009a).",
"Figure REF shows the comparison of SN 2012fr with some well-observed SNe Ia with both optical and UV observations, including SN 2005cf ($\\Delta m_{15}$ = 1.05 mag; Wang et al.",
"2009b), SN 2007S ($\\Delta m_{15}$ = 0.84 mag; Brown et al.",
"2009), SN 2009ig ($\\Delta m_{15}$ = 1.05 mag; Foley et al.",
"2012, and Marion et al.",
"2013), and SN 2011fe ($\\Delta m_{15}$ = 1.06 mag; Brown et al.",
"2012, and Wang et al.",
"2014 in prep.).",
"The sample selected for comparison have similar values of $\\Delta m_{15}$ (B).",
"Of this sample, SN 2005cf and SN 2011fe can be put into the NV (or LVG) subclass, while SN 2007S and SN 2009ig can be put into the 91T-like (or SS) subclass and HV subclass, respectively.",
"In the UV bands, the light curve peaks of SN 2012fr are similar to that of the comparison sample except for SN 2011fe which has an apparently faster post-maximum decline rate.",
"Difference also exists in the optical bands where the tail of SN 2011fe appears to be fainter than SN 2012fr and other comparison SNe by 0.3–0.5 mag, depending on the wavebands.",
"We further noticed that SN 2012fr perhaps has a more prominent shoulder feature at about 30 days after the maximum brightness in comparison with other SNe Ia of our sample.",
"This feature is stronger in SN 2012fr than the others in the $I$ band, bears about the same strength as the others in the $R$ band, and may be a little stronger than the others in the $V$ band.",
"In observations, the relative strength of the $I$ -band secondary shoulder is found to correlate with decline rate (and hence peak luminosity), being more prominent and occurring later in more luminous SNe Ia.",
"This observed behavior may be taken as further evidence that SN 2012fr has a larger production of nickel (and hence iron group elements) in the explosion (Kasen 2006), consistent with the $^{56}$ Ni estimated in Section 5.1.",
"In general, the overall light curve evolution of SN 2012fr resembles closely to those of SN 2007S and SN 2009ig, although SN 2007S is not well sampled in the UV bands.",
"SN 2011fe matches well with SN 2012fr near the maximum phase, but they show noticeable differences in the later phases (especially at shorter wavelengths).",
"ccccc[!th] 0pt Light-Curve Parameters of SN 2012fr Band $\\lambda _{central}$ t$_{max}$ a m$_{peak}$ b $\\Delta $ m$_{15}$ b (Å) (-240000) (mag) (mag) uvw2 1928 56243.40(40) 15.00(04) 1.03(06) uvm2 2246 56246.10(60) 15.81(05) 0.98(07) uvw1 2600 56240.60(50) 13.27(03) 1.09(05) U 3650 56242.10(40) 11.89(03) 1.09(05) B 4450 56243.50(30) 12.01(03) 0.85(05) V 5500 56245.00(30) 11.99(03) 0.59(05) R 6450 56244.90(80) 11.89(10) 0.62(10) I 7870 56242.50(80) 12.28(10) 0.55(10) aUncertainties of peak-light dates, in units of 0.01 day, are 1 $\\sigma $ bUncertainties of magnitudes, in units of 0.01 mag, are 1 $\\sigma $"
],
[
"Color Curves and Interstellar Extinction",
"Figure REF shows the color curves of SN 2012fr, corrected for the Galactic reddening from Schlegel et al.",
"(1998) and the host-galaxy reddening derived from the following analysis.",
"Overplotted are the color curves of SN 2005cf, SN 2007S, SN 2009ig, and SN 2011fe.",
"The overall color evolution of SN 2012fr is similar to those of the selected SNe Ia, but scatter presents in the colors at shorter wavelengths (i.e., $U - B$ and $B - V$ ).",
"In $U - B$ , SN 2012fr reached at the bluest color at about one week before $B$ -band maximum light, as similarly seen in SN 2009ig and SN 2007S.",
"Of these sample, SN 2007S seems to have a relatively bluer color at this stage.",
"SN 2005cf and SN 2011fe reached at their bluest colors a few days later and they are also redder relative to the other three sample.",
"In $B - V$ , SN 2012fr is redder than the comparison SNe Ia by $\\sim $ 0.2 mag near the maximum light.",
"During the period from t $\\sim $ 40 days to t $\\sim $ 80 days, the $B - V$ color of SN 2012fr appears somewhat bluer than that of SN 2005cf and SN 2011fe, with a slope steeper than the $Lira-Phillips$ relation (Phillips et al.",
"1999).",
"A similar feature of faster $B - V$ evolution is also seen in the HV SNe Ia at comparable phases (e.g., Wang et al.",
"2008).",
"The $V - R$ and $V - I$ color curves of SN 2012fr exhibits a behavior that is very similar to those of the comparison SNe.",
"The Galactic reddening of SN 2012fr is $E()_{Gal}$ = 0.02 mag [55], corresponding to an extinction of 0.06 mag adopting the standard reddening law of [10].",
"The reddening due to the host galaxy can be estimated by several empirical methods.",
"A value of $\\sim $ 0.1 mag can be derived for E$(B - V)_{host}$ by using the empirical relation established between the intrinsic $B_{max} - V_{max}$ color and $\\Delta m_{15}$ (i.e., Phillips et al.",
"1999; Wang et al.",
"2009b).",
"The comparison of the late-time $B - V$ color with that predicted by the $Lira-Phillips$ relation (see the dashed line in Figure 5) gives a host-galaxy reddening of E$(B - V)_{host}$ = $-$ 0.07 mag for SN 2012fr.",
"The negative reddening is apparently unphysical.",
"This inconsistency indicates that the intrinsic color is not well understood yet for SNe Ia of different subtypes; and any empirical method should be used with caution for any individual SN Ia.",
"On the other hand, the high-resolution spectrum of SN 2012fr does not show any significant signature of Na1 D absorption feature (C13), suggesting a low reddening.",
"C13 therefore placed an upper limit, with $E() < 0.015$ , for the host-galaxy reddening of SN 2012fr.",
"In the following analysis, we assume the total line-of-sight reddening of SN 2012fr as $E()$ = 0.03 mag.",
"Figure: Comparison of the color curves between SN 2012fr and some representative SNe Ia (see text for references).The dashed line in the panel shows the Lira-PhillipsLira-Phillips loci (Phillips et al.",
"1999)."
],
[
"Optical Spectra",
"A total of 20 spectra of SN 2012fr were obtained with the LJT and YFOSC spectrograph, which covers a phase from $t\\approx -7$ to $t\\approx +71$ days relative to the $B$ -band maximum light.",
"The complete spectral evolution of SN 2012fr is presented in Figure REF , where the three earliest spectra obtained through ARAS are also overplotted.",
"The early-time ARAS spectra show strong absorption features at $5900\\sim 6000$ Å , perhaps due to the largely blueshifted Si2 $\\lambda $ 6355.",
"At around the maximum brightness, the spectral evolution of SN 2012fr generally follows that of normal SNe Ia but showing narrower and weaker absorption features of Si2 6355.",
"The detailed spectral evolution is discussed in the following subsections.",
"Figure: Optical spectral evolution of SN 2012fr.",
"The spectra have been corrected for the redshift of the host galaxy (V hel V_{hel} = 1636 km s -1 ^{-1}) and telluric lines.",
"They have been shifted vertically by arbitrary amounts for clarity; the numbers on the right-hand side mark the epochs of the spectra in days after BB maximum."
],
[
"Spectra Temporal Evolution",
"The spectral comparisons between SN 2012fr and other well-observed SNe Ia are shown in Figures REF and REF .",
"All the spectra have been corrected for redshift but not for the reddening.",
"Figure REF displays the spectrum of SN 2012fr taken at $t=-13$ days, with typical features of $W$ -shaped sulfur lines and iron lines at 4500$\\sim $ 5000Å.",
"The most notable feature is the strong absorption trough at $\\sim $ 5900Å , which can be attributed to the Si2 $\\lambda $ 6355 absorption formed in regions above the photosphere.",
"Such a HVF is similarly seen in SN 2009ig and SN 2005cf, although SN 2005cf shows a relatively lower velocity.",
"Note that this detached HVF is not observed in the spectrum of SN 2011fe, suggesting that it may be not a common feature for the spectroscopically normal SNe Ia.",
"As it can be seen from the plot, the Si2 feature is barely detected in SN 1991T at such an early phase (Mazzali et al.",
"1995).",
"In addition, we notice that the C2 6580 absorption can be detected in SN 2005cf and SN 2011fe, but not in SN 1991T and SN 2009ig.",
"It is not clear whether SN 2012fr shows the C2 $\\lambda $ 6580 absorption at this phase because of a relatively poor S/N ratio for the spectrum (but see discussions below).",
"Figure: Spectrum of SN 2012fr at t=-13t = -13 days from BB-band maximum, overplotted with the spectra of SN 1991T, SN 2005cf, SN 2009ig, and SN 2011fe at similar phases (see text for references).",
"All spectra presented here have been corrected for redshift and reddening.",
"The dashed line marks the absorption minimum of Si2 λ\\lambda 6355 for SN 2012fr.Figure: Spectra of SN 2012fr at t ≈\\approx --7, 0, +7 and +30 days after the BB-band maximum.",
"Overplotted are the comparable-phase spectraof SN 2001eh, SN 2005cf, SN 2007S, SN 2009ig, and SN 2011fe (see text for references).",
"The green line in panel (b) is the SYNOW fitto the t≈\\approx 0 day spectrum of SN 2012fr.Figure REF shows the spectroscopic comparison of SN 2012fr with some SNe Ia at several epochs.",
"The comparison sample was selected because they are well-observed or show similarities to SN 2012fr in some respects, and includes SN 2001eh (Silverman et al.",
"2012), SN 2005cf (Wang et al.",
"2009b), SN 2007S (Blondin et al.",
"2012), SN 2009ig (Foley et al.",
"2012), and SN 2011fe (Pereira et al.",
"2013; Wang et al.",
"2014 in preparations).",
"At one week before the maximum, the spectrum is characterized by lines of singly ionized intermediate-mass elements (IMEs, e.g., Si, S, Mg and Ca).",
"The absorption feature at 6000Å is dominated by the photospheric component of Si2 $\\lambda $ 6355; and the HVF becomes nearly invisible in all of our sample.",
"There is no significant signature of C II 6580 absorption in the t = $-$ 7 day spectrum of SN 2012fr, consistent with the conclusion reached by C13 from their earlier spectra.",
"We noticed that such a C II feature is not visible in SN 2001eh, SN 2007S, and 2009ig.",
"By $t = -7 $ days, the HVFs of Ca2 IR triplet are still prominent and are stronger than the photospheric components in SN 2012fr and SN 2005cf, while they are barely visible in SN 2001eh and SN 2011fe.",
"This indicates that the distribution of Ca2 in the outer layer of the ejecta may provide another significant signature to identify the diversity of SN Ia explosions.",
"A minor absorption at $\\sim $ 5750 Å is likely due to Si2 5972, which is absent in the early spectra; the weak strength suggests a relatively high temperature for the photosphere with respect to normal SNe Ia (Nugent et al.",
"1997).",
"Around maximum brightness, the absorption feature of Si2 $\\lambda $ 6355 in SN 2012fr evolves as a relatively normal profile.",
"Nevertheless, the line profile of Si2 $\\lambda $ 6355 appears apparently narrower compared to those of the normal SNe Ia like SN 2005cf and SN 2011fe.",
"The HVFs of Ca2 IR triplet absorption feature are comparably strong in SN 2012fr and SN 2005cf; but they are nearly invisible seen in SN 2011fe where the absorption trough is dominated by the photospheric component.",
"The O1 $\\lambda $ 7773 feature strengthens in all cases for our sample, while it is not clear for SN 2007S because of the limited wavelength coverage.",
"The line-strength ratio of Si2 5972 to Si2 6355, known as $R$ (Si2) (Nugent et al.",
"1997), is an approximate indicator of the photospheric temperature, with a larger value corresponding to a lower temperature.",
"We measured this parameter to be 0.07$\\pm $ 0.02 for SN 2012fr near the maximum light, which is noticeably smaller than the corresponding value for the standard Ia SN 2005cf, suggesting a higher photospheric temperature for SN 2012fr.",
"At $t \\sim +7$ days, our sample generally exhibit a similar spectral evolution, with the HVFs becoming undetectable in the main spectral lines.",
"One interesting feature is that the absorption trough of Ca2 IR feature clearly splits into two components in SN 2012fr, SN 2001eh, and SN 2007S.",
"This behavior is also consistent with the narrow Si2 profiles seen in these three events, suggesting that the ejecta produced from their explosions may be confined into a smaller range of velocity compared to SN 2005cf, SN 2009ig, and SN 2011fe.",
"At $t\\sim 1$ month (Figure REF d), the spectra are dominated by iron lines and a strong Ca2 IR triplet absorption trough.",
"The double absorption features of Ca2 IR triplet become more prominent in SN 2012fr, SN 2001eh, and SN 2007S relative to the rest comparison SNe Ia.",
"The above comparison suggests that SN 2012fr shares similar features with SN 2001eh and SN 2007S in the near-maximum and post-maximum phases; and it appears to be more similar to the high-velocity SN Ia 2009ig in the very early phase.",
"To examine the mechanism forming the narrower Si2 and Ca2 lines, we adopted the parameterized resonance scattering synthetic-spectrum code SYNOW (Fisher et al.",
"1999; Branch et al.",
"2005) to fit the t = 0 day spectrum of SN 2012fr (see green line in Figure 8b).",
"The radial dependence of the line optical depth is chosen to be exponential with an e-folding velocity v$_{e}$ (i.e., $\\tau \\propto $ exp($-$ v/v$_{e}$ )).",
"We found that the v$_{e}$ should be smaller than 1000 km s$^{-1}$ in order to fit the narrow Si2 absorption at 6100Å in SN 2012fr, suggesting that the silicon layer (and perhaps the Ca2 layer) becomes optically thin near the photosphere."
],
[
"The Ejecta Velocity",
"In this subsection, we examine the ejecta velocity of SN 2012fr via the absorption features of some spectral lines.",
"The location of the blueshifted absorption minimum was measured by using both the Gaussian fit routine and the direct measurement of the center of the absorption, and the results were averaged.",
"Figure: The evolution of HVFs of Si2 and Ca2 in the spectra of SN 2012fr.",
"Panel (a): Si2 λ\\lambda 6355; Panel (b): Ca2 H&K; Panel (c): Ca2 IR triplet.",
"In panels (a) and (b), the blue and red dashed lines represent the Gaussian fit to the detached high-velocity and the photospheric components, respectively; green dotted line denotes the sum of these two components.",
"The blue dashed lines in panel (c) represent the triple-Gaussian fit to the HVFs of Ca2 IR absorption, and the green dotted line is the sum of the Gaussian fit.",
"The wavelength positions of the Ca II H&K and Ca II IR lines at different velocities are marked as a guide to the eyes.Figure REF shows the evolution of the line profiles of Si2 $\\lambda $ 6355, Ca2 IR triplet, and Ca2 H$\\&$ K. For these lines, the blue-side absorption feature gradually weakens with the emergence of the red-side component; such an overall evolution clearly suggests the presence of another HV component in these spectral features.",
"Adopting the double-Gaussian fit and attributing the absorption on the blue side to the high-velocity component, we estimate from the t = $-$ 7 day spectrum that the detached components of Si2 $\\lambda $ 6355 and Ca2 H&K have a velocity of $\\sim $ 22,000 km s$^{-1}$ and $\\sim $ 25,000 km s$^{-1}$ , respectively.",
"These velocities are much higher than the corresponding photospheric velocities (e.g., $v_{phot}$$\\sim $ 12,000 km s$^{-1}$ ; see Figure REF ).",
"Inspection of Figure REF (a) reveals that the detached high-velocity component of the Si2 line almost disappears in the t = $-$ 7 days spectrum, indicating that the HVF of Si2 line is detectable for only a very short time in some SNe Ia.",
"In contrast, the corresponding HVFs of the Ca2 IR triplet can last for a longer time and is still detectable in the t = + 7 days spectrum of SN 2012fr.",
"The duration for the HVFs of Ca2 H&K lines, as shown is Figure REF (b), is unclear due to the possible contamination of the Si2 $\\lambda $ 3860 line.",
"This needs to be clarified, as the Ca2 H&K absorption feature may be a potential indicator of SN Ia diversity (e.g., Maguire et al.",
"2012).",
"As shown in Figure REF (b), the fits using detached high-velocity and photospheric Ca II match well with the observed spectra; the HVF slows down with time and the photospheric component remains at a constant velocity of 12,000 km s$^{-1}$ (see also Fig.",
"10).",
"In contrast, neither the high-velocity nor the photospheric component of Si2$\\lambda $ 3860 (see the dashed lines in the plot) match the two absorption features of Ca II H&K lines before t = +7 days.",
"Thus there does not seem to be any need for Si2 3860 in those profiles of earlier spectra (but see [22]).",
"However, the Si2 $\\lambda $ 3860 may contribute to the Ca2 H&K absorption trough in the later spectra.",
"For example, the absorption near 3720Å in the $t\\approx +17$ spectrum might be due to the photospheric Si2 $\\lambda $ 3860 line (see also Table REF ), as the HVFs of Ca II should have disappeared at this late phase (see also REF (c)).",
"Figure: Velocity evolution of different elements inferred from the spectra of SN 2012fr.",
"The inset plot shows the velocity evolution of the detached, high-velocity components.",
"All of the velocity data are listed in Table .Figure REF (c) shows the evolution of Ca2 IR triplet absorption.",
"The HVFs dominate the line profiles in the earlier phase (see also C13 ), but the substructures of Ca2 $\\lambda $ 8498, $\\lambda $ 8542Å, and $\\lambda $ 8662 absorptions are hardly separated because of the blending.",
"In the $t = -4$ days spectrum, the photospheric components are visible and the weak absorption features near 8200Å and 8350Å can be identified as Ca2 8542 (at $\\sim $ 12,100 km s$^{-1}$ ) and Ca2 8662 (at $\\sim $ 12,100 km s$^{-1}$ ), respectively.",
"The Ca2 $\\lambda $ 8498 absorption is marginally detected at $\\sim $ 8160Å in the t = +3 day spectrum, but it seems to become progressively strong with time along with Ca2 8542 and Ca2 8662 in the later spectra.",
"It is worth mentioning that the three components of Ca2 IR triplet absorptions show a clear separation in the t = +71 day spectrum.",
"The narrow line wings indicate that the ejecta is perhaps confined into a narrow range of velocity.",
"ccccccccc[!th] 0pt Velocity Evolution of Different Lines from SN 2012fr Epoch Si II HVFs Ca II S II HVFs Si II HVFs Ca II (days) ($\\lambda $ 3860) (H$\\&$ K) (H$\\&$ K) ($\\lambda $ 5640) ($\\lambda $ 6355) ($\\lambda $ 6355) (IRT) (IRT) -13.43 -23.15(25)a -15.14(20)a -12.51 -22.11(25)a -13.64(15)a -11.53 -21.61(30)a -12.94(15)a -7.25 -25.50(15) -12.22(25) -12.11(15) -20.41(30)a -12.29(08) -12.01(25) -24.98(20) -6.29 -24.33(15) -12.20(25) -12.05(12) -19.97(30)a -12.18(08) -11.94(25) -24.28(20) -4.28 -23.31(15) -12.14(25) -11.91(10) -12.19(08) -11.95(20) -23.37(20) -2.28 -22.09(15) -12.18(25) -11.81(10) -12.14(06) -11.94(20) -22.54(25) -0.27 -21.01(15) -12.13(20) -11.75(08) -12.13(06) -11.94(20) -21.61(25) 2.74 -20.07(25) -12.19(20) -11.76(08) -12.13(08) -11.95(15) -20.78(25) 3.72 -20.16(20) -12.24(20) -11.77(12) -12.15(08) -11.99(15) -20.54(30) 6.74 -19.27(20) -11.89(20) -11.74(12) -12.13(06) -12.02(15) 8.72 -12.06(30) -18.80(25) -12.03(20) -12.12(08) -12.02(15) 10.68 -12.01(25) -18.19(25) -12.12(20) -12.06(06) -12.14(12) 12.7 -11.99(20) -12.19(15) -12.12(08) -12.16(12) 16.67 -11.92(20) -12.02(15) -11.99(10) -12.10(15) 22.67 -11.89(20) -12.17(18) -11.78(10) -12.00(12) 29.64 -12.11(15) -11.81(10) -11.85(12) 33.61 -12.08(18) -11.76(10) -11.90(12) 36.63 -11.91(15) -11.78(10) -11.94(15) 46.6 -11.73(15) -11.98(12) 54.6 -11.72(15) -11.98(12) 62.55 -11.65(15) -11.86(15) 70.54 -11.63(15) -11.85(15) Uncertainties, in units of 10 km s$^{-1}$ , are 1 $\\sigma $ .",
"aDeduced from double-Gaussian fit.",
"The derived velocities of the intermediate-mass elements (IMEs) such as S, Si, and Ca as a function of time are shown in Figure REF and Table REF .",
"Overplotted are the corresponding velocities obtained by C13.",
"One can see that our results agree with theirs to within 100 km s$^{-1}$ at similar phases.",
"We also notice that the photospheric velocities of Ca2 IR triplet derived in C13 are slower than ours at t $<$ +6 days.",
"This difference might be due to the uncertainties in the fitting process of the line profiles, as the photospheric components of Ca2 IR triplet are relatively weak in the earlier phases and may not be appropriately indicated with the Gaussian profiles.",
"In SN Ia spectra, the Si2 $\\lambda $ 6355 line can better trace the velocity of the photosphere because this feature suffers less blending with other lines.",
"After the maximum light, the velocity of Si2 6355 remains at $\\sim $ 12,000 km s$^{-1}$ for over a month, with a velocity gradient of $2\\pm 13$ km s$^{-1}$ day$^{-1}$ .",
"Such a low velocity gradient is consistent with that derived from C13 at similar phases (e.g., 0.3$\\pm $ 10 km s$^{-1}$ day$^{-1}$ ).",
"Such a low velocity gradient is similarly seen for the velocity evolution of Ca2, which clearly puts SN 2012fr into the LVG category of SNe Ia according to the classification scheme of Benetti et al (2005).",
"As it can be seen from Figure REF , the IMEs of SN 2012fr generally have similar expansion velocities (e.g., $\\sim $ 12,000 km s$^{-1}$ ), suggestive of a relatively uniform distribution of the burning products in the ejecta.",
"Table REF collects the basic information for SN 2012fr and its host galaxy."
],
[
"The Peak Luminosity and the Nickel Mass",
"To further examine the properties of SN 2012fr, it is important to know the luminosity of SN 2012fr and the nickel mass produced in the explosion.",
"The distance to SN 2012fr is important for deriving these two quantities.",
"Direct measurements to its host galaxy NGC 1365 are available through several methods.",
"In our analysis, we adopt the most recent measurement from the Tully-Fisher relation which gives a distance modulus $\\mu $ = 31.38 $\\pm $ 0.06 mag (Tully et al.",
"2009) or a distance of 18.9 Mpc.",
"Adopting this distance and correcting for the galactic extinction, we derive the absolute $B$ - and $V$ -band peak magnitudes as M$_B = -19.49 \\pm $ 0.06 mag and M$_V = -19.48 \\pm $ 0.06 mag, respectively.",
"This indicates that SN 2012fr is brighter than a typical SN Ia (e.g., M$_{V}$ = $-$ 19.27 mag from Wang et al.",
"2006) by $\\sim $ 20%.",
"cc[!th] 0pt Relevant Parameters for NGC 1365 and SN 2012fr Parameter Value Parameters for NGC 1365 Galaxy Morphology SBb Activity type Seyfert 1.8 m-M 31.38 $\\pm $ 0.06 maga $E()_{Gal}$ 0.02b V$_{hel}$ 1636 $\\pm $ 1 km s$^{-1}$ Parameters for SN 2012fr Discovery Date Oct. 27 2012 Epoch of B Maximum 2456243.50(Nov. 12 2012) m$_B$ 12.01 $\\pm $ 0.03 mag m$_V$ 11.99 $\\pm $ 0.03 mag $\\Delta $ m$_{15}(B)$ 0.85 $\\pm $ 0.05 mag M$_{B}$ -19.49 $\\pm $ 0.06 mag M$_{V}$ -19.48 $\\pm $ 0.06 mag L$^{max}_{bol}$ (1.82$\\pm $ 0.15) $\\times 10^{43}$ erg s$^{-1}$ $^{56}$ Ni 0.88 $\\pm $ 0.08 M$_{}$ $v_{max}$ 12,120 $\\pm $ 70 km s$^{-1}$ $v_{10}$ 12,100 $\\pm $ 100 km s$^{-1}$ $\\dot{v}_{10}$ 2 $\\pm $ 13 km s$^{-1}$ day$^{-1}$ $R$ (Si) 0.07 $\\pm $ 0.02 $v_{max}$ is the velocity of Si2 $\\lambda $ 6355 measured at the $B$ -band maximum; $v_{10}$ (Si2) represents the value obtained at t = +10 days.",
"The velocity gradient during this period is defined as $\\dot{v}_{10}$ (Benetti et al.",
"2005); $R$ (Si) represents the ratio of Si2 5972 and Si2 6355 lines at the $B$ -band maximum.",
"aMeasured by Tully-Fisher relation (Tully et al.",
"2009).",
"b[55] The bolometric luminosity of SN 2012fr is derived from the UV and optical photometric observations.",
"The near-infrared (NIR) emission is corrected according to the flux ratio derived by Wang et al.",
"(2009b).",
"They found that the ratio of the NIR-band emission (9000–24,000 Å) to the optical (3200–9000Å) lies between 5-20% for normal SNe Ia like SN 2005cf, depending on the supernova phase.",
"An uncertainty of 10% is assumed for the NIR corrections in the derivation of bolometric luminosity.",
"Considering the SN Ia emission as a blackbody radiation, the missing UV flux at wavelengths shorter than $Swift$ UV filters (e.g., $<$ 1600 Å) is estimated to be $<$ 3.0% based on the spectral template of Hsiao et al.",
"(2007), we thus ignored its contribution in the calculation.",
"The peak bolometric luminosity of SN 2012fr is estimated to be (1.82 $\\pm $ 0.15) $\\times $ 10$^{43}$ erg s$^{-1}$ .",
"The uncertainty includes the errors in the distance modulus, the observed magnitudes, and the NIR corrections.",
"With the derived bolometric luminosity, the synthesized $^{56}$ Ni mass can be estimated using the Arnett law (Arnett 1982; Stritzinger & Leibundgut 2005): $L_{max}=(6.45e^{\\frac{-t_r}{8.8d}}+1.45e^{\\frac{-t_r}{111.3d}})(\\frac{M_{Ni}}{M_{}})\\times 10^{43} erg~s^{-1},$ where $t_r$ is the rise time of the bolometric light curve, and $M_{Ni}$ is the $^{56}$ Ni mass (in units of solar masses, M$_{}$ ).",
"From the discovery information [11] of SN 2012fr, we estimate the rise time to be about 18 days in the $V$ band.",
"Inserting this value and the maximum bolometric luminosity into Equation (1), we derive a nickel mass of 0.88 $\\pm $ 0.08M$_{}$ for SN 2012fr.",
"This value is smaller than that of SN 1991T (e.g., 1.1M$_{}$ ; Contardo et al.",
"2006) and larger than that of SN 2005cf (e.g., 0.78 M$_{}$ ; Wang et al.",
"2009b), SN 2009ig (e.g., 0.80 M$_{}$ ; Sahu et al.",
"2011), and SN 2011fe (e.g., 0.53 M$_{}$ ; Pereira et al.",
"2013).",
"Figure: Comparison of various spectroscopic indicators from SN 2012fr with those from other SNe Ia as measured by Blondin et al.",
"(2012), Silverman et al.",
"(2012), Wang et al.",
"(2009a), and this paper.",
"The selected sample have spectra within ±3\\pm 3 days from BB-band maximum.",
"Panel (a): the Si2 λ\\lambda 5972 vs. Si2 λ\\lambda 6355 at maximum light with spectroscopic subclasses defined by Branch et al.",
"(2009); panel (b): the EWEW vs. velocity of Si2 6355 at maximum light with subclasses defined by Wang et al.",
"(2009a); Panels (c) and (d): the EWEW and V Si II λ6355V_\\mathrm {Si~II \\lambda 6355} vs. Δm 15 \\Delta m _{15}(B) at maximum light."
],
[
"The Spectroscopic Classification",
"[4] and [57] presented detailed comparison study of different classification schemes based on the large SNe Ia spectral datasets from the CfA and Berkeley SuperNova Ia Program, respectively.",
"From the near-maximum light spectrum of SN 2012fr, the equivalent width of Si2 $\\lambda $ 6355 and Si2 $\\lambda $ 5972 lines is measured to be about 60Å and 4Å (see Table 8), respectively.",
"The weak Si2 absorption shows some similarities to the SS/91T subclass of SNe Ia, though its location is also close to the core-normal subclass in the Branch diagram, as shown in Figure REF a and c. On the other hand, SN 2012fr can be clearly put into the LVG group of the classification system of Benetti et al.",
"(2005) in terms of the lower velocity gradient.",
"In the Wang et al.",
"diagram, however, SN 2012fr resides between the HV and the 91T-like subclasses (Figure REF b and d).",
"SN 2012fr has an expansion velocity slightly higher than a normal SN Ia, but it bears many properties that are not seen in the HV subclass.",
"The weak Si2 absorption (see Table REF ) near the maximum light, the lower velocity gradient, and the higher luminosity are all reminiscent of the properties of the 91T-like objects.",
"Although the HVFs of Ca2 IR are also found to be relatively strong for those events [13], the absolute strength of these HVFs are usually very weak in the 91T-like objects (e.g., with EW $<$ 50Å ).",
"Moreover, SN 2012fr has very strong HVF of Si2$\\lambda $ 6355 but this feature is rarely detected in the 91T-like events.",
"This discrepancy further complicates the classification of SN 2012fr.",
"cccc[!th] 0pt The Pseudo EW of Si2 Absorption Line of SN 2012fr Epoch $\\lambda $ 5972 $\\lambda $ 6355 HVFs($\\lambda $ 6355) -13.43 21.6 103.8 -12.51 26.0 138.1 -11.53 32.2 98.6 -7.25 4.8 50.5 20.86 -6.29 7.2a 51.6 18.8 -4.28 7.7a 57.9 -2.28 4.3 62.0 -0.27 4.1 61.8 +2.74 9.1a 67.8 +3.72 10.1a 62.4 +6.74 7.9 62.2 +8.72 6.2 65.8 +10.68 62.5 +12.70 60.5 +16.67 65.5 The pEW of Si2 $\\lambda $ 5972, $\\lambda $ 6355 absorption in the spectra of SN 2012fr, in units of Å. aBlended with bluer absorption line.",
"To examine whether SN 2012fr forms a separate subgroup of SNe Ia, we attempt to collect a sample like SN 2012fr to get a better statistical study of their photometric and spectroscopic behaviors.",
"Figure REF shows the distribution of SNe IaThe spectral and photometric data used in the analysis are taken from Blondin et al.",
"2012, Silverman et al.",
"2012, Hicken et al.",
"2009, and Ganeshalingam et al.",
"2010.",
"We found that SN 2012fr and a few other SNe Ia (including SNe 2001eh, 2004br, 2005eq, 2006cm, and 2007bz) cluster tightly together in the plot, all showing unusually narrow line profiles of Si2$\\lambda $ 6355 and Ca II IR triplet.",
"Another common feature for these objects is that they all have smaller light-curve decline rates, with $\\Delta m_{15}$ $\\lesssim $ 1.0 mag.",
"It is notable that most of the Narrow-Lined (NL) SNe Ia can be also put into the SS/91T-like subclass.",
"Inspection of Figure REF d reveals that the NL subclass also overlap largely with the 91T-like subclass, with 10, 000 km s$^{-1}$$\\lesssim v_\\mathrm {Si~II}$ $\\lesssim $ 12,000 km s$^{-1}$ .",
"The above analysis indicates that SN 2012fr could be a subset of the SS/91T-like subclass.",
"The differences seen in the earlier phases, such as the HVFs of Si2 and Ca2, may be explained by the viewing-angle effect.",
"Maund et al.",
"(2013) obtained the spectropolarimetric measurements of SN 2012fr and found the high-velocity components are highly polarized in the early phases, with the degree of polarization being 0.40% for Si II line and 0.85% for Ca II lines.",
"At around the maximum light, the polarizations for the photospheric components of Si II and Ca II are found to be 0.65% and 0.54%, respectively.",
"On the other hand, the line polarizations of 91T-like SNe Ia are usually at low levels (e.g., $<$ 0.20%) according to Wang et al.",
"(2007).",
"We therefore propose that SN 2012fr may be a counterpart of the 91T-like SNe Ia, but it was viewed in a direction where the ejecta has a clumpy or shell-like structure."
],
[
"Conclusion",
"We have presented the ultraviolet and optical observations of SN 2012fr from the $Swift$ UVOT and the Li-Jiang 2.4-m telescope.",
"Our observations show that SN 2012fr is a luminous SN Ia.",
"The maximum bolometric luminosity deduced from the UV and optical light curves is $(1.82\\pm 0.15) \\times 10^{43}$ erg s$^{-1}$ , corresponding to a synthesized nickel mass of 0.88 $\\pm $ 0.08 M$_{}$ .",
"Generally, the spectral evolution of SN 2012fr show some similarities to the HV SNe Ia in the early phase because of showing detached HVFs but it becomes more similar to the 91T-like subclass in the near-maximum and later phases.",
"In the very earlier phases, strong HVFs are present in the Ca2 IR triplet, Ca2 H&K, and Si2 $\\lambda $ 6355 lines at velocities of 22,000–25,000 km s$^{-1}$ .",
"The absorption of Si2 and Ca2 formed from the photosphere has a velocity of 12,000 km s$^{-1}$ and exhibits an unusually narrow line profile.",
"A comparison with other SNe Ia indicates that SN 2012fr is characterized by narrow-lined, photospheric component near the maximum light.",
"We found that the SNe Ia similar to SN 2012fr are usually slow-decliners with $\\Delta m_{15}$ (B)$\\lesssim $ 1.0 mag, and they show a large overlap with the members of the shallow-silicon/SN 1991T-like subclasses in the Branch et al.",
"and Wang et al.",
"classification schemes.",
"These results, together with the asymmetric high-velocity material, suggest that SN 2012fr may represent a subset of 91T-like SNe Ia viewed at different angles.",
"A larger sample of SN 2012fr-like explosions with very early spectral observations as well as polarization measurements will help us understand the frequency of objects that show detached HVFs and their geometry (e.g., Childress et al.",
"2014), which will finally enable us to set stringent constraints on the nature of the progenitor system for some particular types of SNe Ia.",
"We thank very much the anonymous referee for his/her constructive suggestions which helped to improve the paper a lot.",
"We are also extremely grateful to the staff of Li-Jiang Observatory, Yunnan Observatories of China for the observation and technological support, particularly Yu-Feng Fan, Wei-Min Yi, Chuan-Jun Wang, Yu-Xin Xin, Jiang-Guo Wang, Liang Chang and Shou-Sheng He.",
"We acknowledge the use of public data from the $Swfit$ data archive and thanks to Bryan Irby of NASA Goddard Space Flight Center for the kind help of HEASoft installation.",
"Thanks goes to Yan Gao (YNAO) for the literary suggestion and modification.",
"And we are also very grateful to Bernard Heathcote ($t\\approx -13$ days), and Terry Bohlsen ($t\\approx -12, -11$ days) who uploaded their earlier-phase spectra of SN 2012fr to ARAS.",
"X. Wang is supported by the Major State Basic Research Development Program (2013CB834903), the National Natural Science Foundation of China (NSFC grants 11073013, 11178003, 11325313) and the Foundation of Tsinghua University (2011Z02170).",
"The work of J. M. Bai is supported by the National Natural Science Foundation of China (NSFC grants 11133006, 11361140347) and the Strategic Priority Research Program “The Emergence of Cosmological Structures\" of the Chinese Academy of Sciences (grant No.",
"XDB09000000).",
"B. Wang is supported by National Natural Science Foundation of China (NSFC grants 11322327 and 11103072).",
"T. Zhang is supported by National Natural Science Foundation of China (NSFC grant 11203034).",
"Funding for the LJ 2.4-m telescope has been provided by CAS and the People's Government of Yunnan Province."
]
] | 1403.0398 |
[
[
"Modeling the emergence of modular leadership hierarchy during the\n collective motion of herds made of harems"
],
[
"Abstract Gregarious animals need to make collective decisions in order to keep their cohesiveness.",
"Several species of them live in multilevel societies, and form herds composed of smaller communities.",
"We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected sub-groups (e.g.",
"harems) by combining self organization and social dynamics.",
"It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision.",
"Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortob\\'agy, Hungary).",
"We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal."
],
[
"12em"
],
[
"1.14em 1,2]K. Ozogány 1,2]T. Vicsek [1]Department of Biological Physics, Eötvös University, Pázmány Péter s. 1/A, 1117 Budapest, Hungary [2]MTA-ELTE Statistical and Biological Physics Research Group, Pázmány Péter s. 1/A, 1117 Budapest, Hungary Gregarious animals need to make collective decisions in order to keep their cohesiveness.",
"Several species of them live in multilevel societies, and form herds composed of smaller communities.",
"We present a model for the development of a leadership hierarchy in a herd consisting of loosely connected sub-groups (e.g.",
"harems) by combining self organization and social dynamics.",
"It starts from unfamiliar individuals without relationships and reproduces the emergence of a hierarchical and modular leadership network that promotes an effective spreading of the decisions from more capable individuals to the others, and thus gives rise to a beneficial collective decision.",
"Our results stemming from the model are in a good agreement with our observations of a Przewalski horse herd (Hortobágy, Hungary).",
"We find that the harem-leader to harem-member ratio observed in Przewalski horses corresponds to an optimal network in this approach regarding common success, and that the observed and modeled harem size distributions are close to a lognormal."
],
[
"Keywords",
"Collective animal behaviour $\\cdot $ Leadership hierarchy $\\cdot $ Multilevel societies $\\cdot $ Collective decision making $\\cdot $ Modular hierarchy"
],
[
"Introduction",
"Like in human communities, several unique species of gregarious animals have developed social structures based on multiple levels of hierarchical organization [12], [13].",
"Small groups of closely related individuals can unite in clans which can form bands or loose aggregations.",
"This phenomenon appears in several different taxonomical orders, common examples range from primates [18], [1], through elephants [32] and whales [31], [4] to equids [28], [9].",
"The smallest stable sub-unit where strong bonds exist between members can be a family group based on kinship.",
"One basic unit form is a matrilineal family group consisting of one matriarch and her descendants (african elephant [32], sperm whale[31], killer whale [4]).",
"Another basic form is a one-male reproductive unit, a harem that consists of several breeding females, their subadult descendants, and is dominated and guarded by only one male (Przewalski horses [6], plains zebras [28]), or it sometimes includes several non-dominant males as well (hamadryas baboons [18], geladas [7]).",
"These highly social animals build stable, sometimes lifetime long communities, based on a complex relationship network and a strong hierarchical order.",
"When a group of animals moves together or makes a collective decision a consistent leadership hierarchy may be observed in many cases, that can serve for a facile flow of information, as it was demonstrated in pigeons [21].",
"The leadership hierarchy in a group may be completely independent from dominance [22].",
"However, in social animals dominance can be a determining factor in leadership.",
"This is the case e.g.",
"in chacma baboon [16].",
"Regarding the common success the whole group would gain if led by the one with the best knowledge about the good direction to food resources, but this type of leadership was observed only in few cases, e.g.",
"in bottlenose dolphins [19].",
"Some other traits are empirically shown to affect the individual's chance of becoming a leader, such as central position in the social network [29], increased nutrient requirements [10], [30] or age [33].",
"The dynamics of leadership in a complex society can be well described in terms of hierarchical networks, where the leader-follower relationships between group members are associated with directed connections between the nodes.",
"This approach is very useful in understanding collective behaviour.",
"A collective decision is often based on copying the groupmates$^{\\prime }$ individual decisions [25], and in the network view the spreading of the copied behaviour can be interpreted as an information flow through the directed edges of the network.",
"In principle a network can have a modular structure in addition to being hierarchical.",
"In the past a few models have been proposed to display these features, however, in those cases the edges between the units were not directed [27], [11], thus, did not correspond to leader-follower relationships.",
"Our aim is to construct a model which reproduces the emergence of a modular and hierarchical leadership network, that is similar to the above introduced phenomenon of the \"group of groups\" in a collective decision-making context.",
"We consider the case when leadership is mediated by social relations and dominance.",
"We assume that the leader-follower connections are more dense inside the sub-units, as the social bonds are also usually more tight between the sub-unit members, and that the dominant individuals are able to affect the decisions (or movement directions) of their groupmates.",
"In the model we try to find simple rules which cause the emergence of smaller sub-units inside a group.",
"First, we suppose that every individual has an upper limit for the number of bonds he is able to maintain due to the cost of sociality.",
"Developing a social bond can require remarkable time, or maintaining a harem of a given number of females is a costly task for a male.",
"This upper limit of possible bonds introduces a typical sub-unit size.",
"Besides the limitation in bond number, there is also a need for intra-unit cohesive forces that give rise to higher connectivity inside the sub-units.",
"Figure: The motivation of our model is the understanding of the leadership hierarchy in Przewalski horses.",
"Photo by K. OzogányIn large animal groups (bird flocks, fish schools, insect swarms), where individuals can communicate only locally and individual identification is costly, self organizing can be the main driving mechanism of collective decision making processes.",
"In small stable groups, where global communication is possible and complex social relationships can develop between members, sociality is more important in leadership [17], [15].",
"Our model on the leadership hierarchy is a combination both of the above mechanisms, since this can be a suitable approach for the case of social animals living in big groups [26].",
"As a result, it leads to the spontaneous emergence of a modular hierarchical network underlying a group composed of sub-groups.",
"The observation of the collective movements of a free-ranged herd of Przewalski horses (Equus ferus przewalskii, Fig.",
"REF ) helps us in defining the rules of the algorithm.",
"This herd consists of stable and non-overlapping harems, each of them being guarded and herded by one stallion.",
"The observed collective motion pattern of the herd shows the borderlines of the individual harems through the harem members$^{\\prime }$ cohesive motion (Fig.",
"REF ).",
"Thus, it can be assumed that the leadership is affected by the horses$^{\\prime }$ social bonds.",
"We aim to reproduce the special case of the leadership hierarchy of a wild horse herd.",
"Some aspects of the behaviour of harem-living animals are used to formulate social rules, thus our model is likely to be applicable for other species as well."
],
[
"Observations",
"It was found in plains zebras that movement initiations inside the harems are determined by a consistent hierarchy of the individuals, and the position during travelling correlates with the initiation order [10].",
"In addition, it was shown that movements of the herd are dominated by lactating females at two different levels of social organization.",
"The direction of motion of individual harems were initiated by lactating females, while the motion of the herd was likely to be determined by harems containing more lactating females [10].",
"Both of plains zebras and Przewalski horses live in similar social organization, in a fission-fusion system of harems and bachelor groups.",
"Hence, we suppose that a consistent leadership hierarchy can exist over movements among the harems of a Przewalski horse herd, as well.",
"In order to get an insight into this phenomenon we make an estimation based on aerial images, using the assumption that the harems$^{\\prime }$ relative position occupied in the herd while moving can be an appropriate indicator of the rank in the leadership hierarchy.",
"Figure: Przewalski horse herd during movement (n≈150n\\approx 150).",
"The spatial distribution of the herd roughly shows the borderlines of the harems.",
"Colored areas emphasize the individual harems identified due to the harem members ' ^{\\prime } cohesive motion.",
"White edges point from the leader to the follower harem, and are defined between the neighbouring harems within a given range.",
"As the herd takes on a V formation, from two harems connected by an edge, the one closer to the tip of the V is identified as leader.",
"If both are a similar distance from the tip then the more centered one is the leader.",
"We base this leadership definition on the observation that in plains zebras the individuals in the front are more likely to lead .",
"The black arrow shows the direction of motion.",
"The bachelor groups in the bottom left corner and on the right, and the oxen in the top right corner are not considered.We use small flying robots and a blimp filled by helium to capture aerial records of a cohesive herd of Przewalski horses living under semi-reserve conditions in the Hortobágy National Park, Hungary.",
"The herd has around 240 individuals, and consists of stable and non-overlapping harems, with sizes ranging from 2 to 19, including the harem stallion.",
"There are less stable bachelor groups surrounding the herd of harems, that are not considered.",
"The spatial distribution of the horses roughly shows their social organization, since harem members keep closer to each other than the typical distance between the harems (Fig.",
"REF ).",
"During the collective movements the harems remain as cohesive units inside the herd and the leadership roles among them seem to hold on for several minutes (Movie S1).",
"At longer time scale the leadership may fluctuate to a certain extent, but it may have a well defined structure on average, similarly to the individual hierarchies reported in other species [21], [3].",
"The estimated leadership network of the harems is demonstrated in Fig.",
"REF ., and its hierarchical layout is visualized on the left side of Fig.",
"REF (a) using the reaching centrality method [20].",
"We have analysed several images recorded on different days, and they show a sign of some consistency (Fig.",
"REF and REF ).",
"Our definitions of the nodes and edges of the harem network are the following.",
"A node represents a harem.",
"Individuals who are within one horse length of each other, or keep together during moving, belong to the same harem.",
"As a verification of our harem definition the number of adult and infant horses in each harem was compared with the catalogue of harems established by the national park.",
"Harems which are within a given interaction range of each other, and are seen directly (i.e.",
"not covered by another harem) by at least one of the pair, are linked with an edge.",
"The envelope of the herd is a V formation, pointed in the direction of the movement.",
"Thus from two harems we define as leader the one which is closer to the tip of the V. In the case when both of them are roughly the same distance from the tip, then the more central one is defined as leader.",
"The direction of an edge points from the leader to the follower harem.",
"This is only a first estimation of the precise leadership hierarchy, which is to be determined through a deliberate analysis of individual tracks.",
"The identification of the harems includes some uncertainty as well, since harem members disperse sometimes for short time intervals making the finding of the borderlines of the harems in a still image difficult.",
"The right side of Fig.",
"REF (a) shows two enlarged harems including their estimated internal leadership structure of the individuals.",
"The edges between the individuals are defined in a similar way as the ones of the harem network, based on the individuals$^{\\prime }$ position occupied in the particular harem on the still picture.",
"We denote as the leader of the harem (L) the individual that is closest to the tip of the harem, and denote as follower harem members (F) all the other individuals further from the tip.",
"We define an edge between two individuals if they are seen directly by at least one of the pair, with the direction pointing from the leader individual closer to the front of the harem towards the follower individual."
],
[
"Model description",
"The starting point of our model is the agent-based model of [23].",
"It consists of individuals facing a problem solving situation iteratively in each time step.",
"They have an environment that changes slowly in time over several discrete states.",
"The individuals aim to guess the actual state of the environment, and the ones that guess correctly gain score.",
"They have diverse abilities to guess the state of the environment and are also allowed to seek advice from other groupmates.",
"In the context of collective motion the environment can represent the seasonally changing habitat of the group, including the actually accessible drinking or feeding places, and a good choice represents a good direction towards it.",
"More generally, the environment can be any decision making situation, where individuals can choose from several discrete decisions.",
"The proper choice is the appropriate behaviour, which enables to benefit from the situation.",
"In order to keep the herd$^{\\prime }$ s cohesiveness, the individuals should synchronize their behaviour, which can be achieved by copying groupmates.",
"Every individual in the model is likely to consider the decision of several other groupmates and ponders over them, taking into account its own guess as well.",
"Finally it decides on its own, or copies a groupmate$^{\\prime }$ s decision, who seems to have better ability in guessing.",
"Each individual estimates the perceived ability of others from whom it already copied a decision based on the guessing success, and tries to follow those who seem to be the best.",
"As the model proceeds stable leader-follower relationships emerge between the individuals, where the follower individual considers the decision of the leader individual when making his own decision.",
"In other words, the model leads to spontaneous emergence of a leadership hierarchy, where the leadership hierarchy is defined through the information flow: we assign an edge where information exchange is between two individuals, with the arrow directed from the source of information to the target.",
"In order to arrive at a herd which consists of loosely connected harems, we introduce two types of individuals into the model showing different behaviour.",
"The two types are called leader-type (L) and follower-type (F) individuals.",
"The leader-type individuals have some properties that make them likely to become the leader of a sub-unit.",
"In addition, they have organizational skills, which increase the sub-unit$^{\\prime }$ s cohesiveness.",
"In Przewalski horses the stallions play a dominant role by keeping the harem together, but the direction of motion can also be strongly influenced by a dominant mare.",
"For the sake of simplicity, in the model we assign these two particular roles (leading in movements and sub-unit managing role or herding) to one special individual type, the leader-type individual.",
"It is important to note, that the model does not assume that in horses the stallion leads the harem, only the fact that there is one individual in each harem that has more influence in leading than the others, and that there are intra-unit cohesive forces in the harems that give rise to higher connectivity among members.",
"The leader-type to follower-type ratio is preferably unequal.",
"Every individual of both types tries to maximize its own guessing success by following those who seem to be better in finding the good answer.",
"Leader-type individuals, in addition, try to collect followers and build non-overlapping harems of follower-type individuals due to herding.",
"Three herding behaviours are integrated into the model.",
"First, the leader-types do not favour if their harem members follow individuals from another harems.",
"They try to prevent these leader-follower relationships and thus try to prevent their followers to leave their harem.",
"This behaviour serves as a cohesive force inside the harems.",
"The leader-types also try to prevent other leader-types to follow their harem members.",
"Finally, every leader-type individual tries to increase its harem size by herding those follower-type individuals who are not belonging to any harem yet.",
"The two types of individuals differ in their typical ability in guessing the right state of the environment.",
"The largest ability values are matched to leader-type individuals enabling them to assume higher positions within the hierarchy.",
"Costly tasks are incorporated as limitations in the model.",
"The cost of having links with other individuals is included such a way, that every individual has a link capacity, which is an upper bound of outgoing edges they are able to establish.",
"The two types do not differ in their typical link capacities.",
"For the leader-types, its value also determines the fitness of the individual, since the frequency that the individual shows herding behaviour and the probability that it succeeds in herding is proportional to its link capacity.",
"On the other hand, it determines the maximum harem size of a given leader-type individual.",
"Thus the cost of fighting and herding is included in the link capacity as well.",
"The model starts from naive individuals without relationships, who do not know the abilities of others.",
"As it proceeds the individuals learn about each others$^{\\prime }$ abilities and try to copy decisions from the ones who seem to be more successful in guessing (Movie S2).",
"The resulting hierarchy of a single run is a realization of a leadership structure between the given individuals, which emerges spontaneously, and develops to a stable and beneficial configuration.",
"It is a model of a natural animal leadership structure achieved during a given time of living together and remaining quite consistent."
],
[
"Individuals and their environment",
"The model contains $n$ individuals, of which $m$ are assigned as leader-type and $n-m$ as follower-type.",
"They are embedded in an environment that changes in an unpredictable way, and the individuals have to guess its state to gain benefit.",
"The environment is always in exactly one of the $l$ possible discrete states.",
"The simulation consists of steps, and the state of the environment is constant within a step but may change between steps with probability $p$ .",
"Thus, the characteristic time between state-flips is about $1/p$ steps, during which the individuals can \"learn\" about the environment$^{\\prime }$ s actual value.",
"When environment changes state, the given state is replaced uniformly by a randomly chosen other one.",
"Within each step, individuals must make a decision in the sense, that they must guess the actual state of the environment.",
"Each individual has a predefined ability (e.g.",
"$a_{i}$ denotes the ability of individual $i$ ), which is the probability that the individual can make a proper guess of the state of the environment without any external information.",
"If individual $i$ decides on its own (without copying anyone), than it chooses the current state of the environment with probability $a_i$ , or chooses any other state uniformly from the non-correct states of the environment with probability $1-a_i$ .",
"Each individual has the information about its own ability, but does not know a priori the abilities of other individuals.",
"The final decision of the individuals concerning the actual state of the environment in each round is composed of their own guess and from some copied decisions of other individuals.",
"The number of other individuals a given person tries to copy in a step is a predefined value and depends on its own ability.",
"Formally, individual $i$ considers the decision of $q(1-a_i)$ other individuals (rounded up to the nearest integer), where $a_i$ denotes its own ability and $q$ denotes the maximal number of groupmates someone can ask in a round.",
"Individuals with higher abilities will try to copy a smaller number of other individuals, since they confide in their own guess.",
"Each individual has an upper limit of the number of other individuals whom he can provide information in a single round.",
"This upper bound is called link capacity (e.g.",
"$c_{i}$ denotes the link capacity of the individual $i$ ).",
"The individuals have information about the type, but not about the ability and link capacity of others.",
"They can estimate the ability of those whom they have already copied.",
"The estimation of another$^{\\prime }$ s ability can be interpreted as a trust in someone$^{\\prime }$ s decision.",
"The values of the estimated or perceived ability of others are stored in a matrix, where the element $t_{ij}$ denotes the ability score of individual $j$ as perceived by individual $i$ , and is calculated by using the rule of succession $t_{ij} = \\frac{s_{ij} + 1}{n_{ij}+l},$ where $s_{ij}$ is the number of rounds in which individual $i$ received a correct answer from individual $j$ and $n_{ij}$ is the number of rounds when individual $i$ received any information from individual $j$ .",
"Thus, $t_{ij}$ is an estimation by individual $i$ of success rate of individual $j$ in finding the correct answer.",
"The $t_{ii}$ diagonal element of the perceived ability matrix is equal to $a_i$ , since every individual knows its own ability exactly.",
"In each round individual $i$ considers copying those who it perceives as the best (i.e.",
"$q(1-a_i)$ other individuals with the highest $t_{ij}$ , $j=1,2,...,n$ perceived ability scores).",
"The perceived ability score also determines the weight of the copied answer in a person$^{\\prime }$ s final decision.",
"The leadership hierarchy emerges from the information flow between the individuals, where a directed edge of the hierarchical structure conveys a copied answer from an individual to another.",
"The precise mechanism of individual decisions is described in the next paragraph.",
"Individuals who can infer properly the current state, gain a positive feedback at the end of the step (they gain 1 score), the ones who do not infer the state, gain a negative feedback (they gain 0 score).",
"The total score of an individual is the exponential moving average of all the feedbacks gained during the simulation with a half life of 50 steps (this means that in the average the weight of a feedback vanishes exponentially with time, e.g.",
"a feedback received 50 steps ago has a weight of $0.5$ ).",
"Both the abilities and the link capacities of the individuals are diverse, and are random variables chosen from a given distribution.",
"The abilities are drawn from a bounded Pareto distribution, since the fat-tailed distributions maximize group performance [34] and promote the emergence of hierarchy [23].",
"From the ordered abilities the $m$ largest values are assigned to leader-type individuals and the rest to follower-types.",
"The higher ability values enable the leader-type individuals to assume higher ranking positions in the hierarchy.",
"If the abilities of the leader-type and follower-type individuals would be drawn from different distributions, then when changing their ratio, the overall ability distribution of the group would change.",
"The results would not be comparable to each other, because the ability distribution has a notable effect on the overall success [23].",
"The link capacities are drawn from several different distributions to test their effect on the sizes of the emerging sub-units, namely uniform, Poisson, delta and lognormal distributions with varying parameters.",
"The short descriptions and the default values of the input parameters used in the simulations are listed in Table REF .",
"Table: Description and default values of the input parameters used in the simulations"
],
[
"Decision making",
"Each single round of the simulation consists of the following phases: The state of the environment is calculated: the state of the previous round remains with probability $1-p$ , or changes to a different state with probability $p$ .",
"Individuals make their own guesses about the state of the environment based on their abilities.",
"The leader-type individuals do their herding behaviour which is described in the next section.",
"$rn$ herding attempts take place in each round, where $r$ denotes the herding frequency, ranging from 0 to 1, and $n$ denotes the number of all individuals.",
"Every individual nominates $q(1-a_i)$ other individuals (rounded up to the nearest integer) whom it wishes to copy in the current round.",
"Individual $i$ considers the $i$ th row of the perceived ability matrix (the $t_{ij}$ elements, $j=1,2,...,n$ ), and nominates the individuals with the $q(1-a_i)$ highest perceived ability values.",
"Every individual accepts at most $c_i$ other individuals from those who nominated him, preferring those whom he has already passed information in the previous rounds.",
"Individuals propagate their decision of the environmental state from the previous round to the accepted ones.",
"It is important that the decisions from previous round are passed, because the propagation of decisions from the current round could not be solved in a synchronical way.",
"The remaining (not accepted) individuals who nominated the given person will not get any information in this round from him.",
"Individuals maintain a taboo list, and do not nominate again in the next round the ones who can not propagate an answer to them.",
"Individuals make their decisions by calculating the majority opinion taking into account the answers of the copied individuals and their own guess.",
"Individual $i$ weights the informations about the environmental state, where the weight of a copied answer from individual $j$ is the $t_{ij}$ perceived ability score, and the weight of its own guess is its $t_{ii}=a_i$ ability.",
"Then it chooses the answer with the highest summarized weight.",
"Individuals get a feedback about the current state of the environment, and they re-calculate the perceived abilities of others by updating the $n_{ij}$ and $s_{ij}$ counters in (REF ).",
"The above described single round repeats iteratively in the simulation.",
"In each round a directed network is defined between the individuals, where the edges refer to copied decisions.",
"In other words, if individual $j$ considered the decision of individual $i$ with some weight in his choice than an edge is added to the network with $i$ being the source and $j$ being the target node (i.e.",
"$i$ being the leader and $j$ being the follower).",
"The leader-follower relationships occur spontaneously from the small initial perturbation of the perceived ability matrix and gradually develop to a stable configuration.",
"A random noise is included into the model.",
"Before nominating other individuals for copying we add Gaussian noise with zero expected value and a given standard deviation to the perceived ability values."
],
[
"Leader-type individuals",
"Every individual has a predefined type, they are either leader-type (L) or follower-type (F).",
"Individuals know not only their own type, but also the type of the others.",
"The leader-type individuals are responsible for the forming of sub-units or harems.",
"The harems are identified through the leader-type individuals, e.g.",
"the $h_i$ harem denotes the harem led by the $L_i$ leader-type individual.",
"Each individual is aware of which harems it belongs to.",
"An $L_i$ leader-type individual belongs to its own $h_i$ harem.",
"An $F_j$ follower-type individual belongs to the $h_i$ harem, if $F_j$ considered the decision of $L_i$ in the current round (in other words $F_j$ followed $L_i$ ).",
"Thus, the $h_i$ harem is the union of the $L_i$ leader-type individual and all the follower-type individuals that followed $L_i$ in the current round.",
"These individuals are also called the members of the $h_i$ harem.",
"The size of a harem is the number of individuals belonging to the harem (including the leader as well).",
"Note, that with this definition a follower-type individual can belong to one, more or zero harems.",
"In this latter case it is called a lonely individual.",
"Individuals have a predefined $s_i$ fitness parameter, which in the case of the leader-types, characterizes their potential of maintaining a harem, in the sense that the frequency and success rate of their herding attempts is proportional to their fitness.",
"The fitness of $L_i$ is equal to its link capacity, $s_i=c_i$ .",
"Thus, the link capacity parameter determines, on one hand, $L_i$$^{\\prime }$ s success rate in herding, and on the other hand, an upper bound for the possible size of the harem that $L_i$ is able to keep together.",
"A herding attempt consists of the following three steps: An $L_i$ leader-type individual is chosen randomly for herding with probability proportional to its $s_i$ fitness.",
"For $L_i$ chosen for herding, a list of follower-type individuals is identified.",
"The list includes those members of harem $h_i$ who also follow an individual from a different $h_k$ harem (either an L or an F individual), or are followed by another $L_m$ leader-type individual.",
"The list includes also all the lonely individuals in the case if the herding $L_i$ has free links.",
"In other words, all the possible edge operations are identified, including the adding of new edges and the deleting of undesired edges.",
"A possible edge operation is chosen randomly with uniform probability.",
"If an edge is chosen for adding, directed towards a lonely individual, than the herding $L_i$ reinforces its ability score perceived by the $F_j$ lonely individual.",
"Formally, it means that both the $s_{ji}$ and $n_{ji}$ values in (REF ) are increased by 1 in the $t_{ji}$ perceived ability score.",
"This has an equivalent effect to $L_i$ passing to $F_j$ a proper answer about the state of the environment in the decision making process described in the previous section.",
"Since after this attempt the ability score of $L_i$ perceived by $F_j$ increases, $F_j$ is more probable to follow $L_i$ in the next rounds.",
"If an edge is chosen for deletion, three cases can occur.",
"In the first case, $F_j$ from the $h_i$ harem follows $F_k$ from a different harem.",
"$L_i$ succeeds in breaking this edge randomly with a probability proportional to its $s_{i}$ fitness.",
"Breaking a relationship between a follower $F_j$ and a leader $F_k$ means to reset the perceived ability score $t_{jk}$ to its initial value.",
"In the second case, the undesired edge points from another $L_k$ to an $F_j$ from the $h_i$ harem (in other words $F_j$ follows another $L_k$ beside $L_i$ ).",
"This time $L_i$ and $L_k$ $\"$ fight$\"$ for $F_j$ , and $L_i$ wins over $L_k$ with probability $s_{i}/(s_{i}+s_{k})$ and looses with probability $s_{k}/(s_{i}+s_{k})$ .",
"Then the winner individual reinforces his ability score perceived by $F_j$ , and breaks the edge between $F_j$ and the losing individual.",
"In the third case, the undesired edge points from $F_j$ from the $h_i$ harem to another $L_k$ ($F_j$ is followed by $L_k$ ).",
"$L_i$ and $L_k$ $\"$ fight$\"$ for $F_j$ as in the previous case."
],
[
"Group performance and structural properties of the network",
"To characterize the overall success of the herd, we quantify the common performance of the group through the number of times when the individuals could infer the actual state of the environment correctly.",
"Each individual in each time step scores 1 if successfully guessing the environmental state and 0 if not.",
"The $p_i$ performance of an individual is the exponential moving time average of his score history, with a half life of 50 steps.",
"This means that the scores of the past rounds are weighted with an exponentially decaying factor, where the score of the current round has a weight of 1, while the score 50 steps ago has a weight of $1/2$ .",
"Individual performance is thus simply the ratio of proper guesses of someone, where past guesses have less weight as time passes.",
"The performance of the whole society is the average performance of the individuals.",
"Without the possibility of copying the performance of an individual would be his $a_i$ ability on average.",
"According to this, the relative improvement of the overall performance - compared to a group without copying - can be defined as the common performance minus the average ability, divided by the average ability expressed in percentages.",
"Formally the relative performance improvement which we use to measure the success of the society is: $P=\\frac{\\left\\langle p_i\\right\\rangle - \\left\\langle a_i\\right\\rangle }{\\left\\langle a_i\\right\\rangle }\\cdot 100\\%,$ where $a_i$ are the abilities, and $\\left\\langle \\right\\rangle $ denotes averaging over individuals.",
"A relative performance improvement of $0\\%$ thus means, that the group is performing at the same level as a group without copying, and a relative performance improvement of $100\\%$ means that it performs twice as well.",
"In order to characterize the extent of the hierarchical organization of the network we use three measures, the fraction of noncyclic edges, global reaching centrality [20] and the fraction of the largest cycle-free arc set [8].",
"The reaching centrality method is based on the assumption that the rank of the nodes is related to their impact on the whole network.",
"A node$^{\\prime }$ s impact can be quantified with its local reaching centrality, which is the proportion of all nodes reachable from it via outgoing edges.",
"The global reaching centrality (GRC) of a network is related to the heterogeneity of the local reaching centrality distribution of its nodes which is wider for hierarchical structures.",
"Thus, the quantity GRC measures the level of hierarchy of a network, with a value close to zero corresponding to no hierarchy, while GRC being about 1 signalling a highly hierarchical network.",
"In order to get a qualitative insight of the layout of the emerging leadership network, we visualize the individual and the harem network with the reaching centrality method [20].",
"This visualization method is based on the local reaching centralities of the nodes, where nodes with similar values lie in the same layer, and the one with the highest value is on the top.",
"To reveal modular structure and find the communities we use the clique percolation method and the CFinder software for visualization [2], [24].",
"The convergence is indicated by the fraction of changed edges in a time step going to zero, and the harem sizes converging to stable values."
],
[
"Results",
"As the model proceeds iteratively, a network of leadership emerges.",
"The individual leadership network consists of nodes representing the individuals, and the directed edges between them show whose decision was copied by whom in previous rounds.",
"Since as a consequence of herding sub-groups are expected to emerge led by a leader-type individual, we define the harem leadership network in the simplest way, by considering only the subgraph of the leader-type individuals.",
"In the harem leadership network the leader-type individuals are the nodes, and the connections between them are the edges.",
"Note that the edges in the empirical network of Fig.",
"REF denote the possible leader-follower relationships, and not the realized ones, by definition.",
"A given harem $i$ has an incoming edge pointing from harem $j$ , if $j$ can be seen from and thus followed by $i$ , but it does not necessarily mean that $i$ followed $j$ .",
"The situation is the same in the model, an individual (or a harem) $i$ has an incoming edge from $j$ if its decision is considered by $i$ .",
"But in making its own decision $i$ weights all the considered decisions, and may finally choose a different decision than $j$ .",
"Figure: A typical leadership hierarchy among harems (a) in the experiment of the Przewalski horses based on Fig.",
"and (b) a very similar network resulting from our model.",
"The nodes with names starting with L denote harems of a given harem leader, and the directed edges point from the leader to the follower harem, showing thus the flow of information.",
"Visualization is performed by the reaching centrality layout with z=0.1z = 0.1.",
"The global reaching centrality values of the two harem networks are similar, 0.65 and 0.67 for the experimental, and for the model, respectively, which indicates a similar extent of hierarchy.",
"The enlargements of the blue square areas reveal the internal connections of individuals inside some harems.",
"The pink nodes with names starting with F denote the harem members, and orange emphasizes the harem leaders whose harem is enlarged.",
"Light blue areas denote communities identified with CFinder .",
"Relationships between two harem member individuals connecting two different harems are indicated in gray.",
"The number of harems in the simulation is m=17m=17, and the number of all individuals is n=150n=150, as it is in the experiment, the number of edges between the harems is 26 and 36 in the model and in the experiment, respectively.One outcome of our model is that the leadership network, both on the level of individuals and on the level of harems, converges to a stable state.",
"Convergence is reached after several hundred time steps and it is indicated by the convergence of the values of the underlying perceived ability matrix, and by the fact that the fraction of changed edges over a time step goes to zero.",
"In the presence of a small amount of noise the resulting network can fluctuate slightly around an average, similarly to a real group where leadership roles can be flexible.",
"The typical structure of the converged networks is qualitatively similar to a herd consisting of harems.",
"Examining them with CFinder [24], [2], which uses the clique percolation method, communities can be revealed that are associated with more highly inter-connected subgraphs, typically made of $k=3$ and $k=4$ cliques.",
"Two typical community type occurs in the networks.",
"The first typical community consists of one leader-type and some follower-type individuals.",
"The same leader-type individual is often participating in more of such communities, thus the union of these communities can be viewed as a harem that is led by the leader-type individual.",
"The communities forming a big harem are often overlapping with many shared nodes, most commonly one or two 4-clique communities are embedded in a 3-clique community.",
"The harem leaders (besides their harem forming communities) often form a second type of communities that consists only of leader-type individuals.",
"These communities are the basis of the information flow between the different harems, thus they can be viewed rather as alliances between harems than real communities.",
"Overlaps between different harems, led by different leaders, are very rare.",
"Most of the harems are connected with each other through the leader of the harem, but they are connected through a few edges between two follower-type individuals, as well.",
"If the summarized link capacity of all leader-types is abundant, than communities, that consist only of follower-type individuals, do not survive.",
"Follower-type individuals tend to follow only one leader-type individual, despite the fact that their followings are basically spontaneous.",
"Therefore in the model we define a harem as the list of the follower-type individuals, who consider the choice of the given leader-type individual, including the leader as well.",
"With this definition a harem can contain some follower-type individuals as well, who do not belong to the harem forming communities.",
"As the network converges, the difference between the two harem definitions decreases.",
"However, after 1000 time steps some differences remain, and there are some individuals who are not participating in any community.",
"The network inside a harem is very hierarchical, with one leader-type individual on the top level, and strong hierarchy exists between the harem members as well.",
"There is also a tree-like hierarchy between harems typically with one single leader harem on the top.",
"It is identified as the hierarchy of harem-leaders and represents the next level of organization.",
"The whole herd keeps cohesive, since the network of individuals forms one connected cluster.",
"The network of harems in most of the cases also remains cohesive, but if the number of leader-type individuals is relatively low, it can break up into smaller networks, as discussed later.",
"In analysing the model, we identified some input parameters, such as the ratio of leader-type individuals to all individuals, and the typical value of link capacities, which influence the results more sensitively.",
"Other parameters, like group size, frequency of herding behaviour in a round and the number of considered decisions by an individual have less considerable effect.",
"The frequency of herding behaviour affects first of all the rate of convergence to the stable state, but not the resulting network.",
"The higher the frequency, the faster the convergence is.",
"Figure: Heatmaps of the relative performance improvement () (left column) and global reaching centrality (GRC) (right column) of model networks, as the function of time and ratio of leader-type individuals to the number of all individuals.",
"An optimal harem leader to harem member ratio can be observed, where performance and hierarchy is maximized.",
"The location of the optimum is robust to changes in the shape of the link capacity distribution (LCD) if its average is fixed: (a)-(b) is for a Poisson LCD with λ=20\\lambda = 20, and (c)-(d) for a lognormal LCD with μ=20\\mu =20 mean and σ=20\\sigma =20 standard deviation.",
"For lower average LC values the optimal range is shifted: (e)-(f) is for a Poisson LCD with λ=10\\lambda = 10, and (g)-(h) for a (3,17)(3,17) uniform LCD.",
"The number of all individuals in the simulations is n=200n = 200, their ability distribution is bounded Pareto with 0.250.25 expected value and 1/481/\\sqrt{48} standard deviation, each data point is averaged over 1000 runs."
],
[
"Experiment and model",
"Carrying out simulations with similar input parameters as those observed in the wild horses, very similar networks can result both qualitatively and quantitatively on average.",
"Based on the aerial images (e.g.",
"Fig.",
"REF ) and the network definition in Sect.",
", the hierarchy of harems can be established (Fig.",
"REF (a) and Fig.",
"REF , REF ).",
"The number of identified harems in Fig.",
"REF and thus the number of nodes is 17.",
"The harems contain roughly a total number of 150 horses.",
"A typical harem network of the model is shown in Fig.",
"REF (b).",
"The number of all individuals is $n=150$ , and the number of leader-type individuals is $m=17$ in the simulation, in order to fit the experimental parameters.",
"Both the experimental and model networks are visualized with reaching centrality method [20].",
"At first glance one can see the similar pyramid-like layout of the networks and the common features, such as the presence of one single leader, several nodes in higher layers, and many at the bottom layer.",
"There are many layers that indicates the varying roles, and the edges can connect distant layers.",
"The global reaching centrality (GRC) values of the model networks are very close to those of the experimental networks, on average, quantifying a similar level of hierarchy in the two cases.",
"Particularly, in Fig.",
"REF GRC takes the values $0.65$ and $0.67$ for the experimental and model networks, respectively.",
"However, the number of edges is less in the model (26) than in the experiment (36) in this case.",
"The enlarged areas in the right display the internal structure in some harems."
],
[
"Optimal harem leader to harem member ratio",
"In order to find the main features of the model networks we carried out simulations with a range of input parameters and calculated network properties (group performance and hierarchical measures) in every case by averaging over 1000 runs.",
"The investigation shows that the presence of the leader-type individuals results in an increment in relative performance improvement (REF ) from about $135\\%$ to as high as $170\\%$ .",
"In addition, the resulting network is more hierarchical according to all the three measures studied (fraction of noncyclic edges, global reaching centrality and fraction of forward arcs).",
"For example GRC, can increase from $0.25$ up to $0.9$ for particular input parameters.",
"The despotic approach of our model removes many of the non-efficient cycles.",
"As a consequence, the decisions of top ranking individuals spread more effectively to lower ranking individuals, thereby improving overall success.",
"The ratio of leader-type individuals to all individuals (abbreviated as leader ratio) plays an important role in determining the quality of the resulting network.",
"The simulations indicate that there is an optimum in the value of this parameter (Fig.",
"REF ), where performance and hierarchy is maximized.",
"For example, for Poisson link capacity distribution (LCD) with $\\lambda =20$ the optimal leader ratio lies around $1:10$ , where the relative performance improvement reaches $180\\%$ after several hundred steps, and networks producing performance above $170\\%$ lie in the range from $1:20$ to $1:5$ leader ratio, while outside this region performance does not exceed $160\\%$ (Fig.",
"REF (a)).",
"The GRC has an optimal region as well, where its value reaches $0.9$ , in contrast to $0.4$ outside this region (Fig.",
"REF (b)).",
"However, the optimal region of the GRC is narrower than the one of the performance, it lies between $1:20$ and $1:10$ .",
"The overlap between them, and thus the optimum from the point of view of performance and degree of hierarchy together, is between $1:20$ and $1:10$ .",
"The location of the optimal range is robust to changes in the shape of the link capacity distribution, if its average value is fixed.",
"We examine uniform, delta, Poisson and lognormal link capacity distributions with averages ranging from 5 to 20.",
"Fig.",
"REF (c)-(d) shows results for lognormal LCD with $\\mu =20$ expected value and $\\sigma =20$ standard deviation.",
"In comparison with Fig.",
"REF (a)-(b), the optimal range remains the same, and the performance is maximized around $1:10$ leader ratio, similarly as for the underlying Poisson LCD with $\\mu =20$ average.",
"For lower average link capacity values a minimum of performance appears below the optimal area, see Fig.",
"REF (e) and (g) for a Poisson and uniform LCD with $\\mu =10$ mean.",
"If the number of leader-type individuals is decreased to the extent that $m\\left\\langle c_i \\right\\rangle < n-m$ , i.e.",
"the sum of all link capacities becomes less than the number of follower-type individuals, then link capacities start to limit the free formation of harems.",
"If this threshold is reached the network of harems starts to break apart, since the edges connecting leader-type individuals also start to split up, giving rise to separated harems, and this can cause a decrease in overall performance.",
"Decreasing $m$ still further can slightly increase performance, this can occur because the size of the separated communities increases.",
"The optimal range is shifted upwards, probably because an optimal network structure can emerge when the total number of link capacities is abundant.",
"For an underlying LCD with $\\mu =10$ average, the optimal leader to harem-member ratio is around $1:8$ .",
"Comparison of simulations with different average link capacities leads to the conclusion that the optimal region lies around $m\\approx 1.5 n/\\left\\langle c_i \\right\\rangle $ .",
"The observed maximal harem size in wild horses is around 20 in our experiment and the average size is 9.",
"Therefore the theoretical average link capacity (corresponding to the case of the wild horses) can be between 10 and 20, giving rise to an optimal harem-leader to harem-member ratio of roughly between $1:8$ and $1:10$ , from the point of view of the common success.",
"It is very interesting that the $1:9$ empirical ratio observed in wild horses is close to this model result."
],
[
"Harem size distribution",
"Since our model aims to simulate a group of groups, it is natural to ask what kind of cluster size distributions (in our case, harem size distributions) characterize the resulting network.",
"The harem sizes are defined as the number of follower-type individuals following the given harem leader, and the harem leader is also counted in the size.",
"When the network, and thus the harem sizes, can be considered as converged, we build a histogram and investigate it for different LC distributions.",
"In order to have a roughly optimal performing network in all cases, when simulating with LC distributions with average values ranging from 10 to 25, the leader ratio is set to $1:8$ .",
"We find that the distribution of harem sizes is an asymmetric heavy-tailed distribution, and can be well fitted by a lognormal (Fig.",
"REF (a)).",
"Similarly to the optimal leader ratio, it seems to be independent of the shape of the link capacity distribution.",
"It is very similar for uniform, delta and Poisson distributed link capacities (Fig.",
"REF (a)).",
"However, this only holds if the predefined LCD of the individuals does not limitate the harem formation.",
"Again, if $m\\left\\langle c_i \\right\\rangle < n-m$ , leader-type individuals fill up all their links and a trivial harem size distribution emerges, with a shape determined by the link capacity distribution.",
"As the total number of links is increased, the harem size distribution approaches a lognormal.",
"Figure: Harem size distribution.",
"(a) Model distributions tend to follow a lognormal distribution, provided the total number of link capacities is large enough, and it does not limitate the harem formation.",
"It does not depend on the shape of the link capacity distribution: harem size distributions with underlying delta(25), uniform(15,3515,35) and Poisson(λ=25\\lambda =25) LCDs are shown on the plot.",
"A lognormal distribution is fitted to the case of the Poisson LCD with μ=9.61\\mu =9.61 mean and σ=6.65\\sigma =6.65 standard deviation.",
"(b) The experimental harem size distribution is plotted together with the model distribution (with an underlying Poisson LCD with λ=25\\lambda = 25) and the lognormal fitted to the model.",
"Using the Kolmogorov-Smirnov test the fitted lognormal is accepted as the theoretical distribution of the experimental sample at p=0.97p=0.97 significance level.",
"The experimental distribution is based on data from m=21m = 21 wild horse harems including n=188n = 188 individuals.",
"The number of all individuals in the simulation is n=200n = 200, and m=25m = 25 of which are leader-type individuals.",
"Each data point is averaged over 1000 runs with harem sizes measured at the t=1000t = 1000 time step.",
"Semi-log plots are shown in the top right corner.The comparison of the model with the experiment also shows encouraging agreement.",
"The experimental harem size distribution is based on the data from $m = 21$ wild horse harems including $n=188$ individuals, and its histogram is shown in Fig.",
"REF (b).",
"Despite the considerable error due to the small sample size, an accordance can be proposed with the lognormal harem size distribution coming from the model.",
"We assume the null hypothesis, that the theoretical distribution of the observed harem sizes is the lognormal fitted to the model results for a Poisson ($\\lambda =25$ ) LCD, with fitting parameters $\\mu =9.61$ mean and $\\sigma =6.65$ standard deviation.",
"Using the Kolmogorov-Smirnov test it can be accepted at $p=0.97$ significance level that the experimental sample comes from this theoretical distribution."
],
[
"Discussion",
"The endangered status of the Przewalski horses calls for more profound studies of the overall behaviour of this species [14], and indeed, their collective movements have attracted interest as well [5].",
"However, the different patterns of their collective motion, and the leadership structure behind it, has not been investigated in detail yet.",
"A proper analysis of the individual horse tracks would clarify our estimated picture, and would deepen the understanding of the organization and cohesion of harems.",
"Although, our estimation can give a valuable insight into the leadership hierarchy of the horse harems, since it reflects the possible leader-follower relationships provided that the horses use visual perception when following others.",
"Our model simulates the emergence of a leadership hierarchy in a three-level herd starting from unfamiliar individuals, and results in a stable state representing a consistent leadership network of a natural group.",
"It is based on the individuals copying the behaviour of some of the herd mates when making individual decisions.",
"Collective decision emerges from these individual ones.",
"The specific feature of our present approach is that in line with the spontaneously developing leader-follower relationships between the individuals, dense sub-groups are forming inside the herd that are loosely connected with each other, similarly to the social structure of the wild horses.",
"In the converged network the hierarchy of the sub-groups is similar to the hierarchy of the individuals inside a sub-group, typically including a single leader.",
"Thus, our model accounts for three basic levels of social organizations in a decision-making context.",
"Modeling shows that the harem-leader to harem-member ratio observed in the herd of Przewalski horses corresponds to an optimal network in the sense that the overall success of the group is maximized.",
"In addition, hierarchy is maximized with this ratio as well, suggesting that the degree of hierarchy in a group correlates with common success.",
"The observed distribution of harem sizes is consistent with the lognormal distribution obtained when applying our model.",
"The above results are not too model specific.",
"The model behaves similarly for different values of the main parameters - like group size, the typical number of other group mates whose behaviour a member considers, before making its decision, and the distribution of the maximal possible harem size a leader is able to keep together.",
"A better statistics for the harem sizes of horses and a generalization to other species would improve the understanding of the underlying processes.",
"Finally, we find that a modularly structured leadership hierarchy is more beneficial than a non-modular one when making a collective decision.",
"The spreading of advantageous decisions of higher ability individuals is more facile through mid-level leaders, thus approving the overall success of the herd.",
"This observation suggests that multilevel societies, beside kin relationships and the formation of breeding groups, may be the results of alternative optimization factors, as well."
],
[
"Acknowledgements",
"We are grateful to Kristin Brabender and the Directorate of Hortobágy National Park for providing us their data as well as officially authorizing the carrying out of our observations and to Waltraut Zimmermann, Kölner Zoo for cooperation.",
"We thank Gábor Vásárhelyi and Gergő Somorjai for helping our observations with flying robots.",
"This research was partially supported by the EU ERC COLLMOT project (grant no.",
"227878)."
],
[
"Supplementary Information",
"Additional figures of the Przewalski horse herd (Hortobágy National Park, Hungary) during collective movements Figure: Aerial pictures taken on 22.07.2013 (top) and on 21.05.2014 (bottom).",
"The highlighted areas indicate the individual harems.",
"The black arrow shows the direction of motion.",
"The color-codes match the same stallions ' ^{\\prime } harems on both pictures.",
"The identification is based on the number of adult and infant horses in the harems compared with the up-to-date catalogue of the reserve (note that these numbers can differ on the two pictures due to the elapsed time).",
"The violet, red, blue, yellow and green harems move at the front of the V-formation of the herd, and thus are supposed to play a leader role, in both pictures.",
"On the basis of the above figures it is reasonable to assume that - on average - the leadership roles of the harems remained about the same over the 10 months indicated above.",
"The figures on the right show the hierarchical layout of the estimated leadership network of harems visualized with the reaching centrality method .Figure: Aerial pictures taken on 05.06.2014 (top) and 06.07.2014 (bottom).",
"The blue and violet harems that move at the front area of the herd on 21.05.2014 (Fig.",
"S1/c) are still in leading positions on 05.06.2014 (a), but later on 06.07.2014 (c) they can be found in the middle area.",
"The harems marked with red and green in Fig.",
"S1 could not be identified here.",
"The yellow harem that could be identified only in (a), moves further in the herd as time passes (Fig.",
"S1/a, S1/c and S2/a).",
"The orange and black harems that move at the front both on 05.06.2014 (a) and 06.07.2014 (c), are middle positioned on Fig.",
"S1/a and S1/c, respectively.",
"In comparison with the situation on 21.05.2014 (Fig.",
"S1/c) the leadership roles have somewhat changed.",
"We conjecture that the changes which can be detected on Fig.",
"S1/c, S2/a and S2/c and refer to the supposed leadership structure between 21.05.2014 and 06.07.2014 are likely to be due to the breeding season that takes place in the late spring."
]
] | 1403.0260 |
[
[
"Additive Spanners: A Simple Construction"
],
[
"Abstract We consider additive spanners of unweighted undirected graphs.",
"Let $G$ be a graph and $H$ a subgraph of $G$.",
"The most na\\\"ive way to construct an additive $k$-spanner of $G$ is the following: As long as $H$ is not an additive $k$-spanner repeat: Find a pair $(u,v) \\in H$ that violates the spanner-condition and a shortest path from $u$ to $v$ in $G$.",
"Add the edges of this path to $H$.",
"We show that, with a very simple initial graph $H$, this na\\\"ive method gives additive $6$- and $2$-spanners of sizes matching the best known upper bounds.",
"For additive $2$-spanners we start with $H=\\emptyset$ and end with $O(n^{3/2})$ edges in the spanner.",
"For additive $6$-spanners we start with $H$ containing $\\lfloor n^{1/3} \\rfloor$ arbitrary edges incident to each node and end with a spanner of size $O(n^{4/3})$."
],
[
"Introduction",
"Additive spanners are subgraphs that preserve the distances in the graph up to an additive positive constant.",
"Given an unweighted undirected graph $G$ , a subgraph $H$ is an additive $k$ -spanner if for every pair of nodes $u,v$ it is true that $d_G(u,v) \\le d_H(u,v) \\le d_G(u,v)+k$ In this paper we only consider purely additive spanners, which are $k$ -spanners where $k = O(1)$ .",
"Throughout this paper every graph will be unweighted and undirected.",
"Many people have considered a variant of this problem, namely multiplicative spanners and even mixes between additive and multiplicative spanners [5], [4], [6].",
"The problem of finding a $k$ -spanner of smallest size has received a lot of attention.",
"Most notably, given a graph with $n$ nodes Dor et al.",
"[3] prove that it has a 2-spanner of size $O(n^{3/2})$ , Baswana et al.",
"[1] prove that it has a 6-spanner of size $O(n^{4/3})$ , and Chechik [2] proves that it has a 4-spanner of size $O(n^{7/5}\\log ^{1/5}n)$ .",
"Woodrufff [7] shows that for every constant $k$ there exist graphs with $n$ nodes such that every $(2k-1)$ -spanner must have at least $\\Omega (n^{1+1/k})$ edges.",
"This implies that the construction of 2-spanners are optimal.",
"Whether there exists an algorithm for constructing $O(1)$ -spanners with $O(n^{1+\\varepsilon })$ edges for some $\\varepsilon < 1/3$ is unknown and is an important open problem.",
"Let $G$ be a graph and $H$ a subgraph of $G$ .",
"Consider the following algorithm: As long as there exists a pair of nodes $u,v$ such that $d_H(u,v) > d_G(u,v) + k$ , find a shortest path from $u$ to $v$ in $G$ and add the edges on the path to $H$ .",
"This process will be referred to as $k$ -spanner-completion.",
"After $k$ -spanner-completion, $H$ will be a $k$ -spanner of $G$ .",
"Thus, given a graph $G$ , a general way to construct a $k$ -spanner for $G$ is the following: Firstly, find a simple subgraph of $G$ .",
"Secondly use $k$ -spanner-completion on this subgraph.",
"The main contribution of this paper is: Theorem 1.1 Let $G$ be a graph with $n$ nodes and $H$ the subgraph containing all nodes but no edges of $G$ .",
"For each node add $\\left\\lfloor n^{1/3} \\right\\rfloor $ edges adjacent to that node to $H$ (or, if the degree is less, add all edges incident to that node).",
"After 6-spanner-completion $H$ will have at most $O(n^{4/3})$ edges.",
"It is well-known that a graph with $n$ nodes has a 6-spanner of size $O(n^{4/3})$ [1].",
"The techniques employed in our proof of correctness are similar to those in [1].",
"The creation of the initial graph $H$ corresponds to the clustering in [1] and the 6-spanner-completion corresponds to their path-buying algorithm.",
"For completeness we show that the same method gives a 2-spanner of size $O(n^{3/2})$ .",
"This fact is already known due to [3] and is matched by a lower bound from [7].",
"Theorem 1.2 Let $G$ be a graph with $n$ nodes and $H$ the subgraph where all edges are removed.",
"Upon 2-spanner-completion $H$ has at most $O(n^{3/2})$ edges."
],
[
"Creating a 6-spanner",
"The algorithm for creating a 6-spanner was described in the abstract and the introduction.",
"For a given graph $G$ , a 6-spanner of $G$ can be created by strating with some subgraph $H$ of $G$ and applying 6-spanner-completion to $H$ .",
"6-spanner-thm states that for a suitable starting choice of $H$ we get a spanner of size $O(n^{4/3})$ .",
"The purpose of this section is to show that the 6-spanner created has no more than $O(n^{4/3})$ edges.",
"This will imply that the construction (in terms of the size of the 6-spanner) matches the best known upper bound [1].",
"[of 6-spanner-thm] Inserting (at most) $\\left\\lfloor n^{1/3} \\right\\rfloor $ edges per node will only add $n\\left\\lfloor n^{1/3} \\right\\rfloor = O(n^{4/3})$ edges to $H$ .",
"Therefore it is only necessary to prove that 6-spanner-completion adds no more than $O(n^{4/3})$ edges.",
"Let $v(H)$ and $c(H)$ be defined by: $v(H) =\\sum _{u,v \\in V(G)}\\max \\left\\lbrace 0, d_G(u,v)-d_H(u,v)+5 \\right\\rbrace ,\\quad c(H) = \\# E(H)$ Say that a shortest path, $p$ , from $u$ to $v$ is added to $H$ , and let $H_0$ be the subgraph before the edges are added.",
"Let the path consist of the nodes: $u = w_0, w_1, \\ldots , w_r = v, r \\in \\mathbb {N}$ Let $u^{\\prime } = w_i$ be the node $w_i$ with the smallest $i$ such that $\\deg _{H_0}(w_i) \\ge \\left\\lfloor n^{1/3} \\right\\rfloor $ .",
"Likewise let $v^{\\prime } = w_j$ be the node $w_j$ the largest $j$ such that $\\deg _{H_0}(w_j) \\ge \\left\\lfloor n^{1/3} \\right\\rfloor $ .",
"Remember that if $\\deg _{H_0}(w_i) < \\left\\lfloor n^{1/3} \\right\\rfloor $ then all the edges adjacent to $w_i$ are already in $H_0$ .",
"This implies that $d_{H_0}(u^{\\prime },v^{\\prime }) > d_G(u^{\\prime },v^{\\prime })+6$ since $d_{H_0}(u,v) > d_G(u,v)+6$ .",
"Say that $t$ new edges are added to $H$ .",
"Then there must be at least $t$ nodes on $p$ with degree $>n^{1/3}$ .",
"Since every node can be adjacent to no more than 3 nodes on $p$ (since it is a shortest path) there must be $\\Omega (n^{1/3}t)$ nodes adjacent to $p$ in $H$ .",
"Let $z$ and $w$ be neighbours to $u^{\\prime }$ and $v^{\\prime }$ in $H$ respectively and let $r$ be any node adjacent to $p$ in $H$ .",
"Let $s$ be a node on $p$ such that $r$ and $s$ are adjacent in $H$ .",
"See 6spanner-img for an illustration.",
"Figure: The dashed line denotes the shortest path from uu to vv.",
"The solid lines denote edges.By the triangle inequality we see that: $d_H(z,r) + d_H(r,w) \\le d_G(u^{\\prime },v^{\\prime })+4$ But on the other hand: $d_{H_0}(z,r) + d_{H_0}(r,w) \\ge d_{H_0}(z,w) \\ge d_{H_0}(u^{\\prime },v^{\\prime }) - 2 >d_G(u^{\\prime },v^{\\prime }) + 4$ Combining these two inequalities we obtain $d_{H_0}(z,r) > d_H(z,r)$ or $d_{H_0}(r,w) > d_H(r,w)$ .",
"And from the triangle inequality $d_G(z,r)+5 > d_H(z,r)$ and $d_G(r,w)+5 > d_H(r,w)$ .",
"Since $u^{\\prime }$ and $v^{\\prime }$ have at least $n^{1/3}$ neighbours and there are $\\Omega (n^{1/3}t)$ nodes in $H$ adjacent to $p$ , the definition of $v(H)$ implies that: $v(H) - v(H_0) \\ge \\Omega (t(n^{1/3})^2)$ And since $c(H) - c(H_0) = t$ : $\\frac{v(H)-v(H_0)}{c(H)-c(H_0)} \\ge \\Omega (n^{2/3})$ Since $v(H) \\le O(n^2)$ this implies that $c(H)$ increases with no more than $O(n^2/n^{2/3}) = O(n^{4/3})$ in total when all shortest paths are inserted.",
"Hence $c(H) = O(n^{4/3})$ when the 6-spanner-completion is finished which yields the conclusion."
],
[
"Creating a 2-spanner",
"For completeness we show that 2-spanner-completion gives spanners with $O(n^{3/2})$ edges.",
"This matches the upper bound from [3] and the lower bound from [7].",
"[of 2-spanner-thm] Let $G$ be a graph with $n$ nodes.",
"Whenever $H$ is a spanner of $G$ , define $v(H)$ and $c(H)$ as: $v(H) =\\sum _{u,v \\in V(G)}\\max \\left\\lbrace 0, d_G(u,v) - d_H(u,v) + 3 \\right\\rbrace ,\\quad c(H) = \\sum _{v \\in V(G)} \\left(\\deg _H(v)\\right)^2$ It is easy to see that $0 \\le v(H) \\le 3n^2$ and by Cauchy-Schwartz's inequality $\\sqrt{c(H) \\cdot n} \\ge 2\\# E(H)$ .",
"The goal will be to prove that when the algorithm terminates $c(H) = O(n^2)$ , since this implies that $\\# E(H) = O(n^{3/2})$ .",
"This is done by proving that in each step of the algorithm $c(H) - 12 v(H)$ will not increase.",
"Since $v(H) = O(n^2)$ this means that $c(H) = O(n^2)$ which ends the proof.",
"Therefore it is sufficient to check that $c(H) - 12v(H)$ never increases.",
"Consider a step where new edges are added to $H$ on a shortest path from $u$ to $v$ of length $t$ .",
"Let $H_0$ be the subgraph before the edges are added.",
"Assume that $u,v$ violates the 2-spanner condition in $H_0$ , i.e.",
"$d_{H_0}(u,v) > d_G(u,v)+2$ .",
"Let the shortest path consist of the nodes: $u = w_0, w_1, \\ldots , w_{t-1}, w_t = v$ It is obvious that: $c(H) - c(H_0) \\le \\sum _{i=0}^t (\\deg _H(w_i))^2-(\\deg _H(w_i)-2)^2\\le 4\\sum _{i=0}^t \\deg _H(w_i)$ Every node cannot be adjacent to more than 3 nodes on the shortest path, since otherwise it would not be a shortest path.",
"Using this insight we can bound the number of nodes which in $H$ are adjacent to or on the shortest path from below by: $\\frac{1}{3} \\sum _{i=0}^t \\deg _H(w_i)$ Now let $z$ be a node in $H$ adjacent or on to the shortest path.",
"Obviously: $d_{H}(u,z) + d_{H}(z,v) \\le d_G(u,v)+2$ Furthermore $d_{H_0}(u,z) + d_{H_0}(z,v) > d_G(u,v)+2$ since otherwise there would exist a path from $u$ to $v$ in $H_0$ of length $\\le d_G(u,v)+2$ .",
"Hence: $d_{H}(u,z) + d_{H}(z,v) <d_{H_0}(u,z) + d_{H_0}(z,v)$ Now let $z$ be a node on the shortest path which is adjacent to $w_i$ in $H$ (every node on the path will also be adjacent in $H$ to such a node).",
"Then by the triangle inequality: $d_H(u,z) & \\le d_H(u,w_i) + d_H(w_i,z) & & =d_G(u,w_i) + 1 \\\\&\\le d_G(u,z) + d_G(z,w_i) + 1 & & =d_G(u,z) + 2$ And likewise $d_H(z,v) \\le d_G(z,v)+2$ .",
"Combining these two observations yields: $\\sum _{w \\in V}\\max \\left\\lbrace 0, d_G(z,w) - d_H(z,w) + 3 \\right\\rbrace <\\sum _{w \\in V}\\max \\left\\lbrace 0, d_G(z,w) - d_{H_0}(z,w) + 3 \\right\\rbrace $ Since this holds for every node in $H$ adjacent to or on the shortest path this means that: $v(H) - v(H_0) \\ge \\frac{1}{3} \\sum _{i=0}^t \\deg _H(w_i)$ Combining this with the bound on $c(H) - c(H_0)$ gives: $(c(H) - 12 v(H)) - (c(H_0) - 12 v(H_0)) \\le 0$ which finishes the proof."
]
] | 1403.0178 |
[
[
"The Effect of Block-wise Feedback on the Throughput-Delay Trade-off in\n Streaming"
],
[
"Abstract Unlike traditional file transfer where only total delay matters, streaming applications impose delay constraints on each packet and require them to be in order.",
"To achieve fast in-order packet decoding, we have to compromise on the throughput.",
"We study this trade-off between throughput and in-order decoding delay, and in particular how it is affected by the frequency of block-wise feedback to the source.",
"When there is immediate feedback, we can achieve the optimal throughput and delay simultaneously.",
"But as the feedback delay increases, we have to compromise on at least one of these metrics.",
"We present a spectrum of coding schemes that span different points on the throughput-delay trade-off.",
"Depending upon the delay-sensitivity and bandwidth limitations of the application, one can choose an appropriate operating point on this trade-off."
],
[
"Motivation",
"A recent report [1] shows that $62 \\%$ of the Internet traffic in North America comes from real-time streaming applications such as NetFlix ($28.88\\%$ ) and YouTube ($15.43 \\%$ ).",
"Streaming traffic consumes such a large fraction of Internet bandwidth because video files inherently have a larger size than other forms of data.",
"Thus, there is a need to develop transmission schemes which can ensure a high quality of experience to the user, with efficient use of available bandwidth.",
"Unlike traditional file transfer where only total delay matters, streaming imposes delay constraints on each individual packet.",
"Further, many applications require in-order playback of packets at the receiver.",
"Packets received out of order are buffered until the missing packets in the sequence are successfully decoded.",
"In audio and video applications some packets can be dropped without affecting the streaming quality.",
"However, other applications such as remote desktop, and collaborative tools such as Dropbox and Google Docs have strict order constraints on packets, where packets represent instructions that need to be executed in order at the receiver.",
"To ensure that packets are decoded in order, the transmission scheme must give higher priority to older packets that were delayed, or received in error due to channel noise.",
"However, repeating old packets instead of transmitting new packets results in a loss in the overall rate at which packets are delivered to the user, that is, the throughput.",
"Thus there is a fundamental trade-off between throughput and in-order decoding delay.",
"The throughput loss incurred to achieve low in-order decoding delay can be significantly reduced if the source receives feedback about packet losses, and thus can adapt its future transmission strategy to strike the right balance between old and new packets.",
"We study this interplay between feedback and the throughput-delay trade-off."
],
[
"Previous Work",
"Only a few papers in literature have analyzed streaming codes.",
"Fountain codes [2] are capacity-achieving erasure codes, but they are not suitable for streaming because the decoding delay is proportional to the size of the data.",
"Streaming codes without feedback for constrained channels such as adversarial and cyclic burst erasure channels were first proposed in [3], and also extensive explored in [4], [5].",
"The thesis [3] also proposed codes for more general erasure models and analyzed their decoding delay.",
"Decoding delay has also been analyzed studied in [6], [7] in a multicast scenario with immediate feedback to the source.",
"However, decoding delay does not capture in order packet delivery which is required for streaming applications.",
"This aspect is captured in the delay metrics in [8] and [9], which consider that packets are played in-order at the receiver.",
"The authors in [8] analyze the throughput-delay trade-off for uncoded packet transmission over a channel with long feedback delay.",
"In [9] we propose coding schemes that minimize playback delay in point-to-point streaming for the no feedback and immediate feedback cases.",
"However, the case of block-wise feedback to the source remains to be explored."
],
[
"Our Contributions",
"In this paper we consider this unexplored problem of how to effectively utilize block-wise feedback to the source to ensure in-order packet delivery to the user.",
"In contrast to playback delay considered in [8] and [9], we propose a more versatile delay metric called the in-order decoding exponent.",
"This metric captures the burstiness in the in-order decoding of packet for applications which require packets in-order, but do not necessarily play them at a constant rate.",
"When there is immediate feedback, we can achieve the best throughput-delay trade-off.",
"But when the feedback comes in blocks, we have to compromise on the throughput to ensure fast in-order decoding.",
"We present a spectrum of coding schemes that span different points on the throughput-delay trade-off.",
"Depending upon the delay-sensitivity, and bandwidth limitations of the application, one can choose an appropriate operating point on this trade-off.",
"The proposed codes can be shown to be optimal over a broad class of schemes for the no feedback, and small feedback delay cases."
],
[
"System Model",
"We consider a point-to-point packet streaming scenario where the source has a large stream of packets $s_1, s_2, \\cdots , s_n$ .",
"The encoder creates a coded packet $y_{n} = f(s_1,\\,\\,s_2 \\,\\,..s_n)$ in each slot $n$ and transmits it over the channel.",
"The encoding function $f$ is known to the receiver.",
"For example, if $y_{n}$ is a linear combination of the source packets, the coefficients are included in the transmitted packet so that the receiver can use them to decode the source packets from the coded combination.",
"Without loss of generality, we can assume that $y_n$ is a linear combination of the source packets.",
"We consider an i.i.d.",
"packet erasure channel where every transmitted packet is correctly received with probability $p$ , and otherwise received in error and discarded.",
"An erasure channel is a good model when encoded packets have a set of checksum bits that can be used to verify with high probability whether the received packet is error-free.",
"The receiver application requires the stream of packets to be in order.",
"Packets received out of order are buffered until the missing packets in the sequence are decoded.",
"Due to this in-order property, the transmitter can stop including $s_k$ in coded packets when it knows that the receiver can decode $s_k$ once all $s_i$ for $i < k$ are decoded.",
"We refer such packets as “seen\" packets.",
"The notion of “seen\" is defined formally as follows.",
"Definition 1 (Seen Packets) A packet $s_k$ is said to be “seen\" by the transmitter when it knows that a coded combination that only includes $s_k$ and packets $s_i$ for $1 \\le i \\le k$ is received successfully.",
"We consider that the source receives block-wise feedback about channel erasures after every $d$ slots.",
"Thus, before transmitting in slot $kd+1$ , for all integers $k \\ge 1$ , the source knows about the erasures in slots $(k-1)d +1$ to $kd$ .",
"It can use this information to adapt its transmission strategy in slot $kd +1$ .",
"Block-wise feedback can be used to model a half-duplex communication channel where after every $d$ slots of packet transmission, the channel is reserved for the receiver to send feedback about the status of decoding.",
"Note that the feedback can be used to estimate $p$ , the probability of success of the erasure channel, when it is unknown to the source."
],
[
"Throughput and Delay Metrics",
"We consider two metrics to measure the quality of streaming, the throughput $\\tau $ and in-order decoding exponent $\\lambda $ .",
"The throughput is the rate at which “innovative\" coded packets are received.",
"A coded packet is said to be “innovative\" if it is linear independent with respect to the coded packets received until then.",
"The bandwidth required is proportional to $1/ \\tau $ .",
"The throughput captures the overall rate at which packets go through the channel, irrespective of the order.",
"The in-order decoding aspect is captured by a metric called the in-order decoding exponent $\\lambda $ which is defined as follows.",
"Definition 2 (In-order Decoding Exponent) Let $T$ be the time between two successive instants of decoding one or more packets in-order.",
"Then the in-order decoding exponent $\\lambda $ is $\\lambda \\triangleq -\\lim _{n \\rightarrow \\infty } \\frac{\\log \\Pr (T>n)}{n}.",
"$ The relation (REF ) can also be stated as $\\Pr (T>n) \\doteq e^{-n \\lambda }$ where $\\doteq $ stands for asymptotic equality defined in [10].",
"The in-order decoding exponent captures the burstiness in packet decoding.",
"For example, if the streaming application plays one in-order packet in every slot, and if there are $b$ packets in the receiver buffer, then the probability of an interruption in playback is proportional to $e^{-\\lambda b}$ .",
"In this paper we analyze how the trade-off between $\\tau $ and $\\lambda $ is affected by the block-wise feedback delay $d$ .",
"We first consider the extreme cases of immediate feedback $(d=1)$ and no feedback $(d=\\infty )$ in Section and Section respectively.",
"This gives us insights into the analysis of the $(\\tau ,\\lambda )$ trade-off for general $d$ in Section ."
],
[
"Immediate Feedback",
"In the immediate feedback $(d=1)$ case, the source has complete knowledge of past erasures before transmitting each packet.",
"We can show that a simple automatic-repeat-request (ARQ) scheme is optimal in both $\\tau $ and $\\lambda $ .",
"In this scheme, the source transmits the lowest index unseen packet, and repeats it until the packet successfully goes through the channel.",
"Since a new packet is received in every successful slot, the throughput $\\tau =p$ , the success probability of the erasure channel.",
"The ARQ scheme is throughput-optimal because the throughput $\\tau = p$ is equal to the information-theoretic capacity of the erasure channel [10].",
"Moreover, it also gives the optimal the in-order decoding exponent $\\lambda $ because one in-order packet is decoded in every successful slot.",
"To find $\\lambda $ , first observe that the tail distribution of the time $T$ , the interval between successive in-order decoding instants is, $\\Pr (T>n) &= (1-p)^{n}$ Substituting this in Definition REF we get the exponent $\\lambda = -\\log (1-p)$ .",
"Based on this analysis of the immediate feedback case, we can find limits on the range of achievable $(\\tau ,\\lambda )$ for any feedback delay $d$ as follows.",
"Lemma 1 The throughput and delay metrics $(\\tau ,\\lambda )$ achievable for any feedback delay $d$ lie in the region $0 \\le \\tau \\le \\rho $ , and $0 \\le \\lambda \\le -\\log (1-p)$ .",
"When feedback is received after blocks of $d > 1$ slots, the source has less knowledge about past erasures than in the immediate feedback ($d=1$ ) case.",
"Thus, the $(\\tau , \\lambda )$ trade-off when $d > 1$ is always worse than $(\\tau , \\lambda ) = (p,-\\log (1-p))$ the optimal trade-off for the immediate feedback ($d=1$ ) case."
],
[
"No Feedback",
"Now we consider the other extreme case $(d = \\infty )$ , where there is no feedback to the source about channel erasures.",
"We propose a coding scheme and prove that it gives the best $(\\tau , \\lambda )$ trade-off among a class of codes called full-rank codes which are defined as follows.",
"Definition 3 (Full-rank Codes) In slot $n$ we transmit a linear combination of all packets $s_1$ to $s_{V[n]}$ , where the coefficients are chosen from a large enough field such that the coded combinations are independent with high probability.",
"We refer to $V[n]$ as the transmit index in slot $n$ .",
"Conjecture 1 Since the packets are required in-order at the receiver, we believe that given transmit index $V[n]$ , there is no loss of generality in including all packets $s_1$ to $s_{V[n]}$ .",
"Hence we believe that there is no loss of generality in restricting our attention to full-rank codes.",
"Theorem 1 The optimal throughput-delay trade-off among full-rank codes is $(\\tau , \\lambda ) = (r , D(r|| p))$ for all $0 \\le r \\le p$ .",
"It is achieved by the coding scheme with $V[n] = \\lceil rn \\rceil $ for all $n$ .",
"The term $D(r||p)$ is the binary information divergence function which is defined for $0 < p,r < 1$ as $D(r || p ) = r \\log \\frac{r}{p} + (1-r) \\log \\frac{1-r}{1-p}.$ Note that as $r \\rightarrow 0$ , $D(r||p)$ converges to $-\\log (1-p)$ , which is the optimal $\\lambda $ , as given by Lemma REF .",
"To prove Theorem REF , we first show that the scheme with transmit index $V[n] = \\lceil rn \\rceil $ in time slot $n$ achieves the trade-off $(\\tau , \\lambda ) = (r , D(r|| p)$ .",
"Then we prove the converse by showing that no other full-rank scheme gives a better trade-off.",
"[Proof of Achievability] Consider the scheme with transmit index $V[n] = \\lceil r n\\rceil $ , where $r$ represents the rate of adding new packets to the transmitted stream.",
"The rate of adding packets is below the capacity of the erasure channel.",
"Thus it is easy to see that the throughput $\\tau = r$ .",
"Let $E[n]$ be the number of combinations, or equations received until time $n$ .",
"It follows the binomial distribution with parameter $p$ .",
"All packets $s_1 \\cdots s_{V[n]}$ are decoded when $E[n] \\ge V[n]$ .",
"Define the event $G_n = \\lbrace E[n] < V[n] \\text{ for all } 1 \\le j \\le n\\rbrace $ , that there is no packet decoding until slot $n$ .",
"The tail distribution of time $T$ between successive in-order decoding instants is, $\\Pr (T>n) &= \\sum _{k=0}^{\\lceil n r \\rceil - 1} \\Pr (E[n]=k) \\Pr (G_n | E[n] = k),\\nonumber \\\\&= \\sum _{k=0}^{\\lceil n r \\rceil - 1} \\binom{n}{k} p^{k} (1-p)^{n-k} \\Pr (G_n | E[n] = k), \\nonumber \\\\& \\doteq \\sum _{k=0}^{\\lceil n r \\rceil - 1} \\binom{n}{k} p^{k} (1-p)^{n-k}, \\\\& \\doteq \\binom{n}{\\lceil n r \\rceil - 1} p^{\\lceil n r \\rceil - 1} (1-p)^{n-\\lceil n r \\rceil + 1}, \\\\& \\doteq e^{-n D(r||p)}, $ where in (REF ), we remove the $\\Pr (G_n | E[n] = k)$ when we take the asymptotic equality $\\doteq $ because, by the Generalized Ballot theorem from [11], we can show that $\\Pr (G_n | E[n] = k)$ is $\\omega (1/n)$ .",
"Hence it is sub-exponential and does not affect the exponent of $\\Pr (T> n)$ .",
"In (), we only retain the $k = \\lceil n r \\rceil - 1$ term from the summation because for $r \\le p$ , that term asymptotically dominates other terms.",
"Finally, we use the Stirlings approximation of the binomial coefficient $\\binom{n}{k} \\approx e^{n H(k/n)}$ to obtain ().",
"Hence we have proved that the scheme with $V[n] = \\lceil rn \\rceil $ achieves the throughput-delay trade-off $(\\tau , \\lambda ) = (r , D(r|| p)$ .",
"[Proof of Converse] First let us show that the transmit index $V[n]$ of the optimal full-rank scheme should be non-decreasing in $n$ .",
"Given a scheme which does not satisfy the non-decreasing property, we can permute the order of transmitting the coded packets such that $V[n]$ is non-decreasing in $n$ .",
"Changing the order of the transmitted packets will not affect the throughput $\\tau $ .",
"And it can in fact improve the in-order decoding exponent $\\lambda $ because decoding can occur sooner when the initial coded packets include fewer source packets.",
"In the proposed scheme with $V[n] = \\lceil rn \\rceil $ , we add new packets to the transmitted stream at a constant rate $r$ .",
"But in general a full-rank scheme can vary the rate of adding packets.",
"Suppose it uses rate $r_i$ for $n_i$ slots for all $1 \\le i \\le L$ , such that $\\sum _{i=0}^{L} n_i = n$ and $\\sum _{i=1}^{L} n_i r_i = nr$ .",
"Then, the tail distribution of time $T$ between successive in-order decoding instants is, $\\Pr (T>n) &= \\sum _{k=0}^{\\lceil \\sum _{i=1}^{L} n_i r_i \\rceil - 1} \\Pr (E[n]=k) \\Pr (G_n | E[n] = k),\\\\&\\doteq \\sum _{k=0}^{\\lceil n r \\rceil - 1} \\binom{n}{k} p^{k} (1-p)^{n-k}, \\\\& \\doteq e^{-n D(r||p)}.",
"$ Varying the rate of adding packets affects the term $\\Pr (G_n | E[n] = k)$ in (REF ), but it is still $\\omega (1/n)$ and we can eliminate it when we take the asymptotic equality in ().",
"As a result, the in-order delay exponent is same as that if we had a constant rate $r$ of adding new packets to the transmitted stream.",
"Hence we have proved that no other full-rank scheme can achieve a better $(\\tau , \\lambda )$ trade-off than $V[n] = \\lceil nr \\rceil $ for all $n$ .",
"Fig.",
"REF shows the $(\\tau , \\lambda )$ trade-off for the immediate feedback and no feedback cases, with success probability $p = 0.6$ .",
"The optimal trade-off with any feedback delay $d$ lies in between these two extreme cases.",
"Figure: The trade-off between in-order decoding exponent λ\\lambda and throughput τ\\tau with success probability p=0.6p = 0.6 for the immediate feedback (d=1)(d= 1) and no feedback (d=∞)(d = \\infty ) cases."
],
[
"General Block-wise Feedback",
"In Section and Section we considered the extreme cases of immediate feedback $(d=1)$ and no feedback $(d = \\infty )$ respectively.",
"We now analyze the $(\\tau , \\lambda )$ trade-off with general block-wise feedback delay of $d$ slots.",
"We restrict our attention to a class of coding schemes called time-invariant schemes, which are defined as follows.",
"Definition 4 (Time-invariant schemes) A time-invariant scheme is represented by a vector $\\mathbf {x} = [x_1, \\cdots x_d]$ where $x_i$ , for $ 1 \\le i \\le d$ , are non-negative integers such that $\\sum _{i} x_i = d$ .",
"In each block we transmit $x_i$ linear combinations of the $i$ lowest-index unseen packets in the stream.",
"The above class of schemes is referred to as time-invariant because the vector $\\mathbf {x}$ is fixed across all blocks.",
"Observe that as $d \\rightarrow \\infty $ , the class of time-invariant schemes are equivalent to full-rank codes defined in Definition REF .",
"Conjecture 2 For any coding scheme, there exists a corresponding time-sharing policy between time-invariant schemes that gives the same or strictly better $(\\tau , \\lambda )$ trade-off.",
"We believe this conjecture is true because, it can be shown that any full-rank code can be expressed a time-sharing time-invariant scheme.",
"By Conjecture REF it follows that there is no loss of generality in focusing on time-invariant schemes.",
"There is also no loss of generality in restricting the length of the vector $\\mathbf {x}$ to $d$ .",
"This is because we are still transmitting $d$ independent coded packets.",
"And adding fewer source packets to the coded combinations, can only increase the exponent $\\lambda $ ."
],
[
"Analyzing the $(\\tau , \\lambda )$ of time-invariant schemes",
"Given a vector $\\mathbf {x}$ , define $p_d$ , as the probability of decoding the first unseen packet during the block, and $S_d$ as the number of innovative coded packets that are received during that block.",
"We can express $\\tau _{\\mathbf {x}}$ and $\\lambda _{\\mathbf {x}}$ in terms of $p_d$ and $S_d$ as, $(\\tau _{\\mathbf {x}}, \\lambda _{\\mathbf {x}}) &= \\left(\\frac{\\mathbb {E}[S_d]}{d}, -\\frac{1}{d}\\log (1-p_d) \\right),$ where we get throughput $\\tau _{\\mathbf {x}}$ by normalizing the $\\mathbb {E}[S_d]$ by the number of slots in the slots.",
"We can show that the probability $\\Pr (T>kd)$ of no in-order packet being decoded in $k$ blocks is equal $(1-p_d)^k$ .",
"Substituting this in (REF ) we get $\\lambda _{\\mathbf {x}}$ .",
"Example 1 Consider the time-invariant scheme $\\mathbf {x} = [1, 0, 3, 0]$ where block size $d=4$ .",
"That is, we transmit 1 combination of the first unseen packet, and 3 combinations of the first 3 unseen packets.",
"Fig.",
"REF illustrates this scheme for one channel realization.",
"The probability $p_d$ and $\\mathbb {E}[S_d]$ are, $p_d &= p + (1-p)\\binom{3}{3} p^3 (1-p)^0 = p + (1-p) p^3, \\\\\\mathbb {E}[S_d] &= \\sum _{i=1}^{3} i \\cdot \\binom{4}{i} p^i (1-p)^{4-i} + 3p^4 = 4p-p^4, $ where in (), we get $i$ innovative packets if there are $i$ successful slots for $1 \\le i \\le 3$ .",
"But if all 4 slots are successful we get only 3 innovative packets.",
"We can substitute (REF ) and () in (REF ) to get the $(\\tau , \\lambda )$ trade-off.",
"Figure: Illustration of the time-invariant scheme 𝐱=[1,0,3,0]\\mathbf {x} = [1,0,3,0] with block size d=4d=4.",
"Each bubble represents a coded combination, and the numbers inside it are the indices of the source packets included in that combination.",
"The check and cross marks denote successful and erased slots respectively.",
"The packets that are “seen\" in each block are not included in the coded packets in future blocks.Remark 1 Time-invariant schemes with different $\\mathbf {x}$ can be equivalent in terms of the $(\\tau , \\lambda )$ .",
"In general, given $x_1 \\ge 1$ , if any $x_i = 0$ , and $x_{i+1} = x \\ge 1$ , then the scheme is equivalent to setting $x_i = 1$ and $x_{i+1} = x-1$ , keeping all other elements of $\\mathbf {x}$ the same.",
"For example, $\\mathbf {x} = [ 1, 1, 2,0]$ gives the same $(\\tau , \\lambda )$ as $\\mathbf {x} = [ 1,0,3,0]$ ."
],
[
"Cost of Achieving Optimal $\\tau $ or {{formula:c5aeaca7-cda5-4d68-9669-dd4890081673}}",
"In Section we saw that for the immediate feedback case, we can achieve $(\\tau , \\lambda ) = (p , -\\log (1-p))$ .",
"However, when the feedback is delayed we can achieve optimal $\\tau $ (or $\\lambda $ ) only at the cost of sacrificing the optimality of the other metric.",
"We now find the best achievable $\\tau $ (or $\\lambda $ ) with optimal $\\lambda $ (or $\\tau $ ).",
"Lemma 2 (Cost of Optimal Exponent $\\lambda $ ) For a feedback delay of $d$ slots, the best achievable throughput is $\\tau = (1-(1-p)^d)/d$ , when the in-order decoding exponent $\\lambda = -\\log (1-p)$ .",
"If we want to achieve $\\lambda = -\\log (1-p)$ , we require $p_d$ in (REF ) to be equal to $1-(1-p)^d$ .",
"The only scheme that can achieve this is $\\mathbf {x} = [d, 0, \\cdots , 0]$ , where we transmit $d$ copies of the first unseen packet.",
"The number of innovative packets $S_d$ received in every block is 1 with probability $1-(1-p)^d$ , and zero otherwise.",
"Hence, the best achievable throughput is $\\tau = (1-(1-p)^d)/d$ with optimal $\\lambda = -\\log (1-p)$ .",
"This result gives us insight on how much bandwidth (which is proportional to $1/\\tau $ ) is needed for a highly delay-sensitive application which needs $\\lambda $ to be as large as possible.",
"Lemma 3 (Cost of Optimal Throughput $\\tau $ ) For a feedback delay of $d$ slots, the best achievable in-order decoding exponent is $\\lambda = -\\log (1-p)/d$ , when the throughput $\\tau = p$ .",
"If we want to achieve $\\tau = p$ , we need to guarantee an innovation packet in every successful slot.",
"The only time invariant scheme that achieve this is $\\mathbf {x} = [ 1,1, \\cdots 1]$ , and the vectors $\\mathbf {x}$ that are equivalent to it as given by Remark REF .",
"With $\\mathbf {x} = [ 1,1, \\cdots 1]$ , the probability of decoding the first unseen packet is $p_d = p$ .",
"Substituting this in (REF ) we get $\\lambda =-{\\log (1-p)}{d}$ , the best achievable $\\lambda $ when $\\tau = p$ .",
"Fig.",
"REF shows the best achievable $\\tau $ and $\\lambda $ versus $d$ , when the other metric is at its optimal value.",
"The plots in Fig.",
"REF correspond to moving leftwards and downwards respectively from the optimal trade-off $(p, -\\log (1-p))$ in Fig.",
"REF .",
"Figure: Plot of the best achievable τ\\tau (or λ)\\lambda ) versus dd, while maintaining the optimal value of the other metric λ\\lambda (or τ\\tau ), for channel success probability p=0.6p=0.6."
],
[
"Finding Optimal $(\\tau , \\lambda )$ Trade-off",
"For any given throughput $\\tau $ , our aim is to find the transmission scheme that achieves the maximum $\\lambda $ .",
"We first prove that any convex combination of achievable points $(\\tau , \\lambda )$ can be achieved.",
"Theorem 2 (Convex Combinations of Time-invariant Schemes) Given time-invariant schemes $\\mathbf {x}^{(i)}$ for $1 \\le i \\le B$ , we can achieve the throughput-delay trade-off given by any convex combination of the points $(\\lambda _{\\mathbf {x}^{(i)}}, \\tau _{\\mathbf {x}^{(i)}})$ by time-sharing between the schemes.",
"Here we prove the result for $B=2$ , that is time-sharing between two schemes.",
"It can be extended to general $B$ using induction.",
"Given two time-invariant schemes $\\mathbf {x}^{(1)}$ and $\\mathbf {x}^{(2)}$ which achieve the throughput-delay trade-offs $(\\lambda _{\\mathbf {x}^{(1)}}, \\tau _{\\mathbf {x}^{(1)}})$ and $(\\lambda _{\\mathbf {x}^{(2)}}, \\tau _{\\mathbf {x}^{(2)}})$ respectively, consider a time-sharing strategy where, in each block we use the scheme $\\mathbf {x}^{(1)}$ with probability $\\mu $ and scheme $\\mathbf {x}^{(2)}$ otherwise.",
"Then, it is easy to see that the throughput on the new scheme is $\\tau = \\mu \\tau _{\\mathbf {x}^{(1)}} + (1-\\mu ) \\tau _{\\mathbf {x}^{(2)}}$ .",
"Now we prove the in-order decoding exponent $\\lambda $ is also a convex combinations of $\\lambda _{\\mathbf {x}^{(1)}}$ and $\\lambda _{\\mathbf {x}^{(2)}}$ .",
"Let $p_{d_1}$ and $p_{d_2}$ be the probabilities of decoding the first unseen packet in a block using scheme $\\mathbf {x}^{(1)}$ and $\\mathbf {x}^{(2)}$ respectively.",
"Suppose in an interval with $k$ blocks, we use scheme $\\mathbf {x}^{(1)}$ for $h$ blocks, and scheme $\\mathbf {x}^{(2)}$ in the remaining blocks, we have $\\Pr (T > kd) = (1-p_{d_1})^{h} (1-p_{d_2})^{k-h}.$ Using this we can evaluate $\\lambda $ as, $\\lambda &= \\lambda _{\\mathbf {x}^{(1)}} \\lim _{k \\rightarrow \\infty } \\frac{h}{k} +\\lambda _{\\mathbf {x}^{(2)}} \\lim _{k \\rightarrow \\infty } \\frac{k-h}{k} \\\\&= \\mu \\lambda _{\\mathbf {x}^{(1)}} + (1-\\mu ) \\lambda _{\\mathbf {x}^{(2)}}$ where we get (REF ) using (REF ).",
"As $k \\rightarrow \\infty $ , by the weak law of large numbers, the fraction $h/k$ converges to $\\mu $ .",
"Hence, we have shown that we can interpolate between the $(\\tau , \\lambda )$ trade-off of two policies by time-sharing between them.",
"The main implication of Theorem REF is that, to find the optimal $(\\tau ,\\lambda )$ trade-off, we only have to find the points $(\\tau _{\\mathbf {x}} ,\\lambda _{\\mathbf {x}})$ that lie on the convex envelope of the achievable region spanned by all possible $\\mathbf {x}$ .",
"We determine this optimal trade-off for $d=2, 3$ in Lemma REF and Lemma REF below.",
"Lemma 4 (Optimal Trade-off for $d=2$ ) The optimal $(\\tau , \\lambda )$ trade-off is the line joining points $\\left( (1-(1-p)^2)/2, -\\log (1-p)\\right)$ and $\\left(p, -\\log (1-p)/2\\right)$ .",
"When $d=2$ there are only two possible time-invariant schemes $\\mathbf {x} = [2,0]$ and $[1, 1]$ that give unique $(\\tau , \\lambda )$ .",
"By Remark REF , all other valid vectors $\\mathbf {x}$ are equivalent to one of these schemes.",
"From Lemma REF and Lemma REF we know that the $(\\tau ,\\lambda )$ for these schemes are $( (1-(1-p)^2)/2, -\\log (1-p))$ and $(p, -\\log (1-p)/2)$ respectively.",
"By Theorem REF we can achieve all $(\\tau , \\lambda )$ on the line joining these two points by time-sharing between the two policies.",
"Lemma 5 (Optimal Trade-off for $d=3$ ) The optimal $(\\tau , \\lambda )$ trade-off when $d=3$ is the piecewise linear curve joining points $(\\tau _A, \\lambda _A) &= \\left( \\frac{1-(1-p)^3}{3}, -\\log (1-p)\\right), \\\\(\\tau _B, \\lambda _B) &= \\left( \\frac{2p(2-p)}{3}, -\\frac{2}{3}\\log (1-p)\\right), \\\\(\\tau _C, \\lambda _C) &= \\left(p, -\\frac{\\log (1-p)}{3}\\right).",
"$ When $d=3$ there are four time-invariant schemes $\\mathbf {x}^{(1)} = [3, 0, 0], \\mathbf {x}^{(2)} = [2,1,0],\\mathbf {x}^{(3)} =[1,2,0]$ and $\\mathbf {x}^{(4)} = [1,1,1]$ that give unique $(\\tau , \\lambda )$ , as given by Definition REF and Remark REF .",
"From Lemma REF and Lemma REF we know that $(\\tau _{\\mathbf {x}^{(1)}},\\lambda _{\\mathbf {x}^{(1)}}) =(\\tau _A, \\lambda _A)$ and $(\\tau _{\\mathbf {x}^{(4)}},\\lambda _{\\mathbf {x}^{(4)}}) =(\\tau _C, \\lambda _C)$ .",
"For the other two schemes, we first evaluate $p_d$ and $\\mathbb {E}[S_d]$ and substitute them in (REF ) to get, $(\\tau _{\\mathbf {x}^{(2)}},\\lambda _{\\mathbf {x}^{(2)}}) =(\\tau _B, \\lambda _B)$ and $ (\\tau _{\\mathbf {x}^{(3)}},\\lambda _{\\mathbf {x}^{(3)}}) =\\left( (3p-p^3)/3, -(\\log (1-p)^2(1+p))/3\\right).",
"$ We can show that $\\mathbf {x}^{(2)}$ gives a better trade-off than $\\mathbf {x}^{(3)}$ by showing that for all $p$ , the slopes of the lines joining $(\\tau _{\\mathbf {x}^{(i)}},\\lambda _{\\mathbf {x}^{(i)}})$ for $i = 1, \\cdots 4$ satisfy, $\\frac{\\lambda _{\\mathbf {x}^{(1)}} - \\lambda _{\\mathbf {x}^{(2)}}}{ \\tau _{\\mathbf {x}^{(1)}} - \\tau _{\\mathbf {x}^{(2)}}} &\\ge \\frac{\\lambda _{\\mathbf {x}^{(1)}} - \\lambda _{\\mathbf {x}^{(3)}}}{ \\tau _{\\mathbf {x}^{(1)}} - \\tau _{\\mathbf {x}^{(3)}}} \\\\\\frac{\\lambda _{\\mathbf {x}^{(2)}} - \\lambda _{\\mathbf {x}^{(4)}}}{ \\tau _{\\mathbf {x}^{(2)}} - \\tau _{\\mathbf {x}^{(4)}}} &\\le \\frac{\\lambda _{\\mathbf {x}^{(3)}} - \\lambda _{\\mathbf {x}^{(4)}}}{ \\tau _{\\mathbf {x}^{(3)}} - \\tau _{\\mathbf {x}^{(4)}}}.$ The trade-off for $d=2$ and $d=3$ with $p= 0.6$ is shown in Fig.",
"REF .",
"The point below the piece-wise linear curve for $d=3$ , corresponding to the sub-optimal scheme $\\mathbf {x}^{(3)}=[1,2,0]$ .",
"We observe that the optimal trade-off becomes significantly worse are $d$ increases.",
"From this we can imply that frequent feedback to the source is important in delay-sensitive applications to ensure fast in-order decoding of packets.",
"Figure: The throughput-delay trade-off with p=0.6p=0.6 for d=2,3d=2,3 which can be shown to be optimal over all convex combinations of time-invariant schemes.",
"The point just below the piece-wise linear curve for d=3d=3, corresponding to the sub-optimal scheme 𝐱=[1,2,0]\\mathbf {x} = [1, 2, 0].For general $d$ , it is hard to search for the $(\\tau _{\\mathbf {x}}, \\lambda _{\\mathbf {x}})$ that lie on the optimal trade-off.",
"We suggest a set of time-invariant schemes which are easy to analyze and they give a good $(\\tau , \\lambda )$ trade-off.",
"Definition 5 (Suggested Schemes for General $d$ ) For general $d$ we suggest schemes with $x_1 = a$ and $x_{d-a+1} = d-a$ , for $a = 1, \\cdots d$ .",
"They give the throughput-delay trade-off $(\\tau ,\\lambda ) &=\\left( \\frac{1-(1-p)^a + (d-a)p}{d}, -\\frac{a}{d} \\log (1-p) \\right).$ Fig.",
"REF shows the trade-off given by (REF ) for different values of $d$ .",
"Observe that for $d=2$ and $d=3$ the suggested schemes coincide with the optimal trade-off we derived in Lemma REF and Lemma REF and shown in Fig.",
"REF .",
"As $d \\rightarrow \\infty $ , and $a = \\alpha d$ , the trade-off converges to $( (1-\\alpha )p, -\\alpha \\log (1-p))$ for $ 0 \\le \\alpha \\le 1$ , which is the line joining $(0, -\\log (1-p))$ and $(p, 0)$ .",
"Numerical results suggest that for small $d$ this class of schemes gives the best trade-off among all possible time-invariant schemes $\\mathbf {x}$ , and close to optimal in general.",
"Figure: The throughput-delay trade-off of the suggested coding schemes in Definition with p=0.6p=0.6 and different values of feedback delay dd.",
"Numerical results suggest that this trade-off is optimal all convex combinations of time-invariant schemes for small dd."
],
[
"Concluding Remarks",
"In this paper we analyze how block-wise feedback affects the trade-off between throughput $\\tau $ and in-order decoding exponent $\\lambda $ , which measures the burstiness in-order packet decoding in streaming communication.",
"When there is immediate feedback, we can simultaneously achieve the optimal $\\tau $ and $\\lambda $ .",
"But as the block size increases, and the frequency of feedback reduces, we have to compromise on at least one of these metrics.",
"Our analysis gives us the insight that frequent feedback is crucial to ensure in-order packet delivery in delay-sensitive applications.",
"Given that feedback comes in blocks of $d$ slots, we present a spectrum of coding schemes that span different points on the $(\\tau , \\lambda )$ trade-off as shown in Fig.",
"REF .",
"Depending upon the delay-sensitivity and bandwidth limitations of the applications, these codes provide the flexibility to choose a suitable operating point on trade-off.",
"The proposed codes can be shown to be optimal over the broad class of full-rank codes for small feedback delay $d$ , and when there is no feedback."
]
] | 1403.0259 |
[
[
"Spin effects in thermoelectric properties of Al and P doped zigzag\n silicene nanoribbons"
],
[
"Abstract Electric and thermoelectric properties of silicene nanoribbons doped with Al and P impurity atoms are investigated theoretically for both antiparallel and parallel orientations of the edge magnetic moments.",
"In the former case, appropriately arranged impurities can lead to a net magnetic moment and thus also to spin thermoelectric effects.",
"In the latter case, in turn, spin thermoelectric effects also occur in the absence of impurities.",
"Numerical results based on ab-initio calculations show that the spin thermopower can be considerably enhanced by the impurities."
],
[
"Introduction",
"Since the discovery of graphene, one can observe increasing interest in other two-dimensional honeycomb structures.",
"One of such materials is silicene – a two-dimensional hexagonal lattice of silicon (Si) atoms.",
"In contrast to graphene, silicene has a buckled atomic structure, where the two triangular sublattices are slightly displaced in opposite directions normal to the atomic plane.",
"Band structure calculations show that silicene is a semimetal with zero energy gap and linear electronic spectrum near the $K$ points of the Brilouine zone [1].",
"Similarly to graphene, electrons in the vicinity of the $K$ points (Dirac points) behave like massless fermions.",
"Apart from strictly two-dimensional crystals of silicene, also silicene nanoribbons have been fabricated recently [2], [3], [4], [5].",
"Theoretical investigations of silicene have revealed a number of its interesting properties, like for instance the spin Hall effect induced by spin-orbit interaction [6] or an electrically tunable energy gap [7], [8].",
"The latter effect appears owing to the buckled atomic structure.",
"Moreover, ab-initio numerical calculations have shown that two electrically-controlled gapped Dirac cones for nearly spin polarized states can exist when silicene is in a perpendicular electric field.",
"Accordingly, an effective silicene-based spin filter has been proposed, where spin polarization of electric current can be switched with external electric field [9].",
"Since silicon plays a crucial role in the present-day electronics, integration of silicene into nanoelectronics seems to be more promising than that of graphene, and this possibility opens new perspectives for this novel material [10], [11].",
"Therefore, detailed description and understanding of physical properties of silicene is currently of great interest.",
"First-principle calculations based on the density functional theory (DFT) have shown that silicene surface is very reactive and can be easily functionalized [12].",
"This functionalization, in turn, can significantly change basic properties of silicene.",
"Structural, electronic, magnetic and vibrational properties of silicene with adsorbed or substituted atoms of various types have been extensively studied in recent years [12], [13].",
"The corresponding results show that the low-buckled lattice is stable in a wide range of doping [14].",
"Apart from this, both ad-atoms and substituted atoms induce characteristic modes in the phonon spectrum of silicene.",
"A significant charge transfer between ad-atoms and silicene has also been reported [12].",
"Especially interesting are functionalization- and doping-induced modifications of transport properties.",
"It has been shown that adsorption of alkali metals can transform silicene into a narrow gap semiconductor.",
"On the other hand, by doping with transition-metal atoms, either semiconducting or metallic behavior can be obtained [13].",
"Furthermore, the band gap in silicene can be tuned when it is functionalized with hydrogen [15], [16].",
"As a result of semi-hydrogenation, i.e.",
"hydrogenation from one side only, ferromagnetic ordering can be induced, and the system exhibits then semiconducting properties with a direct energy gap of the order of 1 eV [17].",
"It is worth noting that itinerant magnetism mediated by holes has been also predicted for AlSi monolayers as well as for AlSi armchair nanoribbons [18].",
"The electronic structure can be also modified by externally induced strain.",
"For instance, it has been predicted that compressive (tensile) strain can move the Dirac points in silicene below (above) the Fermi level, so the system can behave like n-type (p-type) doped one [19].",
"Moreover, a sufficiently strong strain can reduce the thermal conductivity of silicene, as follows from non-equilibrium molecular dynamic simulations [20].",
"Apart from two-dimensional silicene crystals, also nanoribbons of armchair (aSiNRs) or zigzag (zSiNRs) type have been widely studied in view of potential applications in silicon-based electronics and spintronics devices.",
"Similarly to graphene nanoribbons (GNRs), zSiNRs exhibit edge magnetic ordering.",
"Ab-initio calculations for pristine zSiNRs show that antiferromagnetic (AFM) state, where magnetic moments of the two edges of a nanoribbon are antiparallel, corresponds to the lowest energy [21], [22].",
"Ferromagnetic (FM) ordering, in which magnetic moments of the two edges are parallel, has a slightly higher energy and can be stable eg in an external magnetic field.",
"In the FM state, zSiNRs have metallic transport properties, whereas in the AFM configuration a wide gap opens and zSiNRs display semiconducting behavior.",
"Electronic, mechanical and magnetic properties of SiNRs have been studied recently by first principle methods [21], [22], [23], [24], [25].",
"In particular, investigations of electron transport properties of zSiNRs have revealed a giant magnetoresistance effect associated with transition from the FM to AFM configuration [24].",
"It has been also predicted that hydrogen-terminated zSiNRs in the presence of in-plane electrical field can behave like a half-metallic ferromagnet with spin polarization up to 99$\\%$ [26].",
"When a local exchange field affects only one edge of zSINRs, an energy gap can be opened in one spin channel, whereas the second spin channel remains gapless [27].",
"Spin gapless semiconductor behavior with 100$\\%$ spin polarization has been also predicted for zSiNRs doped with B or N atoms in edge positions [28], [29].",
"On the other hand, in aSiNRs with B/N substitutions at the edges, semiconductor-metal transition due to formation of a half-filled impurity band near the Fermi level has been predicted [29].",
"Very recently, the interplay between bulk and edge states induced by Rashba spin-orbit coupling in zSiNRs has been investigated in the presence of electrical field [30].",
"It has been shown that the states with opposite velocities can open spin-dependent subgaps which remarkably influence spin polarized transport properties.",
"Thermoelectric properties of nanoscopic systems are currently of great interest due to the possibility of heat to electrical energy conversion at nanoscale, which is important for applications.",
"Thermoelectric properties of aSiNRs and zSINRs have been investigated by ab-initio methods based on DFT and non-equilibrium Green function formalism [21], [22].",
"It has been shown that thermopower of pristine zSiNRs in the low-energy AFM state can be considerably enhanced due to the presence of energy gaps.",
"Moreover, the thermopower strongly depends on the magnetic configuration and a considerable magnetothermopower related to transition from the AFM to FM state can be observed [22].",
"In ferromagnetic systems, interplay between the spin effects and thermoelectric properties can lead to new spin related thermoelectric phenomena [31], [32], [33], [34].",
"The most spectacular spin related thermoelectric effect is the spin thermopower (spin Seebeck effect) which is a spin analog of the conventional thermopower.",
"A considerable spin thermopower has been predicted for pristine zSiNRs with FM ordering [22].",
"In the present paper we analyze electric and thermoelectric phenomena of zSiNRs doped with Al and P impurity atoms.",
"It is shown that the spin thrmopower can be considerably enhanced by impurities.",
"In section 2 we present transmission function for zSiNRs.",
"Thermoelectric properties in the AFM state are presented and discussed in section 3, while those in the FM state are considered in section 4.",
"Summary and final conclusions are in section 5."
],
[
"Transmission in ${\\rm z}$ S{{formula:67e1141a-cae5-43fb-8727-904035aa4828}} NR{{formula:a7e200ad-1ed2-40e4-994f-8abc868fb5e3}} with A{{formula:f69751bc-f2f9-4a04-8c0b-3c283c2097bc}} and P impurity atoms",
"In this section we will present numerical results on electronic transmission in zSiNRs with impurities, obtained by ab-initio numerical calculations within the DFT Siesta code [35].",
"The nanoribbon edges were terminated with hydrogen atoms to remove the dangling bonds, and pristine zSiNRs as well as those with impurity atoms of Al or P type, localized at different positions with respect to the nanoribbon edges, were considered.",
"The impurities were distributed periodically along the chain and localized (i) at one of the edges (PE configuration), (ii) at the nanoribbon center (PC configuration), and (iii) in the middle between the edge and central atoms of the nanoribbon (PM configuration).",
"The elementary cell was adequately enlarged to include one impurity atom.",
"The spin-resolved energy-dependent transmission $T_\\sigma (E)$ through nanoribbons was determined within the non-equilibrium Green function (NEGF) method as implemented in the Transiesta code [36].",
"The structures were optimized until atomic forces converged to 0.02 eV/A.",
"The atomic double-polarized basis (DZP) was used and the grid mesh cutoff was set equal to 200 Ry.",
"The generalized gradient approximation (GGA) with Perdrew-Burke-Ernzerhof parameterization was applied for exchange-correlation part of the total energy functional [40].",
"The performed calculations, similarly to those presented in Refs [21], [22], show that antiferromagnetic (AFM) state, where magnetic moments at one edge are antiparallel to those at the other edge, is the most stable configuration (ground state) in pristine narrow zSiNRs.",
"Ferromagnetic (FM) state, in which the edge moments are all parallel, corresponds to slightly higher energy.",
"The energy difference between the two magnetic states is equal to 0.02 eV for pristine nanoribbon containing $N$ =6 zigzag chains.",
"Thus, magnetic configuration of pristine zSiNRs can be easily changed from the AFM state to the FM one, for instance by an external magnetic field."
],
[
"Low energy state",
"In Fig.",
"REF we show spin density for zSiNRs in the ground state (referred to in the following as the low energy state) for the three different impurity configurations, i.e.",
"PE, PC and PM ones.",
"In the presence of non-magnetic impurity atoms (Al, P), magnetic moments at the two nanoribbon edges do not fully compensate each other.",
"Moreover, some small moments are also localized on inner atoms (see Fig.",
"REF ).",
"As a result, small net magnetization can be observed in the low energy state.",
"Our calculations also show, that when the impurity concentration at the edge (PE configuration) increases, the low-energy state becomes ferromagnetic, with magnetic moments located only at the impurity-free edge of the nanoribbon (see Fig.",
"REF ).",
"This happens when the distance between the edge impurity atoms is smaller than 19 $\\rm Å$ , which is in agreement with Refs [28], [29].",
"Thus, the low-energy state is antiferromagnetic for pristine nanoribbons, generally ferrimagnetic in nanoribbons with impurities, and ferromagnetic for impurities located at one edge and of a sufficiently large concentration.",
"Since smaller impurity concentrations are easier to be achieved experimentally, we have performed calculations for zSiNRs with longer distance between the impurities.",
"More specifically, the distance is equal to the size of the elementary cell shown in Fig.1.",
"Note, Fig.2 displays more elementary cells, as the distance between impurities is there smaller than in Fig.",
"REF .",
"To be more specific, the elementary cell shown in Fig.",
"REF is the system through which transmission is calculated, while the left and and right semi-infinite parts of the nanoribbon are treated as external electrodes.",
"Figure: (Color online) Spin density in the low energy state, calculated within GGA approximation for zSiNRs with N=6N=6for Al (left panel) and P (right panel) impurities in the PE (a,b), PM (c,d) and PC (e,f) configurations.Black and gray dots represent magnetic moments of opposite directions.Figure: (Color online) Spin density in the one-edge FM state, calculated within GGA approximation for zSiNRs with N=6N=6 for Al (left) and P (right) impurities in the PE configuration.Note, the density of impurity atoms is here larger than in Fig. .",
"Meaning of the black and gray dots same as in Fig.",
".Figure: (Color online) Spin-dependent transmission in zSiNR (NN=6) in the low-energy state as a function of energy, calculated within the GGA approximationfor pristine nanoribbon (inset to a) and for Al (left panel) and P (right panel) impurities in the PE (a,b), PM (c,d), and PC (e,f) configurations, respectively.Insets to Figs.",
"c,e,f show transmission in a narrow energy range, where the Fano antiresonances are well resolved.Figure: (Color online) Spin density in the FM state, calculated within GGA approximation for zSiNRs (NN=6)for Al (left panel) and P (right panel) impurities in the PE (a,b), PM (c,d) and PC (e,f) configurations, respectively.Meaning of the black and gray dots same as in Fig.",
".Transmission function $T_{\\sigma }(E)$ corresponding to the nanoribbons shown in Fig.",
"REF is presented in Fig.",
"REF for both spin orientations.",
"The transmission is shown there as a function of energy measured from the corresponding Fermi energy $E_F$ .",
"One can note that the energy gap at the Fermi level, which exists in pristine zSiNRs in the AFM state (inset to Fig.",
"REF a), survives also in the presence of impurity atoms.",
"However, transmission depends now on the spin orientation because magnetic moments of the two edges do not fully compensate each other in the presence of impurities.",
"Moreover, $T_{\\sigma }(E)$ depends on the type and position of impurities.",
"The transmission is strongly modified near the edges of the energy gap.",
"The impurity atoms can lead to states localized in the gap, which give narrow peaks in the transmission function – especially well visible for Al atoms in the PM configuration (below the Fermi level) and in the PC configuration of P impurities (above the Fermi level).",
"Apart from the wide gap near the Fermi energy $E_F$ , additional gaps appear in the spectrum at other energies.",
"A wide gap opens above the Fermi level in the spin-up channel for Al impurities localized at the nanoribbon edge, and a similar gap appears below the Fermi level in the spin-down channel for P impurities (Fig.",
"REF ).",
"It is interesting to note that spin-down electrons in the former case and spin-up holes in the latter case are less influenced by the impurities, so the corresponding gaps are much narrower.",
"The transmission exhibits also a number of antiresonance dips – mainly for higher values of $|E - E_{F}|$ .",
"In the presence of impurity atoms, quantum interference can lead to Fano antiresonances.",
"Especially interesting results are obtained for the PC impurity configuration, where typical and well defined Fano antiresonances in nanoribbons with P impurity atoms appear in the transmission in the vicinity of $E - E_{F}$ = 0.5 eV (Fig.",
"REF f).",
"Due to the destructive interference, a relatively wide gap appears in this energy region, with transmission close to zero.",
"On the other hand, in nanoribbons with Al impurity atoms in the PC configuration, the Fano antiresonance can be observed near $E - E_{F}\\approx -0.5$ eV as well as for $E - E_{F} \\approx 0.4$ eV.",
"The interference effects are more pronounced for spin-up carriers in nanoribbons with P impurities, and spin-down carriers in nanoribbons with Al impurities, as considerable magnetic moments are then localized on nearby Si atoms.",
"Well defined Fano antiresonances can be also visible for Al impurities in the PM configuration (inset to Fig.",
"REF c) Figure: (Color online) Spin-dependent transmission for zSiNRs (NN=6) in the FM state as a function of energy, calculated within the GGA approximationfor pristine ribbon (inset to a) and for Al (left panel) and P (right panel) impurities in the PE (a,b), PM (c,d), PC (e,f) configurations.Insets to e and f show the transmission in a narrow energy range, where the Fano antiresonances are well resolved."
],
[
"Ferromagnetic state",
"Consider now nanoribbons which display ferromagnetic ordering (FM state) in the absence as well as presence of impurities.",
"As already mentioned above, energy of such a state is only slightly higher than that of the above discussed low-energy (ground) state.",
"Thus, the FM state can be stabilized by an external magnetic field or by other methods.",
"Spin density for the three different configurations of impurity atoms is presented in Fig.",
"REF .",
"Note, magnetic moments localized at the two edges are not equal in the presence of impurities.",
"The corresponding spin-resolved transmission function is shown in Fig.",
"REF .",
"Pristine ferromagnetic zSiNRs exhibits typical metallic behavior with a constant transmission in the vicinity of the Fermi level, see the inset to Fig.",
"REF a.",
"In the presence of impurity atoms localized at one of the edges, the metallic character, with almost constant transmission, is preserved for energies very close to $E_{F}$ .",
"For both kinds of impurities, transmission shows a narrow dip below the Fermi level for spin-up electrons, and similar dip above the Fermi level for spin-down electrons, see Figs REF a,b.",
"Additional and more pronounced dips appear for energies close to $\\pm $ 0.5 eV.",
"These dips appear for both spin orientations, but are slightly separated in energy.",
"It is also interesting to note that in the case of P impurities a wider dip corresponds to holes ($E - E_{F} <0$ ), whereas for Al impurities the well defined and wide dip appears for electrons ($E - E_{F} >0$ ).",
"When the impurities are shifted towards the center of the nanoribbon, some important modifications in the transmission function appear.",
"Interesting behavior can be noticed for the PM impurity configuration (Figs REF c,d).",
"One spin channel (spin-down for Al and spin-up for P) remains then conductive (metallic) in the close vicinity of the Fermi level, whereas the second channel becomes semiconducting due to a wide dip in the transmission at the Fermi level.",
"One may expect that the dip appears due to pronounced Fano antiresonance.",
"This behavior significantly changes in the PC configuration, with the impurity atoms localized in the center of the nanoribbon (Figs.",
"REF e,f).",
"Both spin channels are then conductive in the close vicinity of the Fermi level and metallic character of the system is recovered.",
"However, pronounced dips in the transmission spectrum appear for electrons (P impurities) and holes (Al impurities).",
"It is worth to note that the Fano antiresonances appear for energies well above or below the Fermi level.",
"In particular, for energies close to 0.5 eV positive interference effects in nanoribbons with P defects lead to a very well defined peak, which is followed by a wide gap resulting from destructive interference, see the inset to Fig.",
"REF f. Since the transmission is strongly spin dependent, the resonances for spin-up and spin-down channels are well separated.",
"For Al impurities, narrow Fano dips appear for both spin channels, but now the effect can be observed for energies below the Fermi level and corresponds to the holes.",
"Thus, changing the impurity type one can observe quantum interference effects for electrons or holes.",
"Transport properties of nanoribbons with (and without) defects are fully determined by the transmission function.",
"Thus, having found $T_{\\sigma }(E)$ one can determine not only electric conductance, but also thermoelectric parameters.",
"This will be presented in the following sections, and we begin with the low energy (ground) state."
],
[
"Electric and thermoelectric properties of ${\\rm z}$ S{{formula:43d29f82-64cd-4ef3-b77e-ba1ea30e9b4b}} NR{{formula:c786c753-9611-47b7-8340-4ac846751d13}} in the low energy state",
"In the low-energy state of pristine zSiNRs, the two spin channels are equivalent.",
"However, since a nonzero net magnetization generally appears in the presence of impurities, these two channels are then no longer equivalent.",
"When the spin channels are mixed in the nanoribbon on a distance comparable to the system length, no spin thermopower can be observed and only conventional thermoelectric phenomena can occur.",
"We will consider first this limit.",
"In the linear response regime, the electric $I$ and heat $I_{Q}$ currents flowing through the system from left to right when the difference in electrical potential and temperature of the left and right electrodes is $\\Delta $ V and $\\Delta $ T, respectively, can be written in the matrix form as [22] $\\left(\\begin{array}{c}I \\\\I_{Q} \\\\\\end{array}\\right)=\\left(\\begin{array}{cc}e^2 L_{0} & \\frac{e}{T}L_{1} \\\\eL_{1} & \\frac{1}{T}L_{2} \\\\\\end{array}\\right)\\left(\\begin{array}{c}\\Delta V \\\\\\Delta T \\\\\\end{array}\\right),$ where $e$ is the electron charge, while $L_{n} = \\sum _{\\sigma }L_{n \\sigma }$ , with $L_{n \\sigma } = -\\frac{1}{h} \\int dE\\,T_{\\sigma }(E)\\, (E-\\mu )^{n} \\frac{\\partial f}{\\partial E} $ for $n=0,1,2$ .",
"Here, $T_{\\sigma }$ (E) is the spin-dependent transmission function for the system and $f(E-\\mu )$ is the Fermi-Dirac distribution function corresponding to the chemical potential $\\mu $ and temperature $T$ .",
"Basic transport coefficients can be expressed in terms of $L_{n}$ .",
"The electrical conductance $G$ is given by the formula $G=e^{2}L_{0}$ , whereas the electronic contribution to the thermal conductance, $\\kappa _{e}$ , is equal to $\\kappa _{e}=\\frac{1}{T} \\left(L_{2} - \\frac{L_{1}^{2}}{L_{0}}\\right).$ In turn, the thermopower, $S=-\\Delta V /\\Delta T$ , is expressed by the formula $S=-\\frac{L_{1}}{|e|TL_{0}}.$ In the linear response regime considered in this paper, transport properties are determined by electronic states near the Fermi level.",
"In reality the chemical potential in nanoribbons (measured from the Fermi energy $E_F$ ) can be easily varied with an external gate voltage.",
"This technique offers a unique possibility to realize various positions of the Fermi level in a single sample.",
"Alternatively, the chemical potential can be moved down or up by p-type or n-type doping, which results in $\\mu <$ 0 and $\\mu >0$ , respectively.",
"We assume that this doping does not influence transmission functions calculated above, and the donors/acceptors are different form the substituted P and Al atoms.",
"Significant changes in the chemical potential can be caused by a substrate, too.",
"Using the transmission functions through the system (central part of the nanoribbon) determined in the previous section for the low-energy state of pristine and doped zSiNRs, one can calculate the electrical conductance $G$ and electronic term in the thermal conductance, $\\kappa _{e}$ , as well as the thermopower $S$ .",
"The results are presented in Fig.",
"REF as a function of the chemical potential $\\mu $ for Al and P impurities, and for all the three impurity configurations (PE, PM and PC ones).",
"Figure: (Color online) Conductance (a,b), thermopower SS (c,d) and electronic term in the thermal conductance κ e \\kappa _{e} (e,f) as a function of the chemical potential μ\\mu for zSiNRsin the low-energy state with Al (left panel) and P (right panel) impurities, calculated for T=90T=90 K, N=6, and the three considered impurity configurations.For comparison we also show there the results for pristine nanoribbons.",
"For the temperature $T=90$ K assumed in Fig.",
"REF , the energy gap in the vicinity of $\\mu =0$ is well resolved in the electric and in the thermal conductance.",
"For pristine zSiNRs, both $G$ and $\\kappa _{e}$ rapidly increase near the gap edges reaching maximum, and then become reduced with a further increase in $|\\mu |$ .",
"For higher values of $|\\mu |$ , the conductances increase again.",
"In the presence of impurity atoms, both $G$ and $\\kappa _{e}$ are substantially reduced in the whole region of the chemical potential, as shown in Figs.",
"REF a,b for $G$ and Figs.",
"REF e,f for $\\kappa _{e}$ .",
"When the impurities are localized at the nanoribbon edge (PE configuration), the conductance shape is similar to that in pristine system, especially near the gap, though the conductance is reduced.",
"Moreover, a pronounced dip appears in the region of negative $\\mu $ ($\\mu \\approx -0.5$ eV) for P impurities and for positive $\\mu $ ($\\mu \\approx 0.4$ eV) for Al impurities.",
"Much stronger modifications occur in the PC configuration, with defects localized in the nanoribbon center.",
"The conductances $G$ and $\\kappa _{e}$ are then remarkably reduced near the gap edges, especially for negative $\\mu $ for P impurities and positive $\\mu $ for Al impurities.",
"Figure: (Color online) Charge and spin thermopowers as a function of the chemical potential μ\\mu , calculated for the low-energy state of zSiNRswith Al (left panel) and P (right panel) impurities in the three (PE, PM, and PC) configurations, and for T=90K and N=6.The thermopower $S$ of pristine nanoribbon is considerably enhanced inside the main gap in the spectrum (around $E_F$ ), with maxima (positive and negative) at the chemical potentials of several $kT$ from the left and right gap edges.",
"The maxima appear when either particle transport (for negative $\\mu $ ) or hole transport (for positive $\\mu $ ) becomes suppressed, as discussed in details in Ref. 22.",
"A large value of $|S|$ , exceeding 1mV/K, results from rapid increase in transmission near the gap edges.",
"Similar enhancement of the thermopower also appears in the presence of impurities in the PE configuration.",
"However, for nanoribbons with impurities in the PC configuration, the thermopower is considerably reduced since transmission near the gap edges is diminished and changes more smoothly.",
"Moreover, a kind of damped oscillations of $S$ inside the gap can be observed, which follow from states localized in the gap.",
"It is interesting to note that the global maximum of the thermopower in the presence of P defects in the PC configuration is positive and appears near the left edge of the gap (negative $\\mu $ ), whereas such a global maximum in the presence of Al impurities is negative and appears near the right edge of the gap (positive $\\mu $ ).",
"In the presence of impurities, the thermopower is also enhanced for higher values of $|\\mu |$ .",
"This enhancement, however, is less pronounced than that for $\\mu $ inside the main gap.",
"Situation may change when the spin channels are ether not mixed in the nanoribbon, or they are mixed on a scale much longer than the system's length.",
"The spin effects in thermoelectric properties become then important.",
"The electrical conductance $G_{\\sigma }$ of the spin-$\\sigma $ channel is equal to $e^{2} L_{0\\sigma }$ , $G_{\\sigma } = e^{2} L_{0\\sigma }$ , whereas the total electronic contribution to the thermal conductance, $\\kappa _{e}$ , is given by the formula $\\kappa _{e}=\\frac{1}{T} \\sum _{\\sigma } \\left(L_{2\\sigma } - \\frac{L_{1\\sigma }^{2}}{L_{0\\sigma }}\\right).$ Since the two spin channels are not mixed and spin accumulation is important, one can introduce spin-dependent thermopower, $S_{\\sigma }=-\\Delta V_{\\sigma } /\\Delta T = -L_{1\\sigma } / |e|TL_{0\\sigma }$ , which corresponds to the spin-dependent voltage generated by a temperature gradient [22], [34].",
"The conventional (charge) and spin thermopowers can be then written as $S_c = \\frac{1}{2}(S_{\\uparrow } + S_{\\downarrow })$ and $S_{s} = \\frac{1}{2}(S_{\\uparrow } - S_{\\downarrow })$ , respectively.",
"We assume the transmission through the system (central part of the nanoribbon) is the same for nanoribbons with and without spin relaxation.",
"In Fig.",
"REF we show both charge and spin thermopowers as a function of chemical potential for all the three considered localizations of the Al and P impurity atoms.",
"It is interesting to note, that the spin thermopower is relatively large due to a significant spin dependence of the transmission function in the presence of impurities and for the assumed distance between the impurity atoms.",
"The impurity-induced spin thermopower depends on the location of the impurities and is especially large in the PC and PM configurations.",
"When the impurities are located at one of the edges, the spin thermopower is remarkably smaller.",
"Note, the conventional (charge) thermopower is also modified as follows from comparison of the results shown in Fig.",
"REF with the corresponding ones in Fig.",
"REF .",
"This modification appears due to impurity-induced spin dependence of the transmission function.",
"Some modifications induced by impurities appear also in other transport coefficients, for instance in the electronic contribution to the heat conductance, which however are not presented here explicitly."
],
[
"Electric and thermoelectric properties of ${\\rm z}$ S{{formula:d205cc0c-d4fb-4f69-9d18-0edc8d979da3}} NR{{formula:e17660c3-d19e-4e4d-b02f-4b9c98d308d9}} in the ferromagnetic state",
"By applying an external magnetic field one can stabilize the ferromagnetic (FM) configuration of zSiNRs.",
"This configuration can be also stabilized by exchange coupling to a ferromagnetic substrate or to ferromagnetic contacts.",
"Due to a significant magnetic moment in the FM state, transmission function of a nanoribbon is strongly spin-dependent.",
"Figure: (Color online) Spin-dependent conductance G σ G_{\\sigma } and polarization PP as a function of chemical potential μ\\mu , calculated for the FM state of ZSiNRswith Al (left panel) and P (right panel) impurity atoms in the configurations PE (a,b), PM (c,d), and PC (e,f).",
"The other parameters are T=90T=90K, and N=6N=6.Figure: (Color online) Electronic term in the thermal conductance, κ e \\kappa _{e}, as a function of chemical potential μ\\mu in the FM state of zSiNRs for Al (a) and P (b) impurities, and for T=90T=90K and N=6N=6.Figure: (Color online) Charge S c _{c} and spin S s _{s} thermopower as a function of chemical potential μ\\mu calculated for the FM state for zSiNRs with Al (a,c) and P (b,d) impurities, and for T=90T=90K and N=6N=6.Using the spin-resolved transmission $T_{\\sigma }$ (E) determined in Sec.",
"2B for the ferromagnetic state of pristine nanoribbons as well as of nanoribbons with impurities, we investigate now the spin-polarized transport phenomena.",
"Spin-resolved electrical conductance $G_{\\sigma }$ as a function of chemical potential $\\mu $ is presented in Fig.",
"REF for different positions of the Al and P impurity atoms, whereas the thermal conductance $\\kappa _{e}$ is depicted in Fig.",
"REF .",
"Pristine ferromagnetic nanoribbons as well as those with impurities in the PE configuration show metallic character.",
"The conductance $G_{\\sigma }$ is then constant in a close vicinity of $\\mu =0$ and it is practically the same for both spin channels for P impurities, and weakly depends on spin for Al impurities.",
"Strong modifications appear for higher values of $|\\mu |$ , where due to a pronounced dip in the transmission, one spin channel becomes weakly conductive and a considerable polarization $P$ , defined as $P=\\frac{G_{\\uparrow }-G_{\\downarrow }}{G_{\\uparrow }+G_{\\downarrow }}\\times 100\\%$ , can be observed.",
"It is worth to note that the presence of P atoms at one of the edges leads to a considerable polarization (up to 90$\\%$ ) in a narrow region of negative $\\mu $ , whereas for Al impurities a large polarization can be obtained for positive $\\mu $ .",
"For other values of chemical potential, the spin polarization is much smaller, though in narrow regions of $\\mu $ it can achieve even 50$\\%$ .",
"Interesting results are obtained for the PM impurity configuration, where one spin channel – spin up for Al and spin down for P impurities – becomes nonconductive in a close vicinity of $\\mu $ =0, whereas the second channel exhibits metallic character.",
"The system behaves thus like a half-metallic ferromagnet with practically 100$\\%$ polarization.",
"Such a behavior can be important for potential applications of doped silicene nanoribbons in spintronic devices.",
"Relatively high polarization can be also obtained for PC configuration, but in a narrow region of negative $\\mu $ for Al impurity and positive $\\mu $ for P impurities.",
"Moreover, the conductance of a nanoribbon with central P impurities exhibits pronounced spin-dependent dips in the energy region close to $\\mu $ =0.55 eV, resulting from the well-defined spin-dependent Fano antiresonance in the transmission.",
"On the other hand, narrow spin-dependent Fano dips, which occur in transmission well below the Fermi energy for Al impurities in the PC configuration do not give visible modifications of the conductance.",
"When spin mixing takes place on a distance comparable to the systems's length, then only conventional thermoelectric effects can be observed.",
"In turn, when spin relaxation processes are absent, spin effects become relevant.",
"Below we present numerical results just for this particular case.",
"Thermal conductance $\\kappa _{e}$ of the ferromagnetic nanoribbons with impurities is considerably reduced as compared to that of the pristine ones.",
"Bearing in mind thermoelectric properties, it can be important that $\\kappa _{e}$ is remarkably reduced in the vicinity of $\\mu =0$ for the PM impurity configuration due to the gap occurring for one spin channel.",
"Substantial reduction of $\\kappa _{e}$ is also obtained for negative and positive $\\mu $ for one of the PE or PC configurations (Fig.",
"REF ).",
"The thermal conductance is additionally reduced for P impurities in the PC configuration due to the well-defined Fano antiresonance for chemical potentials close to 0.55 eV.",
"The influence of impurity atoms on the charge and spin thermopowers is presented in Fig.",
"REF .",
"For comparison, the thermopower of a pristine nanoribbon is also depicted there.",
"As a general rule one can state that impurity atoms (Al, P) strongly enhance both charge and spin thermopowers.",
"However, the modifications depend on the position and type of the impurities.",
"Very remarkable changes can be observed for impurity atoms in the PM configuration, in which – due to appearance of energy gap in one of the spin channels – the charge and spin thermopowers are strongly enhanced in a close vicinity of $\\mu =0$ .",
"It should be also noticed that the main contribution to $S_{c}$ and $S_{s}$ in the case of Al atoms comes from the majority (spin-up) carriers, as this spin channel becomes nonconductive in the presence of Al impurities.",
"On the other hand, in nanoribbons with P impurities, both $S_{c}$ and $S_{s}$ show different signs, which indicates that the main contribution corresponds to non-conductive spin-down channel.",
"Totally different results are obtained for pristine nanoribbons.",
"Since pristine ferromagnetic nanoribbons exhibit metallic character for both spin directions, the transmission is constant in the region of small $|\\mu |$ and both $S_{c}$ and $S_{s}$ are negligibly small in this region of chemical potential.",
"Similar behavior can be observed for nanoribbons with impurities in the PE and PC configurations, where both thermopowers are practically equal to zero for small values of $|\\mu |$ .",
"However, a considerable enhancement of S$_{c}$ and S$_{s}$ is then obtained for higher values of $|\\mu |$ , where pronounced dips occur in transmission function.",
"High value of $|S|$ in the vicinity of $\\mu \\approx $ 0.55 eV for P impurities can be related to the Fano antiresonance, and thus to rapid changes in transmission function (Fig.",
"REF ).",
"As the Fano effect in FM state is strongly spin dependent, the remarkable spin thermopower close to 0.15 mV/K can be observed in this situation.",
"Strongly enhanced charge and spin thermopowers are also obtained for P impurities localized near the edges, but they appear for negative $\\mu $ .",
"The Al atoms in the PE and PC configurations also lead to quite remarkable modifications.",
"It should be noticed that the edge position of Al atoms leads to considerable enhancement of $|S|$ for positive $\\mu $ , whereas impurities localized in the center enhance $|S|$ for negative values of chemical potential.",
"The opposite relations are found for P impurities, where $|S|$ is enlarged for the edge configuration in the region of negative $\\mu $ but for PC configuration the enhancement is for positive $\\mu $ .",
"All this shows that type of impurities (Al or P) and their localization in the nanoribbons play an important role."
],
[
"Summary and conclusions",
"We have considered transport and thermoelectric effects in silicene nanoribbons with Al and P impurity atoms.",
"Using ab-initio calculations we have determined transmission function through a nanoribbon.",
"Using the calculated transmission we have determined thermoelectric coefficients in the linear response regime, like Seebeck and spin Seebeck parameters as well as the electronic contribution to the heat conductance.",
"The results have been presented for two different situations corresponding to presence and absence of short-range spin mixing in the nanoribbobs.",
"The calculations have been performed for both antiparallel (AFM, low energy) and parallel (FM) configurations of the edge magnetic moments.",
"The key objective was to determine the role of impurity atoms located in various positions with respect to the nanoribbon center.",
"Numerical results clearly show that the Al and P impurity atoms significantly modify the transmission function, and thus also the thermoelectric coefficients.",
"More specifically, the Siebeck and spin Siebeck coefficients are remarkably enhanced by the impurities.",
"This enhancement depends on position of the impurities.",
"We have considered tree different impurity configurations – PE (impurities at one of the edges), PC (impurities in the center of the nanoribbon), and PM (impurities between the edge and center of the nanoribbon).",
"This work was supported by the National Science Center in Poland as the Project No.",
"DEC-2012/04/A/ST3/00372.",
"Numerical calculations were performed at the Interdisciplinary Centre for Mathematical and Computational Modelling (ICM) at Warsaw University and partly at SPINLAB computing facility at Adam Mickiewicz University."
]
] | 1403.0072 |
[
[
"Complex organic molecules in protoplanetary disks"
],
[
"Abstract (Abridged) Protoplanetary disks are vital objects in star and planet formation, possessing all the material which may form a planetary system orbiting the new star.",
"We investigate the synthesis of complex organic molecules (COMs) in disks to constrain the achievable chemical complexity and predict species and transitions which may be observable with ALMA.",
"We have coupled a 2D model of a protoplanetary disk around a T Tauri star with a gas-grain chemical network including COMs.",
"We compare compare synthesised line intensities and calculated column densities with observations and determine those COMs which may be observable in future.",
"COMs are efficiently formed in the disk midplane via grain-surface chemical reactions, reaching peak grain-surface fractional abundances 1e-6 - 1e-4 that of the H nuclei number density.",
"COMs formed on grain surfaces are returned to the gas phase via non-thermal desorption; however, gas-phase species reach lower fractional abundances than their grain-surface equivalents, 1e-12 - 1e-7.",
"Including the irradiation of grain mantle material helps build further complexity in the ice through the replenishment of grain-surface radicals which take part in further grain-surface reactions.",
"There is reasonable agreement with several line transitions of H2CO observed towards several T Tauri star-disk systems.",
"The synthesised line intensities for CH3OH are consistent with upper limits determined towards all sources.",
"Our models suggest CH3OH should be readily observable in nearby protoplanetary disks with ALMA; however, detection of more complex species may prove challenging.",
"Our grain-surface abundances are consistent with those derived from cometary comae observations providing additional evidence for the hypothesis that comets (and other planetesimals) formed via the coagulation of icy grains in the Sun's natal disk."
],
[
"Introduction",
"Protoplanetary disks are crucial objects in star formation.",
"They dissipate excess angular momentum away from the protostellar system, facilitate the accretion of matter from the natal cloud onto the new star, and contain all the material, dust and gas, which will likely go on to form a surrounding planetary system [160].",
"The study of the detailed chemistry of these objects has gained impetus in recent years driven by the impending completion of the Atacama Large Millimeter/Submillimeter Array (ALMA).",
"ALMA, with its unprecedented sensitivity and spatial and spectral resolution, will reveal, for the first time, the composition of protoplanetary disks on $\\sim $ milliarcsecond scales, probing material $\\lesssim $ 10 AU of the parent star in objects relatively close to Earth ($\\approx $ 140 pc).",
"This spatial resolution will be achievable using the most extended configuration (with maximum baseline, $B$ $\\approx $ 16 km) at its highest operational frequencies ($\\nu $ $>$ 275 GHz).",
"This will allow the study of the detailed composition of the cold molecular material within the `planet-forming' region of nearby disks, which will advance our understanding of the process of planetary system formation, and help answer questions regarding the morphology and composition of our own Solar System.",
"The molecules observed in protoplanetary disks have thus far been restricted to small species and associated isotopologues due to their relatively high abundance and simple rotational spectra leading to observable line emission.",
"The sources in which these molecules have been detected are also limited to a handful of nearby, and thus well-studied, objects.",
"Molecules have been observed at both infrared (IR) and (sub)mm wavelengths with the IR emission originating from the inner warm/hot material (T $\\gtrsim $ 300 K, R $\\lesssim $ 10 AU) and the (sub)mm emission originating from the outer cold disk (T $<$ 300 K, R $\\gtrsim $ 10 AU).",
"The molecules detected at (sub)mm wavelengths include CO, HCO+, CN, HCN, CS, N2H+, SO and C2H [72], [37], [144], [137], [46], [61].",
"Also detected are several isotopologues of the listed species, e.g., $^{13}$ CO, C$^{18}$ O, H$^{13}$ CO$^{+}$ , DCO+ and DCN [142], [137], [104].",
"Several relatively complex molecules have also been observed: H2CO [37], [1], [137], [99], [100], HC3N [20], and $c$ -C3H2 [107].",
"Line emission in the (sub)mm can be observed from the ground and such observations have historically been conducted using single-dish telescopes, e.g., the JCMT (James Clerk Maxwell Telescope), the CSO (Caltech Submillimeter Observatory), the IRAM (Institut de Radioastronomie Millimétrique) 30 m telescope, and APEX (Atacama Pathfinder Experiment).",
"More recently, several interferometers have been available, e.g., the SMA (Submillimeter Array), CARMA (Combined Array for Research in Millimeter-wave Astronomy), and PdBI (Plateau-de-Bure Interferometer).",
"These latter facilities have enabled spatially-resolved mapping of very nearby objects including the archetypical protoplanetary disk, TW Hydrae, located at a distance of $\\approx $ 56 pc [99], [100], [66], [105].",
"Due to its proximity, TW Hydrae was observed during ALMA Science Verification which utilised between six and nine antennae working in conjunction to map the line emission from this source [101], [124].",
"Early results from ALMA also include the first detection of the location of the CO snowline[1] in the disk around HD 163296 using DCO+ line emission [83], and in the disk around TW Hydrae using N2H+ line emission [108].",
"[1]The CO snowline marks the transition zone in the disk midplane ($T$ $\\approx $ 17 K) beyond which CO is depleted from the gas via freezeout onto dust grains.",
"The launch of the Herschel Space Observatory allowed the first detection of ground-state transitions of ortho- and para-H2O (at 557 GHz and 1113 GHz, respectively) in the disk of TW Hydrae [63].",
"[15] also report the first detection of HD in TW Hydrae using Herschel, allowing, for the first time, a direct determination of the disk mass without relying on analysis of dust thermal emission or CO rotational line emission.",
"[15] determine a disk mass $<$ 0.05 M$_\\odot $ confirming that TW Hydrae, although considered a rather old system ($\\sim $ 10 Myr), contains sufficient material for the formation of a planetary system.",
"Also detected in the far-IR using Herschel, is the molecular ion, CH+, in the disk of the Herbig Be star, HD 100546 [138], and multiple lines of OH and warm H2O have also been detected in numerous sources [44], [85], [122].",
"Most detections of line emission in the mid-IR have been conducted with the Spitzer Space Telescope and molecules observed include OH, H2O, C2H2, HCN, CO, and CO2 [79], [21], [127], [117], [10].",
"Species detected at IR wavelengths are also limited to abundant, small, simple molecules with strong rovibrational transitions and/or vibrational modes, which are able to survive the high temperatures encountered in the inner disk.",
"[84] also report the detection of near-IR emission lines of C2H2 and HCN, for the first time, using ground-based observatories (CRIRES on the Very Large Telescope and NIRSPEC on the Keck II Telescope).",
"The greatest chemical complexity (outside of our Solar System) is seen in massive star-forming regions towards the Galactic centre [140] and in objects called `hot cores' and `hot corinos', considered important stages in high-mass ($M_{\\ast }$ $\\gtrsim $ 10 $M_{\\odot }$ ) and low-mass ($M_{\\ast }$ $\\lesssim $ 10 $M_{\\odot }$ ) star formation, respectively [62].",
"Hot cores are remnant, often clumpy, cloud material left over from the explosive process of high-mass star formation which is heated by the embedded massive star.",
"They are warm ($T$ $\\sim $ 100 K), dense ($n$ $\\gtrsim $ 10$^{6}$ cm$^{-3}$ ), relatively large ($R$ $\\sim $ 0.1 pc) objects which are heavily shielded by dust from both the internal stellar radiation and the external interstellar radiation ($A_\\mathrm {v}$ $\\sim $ 100 mag).",
"Hot corinos, considered the equivalent early stage of low-mass star formation, possess similar densities and temperatures to hot cores, yet, are much less massive and smaller in spatial extent (typically, $R$ $\\sim $ 100 AU).",
"The line emission from the hot corino arises from a very compact region on the order of 1\" in size for a source at the distance of Taurus (140 pc).",
"Hence, we are limited to studying a handful of nearby sources [23].",
"Nevertheless, hot corinos are certainly as chemically complex as their more massive counterparts (if not more so), attested by the detection of glycolaldehyde, HOCH2CHO, in IRAS 16293+2422 during ALMA Science Verification [70].",
"Hot cores and corinos are typified by the detection of rotational line emission from complex organic molecules (henceforth referred to as COMs), the formation of which remains one of the great puzzles in astrochemistry.",
"The generally accepted mechanism is that simple ices formed on grain surfaces in the molecular cloud at 10 K, either via direct freezeout from the gas phase or via H-addition reactions on the grain (e.g., CO, H2O, H2CO, CH3OH), undergo warming to $\\approx $ 30 K where they achieve sufficient mobility for grain-surface chemistry to occur via radical-radical association to create more complex ice mantle species (e.g., HCOOCH3).",
"The grain-surface radicals necessary for further molecular synthesis are thought to be produced by dissociation via UV photons created by the interaction of cosmic rays with H2 molecules.",
"Dissociation and/or ionisation via energetic electrons, created along the impact track as a cosmic ray particle penetrates a dust grain, is an alternative scenario [71].",
"Further warming to $T$ $\\gtrsim $ 100 K allows the removal of these more complex species from the ice mantle via thermal desorption thus `seeding' the gas with gas-phase COMs.",
"Typically, the observed rotational line emission is characterised by a gas temperature of $\\gtrsim $ 100 K with COMs observed at abundances $\\sim $ 10$^{-10}$ to $\\sim $ 10$^{-6}$ times that of the H2 number density [62].",
"Comparing the physical conditions in hot cores/corinos with those expected in the midplane and molecular regions of protoplanetary disks, it appears a similar chemical synthesis route to COMs may be possible; however, to date, targeted searches for gas-phase COMs in nearby protoplanetary disks have been unsuccessful [137], [99], [100].",
"The possible reasons for this are severalfold: (i) gas-phase COMs are relatively abundant in disks; however, due to their more complex spectra and resulting weaker emission and the small intrinsic size of disks, existing telescopes are not sufficiently sensitive to detect line emission from COMs on realistic integration time scales, (ii) gas-phase COMs are relatively abundant in disks; however, previous targeted searches have not selected the best candidate lines for detection with existing facilities, and (iii) gas-phase COMs achieve negligible abundances in disks.",
"The latter reason may be related to the major difference between hot cores/corinos and disks: the presence of external UV and X-ray radiation.",
"Certainly, observations using ALMA, with its superior sensitivity and spectral resolution, will elucidate which scenario is correct.",
"The confirmation of the presence (or absence) of COMs in disks is of ultimate astrobiological importance; is it possible for prebiotic molecules to form in the disk and survive assimilation into planets and other objects such as comets and asteroids?",
"Looking at our own Solar System, it appears possible.",
"Many relatively complex molecules have been observed in the comae of multiple comets: H2CO, CH3OH, HCOOH, HC3N, CH3CN, C2H6 [88].",
"The brightest comet in modern times, Hale-Bopp, displayed immense chemical complexity with additional detections of CH3CHO, NH2CHO, HCOOCH3, and ethylene glycol, (CH2OH)2 [28], [29].",
"In addition, the simplest amino acid, glycine (NH2CH2COOH), was identified in samples of cometary dust from comet 81P/Wild 2 returned by the Stardust mission [42].",
"The detection of gas-phase glycine is considered one of the `holy grails' of prebiotic chemistry; however, thus far, searches for gas-phase glycine towards hot cores have been unsuccessful [135].",
"In [151] and [152], henceforth referred to as WMN10 and WNMA12, we calculated the chemical composition of a protoplanetary disk using a gas-phase chemical network extracted from the UMIST Database for Astrochemistry ([161] [161][2]), termed `Rate06', and the grain-surface chemical network from [58] and [59].",
"We included the accretion of gas-phase species onto dust grains and allowed the removal of grain mantle species via both thermal and non-thermal desorption.",
"In WMN10, our aim was to study the effects of cosmic-ray-induced desorption, photodesorption, and X-ray desorption on the chemical structure of the disk, whereas, in WNMA12, we extended our investigations to cover the importance of photochemistry and X-ray ionisation on disk composition.",
"In both works, we focussed our discussions on species detected in disks, the most complex of which, at that time, was formaldehyde, H2CO.",
"[2]http://www.udfa.net Rate06 includes several gas-phase COMs, including methanol (CH3OH), formaldehyde (H2CO), formic acid (HCOOH), methyl formate (HCOOCH3), dimethyl ether (CH3OCH3) and acetone (CH3COCH3).",
"These represent the most simple alcohol, aldehyde, carboxylic acid, ester, ether and ketone, respectively.",
"The network also includes several larger members of these families, e.g., ethanol (C2H5OH) and acetaldehyde (CH3CHO).",
"In WMN10 and WNMA12, we adopted the grain-surface network of [58] and [59] which includes the grain-surface synthesis of several of these more complex species.",
"However, this network concentrates on simple atom-addition reactions, more likely to occur at the lower temperatures encountered in dark clouds.",
"Hence, to date, the grain-surface chemistry that has been included is by no means comprehensive regarding the grain-surface synthesis of COMs.",
"In this work, we study the efficiency of the synthesis of COMs in protoplanetary disks using a chemical network typically used for hot core and hot corino chemical models.",
"In Sect.",
", we describe our protoplanetary disk model (Sect.",
"REF ) and chemical network (Sect.",
"REF ).",
"In Sects.",
"and , we present and discuss our results, respectively, and in Sect.",
"we state our conclusions.",
"Our protoplanetary disk physical structure is calculated according to the methods outlined in [92] with the addition of X-ray heating as described in [93].",
"We model an axisymmetric disk in Keplerian rotation about a typical classical T Tauri star with mass, $M_\\ast $ = 0.5 $M_\\odot $ , radius, $R_\\ast $ = 2 $R_\\odot $ and effective temperature, $T_\\ast $ = 4000 K [75].",
"The surface density distribution is determined by the central star's mass and radius and assuming a constant disk mass accretion rate, $\\dot{M}$ , [121] and we parameterise the kinematic viscosity, $\\nu $ , using the $\\alpha $ -disk model of [132].",
"We use a viscous parameter, $\\alpha $ = 0.01, and a mass accretion rate, $\\dot{M}$ = $10^{-8}$ $M_\\odot $ yr$^{-1}$ , typical values for accretion disks around classical T Tauri stars.",
"We self-consistently solve the equation of hydrostatic equilibrium in the vertical direction and the local thermal balance between the heating and cooling of the gas to model the gas temperature, dust temperature, and density structure of the disk.",
"The heating mechanisms included are grain photoelectric heating by UV photons and heating due to hydrogen ionisation by X-rays.",
"We include gas-grain collisions and line transitions as cooling mechanisms.",
"The UV field in our disk model has two sources: the central star and the interstellar medium.",
"The central star's UV radiation field has three components: black-body radiation at the star's effective temperature, hydrogenic bremsstrahlung emission, and strong Ly-$\\alpha $ line emission.",
"The latter two components are necessary for accurately modelling the excess UV emission often observed towards T Tauri stars, which is thought to arise from an accretion shock as material from the disk impinges upon the stellar surface [69].",
"For the UV extinction, we include absorption and scattering by dust grains.",
"The combined UV spectrum originating from the T Tauri star is displayed in Fig.",
"C.1 in [92] and replicated in Fig.",
"1 in WNMA12.",
"The total UV luminosity is $L_\\mathrm {UV}$ $\\sim $ 10$^{31}$ erg s$^{-1}$ .",
"We model the X-ray spectrum of the T Tauri star by fitting the XMM-Newton spectrum observed towards the classical T Tauri star, TW Hydrae, with a two-temperature thin thermal plasma model [82].",
"The best-fit parameters for the temperatures are $kT_1$ = 0.8 keV and $kT_2$ = 0.2 keV.",
"For the foreground interstellar hydrogen column density, we find $N$ (H2) = 2.7 $\\times $ 10$^{20}$ cm$^{-2}$ .",
"For the X-ray extinction, we include attenuation due to all elements and Compton scattering by hydrogen.",
"The resulting X-ray spectrum is shown in Fig.",
"1 in [93] and is replicated in Fig.",
"1 in WNMA12.",
"The total X-ray luminosity of the star is $L_\\mathrm {X}$ $\\sim $ 10$^{30}$ erg s$^{-1}$ .",
"We assume the dust and gas in the disk are well mixed, and we adopt a dust-grain size distribution which reproduces the extinction curve observed in dense clouds [155].",
"The dust grains are assumed to consist of silicate and carbonaceous material, and water ice.",
"The resulting wavelength-dependent dust absorption coefficient is shown in Fig.",
"D.1 in [92].",
"We acknowledge that this is a simplistic treatment of the dust-grain distribution in protoplanetary disks since it is thought that gravitational settling and grain coagulation (grain growth) will perturb the dust size and density distribution from that observed in dense clouds [35], [36], [33].",
"To keep the chemical calculation computationally tractable, we adopt average values for the dust grain size and density that are consistent with the dust model adopted in the disk structure calculation [14].",
"In Fig.",
"REF , to guide the discussion in the paper, we present the resulting gas temperature (top left panel), total H nuclei number density, $n_{H}$ (top right panel), integrated UV flux (bottom left panel), and integrated X-ray flux (bottom right panel), as a function of disk radius, $R$ , and height, $Z/R$ .",
"The contours in the top left panel represent the dust temperature.",
"Here, we do not discuss the resulting disk structure in detail as this is covered in a series of previous publications [151], [152].",
"In WMN10, we also present the two-dimensional gas and dust temperatures and gas number density, with supporting material and comprehensive discussions in the Appendix of that paper.",
"In WNMA12, we display and discuss the incident UV and X-ray spectra and the resulting two-dimensional wavelength-integrated UV and X-ray fluxes.",
"The physical conditions throughout the disk cover many different regimes that generally differ from the conditions typical of dark clouds and hot cores and corinos.",
"The gas temperature ranges from $\\approx $ 17 K in the outer disk midplane to $>$ 6000 K in the inner disk surface.",
"The H nuclei number density in the dark disk midplane spans many orders of magnitude from $\\sim $ 10$^7$ cm$^{-3}$ in the outer disk (R $\\approx $ 300 AU) to $\\sim $ 10$^{14}$ cm$^{-3}$ in the inner disk (R $\\sim $ 1 AU).",
"We also remind the reader that protoplanetary disks around T Tauri stars are irradiated by X-rays in addition to cosmic rays and UV photons; hence, the ionisation rates and induced photodestruction rates in the molecular regions of protoplanetary disks can reach values much higher than those experienced in molecular clouds and hot cores and corinos (see Fig.",
"REF ).",
"Figure: Gas temperature (top left panel), H nuclei number density (top right panel),integrated UV flux (bottom left panel), and integrated X-ray flux (bottom right panel)as a function of disk radius, RR, and disk height, Z/RZ/R.The contours in the top left panel represent the dust temperature."
],
[
"Chemical Model",
"We include various processes in our chemical network: gas-phase two-body reactions, photoreactions, cosmic-ray and X-ray reactions, gas-grain interactions, grain-surface two-body reactions, grain-surface photoreactions, and grain-surface cosmic-ray-induced and X-ray induced photoreactions.",
"The latter three processes allow the chemical processing of grain-surface material by the radiation field present in the disk.",
"The gas-phase chemistry (including the photochemistry and cosmic-ray chemistry) is from [78] which is based on the OSU chemical network[3] and which includes new possible routes to the formation of methyl formate and its structural isomers.",
"To render the network suitable for protoplanetary disk chemistry, we have added reactions and rate coefficients applicable at higher temperatures [56], [57].",
"We include the direct X-ray ionisation of elements and calculate the X-ray ionisation rate throughout the disk using the prescription outlined in WNMA12.",
"To simulate the X-ray chemistry, we duplicate the set of cosmic-ray-induced photoreactions and scale the rates by the X-ray ionisation rate.",
"We also include the explicit calculation of the photochemical rates using the UV spectrum at each point in the disk for those species for which photodissociation and photoionisation cross sections exist (around 60 species, see [143] [143][4]).",
"In the absence of cross-sectional data, we approximate the rate by scaling the expected interstellar rate by the ratio of the integrated UV flux to that of the ISRF ($\\approx $ 1.6 $\\times $ 10$^{-3}$ erg cm$^{-2}$ s$^{-1}$ ).",
"[3]http://www.physics.ohio-state.edu/$\\sim $eric/ [4]http://www.strw.leidenuniv.nl/$\\sim $ewine/photo For gas-grain interactions, we include accretion from the gas onto dust grains (i.e., freezeout) and desorption from grain surfaces back into the gas phase.",
"For calculating the gas accretion and thermal desorption rates, we use the theory of [58] as in our previous work (see Equations 2, 3, and 4 in WMN10).",
"We assume, for simplicity, that the grains are negatively charged compact spheres with a radius, $a$ = 0.1 $\\mu $ m, and a constant fractional abundance (relative to the H nuclei number density) of $\\sim $ 10$^{-12}$ , equivalent to a gas-to-dust mass ratio of $\\sim $ 100.",
"We assume a sticking coefficient, $S$ $\\sim $ 1, for all species.",
"We model the grain-surface formation of molecular hydrogen by assuming the rate of H2 formation equates to half the rate of arrival of H atoms on the grain surface and use a reduced sticking coefficient for atomic hydrogen ($S$ $\\sim $ 0.3).",
"We adopt binding energies from the work of [78] which originate from [48] and [50] and references therein (also see Table REF ).",
"[48] assume the value for the molecule measured in water ice (which makes up the largest component of the ice mantle) and we maintain this convention [27].",
"We include the dissociative recombination of cations on grain surfaces with the products returned to the gas phase.",
"We adopt the same branching ratios as for the equivalent gas-phase dissociative recombination reaction.",
"Note that in the densest regions of the disk (n $\\gtrsim $ 10$^{10}$ cm$^{-3}$ ), the assumption of negatively charged grains becomes invalid and explicit grain-charging reactions should be included.",
"[141] found that neutral grains dominate the grain population in the midplane in the inner disk ($R$ $<$ 1.5 AU).",
"In this region, we are likely overestimating the recombination of cations on grain surfaces where n(G$^{0}$ )/n(G$^{-}$ ) is of the order of a factor of a few.",
"Because we are concerned with the chemistry occurring in the outer disk (R $\\gg $ 1 AU), the neglect of explicit grain charging will not affect the discussion and conclusions presented in this work.",
"In addition to thermal desorption, we include photodesorption, cosmic-ray-induced desorption (via heating), and reactive desorption.",
"For the photodesorption rates, we use the most recent experimental values for the photodesorption yields [96], [97].",
"In this model, we include a `coverage' factor, $\\theta _s$ , accounting for recent experimental results which suggest that photodesorption occurs from the top few monolayers only [17].",
"For molecules which do not have constrained photodesorption yields, $Y_i$ , we use the early experimental value determined for water ice by [156] of 3 $\\times $ $10^{-3}$ molecules photon$^{-1}$ (see Table 1 and Equation 1 in WNMA12).",
"The photodesorption rate for species $i$ , $k^\\mathrm {pd}_{i}$ , is thus given by $k^\\mathrm {pd}_{i} = F_\\mathrm {UV} \\, Y_i \\, \\sigma _d \\, n_d \\left( \\frac{n^s_i}{n^s_\\mathrm {tot}}\\right) \\theta _s\\qquad \\mathrm {cm}^{-3}\\mathrm {s}^{-1}$ where $F_\\mathrm {UV}$ (photons cm$^{-2}$ s$^{-1}$ ) is the wavelength-integrated UV photon flux, $\\sigma _d$ (cm$^{2}$ ) is the dust-grain cross section, $n_d$ is the number density of dust grains, $n^s_i$ (cm$^{-3}$ ) is the number density of species $i$ on the grain, and $n^s_\\mathrm {tot}$ (cm$^{-3}$ ) is the total number density of grain surface species.",
"The surface coverage factor, $\\theta _s$ is given by $\\theta _s ={\\left\\lbrace \\begin{array}{ll}1 &\\quad \\mathrm {for} \\quad M \\ge 2 \\\\M/2 &\\quad \\mathrm {for} \\quad M < 2,\\end{array}\\right.",
"}$ where $M$ is the total number of monolayers per grain.",
"We include photodesorption by both external and internal UV photons, the latter of which are produced by the interaction of cosmic rays with H2.",
"We adopt a value for the integrated cosmic-ray-induced UV photon flux equal to $\\sim $ 10$^{4}$ photons cm$^{-2}$ s$^{-1}$ [119].",
"In regions where cosmic rays are attenuated, we scale the internal UV photon flux by the corresponding cosmic-ray ionisation rate.",
"In WMN10, we investigated the influence of X-ray desorption on molecular abundances in protoplanetary disks and found it a potentially very powerful mechanism for returning grain-surface molecules to the gas phase.",
"We used the theoretical framework of [81], [90], and [40] to estimate the X-ray absorption cross sections and desorption rates.",
"However, the interaction of X-ray photons with ice is still not well understood and indeed, recent experiments suggest that the picture is somewhat complicated with soft X-rays inducing chemistry in the ice via the production of ionic fragments [6], [68].",
"In general, there is a significant lack of quantitative data on X-ray induced desorption of astrophysical ices, hence, we choose not to include this process explicitly in this work and instead, we treat X-ray photodesorption as we treat UV photodesorption.",
"Note that we also allow X-rays to dissociate and ionise grain mantle material in line with that seen in experiments (see discussion below).",
"For the calculation of the cosmic-ray-induced thermal desorption rates we use the method of [59] (see Equation 5 in WMN10).",
"Here, we also include the process of reactive desorption for the first time.",
"We follow the method of [49] and assume, for each grain-surface reaction which leads to a single product, a proportion of the product will be released into the gas phase.",
"This assumes a probability that, upon reaction, a proportion of the energy released goes into desorbing the molecule from the grain surface.",
"Investigations into the efficacy of reactive desorption in dark cloud models by [49] constrain the value for the probability of desorption to $P_\\mathrm {rd}$ = 0.01 and we adopt this value in our work.",
"Recently, [147] suggested reactive desorption from grain surfaces followed by radiative association in the gas phase as a potential mechanism for the production of several complex molecules recently detected in dark clouds and prestellar cores [9], [24].",
"The species detected include the methoxy radical (CH3O), ketene (CH2CO), acetaldehyde (CH3CHO), methyl formate (HCOOCH3), and dimethyl ether (CH3OCH3).",
"Certainly, the detection of gas-phase complex molecules in regions with a temperature $\\lesssim $ 15 K was unexpected and further adds to the puzzle regarding the chemical origin of COMs.",
"In addition, recent experiments have investigated grain-surface chemistry induced by the irradiation of a single monolayer of O2 ice by a beam of D atoms [34].",
"These experiments suggest that reactive desorption is particularly efficient for the reformation of doubly-deuterated water (D2O) and O2 via the surface reactions, s-D + s-OD and s-O + s-O, respectively.",
"[34] find these reactions release D2O and O2 into the gas phase with efficiencies, $>$ 90 % and $\\approx $ 60 %, respectively We discuss the sensitivity of our results to the assumed probability for reactive desorption in Appendix A.",
"Our grain-surface network is also from [78] which itself is derived from [48] and [50] with the grain-surface reaction rates calculated according to [58].",
"We assume a density of surface sites equal to $\\approx $ 1.5 $\\times $ 10$^{15}$ cm$^{-2}$ and for the barrier between surface sites, $E_b$ $\\approx $ 0.3 $E_D$ , where $E_D$ is the binding (desorption) energy to the grain surface of the reactant of interest.",
"We discuss the sensitivity of our results to the assumed diffusion barrier in in Appendix A.",
"For light reactants i.e., H and H2, the diffusion rate is replaced with the quantum tunnelling rate assuming a barrier thickness of 1 $Å$ (10$^{-8}$ cm).",
"Whether or not there is a quantum component in the diffusion rate of H atoms on grain surfaces remains a controversial topic.",
"Analysis of experimental work on H2 formation on bare grain surfaces concluded there was no quantum diffusion component [115], [116], [74].",
"A reanalysis of this experimental work determined a quantum component is necessary to explain the rate of formation of HD observed in the experiments [22].",
"More recent experiments on H atom diffusion on amorphous solid water (ASW) have proved inconclusive [154].",
"Here, since the bulk of our ice mantle is composed of water ice, we choose the `optimistic' case and allow quantum tunnelling for H and H2.",
"The dissociation and ionisation of grain-surface species via UV photons (originating both externally and internally via cosmic rays) and X-rays are new processes, not included in our previous work.",
"The importance of UV processing for building chemical complexity in interstellar ice analogues has been known for some time [5].",
"An example of a grain-surface photoreaction is the photodissociation of grain-surface methanol, s-CH3OH, into its constituent radicals, s-CH3 and s-OH, which are then available on the grain to take part in further surface-association reactions.",
"Note that grain-surface (ice) species are prefixed with `s-'.",
"For these reactions, we assume the rate of the equivalent gas-phase reaction.",
"This is supported by recent estimates of the grain-surface photodestruction of pure methanol ice which also show that photochemistry occurs deep within the bulk ice [98].",
"The various reaction channels possible are estimated by assuming a grain-surface molecule will likely dissociate into its functional group components (as demonstrated above for methanol), i.e., allowing no rearrangement of the constituent atoms, and allowing reactions involving destruction of the weaker bonds to have higher branching ratios [50].",
"For each ionisation event, we assume that the cation dissociatively recombines with the electron, and the excess energy in the products lost via translation energy on the grain mantle, i.e., the products remain on the grain surface [50].",
"We adopt the same branching ratios for the equivalent gas-phase dissociative recombination reaction.",
"For X-ray induced dissociation and ionisation of grain-surface species, we follow the same formulation as that adopted for the gas-phase X-ray reactions: we duplicate the set of cosmic-ray-induced photoreactions and scale the reaction rates by the X-ray ionisation rate calculated in the disk.",
"In line with experiments, we allow X-ray and UV photodissociation and ionisation to occur throughout the ice mantle [98], [6], [68].",
"In addition to cosmic-ray-induced photoreactions, the direct impact of cosmic ray particles, which, in the Galaxy, consist predominantly of protons and stripped nuclei [133] can penetrate the dust grain and induce cascades of up to 10$^{2}$ suprathermal atoms along the impact track.",
"These atoms, in turn, ionise the molecular material within the ice mantle also creating energetic electrons ($\\sim $ keV) which also dissociate ionise the ice mantle material [71], [12], [13].",
"Here, we simulate this process in the ice by also adopting the direct cosmic-ray ionisation rates for the equivalent gas-phase reaction.",
"Our complete chemical network has $\\approx $ 9300 reactions involving $\\approx $ 800 species and is one of the most complex chemical models of a protoplanetary disk constructed to date."
],
[
"Molecular Line Emission",
"In order to compare our model results with current observations and select potential molecules and line transitions which may be observable in protoplanetary disks, we have calculated the molecular line emission from the disk to determine rotational line transition intensities for molecules of interest in this work.",
"For simplicity, we assume the disk is `face-on', i.e., has an inclination of 0°, and that local thermodynamic equilibrium (LTE) holds throughout.",
"The former assumption allows us to quickly and efficiently calculate the line emission without worrying about geometrical effects due to disk inclination, whilst the latter assumption makes the calculation more computationally tractable and allows us to compute an entire spectrum of a particular molecule in a single calculation.",
"This is important for COMs, in particular, which typically have many energy levels and transitions and thus, relatively complex spectra.",
"In order to perform a non-LTE calculation, we require accurate collisional rate coefficients which are only available for a handful of molecules (see, e.g., the Leiden Atomic and Molecular Database or LAMDA[5]).",
"For molecules considered in this work, collisional data for H2CO, CH3OH, HC3N and CH3CN only are available.",
"We expect the disk conditions to depart from LTE mainly in the outer, colder, more diffuse regions of the disk.",
"However, we also only expect our LTE disk-integrated line intensities to deviate from those calculated assuming non-LTE conditions by no more than a factor of a few [110].",
"Since this work is exploratory in nature, the line intensities we calculate are still very useful for determining which transitions in which molecules may be detectable using ALMA.",
"[5]http://home.strw.leidenuniv.nl/$\\sim $moldata/ There are numerous caveats when assuming LTE.",
"The intense background thermal radiation in warm regions of the disk may radiatively pump particular transitions leading to weakly (or possibly, strongly) masing lines ($T_\\mathrm {rot}$ $>$ $T_{k}$ ).",
"Formaldehyde (H2CO) and methanol (CH3OH) masers have been observed in the local interstellar medium and are commonly associated with massive star forming regions [41].",
"Indeed, there have been suggestions that methanol masers trace the protoplanetary disk material around massive embedded protostars [94].",
"Certainly, the potential for observable maser emission from formaldehyde and methanol in disks around low-mass stars should be explored in future work.",
"The disk-integrated line flux density, $F_\\nu $ , is determined by integrating the solution of the radiative transfer equation in the vertical direction and summing over radial sections of the disk, i.e., $F_\\nu = \\frac{1}{4\\pi \\,D^2}\\int _{r_\\mathrm {min}}^{r_\\mathrm {max}}\\int _{-z_\\mathrm {max}(r)}^{+z_\\mathrm {max}(r)} 2\\pi \\, r \\, \\bar{\\eta }_\\nu (r,z)\\, \\mathrm {d}z\\,\\mathrm {d}r$ where $D$ is the distance to source and $\\bar{\\eta }_\\nu (r,z)$ is the emissivity at a grid point $(r,z)$ times the absorption in the upper disk, i.e., $\\bar{\\eta }_\\nu (r,z) = n_\\mathrm {u}(r,z)\\, A_\\mathrm {ul} \\, \\phi _\\nu \\frac{h\\nu }{4\\pi }\\exp {\\left[ -\\tau _\\nu (r,z)\\right]}.$ Here, $n_\\mathrm {u}$ is the abundance in the upper energy level of the transition, $A_\\mathrm {ul}$ is the Einstein coefficient for spontaneous emission from the upper level, $u$ , to the lower level, $l$ , $\\phi _\\nu $ is the value of the line profile function at the frequency, $\\nu $ (assumed to be Gaussian in shape), and $h$ is Planck's constant.",
"The optical depth, $\\tau _\\nu (r,z)$ , is $\\tau _\\nu (r,z) = \\int _{z}^{z_\\mathrm {max}} \\chi _\\nu (r,z^{\\prime })\\, \\mathrm {d}z^{\\prime },$ where the absorption coefficient, $\\chi _\\nu $ , is given by $\\chi _\\nu = \\rho \\kappa _\\nu + \\left( n_\\mathrm {l}B_\\mathrm {lu} - n_\\mathrm {u}B_\\mathrm {ul} \\right) \\,\\phi _\\nu \\frac{h\\nu }{4\\pi }.$ Here, $\\rho $ is the dust mass density (g cm$^{-3}$ ), $\\kappa _\\nu $ is the dust mass absorption coefficient (cm$^2$ g$^{-1}$ ), $n_\\mathrm {l}$ is the abundance in the lower energy level of the transition and $B_\\mathrm {lu}$ and $B_\\mathrm {ul}$ are the Einstein coefficients for absorption and stimulated emission, respectively.",
"Since we assume LTE holds throughout, our level populations are given by the Boltzmann distribution, i.e., the population of an energy level, $i$ , is determined using $\\frac{n_i}{n} = \\frac{g_i \\exp { \\left( -E_{i}/T\\right)}}{Z(T)}$ where $n_i$ is the number density in level $i$ , $n$ is the total number density of the molecule ($\\sum _i n_i$ ), $g_i$ is the degeneracy of the level, $E_i$ is the energy (in units of K) and $T$ is the gas temperature.",
"We explicitly calculate the rotational partition function, $Z_\\mathrm {rot}(T)$ , by summing over populated energy levels in colder regions, i.e., $Z_\\mathrm {rot}(T) = \\sum _i g_i \\exp {\\left(-E_i/T\\right)}$ , and swapping to the high temperature approximation once the higher energy levels become sufficiently populated, that is, $Z_\\mathrm {rot}(T) \\approx \\sqrt{\\pi }(kT)^{3/2}/(\\sigma \\sqrt{ABC}) $ , where, $k$ is Boltzmann's constant, $\\sigma $ is the symmetry factor, i.e., the number of indistinguishable rotational orientations of the molecule, and $A$ , $B$ , and $C$ are the rotational constants in energy units.",
"In the inner regions of the disk, the gas temperature is sufficiently high that molecules can become vibrationally excited.",
"COMs, in particular, can be vibrationally excited at relatively moderate temperatures, $\\approx $ 300 K; hence, we also include the vibrational partition function in our determination of the total partition function, $Z(T) = Z_\\mathrm {vib}(T)\\times Z_\\mathrm {rot}(T)$ , where $Z_\\mathrm {vib}(T) = \\prod _i \\frac{1}{1 - \\exp {\\left(-h\\nu _i/kT\\right)}}$ with $\\nu _i$ representing the set of characteristic vibrational frequencies for each molecule.",
"Typically, $i$ $\\gg $ 1 for complex molecules, e.g., methanol has 12 characteristic frequencies of vibration.",
"We use the molecular rotational line lists from either the Cologne Database for Molecular Spectroscopy (CDMS, [87] [87][6]) or the Jet Propulsion Laboratory (JPL) molecular spectroscopic database ([111] [111][7]) and molecular constants (rotational constants and vibrational frequencies) from CDMS, JPL, and the NIST database[8]$^{,}$ [9].",
"For line lists from CDMS, we use internally generated Einstein A coefficients provided by the database.",
"For JPL, we convert the listed line intensities, $S_{\\nu }$ , to transition probabilities using the partition function provided by the database at the reference temperature, 300 K [111].",
"We have benchmarked our partition functions with those provided by CDMS and JPL.",
"Our results for H2CO and CH3OH at low temperatures ($T$ $\\le $ 37.5 K) agree with those provided by the databases to $>$ 3 significant figures.",
"At higher temperatures, our results agree to within $\\approx $ 15% of the database values over the temperature range applicable to our disk model ($T$ $\\lesssim $ 150 K, see Fig.",
"REF ).",
"[6]http://www.astro.uni-koln.de/cdms/ [7]http://spec.jpl.nasa.gov/ [8]http://webbook.nist.gov/chemistry [9]http://cccbdb.nist.gov In our calculations, we assume a distance to source, $D$ = 140 pc, the distance to the Taurus molecular cloud complex where many well-studied protoplanetary disks are located.",
"To convert line flux densities to sources at other distances, one must simply scale the values by the square of the ratio of the distances, e.g., for a source at 400 pc, for example, the Orion molecular cloud, the line flux density is reduced by a factor, (140/400)$^2$ = 0.123, whilst for a source at 56 pc, for example, TW Hya, the line flux density is enhanced by a factor, (140/56)$^{2}$ = 6.25.",
"We determine the chemical structure of the disk by calculating the time-dependent chemical evolution at each grid point in our model.",
"Firstly, we investigate the influence of different chemical processes by running a reduced grid at a fixed radius in the outer disk (R = 305 AU).",
"Secondly, we present results from our full disk model, i.e., by running the entire grid ($\\approx $ 10,000 points) as a series of single-point models.",
"We calculate the chemical structure between a radius of $\\approx $ 1 AU and $\\approx $ 305 AU.",
"We concentrate our results and discussions on the outer cold disk ($T$ $\\lesssim $ 100 K) where sufficient freezeout allows grain-surface synthesis to occur.",
"We map the chemical structure of the disk at a time of 10$^6$ yr, the typical age of classical T Tauri stars."
],
[
"Initial Abundances",
"Our initial abundances are extracted from the results of a simple time-dependent dark cloud model with constant physical conditions ($n$ = 10$^{5}$ cm$^{-3}$ , $T$ = 10 K, $A_v$ = 10 mag) at a time of 10$^5$ yr. We use the same chemical network for the generation of the initial abundances as in our full disk model.",
"Our initial elemental abundance ratios for H:He:O:C:N:S:Na:Mg:Si:Cl:Fe are 1:9.75(-2):1.76(-4):7.3(-5):2.14(-5):2(-8):3(-9):3(-9):3(-9):3(-9):3(-9) where $a$ ($b$ ) represents $a$ $\\times $ 10$^b$ [54].",
"The initial abundances for a selection of gas-phase and grain-surface (ice) molecules are shown in Table REF along with their corresponding grain-surface binding energies.",
"The species are ordered by mass and we list the initial fractional abundances and binding energies of the complex molecules of interest in this work separately.",
"The calculations begin with appreciable fractional abundances (relative to total H nuclei number density, $n_\\mathrm {H}$ = $n$ (H) + 2$n$ (H2)) of relatively simple ices, such as, s-H2O ($\\approx $ 1 $\\times $ 10$^{-4}$ ), s-CO ($\\approx $ 3 $\\times $ 10$^{-5}$ ), s-CH4 ($\\approx $ 3 $\\times $ 10$^{-5}$ ), s-NH3 ($\\approx $ 1 $\\times $ 10$^{-5}$ ), and s-N2 ($\\approx $ 4 $\\times $ 10$^{-6}$ ).",
"Formaldehyde (s-H2CO), methylamine (s-CH3NH2), methanol (s-CH3OH), formamide (s-NH2CHO), and ethane (s-C2H6) also achieve relatively high fractional abundances on the grains ($\\gtrsim $ 10$^{-8}$ ).",
"These species are efficiently formed via atom-addition reactions at 10 K. s-C3H4 also reaches an appreciable fractional abundance ($\\approx $ 2 $\\times $ 10$^{-6}$ ).",
"In this network we do not distinguish between isomers of C3H4.",
"Because propyne has the lower zero-point energy and also possesses a rotational spectrum, we choose to treat C3H4 as propyne (CH3CCH) as opposed to allene (CH2CCH2).",
"Under dark cloud conditions, methanol and formaldehyde are formed on the grain via the sequential hydrogenation of CO ice, $s\\mbox{-}{CO} \\xrightarrow{}s\\mbox{-}{HCO} \\xrightarrow{}s\\mbox{-}{H2CO} \\xrightarrow{}\\genfrac{}{}{0.0pt}{}{s\\mbox{-}{CH3O}}{s\\mbox{-}{CH2OH}} \\xrightarrow{}s\\mbox{-}{CH3OH}.$ Methylamine is formed via the hydrogenation of s-CH2NH, $s\\mbox{-}{CH2NH}\\xrightarrow{}\\genfrac{}{}{0.0pt}{}{s\\mbox{-}{CH2NH2}}{s\\mbox{-}{CH3NH}}\\xrightarrow{}s\\mbox{-}{CH3NH2}$ s-CH2NH has multiple formation pathways originating from atom addition to small hydrocarbon radicals, $s\\mbox{-}{CH2} \\xrightarrow{}~s\\mbox{-}{H2CN}~\\xrightarrow{} s\\mbox{-}{CH2NH} \\\\$ and $s\\mbox{-}{CH3} \\xrightarrow{} s\\mbox{-}{CH2NH} .$ Propyne, s-CH3CCH, forms via successive hydrogenation of s-C3 and s-C3H, both of which form readily in the gas phase and subsequently freeze out onto the dust grains.",
"Grain-surface formamide originates from s-OCN e.g., $s\\mbox{-}{OCN}\\xrightarrow{} s\\mbox{-}{HNCO}\\xrightarrow{}s\\mbox{-}{NHCHO}\\xrightarrow{} s\\mbox{-}{NH2CHO}.$ s-OCN can form either on the grain via the reactions $s\\mbox{-}{CN} \\xrightarrow{} s\\mbox{-}{OCN}$ and $s\\mbox{-}{NO} \\xrightarrow{} s\\mbox{-}{OCN},$ or in the gas phase via atom-radical and radical-radical reactions, such as, ${N} + {HCO} \\longrightarrow {OCN} + {H}$ and ${CN} + {OH} \\longrightarrow {OCN} + {H}$ whence it can freeze out onto dust grains.",
"Ethane, s-C2H6, forms on the grain in a similar manner to propyne, via the sequential hydrogenation of s-C2 and s-C2H.",
"Note that radicals, such as, CH3O and CH2OH, can also be formed via the photodissociation of larger species, in this case, CH3OH, as well as on the grain and via ion-molecule chemistry.",
"Table: Initial fractional abundances (with respect to n H n_\\mathrm {H})and molecular binding (desorption) energies, E D E_\\mathrm {D} (K)."
],
[
"Vertical abundance profiles at R = 305 AU",
"Here, we present results from a series of reduced grids to investigate the particular chemical processes responsible for the production and destruction of COMs in the outer disk.",
"We calculated the chemical evolution over a 13-point grid in a single vertical slice of the disk at a fixed radius of 305 AU.",
"The physical conditions in this slice are presented in Fig.",
"REF .",
"The H nuclei number density decreases from a maximum value of $\\approx $ 5 $\\times $ 10$^{7}$ cm$^{-3}$ in the disk midplane to $\\approx $ 6 $\\times $ 10$^{5}$ cm$^{-3}$ at the surface and the gas temperature increases from a minimum of $\\approx $ 16 K in the midplane to $\\approx $ 42 K at the disk surface.",
"The gas and dust temperatures at the disk surface decouple above a height of $\\approx $ 150 AU such that the dust temperature on the disk surface reaches a maximum value of $\\approx $ 28 K. The disk midplane is heavily shielded from both UV and X-ray photons; however, cosmic rays are able to penetrate the entire disk.",
"As expected, the disk surface is heavily irradiated with the UV and X-ray fluxes reaching a value $\\sim $ 10$^{-2}$ erg cm$^{-2}$ s$^{-1}$ , an order of magnitude stronger than the integrated interstellar UV flux (1.6 $\\times $ 10$^{-3}$ erg cm$^{-2}$ s$^{-1}$ ).",
"We present results from five different models which increase incrementally in complexity from Model 1 through to Model 5.",
"Model 1 is the most simple and includes gas-phase chemistry, freezeout onto dust grains, and thermal desorption.",
"In Model 2, we add cosmic-ray-induced thermal desorption and photodesorption by internal and external UV photons and X-ray photons.",
"In Model 3, we include grain-surface chemistry and in Model 4, we also add the cosmic-ray, X-ray, and UV photoprocessing of ice mantle material.",
"The most complex model, Model 5, also includes reactive desorption (see Sect.",
"REF ).",
"In Figs.",
"REF and REF we present the fractional abundance (relative to H nuclei number density) as a function of disk height, $Z$ , at a radius, $R$ = 305 AU, of a selection of gas-phase and grain-surface (ice) COMs, respectively[10].",
"Note that we have used the same scale for the abundance of each gas-phase molecule and analogous grain-surface species to ease the comparison between plots.",
"[10]The data used to plot Figs.",
"REF to REF are available upon request.",
"Figure: Physical conditions as a function of disk heightat a radius, RR = 305 AU.Number density of H nuclei (cm -3 ^{-3}) and gas and dust temperature (K)are shown in the left-hand panel (solid red lines, solid blue lines, and dashed blue lines, respectively).UV and X-ray radiative fluxes (erg cm -2 ^{-2} s -1 ^{-1})and X-ray and cosmic-ray ionisation rates (s -1 ^{-1})are shown in the right-hand panel(solid red lines, dashed red lines, dashed blue lines, and dotted blue lines, respectively).Figure: Fractional abundance (with respect to H nuclei number density)of gas-phase molecules as a functionof disk height, ZZ at a radius, RR = 305 AU.The chemical complexity in the model increases from Model 1 to Model 5(see Sect.",
"for details).Figure: Same as Fig.",
"for grain-surface (ice) species."
],
[
"Model 1: Freezeout and thermal desorption",
"In Model 1 (red lines in Figs.",
"REF and REF ), where we include freezeout and thermal desorption only, a handful of molecules achieve an appreciable fractional abundance ($\\gtrsim $ 10$^{-11}$ ) in the disk molecular layer: H2CO, HC3N, CH3CN, and CH3NH2.",
"These species are depleted in the disk midplane below a height of $\\approx $ 100 AU due to efficient freezeout onto dust grains.",
"Higher in the disk, gas-phase formation replenishes molecules lost via freezeout onto grain surfaces.",
"These species generally retain a similar peak fractional abundance as that achieved under dark cloud conditions.",
"The exception to this is acetonitrile (CH3CN) which increases from an initial fractional abundance of $\\sim $ 10$^{-12}$ to reach a peak value of $\\sim $ 10$^{-10}$ at $Z$ $\\approx $ 200 AU.",
"The fractional abundances decrease towards the disk surface due to increasing photodestruction.",
"None of the other gas-phase species achieve significant peak fractional abundances in the molecular layer ($\\lesssim $ 10$^{-11}$ ), even those which begin with an appreciable initial abundance, i.e., CH3OH and CH3CCH.",
"The fractional abundance of gas-phase CH3OH remains $\\lesssim $ 10$^{-14}$ throughout the disk height.",
"For these species, freezeout onto dust grains wins over gas-phase formation.",
"The more complex molecules, which cannot form in the gas-phase under dark cloud conditions, are also unable to form under the conditions in the outer disk, in the absence of grain-surface chemistry.",
"Regarding the grain-surface results for Model 1 (Fig.",
"REF ), due to the higher binding energies of most species, thermal desorption alone is unable to remove significant fractions of the grain mantle.",
"Hence, the ice remains abundant throughout the vertical extent of the disk with most species retaining their initial fractional abundance.",
"We see enhancements in the fractional abundances of s-H2CO, s-HCOOH, s-HC3N and s-CH3CN around 100 AU.",
"All four species have gas-phase routes to formation; however, under the conditions beyond $Z$ $\\approx $ 100 AU, additional molecules created in the gas phase can accrete onto dust grains thereby increasing their abundance on the grain mantle."
],
[
"Model 2: Non-thermal desorption",
"In Model 2 (green lines in Figs.",
"REF and REF ), we have added cosmic-ray-induced thermal desorption and photodesorption due to external and internal UV photons and X-ray photons.",
"Non-thermal desorption has a powerful effect on both the gas-phase and grain-surface abundances as we also concluded in our previous work (WMN10).",
"There are several noticeable effects: (i) the gas-phase abundances of many molecules are enhanced towards the disk midplane, relative to the results for Model 1, due to cosmic-ray-induced thermal desorption and photodesorption, (ii) the abundances of grain-surface molecules drop significantly towards the disk surface due to photodesorption by external UV photons, and (iii) there is a shift in the position of the gas-phase `molecular layer' towards the midplane.",
"This latter effect is due to a combination of non-thermal desorption and enhanced gas-phase formation lower in the disk, and enhanced destruction higher in the disk due to the release of a significant fraction of the grain mantle back into the gas phase.",
"Non-thermal desorption effectively `seeds' or replenishes the gas with molecules that otherwise would remain bound to the grain, e.g., H2O and its protonated form, H3O+, which then go on to take part in gas-phase reactions which can form (destroy) molecules which would otherwise be depleted (abundant).",
"The fractional abundance of gas-phase H2CO is enhanced to $\\sim $ 10$^{-12}$ in the disk midplane.",
"However, its fractional abundance in the molecular layer and disk surface reaches values similar to that in Model 1 ($\\sim $ 10$^{-10}$ – 10$^{-9}$ ).",
"CH3OH reaches a peak abundance of $\\sim $ 10$^{-10}$ around a height of $\\approx $ 100 AU corresponding to the height where photodesorption by external photons begins to have an effect.",
"Once released from the grain, methanol is efficiently destroyed under the conditions in the upper disk since methanol does not have efficient gas-phase routes to formation at low temperatures.",
"Methanol is also significantly enhanced in the disk midplane by cosmic-ray-induced desorption, reaching a fractional abundance, $\\sim $ 10$^{-13}$ .",
"Other gas-phase species enhanced in the midplane and molecular layer due to non-thermal desorption and reaching appreciable fractional abundances ($\\gtrsim $ 10$^{-11}$ ) include HCOOH, HC3N, CH3CN, CH3CCH, and CH3NH2."
],
[
"Model 3: Grain-surface chemistry",
"In Model 3 (blue lines in Figs.",
"REF and REF ), we have added thermal grain-surface chemistry [58].",
"Grain-surface chemistry is a very important process for building chemical complexity in the ice mantle under the conditions in the outer disk.",
"There are several reasons for this.",
"In addition to reactions involving atoms and other neutrals, which are well known in ice chemistry at 10 K, it appears that a reasonably high abundance of radicals, such as, s-OH and s-HCO, can be achieved without radiation processing of the ice mantle.",
"They form via association reactions on the grain, e.g, s-H + s-CO, or accrete from the gas phase where they are formed via gas-phase chemistry or via the photodissociation of larger species.",
"These radicals can diffuse and react at the dust temperatures in the outer disk (17 K – 30 K).",
"The major effect of the addition of thermal grain-surface chemistry is that the gas-phase fractional abundances of all molecules we consider in this section are enhanced (relative to the results for Model 1 and Model 2).",
"We see around an order of magnitude increase in the peak abundance of formaldehyde in the molecular layer of the disk, from $\\sim $ 10$^{-9}$ to $\\sim $ 10$^{-8}$ .",
"Gas-phase methanol is enhanced throughout the molecular layer and disk surface also reaching a peak fractional abundance of $\\sim $ 10$^{-8}$ at a height of $\\approx $ 150 AU.",
"We see a corresponding rise in the ice abundance indicative that grain-surface formation of methanol is replenishing the grain mantle molecules lost to the gas by non-thermal desorption.",
"Formic acid, HCOOH, reaches a peak abundance of $\\sim $ 10$^{-8}$ in the molecular layer.",
"We see a dramatic rise in the fractional abundance of s-HCOOH throughout the disk height.",
"s-HCOOH can be formed at low temperatures ($\\approx $ 20 K) on grain surfaces via the barrierless reactions, $s\\mbox{-}{H} + s\\mbox{-}{COOH} \\longrightarrow s\\mbox{-}{HCOOH}$ and $s\\mbox{-}{OH} + s\\mbox{-}{HCO} \\longrightarrow s\\mbox{-}{HCOOH}.$ The former reaction requires s-COOH to be sufficiently abundant and this is formed on grain-surfaces via s-OH + s-CO which has a high reaction barrier ($\\approx $ 3000 K).",
"Thus, the latter reaction is the main route to formation in the cold midplane.",
"Although the mobility of both s-OH and s-HCO is relatively low at $\\approx $ 20 K, the long lifetime of the disk, $\\sim $ 10$^6$ years, and the high density in the midplane, $\\sim $ 10$^{7}$ cm$^{-3}$ , allows the sufficient buildup of formic acid ice via this reaction.",
"Gas-phase HC3N experiences a order-of-magnitude enhancement in Model 3 relative to Model 2, reaching a peak fractional abundance of $\\sim $ 10$^{-9}$ .",
"However, we see a drop in the fractional abundance of grain-surface s-HC3N below a height, $Z$ $\\lesssim $ 150 AU, relative to the results for Model 2. s-HC3N can be sequentially hydrogenated up to s-C2H5CN (propionitrile).",
"We also see enhancements, related to grain-surface chemistry, in gas-phase CH3CN, CH3CCH, CH3CHO, NH2CHO, and CH3NH2 in the molecular layer and disk surface, reaching peak fractional abundances of $\\sim $ 10$^{-10}$ – 10$^{-8}$ .",
"s-CH3CN is formed via successive hydrogenation of s-C2N, $s\\mbox{-}{C2N} \\xrightarrow{}s\\mbox{-}{HCCN} \\xrightarrow{}s\\mbox{-}{CH2CN} \\xrightarrow{}s\\mbox{-}{CH3CN}$ In turn, s-C2N can form via s-C + s-CN and s-N + s-C2, or freeze out from the gas phase, where it is formed via ion-molecule and radical-radical reactions.",
"In warmer regions, s-CH3CN can also form via the radical-radical reaction, s-CN + s-CH3.",
"Intermediate species in the above hydrogenation sequence can also form via grain-surface association reactions, e.g., $s\\mbox{-}{CH} + s\\mbox{-}{CN} \\longrightarrow s\\mbox{-}{HCCN}$ and $s\\mbox{-}{CH2} + s\\mbox{-}{CN} \\longrightarrow s\\mbox{-}{CH2CN}.$ s-CH3CCH, forms via the successive hydrogenation of s-C3 and s-C3H, and in warmer gas can also form via the barrierless radical-radical reaction, $s\\mbox{-}{CH} + s\\mbox{-}{C2H3} \\longrightarrow s\\mbox{-}{CH3CCH}.$ In turn, s-C2H3, has several grain-surface formation routes, in addition to the hydrogenation of s-C2H2, e.g., s-C + s-CH3 and s-CH + s-CH2.",
"Acetaldehyde (s-CH3CHO) can form via the association of s-CH3 and s-HCO or via s-CH3CO which, in turn, forms on the grain via s-CH3 + s-CO.",
"This latter reaction has a barrier of $\\approx $ 3500 K and can only proceed on sufficiently warm grains.",
"The formation of formamide (s-NH2CHO) and methylamine (s-CH3NH2) via atom-addition reactions was discussed previously in relation to dark cloud chemistry (see Sect.",
"REF ).",
"In warmer regions, again, there are formation routes via the association of the amine and formyl radicals (s-NH2 + s-HCO), and the methyl and amine radicals (s-CH3 + s-NH2), respectively.",
"In all cases, these precursor radicals are either formed on the grain via atom-addition reactions, or they form in the gas phase via ion-molecule chemistry or photodissociation and freeze out onto dust grains.",
"Moving on to the more complex species, we see much the same effect as for those already discussed; gas-phase abundances are enhanced when grain-surface chemistry is included.",
"Each gas-phase species reaches a peak abundance of $\\sim $ 10$^{-12}$ – 10$^{-10}$ in the molecular layer, which correlates with enhancements in the grain-surface abundances and hence are directly due to synthesis via grain-surface chemistry and subsequent release to the gas phase via non-thermal desorption.",
"The peak fractional abundance attained by the more complex species on the grain surface ranges between $\\sim $ 10$^{-9}$ – 10$^{-7}$ .",
"For those complex species for which we begin with negligible abundances on the grain ($\\lesssim $ 10$^{-13}$ ), i.e., s-C2H5OH, s-CH3OCH3, s-CH3COCH3, s-CH3COOH, s-HCOOCH3, and s-HOCH2CHO, grain-surface chemistry is absolutely necessary for their formation.",
"These species form predominantly via radical-radical association routes and, thus, require significant abundances of precursor radicals.",
"For example, methyl formate (s-HCOOCH3) can form on the grain via the association reaction, $s\\mbox{-}{HCO} + s\\mbox{-}{CH3O} \\longrightarrow s\\mbox{-}{HCOOCH3}.$ Both reactants are steps on the ladder of the sequential hydrogenation of CO to form CH3OH.",
"Similarly, $s\\mbox{-}{CH3} + s\\mbox{-}{CH2OH} \\longrightarrow s\\mbox{-}{C2H5OH},$ $s\\mbox{-}{CH3} + s\\mbox{-}{CH3O} \\longrightarrow s\\mbox{-}{CH3OCH3},$ $s\\mbox{-}{CH3} + s\\mbox{-}{CH3CO} \\longrightarrow s\\mbox{-}{CH3COCH3},$ $s\\mbox{-}{OH} + s\\mbox{-}{CH3CO} \\longrightarrow s\\mbox{-}{CH3COOH},$ and, $s\\mbox{-}{CH2OH} + s\\mbox{-}{HCO} \\longrightarrow s\\mbox{-}{HOCH2CHO}.$ Note that the isomers, s-HCOOCH3 and s-HOCH2CHO (methyl formate and glycolaldehyde), are relatively abundant in the disk midplane.",
"Both species are formed via reactants which are products of the hydrogenation of s-CO.",
"In contrast, the third member of this family, s-CH3COOH, is formed on the grain via s-CH3CO which is formed via the association of s-CH3 and s-CO.",
"This reaction has a barrier of around 3500 K and so cannot proceed at the low temperatures in the disk midplane.",
"Radicals, such as, CH2OH and CH3O, are also formed in the gas via the photodissociation of larger molecules, e.g., CH3OH."
],
[
"Model 4: Radiation processing of ice",
"In Model 4 we added cosmic-ray, X-ray and UV photo-processing of ice mantle material.",
"The behaviour of ice under irradiation is still very uncertain with only a handful of quantitative experiments conducted.",
"Experiments on UV-irradiated pure methanol ice show that chemistry is induced by the process with many gas-phase products, other than methanol, observed [53], [65], [98].",
"Experiments investigating soft X-ray irradiated ice also show a rich chemistry as discussed earlier in the context of X-ray desorption [6], [26], [68].",
"Experiments have also been performed to simulate the chemical consequences of the direct impact of cosmic-ray particles on ice-covered dust grains [71], [12], [13].",
"High energy cosmic rays can fully penetrate dust grains creating suprathermal atoms which, in turn, transfer energy to the ice mantle and ionise molecules creating high-energy electrons ($\\sim $ keV) which induce a further cascade of secondary electrons.",
"[12] simulate this effect by irradiating astrophysical ices (consisting of s-CO and s-CH3OH) with energetic electrons and find that complex molecules, such as glycolaldehyde and methyl formate, can efficiently form, reaching relative abundances commensurate with those observed in hot cores.",
"The radiation processing of grain mantle material should provide an additional means for complex species to build up on the grain since the process allows the replenishment of grain-surface atoms and radicals to take part in further reactions.",
"Indeed, to reproduce the observed abundances of gas-phase COMs in hot cores ($\\sim $ 10$^{-8}$ – 10$^{-6}$ ), models need to include radiative processing of ice to produce the necessary precursor molecules [50].",
"For example, instead of s-CH3OH effectively becoming the `end state' of s-CO via hydrogenation, cosmic-rays, X-rays, and UV photons can break apart s-CH3OH into s-CH3 and s-OH which are then available to either reform methanol or form other species such as s-CH3CN (s-CH3 + s-CN) or s-HCOOH (s-OH + s-HCO).",
"More complex molecules can also be synthesised by the association of surface radicals; a most important case in hot cores is the association of the s-HCO and s-CH3O radicals to form methyl formate (s-HCOOCH3).",
"[78] found that the gas-phase formation of methyl formate contributes, at most, to 1.6 % of the total abundance in hot cores.",
"Another case is the surface formation of dimethyl ether (s-CH3OCH3) via the association of the surface radicals s-CH3 and s-CH3O.",
"These and many other cases are discussed in Sect.",
"REF and in further detail in [50].",
"The results for Model 4 are represented by the yellow-orange lines in Figs.",
"REF and REF .",
"In general, we see drops in abundances of gas-phase molecules in the molecular layer and disk surface ($\\gtrsim $ 100 AU) which correlates with drops in the abundances of grain-surface molecules at a similar height.",
"Conversely, in the disk midplane ($\\lesssim $ 100 AU), we see an increase in the abundances of gas-phase and grain-surface species.",
"This is most noticeable for those species which otherwise are unable to form efficiently on the grain at low temperatures: s-CH3CHO, s-C2H5OH, s-CH3OCH3, s-CH3COCH3, and, s-CH3COOH.",
"The grain-surface fractional abundances of these species are enhanced by between four (e.g., s-CH3CHO) and nine orders of magnitude (e.g., s-CH3COCH3) to values $\\sim $ 10$^{-10}$ – 10$^{-6}$ (relative to the results from Model 3).",
"However, this dramatic increase in grain-surface abundance in the disk midplane does not necessarily translate to an `observable' gas-phase fractional abundance ($\\sim $ 10$^{-17}$ – 10$^{-13}$ ).",
"The internal cosmic-ray-induced photons help to build chemical complexity on the ice in the midplane by breaking apart the more simple species, e.g., methanol, generating radicals which can go on to create more complex species.",
"However, the increasing strength of external UV photons and X-rays towards the disk surface acts to break down this complexity and the grain-surface chemistry favours the production of more simple ice species, e.g., s-CO2 and s-H2O.",
"In fact, we find that the main repository of carbon and oxygen in the molecular layer is s-CO2.",
"In Fig.",
"REF we present the fractional abundance of s-CO2 as a function of disk height for our reduced grid at R = 305 AU for each of our chemical models.",
"In Model 4, s-CO2 reaches a peak fractional abundance of $\\sim $ 10$^{-4}$ at a height of $\\approx $ 150 AU, which corresponds to the point where there is a sharp decrease in the abundance of other C- and O-containing complex molecules.",
"s-CO2 is formed on grain-surfaces via the reaction, $s\\mbox{-}{CO} + s\\mbox{-}{OH} \\longrightarrow s\\mbox{-}{CO2} + s\\mbox{-}{H}.$ We find that the rate for this reaction, under the physical conditions in the molecular layer, is marginally faster than the rate for the rehydrogenation of s-OH.",
"Over time, s-CO2 grows at the expense of s-CO and s-H2O, and indeed, other O- and C-containing species.",
"CO2 has a smaller cross-section for photodissociation than H2O at longer wavelengths [143] and so is more photostable in the molecular layer of the disk where the UV radiation field is softer than that in the upper disk.",
"Historically, the above reaction has been included in chemical networks with a small reaction barrier of $\\approx $ 80 K based on the gas-phase reaction potential energy surface [134], [126], [50], [51].",
"Recent experiments suggest the effective barrier for this reaction is closer to $\\approx $ 400 K [91] and we have adopted the higher barrier in the work presented here.",
"Lowering the barrier to 80 K further increases the production of s-CO2, further decreasing the abundances of other O- and C-containing species.",
"CO2 ice has been observed in many different environments with a typical abundance $\\sim $ 30% that of water ice; however, its exact formation mechanism under cold interstellar conditions remains a puzzle.",
"Recently, [118] observed CO2 ice in absorption towards the low-mass protostar, HOPS-68, and analysis of their data revealed the CO2 was contained within an ice matrix consisting of almost pure CO2 ice ($\\approx $ 90%).",
"The authors postulate that HOPS-68 has a flattened envelope morphology, with a high concentration of material within $\\approx $ 10 AU of the central star, thereby explaining the lack of primordial hydrogen-rich ices along the line of sight.",
"They also propose a scenario where an energetic event has led to the evaporation of the primordial grain mantle and subsequent cooling and recondensation has led to the production of CO2 ice in a H-poor ice mantle.",
"Certainly, our results suggest that the reprocessing of ice species by UV photons may play a role in driving the production of CO2 ice at the expense of other typical grain mantle species, such as, CO and H2O.",
"Concerning commonly observed gas-phase species in disks, we find the inclusion of radiation processing of ice has an effect mainly on those species for which precursor species remain frozen out at this radius.",
"For CO, we do not see a strong effect because CO does not rely on grain-surface chemistry for its formation.",
"In the molecular and surface layers of the disk, CO exists predominantly in the gas phase as it is able to thermally desorb from grain surfaces.",
"We see a small decrease in the CO abundance (on the order of a factor of a few) in Model 4 relative to Model 3 between a height of 50 AU and 150 AU as CO is driven into s-CO2(discussed above).",
"We also do not see a strong effect on the abundance of carbon monosulphide, CS, since this species reaches its peak fractional abundance ($\\sim $ 10$^{-8}$ ) in the surface layers of the disk (above a height of 150 AU) only.",
"In contrast, for CN (and its precursor species, HCN), we do see a similar drop in abundance around 150 AU in Model 4 (compared with Model 3) since HCN mainly exists as ice on the grain at this disk radius.",
"The ethynyl radical, C2H exhibits a similar behaviour to CN which is related to acetylene, C2H2, also existing primarily as ice, albeit at a lower abundance than for more abundant ice species, such as s-HCN and s-H2O.",
"s-C2H2 can be hydrogenated on the grain to form s-C2H3 (and beyond).",
"Figure: Fractional abundance (with respect to H nuclei number density)of s-CO2 as a function of disk height, ZZ at a radius, RR = 305 AU.The chemical complexity in the model increases from Model 1 to Model 5(see Sect.",
"for details)."
],
[
"Model 5: Reactive desorption",
"In Model 5 we also include reactive desorption and the results are represented by the gray lines in Figs.",
"REF and REF .",
"In general, we find reactive desorption helps release further gas-phase COMs in the disk midplane leading to an enhancement of around one to two orders of magnitude.",
"However, this enhancement is not necessarily sufficient to increase the molecular column density to an observable value.",
"For example, acetaldehyde (CH3CHO) is enhanced from a fractional abundance of $\\sim $ 10$^{-13}$ in Model 4 to $\\approx $ 2 $\\times $ 10$^{-12}$ in Model 5.",
"We also see a slight increase (around a factor of a few) in the grain-surface abundances of several species in the disk midplane; s-CH3CHO, s-CH3NH2, s-C2H5OH, s-CH3OCH3, s-CH3COCH3, and s-CH3COOH.",
"At heights $Z$ $\\gtrsim $ 100 AU, the results from Model 4 and Model 5 are similar.",
"The inclusion of reactive desorption also helps increase the abundance of gas-phase CS, CN, HCN, and C2H in the disk midplane.",
"However, this enhancement is not sufficient to significantly increase the observable column density.",
"This is because these species can form in the gas phase and thus are not reliant on grain-surface association reactions.",
"For CO, reactive desorption is unimportant as, again, grain-surface chemistry does not contribute significantly to the formation of CO on the grain.",
"In the disk midplane, cosmic-ray-induced desorption is the primary mechanism for releasing s-CO into the gas phase."
],
[
"Full disk model",
"In Figs.",
"REF and REF , we display the fractional abundances of COMs (relative to total number density) as a function of disk radius, $R$ , and height, $Z/R$ , for gas-phase and grain-surface (ice mantle) species, respectively.",
"In Fig.",
"REF , we display the vertical column densities of gas-phase (red lines) and grain-surface (blue lines) species as a function of disk radius, $R$ .",
"Figures REF to REF show that the grain-surface (ice mantle) abundances and resulting column densities are consistently higher than those of the corresponding gas-phase species.",
"Most of the grain-surface COMs reach maximum fractional abundances of $\\sim $ 10$^{-6}$ to 10$^{-4}$ relative to the local number density and are confined to a layer in the midplane, whereas, we see a large spread in the maximum fractional abundances of gas-phase species from $\\sim $ 10$^{-12}$ to $\\sim $ 10$^{-7}$ .",
"The grain-surface molecules also reach their peak abundance over a much larger region in the disk than their gas-phase analogues.",
"The exceptions to this are formaldehyde, H2CO, cyanoacetylene, HC3N, and acetonitrile, CH3CN, all of which can be formed readily via gas-phase chemistry.",
"These species are abundant throughout most of the upper regions of the disk, whereas, the more complex species reach their peak abundance in the outer disk only in a layer bounded below by freezeout and above by destruction via photodissociation and ion-molecule reactions.",
"Generally, the more complex the molecule, the less abundant it is in the gas phase, and the smaller the extent over which the molecule reaches an appreciable abundance.",
"This indicates that species which can form in the gas phase, e.g., H2CO and HC3N, are less sensitive to the variation in disk physical conditions than the more complex species, such as, HCOOCH3 and HOCH2CHO, which can only form via grain-surface chemistry.",
"Given that the abundance of gas-phase complex species relies on efficient photodesorption and limited photodestruction over a narrow range of the disk, one can imagine that a small increase/decrease in UV flux or X-ray flux in the disk may either inhibit or enhance the abundance of gas-phase complex molecules to a much more significant degree than for the simpler species.",
"In addition, the evolution of dust grains may also be important.",
"Dust-grain settling (or sedimentation) towards the disk midplane can lead to the increased penetration of UV radiation, whereas dust-grain coagulation (or grain growth) leads to bigger grains and a smaller grain-surface area available for the absorption of UV photons [32], [35], [33].",
"We discuss this issue further in Sect. .",
"We intend to explore the effects of dust evolution in future work.",
"Several of the grain-surface species (s-CH3OH, s-HC3N, s-CH3CN, s-CH3CCH, s-CH3NH2, s-C2H5OH, and s-CH3COOH) remain frozen out down to $\\approx $ 2 AU.",
"These species have desorption or binding energies higher than $\\approx $ 4500 K and so we would expect their snow lines to reside at a similar radius to that for water ice ($\\approx $ 2 AU).",
"s-HCOOH, s-NH2CHO, s-HCOOCH3, and s-HOCH2CHO also possess high binding energies ($\\gtrsim $ 4000 K); however, these species have snow lines at $\\approx $ 5 AU.",
"Within 5 AU, the dust temperature is $>$ 70 K and radical-radical association reactions are more important than atom-addition reactions due to the fast desorption rates of atoms at these temperatures.",
"Grain-surface species which depend on atom-addition routes to their formation are not formed as efficiently on warm grains.",
"For example, s-HCOOCH3 is formed either via the hydrogenation of s-COOCH3 or via the reaction between s-HCO and s-CH3O.",
"These latter two species, in turn, are formed via hydrogenation of s-CO on the grain.",
"s-CH3O is also formed via the photodissociation of s-CH3OH by cosmic-ray-induced photons.",
"The radical-radical formation routes of s-HCOOH, s-NH2CHO, and s-HOCH2CHO all rely on the formation of s-HCO which, in turn, is formed mainly via the hydrogenation of s-CO.",
"In contrast, at warmer temperatures, s-CH3OH can efficiently form via the association of s-CH3 and s-OH rather than via the hydrogenation of s-CO.",
"Both these radicals can form in the gas and accrete onto grains, or they are formed via the cosmic-ray induced photodissociation of grain-mantle molecules.",
"A similar argument holds for s-CH3CN (s-CH3 + s-CN), s-CH3CCH (s-C2H3 + s-CH), s-CH3NH2 (s-CH3 + s-NH2), s-C2H5OH (s-CH3 + s-CH2OH), s-CH3COOH (s-CH3 + s-CH3CO).",
"s-CH2OH and s-CH3CO also have radical-radical association formation routes, i.e., s-CH2 + s-OH and s-CH3 + s-CO.",
"In Fig.",
"REF , we display the column density of gas-phase (red lines) and grain-surface (blue lines) species as a function of radius, $R$ .",
"In Table REF , we also display the calculated column densities at radii of 10, 30, 100, and 305 AU.",
"An expanded version of Table REF which includes all data used to plot Fig.",
"REF is available in the electronic edition of the journal (Tables REF and REF ).",
"Most of the gas-phase molecules reach values $\\gtrsim $ 10$^{12}$ cm$^{-2}$ throughout most of the outer disk ($R$ $\\gtrsim $ 50 AU).",
"The exceptions are HC3N and the more complex species considered here, i.e., C2H5OH, CH3OCH3, HCOOCH3, CH3COCH3, HOCH2CHO, and CH3COOH.",
"Most of these species achieve a column density $\\sim $ 10$^{11}$ cm$^{-2}$ throughout the outer disk.",
"CH3COCH3 peaks around 100 AU before decreasing towards larger radii and CH3COOH reaches a peak column density of $\\sim $ 10$^{10}$ cm$^{-2}$ between $\\approx $ 25 and 50 AU before steadily decreasing towards larger radii.",
"Both these species form via s-CH3CO which, in turn, forms on the grain via s-CH3 + s-CO.",
"This latter reaction has an appreciable reaction barrier ($\\approx $ 3500 K) and can only proceed on sufficiently warm grains.",
"We produce very similar gas-phase and grain-surface column densities for the structural isomers, glycolaldehyde (HOCH2CHO) and methyl formate (HCOOCH3), which are formed primarily via the grain-surface association routes, s-CH2OH + s-HCO and s-CH3O + s-HCO, respectively.",
"These species have been detected in the gas phase in the hot core, Sgr B2(N), and in the low-mass protostar, IRAS 16293+2422 [64], [70].",
"In both sources, methyl formate is more than a factor of 10 more abundant than glycolaldehyde.",
"In this work, we have assumed a branching ratio of 1:1 for the production of CH3O and CH2OH via the photodissociation of gas-phase and grain-surface methanol.",
"Hence, the formation rates of both radicals via this mechanism are similar leading to similar abundances of methyl formate and glycolaldehyde.",
"[78] investigated various branching ratios for methanol photodissociation in hot cores and concluded that branching ratios for grain-surface cosmic-ray-induced photodissociation have an influence on the resulting gas-phase abundances.",
"They found that models including ratios favouring the methoxy channel (s-CH3O) agreed best with observed abundances of methyl formate; however, ratios favouring the methyl channel (s-CH3) agreed best with the observed gas-phase abundance of glycolaldehyde.",
"This is in contrast with laboratory experiments which show that formation of the hydroxymethyl radical (s-CH2OH) is the dominant channel [98].",
"The mobilities of the s-CH2OH and s-CH3O radicals can also influence the production rates of grain-surface methyl formate and glycolaldehyde.",
"Here, we have followed [50] and assumed that s-CH2OH is more strongly bound to the grain mantle than s-CH3O ($E_{D}$ = 5080 K and 2250 K, respectively) due to the $-$ OH group which allows hydrogen bonding with the water ice.",
"Hence, we expect s-CH3O to have higher mobility than s-CH2OH.",
"However, the reaction rates for the grain-surface formation of methyl formate and glycolaldehyde at the temperatures found in the disk midplane are dominated by the mobility of the s-HCO radical.",
"This radical has a significantly lower binding energy to the grain mantle ($E_D$ = 1600 K) leading, again, to similar grain-surface formation rates for methyl formate and glycolaldehyde.",
"Table: Column density (cm -2 ^{-2}) of gas-phase and grain-surface organic molecules at radii of 10, 30, 100, and 305 AU fromour full disk model.Figure: Fractional abundance of gas-phase molecules with respect to total H nuclei number density as afunction of disk radius, RR, and height, ZZ.Figure: Same as Fig.",
"for grain-surface species.Figure: Column density (cm -2 ^{-2}) as a function of radius, RR, for gas-phase (red lines)and grain-surface (blue lines) molecules.",
"The corresponding data can be found in Tables and ."
],
[
"Line spectra",
"In Fig.",
"REF we display our disk-integrated line spectra for H2CO and CH3OH up to a frequency of 1000 GHz to cover the full frequency range expected for ALMA `Full Science' operations.",
"We also highlight, in gray, the frequency bands expected to be available at the commencement of `Full Science': band 3 (84 to 116 GHz), band 4 (125 to 163 GHZ), band 6 (211 to 275 GHz), band 7 (275 to 373 GHz), band 8 (385 to 500 GHz), band 9 (602 to 720 GHz), and band 10 (787 to 950 GHz).",
"In our calculations, we assume a distance to source of 140 pc.",
"We find peak flux densities of $\\approx $ 160 mJy for H2CO and $\\approx $ 120 mJy for CH3OH.",
"Our calculations suggest that reasonably strong lines of H2CO and CH3OH which fall into bands 7, 8, and 9 are good targets for ALMA `Early Science' and `Full Science' capabilities.",
"The strongest methanol line in band 7 is the 3$_{12}$ -3$_{03}$ transition of A-CH3OH at 305.474 GHz.",
"The peak line flux density calculated is $\\approx $ 45 mJy.",
"Using the full ALMA array (50 antennae) and a channel width of 0.2 km/s, a sensitivity of 5 mJy can be reached in an integration time of $\\approx $ 30 min.",
"In band 8, a good candidate line is the 2$_{12}$ -1$_{01}$ transition of A-CH3OH at 398.447 GHz.",
"The peak line flux density calculated for this transition is $\\approx $ 60 mJy.",
"Again, using the full ALMA array and a similar channel width, a sensitivity of 10 mJy can be reached in $\\approx $ 60 min.",
"Observations of weak lines in ALMA band 9 are more challenging due to the increasing influence of the atmosphere[11] and the strong continuum emission at higher frequencies.",
"Nevertheless, several methanol line transitions in this band may also be accessible with full ALMA.",
"Under the same conditions, a sensitivity of 25 mJy can be reached in $\\approx $ 120 mins at a frequency of 665.442 GHz, corresponding to the 5$_{24}$ -4$_{14}$ transition of E-CH3OH.",
"We remind the reader that the line intensities calculated here can be considered lower limits to the potential intensities due to the truncation of our disk model at 305 AU.",
"It is possible that methanol has not yet been observed in disks because lower frequency transitions, which we find to be relatively weak, have historically been targeted.",
"A deep search for methanol in nearby well-studied objects should help ascertain whether this species is present: if so, this is a clear indication that our current grain-surface chemistry theory works across different physical regimes and sources.",
"[11]http://almascience.eso.org/about-alma/weather/atmosphere-model We also calculated the line spectra for HCOOH, HC3N, CH3CN, CH3CCH, and NH2CHO and found that the line intensities were negligible in all cases ($\\ll $ 10 mJy).",
"For more complex molecules (e.g., HCOOCH3 and CH3OCH3), line detection and identification in nearby protoplanetary disks may prove challenging, even with the superior sensitivity and spatial resolution of ALMA.",
"Also, millimeter and (sub)millimeter observations of molecular line emission from warm, chemically rich sources (for example, hot cores and massive star-forming regions located towards the galactic centre), often suffer from line overlapping (or blending) which can hinder the solid identification of specific COMs [62].",
"It is also worth noting here that the Square Kilometre Array (SKA[12]), due for completion by 2020 and consisting of some 3000 dishes spread over numerous sites around the world, will have high sensitivity in the 70 MHz to 30 GHz frequency range.",
"This may allow detection of line emission/absorption from complex molecules in nearby disks at lower frequencies than those achievable with ALMA [80].",
"[12]http://www.skatelescope.org/ Figure: Disk-integrated line spectra (Jy) for H2CO (left) and CH3OH (right).The gray boxes indicate the frequency coverage of ALMA `Full Science' receivers:band 3 (84 to 116 GHz), band 4 (125 to 163 GHZ), band 6 (211 to 275 GHz),band 7 (275 to 373 GHz), band 8 (385 to 500 GHz), band 9 (602 to 720 GHz), andband 10 (787 to 950 GHz).In this section, we discuss and compare our results with observations of molecular line emission from protoplanetary disks and cometary comae, and with results from other models with similar chemical complexity.",
"We also discuss the astrobiological significance of our work."
],
[
"Comparison with observations",
"Our exploratory calculations suggest that complex organic molecules may be efficiently formed on grain surfaces in protoplanetary disks.",
"However, it is difficult to observe ice species in disks and indeed, this has only been achieved for water ice in a handful of (almost) edge-on systems [136].",
"Instead, in the cold outer regions of disks, we are limited to observing gas-phase molecules which possess a permanent dipole moment, only.",
"This also presents difficulties if the gas-phase form of the molecule is not present in sufficient quantities and/or also possesses a complex rotational spectrum.",
"The only relatively complex molecules detected in disks, to date, are formaldehyde, H2CO, cyanoacetylene, HC3N, and cyclopropenylidene, $c$ -C3H2.",
"[37] detected several rotational lines of H2CO in the disks of DM Tau and GG Tau deriving a column density of $\\sim $ 10$^{12}$ cm$^{-2}$ .",
"[1] and [137] present detections of formaldehyde in the disk of LkCa 15 determining column densities of 7.2 – 19 $\\times $ 10$^{12}$ cm$^{-2}$ and 7.1 – 51 $\\times $ 10$^{11}$ cm$^{-2}$ , respectively.",
"The large spread in column density is due to the difficulty in using a simple model to derive column densities from observations, even when several lines of the species are detected.",
"From Fig.",
"REF and Table REF , we can see the column density in the outer disk i.e., $>$ 10 AU (10$^{12}$ – 10$^{13}$ cm$^{-2}$ ) compares well with those values constrained from observation.",
"More recently, [99], [100] and [106] present detections of H2CO using the SMA in a selection of protoplanetary disks in the well-studied Taurus region and in the Southern sky.",
"They confirmed the previous detections of H2CO in the disks of DM Tau and LkCa 15, and they also present new detections: one line in the disk of AA Tau, two lines in GM Tau, and two lines in TW Hya.",
"They also detected formaldehyde in the disks of IM Lup, V4046 Sgr, and HD 142527.",
"Their detected lines and line intensities towards T Tauri disks are listed in Table REF .",
"Their values range from $\\sim $ 100 mJy km s$^{-1}$ to $\\sim $ 1 Jy km s$^{-1}$ depending on the source and transition.",
"The authors do not infer any column densities using their data and explain the difficulties in doing so.",
"Instead, they present integrated intensities with which we can compare our calculated line intensities.",
"We also note here the first detection of HC3N in a selection of T Tauri disks by [20].",
"The authors state that their observations are most sensitive to a radius of around 300 AU and derive column densities of $\\lesssim $ 3.5 $\\times $ 10$^{11}$ cm$^{-2}$ , $\\approx $ 8 $\\times $ 10$^{11}$ cm$^{-2}$ and $\\approx $ 13 $\\times $ 10$^{11}$ cm$^{-2}$ for DM Tau, LkCa 15 and GO Tau, respectively.",
"Comparing this with our calculations at 305 AU in Table REF , we determine a column density of 1 $\\times $ 10$^{11}$ cm$^{-2}$ , which is within the upper limit derived for DM Tau, but around one order of magnitude lower than the column densities for the remaining two sources.",
"We also calculated the rotational line spectra for HC3N and found the lines were much weaker than the observed line intensities ($\\ll $ 10 mJy km s$^{-1}$ versus 60 – 100 mJy km s$^{-1}$ ).",
"We note here that LkCa 15 is a particularly peculiar object: the discovery of a large cavity in continuum emission within a radius of $\\approx $ 50 AU has reclassified this object as a transition disk [113] in which planet formation is likely at an advanced stage [77].",
"Analysis of CO line observations also identified the lack of a vertical temperature gradient in this disk [114].",
"In addition, GO Tau hosts a particularly large, massive molecular disk [128].",
"Hence, our disk model is likely not a good analogue for both these sources, providing further explanation for the disagreement between our model results and observations.",
"Recently, [107] reported the detection of cyclopropenylidene, $c$ -C3H2, in a protoplanetary disk for the first time.",
"The authors identified several lines of this species in ALMA Science Verification observations of the disk of HD 163296, a Herbig Ae star.",
"This allowed the authors to derive a column density $\\sim $ 10$^{12}$ – 10$^{13}$ cm$^{-2}$ .",
"Herbig Ae/Be stars are more massive and luminous than T Tauri stars, hence, our disk model is not a suitable analogue for this source.",
"However, it is interesting to consider whether this species may also be detectable in disks around T Tauri stars.",
"Our model predicts a column density of $\\approx $ 1 $\\times $ 10$^{11}$ cm$^{-2}$ at a radius of 100 AU and $\\approx $ 3 $\\times $ 10$^{11}$ cm$^{-2}$ at 305 AU, around two orders of magnitude lower than that derived for HD 163296.",
"Gas-phase methanol, CH3OH, has not yet been detected in a protoplanetary disk.",
"However, there have been multiple searches in several well-studied objects giving well-constrained upper limits to the line intensities and column densities.",
"[137] searched for four lines of methanol (2$_{02}$ -1$_{01}$ A, 4$_{22}$ -3$_{12}$ E, 5$_{05}$ -4$_{04}$ A, 7$_{07}$ -6$_{06}$ A) in the disks of LkCa15 and TW Hya using the IRAM 30 m and JCMT single-dish telescopes.",
"In all cases, upper limits only were determined, leading to derived upper column densities between $\\approx $ 1 $\\times $ 10$^{13}$ and $\\approx $ 4 $\\times $ 10$^{14}$ cm$^{-2}$ .",
"Again, our calculated column densities agree with these values in that we predict column densities generally lower than the upper limits derived from the observations.",
"[99], [100] also included a line transition of methanol in their SMA line survey of protoplanetary disks.",
"They targeted the 4$_{22}$ -3$_{12}$ transition of E-type CH3OH at 218.440 GHz in a range of T Tauri and Herbig Ae/Be disks and were unable to detect the line in all cases.",
"In Table REF , we compare our modelled line intensities with observations towards sources in which H2CO has been detected and in which H2CO and CH3OH upper limits have been determined.",
"We restrict this list to T Tauri stars which possess a substantial gaseous disk.",
"We have rescaled our modelled intensities by the disk size and distance to source using the values listed in Table REF .",
"We have listed the sources roughly in order of decreasing spectral type, from K3 (GM Aur) to M1 (DM Tau).",
"We have converted the IRAM 30 m and JCMT line intensities from [137] using the standard relation $\\left( \\frac{T}{1\\,\\mathrm {K}} \\right)= \\left( \\frac{S_\\nu }{1\\,\\mathrm {Jy}\\,\\mathrm {beam}^{-1}} \\right)\\left[ 13.6 \\left( \\frac{300\\,\\mathrm {GHz}}{\\nu } \\right)^{2}\\left( \\frac{1\"}{\\theta ^2} \\right) \\right]$ where $T$ is the line intensity in K, $S_\\nu $ is the line intensity in Jy beam$^{-1}$ , $\\nu $ is the line frequency in GHz and $\\theta $ is the beam size in arcseconds.",
"The modelled line intensities for H2CO agree reasonably well (within a factor of three) with most transitions towards most sources.",
"For the hotter stars (GM Tau, LkCa 15, V4046 Sgr, and TW Hya) there is better agreement for the higher frequency transitions than for the lower frequency transitions.",
"For the cooler stars (DM Tau and GG Tau), there is also reasonable agreement with the lower frequency transitions.",
"The change in line intensity ratios moving from hotter stars to cooler stars reflects the change in disk temperature structure and thus excitation conditions.",
"For the lines in which we see poor agreement, the calculations tend to underestimate the observed line intensities.",
"We would not expect absolute agreement with any particular source because we have adopted `typical' T Tauri star-disk parameters in our model.",
"However, the level of agreement between our calculations and observations is sufficient for us to conclude that our model is providing a reasonable description of the formation and distribution of H2CO in protoplanetary disks around T Tauri stars and the resulting line emission expected from these objects.",
"Comparing the methanol upper limits and calculated line intensities, we see that our calculations fall well within the upper limits for all sources.",
"Our calculations suggest that the lines of methanol targeted in previous surveys of disks are likely too weak to have been observable.",
"However, our calculations also suggest several potential candidate lines we expect to be strong enough for detection with ALMA (see Sect.",
"REF and Fig.",
"REF ).",
"Table: H2CO and CH3OH rotational transitions in protoplanetary disks."
],
[
"Comparison with other models",
"Here, we compare our results with other protoplanetary disk models, concentrating on work which has published lists of column densities or fractional abundances for relatively complex species.",
"Historically, chemical models of disks have concentrated on simple, abundant species (and isotopologues), since these are readily observed in many systems (e.g., CO, HCO+, CN, CS, and HCN).",
"As we enter the era of ALMA, it is likely that the molecular inventory of protoplanetary disks will significantly increase, requiring much more sophisticated complex chemical models, such as that presented here.",
"In Table REF , we compare column densities of various complex molecules at a radius of $\\approx $ 250 AU with other protoplanetary disk models of comparable chemical complexity and with similar chemical ingredients.",
"[159] presented a chemically complex model of a protoplanetary disk, including a comprehensive deuterium chemistry.",
"We compare our column densities with Model C in that work, which includes both grain-surface chemistry and non-thermal desorption.",
"[131] present results from a disk model which uses a network with a number of chemical reactions ($\\approx $ 7300) approaching the number in the network presented here ($\\approx $ 9300).",
"We compare our results with their `laminar' model in which they neglect turbulent mixing, since we do not consider mixing in this work.",
"We also list the column densities from our most recent paper, WNMA12, which is most similar to the work presented here in that the disk physical model is identical as are the methods used to calculate the chemistry.",
"In WNMA12 we used a chemical network based on `Rate06', the most recent release of the UMIST Database for Astrochemistry (UDfA) available at that time, whereas, here, our network is derived from the Ohio State University (OSU) network and includes a vastly more comprehensive grain-surface chemical network to simulate the build up of complex molecules.",
"The network used in W07 is also derived from UDfA, albeit an earlier version [86], whereas, the network used by SW11 is also based on the OSU network.",
"Care must be taken when comparing results from different protoplanetary disk models, as they often differ in physical ingredients as well as the chemistry.",
"The work presented here generally predicts higher column densities for COMs than those presented in W07 and SW11 despite relatively similar (within an order of magnitude) column densities for CO, H2CO, and HC3N.",
"In this work, we calculate significantly higher column densities for CH3OH, HCOOH, CH3CN, CH3CHO, NH2CHO, HCOOCH3, C2H5OH, CH3OCH3, and CH3COCH3.",
"The network used by SW11 is based on that presented in [48] which does not contain many pathways to the larger species introduced in [50].",
"Also, they adopt $E_{d}$ = 0.77 $E_{D}$ for their grain-surface diffusion rates [125], where $E_{D}$ is the binding energy of the molecules to the grain surface.",
"This is a rather conservative value and partly explains their much lower abundances of complex species.",
"In addition, they do not consider quantum tunnelling of H atoms on grain surfaces, nor through reaction energy barriers [130].",
"The neglect of H atom tunnelling through reaction energy barriers explains the particularly low column density of methanol in SW11 ($\\sim $ 10$^{8}$ cm$^{-2}$ ).",
"W07 include atom-addition grain-surface reactions only and thus neglect radical-radical pathways to form larger COMs.",
"Comparing our results with those from our previous work (WNMA12), we see a significant increase in the column density of CH3OH, HCOOCH3, CH3OCH3, and CH3COCH3 when using the gas-grain network presented here.",
"The higher column density of grain-surface methanol, s-CH3OH, can be attributed to the higher binding energy of CO adopted here.",
"In previous work, we used the value measured for pure CO ice (855 K) as opposed to the value measured in water ice (1150 K).",
"The binding energy regulates the abundance of s-CO on the grain and thus the amount available for conversion to s-CH3OH, as well as the grain-surface radicals, s-HCO, s-CH3O, and s-CH2OH.",
"Regarding the formation of s-HCOOCH3, the grain-surface association reaction, s-HCO + s-CH3O, is included in both models.",
"The difference in column density is due, again, to the different sets of binding energies adopted.",
"The results from our exploratory calculations presented in Sect.",
"REF demonstrate the importance of radiation processing for the production of s-CH3OCH3 and s-CH3COCH3 in the disk midplane.",
"The midplane is the densest region of the disk and thus contributes significantly to the vertical column density.",
"In previous work we did not include the processing of ice mantle material by UV photons and X-rays.",
"We also see a decrease in the column density of gas phase HC3N, and a corresponding increase in the grain-surface column density, compared with our previous values.",
"This is due to the increased binding energy for HC3N adopted here (4580 K compared with 2970 K).",
"Our previous value shows better agreement with the column densities constrained from observations ( $\\sim $ 10$^{12}$ cm$^{-2}$ ).",
"Protoplanetary disks are turbulent environments and the effects of vertical turbulent mixing on disk chemical structure has been investigated by multiple groups [67], [158], [129], [3], [60], [131].",
"[131] conducted a comprehensive investigation of disk chemical structure with and without turbulent mixing and identified a plethora of species which are sensitive to mixing.",
"[131] also used a chemical network including several complex molecules (see Table REF ).",
"Of the gas-phase molecules of interest here, they found that the column densities of HCOOH, HC3N, and CH3CN, were significantly affected by the inclusion of turbulent mixing.",
"However, they concentrated their discussions on species which reached column densities $\\gtrsim $ 10$^{11}$ cm$^{-2}$ .",
"In this work, we assume the dust grains are well mixed with the gas and, for the calculation of the chemical structure, we assume the grains are compact spherical grains with a fixed radius.",
"In reality, the dust grains will have both a size distribution and variable dust-to-gas mass ratio caused by gravitational settling towards the midplane and dust-grain coagulation (grain growth).",
"Several groups have also looked at the effects of dust-grain evolution on protoplanetary disk chemistry [2], [45], [145], [4].",
"A parameterised treatment of grain growth affects the geometrical height of the molecular layer but appears to have little effect on the column densities of gas-phase molecules [2].",
"Larger grains may lead to a reduced volume grain-surface area (for a fixed dust-to-gas mass ratio) which will lower the accretion rate of molecules onto dust grains, thereby potentially lowering the formation rate of COMs.",
"However, this effect depends on the assumed morphology and porosity of the grains.",
"Grain coagulation models suggest that the growing dust grains retain a `fluffy' morphology (with a low filling factor, $\\ll $ 1) such that the volume grain-surface area may not significantly decrease before compression occurs [103], [102], [73].",
"Grain settling towards the midplane allows the deeper penetration of UV radiation leading to warmer grains in the disk midplane.",
"This subsequently results in a smaller depletion (freezeout) zone and a larger molecular layer situated closer to the midplane [45].",
"[4] performed a coupled calculation of the structure of a protoplanetary disk including dust evolution and radiative transfer, and subsequently calculated the chemical evolution.",
"They find that the abundances of several species are enhanced in models including dust evolution, including the relatively complex species, NH2CN and HCOOH.",
"We intend to explore the effects of grain evolution on the formation and distribution of COMs in future models.",
"A final issue to consider is the validity of our set of initial abundances.",
"Disk formation is thought to be a vigorous, energetic, and potentially destructive process.",
"Accretion shocks are thought to occur as material falls from the envelope onto the disk.",
"Heating by the shock may raise the temperature of the dust grains above the sublimation temperature of ices and energised ions may sputter ices from dust grain surfaces [95], [139].",
"Hence, using initial abundances representative of dark cloud (or prestellar) conditions may not be realistic because dust grains may be stripped of ices as they pass through an accretion shock during the disk formation stage.",
"[148] studied the 2D chemical evolution during the protostellar collapse phase to determine the chemical history of simple ices contained within the disk at the end of collapse.",
"They concluded that accretion shocks that occur as material falls from the envelope onto the disk are much weaker than commonly assumed.",
"For the outer disk, the main contribution to heating is via stellar heating [148].",
"Sputtering of dust grains by energetic ions can also occur.",
"[148] also considered this and concluded that the shock velocities experienced by the gas, $\\approx $ 8 km s$^{-1}$ , are not sufficient to energise ions, such as He+, to energies required for the removal of water molecules from grain surfaces.",
"As a result, much of the material contained within the outer disk ($\\gtrsim $ 10 AU) at the end of collapse consists primarily of “pristine” interstellar ice [149].",
"Hence, beginning our simulations with initial molecular abundances representative of prestellar conditions is an appropriate assumption.",
"Table: Calculated column densities (cm -2 ^{-2}) at RR = 250 AU."
],
[
"Complex molecules in comets",
"It is generally accepted that minor bodies in the Solar System, such as asteroids and comets, likely formed in conjunction with the planets in the Sun's primordial disk and can be considered remnant material left over from the process of planet formation.",
"When a comet's orbit is perturbed in such a way that it is injected into the inner Solar System, the gradual warming of the nearing Sun evaporates solid surface material and creates an expansive cometary coma of gaseous volatile material enveloping the comet nucleus.",
"Photolysis of the sublimated material (termed `parent' species) and subsequent chemistry creates ions and radicals and new molecules (termed `daughter' species).",
"It is now understood that comets are complex objects composed of ice (mainly H2O, CO2, and CO), refractory material (such as silicates), and organic matter.",
"To date, more than 20 parent molecules have been observed in cometary comae including the relatively complex species, H2CO, CH3OH, HCOOH, CH3CHO, HC3N, CH3CN, NH2CHO, HCOOCH3, and (HOCH2)2 (ethylene glycol), which are relevant to this work.",
"Of these species, CH3CHO, NH2CHO, HCOOCH3, and (HOCH2)2 have been observed in only a single object, comet Hale-Bopp, with percentage abundances of 0.02 %, 0.015 %, 0.08 %, and 0.25 % (relative to H2O), respectively [19], [28], [29], [30], [88].",
"In Fig.",
"REF we present the range of calculated abundances for each of these grain-surface species relative to water ice (red lines) and compare these with our initial adopted dark cloud ice ratios (green asterisks) and data derived from cometary comae observations (blue asterisks and lines).",
"The fractional abundances from the disk model are determined by the relative vertical column densities at each radius.",
"We restrict our data to $R$ $\\gtrsim $ 20 AU which corresponds to the radius beyond which grain-surface COMs achieve significant column densities (see Fig.",
"REF ).",
"This also correlates with the region where comets are postulated to have originally formed and resided in modern dynamical models of the Solar System [47], [153].",
"The single points and upper limits for the comet observations refer to data derived from observations of comet Hale-Bopp.",
"We find that our range of calculated abundances (relative to water ice) are consistent with those derived from observations, with some overlap between the two datasets for most species.",
"Exceptions include s-CH3CHO and s-NH2CHO for which a single observation only is available.",
"In both cases, our data range is larger than the observed ratio, with the lower limit of our data within a factor of a few of the measured ratio.",
"Another exception is s-CH3CCH, for which an upper limit towards Hale-Bopp only has been derived [29].",
"Again, we find our calculated ratio range is larger than the upper limit.",
"In this case, the lower limit of our data is much further away from that derived from observation, by a factor of $\\approx $ 30.",
"It is also interesting to compare our range of calculated abundances in the disk model with our initial abundances adopted from a dark cloud model (see Table REF ).",
"The s-H2CO/s-CH3OH ratio indicates there is significant chemical processing of the dark cloud grain-surface material within the disk with this ratio decreasing from cloud to disk.",
"For all other species (except s-CH3CCH) the dark cloud abundance is lower than the lower limit reached in the disk model indicating that disk physical conditions are necessary for thermal grain-surface chemistry to efficiently form the complex molecules observed in cometary comae.",
"It certainly appears that our grain-surface chemistry is appropriate for describing the grain-surface formation of most COMs observed in cometary comae, supporting the postulation that comets are formed via the coagulation and growth of icy dust grains within the Sun's protoplanetary disk.",
"One outstanding issue is the high abundance of ethylene glycol ((HOCH2)2) observed towards comet Hale-Bopp, with a percentage abundance of 0.25% relative to water ice.",
"This ratio is similar to that observed for H2CO and around an order of magnitude higher than the ratio derived for CH3CHO and NH2CHO.",
"Also, (HOCH2)2 is observed to be at least 5 times more abundant than the chemically-related species, HOCH2CHO [29].",
"In this network, we include a single barrierless route to the formation of s-(HOCH2)2 via the grain-surface association of two s-CH2OH radicals.",
"Under the conditions throughout much of the disk, the mobility of this radical is significantly slower than smaller radicals, such as, s-CH3 and s-HCO, due to its significantly larger binding energy to the grain mantle ($\\approx $ 5000 K).",
"The large binding energy is due to the presence of the -OH functional group allowing hydrogen bonding of this species with the bulk water ice [50].",
"Hence, the reaction forming s-(HOCH2)2 cannot compete with other barrierless radical-radical association reactions which form, for example, s-C2H5OH and s-HOCH2CHO.",
"We find a negligible abundance of s-(HOCH2)2 is produced throughout our disk model.",
"In the network used here, radical-radical association pathways only have been included for the formation of many COMs, in addition to pathways involving sequential hydrogenation of precursor molecules.",
"However, an alternative grain-surface route to the production of (HOCH2)2 (and other COMs) has been proposed by [25] involving sequential atom-addition reactions.",
"For example, (HOCH2)2 is postulated to form via the hydrogenation of s-OCCHO, which in turn is formed from s-CO via atom-addition reactions, i.e., $s\\mbox{-}{CO} \\xrightarrow{}s\\mbox{-}{HCO} \\xrightarrow{}s\\mbox{-}{HC2O} \\xrightarrow{}s\\mbox{-}{OCCHO}$ followed by $s\\mbox{-}{OCCHO} \\xrightarrow{}s\\mbox{-}{CHOCHO} \\xrightarrow{}s\\mbox{-}{HOCH2CHO} \\xrightarrow{}s\\mbox{-}{(HOCH2)2}.$ In this sequence, 2s-H implies a barrier penetration reaction by a hydrogen atom followed by the exothermic addition of an additional H atom.",
"This sequence of atom-addition reactions is postulated to lead to different ratios of resultant grain-surface COMs relative to the radical-radical network used here and may provide a route to the formation of s-(HOCH2)2.",
"However, as discussed in [62], many of these reaction rates and reaction barriers remain unmeasured.",
"The efficacy of this type of formation route to COMs under protoplanetary disk conditions is yet to be studied and we intend to explore this in future work.",
"Of course, it is also possible that significant processing of the cometary surface by UV photons (and potentially, cosmic rays) over the comet's lifetime may lead to a surface composition which differs from the initial grain mantle composition in the protoplanetary disk.",
"In addition, thermally driven chemical processing of the comet's interior may occur.",
"This may be caused by heating due to radioactive decay of radionuclides, such as $^{26}$ Al [150], [120].",
"Figure: Range of abundances of grain-surface complex molecules relative to water ice from ourmodel calculations (red lines) compared with those derived from observations of cometary comae(blue asterisks and lines) and our initial dark cloud ice ratios (green asterisks).The comet data is from and .The single points and upper limits for the comet ratios represent data derived from observations ofcomet Hale-Bopp ."
],
[
"Implications for astrobiology",
"One of the most complex molecules detected to date is aminoacetonitrile, NH2CH2CN, which was observed towards the hot core in the massive star-forming region, Sgr B2(N), with a fractional abundance $\\sim $ 10$^{-9}$ [11].",
"NH2CH2CN has been postulated as a potential precursor to the simplest amino acid, glycine, NH2CH2COOH.",
"In turn, amino acids are the building blocks of proteins, considered a key component for the commencement of life.",
"Multiple routes to the formation of glycine (and other simple amino acids) under interstellar conditions have been proposed including via Strecker synthesis [112], UV-irradiated ice mantles [16], [89], and gas-phase chemistry [18].",
"Recently, [52] investigated the formation of glycine in hot cores via grain-surface radical-radical reactions, i.e., an extension to the reaction scheme used here, incorporating the ice chemistry proposed in [162] to describe the formation of glycine in UV-irradiated ices.",
"[52] calculated a peak fractional abundance for gas-phase glycine $\\sim $ 10$^{-10}$ – 10$^{-8}$ with the molecule returned to the gas phase at temperatures $\\gtrsim $ 200 K. He also included gas-phase formation of glycine [18] and determined it to have a negligible effect on the resulting abundances.",
"The detection of glycine is considered one of the holy grails of astrochemistry and astrobiology; however, searches for gas-phase glycine, thus far, have been unsuccessful [135].",
"The predictions from [52] are consistent with upper limits derived from these observations.",
"He proposes that due to the high binding energy of glycine, the emission from hot cores is expected to be very compact, and thus, an ideal target for detection with ALMA.",
"Certainly, a similar grain-surface formation route to s-NH2CH2CN and thus, s-NH2CH2COOH, may be possible under protoplanetary disk conditions and should be explored in future models, particularly considering the recent identification of glycine in a sample returned from comet 81P/Wild 2 [42] and the detection of numerous amino acids in meteorites, some of which are either very rare on Earth or, indeed, unknown in terrestrial biochemistry [43].",
"Models would help ascertain whether it is possible for simple amino acids to form on dust grains in the Sun's protoplanetary disk and become incorporated into comets and asteroids.",
"Such models could also provide further evidence for the delivery of prebiotic molecules to Earth via asteroid and/or cometary impact, rather than forming `in situ' early in the Earth's evolution.",
"However, based on our molecular line emission calculations (see Sect.",
"REF ), even if such prebiotic molecules were present in quantities similar to that expected in hot cores, the detection of the gas-phase form of these species in protoplanetary disks would be incredibly challenging, if not impossible, even with ALMA full capabilities."
],
[
"Summary",
"In this work, we have presented the results of exploratory models investigating the synthesis of COMs in a protoplanetary disk around a typical T Tauri star.",
"We used an extensive chemical network, typically adopted in chemical models of hot cores, which includes gas-phase chemistry, gas-grain interactions (freeze out and desorption), grain-surface chemistry, and the irradiation of ice mantle material.",
"We summarise the main results of this work below.",
"COMs can form efficiently on the grain mantle under the physical conditions in the disk midplane via grain-surface chemical reactions, reaching peak fractional abundances, $\\sim $ 10$^{-6}$ to $\\sim $ 10$^{-4}$ that of the H nuclei number density.",
"Gas-phase COMs are released to the gas phase via non-thermal desorption, with photodesorption via external photons being the most important process for increasing the abundances in the `molecular layer', and cosmic-ray-induced desorption being most important in the disk midplane.",
"This mechanism is different to that in hot core models which require a `warm up' phase in which the temperature increases from 10 K to $\\gtrsim $ 100 K over a time scale of $\\approx $ 10$^{5}$ years [48].",
"Most gas-phase COMs reside in a narrow region within the `molecular layer' ranging in peak fractional abundance from $\\sim $ 10$^{-12}$ (e.g., CH3COCH3) to $\\sim $ 10$^{-7}$ (e.g., HCOOH).",
"Generally, the more complex the species, the lower the peak gas-phase fractional abundance and column density.",
"H2CO, HC3N, and CH3CN, are exceptions to the above statement.",
"These species have gas-phase and grain-surface routes to formation and so are relatively abundant throughout the molecular layer and upper disk.",
"Including the irradiation of ice mantle material allows further complexity to build in the ice mantle through the generation of grain-surface radicals which are available for further molecular synthesis.",
"This process increases the abundances of more complex molecules in the disk midplane, further enhancing the abundance of several COMs, e.g., CH3CHO, C2H5OH, CH3OCH3, CH3COCH3, and CH3COOH.",
"However, this increase in grain-surface abundances does not necessarily translate to an `observable' abundance in the gas phase.",
"Reactive desorption provides an additional means for molecules to return to the gas phase in the disk midplane, e.g., CH3CHO, C2H5OH, and CH3OCH3.",
"The calculated column densities for H2CO and CH3OH are consistent with values and upper limits derived from observations.",
"There is reasonably good agreement between the majority of our calculated line intensities for H2CO and those observed towards nearby T Tauri stars.",
"For the hotter stars, we get better agreement with the higher frequency transitions than the lower frequency transitions.",
"For the cooler stars, we also get reasonable agreement with the lower frequency transitions.",
"There is poor agreement with observed HC3N line intensities towards LkCa 15 and GO Tau, which is attributed to our lower calculated column density for this species.",
"This disagreement may also be due to the particularites of these two sources: LkCa 15 is now considered a transition disk with a large gap in continuum emission within $\\approx $ 50 AU and GO Tau hosts a particularly large, massive molecular disk ($R_\\mathrm {CO}$ $\\approx $ 900 AU).",
"The predicted line intensities for methanol line emission lie well below the upper limits determined towards all sources.",
"We suggest strong lines of methanol around $\\approx $ 300 GHz (and higher frequencies) are excellent candidates for observation in nearby protoplanetary disks using ALMA ( for details see Sect.",
"REF and Fig.",
"REF ).",
"The molecular line emission calculations put interesting constraints on the observability of COMs in protoplanetary disks.",
"The calculations suggest that the detection of more complex species, especially those typically observed in hot cores, e.g., CH3CN and HCOOCH3, may prove challenging, even with ALMA `Full Science' operations.",
"Detections of COMs of prebiotic significance, e.g., NH2CH2CN and NH2CH2COOH, in protoplanetary disks, may provide additional challenges, remaining beyond the reach of ALMA.",
"Our grain-surface fractional abundances (relative to water ice) for the outer disk ($R$ $\\gtrsim $ 20 AU) are consistent with abundances derived for comets, suggesting a grain-surface route to the formation of COMs observed in cometary comae.",
"This lends support to the idea that comets formed via the coagulation and growth of icy grains in the Sun's natal protoplanetary disk.",
"Two of the most complex molecules observed in disks, H2CO and HC3N, can both be efficiently synthesised by gas-phase chemistry alone and, thus, are not currently validations of the efficacy of grain-surface chemistry in protoplanetary disks.",
"Observations of molecules which can only be efficiently formed on the grain, e.g., CH3OH, are required in order to determine the degree to which grain-surface chemistry contributes to the chemical content in protoplanetary disks.",
"Methanol is an important molecule in that it is essentially the next `rung' on the `ladder' of molecular complexity following H2CO.",
"It is also a parent molecule of many more complex species.",
"The calculations suggest that the expected line intensities of transitions of methanol lie well below current observational upper limits.",
"Utilising ALMA, with its unprecedented sensitivity and spectral and spatial resolution, and performing a deep search for the strongest transitions of methanol that fall with the observing bands, would confirm whether grain-surface chemistry is an important mechanism in protoplanetary disks.",
"The authors thank an anonymous referee for his or her useful comments which greatly improved the discussion in the paper.",
"C.W.",
"acknowledges support from the European Union A-ERC grant 291141 CHEMPLAN and financial support (via a Veni award) from the Netherlands Organisation for Scientific Research (NWO).",
"Astrophysics at QUB is supported by a grant from the STFC.",
"E. H. thanks the National Science Foundation for support of his program in astrochemistry.",
"He also acknowledges support from the NASA Exobiology and Evolutionary program through a subcontract from Rensselaer Polytechnic Institute.",
"H. N. acknowledges the Grant-in-Aid for Scientific Research 23103005 and 25400229.",
"She also acknowledges support from the Astrobiology Project of the CNSI, NINS (Grant Number AB251002, AB251012).",
"S.L.W.W.",
"and J.C.L.",
"acknowledge support from S.L.W.W.",
"'s startup funds provided by Emory University.",
"The authors thank K. Öberg and D. Semenov for providing observational data and model calculations for inclusion in this paper."
],
[
"On the assumed parameters",
"In this work, we have used a fixed set of parameters for the calculation of the thermal grain-surface reaction rates and desorption rates.",
"Two parameters which may have a strong influence on the grain-surface abundances and subsequent gas-grain balance are the diffusion barriers between surface sites, $E_{b}$ , and the probability for reactive desorption, $P_{rd}$ .",
"We assume the grain-surface diffusion barrier for each species is proportional to its binding (desorption) energy to the grain surface, $E_{D}$ .",
"Here, we assume an optimistic value, $E_{b}$ /$E_{D}$ = 0.3.",
"This value allows the reasonably efficient diffusion of radicals within the grain mantle when $T$ $\\gtrsim $ 15 K, which helps to build chemical complexity in the outer disk.",
"[146] recently explored the effects of the value assumed for $E_{b}$ /$E_{D}$ in a macroscopic Monte Carlo model of hot core chemistry, using a `layer-by-layer' approach to calculate the grain mantle composition.",
"They explored a range of values for $E_{b}$ /$E_{D}$ : 0.3 [58], 0.5 [48], and 0.77 [125].",
"They concluded models using the intermediate value, $E_{b}$ /$E_{D}$ = 0.5, produced ice compositions in better agreement with observations; however, models with $E_{b}$ /$E_{D}$ = 0.3 also gave reasonable agreement for the warm-up phase.",
"In addition, we assume a conservative value for the probability for reactive desorption, $P_{rd}$ = 0.01.",
"This value is that constrained in investigations into the efficacy of reactive desorption in dark cloud chemical models [49].",
"Recently, reactive desorption has been postulated as a potential mechanism for the release of precursor COMs (e.g., H2CO and CH3OH) in cold, dark clouds where they eventually form larger complex organic molecules (e.g., HCOOCH3 and CH3OCH3) in the gas phase via radiative association [147].",
"In addition, as discussed in the main body of the paper, recent experiments suggest that reactive desorption is particularly efficient for the reformation of doubly-deuterated water (D2O) and O2 via the surface reactions, s-D + s-OD and s-O + s-O, with efficiencies, $>$ 90 % and $\\approx $ 60 %, respectively [34].",
"In this appendix, we present results from additional exploratory models to investigate the effects of a higher diffusion barrier and a higher probability for reactive desorption.",
"In Table REF , we list the parameters adopted in four models.",
"In Model A, we assume $E_{b}$ /$E_{D}$ = 0.3 and $P_{rd}$ = 0.01.",
"This model corresponds to our fiducial model, the full results for which are discussed in the main section of this paper.",
"In Model B, we adopt a higher diffusion barrier, $E_{b}$ /$E_{D}$ = 0.5, and in Model C we adopt a higher probability for reactive desorption, $P_{rd}$ = 0.1.",
"We present results from an additional model, Model D, in which we have adopted the higher values for both parameters.",
"In Figures REF and REF we present the fractional abundances of gas-phase and grain-surface COMs, respectively, relative to the H nuclei number density, as a function of disk radius, $R$ .",
"We show and discuss results for the disk midplane only[13].",
"This is the region where grain-surface COMs form most efficiently in our fiducial disk model.",
"In Sects.",
"REF and REF , we discuss the effects and importance of the values adopted for the diffusion barrier and the probability for reactive desorption, respectively.",
"[13]The data used to plot Figs.",
"REF and REF are available upon request.",
"Table: Model parametersFigure: Fractional abundance (with respect to H nuclei number density) of gas-phase moleculesas a function of radius, RR, along the disk midplane.The differences between Models A to D are described in the text and listed inTable .Figure: Same as Fig.",
"for grain-surface species."
],
[
"Diffusion barrier",
"In Figs.",
"REF and REF , the red lines represent results from Model A (our fiducial model) in which we have adopted $E_{b}$ /$E_{D}$ = 0.3 and the green lines represent results from Model B in which we have used $E_{b}$ /$E_{D}$ = 0.5.",
"There are only minor differences (less than an order of magnitude) between the gas-phase and grain-surface abundances of H2CO, CH3OH, HC3N, and CH3CCH calculated using Model A and Model B.",
"These are species which can form either in the gas phase or which depend on hydrogenation reactions for their formation.",
"Lower abundances are calculated for Model B relative to Model A in the outer disk for those grain-surface species which require radical-radical association to enhance their abundance above that achieved in dark clouds.",
"In Model B, the fractional abundances attained in the very outer disk midplane, $\\approx $ 300 AU, for most species are comparable to those achieved under dark cloud conditions (see Table REF and Fig.",
"REF ).",
"We see an enhancement in the fractional abundances of s-C2H5OH, s-CH3OCH3, and s-HCOOCH3 relative to their initial abundances.",
"This is indicative that radical-radical grain-surface chemistry still operates in the outer disk midplane in Model B, albeit significantly slower relative to Model A.",
"Moving inwards along the midplane, the temperature and density both increase and there is a corresponding increase in the fractional abundances of most COMs.",
"This increase is mirrored in the fractional abundances attained by the analogous gas-phase species.",
"s-CH3CHO, s-C2H5OH, s-CH3OCH3, and s-CH3COCH3 exhibit an interesting behaviour between $\\approx $ 50 and 150 AU.",
"Within this region, the fractional abundances of all four species show a `dip' or minimum around 70 AU.",
"The dust temperature within $\\approx $ 150 AU in the midplane is $\\gtrsim $ 22 K allowing thermal desorption of volatile molecules, for example, s-CO ($E_D$ = 1150 K).",
"The species showing this minimum all form via grain-surface reactions involving the relatively volatile methyl radical, s-CH3 ($E_D$ = 1180 K).",
"In Model B, the grain-surface reaction rates are not sufficiently fast to compete with the thermal desorption of s-CH3 until the dust temperature increases to a value which allows grain-surface thermal chemistry to operate efficiently.",
"The mobility of grain-surface species is dependent upon $\\exp (-E_{b}/T)$ = $\\exp (-\\chi E_{D})$ , where $\\chi $ = $(E_{b}/E_{D})/T$ .",
"In Model A, there is efficient mobility of grain-surface radicals and thus efficient grain-surface synthesis when $\\chi $ $\\gtrsim $ 0.02.",
"In Model B this value of $\\chi $ (a measure of the degree of mobility) is attained when T $\\gtrsim $ 28 K which is reached within $\\approx $ 70 AU in the disk midplane.",
"In Model B, relatively high fractional abundances of grain-surface COMs are attained that are comparable with those from Model A.",
"However, the radial range over which they reach their peak fractional abundance is restricted to regions where $T$ $\\gtrsim $ 28 K and where the temperature is also lower than the desorption temperature of each molecule.",
"Results from Model A and Model B are similar within $\\approx $ 50 AU of the central star."
],
[
"Reactive desorption",
"In Figs.",
"REF and REF , the blue lines represent results from Model C in which we have adopted a higher probability for reactive desorption, $P_\\mathrm {rd}$ = 0.1.",
"The increased reactive desorption has little effect on the grain-surface abundances.",
"However, in the outer disk midplane, there is around an order of magnitude enhancement in the gas-phase fractional abundances when using the higher probability.",
"In the inner disk, thermal desorption is the most important mechanism for releasing grain mantle material back into the gas phase so that the results from all models converge at small radii.",
"There is a similar effect seen when comparing results from Model B and Model D, in which the higher diffusion barrier, $E_b/E_D$ = 0.5, has been adopted.",
"Results for Model B and Model D are represented by the green lines and yellow lines, respectively.",
"Again, there is little change in the grain-surface species when the probability for reactive desorption is increased to 0.1.",
"For the gas-phase abundances, in Model D there is the familiar `order-of-magnitude' enhancement when using $P_\\mathrm {rd}$ = 0.1.",
"Note that for the most `optimistic' model, Model C, the gas-phase COMs reach peak fractional abundances between 10$^{-13}$ and 10$^{-9}$ in the outer disk midplane ($R$ $\\gtrsim $ 10 AU).",
"This enhancement in fractional abundance will increase the column densities of COMs; however, the main contribution to the COM gas-phase column density remains the photodesorbed material in the molecular layer.",
"cccccccccccccccc Column densities (cm$^{-2}$ ) of gas-phase molecules as a function of radius.",
"Radius (AU) H2CO CH3OH HCOOH HC3N CH3CN CH3CCH CH3CHO NH2CHO CH3NH2 C2H5OH CH3OCH3 HCOOCH3 CH3COCH3 HOCH2CHO CH3COOH continued.",
"Radius (AU) H2CO CH3OH HCOOH HC3N CH3CN CH3CCH CH3CHO NH2CHO CH3NH2 C2H5OH CH3OCH3 HCOOCH3 CH3COCH3 HOCH2CHO CH3COOH 1.07(0) 7.8(12) 1.9(16) 7.4(14) 8.0(16) 8.2(16) 2.0(18) 5.9(14) 6.6(15) 1.2(17) 2.3(13) 1.4(15) 8.0(15) 2.5(09) 4.8(07) 6.6(10) 1.15(0) 5.8(12) 1.0(16) 1.8(14) 4.7(16) 4.5(16) 1.3(18) 6.0(14) 3.3(15) 6.7(16) 3.2(13) 7.8(14) 3.8(15) 2.2(09) 6.9(05) 5.4(10) 1.23(0) 6.7(12) 2.4(14) 6.0(12) 1.0(16) 8.8(15) 4.5(17) 5.5(14) 1.1(15) 3.3(15) 3.6(13) 1.2(13) 4.7(13) 1.4(08) 1.2(05) 4.1(09) 1.32(0) 6.1(12) 1.1(14) 3.0(12) 3.0(15) 7.4(14) 9.2(16) 5.0(14) 8.2(14) 5.3(13) 3.4(13) 4.3(08) 1.6(11) 1.5(07) 5.7(04) 5.8(08) 1.42(0) 7.3(12) 1.5(14) 1.4(12) 1.1(15) 1.7(14) 3.7(16) 2.0(14) 7.0(14) 3.3(13) 1.3(13) 6.1(08) 9.7(10) 8.2(06) 9.7(04) 4.6(08) 1.52(0) 1.8(15) 1.4(14) 9.3(11) 5.0(14) 1.5(13) 2.1(16) 1.8(14) 8.8(14) 2.6(13) 1.1(13) 4.3(09) 1.3(12) 5.0(11) 1.1(06) 4.0(11) 1.63(0) 3.2(15) 8.2(13) 2.8(12) 2.8(14) 1.6(13) 1.9(16) 1.8(14) 2.1(14) 2.0(13) 8.2(12) 2.3(09) 9.7(11) 1.7(12) 1.3(07) 3.2(12) 1.75(0) 2.3(15) 4.8(14) 8.3(11) 3.3(15) 2.5(15) 3.3(16) 3.8(14) 6.1(13) 3.0(14) 3.4(12) 1.9(11) 3.3(12) 6.7(11) 1.6(04) 2.0(12) 1.87(0) 1.5(15) 4.4(14) 2.1(11) 3.5(15) 2.3(15) 4.0(16) 4.1(14) 1.7(12) 1.4(14) 3.2(11) 4.0(11) 4.4(12) 9.6(10) 2.0(04) 2.2(12) 2.01(0) 2.1(15) 4.0(14) 3.3(09) 2.9(15) 1.1(15) 2.4(17) 2.9(14) 5.9(09) 2.1(14) 4.4(09) 7.8(11) 2.4(12) 1.7(10) 2.5(04) 1.6(10) 2.15(0) 1.5(15) 4.1(12) 1.8(09) 1.0(15) 8.3(14) 1.5(17) 9.3(13) 2.2(07) 4.4(13) 1.2(07) 6.2(11) 1.5(11) 3.1(09) 3.8(04) 3.1(07) 2.31(0) 9.0(14) 2.9(10) 2.3(09) 2.5(14) 2.4(13) 1.2(17) 3.0(13) 6.6(06) 5.3(11) 3.1(04) 7.6(11) 3.3(10) 2.9(09) 3.9(04) 2.2(05) 2.48(0) 1.3(14) 6.0(08) 2.7(09) 2.3(13) 5.4(12) 1.2(16) 1.1(13) 1.6(06) 1.1(10) 1.6(04) 1.2(11) 9.9(08) 4.6(08) 6.2(03) 2.3(05) 2.66(0) 8.1(13) 1.6(08) 3.0(09) 9.6(12) 3.9(12) 2.4(15) 2.2(12) 1.4(06) 1.3(09) 2.3(04) 3.4(11) 2.4(08) 1.0(09) 3.9(03) 4.1(05) 2.85(0) 8.4(13) 1.1(08) 3.0(09) 7.0(12) 3.8(12) 6.0(14) 5.6(11) 1.7(06) 2.3(08) 3.0(04) 1.2(12) 6.2(07) 1.2(09) 3.0(03) 7.6(05) 3.05(0) 6.9(13) 1.4(08) 4.2(09) 6.8(12) 4.8(12) 7.3(14) 5.7(11) 1.8(06) 2.8(08) 2.9(04) 5.9(11) 3.0(07) 7.8(07) 2.4(03) 7.1(05) 3.28(0) 1.5(14) 2.4(08) 8.6(09) 8.7(12) 7.3(12) 8.9(14) 9.2(11) 3.9(06) 4.9(08) 4.0(04) 1.4(12) 5.7(08) 7.4(09) 1.9(04) 8.6(05) 3.51(0) 5.8(13) 1.8(08) 5.2(09) 8.5(12) 4.7(12) 1.2(15) 7.5(11) 2.8(06) 4.6(08) 2.8(04) 7.9(10) 3.3(07) 8.0(06) 3.1(03) 5.9(05) 3.77(0) 5.6(13) 2.3(08) 7.7(09) 8.6(12) 5.5(12) 1.3(15) 8.2(11) 3.3(06) 5.5(08) 2.9(04) 3.1(10) 3.6(07) 4.5(06) 3.5(03) 5.6(05) 4.04(0) 5.4(13) 2.6(08) 9.0(09) 8.9(12) 6.9(12) 1.5(15) 8.8(11) 3.8(06) 6.4(08) 2.9(04) 9.5(09) 4.1(07) 2.4(06) 3.9(03) 5.3(05) 4.33(0) 5.4(13) 2.8(08) 1.0(10) 8.8(12) 6.1(12) 1.7(15) 9.6(11) 4.2(06) 7.0(08) 3.2(04) 3.6(09) 4.7(07) 1.5(06) 4.4(03) 5.1(05) 4.64(0) 6.1(13) 2.6(08) 1.0(10) 7.4(12) 6.1(12) 9.3(14) 4.4(11) 8.7(06) 6.4(08) 3.5(04) 1.3(10) 2.9(07) 1.7(06) 5.0(03) 6.8(05) 4.98(0) 7.2(13) 6.2(08) 9.4(09) 5.2(12) 5.2(12) 1.5(14) 6.7(10) 2.4(07) 5.4(08) 1.4(06) 3.2(11) 9.6(06) 8.2(06) 4.4(03) 2.2(06) 5.34(0) 7.8(13) 5.2(09) 8.5(09) 5.7(12) 5.5(12) 1.4(13) 1.8(10) 1.7(09) 1.4(09) 2.0(07) 2.7(11) 6.6(06) 4.8(08) 1.1(04) 6.5(06) 5.72(0) 6.3(13) 1.1(10) 9.6(09) 4.5(12) 6.0(12) 1.4(12) 8.1(09) 3.5(08) 2.7(09) 1.9(08) 7.3(10) 6.7(07) 3.7(09) 7.6(04) 1.2(07) 6.14(0) 5.7(13) 1.5(10) 9.7(09) 5.5(12) 5.0(12) 6.7(11) 7.4(09) 3.0(10) 3.5(09) 8.2(07) 3.4(10) 1.1(08) 2.7(09) 5.1(05) 2.5(07) 6.58(0) 3.3(13) 6.2(09) 1.1(10) 4.2(12) 6.0(12) 4.5(11) 6.2(09) 6.5(09) 1.6(09) 2.9(07) 1.5(10) 9.9(07) 1.7(09) 1.4(06) 2.8(07) 7.06(0) 6.2(12) 3.3(08) 1.3(10) 2.8(12) 5.1(12) 3.4(11) 5.1(09) 2.1(08) 4.4(08) 5.4(05) 5.7(09) 5.4(07) 9.4(08) 2.7(06) 1.7(07) 7.57(0) 3.6(12) 3.5(08) 1.5(10) 2.6(12) 5.8(12) 3.2(11) 4.8(09) 2.4(08) 5.1(08) 7.0(05) 2.3(09) 2.6(07) 4.6(08) 3.3(06) 1.4(07) 8.11(0) 3.2(12) 5.5(08) 2.0(10) 2.4(12) 5.1(12) 3.3(11) 5.0(09) 3.9(08) 5.5(08) 1.1(06) 8.1(08) 1.9(07) 1.7(08) 6.3(06) 1.5(07) 8.67(0) 3.1(12) 5.4(08) 3.8(10) 2.3(12) 5.6(12) 3.5(11) 6.3(09) 6.7(08) 5.9(08) 1.7(06) 2.2(08) 2.2(07) 4.7(07) 1.2(07) 1.7(07) 9.33(0) 3.4(12) 9.3(08) 5.9(10) 2.1(12) 5.0(12) 4.0(11) 8.4(09) 1.2(09) 6.4(08) 2.6(06) 6.0(07) 3.8(07) 1.4(07) 2.4(07) 1.9(07) 1.00(1) 3.7(12) 1.0(09) 8.1(10) 2.0(12) 5.5(12) 4.5(11) 1.1(10) 1.7(09) 7.2(08) 3.8(06) 2.2(07) 5.8(07) 5.8(06) 4.2(07) 2.1(07) 1.07(1) 4.2(12) 1.2(09) 1.5(11) 1.9(12) 5.1(12) 4.9(11) 1.5(10) 2.7(09) 8.7(08) 5.7(06) 1.1(07) 8.9(07) 2.9(06) 7.6(07) 3.6(07) 1.15(1) 4.8(12) 1.4(10) 3.1(11) 1.8(12) 5.3(12) 5.5(11) 2.7(10) 4.2(09) 1.0(09) 9.6(06) 1.1(07) 2.2(08) 2.1(06) 1.3(08) 6.3(07) 1.23(1) 6.0(12) 5.2(09) 3.7(11) 1.7(12) 4.9(12) 5.7(11) 6.8(10) 6.2(09) 1.3(09) 1.5(07) 1.5(07) 3.0(08) 3.1(06) 2.2(08) 1.3(08) 1.32(1) 9.0(12) 2.6(10) 4.4(11) 1.5(12) 5.0(12) 6.1(11) 7.8(10) 9.1(09) 1.6(09) 2.7(07) 3.0(07) 1.2(09) 8.2(06) 4.6(08) 2.5(08) 1.42(1) 1.6(13) 4.5(10) 5.1(11) 1.4(12) 4.7(12) 6.3(11) 5.6(10) 1.2(10) 2.7(09) 6.5(07) 7.9(07) 3.6(09) 2.4(07) 9.4(08) 3.6(08) 1.52(1) 2.0(13) 2.4(10) 5.6(11) 1.3(12) 4.8(12) 5.8(11) 3.3(10) 1.3(10) 4.2(09) 2.0(08) 1.4(08) 6.4(09) 5.5(07) 1.1(10) 6.1(08) 1.63(1) 2.2(13) 7.2(10) 5.9(11) 1.3(12) 4.6(12) 5.6(11) 1.6(10) 1.3(10) 6.5(09) 2.4(08) 4.0(08) 1.1(10) 1.1(08) 5.7(09) 8.0(08) 1.75(1) 3.0(13) 9.0(10) 6.5(11) 1.2(12) 4.4(12) 5.6(11) 1.1(10) 1.4(10) 1.0(10) 1.0(09) 1.0(09) 4.1(10) 1.9(08) 1.6(10) 9.9(08) 1.87(1) 9.0(13) 5.3(10) 6.7(11) 1.1(12) 4.2(12) 5.6(11) 8.6(09) 1.4(10) 1.8(10) 1.7(09) 1.4(09) 2.8(10) 2.8(08) 4.1(10) 1.5(09) 2.01(1) 2.2(14) 1.4(11) 6.6(11) 1.0(12) 4.1(12) 5.4(11) 1.8(10) 1.3(10) 2.8(10) 4.5(09) 7.5(09) 5.5(10) 3.9(08) 1.5(10) 4.0(09) 2.15(1) 5.1(14) 7.2(10) 6.9(11) 9.5(11) 4.0(12) 4.7(11) 1.2(10) 1.2(10) 4.1(10) 8.7(09) 8.1(09) 7.5(10) 6.6(08) 2.6(10) 5.6(09) 2.31(1) 7.3(14) 8.2(10) 8.1(11) 8.8(11) 3.9(12) 4.3(11) 1.8(10) 1.4(10) 5.4(10) 1.3(10) 1.6(10) 4.5(10) 8.2(08) 2.4(10) 9.3(09) 2.48(1) 5.3(14) 1.1(11) 6.1(11) 8.2(11) 3.7(12) 5.0(11) 2.0(10) 2.0(10) 5.8(10) 1.5(10) 2.1(10) 3.2(10) 1.1(09) 1.0(10) 7.9(09) 2.66(1) 2.9(14) 1.5(11) 7.5(11) 7.8(11) 3.5(12) 6.1(11) 1.7(10) 2.8(10) 6.2(10) 2.1(10) 2.2(10) 5.0(10) 1.4(09) 1.4(10) 8.4(09) 2.85(1) 1.3(14) 2.0(11) 7.1(11) 7.3(11) 3.2(12) 7.2(11) 2.4(10) 4.2(10) 6.5(10) 2.2(10) 2.6(10) 3.7(10) 1.3(09) 1.1(10) 1.0(10) 3.05(1) 5.1(13) 2.2(11) 7.5(11) 6.9(11) 2.9(12) 8.2(11) 2.0(10) 5.6(10) 6.6(10) 1.9(10) 2.2(10) 4.6(10) 1.5(09) 1.2(10) 6.5(09) 3.28(1) 1.8(13) 2.6(11) 7.9(11) 6.5(11) 2.5(12) 8.9(11) 3.5(10) 7.2(10) 6.9(10) 2.3(10) 3.4(10) 5.4(10) 1.2(09) 1.2(10) 5.7(09) 3.51(1) 5.4(12) 3.1(11) 8.7(11) 6.1(11) 2.1(12) 9.6(11) 3.7(10) 8.1(10) 7.2(10) 1.9(10) 2.7(10) 6.7(10) 1.6(09) 1.4(10) 3.8(09) 3.77(1) 2.6(12) 3.7(11) 9.0(11) 5.7(11) 1.8(12) 9.9(11) 4.2(10) 9.1(10) 7.9(10) 2.4(10) 3.7(10) 7.0(10) 2.3(09) 1.3(10) 2.7(09) 4.04(1) 2.1(12) 4.3(11) 9.9(11) 5.4(11) 1.5(12) 1.0(12) 6.1(10) 9.5(10) 8.7(10) 2.6(10) 4.0(10) 8.3(10) 3.1(09) 1.5(10) 3.5(09) 4.33(1) 1.9(12) 5.4(11) 1.0(12) 4.9(11) 1.3(12) 1.1(12) 6.9(10) 1.0(11) 9.5(10) 2.9(10) 4.7(10) 8.7(10) 4.6(09) 1.5(10) 2.9(09) 4.64(1) 1.8(12) 6.5(11) 1.2(12) 4.5(11) 1.2(12) 1.1(12) 7.4(10) 1.0(11) 1.0(11) 2.8(10) 4.1(10) 9.3(10) 6.6(09) 1.6(10) 3.3(09) 4.98(1) 1.8(12) 8.3(11) 1.4(12) 4.5(11) 1.1(12) 1.2(12) 9.2(10) 1.1(11) 1.1(11) 3.6(10) 5.6(10) 1.0(11) 8.5(09) 1.8(10) 3.6(09) 5.34(1) 1.7(12) 9.9(11) 1.7(12) 4.2(11) 9.9(11) 1.2(12) 1.1(11) 1.1(11) 1.1(11) 3.7(10) 5.5(10) 1.1(11) 1.1(10) 1.9(10) 3.6(09) 5.72(1) 1.6(12) 1.3(12) 2.3(12) 4.0(11) 9.5(11) 1.2(12) 1.3(11) 1.2(11) 1.2(11) 3.9(10) 5.8(10) 1.1(11) 1.4(10) 2.0(10) 4.5(09) 6.14(1) 1.6(12) 1.7(12) 2.9(12) 3.6(11) 9.2(11) 1.3(12) 1.4(11) 1.3(11) 1.3(11) 4.1(10) 6.2(10) 1.1(11) 1.7(10) 2.0(10) 5.0(09) 6.58(1) 1.5(12) 2.0(12) 4.0(12) 3.0(11) 8.8(11) 1.2(12) 1.7(11) 1.6(11) 1.4(11) 4.3(10) 6.6(10) 1.1(11) 2.2(10) 2.1(10) 5.4(09) 7.06(1) 1.4(12) 2.5(12) 5.4(12) 2.4(11) 8.5(11) 1.1(12) 2.2(11) 1.9(11) 1.6(11) 5.0(10) 7.4(10) 1.1(11) 3.0(10) 2.4(10) 6.4(09) 7.57(1) 1.4(12) 3.1(12) 6.6(12) 2.2(11) 8.4(11) 1.0(12) 3.2(11) 2.2(11) 1.9(11) 6.1(10) 8.7(10) 1.1(11) 4.4(10) 2.6(10) 7.8(09) 8.11(1) 1.4(12) 3.6(12) 7.5(12) 2.2(11) 8.1(11) 9.7(11) 4.6(11) 2.5(11) 2.3(11) 7.6(10) 1.0(11) 1.1(11) 6.1(10) 2.9(10) 8.8(09) 8.70(1) 1.4(12) 4.2(12) 8.0(12) 2.2(11) 7.5(11) 9.8(11) 6.9(11) 3.0(11) 3.3(11) 1.2(11) 1.2(11) 1.1(11) 1.0(11) 3.2(10) 8.2(09) 9.33(1) 1.5(12) 5.0(12) 8.5(12) 2.2(11) 7.1(11) 1.0(12) 7.7(11) 3.8(11) 5.8(11) 1.4(11) 1.3(11) 1.1(11) 1.6(11) 3.7(10) 7.2(09) 1.00(2) 1.5(12) 5.8(12) 9.1(12) 2.1(11) 6.9(11) 1.1(12) 7.2(11) 5.1(11) 8.4(11) 1.4(11) 1.4(11) 1.3(11) 1.7(11) 4.3(10) 6.6(09) 1.07(2) 1.5(12) 6.8(12) 9.8(12) 2.0(11) 6.7(11) 1.3(12) 6.6(11) 6.8(11) 9.7(11) 1.3(11) 1.4(11) 1.4(11) 1.4(11) 4.8(10) 6.0(09) 1.15(2) 1.6(12) 7.8(12) 1.1(13) 1.8(11) 7.0(11) 1.5(12) 6.3(11) 8.4(11) 9.8(11) 1.2(11) 1.3(11) 1.5(11) 1.3(11) 5.3(10) 5.7(09) 1.23(2) 1.6(12) 8.5(12) 1.5(13) 8.4(10) 5.1(11) 1.6(12) 6.1(11) 9.7(11) 8.1(11) 1.1(11) 1.3(11) 1.6(11) 1.1(11) 5.5(10) 5.4(09) 1.32(2) 1.6(12) 8.9(12) 1.3(13) 2.9(10) 2.7(11) 1.8(12) 5.1(11) 6.4(11) 6.9(11) 9.5(10) 1.2(11) 1.7(11) 8.6(10) 5.4(10) 4.6(09) 1.42(2) 1.8(12) 9.7(12) 1.4(13) 2.5(10) 2.7(11) 1.9(12) 4.8(11) 6.1(11) 5.0(11) 8.5(10) 1.2(11) 1.7(11) 6.9(10) 5.3(10) 4.5(09) 1.52(2) 2.1(12) 1.0(13) 1.5(13) 2.5(10) 3.0(11) 2.0(12) 4.3(11) 6.5(11) 3.8(11) 7.5(10) 1.1(11) 1.7(11) 5.3(10) 5.3(10) 4.1(09) 1.63(2) 2.4(12) 1.1(13) 1.7(13) 2.7(10) 3.6(11) 2.1(12) 3.9(11) 7.3(11) 3.6(11) 7.1(10) 1.0(11) 1.8(11) 4.2(10) 5.4(10) 3.8(09) 1.75(2) 2.8(12) 1.2(13) 1.9(13) 3.3(10) 4.7(11) 2.1(12) 3.7(11) 7.9(11) 3.8(11) 6.8(10) 1.0(11) 1.8(11) 3.5(10) 5.6(10) 3.7(09) 1.87(2) 3.4(12) 1.3(13) 2.0(13) 4.5(10) 6.2(11) 2.0(12) 3.5(11) 8.8(11) 4.2(11) 6.7(10) 1.0(11) 1.9(11) 2.5(10) 5.8(10) 3.6(09) 2.01(2) 4.3(12) 1.4(13) 2.2(13) 7.4(10) 8.5(11) 1.9(12) 3.4(11) 9.7(11) 5.1(11) 6.7(10) 1.0(11) 2.0(11) 1.9(10) 6.3(10) 3.6(09) 2.15(2) 5.6(12) 1.5(13) 2.3(13) 1.4(11) 1.1(12) 1.9(12) 3.4(11) 1.0(12) 6.5(11) 6.5(10) 9.2(10) 2.2(11) 1.2(10) 7.2(10) 3.3(09) 2.31(2) 6.6(12) 1.6(13) 2.1(13) 1.9(11) 1.0(12) 1.8(12) 3.4(11) 1.0(12) 7.5(11) 5.9(10) 7.8(10) 2.4(11) 6.7(09) 8.4(10) 2.5(09) 2.48(2) 7.2(12) 1.7(13) 1.6(13) 1.7(11) 7.3(11) 1.8(12) 3.4(11) 9.2(11) 7.9(11) 6.1(10) 7.6(10) 2.7(11) 4.2(09) 9.8(10) 1.9(09) 2.66(2) 8.0(12) 1.7(13) 1.2(13) 1.6(11) 6.0(11) 1.8(12) 3.3(11) 8.1(11) 8.5(11) 6.0(10) 7.5(10) 2.9(11) 2.9(09) 1.1(11) 1.6(09) 2.85(2) 8.2(12) 1.7(13) 9.6(12) 1.3(11) 4.8(11) 1.8(12) 3.3(11) 7.4(11) 9.4(11) 6.2(10) 7.7(10) 3.2(11) 2.6(09) 1.2(11) 1.3(09) 3.05(2) 8.3(12) 1.7(13) 8.2(12) 9.8(10) 4.1(11) 1.8(12) 3.9(11) 7.0(11) 1.1(12) 6.8(10) 8.6(10) 3.5(11) 5.7(09) 1.3(11) 1.1(09) $a(b)$ represents $a\\times 10^b$ cccccccccccccccc Column densities (cm$^{-2}$ ) of grain-surface (ice) molecules as a function of radius.",
"Radius (AU) H2CO CH3OH HCOOH HC3N ceCH3CN CH3CCH CH3CHO NH2CHO CH3NH2 C2H5OH CH3OCH3 HCOOCH3 CH3COCH3 HOCH2CHO CH3COOH continued.",
"Radius (AU) H2CO CH3OH HCOOH HC3N CH3CN CH3CCH CH3CHO NH2CHO CH3NH2 C2H5OH CH3OCH3 HCOOCH3 CH3COCH3 HOCH2CHO CH3COOH 1.07(0) 4.5(01) 7.6(11) 2.2(10) 8.6(10) 9.7(10) 4.7(11) 2.4(05) 1.1(12) 7.7(11) 1.1(12) 9.8(06) 1.9(08) 1.3(01) 1.1(05) 2.1(08) 1.15(0) 1.9(01) 8.9(11) 1.7(10) 6.5(10) 8.3(10) 4.5(11) 2.9(05) 9.0(11) 9.4(11) 1.2(12) 1.1(07) 1.8(08) 1.3(01) 2.6(04) 2.2(08) 1.23(0) 1.9(01) 1.3(11) 3.6(09) 3.7(10) 4.6(10) 3.6(11) 3.1(05) 8.0(11) 1.6(11) 1.4(12) 4.3(05) 6.5(06) 2.1(00) 1.9(04) 8.6(07) 1.32(0) 1.7(01) 8.0(10) 2.2(09) 1.4(10) 4.9(09) 1.1(11) 2.7(05) 6.6(11) 4.6(09) 1.4(12) 2.1(01) 4.4(04) 0.4(00) 1.6(04) 2.7(07) 1.42(0) 3.7(01) 1.5(11) 1.4(09) 6.8(09) 1.3(09) 5.0(10) 9.8(04) 8.0(11) 3.9(09) 8.0(11) 2.8(01) 2.7(04) 0.2(00) 9.3(03) 3.4(07) 1.52(0) 9.5(04) 1.0(13) 7.5(10) 1.0(11) 1.3(09) 8.4(11) 8.0(05) 7.9(13) 1.8(11) 7.6(13) 4.6(03) 8.7(06) 2.0(05) 5.3(06) 4.0(12) 1.63(0) 7.3(05) 7.5(14) 4.4(13) 3.2(12) 3.6(11) 3.3(13) 6.7(06) 5.8(14) 2.4(12) 1.1(16) 1.4(04) 1.2(08) 3.9(06) 1.9(10) 1.3(16) 1.75(0) 4.4(06) 6.2(18) 3.0(13) 5.6(16) 1.5(17) 3.4(16) 7.9(08) 4.3(14) 8.7(17) 2.3(16) 1.7(09) 1.6(11) 2.5(07) 6.8(11) 1.2(17) 1.87(0) 1.5(07) 8.0(18) 3.3(13) 5.9(17) 9.5(17) 3.5(17) 4.0(09) 3.4(14) 1.7(18) 2.1(16) 2.3(10) 4.0(11) 5.6(07) 1.0(12) 1.4(17) 2.01(0) 9.5(07) 1.0(19) 1.1(13) 2.9(18) 1.3(18) 3.5(19) 4.9(09) 2.0(14) 2.4(18) 1.5(16) 2.6(11) 1.7(12) 2.3(08) 2.8(12) 6.8(16) 2.15(0) 1.1(08) 7.2(18) 2.5(13) 4.9(18) 1.0(18) 1.0(20) 4.2(09) 2.1(14) 1.9(18) 5.1(15) 8.5(11) 1.9(12) 4.5(08) 2.0(12) 2.4(16) 2.31(0) 1.5(08) 1.1(19) 1.0(14) 7.7(18) 1.2(18) 2.4(20) 1.5(10) 5.0(14) 3.2(18) 6.2(15) 5.2(12) 4.8(12) 2.9(09) 4.0(12) 3.4(16) 2.48(0) 6.8(07) 1.1(19) 8.3(13) 5.7(18) 9.0(17) 2.4(20) 1.5(10) 1.5(14) 3.3(18) 7.0(15) 4.2(12) 1.4(12) 2.1(09) 1.2(12) 4.2(16) 2.66(0) 1.1(08) 1.1(19) 2.3(14) 4.6(18) 7.6(17) 2.3(20) 8.4(09) 4.4(14) 3.4(18) 8.7(15) 8.2(13) 3.3(12) 3.2(10) 3.5(12) 6.9(16) 2.85(0) 2.3(08) 1.0(19) 5.1(14) 4.0(18) 5.5(17) 2.2(20) 6.8(09) 5.8(14) 3.3(18) 9.5(15) 1.0(15) 3.6(12) 1.3(11) 4.9(12) 1.1(17) 3.05(0) 1.3(08) 9.0(18) 3.5(14) 3.6(18) 5.0(17) 1.9(20) 4.7(09) 1.1(14) 2.7(18) 6.9(15) 3.6(14) 7.7(11) 6.0(09) 8.4(11) 7.7(16) 3.28(0) 4.0(08) 1.3(19) 8.6(14) 4.9(18) 7.4(17) 2.6(20) 7.7(09) 4.5(15) 4.2(18) 1.1(16) 9.3(14) 2.9(13) 6.2(11) 3.6(13) 1.2(17) 3.51(0) 5.2(07) 7.1(18) 1.8(14) 3.0(18) 4.2(17) 1.6(20) 2.8(09) 2.3(13) 1.9(18) 3.8(15) 2.4(13) 2.0(11) 2.8(08) 1.5(11) 3.6(16) 3.77(0) 3.9(07) 6.4(18) 1.5(14) 2.8(18) 3.9(17) 1.5(20) 2.4(09) 1.6(13) 1.7(18) 3.0(15) 7.3(12) 1.4(11) 1.2(08) 9.1(10) 2.7(16) 4.04(0) 2.8(07) 5.6(18) 1.2(14) 2.6(18) 3.4(17) 1.3(20) 2.0(09) 1.1(13) 1.4(18) 2.2(15) 1.7(12) 9.6(10) 4.9(07) 5.6(10) 1.9(16) 4.33(0) 2.1(07) 5.0(18) 9.6(13) 2.4(18) 3.0(17) 1.2(20) 1.7(09) 9.0(12) 1.2(18) 1.7(15) 4.9(11) 7.6(10) 2.4(07) 4.0(10) 1.4(16) 4.64(0) 2.5(07) 4.5(18) 1.3(14) 2.1(18) 2.6(17) 1.1(20) 1.1(09) 1.2(13) 1.1(18) 1.5(15) 3.0(12) 1.0(11) 4.3(07) 5.4(10) 1.4(16) 4.98(0) 8.7(07) 4.3(18) 3.9(14) 1.7(18) 1.7(17) 1.0(20) 7.5(08) 6.2(13) 1.0(18) 1.7(15) 3.9(14) 5.0(11) 1.1(09) 3.5(11) 3.0(16) 5.34(0) 2.7(08) 4.0(18) 1.6(15) 1.5(18) 8.7(16) 8.4(19) 9.8(08) 6.4(14) 9.8(17) 2.2(15) 2.1(15) 4.2(12) 4.7(11) 4.3(12) 4.6(16) 5.72(0) 7.4(08) 3.9(18) 3.0(16) 1.6(18) 6.3(16) 7.6(19) 2.6(09) 1.5(17) 1.1(18) 4.1(15) 5.0(15) 1.2(15) 3.3(13) 1.5(15) 5.0(16) 6.14(0) 7.6(08) 3.6(18) 1.2(17) 1.5(18) 5.8(16) 6.9(19) 4.5(09) 5.9(17) 1.1(18) 5.1(15) 6.3(15) 6.2(15) 6.0(13) 6.3(15) 4.2(16) 6.58(0) 5.4(08) 3.4(18) 3.1(17) 1.5(18) 5.8(16) 6.3(19) 8.9(09) 1.5(18) 1.2(18) 6.9(15) 9.0(15) 1.8(16) 1.1(14) 1.8(16) 3.3(16) 7.06(0) 3.1(08) 3.2(18) 5.8(17) 1.8(18) 6.4(16) 5.8(19) 2.1(10) 2.7(18) 1.3(18) 9.8(15) 1.4(16) 3.6(16) 2.1(14) 3.5(16) 2.6(16) 7.57(0) 3.1(08) 3.0(18) 7.7(17) 2.1(18) 7.9(16) 5.4(19) 5.0(10) 3.8(18) 1.4(18) 1.3(16) 2.0(16) 5.0(16) 3.3(14) 5.1(16) 2.1(16) 8.11(0) 5.1(08) 2.8(18) 9.1(17) 2.1(18) 1.0(17) 5.2(19) 1.4(11) 4.7(18) 1.4(18) 1.6(16) 2.3(16) 6.2(16) 3.8(14) 6.6(16) 1.7(16) 8.67(0) 1.2(09) 2.6(18) 1.0(18) 2.0(18) 1.2(17) 5.1(19) 5.3(11) 5.6(18) 1.5(18) 2.0(16) 2.4(16) 7.5(16) 3.7(14) 8.3(16) 1.5(16) 9.33(0) 3.1(09) 2.5(18) 1.1(18) 1.9(18) 1.3(17) 5.1(19) 2.0(12) 6.6(18) 1.7(18) 2.5(16) 2.8(16) 8.9(16) 4.6(14) 1.0(17) 1.5(16) 1.00(1) 6.7(09) 2.3(18) 1.1(18) 1.7(18) 1.2(17) 4.7(19) 5.9(12) 6.9(18) 1.8(18) 2.9(16) 3.1(16) 9.5(16) 6.1(14) 1.1(17) 1.5(16) 1.07(1) 1.7(10) 2.1(18) 1.1(18) 1.5(18) 1.1(17) 4.5(19) 2.1(13) 7.2(18) 2.0(18) 3.4(16) 4.0(16) 1.1(17) 1.0(15) 1.3(17) 1.7(16) 1.15(1) 4.8(10) 2.0(18) 1.1(18) 1.3(18) 1.0(17) 4.1(19) 1.6(14) 7.4(18) 2.2(18) 4.0(16) 5.3(16) 1.2(17) 2.8(15) 1.4(17) 2.3(16) 1.23(1) 1.6(11) 1.9(18) 1.1(18) 1.0(18) 6.9(16) 3.8(19) 1.7(15) 7.6(18) 2.6(18) 5.0(16) 7.6(16) 1.5(17) 9.9(15) 1.6(17) 5.5(16) 1.32(1) 7.8(11) 1.8(18) 1.0(18) 7.1(17) 4.7(16) 3.3(19) 6.8(15) 7.7(18) 3.2(18) 6.4(16) 1.2(17) 1.9(17) 3.2(16) 1.8(17) 1.1(17) 1.42(1) 5.0(12) 1.8(18) 1.0(18) 5.1(17) 4.5(16) 2.8(19) 1.9(16) 7.8(18) 4.3(18) 8.5(16) 2.0(17) 2.8(17) 6.9(16) 2.3(17) 1.2(17) 1.52(1) 2.1(13) 1.7(18) 9.6(17) 3.8(17) 7.4(16) 2.4(19) 3.8(16) 7.6(18) 5.0(18) 1.1(17) 2.9(17) 3.6(17) 1.0(17) 2.9(17) 1.0(17) 1.63(1) 8.1(13) 1.6(18) 8.6(17) 2.9(17) 2.0(17) 2.1(19) 7.4(16) 7.2(18) 5.8(18) 1.5(17) 3.8(17) 4.0(17) 1.2(17) 3.7(17) 5.9(16) 1.75(1) 5.6(14) 1.6(18) 6.9(17) 2.0(17) 3.6(17) 1.9(19) 1.4(17) 6.3(18) 6.9(18) 2.4(17) 4.9(17) 4.0(17) 1.2(17) 4.1(17) 3.5(16) 1.87(1) 1.0(16) 1.7(18) 5.2(17) 1.3(17) 5.4(17) 1.8(19) 2.6(17) 4.8(18) 9.0(18) 4.0(17) 6.5(17) 3.2(17) 8.9(16) 3.6(17) 2.3(16) 2.01(1) 8.0(16) 1.8(18) 4.0(17) 8.4(16) 5.4(17) 1.6(19) 3.0(17) 3.2(18) 1.0(19) 5.6(17) 7.6(17) 2.4(17) 7.9(16) 3.0(17) 1.8(16) 2.15(1) 6.7(17) 2.1(18) 3.1(17) 4.7(16) 4.8(17) 1.4(19) 3.8(17) 2.0(18) 1.1(19) 7.7(17) 8.7(17) 1.7(17) 8.9(16) 2.5(17) 1.5(16) 2.31(1) 4.0(18) 2.0(18) 3.0(17) 2.2(16) 4.3(17) 1.3(19) 7.4(17) 2.3(18) 1.1(19) 6.8(17) 8.6(17) 2.0(17) 1.0(17) 2.2(17) 1.3(16) 2.48(1) 6.9(18) 1.5(18) 2.8(17) 1.1(16) 3.4(17) 1.3(19) 7.2(17) 3.6(18) 8.2(18) 4.4(17) 7.2(17) 2.8(17) 8.5(16) 1.9(17) 1.2(16) 2.66(1) 7.8(18) 1.2(18) 2.5(17) 5.7(15) 2.7(17) 1.2(19) 6.0(17) 4.6(18) 6.2(18) 3.0(17) 6.0(17) 3.7(17) 6.2(16) 1.9(17) 9.0(15) 2.85(1) 7.5(18) 9.9(17) 2.4(17) 2.6(15) 2.3(17) 1.1(19) 5.9(17) 5.1(18) 4.9(18) 2.2(17) 5.2(17) 4.4(17) 4.8(16) 2.0(17) 8.6(15) 3.05(1) 6.4(18) 8.4(17) 2.4(17) 1.3(15) 2.1(17) 1.0(19) 6.2(17) 5.1(18) 4.1(18) 1.8(17) 4.7(17) 4.8(17) 5.1(16) 2.1(17) 6.5(15) 3.28(1) 5.1(18) 7.3(17) 2.3(17) 6.7(14) 2.0(17) 9.4(18) 6.5(17) 4.7(18) 3.6(18) 1.7(17) 4.4(17) 4.9(17) 6.3(16) 2.3(17) 3.5(15) 3.51(1) 3.9(18) 6.6(17) 2.1(17) 3.4(14) 1.8(17) 8.7(18) 6.7(17) 4.3(18) 3.2(18) 1.7(17) 4.2(17) 5.1(17) 8.8(16) 2.4(17) 2.7(15) 3.77(1) 2.8(18) 6.2(17) 1.9(17) 1.6(14) 1.7(17) 8.0(18) 6.6(17) 3.8(18) 2.8(18) 1.7(17) 4.0(17) 5.1(17) 1.2(17) 2.6(17) 1.7(15) 4.04(1) 2.1(18) 5.9(17) 1.8(17) 1.5(14) 1.5(17) 7.2(18) 6.1(17) 3.2(18) 2.5(18) 1.6(17) 3.8(17) 4.9(17) 1.5(17) 2.5(17) 1.7(15) 4.33(1) 1.4(18) 5.8(17) 1.6(17) 8.0(13) 1.3(17) 6.6(18) 5.3(17) 2.7(18) 2.2(18) 1.6(17) 3.5(17) 4.7(17) 1.8(17) 2.5(17) 1.4(15) 4.64(1) 9.8(17) 5.9(17) 1.4(17) 6.9(13) 1.2(17) 6.0(18) 4.4(17) 2.1(18) 1.9(18) 1.5(17) 3.3(17) 4.4(17) 2.2(17) 2.3(17) 1.3(15) 4.98(1) 7.4(17) 6.1(17) 1.3(17) 6.0(13) 1.0(17) 5.5(18) 3.7(17) 1.6(18) 1.7(18) 1.4(17) 3.1(17) 4.1(17) 2.4(17) 2.1(17) 1.3(15) 5.34(1) 5.8(17) 6.3(17) 1.2(17) 5.3(13) 9.1(16) 5.1(18) 3.1(17) 1.3(18) 1.5(18) 1.3(17) 2.8(17) 3.7(17) 2.4(17) 1.9(17) 1.1(15) 5.72(1) 4.7(17) 6.6(17) 1.1(17) 3.8(13) 7.7(16) 4.7(18) 2.6(17) 9.5(17) 1.2(18) 1.1(17) 2.6(17) 3.3(17) 2.4(17) 1.6(17) 1.1(15) 6.14(1) 4.1(17) 7.0(17) 1.0(17) 3.0(13) 6.4(16) 4.3(18) 2.2(17) 7.1(17) 1.0(18) 1.0(17) 2.3(17) 3.0(17) 2.3(17) 1.4(17) 1.1(15) 6.58(1) 3.6(17) 7.5(17) 9.5(16) 2.0(13) 5.2(16) 3.9(18) 1.9(17) 5.4(17) 8.3(17) 9.1(16) 2.0(17) 2.7(17) 2.1(17) 1.3(17) 9.0(14) 7.06(1) 3.5(17) 7.9(17) 8.6(16) 1.4(13) 4.2(16) 3.5(18) 1.8(17) 4.1(17) 6.8(17) 8.2(16) 1.8(17) 2.4(17) 1.8(17) 1.1(17) 7.0(14) 7.57(1) 3.5(17) 8.2(17) 7.8(16) 1.2(13) 3.4(16) 2.9(18) 1.7(17) 3.2(17) 5.6(17) 7.5(16) 1.6(17) 2.2(17) 1.7(17) 1.0(17) 5.1(14) 8.11(1) 3.4(17) 8.2(17) 7.4(16) 1.0(13) 2.9(16) 2.3(18) 1.8(17) 2.6(17) 4.9(17) 7.0(16) 1.5(17) 1.9(17) 1.7(17) 9.2(16) 4.2(14) 8.70(1) 3.5(17) 8.6(17) 7.3(16) 9.3(12) 2.7(16) 1.7(18) 2.5(17) 2.1(17) 5.0(17) 8.0(16) 1.4(17) 1.8(17) 1.8(17) 8.7(16) 3.7(14) 9.33(1) 3.5(17) 1.0(18) 8.2(16) 8.3(12) 2.8(16) 1.3(18) 4.2(17) 1.7(17) 8.4(17) 1.3(17) 1.6(17) 1.5(17) 2.4(17) 8.0(16) 4.5(14) 1.00(2) 3.4(17) 1.1(18) 1.1(17) 8.2(12) 2.7(16) 1.1(18) 4.1(17) 2.0(17) 1.3(18) 1.4(17) 1.6(17) 1.3(17) 2.6(17) 7.6(16) 7.1(14) 1.07(2) 3.1(17) 1.1(18) 1.5(17) 8.2(12) 2.5(16) 9.5(17) 2.9(17) 3.0(17) 1.5(18) 1.2(17) 1.5(17) 1.2(17) 2.2(17) 7.2(16) 9.7(14) 1.15(2) 2.9(17) 1.0(18) 1.9(17) 7.4(12) 2.1(16) 8.5(17) 2.2(17) 4.1(17) 1.4(18) 9.0(16) 1.2(17) 1.1(17) 1.8(17) 6.8(16) 1.0(15) 1.23(2) 2.9(17) 9.7(17) 1.9(17) 5.7(12) 1.8(16) 7.7(17) 2.0(17) 3.6(17) 1.1(18) 7.2(16) 1.0(17) 9.6(16) 1.6(17) 6.1(16) 1.1(15) 1.32(2) 2.9(17) 9.1(17) 1.8(17) 5.1(12) 1.6(16) 7.0(17) 1.7(17) 2.8(17) 8.5(17) 5.7(16) 8.6(16) 8.4(16) 1.3(17) 5.3(16) 9.2(14) 1.42(2) 2.9(17) 8.6(17) 1.5(17) 4.7(12) 1.3(16) 6.4(17) 1.5(17) 1.9(17) 6.6(17) 4.6(16) 7.2(16) 7.1(16) 1.1(17) 4.5(16) 7.0(14) 1.52(2) 2.9(17) 8.5(17) 1.3(17) 4.4(12) 1.1(16) 5.8(17) 1.3(17) 1.1(17) 4.9(17) 3.8(16) 6.2(16) 5.9(16) 9.4(16) 3.7(16) 5.5(14) 1.63(2) 2.8(17) 8.4(17) 1.1(17) 4.2(12) 9.7(15) 5.3(17) 1.1(17) 7.1(16) 3.7(17) 3.1(16) 5.3(16) 5.0(16) 7.6(16) 3.0(16) 4.4(14) 1.75(2) 2.8(17) 8.5(17) 1.0(17) 4.0(12) 8.2(15) 4.8(17) 9.4(16) 4.8(16) 2.7(17) 2.5(16) 4.4(16) 4.1(16) 5.8(16) 2.5(16) 3.7(14) 1.87(2) 2.7(17) 8.8(17) 8.9(16) 3.8(12) 6.9(15) 4.4(17) 7.6(16) 3.4(16) 1.9(17) 2.0(16) 3.5(16) 3.4(16) 3.9(16) 2.0(16) 3.2(14) 2.01(2) 2.7(17) 8.7(17) 8.0(16) 3.7(12) 5.8(15) 4.0(17) 6.3(16) 2.6(16) 1.4(17) 1.6(16) 2.8(16) 2.8(16) 2.6(16) 1.6(16) 2.7(14) 2.15(2) 2.7(17) 8.5(17) 7.4(16) 3.5(12) 4.6(15) 3.6(17) 5.2(16) 2.0(16) 1.0(17) 1.2(16) 2.2(16) 2.3(16) 1.7(16) 1.3(16) 2.1(14) 2.31(2) 2.7(17) 8.1(17) 6.9(16) 3.6(12) 3.4(15) 3.2(17) 4.4(16) 1.6(16) 7.7(16) 9.3(15) 1.6(16) 1.8(16) 1.1(16) 1.1(16) 1.4(14) 2.48(2) 2.9(17) 7.7(17) 6.1(16) 4.0(12) 2.5(15) 2.9(17) 3.7(16) 1.4(16) 5.5(16) 6.9(15) 1.2(16) 1.4(16) 6.2(15) 8.3(15) 8.9(13) 2.66(2) 3.9(17) 8.0(17) 5.7(16) 4.5(12) 2.1(15) 2.7(17) 3.3(16) 1.2(16) 3.8(16) 5.0(15) 8.5(15) 1.0(16) 3.1(15) 6.4(15) 6.3(13) 2.85(2) 5.2(17) 8.4(17) 4.6(16) 5.0(12) 1.9(15) 2.6(17) 4.9(16) 8.9(15) 3.3(16) 5.4(15) 9.1(15) 7.0(15) 2.6(15) 4.4(15) 4.6(13) 3.05(2) 6.0(17) 8.8(17) 3.3(16) 5.5(12) 2.0(15) 2.5(17) 9.4(16) 6.7(15) 4.0(16) 9.5(15) 1.5(16) 4.2(15) 5.7(15) 2.7(15) 3.2(13) $a(b)$ represents $a\\times 10^b$"
]
] | 1403.0390 |
[
[
"Area law in one dimension: Degenerate ground states and Renyi\n entanglement entropy"
],
[
"Abstract An area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in one-dimensional gapped quantum systems.",
"Suppose in a chain of $n$ spins the ground states of a local Hamiltonian with energy gap $\\epsilon$ are constant-fold degenerate.",
"Then, the Renyi entanglement entropy $R_\\alpha(0<\\alpha<1)$ of any ground state across any cut is upper bounded by $\\tilde O(\\alpha^{-3}/\\epsilon)$, and any ground state can be well approximated by a matrix product state of subpolynomial bond dimension $2^{\\tilde O(\\epsilon^{-1/4}\\log^{3/4}n)}$."
],
[
"Introduction",
"The area law states that for a large class of “physical” quantum many-body states the entanglement of a region scales as its boundary (area) [6].",
"This is in sharp contrast to the volume law for generic states [11]: the entanglement of a region scales as the number of sites in (i.e., the volume of) the region.",
"In one dimension (1D), the area law is of particular interest for it characterizes the classical simulability of quantum systems.",
"Specifically, bounded (or even logarithmic divergence of) Renyi entanglement entropy across all cuts implies efficient matrix product state (MPS) representations [18], which underlie the (heuristic) density matrix renormalization group (DMRG) algorithm [19], [20].",
"Since MPS can be efficiently contracted, the 1D local Hamiltonian problem with the restriction that the ground state satisfies area laws is in NP.",
"Furthermore, a structural result (Lemma REF and see also [10], [1]) from the proof of the area law for the ground state of 1D gapped Hamiltonians is an essential ingredient of the (provably) polynomial-time algorithm [14] for computing such states, establishing that the 1D gapped local Hamiltonian problem is in P. The area law is now a central topic in the emerging field of Hamiltonian complexity [15].",
"We start with the definition of entanglement entropy.",
"Definition 1 (Entanglement entropy) The Renyi entanglement entropy $R_\\alpha (0<\\alpha <1)$ of a bipartite (pure) quantum state $\\rho _{AB}$ is defined as $R_\\alpha (\\rho _A)=(1-\\alpha )^{-1}\\log \\mathrm {tr}\\rho _A^\\alpha ,$ where $\\rho _A=\\mathrm {tr}_B\\rho _{AB}$ is the reduced density matrix.",
"The von Neumann entanglement entropy is defined as $S(\\rho _A)=-\\mathrm {tr}(\\rho _A\\log \\rho _A)=\\lim _{\\alpha \\rightarrow 1^-}R_\\alpha (\\rho _A).$ Here are three arguments why Renyi entanglement entropy is more suitable than von Neumann entanglement entropy for formulating area laws, although the latter is the most popular entanglement measure (for pure states) in quantum information and condensed matter theory.",
"1 (conceptual, classical simulability).",
"In 1D, (unlike bounded Renyi entanglement entropy) bounded von Neumann entanglement entropy across all cuts does not necessarily imply efficient MPS representations; see [16] for a counterexample.",
"Although slightly outside the scope of the present paper, related results are summarized in Table REF (see also [8]).",
"2 (conceptual, quantum computation).",
"Quantum states with little von Neumann entanglement entropy across all cuts support universal quantum computation, while an analogous statement for Renyi entanglement entropy is expected to be false [17].",
"3 (technical).",
"An area law for Renyi entanglement entropy implies that for von Neumann entanglement entropy (Table REF ), as $R_\\alpha $ is a monotonically decreasing function of $\\alpha $ .",
"Table: Relations between various conditions in 1D: unique ground state of a gapped local Hamiltonian (Gap), exponential decay of correlations (Exp), area law for Renyi entanglement entropy R α ,∀αR_\\alpha ,\\forall \\alpha (AL-R α R_\\alpha ), area law for von Neumann entanglement entropy (AL-SS), efficient matrix product state representation (MPS).",
"A check (cross) mark means that the item in the row implies (does not imply) the item in the column.",
"The asterisk marks one contribution of the present paper.",
"It is an open problem whether exponential decay of correlations implies area laws for Renyi entanglement entropy R α ,∀αR_\\alpha ,\\forall \\alpha : Indeed, Theorem 4 in (or Theorem 1 in ) may lead to divergence of R α R_\\alpha if α\\alpha is small.Hastings first proved an area law for the ground state of 1D Hamiltonians with energy gap $\\epsilon $ : The von Neumann entanglement entropy across any cut is upper bounded by $2^{O(\\epsilon ^{-1})}$ [10], where the local dimension of each spin (denoted by “$d$ ” in qudits) is assumed to be an absolute constant.",
"The Renyi entanglement entropy $R_\\alpha $ for $\\alpha _0<\\alpha <1$ was also discussed, where $\\alpha _0$ is $\\epsilon $ -dependent and $\\lim _{\\epsilon \\rightarrow 0^+}\\alpha _0=1$ .",
"The bound on the von Neumann entanglement entropy was recently improved to $\\tilde{O}(\\epsilon ^{-3/2})$ [1] (see Section for an explanation of this result), where $\\tilde{O}(x):=O(x~\\mathrm {poly}\\log x)$ hides a polylogarithmic factor.",
"These proofs of area laws assume a unique (nondegenerate) ground state.",
"Ground-state degeneracy is an important physical phenomenon often associated with symmetry breaking (e.g., the transverse field Ising chain) and/or topological order (e.g., the Haldane/AKLT chain with open boundary conditions).",
"Since not all degenerate ground states of 1D gapped Hamiltonians have exponential decay of correlations, it may not be intuitively obvious to what extent they satisfy area laws.",
"In the present paper, an area law is proved for the Renyi entanglement entropy of possibly degenerate ground states in 1D gapped systems.",
"Since in this context the standard bra-ket notation may be cumbersome, quantum states and their inner products are simply denoted by $\\psi ,\\phi \\ldots $ and $\\langle \\psi ,\\phi \\rangle $ , respectively, cf.",
"$\\Vert |\\psi \\rangle -|\\phi \\rangle \\Vert $ versus $\\Vert \\psi -\\phi \\Vert $ .",
"Suppose in a chain of $n$ spins the ground states are constant-fold degenerate.",
"Theorem 1 (a) The Renyi entanglement entropy $R_\\alpha (0<\\alpha <1)$ of any ground state across any cut is upper bounded by $\\tilde{O}(\\alpha ^{-3}/\\epsilon )$ ; (b) Any ground state $\\psi $ can be approximated by an MPS $\\phi $ of subpolynomial bond dimension $2^{\\tilde{O}(\\epsilon ^{-1/4}\\log ^{3/4}n)}$ such that $|\\langle \\psi ,\\phi \\rangle |>1-1/\\mathrm {poly}(n)$ .",
"Remark The proof of Theorem REF assumes constant-fold exact ground-state degeneracy and open boundary conditions (with one cut).",
"It should be clear that a minor modification of the proof leads to the same results in the presence of an exponentially small $2^{-\\Omega (n)}$ splitting of the ground-state degeneracy (as is typically observed in physical systems) and works for periodic boundary conditions (with two cuts).",
"However, it is an open problem to what extent degenerate ground states satisfy area laws if the degeneracy grows with the system size.",
"Theorem REF (b) is a theoretical justification of the practical success of DMRG as a (heuristic) variational algorithm over MPS to compute the ground-state space in 1D gapped systems with ground-state degeneracy, and paves the way for a (provably) polynomial-time algorithm to compute the ground-state space.",
"As an important immediate corollary of Theorem REF (a), the von Neumann entanglement entropy of a unique ground state is upper bounded by $\\tilde{O}(\\epsilon ^{-1})$ , which even improves the result of [1] and may possibly be tight up to a polylogarithmic factor.",
"An example with the von Neumann entanglement entropy $S=\\tilde{\\Omega }(\\epsilon ^{-1/4})$ was constructed in [7]; see also [13] for a translationally invariant construction with $S=\\Omega (\\epsilon ^{-1/12})$ .",
"We loosely follow the approach in [1] with additional technical ingredients.",
"Approximate ground-space projection (AGSP) [3] is a tool for bounding the decay of Schmidt coefficients: An “efficient” family of AGSP imply an area law.",
"Section is devoted to perturbation theory, which is necessary to improve the efficiency of AGSP.",
"As a technical contribution, the analysis in Section 6 of [1] is improved (and simplified), resulting in a tightened upper bound $\\tilde{O}(\\epsilon ^{-1})$ (versus $\\tilde{O}(\\epsilon ^{-3/2})$ given in [1]) on the (von Neumann) entanglement entropy.",
"Although the perturbation theory is developed in 1D, generalizations to higher dimensions may be straightforward but are not presented in the present paper.",
"In Section , a family of AGSP are constructed in 1D systems with nearly degenerate ground states.",
"Although the ground-state degeneracy of the original Hamiltonian is assumed to be exact, perturbations may lead to an exponentially small splitting of the degeneracy.",
"Then, “fine tunning” using Lagrange interpolation polynomials appears necessary to repair this splitting at the level of AGSP.",
"In Section , an area law is derived from AGSP for any ground state by constructing a sequence of approximations to a set of basis vectors of the ground-state space (it requires new ideas to keep track of such a set of basis vectors).",
"The construction is more efficient than the approach (Corollary 2.4 and Section 6.2) in [1], resulting in an area law for the Renyi entanglement entropy.",
"Finally, efficient MPS representations follow from the decay of the Schmidt coefficients."
],
[
"Perturbation theory",
"Assume without loss of generality that the original 1D Hamiltonian is $H^{\\prime }=\\sum _{i=-n}^nH^{\\prime }_i$ , where $0\\le H^{\\prime }_i\\le 1$ acts on the spins $i$ and $i+1$ .",
"Consider the middle cut.",
"Let $\\epsilon _0(\\cdot )$ denote the ground-state energy of a Hamiltonian.",
"Define $H=H_L+H_{-s}+H_{1-s}+\\cdots +H_{s-1}+H_s+H_R$ as (i) $H_L=H^{\\prime }_L-\\epsilon _0(H^{\\prime }_L)$ and $H_R=H^{\\prime }_R-\\epsilon _0(H^{\\prime }_R)$ , where $H^{\\prime }_L:=\\sum _{i=-n}^{-s-1}H^{\\prime }_i$ and $H^{\\prime }_R:=\\sum _{i=s+1}^{n}H^{\\prime }_i$ ; (ii) $H_i=H^{\\prime }_i$ for $i=\\pm s$ ; (iii) $H_i=H^{\\prime }_i-\\epsilon _0(H^{\\prime }_M)/(2s-1)$ for $1-s\\le i\\le s-1$ , where $H^{\\prime }_M:=\\sum _{i=1-s}^{s-1}H^{\\prime }_i$ .",
"Hence, (a) $H_L\\ge 0,~H_R\\ge 0,$ and $\\epsilon _0(H_L)=\\epsilon _0(H_R)=0$ ; (b) $0\\le H_i\\le 1$ for $i=\\pm s$ ; (c) $0\\le \\sum _{i=1-s}^{s-1}H_i\\le 2s-1$ and $\\epsilon _0(\\sum _{i=1-s}^{s-1}H_i)=0$ ; (d) $H=H^{\\prime }-\\epsilon _0(H^{\\prime }_L)-\\epsilon _0(H^{\\prime }_M)-\\epsilon _0(H^{\\prime }_R)$ so that the (degenerate) ground states and the energy gap are preserved.",
"Suppose the ground states of $H$ are $f$ -fold degenerate, where $f=O(1)$ is assumed to be an absolute constant.",
"Let $0\\le \\epsilon _0=\\epsilon _1=\\cdots =\\epsilon _{f-1}<\\epsilon _f\\le \\epsilon _{f+1}\\le \\cdots $ be the lowest energy levels of $H$ with the energy gap $\\epsilon :=\\epsilon _f-\\epsilon _0$ .",
"Define $H_L^{\\le t}=H_LP^{\\le t}_L+t(1-P^{\\le t}_L),$ where $P^{\\le t}_L$ is the projection onto the subspace spanned by the eigenstates of $H_L$ with eigenvalues at most $t$ .",
"$H_R^{\\le t}$ is defined analogously.",
"Let $H^{(t)}:=H_L^{\\le t}+H_{-s}+H_{1-s}+\\cdots +H_{s-1}+H_s+H_R^{\\le t}\\le 2t+2s+1$ be the truncated Hamiltonian with the lowest energy levels $0\\le \\epsilon ^{\\prime }_0\\le \\epsilon ^{\\prime }_1\\le \\cdots $ and the corresponding (orthonormal) eigenstates $\\phi _0^{(t)},\\phi _1^{(t)},\\ldots $ .",
"Note that all states are normalized unless otherwise stated.",
"Define $\\epsilon ^{\\prime }=\\epsilon ^{\\prime }_f-\\epsilon ^{\\prime }_0$ as the energy gap of $H^{(t)}$ .",
"Let $B:=H_{-s}+H_s$ be the sum of boundary terms, and $P_t$ be the projection onto the subspace spanned by the eigenstates of $H-B$ with eigenvalues at most $t$ so that $ H_LP_t=H_L^{\\le t}P_t,H_RP_t=H_R^{\\le t}P_t~\\Rightarrow ~HP_t=H^{(t)}P_t.$ Lemma 1 $0\\le \\epsilon ^{\\prime }_0\\le \\epsilon _0\\le 2$ and $\\epsilon ^{\\prime }_f\\le \\epsilon _f\\le [\\log _2f]+4=O(1)$ .",
"Let $\\psi _0,\\psi _L,\\psi _M,\\psi _R$ be the ground states of $H,H_L,\\sum _{i=1-s}^{s-1}H_i,H_R$ , respectively.",
"$&&\\epsilon _0\\le \\langle \\psi _L\\psi _M\\psi _R,H\\psi _L\\psi _M\\psi _R\\rangle \\nonumber \\\\&&=\\langle \\psi _L,H_L\\psi _L\\rangle +\\left\\langle \\psi _M,\\sum _{i=1-s}^{s-1}H_i\\psi _M\\right\\rangle +\\langle \\psi _R,H_R\\psi _R\\rangle +\\langle \\psi _L\\psi _M\\psi _R,B\\psi _L\\psi _M\\psi _R\\rangle \\le \\Vert B\\Vert \\le 2.$ Let $f^{\\prime }=[\\log _2f]+1$ and $\\phi _R$ be the ground state of $\\sum _{i=f^{\\prime }-s+1}^sH_i+H_R$ .",
"For any state $\\phi _M$ of the spins $1-s,2-s,\\cdots ,f^{\\prime }-s$ , $&&\\langle \\psi _L\\phi _M\\phi _R,H\\psi _L\\phi _M\\phi _R\\rangle \\nonumber \\\\&&=\\langle \\psi _L,H_L\\psi _L\\rangle +\\left\\langle \\psi _L\\phi _M\\phi _R,\\sum _{i=-s}^{f^{\\prime }-s}H_i\\psi _L\\phi _M\\phi _R\\right\\rangle +\\left\\langle \\phi _R,\\left(\\sum _{i=f^{\\prime }+1-s}^sH_i+H_R\\right)\\phi _R\\right\\rangle \\nonumber \\\\&&\\le \\langle \\psi ,H_L\\psi \\rangle +\\left\\langle \\psi ,\\sum _{i=-s}^{f^{\\prime }-s}H_i\\psi \\right\\rangle +f^{\\prime }+1+\\left\\langle \\psi ,\\left(\\sum _{i=f^{\\prime }+1-s}^sH_i+H_R\\right)\\psi \\right\\rangle \\nonumber \\\\&&\\le \\langle \\psi ,H\\psi \\rangle +f^{\\prime }+1=\\epsilon _0+f^{\\prime }+1\\le f^{\\prime }+3~\\Rightarrow \\epsilon _f\\le f^{\\prime }+3=[\\log _2f]+4.$ Let $\\phi ^{(r)}$ be an eigenstate of $H^{(r)}$ with eigenvalue $\\epsilon ^{(r)}$ .",
"Lemma 2 For $r,t>\\epsilon ^{(r)}$ , $\\Vert (1-P_t)\\phi ^{(r)}\\Vert ^2\\le |\\langle \\phi ^{(r)},(1-P_t)BP_t\\phi ^{(r)}\\rangle |/(\\min \\lbrace r,t\\rbrace -\\epsilon ^{(r)}).$ It follows from $&&\\epsilon ^{(r)}=\\langle \\phi ^{(r)},H^{(r)}\\phi ^{(r)}\\rangle \\nonumber \\\\&&=\\langle \\phi ^{(r)},(1-P_t)H^{(r)}(1- P_t)\\phi ^{(r)}\\rangle +\\langle \\phi ^{(r)},P_tH^{(r)}\\phi ^{(r)}\\rangle +\\langle \\phi ^{(r)},(1-P_t)H^{(r)}P_t\\phi ^{(r)}\\rangle \\nonumber \\\\&&\\ge \\langle \\phi ^{(r)},(1-P_t)(H^{(r)}-B)(1- P_t)\\phi ^{(r)}\\rangle +\\epsilon ^{(r)}\\Vert P_t\\phi ^{(r)}\\Vert ^2\\nonumber \\\\&&+\\langle \\phi ^{(r)},(1-P_t)(H^{(r)}-B)P_t\\phi ^{(r)}\\rangle +\\langle \\phi ^{(r)},(1-P_t)BP_t\\phi ^{(r)}\\rangle \\nonumber \\\\&&\\ge \\min \\lbrace r,t\\rbrace \\Vert (1- P_t)\\phi ^{(r)}\\Vert ^2+\\epsilon ^{(r)}(1-\\Vert (1-P_t)\\phi ^{(r)}\\Vert ^2)-|\\langle \\phi ^{(r)},(1-P_t)BP_t\\phi ^{(r)}\\rangle |.$ Suppose $\\epsilon ^{(r)}=O(1)$ and $r\\ge \\epsilon ^{(r)}+100=O(1)$ .",
"Lemma 3 $\\Vert (1-P_t)\\phi ^{(r)}\\Vert \\le 2^{-\\Omega (t)}.$ Let $t_0=\\epsilon ^{(r)}+100=O(1)$ .",
"We show that there exists $c=O(1)$ such that $ \\Vert (1-P_{t_i})\\phi ^{(r)}\\Vert \\le 2^{-i}$ for $t_i=t_0+ci$ .",
"The proof is an induction on $i$ with fixed $r$ .",
"Clearly, (REF ) holds for $i=0$ .",
"Suppose (REF ) holds for $i=0,1,\\ldots ,j-1$ .",
"Let $P_{t_{-1}}=0$ for notational convenience.",
"Lemma REF implies $&&\\Vert (1-P_{t_j})\\phi ^{(r)}\\Vert ^2\\le |\\langle \\phi ^{(r)},(1-P_{t_j})BP_{t_j}\\phi ^{(r)}\\rangle |/(\\min \\lbrace r,t_j\\rbrace -\\epsilon ^{(r)})\\nonumber \\\\&&\\le \\left|\\left\\langle \\phi ^{(r)},(1-P_{t_j})B\\sum _{i=0}^j(P_{t_i}-P_{t_{i-1}})\\phi ^{(r)}\\right\\rangle \\right|/100\\nonumber \\\\&&\\le \\Vert (1-P_{t_j})\\phi ^{(r)}\\Vert \\sum _{i=0}^j\\Vert (1-P_{t_j})B(P_{t_i}-P_{t_{i-1}})\\Vert \\Vert (P_{t_i}-P_{t_{i-1}})\\phi ^{(r)}\\Vert /100\\nonumber \\\\&&\\Rightarrow \\Vert (1-P_{t_j})\\phi ^{(r)}\\Vert \\le \\sum _{i=0}^j\\Vert (1-P_{t_j})BP_{t_i}\\Vert \\Vert (1-P_{t_{i-1}})\\phi ^{(r)}\\Vert /100\\le \\sum _{i=0}^je^{(t_i-t_j)/8}2^{-i}/10,$ where we have used the induction hypothesis (REF ) and the inequality $\\Vert (1-P_{t_j})BP_{t_i}\\Vert \\le 4e^{(t_i-t_j)/8}$ (Lemma 6.6(2) in [1]).",
"Hence (REF ) holds for $i=j$ by setting $c=16\\ln 2$ .",
"Let $\\Phi ^{(t)}:=P_t\\phi ^{(t)}/\\Vert P_t\\phi ^{(t)}\\Vert $ .",
"Lemma 4 $\\langle \\Phi ^{(t)},H\\Phi ^{(t)}\\rangle \\le \\epsilon ^{(t)}+2^{-\\Omega (t)}.$ (REF ) implies $&&\\epsilon ^{(t)}=\\langle \\phi ^{(t)},H^{(t)}\\phi ^{(t)}\\rangle \\nonumber \\\\&&\\ge \\langle \\phi ^{(t)},P_tH^{(t)}P_t\\phi ^{(t)}\\rangle +\\langle \\phi ^{(t)},P_tH^{(t)}(1-P_t)\\phi ^{(t)}\\rangle +\\langle \\phi ^{(t)},(1-P_t)H^{(t)}P_t\\phi ^{(t)}\\rangle \\nonumber \\\\&&=\\langle \\phi ^{(t)},P_tHP_t\\phi ^{(t)}\\rangle +\\langle \\phi ^{(t)},P_tB(1-P_t)\\phi ^{(t)}\\rangle +\\langle \\phi ^{(t)},(1-P_t)BP_t\\phi ^{(t)}\\rangle \\nonumber \\\\&&\\ge \\langle \\phi ^{(t)},P_tHP_t\\phi ^{(t)}\\rangle -2\\Vert BP_t\\phi ^{(t)}\\Vert \\cdot \\Vert (1-P_t)\\phi ^{(t)}\\Vert \\ge \\langle \\phi ^{(t)},P_tHP_t\\phi ^{(t)}\\rangle -2^{-\\Omega (t)}\\nonumber \\\\&&\\Rightarrow \\langle \\Phi ^{(t)},H\\Phi ^{(t)}\\rangle \\le (\\epsilon ^{(t)}+2^{-\\Omega (t)})/\\Vert P_t\\phi ^{(t)}\\Vert ^2=(\\epsilon ^{(t)}+2^{-\\Omega (t)})/(1-2^{-\\Omega (t)})=\\epsilon ^{(t)}+2^{-\\Omega (t)}.$ Remark Suppose $r\\ge t$ .",
"A very minor modification of the proof implies $\\langle \\Phi ^{(r),t},H\\Phi ^{(r),t}\\rangle \\le \\epsilon ^{(r)}+2^{-\\Omega (t)}~\\mathrm {for}~\\Phi ^{(r),t}:=P_t\\phi ^{(r)}/\\Vert P_t\\phi ^{(r)}\\Vert .$ Since the proofs of Lemmas REF , REF , REF , REF do not require an energy gap, these lemmas also hold in gapless systems.",
"Let $G$ be the ground-state space of $H$ .",
"Lemma 5 For any state $\\psi $ with $\\langle \\psi ,H\\psi \\rangle \\le \\epsilon _0+\\varepsilon $ , there exists a state $\\psi _g\\in G$ such that $\\Vert \\psi -\\psi _g\\Vert ^2\\le 2\\varepsilon /\\epsilon .$ The state $\\psi $ can be decomposed as $\\psi =c_g\\psi _g+c_e\\psi _e,~c_g,c_e\\ge 0,~c_g^2+c_e^2=1,$ where $\\psi _g\\in G$ and $\\psi _e\\perp G$ .",
"Then, $c_g^2\\epsilon _0+c_e^2\\epsilon _f\\le \\langle \\psi ,H\\psi \\rangle \\le \\epsilon _0+\\varepsilon \\Rightarrow c_e^2\\le \\varepsilon /\\epsilon \\Rightarrow \\Vert \\psi -\\psi _g\\Vert ^2=2-2c_g\\le 2\\varepsilon /\\epsilon .$ Theorem 2 For $t\\ge O(\\log \\epsilon ^{-1})$ , (a) $0\\le \\epsilon _0-\\epsilon ^{\\prime }_{f-1}\\le \\epsilon _0-\\epsilon ^{\\prime }_{f-2}\\le \\cdots \\le \\epsilon _0-\\epsilon ^{\\prime }_0\\le 2^{-\\Omega (t)}$ ; (b) there exists $\\psi _i^{(t)}\\in G$ such that $\\Vert \\psi _i^{(t)}-\\phi _i^{(t)}\\Vert ^2\\le 2^{-\\Omega (t)}$ for $i=0,1,\\ldots ,f-1$ ; (c) $\\epsilon ^{\\prime }\\ge \\epsilon /10$ .",
"Lemma REF implies $&&\\epsilon ^{\\prime }_0\\le \\epsilon ^{\\prime }_1\\le \\cdots \\le \\epsilon ^{\\prime }_{f-1}\\le \\epsilon _0\\le \\langle \\Phi ^{(t)}_0,H\\Phi ^{(t)}_0\\rangle \\le \\epsilon ^{\\prime }_0+2^{-\\Omega (t)},\\\\&&\\langle \\Phi ^{(t)}_f,H\\Phi ^{(t)}_f\\rangle \\le \\epsilon ^{\\prime }_f+2^{-\\Omega (t)}=\\epsilon ^{\\prime }_0+\\epsilon ^{\\prime }+2^{-\\Omega (t)}\\le \\epsilon _0+\\epsilon ^{\\prime }+2^{-\\Omega (t)}.$ (a) follows from (REF ).",
"Using Lemma REF , there exists $\\psi ^{(t)}_0,\\psi ^{(t)}_1,\\ldots ,\\psi ^{(t)}_f\\in G$ such that $ \\Vert \\Phi ^{(t)}_i-\\psi ^{(t)}_i\\Vert ^2\\le 2^{-\\Omega (t)}/\\epsilon =2^{-\\Omega (t)+\\log \\epsilon ^{-1}}$ for $i=0,1,\\ldots ,f-1$ and $ \\Vert \\Phi ^{(t)}_f-\\psi ^{(t)}_f\\Vert ^2\\le \\epsilon ^{\\prime }/\\epsilon +2^{-\\Omega (t)}/\\epsilon .$ Lemma REF implies $ \\Vert \\phi ^{(t)}_i-\\Phi ^{(t)}_i\\Vert ^2\\le 2^{-\\Omega (t)}.$ (b) follows from (REF ), (REF ) as $t\\ge O(\\log \\epsilon ^{-1})$ .",
"(c) follows from (REF ), (REF ), (REF ), because $\\phi ^{(t)}_0,\\phi ^{(t)}_1,\\ldots ,\\phi ^{(t)}_f$ are pairwise orthogonal while $\\psi ^{(t)}_0,\\psi ^{(t)}_1,\\ldots ,\\psi ^{(t)}_f$ are linearly dependent."
],
[
"Approximate ground-space projection",
"Recall that $H^{(t)}$ is the truncated Hamiltonian with the lowest energy levels $0\\le \\epsilon ^{\\prime }_0\\le \\epsilon ^{\\prime }_1\\le \\cdots $ and the corresponding (orthonormal) eigenstates $\\phi _0^{(t)},\\phi _1^{(t)},\\ldots $ .",
"Theorem REF implies that the lowest $f$ energy levels are nearly degenerate: $\\epsilon ^{\\prime }_0\\approx \\epsilon ^{\\prime }_{f-1}$ , and $\\epsilon ^{\\prime }=\\epsilon ^{\\prime }_f-\\epsilon ^{\\prime }_0$ is the energy gap.",
"Let $G^{\\prime }:=\\mathrm {span}\\lbrace \\phi _i^{(t)}|i=0,1,\\ldots ,f-1\\rbrace $ be the ground-state space of $H^{(t)}$ .",
"Let $R(\\psi )$ denote the Schmidt rank of a state $\\psi $ across the middle cut.",
"Definition 2 (Approximate ground-space projection (AGSP) [3]) A linear operator $A$ is a $(\\Delta ,D)$ -AGSP if (i) $A\\psi =\\psi $ for $\\forall \\psi \\in G^{\\prime }$ ; (ii) $A\\psi \\perp G^{\\prime }$ and $\\Vert A\\psi \\Vert ^2\\le \\Delta $ for $\\forall \\psi \\perp G^{\\prime }$ ; (iii) $R(A\\psi )\\le DR(\\psi )$ for $\\forall \\psi $ .",
"Let $\\epsilon ^{\\prime }_\\infty :=2s+2t+1$ be an upper bound on the maximum eigenvalue of $H^{(t)}$ .",
"Lemma 6 Suppose $l^2(\\epsilon ^{\\prime }_{f-1}-\\epsilon ^{\\prime }_0)/(\\epsilon ^{\\prime }_\\infty -\\epsilon ^{\\prime }_f)\\le 1/10$ .",
"Then there exists a polynomial $C_l$ of degree $fl$ such that (i) $C_l(\\epsilon ^{\\prime }_0)=C(\\epsilon ^{\\prime }_1)=\\cdots =C(\\epsilon ^{\\prime }_{f-1})=1$ ; (ii) $C_l^2(x)\\le 2^{2f+4}e^{-4l\\sqrt{\\epsilon ^{\\prime }/\\epsilon ^{\\prime }_\\infty }}$ for $\\epsilon ^{\\prime }_f\\le x\\le \\epsilon ^{\\prime }_\\infty $ .",
"The Chebyshev polynomial of the first kind of degree $l$ is defined as $T_l(x)=\\cos (l\\arccos x)=\\cosh (ly),~y:=\\mathrm {arccosh}x.$ By definition, $|T_l(x)|\\le 1$ for $|x|\\le 1$ .",
"For $x\\ge 1$ , $T_l(x)$ is monotonically increasing function of $x$ , and $T_l(x)\\ge e^{ly}/2\\ge e^{2l\\tanh (y/2)}/2=e^{2l\\sqrt{(x-1)/(x+1)}}/2,~\\frac{T^{\\prime }_l(x)}{T_l(x)}=\\frac{l\\tanh (ly)}{\\sinh y}\\le \\frac{l(l y)}{y}=l^2.$ Let $g(x):=(\\epsilon ^{\\prime }_\\infty +\\epsilon ^{\\prime }_f-2x)/(\\epsilon ^{\\prime }_\\infty -\\epsilon ^{\\prime }_f)$ such that $g(\\epsilon ^{\\prime }_\\infty )=-1$ and $g(\\epsilon ^{\\prime }_f)=1$ .",
"Define $S_l(x)=T_l(g(x))$ as a polynomial of degree $l$ .",
"Clearly, $|S_l(x)|\\le 1$ for $\\epsilon ^{\\prime }_f\\le x\\le \\epsilon ^{\\prime }_\\infty $ and $S_l(\\epsilon ^{\\prime }_0)=T_l(g(\\epsilon ^{\\prime }_0))\\ge e^{2l\\sqrt{(g(\\epsilon ^{\\prime }_0)-1)/(g(\\epsilon ^{\\prime }_0)+1)}}/2\\ge e^{2ł\\sqrt{\\epsilon ^{\\prime }/\\epsilon ^{\\prime }_\\infty }}/2.$ There exists $\\epsilon ^{\\prime }_0\\le \\xi \\le \\epsilon ^{\\prime }_{f-1}$ such that $&&S_l(\\epsilon ^{\\prime }_{f-1})=S_l(\\epsilon ^{\\prime }_0)+(\\epsilon ^{\\prime }_{f-1}-\\epsilon ^{\\prime }_0)S^{\\prime }_l(\\xi )\\ge S_l(\\epsilon ^{\\prime }_0)(1+(\\epsilon ^{\\prime }_{f-1}-\\epsilon ^{\\prime }_0)T^{\\prime }_l(g(\\xi ))g^{\\prime }(\\xi )/T_l(g(\\xi )))\\nonumber \\\\&&\\Rightarrow S_l(\\epsilon ^{\\prime }_{f-1})/S_l(\\epsilon ^{\\prime }_0)\\ge 1-2l^2(\\epsilon ^{\\prime }_{f-1}-\\epsilon ^{\\prime }_0)/(\\epsilon ^{\\prime }_\\infty -\\epsilon ^{\\prime }_f)\\ge 4/5.$ Assume without loss of generality that $\\epsilon ^{\\prime }_0,\\epsilon ^{\\prime }_1,\\ldots ,\\epsilon ^{\\prime }_{f-1}$ are pairwise distinct.",
"Let $L(x)=\\sum _{i=1}^fa_ix^i$ be the Lagrange interpolation polynomial of degree $f$ such that $L(0)=0$ and $L(S_l(\\epsilon ^{\\prime }_0))=L(S_l(\\epsilon ^{\\prime }_1))=\\cdots =L(S_l(\\epsilon ^{\\prime }_{f-1}))=S_l(\\epsilon ^{\\prime }_0)$ .",
"For each $i=1,2,\\ldots ,f-1$ , there exists $S_l(\\epsilon ^{\\prime }_{i-1})>\\xi _i>S_l(\\epsilon ^{\\prime }_i)$ such that $L^{\\prime }(\\xi _i)=0$ .",
"Then, $L^{\\prime }(x)=a_1\\prod _{i=1}^{f-1}(1-x/\\xi _i).$ Clearly, $a_1>0$ and $L^{\\prime }(x)>0$ for $x<S_l(\\epsilon ^{\\prime }_{f-1})$ .",
"Hence, $&&S_l(\\epsilon ^{\\prime }_0)=L(S_l(\\epsilon ^{\\prime }_{f-1}))=\\int _0^{S_l(\\epsilon ^{\\prime }_{f-1})}L^{\\prime }(x)\\mathrm {d}x\\ge a_1\\int _0^{S_l(\\epsilon ^{\\prime }_{f-1})}(1-x/S_l(\\epsilon ^{\\prime }_{f-1}))^{f-1}\\mathrm {d}x\\nonumber \\\\&&=a_1S_l(\\epsilon ^{\\prime }_{f-1})/f\\Rightarrow a_1\\le 5f/4.$ For $|x|\\le 1$ , $\\xi _1>\\xi _2>\\cdots >\\xi _{f-1}>S_l(\\epsilon ^{\\prime }_f)=1\\Rightarrow |L^{\\prime }(x)|\\le a_1(1+|x|)^{f-1}\\Rightarrow |L(x)|\\le 2^{f+1}.$ Finally, $C_l(x):=L(S_l(x))/S_l(\\epsilon ^{\\prime }_0)$ is a polynomial of degree $fl$ .",
"Lemma 7 (Lemma 4.2 in [1]) For any polynomial $p_l$ of degree $l\\le s^2$ and any $t,\\psi $ , $R(p_l(H^{(t)})\\psi )\\le l^{O(\\sqrt{l})}R(\\psi ).$ Let $l=s^2/f$ and $t=\\Omega (s)$ .",
"The assumption $1/10\\ge l^2(\\epsilon ^{\\prime }_{f-1}-\\epsilon ^{\\prime }_0)/(\\epsilon ^{\\prime }_\\infty -\\epsilon ^{\\prime }_f)=O(s^42^{-\\Omega (t)}/(s+t))=O(s^32^{-\\Omega (s)})$ is satisfied with sufficiently large $s>O(1)$ .",
"Lemmas REF , REF imply a $(\\Delta ,D)$ -AGSP $A=C_l(H^{(t)})$ for $H^{(t)}$ with $\\Delta =2^{2f+4}e^{-4l\\sqrt{\\epsilon ^{\\prime }/\\epsilon ^{\\prime }_\\infty }}=2^{-\\Omega (s^2\\sqrt{\\epsilon /t})},~D=(s^2)^{O(\\sqrt{s^2})}=s^{O(s)}.$ In particular, the condition $1/100\\ge \\Delta D^2=2^{-\\Omega (s^2\\sqrt{\\epsilon /t})}s^{O(s)}\\Rightarrow 1/100\\ge \\Delta D$ can be satisfied by fixing $t=t_0=\\Theta (s_0)$ and $s=s_0=\\tilde{O}(\\epsilon ^{-1})$ so that $\\Delta =2^{-\\tilde{\\Omega }(\\epsilon ^{-1})}$ and $D=2^{\\tilde{O}(\\epsilon ^{-1})}$ ."
],
[
"Area law",
"Hereafter $f=2$ is assumed for ease of presentation.",
"It should be clear that a very minor modification of the proof works for any $f=O(1)$ .",
"Suppose $s=s_0$ and $t=t_0$ as given above so that $A$ is a $(\\Delta ,D)$ -AGSP for $H^{(t_0)}$ with $\\Delta D^2\\le 1/100$ .",
"Recall that $\\phi _0^{(t_0)},\\phi _1^{(t_0)}$ are the lowest two eigenstates and $G^{\\prime }=\\mathrm {span}\\lbrace \\phi _0^{(t_0)},\\phi _1^{(t_0)}\\rbrace $ is the ground-state space of $H^{(t_0)}$ .",
"Lemma 8 There exist $\\varphi _0,\\varphi _1\\in G^{\\prime }$ and $\\psi _0,\\psi ^{\\prime }_0$ such that (i) $\\varphi _0\\perp \\varphi _1$ ; (ii) $|\\langle \\varphi _0,\\psi _0\\rangle |^2\\ge 24/25$ ; (iii) $R(\\psi _0)=2^{\\tilde{O}(\\epsilon ^{-1})}$ ; (iv) $|\\langle \\varphi _1,\\psi ^{\\prime }_0\\rangle |^2\\ge 24/25$ ; (v) $R(\\psi ^{\\prime }_0)=2^{\\tilde{O}(\\epsilon ^{-1})}$ .",
"Let $P^{\\prime }$ be the projection onto $G^{\\prime }$ .",
"Consider $ \\max _{R(\\psi )=1}\\Vert P^{\\prime }\\psi \\Vert ^2.$ As the set $\\lbrace \\psi |R(\\psi )=1\\rbrace $ of product states is compact, the optimal state exists and is still denoted by $\\psi $ .",
"This state and $\\phi :=A\\psi $ can be decomposed as $\\psi =c_g\\psi _g+c_e\\psi _e,~\\phi =c^{\\prime }_g\\phi _g+c^{\\prime }_e\\phi _e,$ where $\\psi _g,\\phi _g\\in G^{\\prime }$ and $\\psi _e,\\phi _e\\perp G^{\\prime }$ .",
"The definition of AGSP implies $c_g=c^{\\prime }_g,~\\psi _g=\\phi _g,~|c^{\\prime }_e|^2\\le \\Delta ,~R(\\phi )\\le D.$ The Schmidt decomposition of the unnormalized state $\\phi $ implies $\\phi =\\sum _{i=1}^{R(\\phi )}\\lambda _iL_i\\otimes R_i\\Rightarrow \\sum _{i=1}^{R(\\phi )}\\lambda _i^2=\\Vert \\phi \\Vert ^2=|c^{\\prime }_g|^2+|c^{\\prime }_e|^2\\le |c_g|^2+\\Delta .$ Since $|c_g|^2$ is the optimal value in (REF ), $&&|c_g|=|\\langle \\psi _g,\\phi \\rangle |\\le \\sum _{i=1}^{R(\\phi )}\\lambda _i|\\langle \\psi _g,L_i\\otimes R_i\\rangle |\\le \\sum _{i=1}^{R(\\phi )}\\lambda _i\\Vert P^{\\prime }L_i\\otimes R_i\\Vert \\le |c_g|\\sum _{i=1}^{R(\\phi )}\\lambda _i\\Rightarrow 1\\le \\left(\\sum _{i=1}^{R(\\phi )}\\lambda _i\\right)^2\\nonumber \\\\&&\\le R(\\phi )\\sum _{i=1}^{R(\\phi )}\\lambda _i^2\\le D(|c_g|^2+\\Delta )\\le D|c_g|^2+1/100\\Rightarrow |c_g|^2\\ge 99D^{-1}/100\\ge 99\\Delta .$ Applying the AGSP twice, the state $\\psi _0:=A^2\\psi /\\Vert A^2\\psi \\Vert $ satisfies $ \\Vert P^{\\prime }\\psi _0\\Vert ^2\\ge 1-\\Delta /50,~R(\\psi _0)=D^2=2^{\\tilde{O}(\\epsilon ^{-1})}.$ Define $\\varphi _0=P^{\\prime }\\psi _0/\\Vert P^{\\prime }\\psi _0\\Vert \\in G^{\\prime }$ and $\\varphi _1\\in G^{\\prime }$ such that $\\varphi _0\\perp \\varphi _1$ .",
"Clearly, $|\\langle \\varphi _0,\\psi _0\\rangle |^2\\ge 1-\\Delta /50,~\\langle \\varphi _1,\\psi _0\\rangle =0,~|\\langle \\varphi _e,\\psi _0\\rangle |^2\\le \\Delta /50~\\mathrm {for}~\\forall ~\\varphi _e\\perp G^{\\prime }.$ Consider $\\max _{R(\\psi ^{\\prime })=1}|\\langle \\varphi _1,\\psi ^{\\prime }\\rangle |^2.$ As the set $\\lbrace \\psi ^{\\prime }|R(\\psi ^{\\prime })=1\\rbrace $ of product states is compact, the optimal state exists and is still denoted by $\\psi ^{\\prime }$ .",
"This state and $\\phi ^{\\prime }:=A\\psi ^{\\prime }-\\langle \\psi _0,\\psi ^{\\prime }\\rangle \\psi _0$ can be decomposed as $\\psi ^{\\prime }=c_0\\varphi _0+c_1\\varphi _1+c_e\\varphi _e,~\\phi ^{\\prime }=c_1\\varphi _1+c_r\\varphi _r,$ where $\\varphi _e\\perp G^{\\prime }$ and $\\varphi _r\\perp \\varphi _1$ .",
"Specifically, $&&c_r\\varphi _r=c_0(A\\varphi _0-\\langle \\psi _0,\\varphi _0\\rangle \\psi _0)-c_1\\langle \\psi _0,\\varphi _1\\rangle \\psi _0+c_e(A\\varphi _e-\\langle \\psi _0,\\varphi _e\\rangle \\psi _0)\\nonumber \\\\&&\\Rightarrow |c_r|\\le 0.2|c_0|\\sqrt{\\Delta }+1.2|c_e|\\sqrt{\\Delta }\\le 1.4\\sqrt{\\Delta }~\\mathrm {and}~R(\\phi ^{\\prime })\\le D+R(\\psi _0)\\le D+D^2\\le 2D^2.$ The Schmidt decomposition of the unnormalized state $\\phi ^{\\prime }$ implies $\\phi ^{\\prime }=\\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda ^{\\prime }_iL^{\\prime }_i\\otimes R^{\\prime }_i\\Rightarrow \\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda _i^{\\prime 2}=\\Vert \\phi ^{\\prime }\\Vert ^2=|c_1|^2+|c_r|^2\\le |c_1|^2+2\\Delta .$ Since $\\psi ^{\\prime }$ is the optimal state, $&&|c_1|=|\\langle \\varphi _1,\\phi ^{\\prime }\\rangle |\\le \\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda ^{\\prime }_i|\\langle \\varphi _1,L^{\\prime }_i\\otimes R^{\\prime }_i\\rangle |\\le \\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda ^{\\prime }_i|\\langle \\varphi _1,\\psi ^{\\prime }\\rangle |=|c_1|\\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda ^{\\prime }_i\\Rightarrow 1\\le \\left(\\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda ^{\\prime }_i\\right)^2\\nonumber \\\\&&\\le R(\\phi ^{\\prime })\\sum _{i=1}^{R(\\phi ^{\\prime })}\\lambda _i^{\\prime 2}\\le 2D^2(|c_1|^2+2\\Delta )\\le 2D^2|c_1|^2+1/25\\Rightarrow |c_1|^2\\ge 12D^{-2}/25\\ge 48\\Delta .$ Hence $\\psi ^{\\prime }_0=\\phi ^{\\prime }/\\Vert \\phi ^{\\prime }\\Vert $ is a state with $R(\\psi ^{\\prime }_0)=R(\\phi ^{\\prime })\\le 2D^2=2^{\\tilde{O}(\\epsilon ^{-1})}$ and $|\\langle \\varphi _1,\\psi ^{\\prime }_0\\rangle |^2\\ge 24/25$ .",
"Recall that $G$ is the ground-state space of $H$ .",
"Lemma 9 For any $\\Psi \\in G$ , there is a sequence of approximations $\\lbrace \\Psi _i\\rbrace $ such that (a) $|\\langle \\Psi _i,\\Psi \\rangle |\\ge 1-2^{-\\Omega (i)}$ ; (b) $R_i:=R(\\Psi _i)=2^{\\tilde{O}(\\epsilon ^{-1}+\\epsilon ^{-1/4}i^{3/4})}$ .",
"Let $t_i=t_0+i$ .",
"Theorem REF (b) is a quantitative statement that $G$ and $\\mathrm {span}\\lbrace \\phi _0^{(t_i)},\\phi _1^{(t_i)}\\rbrace $ are exponentially close.",
"In particular, setting $t_0$ to be a sufficiently large constant implies that $G^{\\prime }$ and $\\mathrm {span}\\lbrace \\phi _0^{(t_i)},\\phi _1^{(t_i)}\\rbrace $ are close up to a small constant.",
"Hence Lemma REF (ii) implies $ |\\langle \\phi _0^{(t_i)},\\psi _0\\rangle |^2+|\\langle \\phi _1^{(t_i)},\\psi _0\\rangle |^2\\ge 9/10.$ Let $l_i=s_i^2/2=\\Theta (\\sqrt{t_i^3/\\epsilon })=O(t_i^2)$ such that the assumption $1/10\\ge l_i^2(\\epsilon ^{\\prime }_1-\\epsilon ^{\\prime }_0)/(\\epsilon ^{\\prime }_\\infty -\\epsilon ^{\\prime }_2)=O(s_i^32^{-\\Omega (s_i)})$ is satisfied with sufficiently large $s_i>O(1)$ .",
"Lemmas REF , REF imply a $(\\Delta _i,D_i)$ -AGSP $A_i=C_{l_i}(H^{(t_i)})$ for $H^{(t_i)}$ with $\\Delta _i=2^{-\\Omega (s_i^2\\sqrt{\\epsilon /t_i})}=2^{-\\Omega (t_i)},~D_i=s_i^{O(s_i)}=2^{\\tilde{O}(\\epsilon ^{-1/4}t_i^{3/4})}.$ Hence the sequence of operators $\\lbrace A_i\\rbrace _{i=1}^{+\\infty }$ converges exponentially due to Theorem REF (b).",
"Clearly, $A_\\infty :=\\lim _{i\\rightarrow +\\infty }A_i$ is just the projection onto $G$ .",
"Let $\\psi _i:=A_i\\psi _0/\\Vert A_i\\psi _0\\Vert $ with $\\psi _\\infty \\in G$ such that $R(\\psi _i)\\le R(\\psi _0)D_i\\le 2^{\\tilde{O}(\\epsilon ^{-1}+\\epsilon ^{-1/4}t_i^{3/4})},~|\\langle \\psi _i,\\psi _\\infty \\rangle |\\ge 1-2^{-\\Omega (t_i)}.$ Similarly, Let $\\psi ^{\\prime }_i:=A_i\\psi ^{\\prime }_0/\\Vert A_i\\psi ^{\\prime }_0\\Vert $ with $\\psi ^{\\prime }_\\infty \\in G$ such that $R(\\psi ^{\\prime }_i)\\le 2^{\\tilde{O}(\\epsilon ^{-1}+\\epsilon ^{-1/4}t_i^{3/4})},~|\\langle \\psi ^{\\prime }_i,\\psi ^{\\prime }_\\infty \\rangle |\\ge 1-2^{-\\Omega (t_i)}.$ (REF ) with $i=+\\infty $ is a quantitative statement that $\\psi _0$ is close to $G$ , and hence $\\psi _0$ and $\\psi _\\infty $ are close up to a small constant.",
"Since $\\psi _0$ and $\\varphi _0$ are close up to a small constant, $\\psi _\\infty $ and $\\varphi _0$ are also close.",
"The same arguments imply that $\\psi ^{\\prime }_\\infty $ and $\\varphi _1$ are close.",
"Hence, $\\psi _\\infty $ and $\\psi ^{\\prime }_\\infty $ are almost orthogonal.",
"Any state $\\Psi \\in G$ can be decomposed as $\\Psi =c\\psi _\\infty +c^{\\prime }\\psi ^{\\prime }_\\infty ,~|c|=O(1),~|c^{\\prime }|=O(1).$ Then, $\\lbrace \\Psi _i:=c\\psi _i+c^{\\prime }\\psi ^{\\prime }_i\\rbrace _{i=0}^{+\\infty }$ is a sequence of approximations to $\\Psi $ with (b) $R(\\Psi _i)=2^{\\tilde{O}(\\epsilon ^{-1}+\\epsilon ^{-1/4}t_i^{3/4})}$ .",
"(a) also follows immediately.",
"[Proof of Theorem REF ] (a) Let $\\Lambda _i$ be the Schmidt coefficients of $\\Psi $ across the middle cut.",
"Then, $1-p_i:=\\sum _{j=1}^{R_i}\\Lambda _j^2\\ge |\\langle \\Psi _i,\\Psi \\rangle |^2\\ge 1 - 2^{-\\Omega (i)}.$ The Renyi entanglement entropy of $\\Psi $ is upper bounded by $&&\\frac{\\log \\left(R^{1-\\alpha }_0+\\sum _{i=0}^{+\\infty }p_i^\\alpha (R_{i+1}-R_i)^{1-\\alpha }\\right)}{1-\\alpha }\\le \\frac{\\log \\left(2^{(1-\\alpha )\\tilde{O}(\\epsilon ^{-1})}+\\sum _{i=0}^{+\\infty }2^{(1-\\alpha )\\tilde{O}(\\epsilon ^{-1}+\\epsilon ^{-1/4}i^{3/4})-\\alpha \\Omega (i)}\\right)}{1-\\alpha }\\nonumber \\\\&&=\\tilde{O}(\\epsilon ^{-1})+\\frac{\\log (O(1)+2^{(1-\\alpha )\\tilde{O}((1-\\alpha )^3\\alpha ^{-3}/\\epsilon )})}{1-\\alpha }=\\tilde{O}(\\epsilon ^{-1}+(1-\\alpha )^3\\alpha ^{-3}/\\epsilon )=\\tilde{O}(\\alpha ^{-3}\\epsilon ^{-1}).$ (b) Finally we sketch the proof that $\\Psi $ is well approximated by an MPS of small bond dimension.",
"We first express it exactly as an MPS of possibly exponential (in $n$ ) bond dimension and then truncate the MPS cut by cut.",
"It is shown in [18] the error accumulates at most additively: If an inverse polynomial overall error $1/p(n)=1/\\mathrm {poly}(n)$ is allowed, it suffices that the error of truncating each cut is $1/(np(n))=1/\\mathrm {poly}(n)$ .",
"We require that $1/\\mathrm {poly}(n)=p_i\\Rightarrow i=O(\\log n),$ and hence the bond dimension is $2^{\\tilde{O}(\\epsilon ^{-1/4}\\log ^{3/4}n)}$ ."
],
[
"Notes",
"For nondegenerate systems ($f=1$ ), the upper bound claimed in [1] on the von Neumann entanglement entropy is $\\tilde{O}(\\epsilon ^{-1})$ .",
"However, the proof in [1] of this claim appears incomplete.",
"Specifically, in Lemma 6.3 in [1] $t_0$ should be at least $O(\\epsilon _0/\\epsilon ^2+\\epsilon ^{-1})$ in order that the robustness theorem (Theorem 6.1 in [1]) applies to $H^{(t_0)}$ , i.e., the robustness theorem does not guarantee that $H^{(t_0)}$ is gapped if $t_0=O(1)$ .",
"Then $s=\\tilde{O}(\\epsilon ^{-1})$ (and $l=s^2$ ) does not give an AGSP for $H^{(t_0)}$ with $\\Delta D\\le 1/2$ , but $s=\\tilde{O}(\\epsilon ^{-3/2})$ does.",
"A straightforward calculation shows that the upper bound $\\tilde{O}(\\epsilon ^{-3/2})$ on the von Neumann entanglement entropy follows from the proof in [1].",
"Nevertheless, in the present paper I have shown that the claim in [1] is correct, because Theorem REF (as a stronger version of the robustness theorem) only requires $t\\ge O(\\log \\epsilon ^{-1})$ .",
"After the appearance of the present paper on arXiv [12], Section (perturbation theory) was extended to higher dimensions [2].",
"In particular, Theorems 4.2, 4.6 in [2] are generalizations of Lemmas REF , REF , respectively."
]
] | 1403.0327 |
[
[
"Identification campaign of supernova remnant candidates in the Milky\n Way. II. X-ray studies of G38.7-1.4"
],
[
"Abstract We report on XMM-Newton and Chandra observations of the Galactic supernova remnant candidate G38.7-1.4, together with complementary radio, infrared, and gamma-ray data.",
"An approximately elliptical X-ray structure is found to be well correlated with radio shell as seen by the Very Large Array.",
"The X-ray spectrum of G38.7-1.4 can be well-described by an absorbed collisional ionization equilibrium plasma model, which suggests the plasma is shock heated.",
"Based on the morphology and the spectral behaviour, we suggest that G38.7-1.4 is indeed a supernova remnant belongs to a mix-morphology category."
],
[
"Introduction",
"Supernovae (SNe) and their remnants play a crucial role in driving the dynamical and chemical evolution of galaxies (Woosley & Weaver 1995).",
"Each SN produces bulks of heavy elements and disperses them throughout the interstellar medium (ISM) (e.g.",
"Thielemann et al.",
"1996; Chieffi & Limongi 2004).",
"The shock waves from the SN explosions may also trigger star formation in molecular clouds (Boss 1995).",
"In addition, the blast waves in supernova remnants (SNRs) can accelerate particles to relativistic energies via Fermi-I acceleration (Reynolds 2008), which has long been suggested as a promising acceleration mechanism for Galactic cosmic rays (GCRs).",
"For investigating the role of SNRs as GCR accelerators, it is necessary to ask whether they can account for the entire energy density of CRs in the Milky Way.",
"This is related to the mechanical power provided by the SNe, which in turn is associated with their event rate.",
"In our Milky Way, the currently known SNR population is far below the expectation based on a event rate of 2 SNe/century (Tammann et al.",
"1994) and a typical evolution timescale of $\\sim 10^{5}$ yrs (see Hui et al.",
"2012; Hui 2013).",
"Therefore, deeper searches for Galactic SNRs are certainly needed.",
"With the much improved spatial and spectral resolution and enlarged effective area, state-of-the-art X-ray observatories like Chandra and XMM-Newton provide powerful tools for studying the shock-heated plasma and the non-thermal emission from the leptonic acceleration in SNRs (Hui 2013; Kang 2013).",
"However, the number of X-ray detected SNRs is still significantly smaller than the corresponding number of detections in the radio.",
"Until now there are 302 SNRs that have been uncovered in the Milky Way: 274 objects recorded in Green (2009) plus 28 objects reported in Ferrand & Safi-Harb (2012), while the number of Galactic SNRs detected in X-rays is about 100http://www.physics.umanitoba.ca/snr/SNRcat/.",
"Therefore, enlarging the sample of the X-ray detected SNRs would be valuable.",
"Recently, we have initiated an observational campaign for searching and identifying new Galactic SNRs with X-ray telescopes (Hui et al.",
"2012).",
"Here we report our detailed X-ray analysis of the another SNR candidate G38.7–1.4.",
"G38.7–1.4 was first detected as an unidentified object in ROSAT All-Sky Survey (RASS) with an extent of about $12^{\\prime } \\times 8^{\\prime }$ .",
"This object has a centrally-peaked morphology in X-rays.",
"Cross-correlating with the Effelsberg Galactic Plane 11 cm survey data, G38.7–1.4 is found to positionally coincide with an incomplete radio shell (cf.",
"Figure 1d in Schaudel et al.",
"2002).",
"A follow-up observation with the Effelsberg telescope at a wavelength of 6 cm revealed a non-thermal radio emission with a spectral index of $\\alpha =-0.79\\pm 0.23$ .",
"Furthermore, the existence of polarization in the radio shell was reported by Schaudel et al.",
"(2002).",
"All these evidences suggest that G38.7–1.4 is very likely to be a SNR with center-filled X-ray (mixed) morphology.",
"However, the poor spatial resolution ($\\sim 96$ \") and the limited photon statistic ($\\sim 50$ source counts) of RASS data do not allow one to unambiguously confirm its physical nature.",
"Furthermore, the limited energy bandwidth of ROSAT (0.1-2.4 keV) does not allow one to determine whether a possible hard X-ray ($>$ 2 keV) component, arising from the interactions of the reflected shocks with the dense ambient medium or alternatively from the synchrotron emission radiated by relativistic leptons, is present in the hard X-ray band.",
"This motivates us to carry out a detailed X-ray studies of G38.7–1.4 with XMM-Newton and Chandra.",
"Considering the composition of GCRs, leptons only constitute a small proportion.",
"A large fraction of the observed GCRs are hadrons (i.e proton and heavy ions; cf.",
"Sinnis et al.",
"2009).",
"Due to the large masses of hadrons, they are not efficient synchrotron emitters.",
"X-ray and radio observations are generally difficult to constrain their presence.",
"On the other hand, the collision of relativistic hadrons can lead to the production of neutral pions which can subsequently decay into $\\gamma -$ rays (Caprioli 2011).",
"For complementing the aforementioned X-ray investigation of G38.7–1.4 as a possible acceleration site of GCRs, we have also conducted a search for $\\gamma -$ ray emission at the location of G38.7–1.4 wtih the Large Area Telescope (LAT) on board the Fermi Gamma-Ray Space Telescope.",
"In this paper, we report a detailed high energy investigation of G38.7–1.4.",
"The observations and the data reduction of XMM-Newton and Chandra observatories are described in section 2.",
"Sections 3 and 4 present the results of the X-ray spatial and spectral analysis respectively.",
"In section 5, we describe a deep search of $\\gamma -$ ray emission with Fermi.",
"Finally, we discuss the physical implications of the results and summarize our conclusions in sections 6 and 7 respectively.",
"We have observed G38.7–1.4 with XMM-Newton on April 19, 2012 (ObsID: 0675070401) for a $\\sim $ 20 ks total exposure.",
"In this observation, the EPIC MOS1/2 and PN instruments were operated in full-frame mode using the medium filter to block optical stray light.",
"All the data were processed with the XMM-Newton Science Analysis Software (SAS) package (Version 11.0.0).",
"Calibrated event files for the MOS1, MOS2, and PN detectors were produced using the SAS task emchain and epchain, following standard procedures.",
"Events spread at most in two contiguous pixels for the PN (i.e., pattern = 0–4) and in four contiguous pixels for the MOS (i.e., pattern = 0–12) have been selected.",
"We further cleaned the data by accepting only the good time intervals (GTIs) when the sky background was low for the whole camera ($<2.6$ counts s$^{-1}$ for MOS1 and MOS2, $<20$ counts s$^{-1}$ for PN).",
"The effective exposure times after background cleaning for MOS1, MOS2, and PN are 11.0 ks, 12.3 ks, and 11.2 ks, respectively.",
"Data analyses were restricted to the 0.5–10.0 keV energy band."
],
[
"Chandra Observation",
"We have also observed G38.7–1.4 with Chandra on 2012 June 9-10 (ObsID 13770) using Advanced CCD Imaging Spectrometer (ACIS-I) with a frame time of 3.2 s. The total exposure time is $\\sim 28$ ks.",
"The data was configured in the VFAINT telemetry mode.",
"Data reduction and analysis were processed with Chandra Interactive Analysis Observations (CIAO) version 4.5 software and the Chandra Calibration Database (CALDB) version 4.5.5.1.",
"The level-2 data with background cleaning was used in our study.",
"Data analysis was restricted to the energy range of $0.5-8.0$ keV."
],
[
"Spatial Analysis",
"Figure REF and Figure REF display the X-ray color image of G38.7–1.4 as observed by Chandra in the energy range of 0.5–8 keV and XMM-Newton in the energy range of 0.5–10 keV respectively.",
"These two images were created by using an adaptive smoothing algorithm with a Gaussian radius of $\\le 10^{\\prime \\prime }$ in order to probe the detailed structure of the diffuse emission.",
"A center-filled X-ray morphology has been revealed.",
"The angular extent of G38.7–1.4 as seen in X-ray is $\\sim 8^{\\prime }\\times 6.6^{\\prime }$ (major axis and minor axis of the dashed ellipse illustrated in Fig.",
"REF ).",
"Figure: A 25 ' ×25 ' 25^{\\prime }\\times 25^{\\prime } X-ray color image of G38.7–1.4 as observedby Chandra ACIS-I (red: 0.5-1 keV green: 1-2 keV blue: 2-8 keV).Adaptively smoothing with a Gaussian kernal of σ<10\"\\sigma <10\" has been applied.",
"21 X-ray point-like sources are detected inthis field.",
"The properties of these sources are summarized in Table 1.The dashed ellipse illustrates the extraction regionfor the remnant spectrum in both XMM-Newton and Chandra data (see Sec.",
"4).Figure: Same field-of-view as Fig as observed by XMM-Newton.",
"The X-ray colorimage (red: 0.5-1 keV green: 1-2 keV blue: 2-10 keV) is created by combining the data from all three cameras andhas been adaptively smoothed a Gaussian kernal of σ<10\"\\sigma <10\".",
"The dashed ellipse illustrates the extraction regionfor the remnant spectrum in both XMM-Newton and Chandra data (see Sec.",
"4).The only confirmed detection of point source in this XMM-Newton observation islabelled as “Source A\" which is consistent with “Source 20\" detected in the Chandra field.Figure: Left panel: 1.4 GHz VLA image of G38.7–1.4.",
"The ellipse in the lower rightcorner indicates the restored beam size of 58 '' ×54 '' 58^{\\prime \\prime } \\times 54^{\\prime \\prime }.Right panel: Smoothed XMM-Newton MOS1/2 image of G38.7–1.4 superimposed with theradio contour determined as in the left panel.For investigating the radio counterpart of G38.7–1.4 we extracted the radio data taken with the Very Large Array (VLA) on August 18, 1996 from the NRAO Science Data Archive.",
"These continuum observations were collected at a central frequency of 1.4 GHz in the VLA's D configuration.",
"After standard cleaning process, a calibrated radio image of G38.7–1.4 is shown in Figure REF left panel.",
"This incomplete radio shell along the southern part of G38.7–1.4 is consistent with that taken with the Effelsberg radio telescope at 11 cm (Schaudel et al.",
"2002).",
"In this VLA radio map, apart from the diffuse emission, we noted an extended bright radio feature around the center of the FoV.",
"For comparing the radio and X-ray features of G38.7–1.4, we overlaid the radio contours on the XMM-Newton MOS1/2 image (see Figure REF right panel).",
"For searching the counterpart of G38.7–1.4 in the other wavelengths, we have also explored the infrared data obtained by Wide-field Infrared Survey Explorer (WISE; Wright et al.",
"2010) and H$\\alpha $ image downloaded from the Southern H-alpha Sky Survey Atlas (SHASSA; Gaustad et al.",
"2001).",
"No possible diffuse infrared and H$\\alpha $ emission associated with G38.7–1.4 was found in our study.",
"The sub-arcsecond resolution of Chandra enables us to search for the possible stellar remnant associated with G38.7–1.4.",
"By means of the wavelet source detection algorithm (CIAO tool: wavdetect), we searched for the point sources in the whole ACIS-I data.",
"The exposure variation across the detector was accounted by the exposure map.",
"We set the detection threshold such that no more than one false detection caused by background fluctuation is in the whole field.",
"The lower limit of source significance was set to be 4$\\sigma _{G}$ , where $\\sigma _{G}$ is Gehrels error.",
"21 sources were detected and marked by white circles in Figure REF .",
"The properties of these sources are given in Table .",
"Based on the ratio between the source extents and the estimates of the PSF sizes at their position reported by wavdetect, most of these sources are found to be point-like except for Sources 13, 17 and 20.",
"However, in view of their coincidence with the diffuse remnant emission, such ratios can possibly be overestimated.",
"To investigate if these X-ray sources are promising isolated neutron star candidates, we focused on those coinciding with the remnant emission (Source IDs: 13, 17, 19, 20) and proceeded to search for their possible optical counterparts since the X-ray-to-optical flux ratio ($f_{x}/f_{opt}$ ) provides a rudimentary parameter for discriminating the nature of the source.",
"For an isolated neutron star, $f_{x}/f_{opt}$ is typically larger than 1000 (cf.",
"Haberl 2007) while for field stars and active galactic nuclei the ratio are much lower than the typical $<0.3$ and $<50$ , respectively (Maccacaro et al.",
"1988; Stocke et al.",
"1991).",
"By cross-correlating the X-ray sources with the SIMBAD and NED databases and the VizieR catalogue service, we searched for the possible optical counterparts within a search radius of 2 arcsec around each source.",
"Among all these four sources, only Source 19 has an optical counterpart (USNO-B1.0 0944-0406013) identified.",
"For the other three sources without optical counterparts found in this search, we computed their limiting X-ray to optical flux ratios.",
"The X-ray to optical flux ratio is defined as log($f_{x}/f_{opt}$ )=log$f_{x}$ +5.67+0.4R (Green et al.",
"2004), where R is the R-band magnitude and $f_{x}$ is derived in the 0.5–2.0 keV.",
"Since these X-ray sources are too faint for spectral fitting, we crudely estimated their X-ray fluxes in the 0.5–2.0 keV energy range from the count rates with the aid of WebPIMMS by assuming a weighted average column density of N$_{H}=1.0\\times 10^{22}$ cm$^{-2}$ toward G38.7–1.4 from the HI survey by Kalberla et al.",
"(2005) and an absorbed power-law spectrum with $\\Gamma =1.8$ .",
"Taking the limiting magnitude of the USNO-B1.0 catalog (i.e.",
"R$>$ 21), the limiting X-ray to optical flux ratios of Sources 13, 17 and 20 are found to be $>0.25$ , $>0.26$ and $>0.32$ respectively.",
"These values are not constraining in determining their emission nature.",
"Dedicated optical observations of these sources are encouraged for further investigation.",
"We have also computed their hardness ratios defined as HR=$(C_{2.5-8.0keV}-C_{0.5-2.5keV})/(C_{2.5-8.0keV}+C_{0.5-2.5keV}$ ), where $C_{0.5-2.5keV}$ and $C_{2.5-8.0keV}$ are the net counts in the 0.5–2.5 keV and 2.5–8.0 keV energy bands.",
"The hardness ratio of Source 13, 17 and 20 are -0.46, 0.03 and -0.82 respectively, which are rather soft.",
"If any of these sources are indeed isolated neutron stars, their emission should be thermal dominant.",
"We have also attempted to search for the point sources with XMM-Newton data.",
"A corresponding set of exposure maps was generated to account for spatial quantum efficiency, mirror vignetting, and field of view of each instrument by running the task eexpmap.",
"Utilizing the SAS task edetect_chain, we performed the source detection on MOS1, MOS2 and PN images individually.",
"We set the threshold of detection likelihood to be mlmin=10 throughout the search, which corresponds to a detection significance of $\\gtrsim $ 4-$\\sigma $ .",
"Only one source, Source A (see Fig.",
"REF ), can be detected by all three cameras.",
"Its position consistent with Source 20 found by Chandra (see Fig.",
"REF ).",
"The inferior performance of source detection with XMM-Newton can be ascribed to its relatively poor resolution and high instrumental background.",
"We also noted that the brightest part of the radio emission as seen by VLA conicides with Source A (see Figure REF right panel).",
"lcccccccc 0pc X-ray sources in the field of view of Chandra ACIS ID RA (J2000) Dec (J2000) $\\delta $ RAa $\\delta $ Deca S/Nb Net counts Photon fluxc PSF Ratiod (h:m:s) (d:m:s) (arcsec) (arcsec) $\\sigma _{G}$ (counts) (10$^{-6}$ photon cm$^{-2}$ s$^{-1}$ ) 1 19:07:10.090 +04:40:07.45 1.01 0.89 4.27 $21\\pm 6$ $1.56\\pm 0.44$ 1.74 2 19:07:35.220 +04:35:42.74 0.76 0.75 4.33 $23\\pm 6$ $1.81\\pm 0.50$ 1.20 3 19:06:31.099 +04:34:20.07 0.72 0.63 5.27 $31\\pm 7$ $2.54\\pm 0.60$ 0.95 4 19:06:46.800 +04:29:32.99 0.14 0.28 6.68 $40\\pm 8$ $3.50\\pm 0.71$ 1.45 5 19:07:37.783 +04:33:47.60 1.35 1.01 4.09 $25\\pm 7$ $1.83\\pm 0.52$ 2.59 6 19:06:40.222 +04:27:06.79 0.89 1.05 4.01 $22\\pm 6$ $1.73\\pm 0.51$ 2.98 7 19:07:04.356 +04:36:54.85 0.96 0.83 4.75 $23\\pm 6$ $1.73\\pm 0.46$ 3.49 8 19:07:35.777 +04:31:00.48 0.54 1.13 5.03 $26\\pm 6$ $1.96\\pm 0.49$ 1.92 9 19:06:34.817 +04:28:31.57 0.68 0.69 4.24 $24\\pm 7$ $1.97\\pm 0.55$ 1.70 10 19:06:43.714 +04:27:53.03 0.42 0.36 4.27 $22\\pm 6$ $1.59\\pm 0.45$ 1.17 11 19:07:28.884 +04:35:12.89 0.62 0.62 5.77 $33\\pm 7$ $2.36\\pm 0.53$ 1.83 12 19:07:32.844 +04:31:12.61 0.65 0.76 4.97 $26\\pm 7$ $2.03\\pm 0.51$ 2.20 13 19:06:58.742 +04:32:03.27 0.92 0.97 4.20 $26\\pm 7$ $1.70\\pm 0.47$ 28.21 14 19:06:48.403 +04:38:06.99 0.86 0.77 6.59 $36\\pm 7$ $2.58\\pm 0.53$ 2.54 15 19:07:22.577 +04:31:23.01 0.44 0.34 5.13 $27\\pm 7$ $2.01\\pm 0.50$ 1.92 16 19:07:15.862 +04:36:14.94 0.62 0.58 6.22 $28\\pm 6$ $1.87\\pm 0.42$ 2.82 17 19:07:07.550 +04:31:44.75 0.70 0.62 5.07 $29\\pm 7$ $2.00\\pm 0.49$ 19.69 18 19:07:14.371 +04:36:03.77 0.36 0.34 7.65 $43\\pm 8$ $3.14\\pm 0.58$ 1.66 19 19:07:14.232 +04:28:13.24 0.34 0.27 4.29 $25\\pm 7$ $1.66\\pm 0.46$ 2.14 20 19:07:16.094 +04:32:30.02 0.55 0.59 6.69 $34\\pm 7$ $2.24\\pm 0.46$ 7.55 21 19:06:37.682 +04:31:33.33 0.34 0.39 8.77 $49\\pm 8$ $3.66\\pm 0.62$ 1.68 aPosition uncertainty.",
"bEstimates of source significance in units of Gehrels error: $\\sigma _{G}=1+\\sqrt{C_{B}+0.75}$ where $C_{B}$ is the background counts.",
"cAbsorbed photon fluxes in the energy range of 0.5-8.0 keV.",
"dThe ratios between the source extents and the estimates of the PSF sizes."
],
[
"Spectral Analysis",
"We extracted the spectrum of the remnant emission from the Chandra data within the elliptical region illustrated in Fig.",
"REF .",
"All the detected point-like sources within the selected region (i.e.",
"Sources 4, 13 & 17) were removed before extraction.",
"We utilized the CIAO tool specextract to extract the spectra and to compute the response files.",
"In view of low signal-to-noise ratio, the intrinsic extent of G38.7–1.4 in X-ray is uncertain.",
"Therefore, the background spectrum is sampled from the blank-sky events.",
"After background subtraction there are $\\sim $ 2166 net counts available in 0.5-8 keV for the spectral analysis.",
"We binned the spectrum extracted from Chandra so as to have at least 20 counts per bin.",
"We utilized the XMM-Newton SAS tool evselect to extract the remnant spectrum from the XMM-Newton data within the same region adopted in the Chandra analysis.",
"Although Sources 4, 13 and 17 were not detected by XMM-Newton, we have removed the photons within 15\" from their positions in order to minimize the contamination.",
"Similarly, the background spectra for each camera were sampled from the blank-sky events.",
"Response files were constructed by using the XMM-Newton SAS tasks rmfgen and arfgen.",
"After background subtraction, there are 989, 1532 and 5990 net counts in 0.5-10 keV from MOS1, MOS2 and PN respectively.",
"The extracted spectra were binned to have at least 15 for MOS1/2 and 30 for PN source counts per bin.",
"All the spectral fits were performed with the XSPEC software package (version: 12.7.0).",
"All quoted errors are 1$\\sigma $ for 2 parameters of interest.",
"For accounting the photoelectric absorption, we used the Wisconsin cross-sections throughout the analysis (Morrison & McCammon 1983; XSPEC model: WABS).",
"We began with examining the spectrum with an absorbed collisional ionization equilibrium (CIE) plasma model (XSPEC model: VEQUIL).",
"First, we fixed the abundance of metals at the solar abundances in order to minimize the number of free parameters.",
"We fitted the spectrum obtained from individual camera separately for checking whether the results are consistent.",
"While the spectrum obtained from Chandra can be fitted reasonably well with the CIE model, residuals have been seen in all XMM-Newton spectra at energies $\\gtrsim 2$ keV.",
"These residuals can be modeled by including an additional power-law model in the fitting which yields a photon index of $\\sim 1.1$ , $\\sim 0.3$ and $\\sim 0.7$ for MOS1, MOS2 and PN respectively.",
"The steepness of these photon indices are inconsistent with being from a non-thermal component.",
"Therefore, we speculate that these residuals were possibly resulted from some residual soft proton contamination in individual cameras after the data screening.",
"In order to tightly constrain the spectral parameters, we fitted the data obtained from all cameras simultaneously with an untied power-law component to account for the residual soft proton contamination in the XMM-Newton data.",
"The best-fit CIE model yields a column density of $N_{H}=6.85^{+0.33}_{-0.35}\\times 10^{21}$ cm$^{-2}$ and a plasma temperature of $kT=0.64\\pm 0.02$ keV (with $\\chi ^{2}$ = 842.47 for 704 degrees of freedom; hereafter dof).",
"To examine whether the metal abundance of G38.7–1.4 deviates from the solar values, we thawed the corresponding parameters individually to see if the goodness-of-fit can be improved.",
"With the abundances of oxygen (O) and neon (Ne) as free parameters, the fit is found to be somewhat improved ($\\chi ^{2}$ = 762.43 for 702 dof).",
"Both O and Ne are suggested to be overabundant with respect to their solar values.",
"The spectral parameters are tabulated in Table REF .",
"The observed spectra with the CIE fit are display in Figure REF .",
"We also examined these spectra with an absorbed non-equilibrium ionization model of a constant temperature and a single ionization timescale (XSPEC model: VNEI) with the abundances of O and Ne as free parameters.",
"The best-fit plasma temperature and the line-of-sight absorption are found to be $kT=0.65^{+0.03}_{-0.01}$ keV and $N_\\mathrm {H}=5.44^{+0.33}_{-0.34}\\times 10^{21}$ cm$^{-2}$ .",
"The abundances of O and Ne are found to be $7.63^{+1.41}_{-1.30}$ and $3.39^{+0.84}_{-0.77}$ of their solar values.",
"All these parameters are similar to those inferred from the CIE model.",
"The inferred ionization timescale $\\tau _{ion}$ is found to be $\\sim 6.8\\times 10^{12}$ s cm$^{-3}$ , which suggests the system can possibly reach the condition for CIE already.",
"Also, statistically, the NEI model does not provide a better description of the data than the corresponding CIE fit ($\\chi ^{2}$ = 815.44 for 701 dof).",
"Therefore, we will not consider the NEI model in all subsequent discussions.",
"As the line features are not prominent in the observed energy spectra (see Fig.",
"REF ), we have also attempted to fit the data with a simple absorbed power-law model.",
"Nevertheless, this fit yields an undesirable goodness-of-fit ($\\chi ^{2}$ = 1507.73 for 710 dof) and an unreasonable photon index of $\\Gamma \\sim 4.8$ .",
"And hence, this emission scenario will not be further considered.",
"In addition to the diffuse X-ray emission, we also examined the spectrum for the only point source detected by XMM-Newton, namely source A (see Fig.",
"REF ).",
"The source spectrum has been extracted from a circular region with a radius of 15$^{\\prime \\prime }$ centered at the position reported by the source detection algorithm.",
"This choice of extraction regions corresponds to the encircled energy fraction of $\\sim $ 70%.",
"For the background subtraction, we sampled from the nearby low-count regions in the individual camera.",
"There are $\\sim $ 76 net counts in total available for the spectral fitting.",
"In view of its softness (see Sec.",
"3), we first examined the spectrum of Source A wtih a blackbody model (XSPEC model: BBODYRAD) which yields $N_\\mathrm {H}=8.20^{+3.19}_{-2.17}\\times 10^{21}$ cm$^{-2}$ and $kT=0.30^{+0.10}_{-0.07}$ keV with an acceptable goodness of fit $\\chi ^{2}$ = 11.37 for 10 dof.",
"The normalization implies an emitting region with a radius of $R< 151.45D_\\mathrm {kpc}$ m, where $D_\\mathrm {kpc}$ is the source distance in units of kpc.",
"The unabsorbed flux of source A is found to be $f_{x} \\simeq 3.9 \\times 10^{-14}$ ${\\rm erg}\\, {\\rm cm}^{-2}\\, {\\rm s}^{-1}$ in the energy range of 0.5-10.0 keV.",
"We have also attempted to fit the spectrum with a power-law which results in similar a goodness of fit as the blackbody fit ($\\chi ^{2}$ = 11.89 for 10 dof).",
"But we noted that the best-fit photon index ($\\Gamma =6.03^{+1.77}_{-1.33}$ ) is apparently steeper than a conventional range ($\\Gamma \\lesssim 3$ ) of acceleration processes.",
"We have also tested a scenario that it is a clump of the diffuse emission of G38.7–1.4 by fitting a VEQUIL model to its spectrum.",
"However, the goodness of fit is worse than the other two tested models ($\\chi ^{2}$ = 12.29 for 10 dof).",
"Table: X-ray spectral properties of G38.7–1.4.",
"All quoted errors are 1σ\\sigma for 2 parameters of interest.Figure: (upper panel) The X-ray energy spectra of G38.7–1.4 obtained from the XMM-Newton MOS1 (black), MOS2 (red), PN (green)and Chandra ACIS-I (blue)are simultaneously fitted with an absorbed collisional ionization equilibrium plasma model.",
"Additional power-law componenthave been applied to account for the residual soft proton contamination in the individual XMM-Newton camera.",
"(lower panel) Contributions to the χ 2 \\chi ^{2} fit statistic are shown."
],
[
"$\\gamma $ -ray Observation & Data Analysis",
"In order to further probe if G38.7–1.4 is a site for GCR acceleration, we searched for the $\\gamma $ -ray emission at its location.",
"The Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope is able to detect $\\gamma $ -rays with energies between $\\sim $ 100 MeV and $\\sim $ 300 GeV (Atwood et al.",
"2009).",
"Data used in this work were obtained between 2008 August 4 and 2012 November 21, which are available from the Fermi Science Support Centerhttp://fermi.gsfc.nasa.gov/ssc/data/analysis/software/.",
"We used the Fermi Science Tools “v9r23p1” package to reduce and analyse the data in the vicinity of G38.7–1.4.",
"Throughout this paper we used Pass 7 data with events in the “Source” class (i.e., event class 2) only.",
"The corresponding instrument response functions (IRFs) “P7SOURCE_V6\" (Atwood et al.",
"2009), which has been recommended for most analysis, was adopted.",
"Photon energies were restricted to 200 MeV to 300 GeV.",
"To reject atmospheric $\\gamma $ -rays from the Earths limb, we excluded the events with zenith angles larger than 100$$ .",
"In order to reduce systematic uncertainties and achieve a better background modelling, a circular region-of-interest (ROI) with a diameter of 10 centered at the nominal position of G38.7–1.4 was selected in our analysis.",
"To investigate the $\\gamma $ -ray spectral characteristic of G38.7–1.4, we performed an unbinned likelihood analysis with the aid of gtlike, by putting a point source with a PL model at the nominal position of G38.7–1.4 (i.e RA=$19^{\\rm h}07^{\\rm m}05^{\\rm s}$ Dec=$+04^{\\circ }31^{^{\\prime }}11^{^{\\prime \\prime }}$ (J2000)) .",
"For the background model, we included the Galactic diffuse model (gal_2yearp7v6_v0.fits), the isotropic background (iso_p7v6source.txt), as well as all point sources reported in the Fermi LAT 2-Year Source Catalog (2FGL) within 10 from the center of the ROI.",
"All these 2FGL sources were assumed to be point sources which have specific spectra suggested by the 2FGL catalog (Nolan et al.",
"2012).",
"While the spectral parameters of the 2FGL sources locate within the ROI were set to be free, we kept the parameters for those lying outside our adopted ROI fixed at the values given in 2FGL (Nolan et al.",
"2012).",
"We allowed the normalizations of diffuse background components to be free.",
"We found that there is no $\\gamma $ -ray detection of G38.7–1.4 in our study (TS=$-3\\times 10^{-3}$ at the nominal position of G38.7–1.4).",
"With the statistical uncertainties of both photon index and the prefactor concerned, the $1\\sigma $ limiting photon flux at energies $>100$ MeV is constrained to be $<1.3\\times 10^{-9}$ photons cm$^{-2}$ s$^{-1}$ ."
],
[
"Discussion",
"Based on the best-fit X-ray spectral parameters, we discuss the physical properties of G38.7–1.4.",
"Our analysis suggests a plasma temperature of $\\sim 7.5\\times 10^{6}$ K which allows us to estimate the shock velocity $v^{2}_{s}=16kT/(3m_{p} \\mu )$ (Reynolds 2008), where $k$ is the Boltzmann constant, $m_{p}$ is the proton mass, $\\mu $ is the mean mass per particle.",
"For a fully ionized plasma of cosmic abundances ($\\mu \\sim $ 0.6), the shock velocity is estimated to be $\\sim 745$ km s$^{-1}$ .",
"Assuming the shocked densities of hydrogen $n_\\mathrm {H}$ and electrons $n_\\mathrm {e}$ are uniform in the extraction region, the normalization of the CIE model can be approximated by $10^{-14} n_{e} n_\\mathrm {H} V/4 \\pi D^{2}$ , where $D$ is the distance to G38.7–1.4 in cm and $V$ is the volume of interest in units of cm$^{3}$ .",
"Assuming a geometry of an ellipsoid for the extraction region, the volume of interest for the SNR is $\\sim 1.29 \\times 10^{56} D^{3}_\\mathrm {kpc}$ cm$^{3}$ , where $D_\\mathrm {kpc}$ is the remnant distance in units of kpc.",
"Assuming a completely ionized plasma with $\\sim 10\\%$ He ($n_{e} \\sim 1.2 n_\\mathrm {H}$ ), the 1-$\\sigma $ confidence interval of the normalization implies that the shocked hydrogen and electron densities are in the ranges of $n_\\mathrm {H} \\simeq (0.21-0.24) D^{-0.5}_\\mathrm {kpc}$ cm$^{-3}$ and $n_{e} \\simeq (0.25-0.29) D^{-0.5}_\\mathrm {kpc}$ cm$^{-3}$ , respectively.",
"To determine the remnant age, we assume that G38.7–1.4 is in the Sedov phase and the shocked plasma is fully ionized with a single temperature.",
"The shock temperature can be estimated by $T \\simeq 8.1 \\times 10^{6} E^{2/5}_{51} n^{−2/5}_\\mathrm {ISM_{-1}} t^{−6/5}_{4}$ K (Hui et al.",
"2012), where $t_{4}$ , $E_{51}$ , and $n_\\mathrm {ISM_{-1}}$ are the time after the explosion in units of $10^{4}$ years, the released kinetic energy in units of $10^{51}$ erg, and the ISM density of 0.1 cm$^{-1}$ , respectively.",
"Assuming it is a strong shock, $n_\\mathrm {ISM}$ is estimated to be 0.25$n_{H}$ .",
"Taking the 1-$\\sigma $ uncertainties of the temperature inferred from the spectral fitting and assuming $E_{51}=1$ and a distance of 4 kpc, the age of G38.7–1.4 is constrained to be $(1.1 - 1.2) D^{1/6}_\\mathrm {kpc} \\times 10^{4}$ years.",
"Since the distance plays a crucial role in determining the physical properties of a SNR, to estimate the distance of G38.7–1.4 is essential in our study.",
"We first tried to obtain the distance at the lower side via the optical extinction.",
"Lucke (1978) built contour plots of equal mean reddening up to 2 kpc and found the the mean color excess $E_\\mathrm {B-V}=0.25$ mag kpc$^{-1}$ by using color excess and photometric distances in the UBV system for 4000 OB stars.",
"From studies using ultraviolet spectroscopy of reddened stars and X-ray scattering halos in our Galaxy, Predehl & Schmitt (1995) found the relationship between N$_\\mathrm {H}$ and the total extinction A(V) to be approximately N$_\\mathrm {H}$ /A(V)=$1.79\\times 10^{21}$ cm$^{-2}$ .",
"Adopted the N$_\\mathrm {H} \\sim 5.3\\times 10^{21}$ inferred from the spectral fitting (cf.",
"Table REF ) and a typical value of 3.1 for A(V)/$E_\\mathrm {B-V}$ in the Milky Way, we can crudely estimate a distance of $\\sim $ 4 kpc.",
"A dedicated HI observation of this newly identified SNR is encouraged for a more reliable distance estimation.",
"Non-detection of $\\gamma -$ ray emission in the energy range of $0.2-300$ GeV at the nominal position of G38.7–1.4 was reported in our study.",
"We have placed a limiting photon flux of $F$ ($\\ge $ 100 MeV)$<1.3\\times 10^{-9}$ photons cm$^{-2}$ s$^{-1}$ .",
"Using the parameters inferred from the aforementioned X-ray analysis, we can estimate what is the expected intensity of the $\\gamma -$ ray flux from G38.7–1.4.",
"From Drury et al.",
"(1994), the theoretical $\\gamma -$ ray flux can be estimated as $F$ ($\\ge $ 100 MeV)$\\approx 4.4\\times 10^{-7}\\theta \\ (\\frac{\\rm E_{SN}}{10^{51}{\\rm erg}})(\\frac{d}{\\rm 1~kpc})^{-2}(\\frac{n}{\\rm 1~cm^{-3}}) {\\rm cm}^{-2}~{\\rm s}^{-1}$ .",
"Adopting a distance of $d\\sim 4$ kpc, the ambient density of $n=n_{\\rm ISM}\\sim 0.05$ cm$^{-3}$ as inferred from X-ray spectral fit and assuming a canonical explosion energy of $E_{\\rm SN}\\sim 10^{51}$ ergs which converts $\\theta =10\\%$ into GCR energy (cf.",
"Kang 2013), a flux of $F$ ($\\ge $ 100 MeV)$\\sim 1.4\\times 10^{-10}$ ${\\rm cm}^{-2}~{\\rm s}^{-1}$ is expected from G38.7–1.4.",
"A deeper $\\gamma -$ ray search is encouraged for further probing whether G38.7–1.4 is an acceleration site of GCRs indeed."
],
[
"Summary & Conclusion",
"We have performed a detailed spectro-imaging X-ray study of the supernova remnant candidate G38.7–1.4 with XMM-Newton and Chandra.",
"A central-filled X-ray structure correlated with an incomplete radio shell has been revealed.",
"Its X-ray spectrum is thermal dominated and has shown the presence of a hot plasma accompanied with metallic emission lines.",
"These observed properties indicate that G38.7–1.4 is a SNR belong to a mix-morphology category.",
"The enhanced abundances of O and Ne suggest G38.7–1.4 might be resulted from a core-collapsed SN.",
"We have also searched for the possible $\\gamma $ -ray emission from G38.7–1.4 with Fermi LAT data.",
"With the adopted $\\sim 4.3$ yrs data span in this study, we report a non-detection of any $\\gamma -$ ray emission in the energy range of $0.2-300$ GeV.",
"This project is supported by the National Science Council of the Republic of China (Taiwan) through grant NSC100-2628-M-007-002-MY3 and NSC100-2923-M-007-001-MY3.",
"CYH and KAS are supported by the National Research Foundation of Korea through grant 2011-0023383.",
"LT would like to thank the German Deutsche Forschungsgemeinschaft (DFG) for financial support in project SFB TR 7 Gravitational Wave Astronomy."
]
] | 1403.0474 |
[
[
"The helium abundance in the metal-poor globular clusters M30 and NGC6397"
],
[
"Abstract We present the helium abundance of the two metal-poor clusters M30 and NGC6397.",
"Helium estimates have been obtained by using the high-resolution spectrograph FLAMES at the ESO Very Large Telescope and by measuring the HeI line at 4471 A in 24 and 35 horizontal branch stars in M30 and NGC6397, respectively.",
"This sample represents the largest dataset of He abundances collected so far in metal-poor clusters.",
"The He mass fraction turns out to be Y=0.252+-0.003 (sigma=0.021) for M30 and Y=0.241+-0.004 (sigma=0.023) NGC6397.",
"These values are fully compatible with the cosmological abundance, thus suggesting that the horizontal branch stars are not strongly enriched in He.",
"The small spread of the Y distributions are compatible with those expected from the observed main sequence splitting.",
"Finally, we find an hint of a weak anticorrelation between Y and [O/Fe] in NGC6397 in agreement with the prediction that O-poor stars are formed by (He-enriched) gas polluted by the products of hot proton-capture reactions."
],
[
"Introduction",
"Helium is the most abundant among the few chemical elements ($^{3}$ He, $^{4}$ He, D, $^6$ Li, $^7$ Li, $^9$ Be, $^{10}$ B and $^{11}$ B) synthesized directly in the primordial furnax of the Big Bang.",
"The most recent determination of the primordial He mass fraction provides an initial value $Y_{P}$ = 0.254$\\pm $ 0.003 [37].",
"The study of the He content of stars in globular clusters (GCs) is still a challenging task but it is crucial for a number of aspects of the stellar astrophysics.",
"First of all, the He content in Galactic GC stars is thought to be a good tracer of the primordial He abundance because these are among the first generations of stars formed in the Universe and the mixing episodes occurring during their evolution only marginally affect their surface He abundance [68].",
"Moreover, the He content is usually invoked as one of the possible second parameter [29], [21], [15], [51], to explain the observed distribution of stars along the horizontal branch (HB), being the overall metallicity the first parameter.",
"Finally, observational evidence reveal the presence of multiple stellar generations in GCs, formed in short timescales ($\\sim $ 100 Myr) after the initial star-formation burst, from a pristine gas polluted by the products of hot proton-capture processes [30].",
"Thus, these new stars are expected to be characterized by (mild or extreme) He enhancement with respect to the first ones, together with enhancement of Na and Al, and depletion of O and Mg.",
"Despite such an importance, however, the intrinsic difficulties in the derivation of He abundances in low-mass stars have prevented a detailed and systematic investigation of He in GCs.",
"Only few photospheric He transitions are available in the blue-optical spectral range ($<$ 5900 $\\mathring{A}$ ) and they are visible only at high effective temperatures ($T_{eff}$ ).",
"Therefore, He lines in GC stars can be detected only among the HB stars hotter than $\\sim $ 9000 K (the precise boundary also depends on the available signal-to-noise ratio of the spectra, SNR).",
"Instead, the measure of the He abundance in FGK-type stars is limited only to the use of the chromospheric line at 10830 $\\mathring{A}$ , while no photospheric He line is available in these stars.",
"Unfortunately, this transition is extremely weak and very high SNR and spectral resolution are required for a proper measurement.",
"Moreover, the precise He abundance heavily depends on the modeling of the chromosphere.",
"However, this line can provide differential measures of the He abundance, as performed by [57] in two giants in NGC2808, [22] in 12 giant in Omega Centauri and [23] in two giants in Omega Centauri.",
"[57] point out a Y difference of at least 0.17 between the two stars.",
"A similar difference has been suggested by [23] for giants in Omega Centauri.",
"A further complication in the measurement of the He abundance in HB stars is provided by diffusion processes, like radiative levitation and gravitational settling, occurring in the radiative atmospheres of HB stars hotter than $\\sim $ 11000-12000 K, corresponding to the so-called Grundahl Jump [34].",
"These phenomena lead to a substantial modification of the surface chemical composition, and in particular to a decrease of the He abundance [3] and an enhancement of the iron-peak element abundances.",
"As a consequence, only HB stars in the narrow $T_{eff}$ range between $\\sim $ 9000 and $\\sim $ 11000 K can be used as reliable diagnostics of the He content of the parent cluster.",
"At present, determinations of the He mass fraction (Y) in GC HB stars not affected by diffusion processes have been obtained only for some metal-intermediate ([Fe/H]$\\sim $ –1.5/–1.1) GCs: NGC6752 [70], M4 [71], NGC1851 [31], M5 [32], NGC2808 [47] and M22 [33].",
"All these analyses are based on the photospheric He I line at 5875 $\\mathring{A}$ .",
"Some evidence suggest that the variation of He in GC stars is linked to different chemical compositions.",
"The differential analysis performed by [57] on two giants in NGC2808 with different Na content highlights that the Na-rich star is also He enriched at odds with the Na-poor one.",
"[70] and [71] derived He, Na and O abundances for HB stars in NGC6752 and M4, respectively, finding that the stars along the reddest part of the HB of NGC6752 have a standard He content, as well as Na and O abundances compatible with the first generation, while the stars in the bluest part of the HB of M4 are slightly He-enhanced (by $\\sim $ 0.05), with Na and O abundance ratios compatible with the second stellar generation.",
"In a similar way, [47] found a clear evidence of He enhancement (by $\\sim $ 0.09) among the bluest HB stars in NGC2808, that are also all Na-rich.",
"Further spectroscopic evidence (not including the measure of He abundances) strengthen the connection between the HB morphology and the chemical composition, pointing out that the bluest portion of the HB (before the onset of the radiative levitation) is populated mainly by second generation stars, while the reddest part of the sequence is dominated by first generation stars [45] or by a mixture of first and second generation stars [47].",
"In this paper we present the first determination of the He abundance in HB stars of the metal-poor GCs M30 and NGC6397 ([Fe/H]= –2.28$\\pm $ 0.01 and [Fe/H]= –2.12$\\pm $ 0.01, [41] and [42], respectively)."
],
[
"Observations",
"In this work we analyzed a set of high-resolution spectra acquired with the multi-object spectrograph FLAMES in the MEDUSA/GIRAFFE mode at the Very Large Telescope of the European Southern Observatory .",
"The spectra are part of a dataset secured within a project aimed at studying the general properties of blue straggler stars [24], [25], [27], [41], [42].",
"The employed GIRAFFE grating is HR5A (4340-4587 $\\mathring{A}$ , with a spectral resolution of $\\sim $ 18000), suitable to sample the He I line at 4471.5 $\\mathring{A}$ .",
"Spectra have been reduced with the standard ESO FLAMES pipeline.",
"Six exposures of 45 min each have been secured in each cluster.",
"The SNR per pixel of the spectra around the He line ranges from $\\sim $ 60 up to $\\sim $ 130 for M30, and from $\\sim $ 75 up to $\\sim $ 220 for NGC6397.",
"Radial velocity, atmospheric parameters and projected rotational velocity ($v_{e}$ sini) of each target have been derived and discussed in [41], [42] and we refer the reader to those papers for a detailed description.",
"Excluding stars with too noisy spectra and/or too low temperatures (for which the He I line is not detectable), we are finally able to measure the He I line in 24 stars of M30 and in 35 of NGC6397.",
"Fig.",
"REF shows the position of the targets in the color-magnitude diagrams of the two clusters (large circles) .",
"Table 1 lists their coordinates and atmospheric parameters.",
"Figure: Color-magnitude diagram of M30 and of NGC6397 (Contreras Ramos et al., 2014, in preparation, right panel):large circles are the FLAMES targets.",
"The superimposed black curve are the theoreticalZAHB models used to infer the atmospheric parameters."
],
[
"Chemical analysis",
"Stellar atmospheric parameters have been derived by [41] and [42] from the photometry.",
"We recall the main information about the atmospheric parameters determination.",
"$T_{eff}$ and logg have been derived by projecting the position of each star in the (V, V-I) plane on the best-fit theoretical Zero-Age Horizontal Branch (ZAHB) model.",
"For NGC6397 the used ZAHB model is from the BaSTI dataset [59], while for M30 the ZAHB model is from the Pisa Evolutionary Library dataset [6].",
"For the latter, the choice of a different database of theoretical models is done for consistency with the analysis by [25].",
"However, we checked the consistency between the two sets of models: ZAHB models of the two databases chosen with the same metallicity well overlap each other both in the observative and theoretical plane.",
"The adoption of a dataset instead of another one leads to negligible changes in the atmospheric parameters, typically smaller than 30-40 K and 0.05 in $T_{eff}$ and logg, respectively (note that [47] found a good agreement by using BasTI and PGPUC [69] ZAHB models).",
"The used ZAHB models are shown in Fig.",
"REF .",
"The He abundance has been obtained for each target by fitting the observed He I line at 4471.5 $\\mathring{A}$ with a grid of synthetic spectra, calculated with the appropriate atmospheric parameters and varying only the He abundance.",
"The use of spectral synthesis (instead of the simple measure of the line equivalent width) is mandatory in the analysis of this line, to properly account for its relevant Stark broadening and to include the forbidden component at 4470 $\\mathring{A}$ [49].",
"Fig.",
"REF shows the spectral region around the He line for one of the hottest and one of the coldest target stars in both clusters, with overplotted three synthetic spectra computed with the best-fit Y abundance and $\\delta $ Y=$\\pm $ 0.1.",
"Figure: Spectral region around the He line for the stars #61653 and #53805 in NGC6397and for the stars #30000038 and $30011110 in M30 (thick grey line),with overplotted synthetic spectra (black thin lines) calculated with the best-fitY abundance and δ\\delta Y=±\\pm 0.1.Synthetic spectra have been computed with the code SYNTHE [66] adopting the line list provided by F. Castelli in her website.",
"They have been convolved with a Gaussian profile in order to properly reproduce the spectral resolution of the HR5 grating and with a rotational profile in order to include the projected rotational velocities derived by [41], [42].",
"In order to properly take into account the contribution of the H and He abundances to the opacity, we calculated all the model atmospheres with the last version of the code ATLAS12http://wwwuser.oat.ts.astro.it/castelli/sources/atlas12.html [11].",
"At variance with the widely used ATLAS9 code (that adopts pre-tabulated opacities calculated for specific chemical mixtures, in particular with standard He mass fraction Y= 0.245), ATLAS12 employs the opacity sampling method [58] and allows one to calculate model atmospheres with arbitrary chemical composition.",
"All the model atmospheres have been computed under the assumption of Local Thermodynamical Equilibrium (LTE) and one-dimensional, plane-parallel geometry.",
"We checked the impact of the use of ATLAS9 and ATLAS12 models on the derived He abundance.",
"For Y around the standard value (Y$\\sim $ 0.25) the two models provide the same result, while for He-enhanced stars (at least up to Y$\\sim $ 0.3), the adoption of ATLAS9 models under-estimates Y of about 0.02.",
"On the other hand, for stars with surface He mass fraction of $\\sim $ 0.10, analysis based on the standard ATLAS9 models overestimate Y by $\\sim $ 0.05.",
"Despite the analyzed He transition can suffer for departures from LTE conditions (relevant for B-type stars), this effect is negligible considering the atmospheric parameters and the metallicities of our targets (P. Bonifacio, private communication).",
"The total uncertainty for each star is derived by adding in quadrature the uncertainty in the fitting procedure and that arising from the adopted parameters.",
"The uncertainty in the fitting procedure has been estimated by using MonteCarlo simulations.",
"For observed spectra, the uncertainty associated to a $\\chi ^2$ -minimization cannot be estimated by using the $\\chi ^2$ theorems, that assume that all the pixels are not correlated each other [13], [5].",
"In fact, in observed spectra the adjacent pixels cannot be considered as independent each other because of the re-binning procedure during the wavelength calibration.",
"For each star, we computed a set of 1000 synthetic spectra, calculated with the appropriate atmospheric parameters, rotational velocities and the best-fit He abundance.",
"Each MonteCarlo spectrum has been obtained by rebinning the best-fit synthetic spectrum to the same pixel-size of the GIRAFFE spectra (0.05 $\\mathring{A}$ /pixel) and then by injecting Poissonian noise, in order to reproduce the SNR of each star around the He line.",
"Thus, this set of synthetic spectra is equivalent to the real one but with the He abundances known a priori.",
"This method allows to take into account simultaneously the main sources of uncertainty in the line fitting, namely the finite size of the pixels, the SNR and the continuum estimate.",
"The same analysis performed for the observed spectra has been done for the synthetic ones and the dispersion of the derived Y abundance distribution has been assumed as 1$\\sigma $ uncertainty in the fitting procedure.",
"These uncertainties depend on the injected SNR, but also on the line strength (thus the temperature and the He abundance) and the rotational velocity.",
"Typical errors in He mass fraction range from 0.01 up 0.05.",
"Because we are interested in possible star-to-star variations of the He content, we estimated the internal uncertainties due to the atmospheric parameters.",
"The total error obtained by adding in quadrature the uncertainties due to the individual atmospheric parameters is an upper limit of the internal error, because it does not take into account the covariance terms occurring among the parameters.",
"In order to take into account the effect of the projection process on the derived $T_{eff}$ and log g, we adopt the following procedure: we re-projected each target on the best-fit ZAHB by including its photometric uncertainty, re-determining simultaneously $T_{eff}$ and log g, in order to include the correlation between the two parameters.",
"With this method we derive variations in $T_{eff}$ between $\\sim $ 70 and $\\sim $ 150 K, with corresponding variations in gravity of the order of 0.02.",
"These relatively small uncertainties in $T_{eff}$ and log g are essentially due to the high internal accuracy of the adopted photometric catalogs, with typical photometric uncertainties of $\\sigma $ (V-I)$\\sim $ 0.01-0.02 mag, obtained by the average of several independent measures [25].",
"Note also that these uncertainties do not represent the total error budget in the adopted parameters, but only the internal star-to-star uncertainty related to the adopted procedure in the parameter derivation.",
"Only to provide the general variation of Y due to this procedure, an uncertainty of $\\pm $ 100 K in $T_{eff}$ (coupled with the corresponding variation in gravity of $\\pm $ 0.02) provides a variation in Y of $\\pm $ 0.01 for the hottest stars ($\\sim $ 11000 K) and of $\\pm $ 0.02 for the coldest targets ($\\sim $ 9000 K), whereas the impact of microturbulent velocity is totally negligible.",
"The error in $v_{e}$ sini (typically 2-3 km/s) provides a contribution at a level of less than 0.005.",
"Note that the 4471 $\\mathring{A}$ He line used in this work is slightly less sensitive to the adopted atmospheric parameters with respect to the line at 5875 $\\mathring{A}$ , adopted in the other papers where the He abundance in GC stars is derived.",
"Finally, systematic effects can be due to the choice of the ZAHB model.",
"As extensively discussed by [47], a possible source of systematic errors is the He abundance of the used ZAHB.",
"The He abundance of our targets is not known a priori, thus we derived the atmospheric parameters adopting ZAHB models computed with standard Y.",
"The adoption of a Y-enhanced ZAHB leads to a decrease of gravity by $\\sim $ 0.1-0.15, with a negligible impact on the temperature.",
"Note that a systematic decrease of 0.1 in log g (keeping $T_{eff}$ fixed) implies an increase of the derived Y smaller than 0.02/0.03.",
"As discussed in Section , the adoption of the standard He content for the used ZAHB models is reasonable in light of the derived He content of our targets, thus we do not need to re-derive the atmospheric parameters by using ZAHB models computed with higher Y.",
"A similar effect is obtained if we consider that the stars leaving the ZAHB locus will be more luminous and with a lower gravity (but basically the same temperature) with respect to the ZAHB position."
],
[
"The He content of M30 and NGC6397",
"Table 1 lists the derived He mass fraction of the targets and their total uncertainty.",
"Fig.",
"REF shows the behavior of Y as a function of the temperature for the stars of M30 (upper panel) and NGC6397 (lower panel), while Fig.",
"REF shows the Y distributions in the two samples of stars represented as generalized histograms [38].",
"In both GCs, all the stars have Y around $\\sim $ 0.24-0.25, with the exception of one star in M30 and two stars in NGC6397, that show very low (Y$<$ 0.1) He abundance.",
"The three stars with low He content also show iron abundances higher than that of the parent cluster [41], [42].",
"This behavior is commonly observed in HB stars hotter than the Grundahl Jump [1], [2], [35], [31] and it is predicted by theoretical models [48], [63] as an effect of radiative levitation (responsible for the metal enhancement) and gravitational settling (responsible for the He depletion).",
"We note that some stars with similar $T_{eff}$ to those of the He-poor stars, but with normal Y, are detected.",
"This difference can be due to the fact that we observe the region close to the Grundahl Jump and not all the stars have still undergone the diffusion processes.",
"Interestingly enough, one star in NGC2808 with a temperature higher than that of the Grundahl Jump does not show any evidence of He depletion [47].",
"Excluding the three Fe-rich and He-poor stars, we find average He mass fractions of Y=0.252$\\pm $ 0.005 ($\\sigma $ =0.021) for M30 and Y=0.241$\\pm $ 0.004 ($\\sigma $ =0.023) for NGC6397.",
"It is worth to notice that these are not only the first determinations of Y for M30 and NGC6397, but they are also the first ones for GCs with [Fe/H]$<$ –2.0 dex [2] and [3] identified one HB star in M92 and one HB star in M15 (both [Fe/H]$<$ –2.0 dex), not affected by levitation and gravitational settling effects.",
"However, their huge uncertainties ($\\sim $ 0.3-0.4 dex) do not allow to firmly establish the real He content of these GCs..",
"Figure: Behavior of the He mass fraction Y as a functionof the temperature for the HB stars of M30 (upper panel) and NGC6397 (lower panel).Solid lines are the linear fits calculated excluding the stars with evidence ofradiative levitation.Figure: Generalized histograms for the He mass fractionof the HB stars in M30 (upper panel) and in NGC6397 (lower panel).Figure: Behaviour of the He mass fraction Y as a function of the V-I color (left panel) andof the V-band magnitude (right panel) for M30 (upper panels) and for NGC6397 (lower panels).Solid lines are the best-fit linear fits (the corresponding slopes and uncertainties are labelled).According to the theoretical models of [59], the surface He mass fraction for a star with 0.8 ${\\rm M}_{\\odot }$ , Z= 0.0003 (corresponding to [Fe/H]=–2.1) and $\\alpha $ -enhanced chemical mixture, increases by only 0.01 with respect to the initial value, after the First Dredge-Up episode.",
"Therefore, the derived He abundances of HB stars in M30 and NGC6397 are totally compatible with the expectations for low-mass evolved stars formed with a primordial He abundance [37].",
"No trend between the He abundances and the corresponding (V-I) color and V-band magnitude is detected for the stars with no evidence of radiative levitation .",
"Fig.",
"REF shows the behaviour of Y as a function of (V-I) and V. The best-fit linear fits are calculated with the routine fitexy by [62] to take into account the uncertainties in both the quantities, whereas the corresponding uncertainties in the slope are calculated with the Jackknife bootstrapping technique.",
"In a similar way, no evident trend between Y and $T_{eff}$ is recognized: Fig.",
"REF shows the linear fits, providing slopes of 6.4$\\cdot 10^{-7}\\pm $ 0.005 and -1.15$\\cdot 10^{-5}\\pm $ 0.008.",
"The observed Y values among the stars of each target GC are compatible within the uncertainties.",
"Thus, we can conclude that the two GCs are not strongly enriched in He, displaying a substantial He uniformity: only small (if any) Y variations could be present in their stellar content.",
"This result agrees with the analysis of NGC6397 by [19], based on the width of the observed main sequence (MS), that predicts a maximum internal variation of $\\sim $ 0.02 in the Y distribution of the cluster MS stars.",
"Further results by [50] revealed the presence of a double MS in the color-magnitude diagram of NGC6397.",
"This can be reproduced with a population (accounting for 30% of the total cluster population) having normal Y and another one with a mild He-enhancement of about 0.01.",
"Again, this is fully consistent with our results.",
"Concerning M30, no study so far has revealed splitting or anomalous broadening of the MS, suggesting a small or null intrinsic dispersion in the He content of this cluster, in agreement with our findings.",
"Finally, the uniform He content that we find in the HB stars of M30 and NGC6397 well agrees with theoretical models that predict only a mild He enhancement for clusters with HB morphologies similar to that of our targets (covering a narrow extension in color, thus in $T_{eff}$ ; see Fig.",
"REF ), at odds with clusters with very extended blue tails for which high He enhancements (Y$\\ge $ 0.30) are predicted [15]."
],
[
"He abundance and self-enrichment process",
"He enrichment in GC sub-populations is expected in light of the self-enrichment processes, thought to occur during the early stages (within $\\sim $ 100 Myr) of GC history.",
"All the GCs studied so far, both in the Milky Way [7] and in other galaxies of the Local Group [39], [53], display well-established chemical patterns, with homogeneous iron-peak element abundances and with anticorrelations between C and N, between O and Na, and (for some clusters) between Mg and Al.",
"The only exceptions are a bunch of peculiar GC-like systems with an intrinsic dispersion in their iron content (with broad and/or multimodal [Fe/H] distributions), namely Terzan 5 [26], Omega Centauri [36], [55], [56], M22 [44], [46] and M54 [4], [9] Note that other GCs are suspected to have small iron dispersions, namely NGC1851 [10], NGC5824 [65] and NGC3201 [67], [54], but there is no general consensus about them..",
"The chemical patterns involving light elements and observed in GCs are commonly interpreted as the signature of material processed through the high temperature extension of the proton-capture reactions (like NeNa and MgAl cycles).",
"Intermediate-mass AGB stars [18] and fast-rotating, massive stars [17], both able to ignite the complete CNO-cycle, have been proposed as main polluters.",
"Whichever the true nature of the polluters is, new cluster stars, formed from pristine gas diluted with material processed in the stellar interiors, are expected to be also enriched in He, with a level of He enrichment varying from cluster to cluster, from very small values ($\\le $ 0.02), as in the case of NGC6397 [19], [50], up to extreme He contents (Y$\\sim $ 0.4), as those proposed to explain the complex MS and/or HB morphologies observed in $\\omega $ Centauri [60], NGC2808 [16], [61], [14] and NGC2419 [20].",
"A first, indirect hint of Y-[O/Fe] anti-correlation has been provided by [70] and [71], who analyse red HB stars of NGC6752 and blue HB stars of M4, respectively.",
"The HB stars in NGC6752 show enhanced [O/Fe] ratios and Y compatible with the cosmological value, while the stars along the blue portion of the HB in M4 have enhanced values of Y (by 0.04-0.05) and [O/Fe] ratios compatible with the second generation stars of the cluster.",
"Even if performed on two different clusters, these results by [70] and [71] suggest that the blue part of the HB is mainly populated by stars formed from gas enriched in He and, generally speaking, by the products of the high temperature proton-capture reactions.",
"Analysis based on other elements and not involving directly the measure of the He abundance, have confirmed the connection between the position of the HB stars and their chemical composition [45], [31].",
"We can use our dataset to probe the existence of any Y-[O/Fe] correlations in the two surveyed clusters.",
"Indeed [41] and [42] measured non-LTE [O/Fe] abundances for several HB stars of the two target clusters from the oxygen triplet at $\\sim $ 7770 $\\mathring{A}$ .",
"Fig.",
"REF shows the behavior of Y as a function of [O/Fe] (excluding the Y-poor stars where the radiative levitation and gravitational settling have modified the surface abundances).",
"Abundances of both [O/Fe] and Y are available for only 12 stars of M30.",
"No correlation between the two abundances is detectable (upper panel in Fig.",
"REF ): a straight line fit, performed with the routine fitexy by [62] provides a slope of -0.011$\\pm $ 0.057 [43].",
"The small probability of correlation is confirmed also by the Spearman rank correlation coefficient ($C_S$ =–0.50), leading to a probability of only 90% that the two abundances are correlated.",
"On the other hand, the sample of 33 stars of NGC6397 for which both O and Y are available displays a mild Y-[O/Fe] anti-correlation (lower panel in Fig.",
"REF ).",
"A linear fit provides a slope of -0.036$\\pm $ 0.010, corresponding to a 3.6$\\sigma $ detection.",
"The Spearman rank correlation coefficient is $C_S$ =–0.54, providing a probability higher than 99.9% of an anti-correlation between the two abundances.",
"The same result is confirmed also by a non-parametric Kendall-$\\tau $ test In a similar way, [52] recognized a very mild anti-correlation between Na and Li abundances among the dwarf stars of M4 (and justified in the framework of the multiple populations in GCs).",
"Even if their abundance distributions do not show evidences of intrinsic scatter (in light of the estimated uncertainties), both parametric and non-parametric rank correlation test highlight an anti-correlation between the two abundances.. An interesting difference between the two clusters is their [O/Fe] distributions, being that of NGC6397 larger than that of M30 and including a component with [O/Fe]$<$ 0.",
"Previous determinations of the O abundance in NGC6397 provide a small range of [O/Fe], with no evidence so far of O-poor stars.",
"Despite its proximity, the number of stars in NGC6397 in which the O abundance has been measured is very small and most of the analysis available so far are based on the forbidden O line at 6300 $\\mathring{A}$ .",
"[12] provided [O/Fe] for 2 (out of 16) giants, finding for both the stars [O/Fe]=+0.15 dex.",
"[7], [8] properly measured O in 12 giants observed with UVES (reaching [O/Fe]=+0.11 dex) and provided upper limits for other 7 giants, while for most of the stars of their GIRAFFE survey no measures at all are provided, because of the low SNR and the radial velocity of the cluster (RV$\\sim $ 20 km/s) that leads to an overlap between the forbidden O line with the sky O emission line.",
"Recently, [40] derived O abundances for 16 giant stars, finding a very small variation of O among their stars, from [O/Fe]=+0.41 up +0.77 dex.",
"Only [28] measured the oxygen triplet at 7770 A for 7 dwarf/subgiants (and an upper limit) finding a range between +0.08 and +0.48 dex.",
"However, we suggest a possible bias in the measure of the O distribution of NGC6397 from giant stars.",
"The derivation of the precise [O/Fe] abundances range in the giant stars of metal-poor globular clusters can be quite complex, because the only available oxygen line is the forbidden one that is very weak at low metallicity.",
"Moreover, the almost zero radial velocity makes impossible to properly detect the O line (in the case of M 30 this effect does not occur because of its radial velocity, -185 km/s, prevents any blending with the sky emission line).",
"We conclude that the giant stars are not the best sample to properly study the O abundance (and in particular to identify the most O-poor stars) in NGC6397.",
"If a [O/Fe] sub-solar component does exist among the star of NGC6397, it cannot be detected from the analysis of its giant stars.",
"On the other hand, the O triplet at 7770 A is well detectable and strong among HB stars, providing a more robust diagnostic.",
"Also, we note the very good match between our [O/Fe] distribution and that by [8] for M30, where the very low radial velocity of this cluster prevents any spurious blending between the forbidden O line and the emission O sky line.",
"We checked whether the impact of the atmospheric parameters uncertainties is able to introduce a spurious anti-correlation between the two abundances.",
"In fact, the increase of $T_{eff}$ (coupled with the corresponding increase of log g) leads to an increase of [O/Fe] and a decrease of Y.",
"However, the slope is significantly steeper (-0.75) than that observed for the stars in NGC6397 (see the arrows in Fig.",
"REF , showing the effects of a change in $T_{eff}$ and log g by –200 K and –0.04, respectively).",
"This slope remains the same for stars with different atmospheric parameters and with difference O abundances.",
"Thus, we can rule out that the observed anti-correlation is an artifact of the uncertainty of the atmospheric parameters.",
"Also, we checked that no correlation does exist between the abundances and $v_{e}$ sini; note that the internal uncertainties in $v_{e}$ sini are not able to introduce a spurious anticorrelation between the abundances.",
"As a sanity check, we roughly divided the Y abundances of NGC6397 in two samples, corresponding to [O/Fe] lower and higher than the solar value, finding $<$ Y$>$ = 0.258$\\pm $ 0.005 ($\\sigma $ = 0.015) and $<$ Y$>$ = 0.233$\\pm $ 0.005 ($\\sigma $ = 0.022), respectively.",
"This small difference [19], [50] corresponds to a 3.5$\\sigma $ detection.",
"A Kolmogorov-Smirnov test provides a $\\sim $ 1% probability that the Y abundances of the stars with sub-solar [O/Fe] abundances are extracted from the same population as the stars with [O/Fe]$>$ 0.0."
],
[
"Summary",
"We have analysed the He mass fraction Y for a sample of 24 and 35 HB stars in M30 and NGC6397, respectively.",
"The main results are: (i) both clusters have an average He content compatible with the primordial He abundance ($<$ Y$>$ =0.252$\\pm $ 0.003 for M30 and $<$ Y$>$ =0.241$\\pm $ 0.004 for NGC6397) and they are not strongly enriched in He; (ii) a weak (but statistically significant) anticorrelation between Y and [O/Fe] among the HB stars of NGC6397 does exist (but it is not detected in M30).",
"We suggest that the O-poor, He-rich stars found in the HB of NGC6397 belong to the second stellar generation of the cluster.",
"Unfortunately Na abundances are not available for these stars.",
"In principle, Y-[O/Fe] anti-correlation is expected in all the GCs displaying the chemical signatures of the self-enrichment processes, even if its very small slope makes its detection very hard.",
"The lack of Y-[O/Fe] anti-correlation for the stars in M30 can be due to several causes, mainly the size of our sample (three times smaller than that secured for NGC6397) and the SNR of the spectra (lower than that of the spectra of NGC6397).",
"Also, we cannot rule out that M30 has undergone a self-enrichment process less efficient with respect to NGC6397, as suggested by their different [O/Fe] distributions (in fact M30 shows a lack of stars with [O/Fe]$<$ 0, instead detected among the stars of NGC6397).",
"Thus, the internal variation of the He content in the stellar population of M30 could be smaller than 0.01.",
"The Y-[O/Fe] anti-correlation observed in NGC6397 seems to confirm the theoretical expectations that the GC stars born after the first burst of star formation are both depleted in O and (mildly) enriched in He, demonstrating that the stars usually labelled as second generation stars show the signatures of hot-temperature proton-capture processes, with a simultaneous O-depletion and a weak He enrichment.",
"Figure: Behavior of the Y abundance as a function of[O/Fe] for the stars in M30 (upper panel) and NGC6397 (lower panel).The arrows show the effects of a change inT eff T_{eff} and logg.",
"Labelled are the values of the Spearman correlationcoefficient.",
"Solid lines are the best-fit straight lines.The authors warmly thanks the anonymous referee for his/her helpful comments that improved the quality of the paper.",
"Also, we are grateful to P. Bonifacio for his useful suggestions about the NLTE corrections.",
"This research is part of the project COSMIC-LAB funded by the European Research Council (under contract ERC-2010-AdG-267675).",
"ccccccc 7 0pc Stellar parameters and He abundances.",
"ID Ra Dec $T_{eff}$ log g Y $\\sigma _{Y}$ (J2000) (J2000) (K) M30 10201925 325.0989565 -23.1613142 10914 3.7 0.229 0.021 10202614 325.0891261 -23.1711663 10069 3.6 0.243 0.041 10203922 325.0904138 -23.1512884 10186 3.6 0.261 0.051 10301333 325.1135774 -23.1751181 10023 3.6 0.216 0.056 10301793 325.1201773 -23.1736029 9376 3.4 0.266 0.036 10400762 325.0974077 -23.1873751 9226 3.4 0.208 0.036 10401890 325.1056998 -23.1838967 9931 3.6 0.216 0.045 Identification numbers, coordinates, temperatures, gravities [41], [42], He mass fractions and uncertainties for the observed stars.",
"A complete version of the table is available in electronic form."
]
] | 1403.0595 |
[
[
"Convexity Properties of Dirichlet Integrals and Picone-type Inequalities"
],
[
"Abstract We focus on three different convexity principles for local and nonlocal variational integrals.",
"We prove various generalizations of them, as well as their equivalences.",
"Some applications to nonlinear eigenvalue problems and Hardy-type inequalities are given.",
"We also prove a measure-theoretic minimum principle for nonlocal and nonlinear positive eigenfunctions."
],
[
"A general overview",
"The aim of this paper is to study three elementary convexity principles which have found many applications in eigenvalue problems and functional inequalities.",
"In particular, we will focus on their mutual relations and prove that they are indeed equivalent.",
"In order to smoothly introduce the reader to the subject and clarify the scopes of the paper, we start with the three basic examples which will serve as a model for the relevant generalizations considered in the sequel: the first one is the convexity of the Hamiltonian function for a system of one free massive particle in classical mechanics $\\frac{1}{2}\\, \\frac{|\\phi |^2}{m},$ which is jointly convex as a function both of the mass $m>0$ and of the conjugate momentum $\\phi \\in \\mathbb {R}^N$ ; the second one is the convexity of the quantity $|\\nabla u|^2$ along curves of the type $\\sigma _t=\\Big ((1-t)\\, u^2 + t\\, v^2\\Big )^\\frac{1}{2},\\qquad t\\in [0,1],$ where $u,v\\ge 0$ are differentiable functions, i.e.",
"$|\\nabla \\sigma _t|^2\\le (1-t)\\,|\\nabla u|^2+t\\,|\\nabla v|^2.$ This is also sometimes called hidden convexity; the third and last one is the so-calledThis formula is called “identity” even if it is an inequality, because the difference of the two terms can be written as $\\left\\langle \\nabla u,\\nabla \\left(\\frac{v^2}{u}\\right)\\right\\rangle - |\\nabla v|^2=-\\left|\\nabla v-\\nabla u\\, \\frac{v}{u}\\right|^2,$ which is indeed non positive.",
"The latter is the equality which appears in the original paper [25] by Mauro Picone, after whom the formula is named.",
"The identity is used there to obtain comparison principles for ordinary differential equations of Sturm-Liouville type.",
"Picone identity $\\left\\langle \\nabla u,\\nabla \\left(\\frac{v^2}{u}\\right)\\right\\rangle \\le |\\nabla v|^2,$ where again $u,v\\ge 0$ are differentiable functions, this time with $u>0$ .",
"As a well-known consequence of the convexity of the previous Hamiltonian, we get that $\\rho \\mapsto \\frac{|\\nabla \\rho (x)|^2}{\\rho (x)},$ is convex for every $x$ .",
"This in turn implies convexity of the Fisher information with respect to a reference probability measureHere $\\mu \\ll \\nu $ means that $\\mu $ is absolutely continuous with respect to $\\nu $ .",
"We denote by $d\\mu /d\\nu $ the Radon-Nykodim derivative of $\\mu $ with respect to $\\nu $ .",
"$\\nu $ .",
"For every probability measure $\\mu $ , this functional is given by $\\mathcal {J}(\\mu \\vert \\nu ) = \\int \\left|\\nabla \\log \\rho \\right|^2\\,\\rho \\, d\\nu =\\int \\frac{|\\nabla \\rho |^2}{\\rho }\\, d\\nu ,\\qquad \\mbox{ if } \\mu \\ll \\nu \\, \\mbox{ and }\\, \\rho =\\frac{d\\mu }{d\\nu },$ and observe that the latter can also be re-written as $\\mathcal {J}(\\mu \\vert \\nu )= 4\\, \\int \\left|\\nabla \\sqrt{\\rho }\\,\\right|^2\\,d\\nu ,\\qquad \\mbox{ if } \\mu \\ll \\nu \\, \\mbox{ and }\\, \\rho =\\frac{d\\mu }{d\\nu }.$ From the previous we thus get for $\\rho _t=(1-t)\\, \\rho _0+t\\,\\rho _1$ $\\int \\left|\\nabla \\sqrt{\\rho _t}\\,\\right|^2\\,d\\nu \\le (1-t)\\, \\int \\left|\\nabla \\sqrt{\\rho _0}\\,\\right|^2\\,d\\nu +(1-t)\\, \\int \\left|\\nabla \\sqrt{\\rho _1}\\,\\right|^2\\,d\\nu .$ Thus in particular if $\\rho _t=(1-t)\\, u^2 + t\\, v^2,$ we then obtain that the Dirichlet integral is convex along curves of the form $\\sigma _t=\\sqrt{(1-t)\\, u^2 + t\\, v^2},\\qquad t\\in [0,1],$ which is the hidden convexity exposed above.",
"This striking convexity property of the Dirichlet integral seems to have been first noticed by Benguria in his Ph.D. dissertation (see [5] for example).",
"Note that along the curve of functions $\\sigma _t$ we have $\\Vert \\sigma _t\\Vert ^2_{L^2}=(1-t)\\, \\Vert u\\Vert _{L^2}^2+t\\, \\Vert v\\Vert ^2_{L^2},$ then if $u,v$ belong to the unit sphere of $L^2$ , the same holds true for $\\sigma _t$ .",
"The latter incidentally happens to be a constant speed geodesic for the metric defined by $d_2(u,v) = \\int \\left|u^2-v^2\\right|\\,dx,\\qquad u,v\\in L^2.$ As one should expect, the geodesic convexity described above is helpful to get uniqueness results in eigenvalue problems.",
"We recall that eigenvalues of the Dirichlet-Laplace operator $-\\Delta $ on an open set $\\Omega \\subset \\mathbb {R}^N$ such that $|\\Omega |<\\infty $ are defined as the critical points of the Dirichlet integral on the manifold $\\mathcal {S}_2(\\Omega )=\\left\\lbrace u\\in W^{1,2}_0(\\Omega )\\, :\\, \\int _\\Omega u^2\\, dx=1\\right\\rbrace .$ This constraint naturally introduces Lagrange multipliers, which by homogeneity are the eigenvalues of the Laplace operator, i.e.",
"any constrained critical point $u$ is a weak solution of $-\\Delta u = \\lambda \\, u, \\quad \\mbox{ in }\\ \\Omega ,\\qquad \\qquad u=0, \\quad \\mbox{ on }\\partial \\Omega .$ One says that the function $u$ is an eigenfunction corresponding to the eigenvalue $\\lambda $ .",
"Then the idea is very simple: for a minimum convex problem, critical points are indeed minimizers.",
"This means that hidden convexity trivializes the global analysis for the Dirichlet energy on $\\mathcal {S}_2(\\Omega )\\cap \\lbrace u\\ge 0\\rbrace $ and there cannot be any constant sign critical point $u$ other than its global minimizer.",
"Since the strong minimum principle states that any constant sign eigenfunction (up to a sign) is strictly positive, this imposes any eigenfunction $v\\ge 0$ to be associated with the least eigenvalue $\\lambda _1(\\Omega ) = \\min _{u\\in W^{1,2}_0(\\Omega )} \\left\\lbrace \\int _\\Omega |\\nabla u|^2\\,dx\\, :\\, \\int _\\Omega |u|^2\\,dx=1\\right\\rbrace ,$ which turns out to be simple as well.",
"Another way to prove the same result would be precisely by means of Picone inequality.",
"Let us call $u_1$ a first eigenfunction of $\\Omega $ , i.e.",
"a function achieving $\\lambda _1(\\Omega )$ .",
"If $v\\ge 0$ is a nontrivial eigenfunction with eigenvalue $\\lambda $ , then by strong minimum principle $v>0$ and by Picone inequality one would get $\\lambda =\\lambda \\,\\int _\\Omega v\\, \\frac{u_1^2}{v}=\\int _\\Omega \\left\\langle \\nabla v,\\nabla \\left(\\frac{u_1^2}{v}\\right)\\right\\rangle \\, dx\\le \\int _\\Omega |\\nabla u_1|^2\\, dx=\\lambda _1(\\Omega ),$ and thus $\\lambda _1(\\Omega )=\\lambda $ since $\\lambda _1(\\Omega )$ is the minimal eigenvalue.",
"Of course, in the case of the Laplace operator $-\\Delta $ simplicity of $\\lambda _1(\\Omega )$ and uniqueness of constant sign eigenfunctions are plain consequences of the Hilbertian structure and of the strong minimum principle for supersolutions of uniformly elliptic equations.",
"Indeed, any first eigenfunction $u_1$ must have constant sign and can never vanish on the interior of the connected set $\\Omega $ .",
"Then any other eigenfunction has to be orthogonal in $L^2(\\Omega )$ to $u_1$ (i.e.",
"it has to change sign) unless it is proportional to $u_1$ ..."
],
[
"Aim of the paper",
"...nevertheless, the advantage of the hidden convexity exposed above is that it does not involve any orthogonality concept and applies to general Dirichlet energies of the form $\\int _\\Omega H(\\nabla u)\\,dx,$ where $z\\mapsto H(z)$ is convex, even and positively homogeneous of degree $p>1$ .",
"Moreover, we prove in Proposition REF that this remains true for the whole class of interpolating curves $\\sigma _t=\\Big ((1-t)\\, u^q+t\\, v^q\\Big )^\\frac{1}{q},\\qquad t\\in [0,1].$ with $1\\le q\\le p$ .",
"We point out that $q=1$ corresponds to convexity in the usual sense and that for $q>p$ the property ceases to be true, see Remark REF .",
"Like in the previous model case $p=q=2$ and $H(z)=|z|^2$ , this permits to infer (see Theorem REF ) that the only constant sign critical points of (REF ) on the manifold $\\mathcal {S}_q(\\Omega )=\\left\\lbrace u\\in W^{1,p}_0(\\Omega )\\, :\\, \\int _\\Omega |u|^q\\, dx=1\\right\\rbrace ,\\qquad 1<q\\le p,$ are indeed the global minimizers, which are unique up to a sign.",
"Observe that these critical points yield the following nonlinear version of Helmoltz equation (REF ), i.e.",
"$-\\mathrm {div\\,}\\nabla H(\\nabla u) = \\lambda \\,\\Vert u\\Vert _{L^q(\\Omega )}^{p-q}\\, |u|^{q-2}\\,u, \\quad \\mbox{ in } \\Omega ,\\qquad \\qquad u=0\\quad \\mbox{ on } \\partial \\Omega .$ We refer the reader to [18] for a detailed account on this nonlinear eigenvalue problem in the case $H(z)=|z|^p$ .",
"The previous general version of the hidden convexity can be seen again as a consequence of the joint convexity of the generalized HamiltonianThe parameters $\\beta $ and $q$ are linked through the relation $\\frac{\\beta }{p}+\\frac{1}{q}=1.$ $(m,\\phi )\\mapsto \\frac{H(\\phi )}{m^\\beta }, \\qquad \\mbox{ for }\\ 0\\le \\beta \\le p-1,$ which in turn gives the convexity of the information functional $\\begin{split}\\mathcal {J}_{H,\\beta }(\\mu \\vert \\nu ) = \\int H\\left(\\nabla \\log \\rho \\right)\\,\\rho ^{p-\\beta }\\,d\\nu , \\qquad \\mbox{ if } \\mu \\ll \\nu \\mbox{ and }\\rho = \\frac{d\\mu }{d\\nu },\\end{split}$ where the latter can also be written as $\\mathcal {J}_{H,\\beta }(\\mu |\\nu )=\\left(\\frac{p}{p-\\beta }\\right)^p\\,\\int H\\left(\\nabla \\rho ^\\frac{p-\\beta }{p}\\right)\\,d\\nu , \\qquad \\mbox{ if } \\mu \\ll \\nu \\mbox{ and }\\rho = \\frac{d\\mu }{d\\nu }.$ Finally, hidden convexity is in turn equivalent (see Section ) to the validity of the following generalized Picone inequality $\\left\\langle \\nabla H(\\nabla u), \\nabla \\left( \\frac{v^{q}}{u^{q-1}}\\right) \\right\\rangle \\le H(\\nabla v)^\\frac{q}{p}\\, H(\\nabla u)^\\frac{q-p}{p},$ for all differentiable functions $u,v\\ge 0$ with $u>0$ , which is proved in Proposition REF .",
"Here again we consider $1< q\\le p$ .",
"We point out that equivalence between Picone-type inequalities and the hidden convexity property seemed to be unknown: indeed, one of the main scopes of this paper is to precise the relation between these two properties.",
"Up to now, we have discussed applications of these convexity principles to uniqueness issues in linear and nonlinear eigenvalue problems.",
"But Picone-type inequalities can be used to prove a variety of different results.",
"Without any attempt of completeness (we refer to the seminal paper of Allegretto and Huang [2] and to the recent paper [21] for a significant account on the topic), we focus on applications to Hardy-type functional inequalities.",
"The idea is that when $u$ solves a quasilinear equation with principal part given by $-\\mathrm {div} \\nabla H(\\nabla u),$ by integrating (REF ) and using the equation one can get a lower bound on $\\int H(\\nabla v)$ which does not depend on derivatives of $u$ .",
"This procedure is now well understood, see the recent paper [12].",
"In Theorem REF this is applied to get a sharp anisotropic version of the Hardy inequality which reads as follows $\\left(\\frac{N+\\gamma -p}{p}\\right)^p\\,\\int _{\\mathbb {R}^N} |v|^p\\, F_*(x)^{\\gamma -p}\\,\\, dx\\le \\int _{\\mathbb {R}^{N}} F(\\nabla v)^p\\, F_*(x)^\\gamma \\, dx,\\quad v\\in C^\\infty _0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace ),$ for $1<p<N$ and $\\gamma >p-N$ .",
"Here $F$ is any $C^1$ strictly convex norm and $F_\\ast $ denotes the corresponding dual norm, see Section REF .",
"For the case $\\gamma =0$ a different proof, based on symmetrization arguments, can be found in [29].",
"The same method can be used to get, for example, the following nonlocal version of the Hardy inequality $C \\int _{\\mathbb {R}^N} \\frac{|v|^p}{ |x|^{s\\,p}}\\,dx \\le \\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} \\frac{\\big | v(x)-v(y)\\big |^p}{|x-y|^{N+s\\,p}}\\,dx\\,dy\\,,\\qquad \\mbox{for all} \\,\\, v\\in W^{s,p}_0(\\mathbb {R}^N)\\setminus \\lbrace 0\\rbrace \\,,$ by means of the following discrete version of Picone inequality $|u(x)-u(y)|^{p-2}\\, (u(x)-u(y))\\,\\left[\\frac{v(x)^p}{u(x)^{p-1}}-\\frac{v(y)^p}{u(y)^{p-1}}\\right]\\le |v(x)-v(y)|^p,$ with the choice $u(x) = |x|^{-s\\,p}$ .",
"The constant $C=C(N,s,p)>0$ is sharp and for the sake of completeness we provide details about its computation in Appendix .",
"We point out that (REF ) was proved by this same method by Frank and Seiringer in [17], which is there called ground state substitution.",
"Other fractional Hardy inequalities have appeared in the literature, see [6], [16].",
"In particular, in the recent paper [11] Davila, del Pino and Wei observed that a suitable fractional Hardy inequality on surfaces plays a role in the stability of nonlocal minimal cones, see [11].",
"Noteworthy, not only does the Picone inequality (REF ) have its discrete counterpart (REF ), but also the hidden convexity of the Dirichlet integral has a nonlocal version.",
"Indeed, the Gagliardo seminorm $\\int \\int \\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\\,p}}\\,dx\\,dy$ turns out to be convex along curves of the type (REF ), whenever $u,v$ are positive.",
"Correspondingly, fractional Picone inequalities (or equivalently hidden convexity) are used to get uniqueness results for positive eigenfunctions of the integro-differential operator defined by the following principal value integral $(-\\Delta _p)^s u(x) =2\\ \\mathrm { p.v.}",
"\\int _{\\mathbb {R}^N} \\frac{|u(x)-u(y)|^{p-2} (u(x)-u(y))}{|x-y|^{N+s\\,p}}\\,dy.$ We point out that in order to get uniqueness results for constant sign nonlocal eigenfunctions, i.e.",
"for solutions of $(-\\Delta _p)^s u=\\lambda \\, u^{q-1},\\quad \\mbox{ in }\\Omega ,\\qquad \\qquad u=0,\\quad \\mbox{ in }\\mathbb {R}^N\\setminus \\Omega ,$ as in the local case, one needs to know that non-negative eigenfunctions are indeed strictly positive, at least for $\\Omega $ connected.",
"A proof of this strong minimum principle for nonlocal eigenfunctions is provided in the appendix and seems to be new.",
"The proof is based on a logarithmic lemma recently established in [13].",
"After the acceptance of the present paper, we were informed about the interesting manuscript [14] which contains the proof of a weak Harnack's inequality for supersolutions (as well as a proper Harnack's inequality for solutions) of the operator $(-\\Delta _p)^s$ .",
"The interested reader may find in that article a more detailed account about nonlocal Harnack's inequalities.",
"Nevertheless, those results are not used in this paper."
],
[
"Plan of the paper",
"In Section we present and prove some generalizations of the three convexity principles discussed above, then Section is devoted to discuss their equivalences.",
"Section deals with the nonlocal/discrete versions of these convexities.",
"Applications are then given in Sections and .",
"The paper is concluded by two Appendices: one contains a new strong minimum principle for positive nonlocal eigenfunctions (Theorem REF ), while the second contains some computations related to the determination of the sharp constant in (REF ).",
"Acknowledgements We thank Agnese Di Castro, Tuomo Kuusi and Giampiero Palatucci for having kindly provided us a copy of their work [13], as well as Enea Parini for pointing out a flaw in a preliminary version of the proof of Theorem A.1.",
"Part of this work has been done during the conferences “Linear and Nonlinear Hyperbolic Equations” and “Workshop on Partial Differential Equations and Applications”, both held in Pisa and hosted by Centro De Giorgi and the Departement of Mathematics of the University of Pisa.",
"We acknowledge the two institutions as well as the organizers for the nice atmosphere and the excellent working environment.",
"The second author has been supported by the ERC Starting Grant No.",
"258685 “AnOptSetCon”."
],
[
"Three convexity principles",
"We start with the a couple of classical results, which will be useful in order to prove some of the results of the paper.",
"We give a proof for the reader's convenience.",
"Lemma 2.1 Let $F:\\mathbb {R}^N\\rightarrow \\mathbb {[}0,+\\infty )$ be a positively $1-$ homogeneous function, i.e.",
"$F(\\lambda \\, z)=\\lambda \\, F(z),\\qquad z\\in \\mathbb {R}^N,\\ \\lambda \\ge 0,$ which is level-convex, i.e.",
"$F((1-t)\\,z+t\\, w)\\le \\max \\lbrace F(z),F(w)\\rbrace ,\\qquad z,w\\in \\mathbb {R}^N,\\ t\\in [0,1].$ Then $F$ is convex.",
"Let $x_0,x_1\\in \\mathbb {R}^N$ , if $F(x_0)=F(x_1)=0$ then by (REF ) $F((1-t)\\,x_0+t\\,x_1)=0,\\qquad t\\in [0,1].$ Let us now suppose for example that $F(x_0)>0$ and take $\\varepsilon >0$ , we define $z=\\frac{x_0}{F(x_0)},\\quad w=\\frac{x_1}{F(x_1)+\\varepsilon }\\quad \\mbox{ and }\\quad t=\\frac{F(x_1)+\\varepsilon }{F(x_0)+F(x_1)+\\varepsilon }.$ By using the $1-$ homogeneity of $F$ , we then obtain $\\begin{split}F((1-t)\\, z+t\\, w)&=\\frac{F(x_0+x_1)}{F(x_0)+F(x_1)+\\varepsilon },\\end{split}$ while $\\max \\lbrace F(z),F(w)\\rbrace =\\max \\left\\lbrace 1,\\frac{F(x_1)}{F(x_1)+\\varepsilon }\\right\\rbrace =1.$ Then (REF ) implies $F(x_0+x_1)\\le F(x_0)+F(x_1)+\\varepsilon ,\\qquad x_0,x_1\\in \\mathbb {R}^N,$ and since $\\varepsilon >0$ is arbitrary, we get $F(x_0+x_1)\\le F(x_0)+F(x_1),\\qquad x_0,x_1\\in \\mathbb {R}^N,$ i.e.",
"$F$ is subadditive.",
"This in turn implies the desired result, as $F((1-t)\\, x_0+t\\, x_1)\\le F((1-t)\\, x_0)+F(t\\, x_1)=(1-t)\\, F(x_0)+t\\, F(x_1),$ which concludes the proof.",
"Lemma 2.2 Let $1<p<\\infty $ and let $H:\\mathbb {R}^N\\rightarrow [0,+\\infty )$ be a $C^1$ positively $p-$ homogeneous convex function.",
"If $H(z)=0$ then we have $\\nabla H(z)=0$ as well.",
"The statement is evident if $z=0$ , thus let us suppose that $z\\ne 0$ .",
"Assume on the contrary that $\\nabla H(z)\\ne 0$ , then there exists $h\\in \\mathbb {R}^N$ with unit norm such that $\\langle \\nabla H(z),h\\rangle =|\\nabla H(z)|$ .",
"The function $g(t)=H(z+t\\,h)$ has the following properties $g\\in C^1(\\mathbb {R}),\\qquad g(t)\\ge 0,\\qquad g^{\\prime }(0)=|\\nabla H(z)|>0=g(0).$ This gives a contradiction, thus $\\nabla H(z)=0$ ."
],
[
"Convexity of generalized kinetic energies",
"The first convexity principle we consider is the following.",
"Proposition 2.3 Let $1<p<\\infty $ and let $H:\\mathbb {R}^N\\rightarrow [0+\\infty )$ be a convex positively $p-$ homogeneous function.",
"For every $0<\\beta \\le p-1$ the function $(m,\\phi )\\mapsto \\frac{H(\\phi )}{m^{\\beta }},\\qquad (m,\\phi )\\in (0,+\\infty )\\times \\mathbb {R}^N,$ is convex.",
"For $\\beta =p-1$ , it is sufficient to observe that $\\frac{H(\\phi )}{m^{p-1}}=\\sup _{(t,\\xi )} \\left\\lbrace t\\,m+\\langle \\xi ,\\phi \\rangle \\, :\\, t+H^*(\\xi )\\le 0\\right\\rbrace ,\\qquad m>0,\\ \\phi \\in \\mathbb {R}^N,$ where $H^*$ denotes the Legendre-Fenchel transform of $H$ .",
"This would give the desired result, since the supremum of affine functions is a convex function.",
"For completeness, we verify formula (REF ): since for every $m>0$ the map $t\\mapsto t\\,m$ is increasing, the maximization in (REF ) is unchanged if we replace the inequality constraint by the condition $t+H^*(\\xi )=0$ .",
"Then the right-hand side of (REF ) is equivalent to $\\sup _{\\xi \\in \\mathbb {R}^N} \\langle \\xi ,\\phi \\rangle -H^*(\\xi )\\, m=m\\, \\left[\\sup _{\\xi \\in \\mathbb {R}^N}\\, \\left\\langle \\xi ,\\frac{\\phi }{m}\\right\\rangle -H^*(\\xi )\\right]=m\\, H^{**}\\left(\\frac{\\phi }{m}\\right),$ which gives the desired conclusion, by using that $H^{**}=H$ and the positive homogeneity of $H$ .",
"For $0<\\beta <p-1$ , let us set for simplicity $\\Phi (m,\\phi )=\\frac{H(\\phi )}{m^{p-1}}\\qquad \\mbox{ and }\\qquad \\psi (m)=m^\\vartheta ,\\qquad m>0,\\ \\phi \\in \\mathbb {R}^N,$ where $\\vartheta =\\beta /(p-1)<1$ , then we can rewrite $\\frac{H(\\phi )}{m^\\beta }=\\Phi (\\psi (m),\\phi ),$ where $\\Phi $ is jointly convex thanks to the first part of the proof and decreasing in its first argument, while $\\psi $ is concave, then it is standard to see that their composition is a convex function.",
"Indeed, for every $t\\in [0,1]$ , $m_0,m_1>0$ and $\\phi _0,\\phi _1\\in \\mathbb {R}^N$ , we get $\\begin{split}\\Phi \\Big (\\psi ((1-t)\\, m_0+t\\, m_1),(1-t)\\, \\phi _0+t\\, \\phi _1\\Big )&\\le \\Phi \\Big ((1-t)\\, \\psi (m_0)+t\\, \\psi (m_1),(1-t)\\, \\phi _0+t\\, \\phi _1\\Big )\\\\&\\le (1-t)\\, \\Phi (m_0,\\phi _0)+t\\, \\Phi (m_1,\\phi _1),\\\\\\end{split}$ which gives the desired result.",
"A couple of comments on the previous result are in order.",
"Remark 2.4 As already recalled in the Introduction, the main instance of functions considered in Proposition REF is the following one $(m,\\phi )\\,\\mapsto \\frac{|\\phi |^2}{m},\\qquad \\phi \\in \\mathbb {R}^N, m>0.$ If one regards the scalar quantity $m$ as a mass and the vector quantity $\\phi $ as the moment of this mass, i.e.",
"if we decompose $\\phi $ as $\\phi =v\\, m$ with $v\\in \\mathbb {R}^N$ (the velocity of the mass particle), then we would have $\\frac{|\\phi |^2}{m}=|v|^2\\, m.$ This simple remark is the crucial ingredient of the so-called Benamou-Brenier formula for the $2-$ Wasserstein distance (see [4], [9]).",
"The latter is a distance on the space of probability measures $\\mathcal {P}(\\Omega )$ over $\\Omega $ , defined by $w_2(\\rho _0,\\rho _1)^2:=\\inf _T\\left\\lbrace \\int _{\\Omega \\times \\Omega } |x-T(x)|^2\\, d\\rho _0\\, :\\, T_\\#\\rho _0=\\rho _1\\right\\rbrace ,$ for every $\\rho _0,\\rho _1\\in \\mathcal {P}(\\Omega )$ .",
"Here $T_\\# \\rho _0$ denotes the push-forward of the measure $\\rho _0$ .",
"The Benamou-Brenier formula asserts that we have $w_2(\\rho _0,\\rho _1)^2=\\inf \\left\\lbrace \\int _0^1 \\int _\\Omega |v|^2\\, \\mu _t\\,dx\\,dt\\, :\\, \\begin{array}{c}\\partial _t \\mu _t+\\mathrm {div} (v_t\\, \\mu _t)=0 \\\\\\mu _0=\\rho _0\\ \\mbox{ and }\\ \\mu _1=\\rho _1\\end{array}\\right\\rbrace .$ The latter consists in minimizing the integral of the total kinetic energy (the action), under a conservation of mass constraint.",
"Thanks to the previous discussion, this dynamical problem can be transformed in a convex variational problem under linear constraint, once we introduce the variable $\\phi _t=v_t\\, \\mu _t.$ For generalizations of this transport problem involving functions of the form $H(\\phi )\\, m^{-\\beta }$ the reader can consult [10] and [15].",
"Remark 2.5 (Sharpness of the condition on $\\beta $ ) The previous convexity property fails to be true in general for $\\beta >p-1$ .",
"Let us fix $\\phi _0\\in \\mathbb {R}^N\\setminus \\lbrace 0\\rbrace $ and $m_0>0$ .",
"We take $\\phi _1=c\\, \\phi _0$ and $m_1=c\\, m_0$ with $c>1$ , then we consider the convex combination $(m_t,\\phi _t)=((1-t)\\, m_0+t\\, m_1, (1-t)\\, \\phi _0+t\\, \\phi _1).$ For every $p-1<\\beta <p$ by strict concavity of the function $\\tau \\mapsto \\tau ^{p-\\beta }$ we have $\\begin{split}\\frac{|\\phi _t|^p}{m_t^\\beta }=(1-t+t\\,c)^{p-\\beta }\\, \\frac{|\\phi _0|^p}{m_0^\\beta }&>(1-t)\\,\\frac{|\\phi _0|^p}{m^\\beta _0}+t\\,c^{p-\\beta }\\, \\frac{|\\phi _0|^p}{m_0^\\beta }\\\\&=(1-t)\\,\\frac{|\\phi _0|^p}{m_0^\\beta }+\\frac{|\\phi _1|^p}{m_1^\\beta }.\\end{split}$"
],
[
"Hidden convexity",
"The next convexity principle has been probably first identified by Benguria in his Ph.D. dissertation in the case $p=q=2$ and $H(z)=|z|^2$ , see [5].",
"See also [3] and [27] for some generalizations in the case $q=p$ .",
"Proposition 2.6 (General hidden convexity) Let $1<p<\\infty $ and $1<q\\le p$ .",
"Let $H:\\mathbb {R}^N\\rightarrow [0+\\infty )$ be a positively $p-$ homogeneous convex function.",
"For every pair of differentiable functions $u_0,u_1\\ge 0$ , we define $\\sigma _t(x)=\\Big [(1-t)\\, u_0(x)^q+t\\, u_1(x)^q\\Big ]^\\frac{1}{q}\\qquad t\\in [0,1],\\,x\\in \\Omega .$ Then there holds $H(\\nabla \\sigma _t)\\le (1-t)\\,H(\\nabla u_0)+t\\, H(\\nabla u_1),\\quad t\\in [0,1].$ The proof for the case $p=q$ can be found for example in [7].",
"In order to consider the case $q<p$ , we observe that by Lemma REF , the function $F=H^{q/p}$ is a positively $q-$ homogeneous convex function.",
"Then the first part of the proof implies $F(\\nabla \\sigma _t)\\le (1-t)\\, F(\\nabla u_0)+t\\, F(\\nabla u_1),$ and by raising to the power $p/q$ and using the convexity of $\\tau \\mapsto \\tau ^{p/q}$ , we end up with (REF ).",
"Remark 2.7 We remark that neither strict convexity of $H$ nor strict positivity of the functions is needed in the previous result, unless one is interested in identification of equality cases in (REF ).",
"Moreover, $H$ is only required to be only positively homogeneous, i.e.",
"it is not necessarily even.",
"Remark 2.8 (Sharpness of the condition on $q$ ) Again, the condition $q\\le p$ is vital.",
"Indeed, by taking a non-constant $u_0\\ne 0$ and $u_1=c\\, u_0$ for $c>1$ , then we have $\\sigma _t(x)=((1-t)\\, u_0^q+t\\, u_1^q)^\\frac{1}{q}=((1-t)+t\\, c^q)^\\frac{1}{q}\\, u_0,$ and $H(\\nabla \\sigma _t)=((1-t)+t\\, c^q)^\\frac{p}{q}\\, H(\\nabla u_0)>(1-t)\\, H(\\nabla u_0)+t\\, H(\\nabla u_1),$ by strict concavity of $\\tau \\mapsto \\tau ^{p/q}$ ."
],
[
"Picone inequalities",
"We now prove a general version of the so-called Picone inequality.",
"The usual one, i.e.",
"$\\left\\langle |\\nabla u|^{p-2}\\, \\nabla u, \\nabla \\left(\\frac{v^p}{u^{p-1}}\\right)\\right\\rangle \\le |\\nabla v|^p,\\qquad v\\ge 0,\\, u>0,$ proved by Allegretto and Huang, see [2] corresponds to taking $p=q$ and $H(z)=|z|^p$ in (REF ) below.",
"Proposition 2.9 (General Picone inequality) Let $1<q\\le p$ and let $H:\\mathbb {R}^N\\rightarrow [0,+\\infty )$ be a $C^1$ positively $p-$ homogeneous convex function.",
"For every pair of positive differentiable functions $u,v$ with $u>0$ , we have $\\begin{split}\\frac{1}{p}\\,\\left\\langle \\nabla H(\\nabla u), \\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle \\, &\\le H(\\nabla v)^\\frac{q}{p}\\, H(\\nabla u)^\\frac{p-q}{p}.\\end{split}$ Let us start with the case $p=q$ .",
"We use the convexity inequality $H(z)\\ge H(w)+\\langle \\nabla H(w),z-w\\rangle ,$ with the choices $z=\\nabla v\\qquad \\mbox{ and }\\qquad w=\\nabla u\\, \\left(\\frac{v}{u}\\right).$ By using the $p-$ homogeneity of $H$ we then get $\\begin{split}H(\\nabla v)&\\ge \\left(\\frac{v}{u}\\right)^p\\, H(\\nabla u)+\\langle \\nabla H(\\nabla u),\\nabla v\\rangle \\,\\left(\\frac{v}{u}\\right)^{p-1}-\\langle \\nabla H(\\nabla u),\\nabla u\\rangle \\,\\left(\\frac{v}{u}\\right)^p\\\\&=\\langle \\nabla H(\\nabla u),\\nabla v\\rangle \\,\\left(\\frac{v}{u}\\right)^{p-1}-\\left(1-\\frac{1}{p}\\right)\\, \\langle \\nabla H(\\nabla u),\\nabla u\\rangle \\,\\left(\\frac{v}{u}\\right)^p\\\\&=\\frac{1}{p}\\, \\left\\langle \\nabla H(\\nabla u),\\nabla \\left(\\frac{v^p}{u^{p-1}}\\right)\\right\\rangle ,\\end{split}$ which concludes the proof of (REF ) for $q=p$ .",
"We now take $1<q<p$ and set $F(z)=H(z)^\\frac{q}{p},\\qquad z\\in \\mathbb {R}^N,$ which is convex and positively $q-$ homogeneousAlso observe that $F\\in C^1(\\mathbb {R}^N)$ .",
"It is sufficient to check that $F$ is differentiable at the origin and that its differential vanishes at $z=0$ .",
"Indeed, by using homogeneity we have $F(h)-F(0)=H(h)^\\frac{q}{p}=|h|^q\\, H\\left(\\frac{h}{|h|}\\right)^\\frac{q}{p}=o(|h|),\\qquad h\\in \\mathbb {R}^N\\setminus \\lbrace 0\\rbrace \\mbox{ such that } |h|\\ll 1.$ , then the first part of the proof implies $\\frac{1}{q}\\,\\left\\langle \\nabla F(\\nabla u),\\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle \\le F(\\nabla v).$ Observe that if $H(\\nabla u)=0$ , then we have $\\nabla H(\\nabla u)=0$ as well (see Lemma REF ) and (REF ) holds true.",
"Thus we can assume $H(\\nabla u)\\ne 0$ .",
"The previous inequality is equivalent to $\\frac{1}{p}\\, H(\\nabla u)^\\frac{q-p}{p}\\, \\left\\langle \\nabla H(\\nabla u),\\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle \\le H(\\nabla v)^\\frac{q}{p}.$ If we multiply the previous by $H(\\nabla )^{(p-p)/p}$ we eventually attains the conclusion.",
"Remark 2.10 (Equivalent form of the Picone inequality) For $p\\ne q$ , as a plain consequence of Young inequality, (REF ) implies $\\frac{1}{p}\\,\\left\\langle \\nabla H(\\nabla u), \\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle \\, \\le \\frac{q}{p}\\,H(\\nabla v)+\\frac{p-q}{p}\\, H(\\nabla u).$ Observe that for $q=1$ , the previous inequality reduces to $H(\\nabla u)+\\langle \\nabla H(\\nabla u),\\nabla v-\\nabla u\\rangle \\le H(\\nabla v),$ which just follows from the convexity of $z\\mapsto H(z)$ .",
"On the other hand, by applying (REF ) with the choices (here $\\varepsilon >0$ ) $U=\\big (\\varepsilon +H(\\nabla u)\\big )^{-\\frac{1}{p}}\\, u\\qquad \\mbox{ and }\\qquad V=\\big (\\varepsilon +H(\\nabla v)\\big )^{-\\frac{1}{p}}\\, v,$ we get $\\frac{1}{p}\\, \\big (\\varepsilon +H(\\nabla u)\\big )^{\\frac{q-p}{p}}\\, \\big (\\varepsilon +H(\\nabla v)\\big )^{-\\frac{q}{p}}\\, \\left\\langle \\nabla H(\\nabla u),\\nabla \\left(\\frac{v^q}{u^{q-1}}\\right) \\right\\rangle \\le 1.$ By multiplying the previous by $(\\varepsilon +H(\\nabla u))^{(p-q)/p}\\, (\\varepsilon +H(\\nabla v))^{\\frac{q}{p}}$ and then letting $\\varepsilon $ goes to 0, we get (REF ).",
"Remark 2.11 (Non-homogeneous functions) All the convexity principles considered in this section have been proven under the assumption that $H$ is positively $p-$ homogeneous.",
"Nevertheless, the results are still true for some $H$ violating this condition.",
"This is the case for example of the anisotopic function $H(z)=\\sum _{i=1}^N H_i(z_i),\\qquad \\mbox{ where }\\quad H_i(t)=|t|^{p_i},$ and $1<p_1\\le \\dots \\le p_N$ .",
"Namely, by applying (REF ) to each $H_i$ and then summing up, we get $\\sum _{i=1}^N |u_{x_i}|^{p_i-2}\\, u_{x_i}\\, \\left(\\frac{v^{q_i}}{u^{q_i-1}}\\right)_{x_i}\\le \\sum _{i=1}^N |v_{x_i}|^\\frac{q_i}{p_i}\\, |u_{x_i}|^\\frac{p_i-q_i}{p_i},$ for every $q_1,\\dots ,q_N$ such that $1<q_i\\le p_i$ .",
"In the very same way from (REF ) we get $\\sum _{i=1}^N \\big |(\\sigma _t)_{x_i}\\big |^{p_i}\\le (1-t)\\, \\sum _{i=1}^N |u_{x_i}|^{p_i}+t\\,\\sum _{i=1}^N |v_{x_i}|^{p_i},\\qquad t\\in [0,1],$ where $\\sigma _t=((1-t)\\, u^q+t\\, v^q)^{1/q}$ and $1<q\\le p_1$ .",
"From Proposition REF we can infer the convexity of the function $(m,\\phi )\\mapsto \\sum _{i=1}^N \\frac{|\\phi |^{p_i}}{m^{\\beta _i}},\\qquad (m,\\phi )\\in (0,+\\infty )\\times \\mathbb {R}^N,$ for every $\\beta _1,\\dots ,\\beta _N$ such that $0<\\beta _i\\le p_i-1$ ."
],
[
"Equivalences",
"In this section we will show that the three convexity principles proved in the previous section are indeed equivalent.",
"In other words, they are just three different ways to look at the same principle."
],
[
"Kinetic energies and Hidden convexity",
"Let $u$ be a differentiable function on an open set $\\Omega \\subset \\mathbb {R}^N$ , which is everywhere positive.",
"We first observe that if in the generalized kinetic energy of Proposition REF we make the choice $m=u\\qquad \\mbox{ and }\\qquad \\phi =\\nabla u,$ then we obtain the functional $u\\mapsto \\frac{H(\\nabla u)}{u^\\beta }.$ which is convex in the usual sense, provided that $0<\\beta \\le p-1$ , thanks to Proposition REF .",
"This convexity is indeed equivalent to (REF ), as we now show.",
"Indeed, let us pick two differentiable functions $u,v$ which are everywhere positive.",
"We observe that by setting $U=u^q$ , $V=v^q$ and $\\gamma _t=(1-t)\\, U+t\\, V$ , we get by Proposition REF $\\frac{H(\\nabla \\gamma _t)}{\\gamma _t^{\\beta }}\\le (1-t)\\, \\frac{H(\\nabla U)}{U^\\beta }+t\\, \\frac{H(\\nabla V)}{V^\\beta }.$ By using the homogeneity of $H$ , the previous is equivalent to $H\\left(\\nabla \\gamma ^\\frac{p-\\beta }{p}\\right)\\le (1-t)\\, H\\left(\\nabla U^\\frac{p-\\beta }{p}\\right)+t\\, H\\left(\\nabla V^\\frac{p-\\beta }{p}\\right).$ If we now choose $\\beta $ in such a way thatObserve that such a choice is feasible, since $p\\, \\left(1-\\frac{1}{q}\\right)\\le p-1 \\quad \\Leftrightarrow \\quad \\frac{p}{p-1}\\le \\frac{q}{q-1}\\quad \\Leftrightarrow \\quad p\\ge q.$ $\\frac{p-\\beta }{p}=\\frac{1}{q},\\qquad \\mbox{ i.e.",
"}\\quad \\beta =p\\,\\left(1-\\frac{1}{q}\\right),$ we see that the previous inequality becomes $H(\\nabla \\sigma _t)\\le (1-t)\\, H(u)+t\\, H(v),$ where $\\sigma _t=\\gamma _t^{1/q}=((1-t)\\, u^q+t\\, v^q)^{1/q}$ as always."
],
[
"Hidden convexity and Picone",
"We first show that $(\\mbox{Hidden convexity})\\qquad \\Longrightarrow \\qquad (\\mbox{Picone}).$ As before, given $u,v$ positive differentiable functions with $u>0$ , we set $\\sigma _t(x)=\\Big [(1-t)\\, u(x)^q+t\\, v(x)^q\\Big ]^\\frac{1}{q}\\qquad t\\in [0,1],\\,x\\in \\Omega .$ Then by using the convexity of $t\\mapsto H(\\nabla \\sigma _t)$ , we easily get $\\frac{H(\\nabla \\sigma _t)-H(\\nabla u)}{t}\\le H(\\nabla v)-H(\\nabla u).$ Observe that again by convexity, the incremental ratio on the left-hand side is monotone, then there exists the limit for $t$ monotonically converging to 0, i.e.",
"we obtain $\\left(\\frac{d}{dt} H(\\nabla \\sigma _t)\\right)_{|t=0}\\le H(\\nabla v)-H(\\nabla u).$ The previous is indeed equivalent to (REF ).",
"To see this, it is sufficient to compute the derivative on the left-hand side.",
"We have $\\nabla \\sigma _t=\\sigma _t^{1-q}\\,\\left[(1-t)\\, \\nabla u\\, u^{q-1}+t\\, \\nabla v\\, v^{q-1}\\right]\\qquad \\mbox{ and }\\qquad \\frac{d}{dt} \\sigma _t=\\frac{1}{q}\\,\\sigma _{t}^{1-q}\\, (v^q-u^q),$ so that we can compute $\\begin{split}\\frac{d}{dt} \\nabla \\sigma _t&=(1-q)\\, \\sigma _t^{-q}\\, \\left[(1-t)\\, \\nabla u\\, u^{q-1}+t\\, \\nabla v\\, v^{q-1}\\right]\\,\\frac{d}{dt} \\sigma _t\\\\&+\\sigma _t^{1-q}\\, \\left[\\nabla v\\, v^{q-1}-\\nabla u\\, u^{q-1}\\right].\\end{split}$ Finally, we get $\\begin{split}\\left(\\frac{d}{dt} \\nabla \\sigma _t\\right)_{|t=0}&=-(q-1)\\, \\frac{\\nabla u}{u}\\,\\left(\\frac{d}{dt} \\sigma _t\\right)_{t=0}+\\nabla v\\, \\left(\\frac{v}{u}\\right)^{q-1}-\\nabla u\\\\&=-\\left(\\frac{q-1}{q}\\right)\\, \\nabla u\\, \\left(\\left(\\frac{v}{u}\\right)^q-1\\right)+\\nabla v\\, \\left(\\frac{v}{u}\\right)^{q-1}-\\nabla u.\\end{split}$ We can now compute the left-hand side of (REF ) and obtain $\\begin{split}\\left(\\frac{d}{dt} H(\\nabla \\sigma _t)\\right)_{|t=0}&=\\left\\langle \\nabla H(\\nabla \\sigma _t),\\frac{d}{dt} \\nabla \\sigma _t\\right\\rangle _{|t=0}\\\\&=\\left\\langle \\nabla H(\\nabla u), \\nabla v\\right\\rangle \\, \\left(\\frac{v}{u}\\right)^{q-1}-\\frac{p\\,(q-1)}{q}\\, H(\\nabla u)\\, \\left(\\frac{v}{u}\\right)^q-\\frac{p}{q}\\, H(\\nabla u).\\end{split}$ By inserting this in (REF ), observing that $\\frac{1}{q}\\,\\left\\langle \\nabla H(\\nabla u), \\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle =\\left\\langle \\nabla H(\\nabla u), \\nabla v\\right\\rangle \\, \\left(\\frac{v}{u}\\right)^{q-1}-\\frac{p\\,(q-1)}{q}\\, H(\\nabla u)\\, \\left(\\frac{v}{u}\\right)^q,$ and multiplying everything by $q/p$ , we eventually get $\\frac{1}{p}\\,\\left\\langle \\nabla H(\\nabla u), \\nabla \\left(\\frac{v^q}{u^{q-1}}\\right)\\right\\rangle \\le \\frac{q}{p}\\,H(\\nabla v)+\\frac{p-q}{p}\\, H(\\nabla u).$ For $p=q$ this is exactly Picone inequality (REF ), while for $q<p$ we just have to observe that by Remark REF the previous is equivalent to (REF ).",
"Let us now show that $(\\mbox{Picone})\\qquad \\Longrightarrow \\qquad (\\mbox{Hidden convexity}).$ As we said, inequality (REF ) is actually equivalent to (REF ), for every $v,u$ and $\\sigma _t$ curve of the form (REF ) connecting them.",
"We now fix $u,v$ and $\\sigma _t$ , then by (REF ) we get $H(\\nabla v)-H(\\nabla \\sigma _t)\\ge \\frac{d}{ds} H(\\nabla \\widetilde{\\sigma }_s)_{|s=0},$ and $H(\\nabla u)-H(\\nabla \\sigma _t)\\ge \\frac{d}{ds} H(\\nabla \\widehat{\\sigma }_s)_{|s=0},$ where $s\\mapsto \\widetilde{\\sigma }_s$ and $s\\mapsto \\widehat{\\sigma }_s$ are the curves of the form (REF ) connecting $\\sigma _t$ to $v$ and $\\sigma _t$ to $u$ respectively.",
"In other words, we have $\\widetilde{\\sigma }_s=\\Big [\\left(1-t-s\\,(1-t)\\right)\\, u^q+\\left(t+s\\,(1-t)\\right)\\, v^q\\Big ]^\\frac{1}{q}=\\sigma _{t+s\\,(1-t)},\\qquad s\\in [0,1],$ and $\\widetilde{\\sigma }_s=\\Big [\\left(1-t+s\\,t\\right)\\, u^q+\\left(t-s\\,t \\right)\\, v^q\\Big ]^\\frac{1}{q}=\\sigma _{t-s\\,t},\\qquad s\\in [0,1].$ Thus we get $\\begin{split}H(\\nabla v)-H(\\nabla \\sigma _t)\\ge \\frac{d}{ds} H(\\nabla \\widetilde{\\sigma }_s)_{|s=0}&=\\frac{d}{ds} H(\\nabla \\sigma _{t+s\\,(1-t})_{|s=0}\\\\&=\\frac{d}{ds} H(\\nabla \\sigma _{s})_{|s=t}\\, (1-t)\\end{split}$ and similarly $H(\\nabla u)-H(\\nabla \\sigma _t)\\ge -\\frac{d}{ds} H(\\nabla \\sigma _{s})_{|s=t}\\, t.$ Keeping the two informations together, we finally get $\\frac{H(\\nabla \\sigma _t)-H(\\nabla u)}{t}\\le \\frac{H(\\nabla v)-H(\\nabla \\sigma _t)}{1-t},$ which is equivalent to $H(\\nabla \\sigma _t)\\le (1-t)\\, H(\\nabla u)+t\\, H(\\nabla v)$ ."
],
[
"The discrete case",
"We now prove analogous results for functions which are not necessarily differentiable.",
"Roughly speaking, we are going to replace derivatives by finite differences.",
"For $1<p<\\infty $ and $0<s<1$ , the resulting convexity properties have applications to nonlocal integrals of the type $\\int _{\\mathbb {R}^N} \\int _{\\mathbb {R}^N} \\frac{|u(x)-u(y)|^p}{|x-y|^{N+s\\,p}}\\, dx\\,dy,\\qquad (\\mbox{\\it Gagliardo seminorm}),$ or $\\sup _{0<|h|} \\int _{\\mathbb {R}^N} \\frac{|u(x+h)-u(x)|^p}{|h|^{s\\,p}}\\, dx,\\qquad (\\mbox{\\it Nikolskii seminorm}),$ and more generally $\\int _{0}^{\\infty } \\left(\\sup _{0<|h|\\le t}\\int _{\\mathbb {R}^N} \\frac{|u(x+h)-u(x)|^p}{t^{s\\,p}}\\, dx\\right)^\\frac{r}{p}\\, \\frac{dt}{t},\\qquad p\\le r<\\infty ,\\quad (\\mbox{\\it Besov seminorm}).$ Proposition 4.1 (Discrete hidden convexity) Let $1<p<\\infty $ and $1<q\\le p$ .",
"For every $u_0,u_1\\ge 0$ , we define $\\sigma _t(x)=\\Big [(1-t)\\, u_0(x)^q+t\\, u_1(x)^q\\Big ]^\\frac{1}{q}\\qquad t\\in [0,1],\\,x\\in \\mathbb {R}^N.$ Then we have $|\\sigma _t(x)-\\sigma _t(y)|^p\\le (1-t)\\, |u_0(x)-u_0(y)|^p+t\\, |u_1(x)-u_1(y)|^p,\\quad t\\in [0,1],\\ x,y\\in \\mathbb {R}^N.$ The proof is as in [19], which deals with the case $p=q$ .",
"We observe that $\\sigma _t=\\left\\Vert \\left((1-t)^\\frac{1}{q}\\,u_0,t^\\frac{1}{q}\\,u_1\\right)\\right\\Vert _{\\ell ^q},$ where we set $\\Vert z\\Vert _{\\ell ^q}=(|z_1|^q+|z_2|^q)^{1/q}$ for $z\\in \\mathbb {R}^2$ .",
"The triangular inequality implies that $\\big | \\Vert z\\Vert _{\\ell ^q}-\\Vert w\\Vert _{\\ell ^q}\\big |^q\\le \\Vert z-w\\Vert _{\\ell ^q}^q,\\qquad z,w\\in \\mathbb {R}^2,$ and by using this with the choices $z=\\left((1-t)^\\frac{1}{q}\\,u_0(x),t^\\frac{1}{q}\\,u_1(x)\\right)\\qquad \\mbox{ and }\\qquad w=\\left((1-t)^\\frac{1}{q}\\,u_0(y),t^\\frac{1}{q}\\,u_1(y)\\right),$ we get $|\\sigma _t(x)-\\sigma _t(y)|^q\\le (1-t)\\, |u_0(x)-u_0(y)|^q+t\\, |u_1(x)-u_1(y)|^q.$ By raising both sides to the power $p/q$ and using the convexity of $\\tau \\mapsto \\tau ^{p/q}$ , we get (REF ).",
"Proposition 4.2 (Discrete Picone inequality) Let $1<p<\\infty $ and $1<q\\le p$ .",
"Let $u,v$ be two measurable functions with $v\\ge 0$ and $u>0$ , then $\\begin{split}|u(x)-u(y)|^{p-2}\\, (u(x)-u(y))\\,& \\left[\\frac{v(x)^q}{u(x)^{q-1}}-\\frac{v(y)^q}{u(y)^{q-1}}\\right]\\\\&\\le |v(x)-v(y)|^q\\,|u(x)-u(y)|^{p-q}.\\end{split}$ We notice at first that is sufficient to prove $\\begin{split}|u(x)-u(y)|^{q-2}\\, (u(x)-u(y))\\,& \\left[\\frac{v(x)^q}{u(x)^{q-1}}-\\frac{v(y)^q}{u(y)^{q-1}}\\right]\\le |v(x)-v(y)|^q,\\end{split}$ since (REF ) then follows by multiplying the previous inequality by $|u(x)-u(y)|^{p-q}$ .",
"At this aim, let us start by observing that if $u(x)=u(y)$ , inequality (REF ) is trivially satisfied.",
"We take then $u(x)\\ne u(y)$ and we can always suppose that $u(x)<u(y)$ , up to exchanging the role of $x$ and $y$ .",
"We further observe that if $v(y)=0$ , inequality (REF ) is again trivially satisfied, since $|u(x)-u(y)|^{q-2}\\, (u(x)-u(y))\\left[\\frac{v(x)^q}{u(x)^{q-1}}-\\frac{v(y)^q}{u(y)^{q-1}}\\right]\\le 0.$ We can thus suppose that $v(y)\\ne 0$ , then we rewrite the left-hand side of (REF ) as $\\begin{split}|u(x)-u(y)|^{q-2}\\,& (u(x)-u(y)) \\left[\\frac{v(x)^q}{u(x)^{q-1}}-\\frac{v(y)^q}{u(y)^{q-1}}\\right]\\\\&=u(x)^q\\,\\left(\\frac{v(y)}{u(y)}\\right)^q\\left[ \\left(1-\\frac{u(y)}{u(x)}\\right)^{q-1}\\, \\left(\\left(\\frac{v(x)\\,u(y)}{v(y)\\,u(x)}\\right)^q-\\frac{u(y)}{u(x)}\\right)\\right]\\end{split}$ while the right-hand side of (REF ) rewrites as $\\begin{split}|v(x)-v(y)|^q=u(x)^q\\,\\left(\\frac{v(y)}{u(y)}\\right)^q\\,\\left|\\left(\\frac{v(x)\\,u(y)}{v(y)\\,u(x)}\\right)-\\frac{u(y)}{u(x)}\\right|^q.\\end{split}$ Then if we set $A=\\frac{v(x)\\,u(y)}{v(y)\\,u(x)}\\qquad \\mbox{ and }\\qquad t=\\frac{u(y)}{u(x)},$ the previous manipulations show that (REF ) is equivalent to the following $(1-t)^{q-1}\\, (A^q-t)\\le |A-t|^{q},\\qquad \\mbox{ for } 0\\le t\\le 1,$ The previous elementary inequality is true (see [17]), thus we get the desired conclusion."
],
[
"Positive eigenfunctions",
"In what follows, we denote by $\\Omega \\subset \\mathbb {R}^N$ an open connected set such that $|\\Omega |<\\infty $ .",
"For $1<p<\\infty $ , as it is customary we denote by $W^{1,p}_0(\\Omega )$ the closure of $C^\\infty _0(\\Omega )$ with respect to the $L^p$ norm of the gradient.",
"We also take $H:\\mathbb {R}^N\\rightarrow [0,\\infty )$ to be a $C^1$ convex $p-$ homogeneous function such that $\\frac{1}{C}\\, |z|^p\\le H(z)\\le C\\, |z|^p,\\qquad z\\in \\mathbb {R}^N,$ for some $C\\ge 1$ .",
"Then for $1<q\\le p$ , we set $\\lambda _{p,q}(\\Omega )=\\min _{u\\in W^{1,p}_0(\\Omega )}\\left\\lbrace \\int _\\Omega H(\\nabla u)\\, dx\\, :\\, \\Vert u\\Vert _{L^q(\\Omega )}=1\\right\\rbrace .$ Theorem 5.1 (Uniqueness of positive eigenfunctions) Let $1<p<\\infty $ and $1<q\\le p$ .",
"Let $\\lambda >0$ be such that there exists a non trivial function $u\\in W^{1,p}_0(\\Omega )$ verifying $-\\frac{1}{p}\\,\\mathrm {div\\,} \\nabla H(\\nabla u)=\\lambda \\, u^{q-1},\\qquad u\\ge 0,\\quad \\mbox{ in }\\Omega .$ Then we have $\\lambda \\, \\left(\\int _\\Omega |u|^q\\, dx\\right)^\\frac{q-p}{q}=\\lambda _{p,q}(\\Omega ),$ and $v=u\\,\\Vert u\\Vert _{L^q(\\Omega )}^{-1}$ is a minimizer of the variational problem in (REF ).",
"We first observe that $u>0$ almost everywhereActually, the much stronger result $\\inf _K u\\ge \\frac{1}{C_K},\\qquad \\mbox{ for every compact }K\\Subset \\Omega ,$ holds.",
"Here we want to point out that the weaker information “$u>0$ almost everywhere” suffices for this argument to work.",
"in $\\Omega $ , by the strong minimum principle (see for example [28]).",
"We also notice that the case $p=q$ is now well-established (see for example [2], [7], [21], [27]), we limit ourselves to consider the case $q<p$ .",
"The proof is just based on an application of Proposition REF .",
"We observe that $v$ is a solution of $-\\frac{1}{p}\\,\\mathrm {div\\,}\\nabla H(\\nabla v)=\\lambda \\,\\Vert u\\Vert _{L^q(\\Omega )}^{q-p}\\, v^{q-1},\\qquad v> 0,\\quad \\mbox{ in }\\Omega .$ Moreover, $v$ is admissible for the variational problems defining $\\lambda _{p,q}(\\Omega )$ , thus by testing the previous equation with $v$ itself and using the homogeneity of $H$ , we get $\\lambda \\, \\Vert u\\Vert _{L^q(\\Omega )}^{q-p}\\ge \\lambda _{p,q}(\\Omega ).$ Let $u_1\\in W^{1,p}_0(\\Omega )$ be a function achieving the minimum in the right-hand side of (REF ).",
"Then we have $\\begin{split}\\lambda \\,\\Vert u\\Vert _{L^p(\\Omega )}^{q-p}=\\lambda \\,\\Vert u\\Vert _{L^p(\\Omega )}^{q-p}\\,\\int _\\Omega u_1^{q}\\, dx&=\\lambda \\,\\Vert u\\Vert _{L^p(\\Omega )}^{q-p}\\, \\int _\\Omega v^{q-1}\\, \\frac{u^q_1}{v^{q-1}}\\, dx\\\\&=\\frac{1}{p}\\,\\int _\\Omega \\left\\langle \\nabla H(\\nabla v),\\nabla \\left(\\frac{u^q_1}{v^{q-1}}\\right)\\right\\rangle \\, dx\\\\&\\le \\int _\\Omega H(\\nabla u_1)^\\frac{q}{p}\\, H(\\nabla v)^\\frac{p-q}{p}\\, dx.\\end{split}$ If we now apply Hölder's and Young's inequality, the previous gives the desired result.",
"Remark 5.2 Of course, a completely equivalent proof of the previous result could use Proposition REF , as in [7].",
"As remarked in the Introduction, we believe that a direct application of hidden convexity provides a cleaner justification of the result, while on the other hand Picone inequality offers a quicker proof.",
"An alternative proof can be found in [24], later refined by Kawohl and Lindqvist in [20].",
"Remark 5.3 (Sharpness of the condition $q\\le p$ ) For a general open set $\\Omega $ with finite measure, the previous result can not hold true for $q>p$.",
"Indeed, let us consider an annular domain $T=\\lbrace x\\in \\mathbb {R}^N\\, :\\, 1<|x|<r\\rbrace $ and take $H(z)=|z|^p$ , then the problem $\\lambda ^{rad}_{p,q}(T)=\\min _{W^{1,p}_0(T)}\\left\\lbrace \\int _T |\\nabla u|^p\\, dx\\, :\\, u \\mbox{ radial function},\\Vert u\\Vert _{L^q(T)}=1\\right\\rbrace ,$ admits a minimizer $u_0\\in W^{1,p}_0(T)$ , which is a positive solution of $-\\Delta _p u=\\lambda ^{rad}_{p,q}(T)\\, u^{q-1},\\qquad \\mbox{ in }T,\\qquad \\ \\mbox{ with }\\ \\int _T |u|^q\\, dx=1.$ On the other hand, Nazarov in [23] has proved that if $q>p$ one can always take $r$ sufficiently close to 1 such that minimizers of (REF ) are not radial.",
"This clearly means that $\\lambda ^{rad}_{p,q}(T)>\\lambda _{p,q}(T).$"
],
[
"Hardy-type inequalities",
"As another application of the general Picone inequality (REF ), we have the following family of sharp inequalities.",
"Theorem 5.4 (Weighted Hardy inequalities with general norms) Let $F:\\mathbb {R}^N\\rightarrow [0,+\\infty )$ be a $C^1$ strictly convex norm.",
"Let $1<p<N$ , for every $\\gamma > p-N$ we have $\\left(\\frac{N+\\gamma -p}{p}\\right)^p\\,\\int _{\\mathbb {R}^N} |v|^p\\,F_*(x)^{\\gamma -p}\\, dx\\le \\int _{\\mathbb {R}^{N}} F(\\nabla v)^p\\,F_*(x)^{\\gamma }\\, dx, \\ \\ v\\in C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace ),$ where $F_*$ is the dual norm defined by $F_*(z)=\\sup _{x\\ne 0} \\left\\langle \\frac{x}{F(x)},z\\right\\rangle ,\\qquad z\\in \\mathbb {R}^N.$ We start proving (REF ) for positive $C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ functions.",
"We take $\\beta >0$ and set $u(x)=F_*(x)^{-\\beta },\\qquad x\\in \\mathbb {R}^N\\setminus \\lbrace 0\\rbrace .$ Observe that $u$ is a $C^1$ function in $\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace $ such that $\\begin{split}-\\mathrm {div}(F_*(x)^\\gamma &\\,F(\\nabla u)^{p-1}\\,\\nabla F(\\nabla u))\\\\&=\\beta ^{p-1}\\,\\Big [N-p-\\beta \\, (p-1)+\\gamma \\Big ]\\, u^{p-1}\\, F_*(x)^{\\gamma -p},\\quad \\mbox{ in }\\mathbb {R}^{N}\\setminus \\lbrace 0\\rbrace .\\end{split}$ Indeed, we recall the following relations between $F$ and $F_*$ (see [26] for example) $F(\\nabla F_*(x))=1\\qquad \\mbox{ and }\\qquad \\nabla F(\\nabla F_*(x))=\\frac{x}{F_*(x)},\\qquad x\\ne 0.$ Of course $\\nabla u=-\\beta \\, F_*(x)^{-\\beta -1}\\,\\nabla F_*(x),\\qquad x\\ne 0,$ by using (REF ) and the homogeneity of $F$ we get $F(\\nabla u)^{p-1}=\\beta ^{p-1}\\, F_*(x)^{-(\\beta +1)\\,(p-1)}=\\beta ^{p-1}\\, F_*(x)^{-\\beta \\,p+\\beta -p},$ and still by (REF ) and the fact that $\\nabla F$ is $0-$ homogeneous, we also have $\\nabla F(\\nabla u)=-\\nabla F(\\nabla F_*(x))=-\\frac{x}{F_*(x)},\\qquad x\\ne 0.$ Thus we get $\\begin{split}-\\mathrm {div}\\left(F_*(x)^\\gamma \\,F(\\nabla u)^{p-1}\\, \\nabla F(\\nabla u)\\right)&=\\beta ^{p-1}\\, \\mathrm {div}\\left(F_*(x)^{-\\beta \\,p+\\beta -p+\\gamma }\\, x\\right)\\\\&=\\beta ^{p-1}\\, [N-\\beta \\,p+\\beta -p+\\gamma ]\\, F_*(x)^{-\\beta \\,(p-1)+\\gamma -p},\\end{split}$ as desired, where we used that $\\langle \\nabla F_*(x),x\\rangle =F_*(x),\\qquad x\\ne 0,$ again by homogeneity.",
"Finally, by using the definition of $u$ , we get $F_*(x)^{-\\beta \\,(p-1)+\\gamma -p}=u^{p-1}\\, F_*(x)^{\\gamma -p}.$ Then $u$ verifies $\\begin{split}C_{N,p}\\, \\int _{\\mathbb {R}^N} u^{p-1}\\, &F_*(x)^{\\gamma -p}\\,\\varphi \\, dx\\\\&=\\int _{\\mathbb {R}^N} F_*(x)^\\gamma \\,F(\\nabla u)^{p-1}\\,\\langle \\nabla F(\\nabla u),\\nabla \\varphi \\rangle \\, dx,\\qquad \\varphi \\in C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace ).\\end{split}$ We test the previous equation with $\\varphi =v^{p}\\, u^{1-p}$ , where $v\\in C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ is positive.",
"We get $\\begin{split}\\beta ^{p-1}\\,\\Big [N-\\beta \\,p+\\beta -p+\\gamma \\Big ]&\\int _{\\mathbb {R}^N} v^p\\,F_*(x)^{\\gamma -p}\\, dx\\\\&=\\int _{\\mathbb {R}^N} F_*(x)^\\gamma \\,F(\\nabla u)^{p-1}\\left\\langle \\nabla F(\\nabla u),\\nabla \\left(\\frac{v^p}{u^{p-1}}\\right)\\right\\rangle \\, dx.\\end{split}$ We can now use Proposition REF with the choice $H(z)=F(z)^p$ , so to obtain $\\beta ^{p-1}\\,\\Big [N-\\beta \\,p+\\beta -p+\\gamma \\Big ]\\,\\int _{\\mathbb {R}^N} v^p\\,F_*(x)^{\\gamma -p}\\, dx\\le \\int _{\\mathbb {R}^N} F(\\nabla v)^p\\, F_*(x)^{\\gamma } \\, dx$ In order to conclude, it is now sufficient to observe that the function $\\beta \\mapsto \\beta ^{p-1}\\,\\Big [N-\\beta \\, p+\\beta -p+\\gamma \\Big ],$ is maximal for $\\beta =(N+\\gamma -p)/p$ .",
"The previous argument gives (REF ) for a positive $v\\in C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ .",
"Of course, the previous proof is still valid for positive Lipschitz functions supported in $\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace $ .",
"The result for a general $v\\in C^1_0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ then follows by writing $v=v_+-v_-$ and observing that $v_+,v_-$ are positive Lipschitz functions with support in $\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace $ .",
"Remark 5.5 By taking $\\gamma =0$ in (REF ), we have the usual Hardy inequality on the whole space with respect to a general norm, i.e.",
"$\\left(\\frac{N-p}{p}\\right)^p\\, \\int _{\\mathbb {R}^N} \\left(\\frac{|v|}{F_*(x)}\\right)^p\\, dx\\le \\int _{\\mathbb {R}^N} F(\\nabla v)^p\\, dx$ A different proof of (REF ) (based on symmetrization techniques) can be found in [29].",
"For $\\gamma \\ne 0$ and $F$ being the Euclidean norm, a related inequality can be found in [1]."
],
[
"Positive eigenfunctions",
"We denote by $\\Omega \\subset \\mathbb {R}^N$ an open connected set, which is now supposed to be bounded.",
"Let $1<p<\\infty $ and $0<s<1$ , in what follows we denote by $W^{s,p}_0(\\Omega )$ the completion of $C^\\infty _0(\\Omega )$ with respect to the norm $\\Vert u\\Vert _{W^{s,p}_0(\\Omega )}=\\left(\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} \\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+s\\,p}}\\, dx\\, dy\\right)^\\frac{1}{p}.$ As before, for $1<q\\le p$ we introduce the first eigenvalue $\\lambda ^s_{p,q}(\\Omega )=\\min _{u\\in W^{s,p}_0(\\Omega )}\\left\\lbrace \\Vert u\\Vert ^p_{W^{s,p}_0(\\Omega )}\\, :\\, \\Vert u\\Vert _{L^q(\\Omega )}=1\\right\\rbrace ,$ the reader is referred to [8], [19], [22] for a more detailed account about the case $q=p$ .",
"We notice that a minimizer $u$ of the previous problem is a weak solution of $(-\\Delta _p)^s u=\\lambda ^s_{p,q}(\\Omega )\\, |u|^{q-2}\\, u,\\qquad \\mbox{ in }\\Omega ,$ which means that $\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} \\frac{|u(x)-u(y)|^{p-2}\\, \\big (u(x)-u(y)\\big )}{|x-y|^{N+s\\,p}}\\, \\big (\\varphi (x)-\\varphi (y)\\big )\\, dx\\, dy=\\lambda ^s_{p,q}(\\Omega )\\, \\int _\\Omega |u|^{q-2}\\, u\\, \\varphi \\, dx,$ for every $\\varphi \\in W^{s,p}_0(\\Omega )$ .",
"Theorem 6.1 (Uniqueness of positive eigenfunctions) Let $1<p<\\infty $ , $0<s<1$ and $1<q\\le p$ .",
"Let $\\lambda >0$ be such that there exists a non trivial function $u\\in W^{s,p}_0(\\Omega )$ verifying $(-\\Delta _p)^s u=\\lambda \\, |u|^{q-2}\\, u,\\qquad u\\ge 0,\\quad \\mbox{ in }\\Omega .$ Then we have $\\lambda \\, \\left(\\int _\\Omega |u|^q\\, dx\\right)^\\frac{q-p}{q}=\\lambda _{p,q}(\\Omega ),$ and $v=u\\,\\Vert u\\Vert _{L^q(\\Omega )}^{-1}$ is a minimizer of the problem in (REF ).",
"At first, it is again crucial to observe that $u>0$ almost everywhere in $\\Omega $ , thanks to the minimum principle of Theorem REF .",
"Then the result for the case $p=q$ follows by using [19].",
"We now consider the case $q<p$ .",
"Again, we observe that $v$ solves $(-\\Delta _p)^s v=\\lambda \\,\\Vert u\\Vert _{L^q(\\Omega )}^{q-p}\\, v^{q-1},\\qquad u> 0,\\quad \\mbox{ in }\\Omega .$ and since $v$ is admissible for the variational problems defining $\\lambda ^s_{p,q}(\\Omega )$ we get $\\lambda \\, \\Vert u\\Vert _{L^q(\\Omega )}^{q-p}\\ge \\lambda ^s_{p,q}(\\Omega ).$ Let $u_1\\in W^{s,p}_0(\\Omega )$ be a function achieving the minimum in the right-hand side of (REF ).",
"Then again we have $\\begin{split}\\lambda \\,\\Vert u\\Vert _{L^q(\\Omega )}^{q-p}&=\\lambda \\,\\Vert u\\Vert _{L^q(\\Omega )}^{q-p}\\, \\int _\\Omega v^{q-1}\\, \\frac{u^q_1}{v^{q-1}}\\, dx\\\\&=\\int _{\\mathbb {R}^N} \\int _{\\mathbb {R}^N} \\frac{|v(x)-v(y)|^{p-2}\\, (v(x)-v(y))}{|x-y|^{N+s\\,p}}\\, \\left[\\frac{u_1(x)^q}{v(x)^{q-1}}-\\frac{u_1(y)^q}{v(y)^{q-1}}\\right]\\, dx\\,dy\\\\&\\le \\int _{\\mathbb {R}^N} \\int _{\\mathbb {R}^N} \\frac{|u_1(x)-u_1(y)|^q}{|x-y|^{N\\, \\frac{q}{p}+s\\,q}}\\,\\frac{|v(x)-v(y)|^{p-q}}{|x-y|^{N\\,\\frac{p-q}{p}+s\\,(p-q)}}\\, dx\\, dy \\\\\\end{split}$ where we used Proposition REF .",
"If we now apply Hölder's and Young's inequalities with exponents $p/q$ and $p/(p-q)$ , the previous gives the desired result."
],
[
"Hardy-type inequalities",
"As in the local case, by means of the discrete Picone inequality (REF ) we can prove a nonlocal Hardy inequality, like in [17].",
"The idea is still to look at power-type positive solutions of $(-\\Delta _p)^s u=\\lambda \\, u^{p-1},\\qquad \\mbox{ in }\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace .$ The latter is the content of the next technical result.",
"Lemma 6.2 Let $1<p<\\infty $ and $\\beta ,r>0$ .",
"Then the function $f(x)=\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\,\\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+r}}\\, dy,\\qquad x\\in \\mathbb {R}^N,$ is radial and $\\beta \\,(1-p)-r$ homogeneous.",
"Let us pick $x_1\\ne x_2$ such that $|x_1|=|x_2|$ .",
"Let us set call $R:\\mathbb {R}^N\\rightarrow \\mathbb {R}^N$ the linear isometry defined by the reflection in the hyperplan $\\pi =\\left\\lbrace x\\in \\mathbb {R}^N\\, :\\, \\langle x,x_1-x_2\\rangle =0\\right\\rbrace .$ Then by changing variables we have $\\begin{split}f(x_1)&=\\int _{\\mathbb {R}^N} \\frac{\\left||x_1|^{-\\beta }-|R\\,z|^{-\\beta }\\right|^{p-2}\\, \\left(|x_1|^{-\\beta }-|R\\,z|^{-\\beta }\\right)}{|x_1-R\\, z|^{N+r}}\\, dz\\\\&=\\int _{\\mathbb {R}^N} \\frac{\\left||x_2|^{-\\beta }-|R\\,z|^{-\\beta }\\right|^{p-2}\\, \\left(|x_2|^{-\\beta }-|R\\,z|^{-\\beta }\\right)}{|R\\, x_2-R\\, z|^{N+r}}\\, dz\\\\&=\\int _{\\mathbb {R}^N} \\frac{\\left||x_2|^{-\\beta }-|z|^{-\\beta }\\right|^{p-2}\\,\\left(|x_2|^{-\\beta }-|z|^{-\\beta }\\right)}{|x_2-z|^{N+r}}\\, dz=f(x_2),\\end{split}$ where we used that $R\\, x_2=x_1$ , that $|R\\, z|=|z|$ and the linearity of $R$ .",
"This shows that $f$ is radial.",
"For the second part, it is sufficient to observe that $\\begin{split}f(t\\, x)&=\\int _{\\mathbb {R}^N} \\frac{\\left|t^{-\\beta }\\,|x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\, \\left(t^{-\\beta }\\, |x|^{-\\beta }-|y|^{-\\beta }\\right)}{|t\\,x-y|^{N+r}}\\, dy\\\\&=t^{-\\beta \\,(p-1)-N-r}\\,\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|z|^{-\\beta }\\right|^{p-2}\\,(|x|^{-\\beta }-|z|^{-\\beta })}{|x-z|^{N+r}}\\, t^N\\, dz\\\\&=t^{-\\beta \\,(p-1)-r} f(x),\\qquad x\\in \\mathbb {R}^N,\\end{split}$ for all $t>0$ , which gives the desired conclusion.",
"We then have the following sharp Hardy inequality for the fractional Sobolev space $W^{s,p}$ , first proved in [17].",
"Theorem 6.3 (Fractional Hardy inequality) Let $s\\in (0,1)$ and $1<p<\\infty $ such that $s\\,p<N$ .",
"Then there exists a constant $C=C(N,s,p)>0$ (see equation (REF ) below) such that $C\\,\\int _{\\mathbb {R}^N} \\frac{|v|^p}{|x|^{s\\,p}}\\,dx \\le \\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} \\frac{\\big | v(x)-v(y)\\big |^p}{|x-y|^{N+s\\,p}}\\,dx\\,dy,$ for all $v\\in C^\\infty _0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ .",
"We recall that a positively $\\alpha -$ homogeneous function $u$ which is radially simmetric, that is $u(x) = \\varphi (|x|)\\, \\qquad x\\in \\mathbb {R}^N\\,,$ is uniquely determined modulo a multiplicative constant, namely $u(x)=\\varphi (1)\\, |x|^\\alpha $ .",
"By this elementary observation, we can deduce from Lemma REF that the function $u(x)=|x|^{-\\beta }$ is a solution of $\\begin{split}\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} &\\frac{\\left|u(x)-u(y)\\right|^{p-2}\\,\\left(u(x)-u(y)\\right)}{|x-y|^{N+s\\,p}}\\, (\\varphi (x)-\\varphi (y))\\, dx\\,dy\\\\&= C(\\beta )\\, \\int _{\\mathbb {R}^N} u^{p-1}\\, |x|^{-s\\,p}\\, \\varphi (x)\\, dx, \\qquad \\mbox{ for all } \\varphi \\in C^\\infty _0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace ),\\end{split}$ where $C(\\beta )=2\\,\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\,\\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+s\\,p}}\\, dy,\\qquad x\\in \\mathbb {S}^{N-1},$ and the previous integral is constant for $x\\in \\mathbb {S}^{N-1}$ , thanks to Lemma REF .",
"Then, if one picks $v\\in C^\\infty _0(\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace )$ positive and plugs in $\\varphi = v^p\\,u^{1-p}$ as a test function in the previous equation, it turns out that $C(\\beta )\\, \\int _{\\mathbb {R}^N} |v|^p\\, |x|^{-s\\,p}\\,dx\\le \\int _{\\mathbb {R}^N} \\int _{\\mathbb {R}^N}\\frac{|v(x)-v(y)|^p}{|x-y|^{N+sp}}\\,dx\\,dy.$ An optimization over $\\beta $ leads to the desired result, see Appendix B for more details."
],
[
"A minimum principle for positive nonlocal eigenfunctions",
"In the following, we provide a proof of a minimum principle for positive weak supersolutions to equation $(-\\Delta _p)^s u=0,\\quad \\mbox{ in }\\Omega ,\\qquad \\qquad u\\equiv 0\\quad \\mbox{ in }\\mathbb {R}^N\\setminus \\Omega .$ i.e.",
"for functions $u\\in W^{s,p}_0(\\Omega )$ such that $\\int _{\\mathbb {R}^N}\\int _{\\mathbb {R}^N} \\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}\\,(\\varphi (x)-\\varphi (y))\\, dx\\,dy \\ge 0, \\quad \\mbox{ for all} \\ \\varphi \\in C_0^\\infty (\\Omega ), \\ \\varphi \\ge 0.$ Let $x_0$ be any fixed point in $\\Omega $ , and for every $r>0$ let $B_r(x_0)$ denote the ball of radius $r$ centered at $x_0$ .",
"The main ingredient of our result is the following lemma, which is a consequence of a more general logarithmic estimate recently established by Di Castro, Kuusi and Palatucci in [13].",
"DKP Logarithmic Lemma Let $1<p<\\infty $ , $s\\in (0,1)$ and $u\\in W^{s,p}_0(\\Omega )$ be a supersolution such that $u\\ge 0$ in $B_{2\\,r}(x_0)\\Subset \\Omega $ .",
"Then for every $0<\\delta <1$ there holds $\\begin{split}\\int _{B_r} \\int _{B_r} \\left|\\log \\left(\\frac{\\delta +u(x)}{\\delta +u(y)}\\right)\\right|^p&\\frac{1}{|x-y|^{N+s\\,p}} \\, dx\\,dy\\\\& \\le C\\, r^{N-s\\,p} \\left\\lbrace \\delta ^{1-p}\\, r^{s\\,p}\\,\\int _{\\mathbb {R}^N\\setminus B_{2\\,r} } \\frac{u_-(y)^{p-1}}{|y-x_0|^{N+sp}}\\,dy+ 1\\right\\rbrace ,\\end{split}$ where $u_-=\\max \\lbrace -u,0\\rbrace $ and $C=C(N,p,s)>0$ is a constant.",
"We then have the following minimum principle.",
"Theorem 1.1 Let $\\Omega \\subset \\mathbb {R}^N$ be an open bounded set, which is connected.",
"Let $s\\in (0,1)$ , $1<p<\\infty $ and $u\\in W^{s,p}_0(\\Omega )$ be a weak supersolution such that $u\\ge 0$ in $\\Omega $ .",
"Let us suppose that $u\\lnot \\equiv 0\\qquad \\mbox{ in } \\Omega .$ Then $u>0$ almost everywhere in $\\Omega $ .",
"We first prove that for every $K\\Subset \\Omega $ compact connected, if $u\\lnot \\equiv 0\\qquad \\mbox{ in } K,$ then $u>0$ almost everywhere in $K$ .",
"Let $K\\Subset \\Omega $ be a connected compact set, then $K\\subset \\lbrace x\\in \\Omega \\, :\\, \\mathrm {dist}(x,\\partial \\Omega )>2\\,r\\rbrace $ for some $r>0$ .",
"We then observe that $K$ can be covered by a finite number of balls $B_{r/2}(x_1),\\dots B_{r/2}(x_k)$ such that $x_i\\in K$ and $|B_{r/2}(x_i)\\cap B_{r/2}(x_{i+1})|>0,\\quad i=1,\\dots , k-1.$ Let us now suppose that $u=0$ on a subset of $K$ with positive measure.",
"Then for some $i\\in \\lbrace 1,\\dots ,k-1\\rbrace $ , we have that the set $Z:=\\lbrace x\\in B_{r/2}(x_i)\\,: \\, u(x)=0\\rbrace ,$ has positive measure.",
"We define $F_\\delta (x)=\\log \\left(1+\\frac{u(x)}{\\delta }\\right),\\qquad x\\in B_{r/2}(x_i),$ for $\\delta >0$ and claim that the following Poincaré inequality holds true $\\int _{B_{r/2}(x_i)} |F_\\delta |^p\\, dx\\le \\frac{r^{N+s\\,p}}{|Z|}\\,\\int _{B_{r/2}(x_i)}\\int _{B_{r/2}(x_i)}\\frac{|F_\\delta (x)-F_\\delta (y)|^p}{|x-y|^{N+s\\,p}}\\, dx\\,dy.$ Indeed, observe that $F_\\delta (x)=0\\qquad \\mbox{ for every } x\\in Z\\,,$ hence for every $x\\in B_{r/2}(x_i)$ and $y\\ne x$ with $y\\in Z$ , we get $|F_\\delta (x)|^p=\\frac{|F_\\delta (x)-F_\\delta (y)|^p}{|x-y|^{N+s\\,p}}\\,|x-y|^{N+s\\,p}\\,.$ Now integrating with respect to $y\\in Z$ gives $|Z|\\, |F_\\delta (x)|^p\\le \\left(\\max _{x,y\\in B_{r/2}(x_i)} |x-y|^{N+s\\,p}\\right)\\,\\int _{B_{r/2}(x_i)}\\frac{|F_\\delta (x)-F_\\delta (y)|^p}{|x-y|^{N+s\\,p}}\\, dy\\,,$ which proves (REF ) up to an integration with respect to $x\\in B_{r/2}(x_i)$ .",
"We now observe that $\\left|\\log \\left(\\frac{\\delta +u(x)}{\\delta +u(y)}\\right)\\right|^p=\\left| F_\\delta (x) - F_\\delta (y)\\right|^p,$ thus if we combine (REF ), (REF ) and observe that $u_-\\equiv 0$ , we get $\\begin{split}\\int _{B_{r/2}(x_i)} \\left|\\log \\left(1+\\frac{u}{\\delta }\\right)\\right|^p&\\, dx\\le C\\, \\frac{r^{2\\,N}}{|Z|},\\end{split}$ with $C$ independent of $\\delta $ .",
"By letting $\\delta $ go to 0 in (REF ), we can then infer $u=0\\qquad \\mbox{ almost everywhere in } B_{r/2}(x_i).$ By using property (REF ), we can repeat the previous argument for the balls $B_{r/2}(x_{i-1})$ and $B_{r/2}(x_{i+1})$ and so on, up to obtain that $u=0$ almost everywhere on $K$ .",
"This clearly contradicts (REF ), thus $u>0$ almost everywhere in $K$ .",
"Let us now assume (REF ).",
"Since $\\Omega $ is connected, there exists a sequence of connected compact sets $K_n\\Subset \\Omega $ such that $|\\Omega \\setminus K_n|<\\frac{1}{n}\\qquad \\mbox{ and }\\quad u\\lnot \\equiv 0\\mbox{ in }K_n.$ Then, by the first part of the proof $u>0$ almost everywhere on each $K_n$ .",
"By letting $n$ go to $\\infty $ , we get the conclusion."
],
[
"Optimal constant for the fractional Hardy inequality",
"In Section we used that $u(x)=|x|^{-\\beta }$ is a solution of $(-\\Delta _p)^s u=C(\\beta )\\, \\frac{u^{p-1}}{|x|^{s\\,p}},\\qquad \\mbox{ in }\\mathbb {R}^N\\setminus \\lbrace 0\\rbrace .$ In this appendix we discuss some features of the constant $C(\\beta )=2\\,\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\,\\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+s\\,p}}\\, dy,\\qquad x\\in \\mathbb {S}^{N-1},$ and determines the best constant in (REF ), see equation (REF ) below.",
"Computations are very much the same as in the paper [17] by Frank and Seiringer, up to some simplifications.",
"For simplicity we focus on the case $N\\ge 2$ and $s\\,p<N$ .",
"Lemma 2.1 Let $N\\ge 2$ , $0<s<1$ and $1<p<\\infty $ .",
"For every $0<\\varrho <1$ the function $\\mathcal {G}(\\beta )=\\left[1-\\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\right]\\,\\left[1-\\varrho ^\\beta \\right]^{p-1},\\qquad \\beta >0,$ is maximal for $\\beta =(N-s\\,p)/p$ .",
"We just have to differentiate the function $\\mathcal {G}$ .",
"Indeed, we have $\\begin{split}\\mathcal {G}^{\\prime }(\\beta )&=(p-1)\\,\\log \\varrho \\, \\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\,\\left[1-\\varrho ^\\beta \\right]^{p-1}\\\\&-(p-1)\\,\\varrho ^\\beta \\, \\log \\varrho \\,\\left[1-\\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\right]\\,\\left[1-\\varrho ^\\beta \\right]^{p-2},\\end{split}$ so that $\\mathcal {G}^{\\prime }(\\beta )\\ge 0\\quad \\Longleftrightarrow \\quad \\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\,\\left[1-\\varrho ^\\beta \\right]-\\varrho ^\\beta \\,\\left[1-\\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\right]\\le 0$ that is if and only if $\\beta $ is such that $\\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\le \\varrho ^\\beta .$ By passing to the logarithm, we obtain the assertion.",
"We can then determine the maximal values of $C(\\beta )$ .",
"Lemma 2.2 Let $N\\ge 2$ , $0<s<1$ and $1<p<\\infty $ .",
"For every $\\beta >0$ we have $0<C(\\beta )\\le C\\left(\\frac{N-s\\,p}{p}\\right).$ We recall that $C(\\beta )=2\\,\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\, \\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+s\\,p}}\\, dy,\\qquad \\mbox{ for every }x\\in \\mathbb {S}^{N-1},$ and the right-hand side is independent of $x\\in \\mathbb {S}^{N-1}$ .",
"Thus we have $C(\\beta )=\\frac{2}{\\mathcal {H}^{N-1}(\\mathbb {S}^{N-1})}\\,\\int _{\\mathbb {S}^{N-1}}\\int _{\\mathbb {R}^N} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\, \\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+s\\,p}}\\, dy\\, d\\mathcal {H}^{N-1}(x).$ We observe that for every $y\\in \\mathbb {R}^N\\setminus \\lbrace 0\\rbrace $ we can write $\\mathbb {S}^{N-1}=\\bigcup _{t\\in [-1,1]} \\Sigma _t(y),$ where $\\Sigma _t=\\left\\lbrace x\\in \\mathbb {S}^{N-1}\\, :\\, \\left\\langle x,\\frac{y}{|y|}\\right\\rangle =t\\right\\rbrace \\simeq \\sqrt{1-t^2}\\,\\mathbb {S}^{N-2}.$ By using this and exchanging the order of integration, we then obtain $\\begin{split}C(\\beta )=\\frac{2}{\\mathcal {H}^{N-1}(\\mathbb {S}^{N-1})}&\\,\\int _{\\mathbb {R}^N}\\int _{\\mathbb {S}^{N-1}} \\frac{\\left||x|^{-\\beta }-|y|^{-\\beta }\\right|^{p-2}\\, \\left(|x|^{-\\beta }-|y|^{-\\beta }\\right)}{|x-y|^{N+s\\,p}}\\, d\\mathcal {H}^{N-1}(x)\\,dy\\\\&=\\frac{\\mathcal {H}^{N-2}(\\mathbb {S}^{N-2})}{\\mathcal {H}^{N-1}(\\mathbb {S}^{N-1})}\\,\\int _{\\mathbb {R}^N}\\Big [|1-|y|^{-\\beta }|^{p-2}\\, (1-|y|^{-\\beta })\\\\&\\times \\,\\int _{-1}^1 \\frac{(1-t^2)^\\frac{N-3}{2}}{(1-2\\,t\\,|y|+|y|^2)^\\frac{N+s\\,p}{2}}\\, \\,dt\\Big ]\\,dy\\end{split}$ where we used that $|x-y|=\\sqrt{1-2\\, t\\,|y|+|y|^2},\\qquad \\mbox{ for every } x\\in \\Sigma _t.$ We now set $\\Phi (\\varrho )=\\mathcal {H}^{N-2}(\\mathbb {S}^{N-2})\\,\\int _{-1}^1 \\frac{(1-t^2)^\\frac{N-3}{2}}{(1-2\\,t\\,\\varrho +\\varrho ^2)^\\frac{N+s\\,p}{2}}\\, dt,$ then by using polar coordinates we get $\\begin{split}C(\\beta )&= 2\\,\\int _0^\\infty \\varrho ^{N-1}\\, \\left|1-\\varrho ^{-\\beta }\\right|^{p-2}\\, (1-\\varrho ^{-\\beta })\\, \\Phi (\\varrho )\\,d\\varrho \\\\&=-2\\,\\int _0^1 \\varrho ^{N-1}\\,\\left|1-\\varrho ^{-\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho \\\\&+2\\,\\int _1^\\infty \\varrho ^{N-1}\\,\\left|1-\\varrho ^{-\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho .\\end{split}$ We now perform the change of variable $\\varrho =r^{-1}$ in the second integral and observe that $\\Phi (1/r)=r^{N+s\\,p}\\,\\Phi (r).$ Thus we obtain $\\begin{split}C(\\beta )&=-2\\,\\int _0^1 \\varrho ^{N-1}\\,\\left|1-\\varrho ^{-\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho +2\\,\\int _0^1 \\varrho ^{-1+s\\,p}\\,\\left|1-\\varrho ^{\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho \\\\&=-2\\,\\int _0^1 \\varrho ^{N-1-\\beta \\,(p-1)}\\,\\left|\\varrho ^\\beta -1\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho +2\\,\\int _0^1 \\varrho ^{-1+s\\,p}\\,\\left|1-\\varrho ^{\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho \\\\&=2\\,\\int _0^1 \\varrho ^{s\\,p-1}\\,\\left[1-\\varrho ^{N-s\\,p-\\beta \\,(p-1)}\\right]\\,\\left|1-\\varrho ^{\\beta }\\right|^{p-1}\\, \\Phi (\\varrho )\\,d\\varrho .\\end{split}$ We now observe that the term into square brackets is positive if $\\beta \\le \\frac{N-s\\,p}{p-1}.$ Moreover, thanks to Lemma REF the integrand is maximal for $\\beta =\\frac{N-s\\,p}{p},$ and thus we get the conclusion.",
"Remark 2.3 Observe that we have $C\\left(\\frac{N-s\\,p}{p}\\right)=2\\,\\int _0^1 \\varrho ^{s\\,p-1}\\,\\left[1-\\varrho ^\\frac{N-s\\,p}{p}\\right]^{p}\\, \\Phi (\\varrho )\\,d\\varrho ,$ where the function $\\Phi $ is defined in (REF ).",
"Thus this is the best constant in the Hardy inequality (REF ) (see [17] for more details)."
]
] | 1403.0280 |
[
[
"Ferromagnetic SrRuO3 thin-film deposition on a spin-triplet\n superconductor Sr2RuO4 with highly conducting interface"
],
[
"Abstract Ferromagnetic SrRuO3 thin films are deposited on the ab-surface of single crystals of the spin-triplet superconductor Sr2RuO4 as substrates using pulsed laser deposition.",
"The films are under a severe in-plane compressive strain.",
"Nevertheless, the films exhibit ferromagnetic order with the easy axis along the c-direction below the Curie temperature of 158 K. The electrical transport reveals that the SrRuO3/Sr2RuO4 interface is highly conducting, in contrast with the interface between other normal-metals and the ab-surface of Sr2RuO4.",
"Our results will stimulate the investigations on proximity effects between a ferromagnet and a spin-triplet superconductor."
],
[
"Ferromagnetic SrRuO$_3$ thin-film deposition on a spin-triplet superconductor Sr$_2$ RuO$_4$ with highly conducting interface M. S. Anwar$^{1*}$ , Y. J. Shin$^{2,3}$ , S. R. Lee$^{2,3}$ , S. J. Kang$^{2,3}$ , Y. Sugimoto$^{1}$ , S. Yonezawa$^{1}$ , T. W. Noh$^{2,3}$ , and Y. Maeno$^{1}$ $^{1}$ Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan $^{2}$ Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 151-747, Republic of Korea $^{3}$ Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Republic of Korea $^{*}$ E-mail: anwar@scphys.kyoto-u.ac.jp;m.s.anwar476@gmail.com Ferromagnetic SrRuO$_3$ thin films are deposited on the ab-surface of single crystals of the spin-triplet superconductor Sr$_2$ RuO$_4$ as substrates using pulsed laser deposition.",
"The films are under a severe in-plane compressive strain.",
"Nevertheless, the films exhibit ferromagnetic order with the easy axis along the c-direction below the Curie temperature of 158 K. The electrical transport reveals that the SrRuO$_3$ /Sr$_2$ RuO$_4$ interface is highly conducting, in contrast with the interface between other normal-metals and the ab-surface of Sr$_2$ RuO$_4$ .",
"Our results will stimulate the investigations on proximity effects between a ferromagnet and a spin-triplet superconductor.",
"Hybrid structures with ferromagnetic metals (FMs) and superconductors are fascinating systems exhibiting unconventional phenomena originating from competition and cooperation between the ferromagnetic order and superconductivity.",
"In the past, two aspects of exotic behavior of the proximity effect between a FM and a spin-singlet superconductor (SSC) have been revealed.",
"Firstly, the amplitude and phase of the spin-singlet order parameter spatially oscillates in the FM due to exchange interaction and even leads to a $\\pi $ -junction [1].",
"The penetration length of this effect is intrinsically limited to the order of a few nanometers.",
"Secondly, a spin-triplet pair amplitude emerges over a length of the order of a micrometer into the FM if suitable magnetic inhomogeneity is present at the FM/SSC interface [2], [3], [4], [5], [6], [7].",
"The resulting superconducting symmetry in the FM is spin-triplet s-wave and thus classified as odd-frequency pairing [8].",
"To induce such a superconductivity into a ferromagnet, half metallic FM CrO$_2$ with multi-fold magnetic anisotropy [3], [6], or alternatively ferromagnetic multilayer X/Co/X, where X can be PdNi, CuNi, Ni, or Ho, with non-collinear magnetization, is used in the SSC/FM/SSC Josephson junctions [4], [5], [7].",
"Another approach to realize novel superconducting junctions with FM is to use a spin-triplet superconductor (TSC) with multiple degrees of freedom of the order parameter [10], [11], [12].",
"Indeed, it has been theoretically predicted that not only charge supercurrent but also spin supercurrent emerges at FM/TSC interface and both can be controlled by the magnetization direction in the FM relative to the direction of the spins of the spin triplet Cooper pairs [11].",
"Thus, FM/TSC junctions, with both controllable charge and spin supercurrents, would initiate a new research area that can be termed as “Superspintronics”.",
"Most likely, Sr$_2$ RuO$_4$ (SRO214) exhibits the chiral p-wave spin-triplet superconducting order parameter analogous to the superfluid $^3$ He-A phase [13], [14].",
"SRO214 has also been attracting much attention as one of the leading candidates for the topological superconductor.",
"Recently, its topological nature has been investigated with superconducting junctions [15], [16], [17].",
"The main challenge in realizing a FM/TSC junction is the availability of thin films of spin triplet superconductors.",
"By overcoming the difficulty in growing high quality SRO214 films, superconductivity has finally been induced in films [18].",
"Nevertheless, the transition temperature ($T_{\\rm c}$ ) is about 1 K, which is noticeably lower than its bulk $T_{\\rm c}$ of 1.5 K. Thus, as another approach to fabricate the FM/TSC junctions, we choose to use a high quality single crystal of SRO214, which is already available.",
"SRO214 crystals can be cleaved along the ab-surface with atomically flat surface.",
"However, there are surface reconstructions associated with RuO$_6$ octahedral rotation about the c-axis, which freezes the associated bulk soft phonon mode into a static lattice distortion.",
"The reconstruction alters the surface electronic state [19] and in most cases leads to a poor electrical contact to the ab-surface.",
"Therefore, it is an important issue to obtain highly conducting interface between a SRO214 crystal and a FM-layer in order to realize FM/TSC junctions.",
"Here, we report the first growth of ferromagnetic epitaxial thin films of SRO113 on the ab-surface of single crystals of SRO214.",
"Transport measurements reveal that the SRO113/SRO214 interface is highly conducting.",
"SRO113 is an itinerant ferromagnet with the Curie temperature $T_{\\rm Curie}$ of 160 K [20], and is the $n=\\infty $ member of the same Ruddlesden-Poppor series Sr$_{n+1}$ Ru$_n$ O$_{3n+1}$ as SRO214 with $n=1$ .",
"SRO113 thin films have already been utilized in superconducting devices with the high temperature superconductor YBa$_2$ Cu$_3$ O$_{7-\\delta }$ (YBCO).",
"Interestingly, the supercurrent in the YBCO/SRO113/YBCO junction abruptly disappears when the thickness of the SRO113 layer exceeds 25 nm [22], [21].",
"In another report, superconducting gap in a YBCO/SRO113(18-nm) bilayer persists only at the ferromagnetic domain-wall regions [23].",
"Our findings of highly conducting interface between SRO113 and SRO214 open up a possibility to develop high-quality FM/TSC junctions to explore the unconventional proximity effects.",
"The a-axis mismatch at 300 K between the bulk SRO113 ($a_{\\rm 113}=3.93$ Å in the pseudo-cubic notation) [24] and the bulk SRO214 ($a_{\\rm 214}=3.871$ Å) [14] is $(a_{\\rm 113}-a_{\\rm 214})/a_{\\rm 214}= 1.5$$\\%$ .",
"This raises a possibility that a SRO113 (001) thin film can be grown on the ab-surface of SRO214 under compressive strain.",
"Through out this paper, we use pseudo-cubic notation for the lattice parameters and crystalline axes of SRO113.",
"We used SRO214 single crystals with a typical size of 3 $\\times $ 3 $\\times $ 0.5 mm$^3$ as substrates to grow SRO113 thin films using the pulsed laser deposition (PLD) technique with substrate-to-target distance of 54 mm.",
"The growth temperature of 500 - 700$^\\circ $ C and the oxygen partial pressure of 100 mTorr were employed with the base pressure of $<$ 3 $\\times $ 10$^{-7}$ Torr.",
"It takes about 20 - 50 sec to grow one unit cell at a laser intensity of 1.5 Jcm$^{-2}$ and a repetition rate of 1 - 3 Hz.",
"We also grow SRO113 thin film on an insulating SrTiO${_3}$ substrate as reference sample, whose residual resistivity ratio (RRR) is approximately 7, manifesting good performance of our growth facilities.",
"The magnetization was measured down to 4 K with a SQUID magnetometer (Quantum Design, MPMS-XL).",
"To measure the resistivity down to 4 K, we made a 1 $\\times $ 0.7 mm$^2$ pad of SRO113 film and used a $^4$ He cryostat (Quantum Design; PPMS).",
"The cleaved ab-surface of SRO214 usually consists of regions of the typical size of 100 $\\times $ 1000 $\\mu $ m$^2$ that look flat by optical microscope.",
"Indeed, within such flat regions, we can easily find atomically flat areas of the size of 10 $\\times $ 10 $\\mu $ m$^2$ under atomic force microscope (AFM).",
"However, we also find several steps higher than the thickness of the film (50 nm) with a scanning electron microscope (SEM).",
"If silver paste touches such a higher step, a direct contact between silver paste and SRO214 through the ac-surface would be created.",
"To avoid such a scenario, we carefully put contacts only at most flat surfaces under optical microscope and absence of the direct contact was confirmed under SEM after measurements.",
"Figure REF (a) represents the X-ray diffraction (XRD) spectrum of a 15-nm-thick SRO113 film grown on SRO214.",
"It shows no impurity peaks but only the $(00l)$ peaks for SRO113 and SRO214, indicating that the ${\\it c}$ axis of SRO113 film is in the same direction as that of SRO214 substrate.",
"It is noted that the SRO113 $(00l)$ peaks are located at smaller angles than those of the bulk (vertical red solid lines), indicating that our film has an elongated lattice along the out-of-plane direction.",
"The out-of-plane lattice constant is estimated as 4.00 Å in the pseudo-cubic notation, which is $1.8\\%$ larger than the corresponding bulk value.",
"AFM topographs of a film shown in the inset of Fig.",
"REF (a) reveal that the film has atomically-flat terraces with the step height equal to the lattice constant.",
"Around the (001) and (002) peaks of SRO113, we find the thickness fringes (as the period of the oscillations well corresponds to the thickness of the films), which also suggest that the atomically flat top and bottom surfaces of SRO113 are formed.",
"In addition, the reflection high-energy electron diffraction (RHEED) patterns of the SRO113 film and the SRO214 substrate with the e-beam aligned along the [100] azimuthal direction are shown in the insets of Fig.",
"REF (b).",
"Orientation and distance of the first-order peaks with respect to the central peaks are invariant before and after the SRO113 film deposition.",
"This invariance indicates that the in-plane crystalline axes of the film and substrate are aligned and thus provide evidence for the epitaxial nature of the film.",
"It is noted that the RHEED oscillations taken in-situ during the film growth as shown in Fig.",
"REF (b) demonstrates the changeover of the film-growth modes from layer-by-layer to step-flow at around 500 sec [25].",
"We also study SRO113/SRO214 hybrid using high resolution transmission electron microscopy (HR-TEM), which reveals that SRO113 films are grown epitaxially.",
"Figure: (a) XRD spectrum on a logarithmic scale of a c-axis oriented 15-nm-thick SRO113 film deposited on a SRO214 substrate.",
"Vertical lines indicate the positions of the corresponding peaks of the bulk materials.",
"Insets show an AFM topographic image giving the average roughness of ≈\\approx 0.5 nm and an atomic-step structure along the line given in the AFM-image.",
"(b) RHEED oscillations during the film growth.",
"Insets show the RHEED patterns before and after the film growth.",
"(c) HR-TEM image of SrRuO 3 _3//Sr 2 _2RuO 4 _4 hybrid.",
"It is taken from [120] SrRuO 3 _3∥\\parallel [120] Sr 2 _2RuO 4 _4 zone axis.Epitaxial growth of SRO113 on various perovskite substrates has been extensively studied: [25], [26] under compressive strain along the in-plane direction, SRO113 films exhibit the c-axis elongation.",
"We found that our SRO113 film has essentially the same peak positions as the one deposited on an NdGaO$_3$ substrate [26], which has an in-plane lattice constant (3.86 Å in the pseudo-cubic notation) very similar to that of SRO214.",
"Anticipating that the strong compressive strain could modify the magnetic behavior of the film, we measured its magnetization.",
"Figure REF (a) shows the temperature dependence of the magnetization of a 50-nm-thick SRO113 film with the field of 10 mT both along the c-axis (out-of-plane) and the a-axis (in-plane) directions on field cooling.",
"Surprisingly, we observe that $T_{Curie}$ is 158 K, which is almost equal to the values for the bulk [27], [28].",
"In contrast, it is reported that high-quality SRO113 thin films deposited on SrTiO$_3$ substrates show the reduction of $T_{\\rm Curie}$ to 150 K, although the a-axis mismatch between SRO113 and SrTiO$_3$ is only about $-0.45\\%$ [20].",
"After careful subtraction of the background signals, we obtain the remanent magnetization of the film at 4 K to be 2.8 $\\mu _{\\rm B}$ /Ru along the c-axis and 2 $\\mu _{\\rm B}$ /Ru along the a-axis, which are substantially larger than the expected values [29].",
"The magnetization loops for both the c- and a-axis direction are presented in Fig.",
"REF (b).",
"These data show that our films exhibit a strong magnetic anisotropy with larger remanent magnetization along the c-axis [30].",
"We obtained essentially the same $T_{\\rm Curie }$ , magnetization values, and anisotropy for a 15-nm-thick SRO113 film.",
"Figure: (a) Temperature dependent remanent magnetization of a 50-nm-thick SRO113 film along both the a-axis (blue squares) and the c-axis (red circles).",
"The sample was cooled down in field of 1 T and its magnetization was measured upon warming after removing field at 4 K. (b) Magnetization loops of the film along the a- and c-axes.",
"In both (a) and (b), the substrate contributions were measured before deposition of the film and were subtracted from data measured after deposition.Figure REF (a) presents the temperature dependent resistance $R$ ($T$ ) between 300 K and 4 K for a 50-nm-thick SRO113 film.",
"We measured $R$ ($T$ ) using a dc four-probe technique with two different channel configurations: first with four contacts on the SRO113 film $R$$_{\\rm 12,34}$ ($T$ ); second with two electrodes on SRO113 film and the other two electrodes on the side of SRO214 substrate $R$$_{\\rm 12,56}$ ($T$ ).",
"A schematic of the sample with the electrode arrangements is shown in Fig.",
"REF (b).",
"Note that the electrodes were placed in a flat area, avoiding steps larger than film thickness.",
"The characteristic slope change around 120 K for both data suggests that the c-axis resistivity $\\rho _{\\rm c}$ of SRO214 [31] contributes significantly to transport for both configurations.",
"It is obvious that $\\rho _{\\rm c}$ contributes more for $R$$_{\\rm 12,56}$ ($T$ ) as expected from the electrode configuration.",
"Thus, $R$ ($T$ ) mainly reflects the behavior of SRO214 in the whole temperature range down to 4 K. This fact suggests that the SRO113 film has a good electrical contact with the SRO214 substrate.",
"A linear current-voltage (I-V) curve measured at 4 K with 12,56 configuration of the electrodes also reveals that we obtain an Ohmic contact between SRO113 and SRO214.",
"Note that the residual resistance ratio (RRR) for 12,34 configuration is 60 which is many folds larger than that of SRO113 thin films, and much smaller than that of SRO214.",
"This RRR value suggests small but non-negligible resistance contribution of the interface in particular at low temperatures.",
"Figure: Resistance vs temperature of a SRO113(50 nm)/SRO214 hybrid system.",
"(a) R 12,34 (T)R_{12,34}(T) (on the ab-surface; blue squares) and R 12,56 R_{\\rm 12,56}(TT) (top to side; red circles).",
"Inset shows a linear I-V curve measured in 12,56 configuration.",
"(b) Schematic illustration of the electrodes.",
"Top electrodes 1 to 4 are placed in a flat area away from the steps larger than the film thickness.",
"(c) Simplified model of the resistance circuit.To further discuss the resistance of the SRO113/SRO214, we use a simplified model circuit shown in Fig.",
"REF (c).",
"In this circuit, $R^{\\parallel }_{113}$ and $R^{\\bot }_{113}$ are the resistances of the SRO113 film parallel and perpendicular to the ab-surface, $R_{\\rm int}$ is the interface resistance, and $R_c$ and $R_{ab}$ are the resistances of SRO214 along the c-axis and along the ab-plane.",
"We should note that this simplified model estimates the upper limits for the measured resistances.",
"Thus, the measured values should be lower than the estimated values.",
"In order to estimate each resistance, we refer to existing resistivity data of SRO113 and SRO214.",
"At 300 K, SRO113 thin films exhibit nearly isotropic resistivity $\\rho _{\\rm 113}$ $\\approx $ 250 $\\mu $$\\Omega $ cm depending on the substrate [20].",
"For any kind of substrate, $\\rho _{\\rm 113}$ decreases linearly with decreasing temperature down to $T_{\\rm Curie}$ and exhibits a sharp kink at $T_{\\rm Curie}$ due to the reduction of scatterings by spin fluctuations.",
"The resistivity of SRO214 along the c-axis is $\\rho _{c}$ $\\approx $ 15 m$\\Omega $ cm at 300 K, which is two orders of magnitude larger than the resistivity along the ab-surface $\\rho _{ab}$ $\\approx $ 120 $\\mu $$\\Omega $ cm [31].",
"Whereas $\\rho _{ab}$ decreases monotonically with decreasing temperature, $\\rho _{c}$ exhibits a broad maximum around 120 K [31].",
"Using these resistivity values and the geometrical factors of the present hybrid system, we performed an order estimate of each contribution of the present system at 300 K: $R^{\\parallel }_{113}$ $\\approx $ 70 $\\Omega $ , $R^{\\bot }_{113}$ $\\approx $ 0.2 $\\mu \\Omega $ , $R_c$ $\\approx $ 10 m$\\Omega $ , and $R_{ab}$ $\\approx $ 3 m$\\Omega $ .",
"We also estimate each contribution at 4 K using the values of $\\rho _{\\rm 113} \\approx $ 10 $\\mu \\Omega $ cm, $\\rho _{c}$ $\\approx $ 1 m$\\Omega $ cm, and $\\rho _{ab}$ $\\approx $ 1 $\\mu $$\\Omega $ cm: we obtain $R^{\\parallel }_{113}$ $\\approx $ 3 $\\Omega $ , $R^{\\bot }_{113}$ $\\approx $ 10 n$\\Omega $ , $R_c$ $\\approx $ 1 m$\\Omega $ , and $R_{ab}$ $\\approx $ 20 $\\mu \\Omega $ .",
"In the present model, $R_{\\rm 12,34}$ should satisfy the relation $1/R_{12,34}=1/R^{\\parallel }_{113}+1/(2R^{\\bot }_{113}+2R_{\\rm int}+2R_c+R_{ab})$ .",
"Since $R^{\\parallel }_{113}$ at 300 K is estimated to be four orders of magnitude larger than the observed value of $R_{\\rm 12,34}=6~$ m$\\Omega $ , the first term on the right hand side should be negligible.",
"In the second term, because $R^{\\bot }_{113}$ is estimated to be very small compared with $R_c$ or $R_{ab}$ , the $R^{\\bot }_{113}$ term has little contribution.",
"Thus, the contributions of the SRO113 film to $R_{\\rm 12,34}$ should be negligible.",
"This is indeed consistent with the fact that resistance anomaly at $T_{\\rm Curie}$ is absent in our results.",
"Now, the relation can be reduced as $R_{\\rm 12,34} \\approx 2R_{\\rm int}+2R_c+R_{ab}$ .",
"We can then notice that the estimated value of $2R_c+R_{ab}$ $\\approx $ 20 m$\\Omega $ at 300 K and 2 m$\\Omega $ at 4 K and the observed $R_{\\rm 12,34}$ value ($\\approx $ 6 m$\\Omega $ ) are on the same order.",
"The observed temperature dependence of $R_{\\rm 12,34}$ is also understood as a certain combination of $R_c$ and $R_{ab}$ .",
"Therefore, $R_{\\rm int}$ should have only a very small contribution.",
"For $R$ ($T$ )$_{\\rm 12,56}$ , because contributions of $R^{\\parallel }_{113}$ and $R^{\\bot }_{113}$ are again ignorable, we obtain $R_{12,56} \\cong R_{\\rm int}+R_c+R_{ab}+R^{^{\\prime }}_c+R^{^{\\prime }}_{ab}$ $(R^{^{\\prime }}_c$ and $R^{^{\\prime }}_{ab}$ are additional bulk resistances), and the same conclusion is deduced.",
"Our conclusion of the highly conducting SRO113/SRO214 interface apparently contradicts with previous experimental works indicating that the SRO214 ab-surface is not a good metal because of the surface reconstruction accompanied by the RuO$_6$ octahedral rotation [19].",
"Indeed, we usually cannot achieve good electrical contact between normal metals and the SRO214 ab-surface.",
"In addition, such a surface reconstruction is thought to destroy the superconductivity in the ab-surface region of SRO214.",
"Thus, the highly conducting interface in the present hybrid system is surprising.",
"This observation indicates that the surface reconstructions might be suppressed under expansive surface strain on SRO214 caused by the epitaxial growth of SRO113 films.",
"The detailed interface investigations are also interesting topics in the future.",
"To summarize, we grow ferromagnetic epitaxial SRO113 thin films by PLD using cleaved single crystals of superconducting SRO214 substrates.",
"The films are under severe compressive strain but with very small reduction in $T_{\\rm Curie}$ compared to the bulk.",
"Resistivity measurements reveal that the interface between SRO113 and SRO214 is highly conducting.",
"The epitaxial growth might relax SRO214 surface reconstructions and make the interface rather conducting.",
"The SRO113/SRO214 hybrid system opens a possibility to study FM/TSC junctions in future.",
"We are thankful to D. Manske, K. Char, S. B. Chung, M.R.",
"Cho, and K. Lahabi for valuable discussions.",
"We acknowledge technical support by M. P. Jimenez S. This work was supported by the \"Topological Quantum Phenomena\" (No.",
"22103002) KAKENHI on Innovative Areas from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, and also supported by the Institute for Basic Science (IBS) in Korea."
]
] | 1403.0345 |
[
[
"Restricted Kac modules of Hamiltonian Lie superalgebras of odd type"
],
[
"Abstract This paper aims to describe the restricted Kac modules of restricted Hamiltonian Lie superalgebras of odd type over an algebraically closed field of characteristic $p>3$.",
"In particular, a sufficient and necessary condition for the restricted Kac modules to be irreducible is given in terms of typical weights."
],
[
"Introduction",
"Over an algebraically closed field of characteristic zero, the finite-dimensional or infinite-dimensional irreducible modules were studied for the finite-dimensional simple Cartan Lie superalgebras $W(n), S(n)$ and $ H(n)$ (cf.",
"[6] and references therein).",
"Over a field of characteristic $p>0$ , the analogs of these simple Cartan Lie superalgebras of characteristic zero are simple restricted Lie supealgebras with respect to the usual $p$ -mapping ($p$ -th associative power).",
"Actually, in the characteristic $p>0$ case, there are more finite-dimensional simple restricted Lie superalgebras, which are analogous to the finite-dimensional simple modular Lie algebras of Cartan type or infinite-dimensional simple Lie superalgebras of vector fields over $\\mathbb {C}$ .",
"Among them are the first four series of finite-dimensional simple graded restricted Lie superalgebras of Cartan type $W, S, H$ or $K$ (cf.",
"[14]), which are analogous to the finite-dimensional simple restricted Lie algebras of the Cartan type $W, S, H$ or $K$ (cf.",
"[10]), respectively.",
"Representations of these four series of restricted Lie superalgebras were studied by Shu, Zhang and Yao (cf.",
"[11], [12], [13], [8], [9]).",
"Additionally, there are four series of finite-dimensional graded simple restricted Lie superalgebras of type $HO, KO, SHO$ or $SKO$ (cf.",
"[1]), which are analogous to the infinite-dimensional simple Lie superalgebras of vector fields over $\\mathbb {C}$ (cf.",
"[2]).",
"As far as we know, irreducible modules of the last four series of modular simple Lie superalgebras have not been studied yet.",
"The aim of this paper is to make an attempt to describe the restricted simple modules for the so-called Hamiltonian Lie superalgebras of odd type over a field of prime characteristic."
],
[
"Preliminaries",
"The ground field $\\mathbb {F}$ is assumed to be algebraically closed of characteristic $p>3$ .",
"Write $\\mathbb {Z}_2= \\lbrace \\bar{0},\\bar{1}\\rbrace $ for the additive group of two elements, $\\mathbb {Z}$ and $\\mathbb {N}$ the sets of integers and nonnegative integers, respectively.",
"For a vector superspace $V=V_{\\bar{0}}\\oplus V_{\\bar{1}}, $ we write $|x|=\\theta $ for the parity of a $\\mathbb {Z}_{2}$ -homogeneous element $x\\in V_{\\theta }, $ $\\theta \\in \\mathbb {Z}_{2}.$ The notation $|x|$ appearing in the text implies that $x$ is a $\\mathbb {Z}_2$ -homogeneous element."
],
[
"Divided power superalgebras",
"Fix a pair of positive integers $m, n$ and write $\\underline{r}=(r_1,\\ldots ,r_m\\mid r_{m+1},\\ldots ,r_{m+n})$ for a $(m+n)$ -tuple of non-negative integers.",
"For an $m$ -tuple $\\underline{N}=(N_1,\\ldots ,N_{m})$ , following [3], [4], we write $\\mathcal {O}(m, \\underline{N}\\mid n)$ for the divided power superalgebra, which is a supercommutative associative superalgebra having a (homogeneous) basis: $\\lbrace x^{(\\underline{r})}\\mid r_i<p^{N_i}\\;\\mbox{for}\\; i\\le m \\;\\mbox{and}\\; r_i=0\\;\\mbox{or}\\;1 \\;\\mbox{for}\\; i> m \\rbrace $ with parities $|x^{(\\underline{r})}|=\\left(\\sum _{i>m}r_{i}\\right)\\bar{1}$ and multiplication relations: $x^{(\\underline{r})} x^{(\\underline{s})}=\\Pi _{i=m+1}^{m+n}\\min (1, 2-r_i-s_i)(-1)^{\\Sigma _{m<i,j\\le m+n}r_js_i}\\left(\\begin{array}{c}\\underline{r}+\\underline{s} \\\\\\underline{r}\\end{array}\\right)x^{(\\underline{r}+\\underline{s})}.$ As superalgebras, $\\mathcal {O}(m, \\underline{N}\\mid n)$ is isomorphic to the tensor product superalgebra of the divided power algebra $\\mathcal {O}(m, \\underline{N})$ with the trivial $\\mathbb {Z}_2$ -grading and the exterior (super)algebra $\\Lambda (n)$ with the usual $\\mathbb {Z}_2$ -grading: $\\mathcal {O}(m, \\underline{N}\\mid n)=\\mathcal {O}(m, \\underline{N})\\otimes _{\\mathbb {F}}\\Lambda (n).$"
],
[
"General vectorial Lie superalgebras",
"Let $\\epsilon _i$ be the $(m+n)$ -tuple with 1 at the $i$ -th slot and 0 elsewhere.",
"For simplicity we write $x_i$ for $x^{(\\epsilon _i)}$ .",
"Define the distinguished partial derivatives $\\partial _i$ with parity $|\\partial _i|=|x_i|$ by letting $\\partial _i\\left(x^{(k\\epsilon _j)}\\right)=\\delta _{ij}x^{((k-1)\\epsilon _j)}\\;\\mbox{for $k<p^{N_{j}}$}.$ The Lie superalgebra of $\\mathcal {O}(m, \\underline{N}\\mid n)$ of all superderivations contains an important subalgebra, called the general vectorial Lie superalgebra of distinguished superderivations (a.k.a.",
"the Lie superlagebra of Witt type), denoted by $\\mathfrak {vect}(m, \\underline{N}\\mid n)$ (a.k.a.",
"$W(m, \\underline{N}\\mid n)$ ), having an $\\mathbb {F}$ -basis (see [3], [4] for more details) $\\left\\lbrace x^{(\\underline{r})}\\partial _k\\mid r_i<p^{N_i}\\;\\mbox{for $i\\le m$; $1\\le k\\le m+n$}\\right\\rbrace .$ Generally speaking, $\\mathfrak {vect}(m, \\underline{N}\\mid n)$ contains various finite-dimensional simple Lie superalgebras, which are analogous to finite-dimensional simple modular Lie algebras or infinite-dimensional simple Lie superalgebras of vector fields over $\\mathbb {C}$ , as mentioned in the introduction."
],
[
"Hamiltonian superalgebras of odd type",
"From now on, suppose $m=n$ .",
"As in [3], [4], write $\\mathrm {De}_{f}=\\sum _{i=1}^{2n}(-1)^{|\\partial _{i}||f|}\\partial _{i}(f)\\partial _{i^{^{\\prime }}},$ where $ i^{\\prime }=\\left\\lbrace \\begin{array}{ll}i+n,&\\mbox{if $i\\le n,$ }\\\\i-n,&\\mbox{if $i>n.$}\\end{array}\\right.$ Note that $|\\mathrm {De}_{f}|=|f|+1$ and $[\\mathrm {De}_{f},\\mathrm {De}_{g}]=\\mathrm {De}_{\\lbrace f,g\\rbrace _{B}}\\;\\mbox{for}\\;f,g\\in \\mathcal {O}(n,\\underline{N}\\mid n),$ where $\\lbrace \\cdot ,\\cdot \\rbrace _{B}$ is the Buttion bracket given by $\\lbrace f,g\\rbrace _{B}=\\mathrm {De}_{f}(g)=\\sum _{i=1}^{2n}(-1)^{|\\partial _{i}||f|}\\partial _{i}(f)\\partial _{i^{^{\\prime }}}(g).$ Then ${le}\\left(n, \\underline{N}\\mid n\\right)=\\lbrace \\mathrm {De}_{f}\\mid f\\in \\mathcal {O}(n, \\underline{N}\\mid n)\\rbrace $ is a finite-dimensional simple Lie superalgebra.",
"We call it the Hamiltonian superalgebra of odd type.",
"This Lie superalgebra is called the odd Hamiltonian superalgebra and denoted by $HO(n,n; \\underline{N})$ in [5], which is analogous to the infinite-dimensional Lie superalgebra $HO(n,n)$ of vector fields over $\\mathbb {C}$ in [2].",
"In the present paper, we adopt the notation in [3], [4]."
],
[
"Restricted Lie superalgebras and restricted modules",
"Recall that a Lie superalgebra $\\mathfrak {g}=\\mathfrak {g}_{\\bar{0}}\\oplus \\mathfrak {g}_{\\bar{1}}$ is restricted if $\\mathfrak {g}_{\\bar{0}}$ as Lie algebra is restricted and $\\mathfrak {g}_{\\bar{1}}$ as $\\mathfrak {g}_{\\bar{0}}$ -module is restricted.",
"For a restricted Lie superalgebra $\\mathfrak {g}$ , the $p$ -mapping of $\\mathfrak {g}_{\\bar{0}}$ is also called the $p$ -mapping of the Lie superalgebra $\\mathfrak {g}$ .",
"Let $(\\mathfrak {g}, [p])$ be a restricted Lie superalgebra.",
"A $\\mathfrak {g}$ -module $M$ is called restricted if $x^{p}\\cdot m=x^{[p]}\\cdot m \\; \\mbox{for all \\;$x\\in \\mathfrak {g}_{\\bar{0}}, m\\in M$}.$ By definition, the restricted enveloping algebra of $\\mathfrak {g}$ is $\\mathbf {u}(\\mathfrak {g})=\\mathrm {U}(\\mathfrak {g})/I$ , where $I$ is the $\\mathbb {Z}_{2}$ -graded two-sided ideal of enveloping algebra $\\mathrm {U}(\\mathfrak {g})$ generated by elements $\\left\\lbrace x^{p}-x^{[p]} \\mid x\\in \\mathfrak {g}_{\\bar{0}}\\right\\rbrace .$ Note that $\\mathbf {u}(\\mathfrak {g})$ has a natural structure of a $\\mathbb {Z}$ -graded superalgebra.",
"Suppose $(e_1,\\ldots , e_m\\mid f_1,\\ldots , f_n)$ is a $\\mathbb {Z}_{2}$ -homogeneous basis of ${g}$ .",
"Then $\\mathfrak {u}(\\mathfrak {g})$ has the following $\\mathbb {F}$ -basis: $\\left\\lbrace f_{1}^{b_{1}}\\cdots f_{n}^{b_{n}}e_{1}^{a_{1}}\\cdots e_{m}^{a_{m}}\\mid 0\\le a_{i}\\le p-1, b_{j}=0\\; or \\;1\\right\\rbrace .$ Note that the $\\mathbf {u}({g})$ -modules are precisely the restricted $\\mathfrak {g}$ -modules.",
"A standard fact is that ${le}(n, \\underline{N}\\mid n)$ is a restricted Lie superalgebra if and only if $\\underline{N}=\\underline{1}:=(1,\\ldots ,1).$ Note that the unique $p$ -mapping of ${le}(n, \\underline{1}\\mid n)$ is the usual (associative) $p$ -power.",
"This paper aims to study the finite-dimensional irreducible restricted modules of ${le}(n, \\underline{1}\\mid n)$ .",
"From now on we abbreviate $\\mathcal {O}(n, \\underline{1}\\mid n)$ to $\\mathcal {O}(n)$ , and ${le}(n, \\underline{1}\\mid n)$ to ${le}(n)$ ."
],
[
"Kac modules and root reflections",
"For a $\\mathbb {Z}$ -graded vector space $V=\\oplus _{i\\in \\mathbb {Z}}V_{[i]}$ , we write $\\mathrm {deg}v=i$ for the $\\mathbb {Z}$ -degree of a $\\mathbb {Z}$ -homogeneous element $v\\in V_{[i]}.$ Put $\\overline{{le}}(n)={le}(n)\\oplus \\mathbb {F}\\sum _{ i=1}^{2n}x_{i}\\partial _{i}.$ By letting $\\mathrm {deg}x_{i}=1=-\\mathrm {deg}\\partial _{i},$ $\\mathfrak {le}(n)$ and $\\overline{{le}}(n)$ become $\\mathbb {Z}$ -graded superalgebras."
],
[
"Triangular decompositions",
"Let $\\bar{\\mathfrak {h}}=\\mathfrak {h}\\oplus \\mathbb {F}\\sum _{ i=1}^{2n}x_{i}\\partial _{i},$ where $\\mathfrak {h}=\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{i}x_{i^{\\prime }}}\\mid i \\le n\\rbrace .$ Then $\\bar{\\mathfrak {h}}$ is a Cartan subalgebra of $\\overline{{le}}(n)_{[0]}$ and $\\overline{{le}}(n)=\\oplus _{\\alpha \\in \\bar{\\mathfrak {h}}^{*}}\\overline{{le}}(n)_{\\alpha },$ where $\\overline{{le}}(n)_{\\alpha }=\\mathrm {Span}_{\\mathbb {F}}\\left\\lbrace x\\in \\overline{{le}}(n)\\mid [h, x]=\\alpha (h)x \\;\\mbox{for} \\; h\\in \\bar{\\mathfrak {h}}\\right\\rbrace .$ We denote the basis of $\\bar{\\mathfrak {h}}^{*},$ $\\varepsilon _{i}=\\left(\\mathrm {De}_{x_{i}x_{i^{\\prime }}}\\right)^{*},\\; \\delta =\\left(\\sum _{j=1}^{2n}x_{j}\\partial _{j}\\right)^{*} \\; \\mbox{for}\\; i \\le n.$ We still write $\\varepsilon _{i}$ for $\\varepsilon _{i}|_{\\mathfrak {h}}$ , if no confusion occurs.",
"Clearly, $ \\mathrm {De}_{x_{i}}\\in \\left\\lbrace \\begin{array}{ll}\\overline{{le}}(n)_{-\\varepsilon _{i}-\\delta }& \\mbox{ if}\\; i \\le n \\\\\\overline{{le}}(n)_{\\varepsilon _{i^{\\prime }}-\\delta }& ~\\mbox{if}\\; i>n.\\end{array}\\right.$ Note that $\\overline{{le}}(n)_{[0]}$ has a standard triangular decomposition $\\overline{{le}}(n)_{[0]}=\\mathfrak {n}_{[0]}^{-}\\oplus \\bar{\\mathfrak {h}}\\oplus \\mathfrak {n}_{[0]}^{+},$ where $\\mathfrak {n}_{[0]}^{-}=\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{i}x_{n+j}}\\mid n\\ge i>j\\rbrace +\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{k}x_{l}}\\mid k, l>n\\rbrace ,$ $\\mathfrak {n}_{[0]}^{+}=\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{i}x_{n+j}}\\mid i <j\\le n \\rbrace +\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{k}x_{l}}\\mid k, l\\le n\\rbrace .$ Then $\\overline{{le}}(n)$ has a standard triangular decomposition $\\overline{{le}}(n)=\\mathfrak {n}^{-}_{0}\\oplus \\bar{\\mathfrak {h}}\\oplus \\mathfrak {n}^{+}_{0},$ where $\\mathfrak {n}^{-}_{0}=\\mathfrak {n}^{-}_{[0]}\\oplus \\overline{{le}}(n)_{[-1]},\\;\\mathfrak {n}^{+}_{0}=\\mathfrak {n}^{+}_{[0]}\\oplus _{i>0}\\overline{{le}}(n)_{[i]}.$"
],
[
"Root reflections",
"We now define a sequence of root reflections in the order: $\\gamma _{-\\varepsilon _{1}-\\delta },\\ldots ,\\gamma _{-\\varepsilon _{n}-\\delta }, \\gamma _{\\varepsilon _{n}-\\delta },\\ldots ,$ $\\gamma _{\\varepsilon _{1}-\\delta }.$ Firstly we define $\\mathfrak {n}^{+}_{1}=\\gamma _{-\\varepsilon _{1}-\\delta }(\\mathfrak {n}^{+}_{0})$ to be obtained by removing the subspace $W_{1}=\\mathrm {Span}_{\\mathbb {F}}\\left\\lbrace \\mathrm {De}_{x^{(\\epsilon _{n+1}+\\epsilon _{i}+\\epsilon _{n+j})}}, \\mathrm {De}_{x^{(\\epsilon _{n+1}+\\epsilon _{n+i}+\\epsilon _{n+j})}}\\mid n\\ge i\\ge j \\right\\rbrace $ from $\\mathfrak {n}^{+}_{0}$ and adding $\\mathbb {F}\\mathrm {De}_{x_{1}}$ .",
"Put $\\mathfrak {n}^{-}_{1}=W_{1}\\oplus \\mathfrak {n}^{-}_{[0]}\\oplus \\sum _{i=2}^{2n}\\mathbb {F}\\mathrm {De}_{x_{i}}.$ Then $\\overline{{le}}(n)=\\mathfrak {n}^{-}_{1}\\oplus \\bar{\\mathfrak {h}} \\oplus \\mathfrak {n}^{+}_{1}$ is a new triangular decomposition.",
"Suppose we have defined $\\mathfrak {n}^{+}_{k-1}=\\gamma _{-\\varepsilon _{k-1}-\\delta }\\cdots \\gamma _{-\\varepsilon _{1}-\\delta }(\\mathfrak {n}^{+}_{0}) \\;\\mbox{for}\\; 2\\le k\\le n.$ Then we define $\\mathfrak {n}^{+}_{k}=\\gamma _{-\\varepsilon _{k}-\\delta }(\\mathfrak {n}^{+}_{k-1})$ to be obtained by removing the subspace $W_{k}$ spanned by all $\\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{n+i}+\\epsilon _{n+j})}} $ and $ \\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{i}+\\epsilon _{n+j})}}$ with $r_{1}=\\cdots =r_{n}=r_{n+k+1}=\\cdots =r_{2n}=0, r_{n+k}= 1, n\\ge i\\ge j$ from $\\mathfrak {n}^{+}_{k-1}$ and adding $\\mathbb {F}\\mathrm {De}_{x_{k}} $ .",
"Put $\\mathfrak {n}^{-}_{k}=\\sum _{i=1}^{k}W_{i}\\oplus \\mathfrak {n}^{-}_{[0]}\\oplus \\sum _{i=k+1}^{2n}\\mathbb {F}\\mathrm {De}_{x_{i}}.$ Then $\\overline{{le}}(n)=\\mathfrak {n}^{-}_{k}\\oplus \\bar{\\mathfrak {h}} \\oplus \\mathfrak {n}^{+}_{k}$ is a new triangular decomposition.",
"Next we define $\\mathfrak {n}^{+}_{n+1}=\\gamma _{\\varepsilon _{n}-\\delta }(\\mathfrak {n}^{+}_{n})$ to be obtained by removing root the space $W_{n+1}$ spanned by $\\lbrace \\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{n+i}+\\epsilon _{n+j})}}, \\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{i}+\\epsilon _{n+j})}} \\mid r_{1}=\\cdots =r_{n-1}=0, r_{n}\\ge 1, n\\ge i\\ge j \\rbrace $ from $\\mathfrak {n}^{+}_{n}$ and adding $\\mathbb {F}\\mathrm {De}_{x_{2n}}$ .",
"Put $\\mathfrak {n}^{-}_{n+1}=\\sum _{i=1}^{n+1}W_{i}\\oplus \\mathfrak {n}^{-}_{[0]}\\oplus \\sum _{i=n+1}^{2n-1}\\mathbb {F}\\mathrm {De}_{x_{i}}.$ Then $\\overline{{le}}(n)=\\mathfrak {n}^{-}_{n+1}\\oplus \\bar{\\mathfrak {h}} \\oplus \\mathfrak {n}^{+}_{n+1}$ is a new triangular decomposition.",
"Suppose we have defined $\\mathfrak {n}^{+}_{n+k}=\\gamma _{\\varepsilon _{n-(k-1)}-\\delta }\\cdots \\gamma _{\\varepsilon _{n}-\\delta }(\\mathfrak {n}^{+}_{n}) \\;\\mbox{for}\\; 2\\le k\\le n.$ Finally we define $\\mathfrak {n}^{+}_{n+k+1}=\\gamma _{\\varepsilon _{n-k}-\\delta }(\\mathfrak {n}^{+}_{n+k})$ to be obtained by removing the space $W_{n+k+1}$ spanned by all $ \\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{n+i}+\\epsilon _{n+j})}}$ and $\\mathrm {De}_{x^{(\\underline{r}+\\epsilon _{i}+\\epsilon _{n+j})}}$ with $r_{1}=\\cdots =r_{n-k-1}=0, r_{n-k}\\ge 1,\\cdots ,r_{n}\\ge 1, n\\ge i\\ge j$ from $\\mathfrak {n}^{+}_{n+k}$ and adding $\\mathbb {F} \\mathrm {De}_{x_{2n-k}}$ .",
"Put $\\mathfrak {n}^{-}_{n+k+1}=\\sum _{i=1}^{n+k+1}W_{i}\\oplus \\mathfrak {n}^{-}_{[0]}\\oplus \\sum _{i=k+1}^{2n-k-1}\\mathbb {F}\\mathrm {De}_{x_{i}}.$ Then $\\overline{{le}}(n)=\\mathfrak {n}^{-}_{n+k+1}\\oplus \\bar{\\mathfrak {h}} \\oplus \\mathfrak {n}^{+}_{n+k+1}$ is a new triangular decomposition.",
"Note that $\\mathfrak {n}^{+}_{2n}=\\mathfrak {n}^{+}_{[0]}\\oplus \\overline{{le}}(n)_{[-1]}, \\; \\mathfrak {n}^{-}_{2n}=\\mathfrak {n}^{-}_{[0]}\\oplus _{i>0}\\overline{{le}}(n)_{[i]}.$"
],
[
"Restricted Vermas module and Kac modules",
"Write $\\mathfrak {g}$ for ${le}(n)$ or $\\overline{{le}}(n).$ Suppose $\\mathfrak {g}$ (resp.",
"$\\mathfrak {g}_{[0]}$ ) has a triangular decomposition $\\mathfrak {g}=N^{-}\\oplus H_{\\mathfrak {g}}\\oplus N^{+}\\;(\\mbox{resp.",
"}\\; \\mathfrak {g}_{[0]}=N^{-}_{[0]}\\oplus H_{\\mathfrak {g}}\\oplus N^{+}_{[0]}),$ where $H_{\\mathfrak {g}}=\\mathfrak {h}\\oplus \\delta _{\\mathfrak {g},\\overline{{le}}(n)}\\mathbb {F}\\sum _{ i=1}^{2n}x_{i}\\partial _{i}$ .",
"Let $V=V_{\\bar{0}}\\oplus V_{\\bar{1}}$ be a $\\mathfrak {g}$ -module (resp.",
"$\\mathfrak {g}_{[0]}$ -module).",
"Suppose for $\\lambda \\in H_{\\mathfrak {g}}^{*}$ there is a nonzero vector $v_{\\lambda }\\in V_{\\bar{0}}\\cup V_{\\bar{1}}$ such that $x\\cdot v_{\\lambda }=0,\\; h \\cdot v_{\\lambda }=\\lambda (h)v_{\\lambda },\\;\\mbox{for\\;} x\\in N^{+} \\;(\\mbox{resp.",
"}\\;x\\in N^{+}_{[0]}),\\; h\\in H_{\\mathfrak {g}}.$ Then $v_{\\lambda }$ is called a highest weight vector of $V$ with respect to $B=H_{\\mathfrak {g}}\\oplus N^{+}$ (resp.",
"$B_{[0]}=H_{\\mathfrak {g}}\\oplus N^{+}_{0}$ ).",
"If $V$ is a restricted $\\mathfrak {g}$ -module (resp.",
"$\\mathfrak {g}_{[0]}$ -module), then $\\lambda \\in \\Lambda _{\\mathfrak {g}}$ , where $&&\\Lambda _{\\mathfrak {g}}=\\mathrm {Span}_{\\mathbb {F}_{p}}\\lbrace \\varepsilon _{1},\\ldots ,\\varepsilon _{n},\\delta _{\\mathfrak {g},\\overline{{le}}(n)}\\delta \\rbrace .$ Conversely, for any given $\\lambda \\in \\Lambda _{\\mathfrak {g}}$ , there is a one-dimensional restricted $B$ -module (resp.",
"$B_{[0]}$ -module) $\\mathbb {F}v_{\\lambda }$ satisfying (REF ).",
"Write $L_{\\mathfrak {g}}^{B}(\\lambda )$ (resp.",
"$L_{\\mathfrak {g}_{[0]}}^{B_{[0]}}(\\lambda )$ ) for the unique irreducible $\\mathbb {Z}_{2}$ -graded quotient of restricted Verma module $\\mathbf {u}(\\mathfrak {g})\\otimes _{\\mathbf {u}(B)}\\mathbb {F}v_{\\lambda }$ (resp.",
"$\\mathbf {u}(\\mathfrak {g}_{[0]})\\otimes _{\\mathbf {u}(B_{[0]})}\\mathbb {F}v_{\\lambda }$ ) of $\\mathfrak {g}$ (resp.",
"$\\mathfrak {g}_{[0]}$ ) with respect to $B$ (resp.",
"$B_{[0]}$ ).",
"Recall that, for $\\lambda \\in \\Lambda _{\\mathfrak {g}}$ , $I_{\\mathfrak {g}}\\left(\\lambda \\right)=\\mathbf {u}(\\mathfrak {g})\\otimes _{\\mathbf {u}(\\oplus _{i\\ge 0}\\mathfrak {g}_{[i]})}L_{\\mathfrak {g}_{[0]}}^{ \\mathfrak {n}_{[0]}^{+}\\oplus H_{\\mathfrak {g}}}\\left(\\lambda \\right)$ is called a restricted Kac module of $\\mathfrak {g}$ .",
"We know that $I_{\\mathfrak {g}}\\left(\\lambda \\right)$ has a unique irreducible quotient module $L_{\\mathfrak {g}}^{\\mathfrak {n}_{0}^{+}\\oplus H_{\\mathfrak {g}}}\\left(\\lambda \\right)$ and $\\left\\lbrace L_{\\mathfrak {g}}^{\\mathfrak {n}_{0}^{+}\\oplus H_{\\mathfrak {g}}}(\\lambda )\\mid \\lambda \\in \\Lambda _{\\mathfrak {g}}\\right\\rbrace $ consists of all the irreducible restricted $\\mathfrak {g}$ -modules.",
"For $\\lambda \\in \\Lambda _{\\mathfrak {le}(n)},$ it is easily seen that $I_{{le}(n)}(\\lambda )$ is irreducible as ${le}(n)$ -module if and only if $I_{\\overline{{le}}(n)}(\\lambda )$ is irreducible as $\\overline{{le}}(n)$ -module.",
"In the below, we shall study the irreducibility of $I_{\\overline{{le}}(n)}(\\lambda )$ , where $\\lambda \\in \\Lambda _{\\overline{\\mathfrak {le}}(n)}$ ."
],
[
"Irreducible quotients of restricted Verma modules",
"For $ i \\le n,$ put $\\mathfrak {h}_{i}=\\mathrm {Span}_{\\mathbb {F}}\\lbrace \\mathrm {De}_{x_{j}x_{j^{\\prime }}}\\mid j\\le n, j\\ne i \\rbrace .$ By the definition of $\\mathfrak {h}_{i}$ , one can easily get $\\mathfrak {h}_{i}=\\lbrace h\\in \\bar{\\mathfrak {h}}\\mid \\varepsilon _{i}(h)=\\delta (h)=0\\rbrace .$ For $0 \\le i \\le 2n,$ put $\\mathfrak {b}_{i}=\\mathfrak {n}^{+}_{i}\\oplus \\bar{\\mathfrak {h}}.$ Then we have the following proposition.",
"Proposition 2.1 Let $ i\\le n,$ and $\\lambda \\in \\Lambda _{\\overline{\\mathfrak {le}}(n)}.$ (1) If $\\lambda (\\mathfrak {h}_{i})=0,$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i}}(\\lambda ).$ (2) If $\\lambda (\\mathfrak {h}_{i})\\ne 0,$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i}}(\\lambda -\\varepsilon _{i}-\\delta ).$ (3) If $\\lambda (\\mathfrak {h}_{n-i+1})\\ne 0,$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i}}(\\lambda +(p-1)(\\varepsilon _{n-i+1}-\\delta )).$ (4) If $\\lambda =a\\delta ,$ where $a\\in \\mathbb {F}_{p},$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i}}(\\lambda ).$ (5) If $\\lambda =a\\varepsilon _{n-i+1}+b\\delta ,$ where $ a, b\\in \\mathbb {F}_{p},$ $a\\ne 0, 1,$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i}}(\\lambda +(p-1)(\\varepsilon _{n-i+1}-\\delta )).$ (6) If $\\lambda =\\varepsilon _{n-i+1}+b\\delta ,$ where $b\\in \\mathbb {F}_{p},$ then $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )\\cong L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i}}(\\lambda +(p-2)(\\varepsilon _{n-i+1}-\\delta )).$ (1) Let $0\\ne v$ be a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{i-1}$ .",
"We claim that $v$ is also a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{i}$ .",
"By the definition of $\\gamma _{-\\varepsilon _{i}-\\delta }$ , we have to check that $\\mathrm {De}_{x_{i}}\\cdot v=0$ .",
"Suppose $\\mathrm {De}_{x_{i}}\\cdot v\\ne 0$ .",
"One can easily get $X\\cdot (\\mathrm {De}_{x_{i}}\\cdot v)=0 \\;\\mbox{for}\\; X\\in \\mathfrak {b}_{i-1}.$ Therefore, $\\mathrm {De}_{x_{i}}\\cdot v$ is a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{i-1}$ , which contradicts the uniqueness of highest weight vector with respect to $\\mathfrak {b}_{i-1}$ (up to proportionality).",
"(2) Let $0\\ne v$ be a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{i-1}$ .",
"In this case there exists $j\\ne i,$ where $ j \\le n$ , such that $\\lambda \\left(\\mathrm {De}_{x_{j}x_{j^{\\prime }}}\\right)\\ne 0.$ Since $\\mathrm {De}_{x^{(\\epsilon _{i^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}}\\cdot \\mathrm {De}_{x_{i}}\\cdot v&=& -\\mathrm {De}_{x_{i}}\\cdot \\mathrm {De}_{x^{(\\epsilon _{i^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}} \\cdot v\\\\&&+\\lambda \\left(\\left[\\mathrm {De}_{x_{i}},\\mathrm {De}_{x^{(\\epsilon _{i^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}} \\right]\\right)v\\\\&=&\\lambda \\left(\\mathrm {De}_{x_{j}x_{j^{\\prime }}}\\right)v\\ne 0,$ we have $\\mathrm {De}_{x_{i}}\\cdot v\\ne 0.$ Then $\\mathrm {De}_{x_{i}}\\cdot v$ is a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{i}.$ (3) In this case there exists $ j\\ne n-i+1$ , where $j \\le n$ , such that $\\lambda \\left(\\mathrm {De}_{x_{j}x_{j^{\\prime }}}\\right)\\ne 0.$ Let $0\\ne v$ be a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i-1}.$ We claim that $ \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-1}\\cdot v\\ne 0.$ Suppose not.",
"Then by applying $\\mathrm {De}_{x^{(\\epsilon _{n-i+1}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}},$ we get $0&=&\\mathrm {De}_{x^{(\\epsilon _{(n-i+1)^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}}\\cdot \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-1}\\cdot v\\\\&=& -\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}\\cdot \\mathrm {De}_{x^{(\\epsilon _{(n-i+1)^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}}\\cdot \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}\\cdot v\\\\&&+\\lambda \\left(\\left[\\mathrm {De}_{x_{(n-i+1)^{\\prime }}},\\mathrm {De}_{x^{(\\epsilon _{(n-i+1)^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}}\\right]\\right)\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}\\cdot v\\\\&=&(1-p)\\lambda \\left(\\mathrm {De}_{x_{j}x_{j^{\\prime }}}\\right)\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}\\cdot v,$ hence $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}\\cdot v=0.$ By repeated applications of $\\mathrm {De}_{x^{(\\epsilon _{(n-i+1)^{\\prime }}+\\epsilon _{j}+\\epsilon _{j^{\\prime }})}},$ we can get $v=0,$ a contradiction.",
"A direct verification shows that $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-1}\\cdot v$ is a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i}.$ (4) The proof is similar to the one of (1).",
"(5) Let $0\\ne v$ be a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i-1}.$ Using the fact that $\\mathrm {De}_{x^{((p-1)\\epsilon _{n-i+1}+\\epsilon _{(n-i+1)^{\\prime }})}}\\cdot \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-2)}\\cdot v=\\lambda \\left(\\mathrm {De}_{x_{n-i+1}x_{(n-i+1)^{\\prime }}}\\right)v=av\\ne 0,$ we get $ \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-2)}\\cdot v\\ne 0.$ Using the fact that $&&\\mathrm {De}_{x^{(2\\epsilon _{n-i+1}+\\epsilon _{(n-i+1)^{\\prime }})}}\\cdot \\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-1)}\\cdot v\\\\&=&\\left((p-1)\\lambda +\\frac{1}{2}(p-1)(p-2)(\\varepsilon _{n-i+1}-\\delta )\\right)\\left(\\mathrm {De}_{x_{n-i+1}x_{(n-i+1)^{\\prime }}}\\right)\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}v\\\\&=&(1-a)\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}v\\ne 0,$ we get $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-1)}\\cdot v\\ne 0.$ Then $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-1}\\cdot v$ is a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i}.$ (6) Let $0\\ne v$ be a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i-1}.$ We claim $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-1)}\\cdot v= 0.$ If not, then $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-1)}\\cdot v$ is a highest weight vector of $\\mathfrak {b}_{n+i-1}$ .",
"Eq.",
"(REF ) implies $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{(p-2)}\\cdot v\\ne 0.$ Then $\\mathrm {De}_{x_{(n-i+1)^{\\prime }}}^{p-2}\\cdot v$ is a highest weight vector of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{n+i-1}}(\\lambda )$ with respect to $\\mathfrak {b}_{n+i}.$"
],
[
"Irreducibility of restricted Kac modules",
"Recall the symplectic supergroup $\\mathrm {SP}(n,\\mathbb {F})=\\left\\lbrace A\\in \\mathrm {GL}(2n,\\mathbb {F})\\mid A^{T}JA=J\\right\\rbrace ,$ where $J=\\left(\\begin{array}{cc}0 & I_{n} \\\\-I_{n} & 0 \\\\\\end{array}\\right).$ The conformal symplectic supergroup $\\mathrm {CSP}(n,\\mathbb {F})$ is a direct product of the symplectic group $\\mathrm {SP}(n, \\mathbb {F})$ with the one dimensional multiplicative supergroup $\\mathbb {F}^{*}.$ Each $\\phi \\in \\mathrm {GL}({\\mathcal {O}(n,\\underline{1})_{[1]},\\mathbb {F}})\\times \\mathrm {GL}({\\Lambda (n)_{[1]},\\mathbb {F}})$ can be extended to a $\\mathbb {Z}$ -homogeneous element of $\\mathrm {Aut}(\\mathcal {O}(n)),$ which is still denoted by $\\phi .$ Now we define $f_{\\phi }(x)=\\phi ^{-1}x\\phi ,\\; \\mbox{for}\\; x\\in \\overline{{le}}(n).$ Let $\\phi \\in \\mathrm {GL}({\\mathcal {O}(n,\\underline{1})_{[1]},\\mathbb {F}})\\times \\mathrm {GL}({\\Lambda (n)_{[1]},\\mathbb {F}}).$ If $\\phi \\in \\mathrm {CSP}(n,\\mathbb {F}),$ then $f_{\\phi }\\in \\mathrm {Aut}(\\overline{{le}}(n)).$ (See I in Appendix)."
],
[
"$(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T})$ -module",
"Let $\\mathfrak {T}$ be the canonical maximal torus of the $\\mathrm {CSP}(n,\\mathbb {F}).$ Then $\\mathfrak {T}\\cong \\left\\lbrace \\mathrm {diag}(tt_{1},\\ldots ,tt_{n},tt_{1}^{-1},\\ldots ,tt_{n}^{-1})\\mid t, t_{i}\\in \\mathbb {F}^{*}\\right\\rbrace $ and the Lie algebra of $\\mathfrak {T}$ coincides with $\\bar{\\mathfrak {h}}.$ Let $X(\\mathfrak {T})$ be the character group of $\\mathfrak {T}.$ Then $X(\\mathfrak {T})=\\sum _{i=1}^{n+1}\\mathbb {Z}\\Lambda _{i},$ where, for $t, t_{i}\\in \\mathbb {F}^{*},$ $ \\Lambda _{i}(\\mathrm {diag}(tt_{1},\\ldots ,tt_{n},tt_{1}^{-1},\\ldots ,tt_{n}^{-1}))=\\left\\lbrace \\begin{array}{ll}t_{i}^{-1} & if\\; 1 \\le i \\le n \\\\t & if\\; i=n+1.\\end{array}\\right.$ By definition, a rational $\\mathfrak {T}$ -module $V$ means that $V=\\oplus _{\\lambda \\in X(\\mathfrak {T})}V_{\\lambda },$ where $V_{\\lambda }=\\lbrace v\\in V\\mid \\overline{t}(v)=\\lambda (\\overline{t})v \\;\\mbox{for}\\; \\overline{t}\\in \\mathfrak {T}\\rbrace .$ Set $\\mathfrak {J}=\\lbrace \\mathrm {diag}(t,\\ldots , t)\\mid t\\in \\mathbb {F}^{*}\\rbrace .$ Then we have a $\\mathbb {Z}$ -graded decomposition for a rational $\\mathfrak {T}$ -module $V=\\oplus _{s\\in \\mathbb {Z}} V_{s}$ with $V_{s}=\\left\\lbrace v\\in V\\mid \\overline{t}(v)=t^{s}v \\;\\mbox{for}\\; \\overline{t}=\\mathrm {diag}(t,\\ldots , t)\\in \\mathfrak {J}\\right\\rbrace .$ Put $\\mathcal {W}_{\\mathfrak {J}}(V)=\\lbrace s\\in \\mathbb {Z}\\mid V_{s}\\ne 0\\rbrace .$ We define the action of $\\mathfrak {T}$ on $\\overline{{le}}(n)$ by $\\overline{t}(a)=f_{\\overline{t}}(a) \\;\\mbox{for}\\; \\overline{t}\\in \\mathfrak {T}, a\\in \\overline{{le}}(n).$ $U(\\overline{{le}}(n))$ and its canonical subalgebras become rational $\\mathfrak {T}$ -modules with the action given by $\\mathrm {Ad}(\\overline{t})(a_{1}\\cdots a_{l})=\\overline{t}(a_{1})\\cdots \\overline{t}(a_{l}),$ where $a_{i}\\in \\overline{{le}}(n)$ and $\\overline{t}\\in \\mathfrak {T}.$ Since $\\mathrm {Ad}(\\overline{t})(x^{[p]})=\\mathrm {Ad}(\\overline{t})(x)^{p} \\;\\mbox{for}\\; x\\in \\overline{{le}}(n), \\overline{t}\\in \\mathfrak {T},$ $\\mathbf {u}(\\overline{{le}}(n))$ is also a rational $\\mathfrak {T}$ -module.",
"A finite dimensional superspace $V=V_{\\bar{0}}\\oplus V_{\\bar{1}}$ is called a $(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T})$ -module if $V$ is a $\\mathbf {u}\\left(\\overline{{le}}(n)\\right)$ -module and each $V_{\\bar{i}}$ ($i=0, 1$ ) is a $\\mathfrak {T}$ -module and satisfies: (1) the actions of $\\mathfrak {h}$ coming from $\\overline{{le}}(n)$ and from $\\mathfrak {T}$ coincide, (2) $\\overline{t}(a\\cdot v)=\\mathrm {Ad}(\\overline{t})(a)\\overline{t}(v),$ for $\\overline{t}\\in \\mathfrak {T}, a\\in \\mathbf {u}(\\overline{{le}}(n)), v\\in V.$ Example 3.1 $I_{\\overline{{le}}(n)}(\\lambda )$ and $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i}}(\\lambda )$ ($0 \\le i\\le 2n$ ) are $\\left(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T}\\right)$ -modules, where $\\lambda \\in \\Lambda _{\\overline{\\mathfrak {le}}(n)}$ (See I in Appendix)."
],
[
"Main results",
"For any $\\left(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T}\\right)$ -module $V,$ we define the length of $V$ as the number $|\\mathcal {W}_{\\mathfrak {J}}(V)|$ minus 1, denoted by $l(V)$ .",
"Write $\\mathfrak {g}$ for ${le}(n)$ or $\\overline{{le}}(n).$ Put $\\Omega _{\\mathfrak {g}}=\\left\\lbrace \\sum _{j=1}^{i-1}\\varepsilon _{j}+a\\varepsilon _{i}+\\delta _{\\mathfrak {g},\\overline{\\mathfrak {le}}(n)}b\\delta ,\\sum _{j=1}^{n}\\varepsilon _{j}+\\sum _{ l=i}^{n}\\varepsilon _{l}+\\delta _{\\mathfrak {g},\\overline{\\mathfrak {le}}(n)}b\\delta \\mid a, b\\in \\mathbb {F}_{p}, i \\le n\\right\\rbrace .$ A weight $\\lambda \\in \\Lambda _{\\mathfrak {g}}$ is said to be atypical if $\\lambda \\in \\Omega _{\\mathfrak {g}};$ Otherwise, $\\lambda $ is said to be typical.",
"For $\\lambda \\in \\Lambda _{\\overline{\\mathfrak {le}}(n)} $ and $i$ ($1\\le i\\le 2n$ ), there is uniquely a weight in $\\Lambda _{\\overline{\\mathfrak {le}}(n)}$ , which is denoted by $\\lambda _{i}$ , such that $ L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i}}(\\lambda _{i})=L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda ).$ Theorem 1 For $\\lambda \\in \\Lambda _{\\overline{\\mathfrak {le}}(n)}$ , the restricted Kac module $I_{\\overline{{le}}(n)}(\\lambda )$ is irreducible if and only if $\\lambda $ is typical.",
"Let $\\upsilon _{0}$ and $\\upsilon _{2n}$ be the highest weight vectors of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )$ with respect to $\\mathfrak {b}_{0}$ and $\\mathfrak {b}_{2n}$ , respectively (see Subsection 2.3), and suppose $\\overline{t}(\\upsilon _{0})=t^{l_{0}}\\upsilon _{0},\\;\\overline{t}(\\upsilon _{2n})=t^{l_{2n}}\\upsilon _{2n} \\;\\mbox{for}\\; \\overline{t}=\\mathrm {diag}(t,\\ldots , t)\\in \\mathfrak {J}.$ We have $l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)=l_{2n}-l_{0}.$ Then $ l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)\\le l\\left(I_{\\overline{{le}}(n)}(\\lambda )\\right)=pn$ and $I_{\\overline{{le}}(n)}(\\lambda )$ is irreducible if and only if $l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)= pn.$ Then it suffices to show that $l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)=pn$ if and only if $\\lambda $ is typical.",
"By Proposition REF , we have the following facts: (a) For $2\\le k\\le n,$ $\\lambda _{k-1}(\\mathfrak {h}_{k})=0$ if and only if $ \\lambda =b\\delta \\;\\mbox{or} \\;\\sum _{j=1}^{k-1}\\varepsilon _{j}+a\\varepsilon _{k}+b\\delta , \\; 0\\ne a, b\\in \\mathbb {F}_{p}.$ (b) For $k= n,$ $\\lambda _{k}(\\mathfrak {h}_{2n-k})=0$ if and only if $\\lambda = b\\delta \\; \\mbox{or}\\;\\sum _{j=1}^{n-1}\\varepsilon _{j}+a\\varepsilon _{n}+b\\delta , 0\\ne a, b\\in \\mathbb {F}_{p}.$ (c) For $k>n,$ $\\lambda _{k}(\\mathfrak {h}_{2n-k})=0$ if and only if $\\lambda = b\\delta \\;\\mbox{or}\\;\\sum _{j=1}^{2n-k}\\varepsilon _{j}+a\\varepsilon _{2n-k}+2\\sum _{ j=2n-k+1}^{n}\\varepsilon _{j}+b\\delta , 0\\ne a, b\\in \\mathbb {F}_{p}.$ (d) If $\\lambda =\\sum _{j=1}^{n-1}\\varepsilon _{j}+a\\varepsilon _{n}+b\\delta $ , where $0\\ne a, b\\in \\mathbb {F}_{p}$ , then $\\lambda _{n}=a\\varepsilon _{n}+(b-n+1)\\delta .$ (e) For $k>n$ and $\\lambda =\\sum _{j=1}^{2n-k}\\varepsilon _{j}+a\\varepsilon _{2n-k}+2\\sum _{ j=2n-k+1}^{n}\\varepsilon _{j}+b\\delta , 0\\ne a, b\\in \\mathbb {F}_{p}$ , $ \\lambda _{k}=a\\varepsilon _{2n-k}+(b-2n+k)\\delta .$ By the facts (a)–(e) and Proposition REF , we have $ l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)=\\left\\lbrace \\begin{array}{ll}0, & if \\; \\lambda =b\\delta , b\\in \\mathbb {F}_{p}\\\\pn-1,& if \\; \\lambda =\\sum _{j=1}^{i-1}\\varepsilon _{j}+a\\varepsilon _{i}+b\\delta , 1\\le i\\le n-1, 0\\ne a, b\\in \\mathbb {F}_{p}\\\\pn-p,& if \\; \\lambda =\\sum _{j=1}^{n-1}\\varepsilon _{j}+a\\varepsilon _{n}+b\\delta , 0\\ne a, b\\in \\mathbb {F}_{p}\\\\pn-1,& if \\;\\lambda =\\sum _{j=1}^{n}\\varepsilon _{j}+\\sum _{ j=i}^{n}\\varepsilon _{j}+b\\delta , b\\in \\mathbb {F}_{p}, i < n\\\\pn,& \\mbox{otherwise}.\\end{array}\\right.$ That is, $l\\left(L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )\\right)=pn$ if and only if $\\lambda $ is typical.",
"The proof is complete.",
"As a corollary of Theorem REF , we have Theorem 2 For $\\lambda \\in \\Lambda _{\\mathfrak {le}(n)},$ $I_{{le}(n)}(\\lambda )$ is irreducible if and only if $\\lambda $ is typical.",
"tocchapterAppendix"
],
[
"I.",
"Let $\\phi (x_{i})=\\left\\lbrace \\begin{array}{ll}\\sum _{j=1}^{n}a_{ji}x_{j},& \\mbox{if} \\; i\\le n \\\\\\sum _{j=n+1}^{2n}a_{ji}x_{j},& \\mbox{if}\\; i>n.\\end{array}\\right.$ If $\\phi =\\mathrm {diag}(t,\\ldots , t)\\in \\mathfrak {J},$ then $f_{\\phi }\\left(\\sum _{i=1}^{2n}x_{i}\\partial _{i}\\right)&=&\\sum _{i=1}^{2n}\\phi (x_{i})f_{\\phi }(\\partial _{i})=\\sum _{i=1}^{2n}tx_{i}t^{-1}\\partial _{i}=\\sum _{i=1}^{2n}x_{i}\\partial _{i}$ and $f_{\\phi }(\\mathrm {De}_{f})=t^{\\mathrm {deg}f-2}\\mathrm {De}_{f} \\;\\mbox{for}\\; f\\in \\mathcal {O}(n).$ If $\\phi \\in \\mathrm {SP}(n, \\mathbb {F}),$ then for $1 \\le i, j\\le n,$ we have $\\sum _{k=1}^{n}a_{ik}a_{j^{\\prime }k^{\\prime }}= \\delta _{ij}.$ Then $\\nonumber f_{\\phi }\\left(\\sum _{i=1}^{2n}x_{i}\\partial _{i}\\right)&=&\\sum _{i=1}^{2n}\\phi (x_{i})f_{\\phi }(\\partial _{i})\\\\\\nonumber &=&\\sum _{i=1}^{n}\\sum _{j=1}^{n}a_{ji}x_{j}\\sum _{k=1}^{n}a_{k^{\\prime }i^{\\prime }}\\partial _{k}+\\sum _{i=n+1}^{2n}\\sum _{j=n+1}^{2n}a_{ji}x_{j}\\sum _{k=n+1}^{2n}a_{k^{\\prime }i^{\\prime }}\\partial _{k}\\\\\\nonumber &=&\\sum _{j,k=1}^{n}\\left(\\sum _{i=1}^{n}a_{ji}a_{k^{\\prime }i^{\\prime }}\\right)x_{j}\\partial _{k}+\\sum _{j,k=n+1}^{2n}\\left(\\sum _{i=n+1}^{2n}a_{ji}a_{k^{\\prime }i^{\\prime }}\\right)x_{j}\\partial _{k}\\\\&=&\\sum _{j,k=1}^{2n}\\delta _{jk}x_{j}\\partial _{k}=\\sum _{i=1}^{2n}x_{i}\\partial _{i}$ and $f_{\\phi }(\\mathrm {De}_{f})=\\mathrm {De}_{\\phi (f)} \\;\\mbox{for}\\; f\\in \\mathcal {O}(n).$ Eqs.",
"(REF ) and (REF ) imply that $f_{\\phi }([\\mathrm {De}_{f}, \\mathrm {De}_{g}])=[f_{\\phi }(\\mathrm {De}_{f}),f_{\\phi }(\\mathrm {De}_{g})] \\;\\mbox{for}\\; f, g\\in \\mathcal {O}(n).$ Using Eqs.",
"(REF –REF ) and the fact that $\\phi $ is a $\\mathbb {Z}$ -homogeneous automorphism of $\\mathcal {O}(n)$ , we have $f_{\\phi }\\left(\\left[\\sum _{i=1}^{n}x_{i}\\partial _{i}, \\mathrm {De}_{f}\\right]\\right)=\\left[f_{\\phi }\\left(\\sum _{i=1}^{n}x_{i}\\partial _{i}\\right),f_{\\phi }(\\mathrm {De}_{f})\\right] \\;\\mbox{for}\\; f\\in \\mathcal {O}(n).$ Therefore, $f_{\\phi }\\in \\mathrm {Aut}\\left(\\overline{{le}}(n)\\right).$ II.",
"As in [7], we have $X(\\mathfrak {T})/ p X(\\mathfrak {T})\\cong \\Lambda _{\\overline{{le}}(n)}.$ Then for $a\\in \\mathbf {u}(\\overline{{le}}(n))$ and a highest weight vector $\\upsilon _{\\lambda }$ of $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{0}}(\\lambda )$ with respect to $\\mathfrak {b}_{0}$ , we can define $\\overline{t}(a\\otimes v_{\\lambda })=\\mathrm {Ad}(\\overline{t})(a)\\otimes \\lambda (\\overline{t})(v_{\\lambda }).$ Clearly, $I_{\\overline{{le}}(n)}(\\lambda )_{\\bar{i}}$ , $i=0, 1$ , is a $\\mathfrak {T}$ -module and $\\overline{t}(a\\cdot v)=\\mathrm {Ad}(\\overline{t})(a)\\overline{t}(v) \\;\\mbox{for}\\;\\overline{t}\\in \\mathfrak {T}, a\\in \\mathbf {u}(\\overline{{le}}(n)), v\\in I_{\\overline{{le}}(n)}(\\lambda ).$ We claim the action of $\\mathfrak {T}$ on $\\overline{{le}}(n)$ coincides with that of $\\bar{\\mathfrak {h}}.$ For $\\overline{t}=\\mathrm {diag}(tt_{1},\\ldots ,tt_{n},tt_{1}^{-1},\\ldots ,tt_{n}^{-1})\\in \\mathfrak {T},\\;x^{(\\underline{r})}\\in \\mathcal {O}(n)\\;\\mbox{and}\\;h\\in \\bar{\\mathfrak {h}},$ we can check the following equations: $\\overline{t}(\\mathrm {De}_{x^{(\\underline{r})}})&=&t^{\\mathrm {deg}x^{(\\underline{r})}-2}t_{1}^{r_{1}}\\cdots t_{n}^{r_{n}}t_{1}^{-r_{n+1}}\\cdots t_{n}^{-r_{2n}}\\mathrm {De}_{x^{(\\underline{r})}}\\\\&=&\\left(\\sum _{i=1}^{n}(r_{i^{\\prime }}-r_{i})\\Lambda _{i}+(\\mathrm {deg}x^{(\\underline{r})}-2)\\Lambda _{n+1}\\right)(\\overline{t})\\mathrm {De}_{x^{(\\underline{r})}},$ $[h,\\mathrm {De}_{x^{(\\underline{r})}}]=\\left(\\sum _{i=1}^{n}(r_{i^{\\prime }}-r_{i})\\varepsilon _{i}+(\\mathrm {deg}x^{(\\underline{r})}-2)\\delta \\right)(h)\\mathrm {De}_{x^{(\\underline{r})}},$ $\\overline{t}\\left(\\sum _{i=1}^{2n}x_{i}\\partial _{i}\\right)=\\sum _{i=1}^{2n}x_{i}\\partial _{i}=0(\\overline{t})\\sum _{i=1}^{2n}x_{i}\\partial _{i},$ $\\left[h,\\sum _{i=1}^{2n}x_{i}\\partial _{i}\\right]=0=0(h)\\sum _{i=1}^{2n}x_{i}\\partial _{i}.$ Summarizing, the action of $\\mathfrak {T}$ on $\\overline{{le}}(n)$ coincides with that on $\\bar{\\mathfrak {h}}.$ Then $I_{\\overline{{le}}(n)}(\\lambda )$ is a $\\left(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T}\\right)$ -module.",
"Similarly, $L_{\\overline{{le}}(n)}^{\\mathfrak {b}_{i}}(\\lambda )$ is also $\\left(\\mathbf {u}(\\overline{{le}}(n)),\\mathfrak {T}\\right)$ -module, $ 0 \\le i\\le 2n$ .",
"Acknowledgements.",
"The authors are grateful to Professor Chaowen Zhang for several conversations and suggestions on this topic."
]
] | 1403.0063 |
[
[
"Equilibrium Fermi-liquid coefficients for the fully screened N-channel\n Kondo model"
],
[
"Abstract We analytically and numerically compute three equilibrium Fermi-liquid coefficients of the fully screened $N$-channel Kondo model, namely $c_B$, $c_T$ and $c_\\varepsilon$, characterizing the magnetic field and temperature dependence of the resisitivity, and the curvature of the equilibrium Kondo resonance, respectively.",
"We present a compact, unified derivation of the $N$-dependence of these coefficients, combining elements from various previous treatments of this model.",
"We numerically compute these coefficients using the numerical renormalization group, with non-Abelian symmetries implemented explicitly, finding agreement with Fermi-liquid predictions on the order of 5% or better."
],
[
"Introduction",
"The Kondo effect was first observed, in the 1930s, for iron impurities in gold and silver [1], [2], as an anomalous rise in the resistivity with decreasing temperature.",
"Kondo[3] showed that this effect is caused by an antiferromagnetic exchange coupling between the localized magnetic impurity spins and the spins of the delocalized conduction electrons [3], and based his arguments on a spin-$\\frac{1}{2}$ , one-band model.",
"While this model undoubtedly captures the essential physics correctly in a qualitative way, it has recently been shown[4], [5] that a quantitatively correct description of the Kondo physics of dilute Fe impurities in Au or Ag requires a fully screened Kondo model involving three channels and a spin-$\\frac{3}{2}$ impurity.",
"This conclusion was based on a comparison of temperature and magnetic field dependent transport measurements[6], [4], [5] to theoretical predictions for fully screened Kondo models with channel number $N$ and local spin $S$ related by $N = 2S$ , with $N = 3$ yielding much better agreement than $N=1$ or 2.",
"The theoretical results in Ref.",
"Hanl2013 were obtained using the numerical renormalization group (NRG),[7], [8], [9], [10] and for $N=3$ various non-Abelian symmetries[11], [5], such as SU(2)$\\times $ U(1)$\\times $ SU$(N)$ , had to be exploited to achieve reliable results at finite magnetic field.",
"The technology for implementing non-Abelian symmetries with $N > 2$ in NRG calculations has been developed only recently.",
"[11], [12] Given the complexity of such calculations, it is desirable to benchmark their quality by comparing their predictions to exact results.",
"The motivation for the present paper was to perform such a comparison for the low-energy Fermi-liquid behavior of fully screened Kondo models, as elaborated upon below.",
"All fully screened Kondo models feature a ground state in which the impurity spin is screened by the conduction electrons into a spin singlet.",
"The low-energy behavior of these models can be described by a phenomenological Fermi-liquid theory (FLT) formulated in terms of the phase shift experienced by conduction electrons that scatter elastically off the screened singlet.",
"Such a description was first devised for the simplest case of $N=1$ by Nozières[13], [14] in 1974, and generalized to the case of arbitrary $N$ by Nozières and Blandin (NB)[15] in 1980.",
"Their results were confirmed and elaborated by various authors and methods, including NRG,[7], [8], [16], [17], [18], [19], [20] field-theoretic calculations,[21], [22] the Bethe Ansatz,[23], [24] conformal field theory (CFT),[25], [26] renormalized perturbation theory,[27] and reformulations[28], [29], [30] and generalizations[31], [32], [33] of Nozières' approach in the context of Kondo quantum dots.",
"In the present paper, we focus on three particular Fermi-liquid coefficients, $c_B$ , $c_T$ and $c_\\varepsilon $ , characterizing the leading dependence of the resistivity on magnetic field ($B$ ) and temperature ($T$ ), and the curvature of the equilibrium Kondo resonance as function of excitation energy $(\\varepsilon $ ), respectively.",
"Explicit formulas for all three of these coefficients are available in the literature for $N=1$ , but for general $N$ only for the case of $c_T$ .",
"Given the wealth of previous studies of fully-screened Kondo models, the lack of corresponding formulas for $c_B$ and $c_\\varepsilon $ was somewhat unexpected.",
"Thus, we offer here a unified derivation of all three Fermi-liquid coefficients, $c_T$ , $c_B$ and $c_\\varepsilon $ .",
"We follow the strategy which Affleck and Ludwig (AL)[26] have used to reproduce Nozières' results[13] for $N=1$ , namely doing perturbation theory in the leading irrelevant operator, and generalize it to the case of arbitrary $N$ .",
"Our formulation of this strategy follows that used by Pustilnik and Glazman (PG)[29] for their discussion of Kondo quantum dots.",
"While all pertinent ideas used here can be found in the literature, we hope that our rather compact way of combining them will be found useful.",
"For our numerical work, we faced two challenges: First, the complexity of the numerical calculations increases rapidly with increasing $N$ ; this was dealt with by exploiting non-Abelian symmetries.",
"Second, numerical calculations do not achieve the scaling limit that is implicitely presumed in analytical calculations; its absence was compensated by using suitable definitions of the Kondo temperature, following Ref. Hanl2013a.",
"The paper is organized as follows.",
"In Sec.",
"we define the model and summarize our key results for the Fermi-liquid coefficients $c_B$ , $c_T$ and $c_\\varepsilon $ .",
"Section compactly summarizes relevant elements of FLT and uses them to calculate these coefficients.",
"Section describes our numerical work and results.",
"Section summarizes our conclusions."
],
[
"Model and Main Results",
"The fully-screened Kondo model for $N$ conduction bands coupled to a single magnetic impurity at the origin is defined by the Hamiltonian $H=H_0 + H_{\\rm loc}$ , with $H_0 & = &\\sum _{km\\sigma }\\xi _k c^{\\dagger }_{km\\sigma }c_{km\\sigma } \\; ,\\\\H_{\\rm loc}& = &J_{\\rm K} \\sum _{kk^{\\prime }m\\sigma \\sigma ^{\\prime }}c_{km\\sigma }^{\\dag }\\frac{\\vec{\\tau }_{\\sigma \\sigma ^{\\prime }}}{2}c_{k^{\\prime }m\\sigma ^{\\prime }}\\vec{S} - BS_z .$ Here $H_0$ describes $N$ channels of free conduction electrons, with spin index $\\sigma =(+,-)=(\\uparrow ,\\downarrow )$ and channel index $m =1, \\dots , N$ .",
"We take the dispersion $\\xi _k = \\varepsilon _k - \\varepsilon _{\\rm F}$ to be linear and symmetric around the Fermi energy, $\\xi _k = k \\hbar v_{\\rm F}$ .",
"Each channel has exchange coupling $J_{\\rm K}$ to a local SU(2) spin of size $S= N/2$ with spin operators $\\vec{S}$ , and $B$ describes a local Zeeman field in the $z$ -direction (we use units $g\\mu _B=1$ ).",
"The overall symmetry of the model[19] is SU(2)$\\times $ Sp$(2N)$ for $B=0$ , and U(1)$\\times $ Sp$(2N)$ for $B\\ne 0$ (see Sec.",
"REF for details).",
"The model is characterized by a low-energy scale, the Kondo temperature, $T_{\\rm K}\\sim \\tilde{D}\\exp \\left[-1/(\\nu J_{\\rm K})\\right]$ , where $\\nu $ is the density of states per channel and spin species and $\\tilde{D}$ is of the order of the conduction electron bandwidth.",
"For a disordered metal containing a dilute concentration of magnetic impurities, the magnetic-impurity contribution to the resisitivity has the form[5], [35] $\\rho (T,B)\\propto \\int d\\varepsilon \\bigl (- \\partial _\\varepsilon f(\\varepsilon ,T)\\bigr )\\sum _{m\\sigma }A_{m\\sigma }(\\varepsilon ,T,B) \\; .$ Here $f(\\varepsilon ,T)$ is the Fermi function, and the impurity spectral function $A_{m\\sigma }(\\varepsilon ) = -\\frac{1}{\\pi }{\\rm Im}{m\\sigma }(\\varepsilon ) $ is the imaginary part of the $T$ matrix ${m\\sigma }(\\varepsilon )$ describing scattering off a magnetic impurity.",
"The latter is defined through[37], [38] ${\\cal G}_{m\\sigma ,{\\mathbf {k}},{\\mathbf {k}^{\\prime }}}^c(\\varepsilon )&={\\cal G}_{m\\sigma ,\\mathbf {k}}^0(\\varepsilon )\\delta ({\\mathbf {k}}-{\\mathbf {k}^{\\prime }})\\nonumber \\\\&+{\\cal G}_{m\\sigma ,\\mathbf {k}}^0(\\varepsilon ){\\cal T}_{m\\sigma }(\\varepsilon ) {\\cal G}_{m\\sigma ,\\mathbf {k}^{\\prime }}^0(\\varepsilon ) \\; ,$ with ${\\cal G}_{m\\sigma ,{\\mathbf {k}},{\\mathbf {k}^{\\prime }}}^c$ and ${\\cal G}_{m\\sigma ,\\mathbf {k}}^0$ the full and bare conduction electron Green's functions, respectively.",
"[For a Kondo quantum dot tuned such that the low-energy physics is described by Eq.",
"(), the conductance $G$ through the dot has a form similar to Eq.",
"(REF ), with $\\rho $ replaced by $G$ .",
"[29]] As mentioned in the Introduction, the ground state of the fully screened Kondo model is a spin singlet, and the regime of low-energy excitations below $T_{\\rm K}$ shows Fermi-liquid behavior.",
"[13], [15] One characteristic Fermi-liquid property is that the leading dependence of the $T$ matrix on its arguments, when they are small relative to $T_{\\rm K}$ , is quadratic, $\\frac{A_{m \\sigma } (\\varepsilon ,T,B)}{ A_{m \\sigma } (0, 0,0)}& = & 1 - \\frac{c_\\varepsilon \\varepsilon ^2 + c^{\\prime }_T T^2 +c_B B^2 }{T_{\\rm K}^2} \\; ,$ (Particle-hole and spin symmetries forbid terms linear in $\\varepsilon $ or $B$ .)",
"This implies the same for the resistivity, $\\frac{\\rho (T,B)}{\\rho (0,0)} & = & 1 -\\frac{c_T T^2 + c_B B^2}{T_{\\rm K}^2} \\; ,$ with $c_T = (\\pi ^2/3)c_\\varepsilon + c^{\\prime }_T$ .",
"The so-called Fermi-liquid coefficients $c_\\varepsilon $ , $c_T$ and $c_B$ are universal, $N$ -dependent numbers, characteristic of the fully screened Fermi-liquid fixed point.",
"For $N=1$ , the coefficients $c_T$ and $c_B$ have recently been measured experimentally in transport studies through quantum dots and compared to theoretical predictions.",
"[39] The coefficient $c_\\varepsilon $ is, in principle, also measurable via the non-linear conductance of a Kondo dot coupled strongly to one lead and very weakly to another.",
"[29] (The latter condition corresponds to the limit of a weak tunneling probe; it ensures that the non-linear conductance probes the equilibrium shape of the Kondo resonance, and hence the equilibrium Fermi-liquid coefficient $c_\\varepsilon $ .)",
"The goal of this paper is twofold: first, to analytically establish the $N$ dependence of $c_\\varepsilon $ , $c_T$ and $c_B$ using Fermi-liquid theory similar to NB; and second, to numerically calculate them using an NRG code that exploits non-Abelian symmetries, in order to establish a benchmark for the quality of the latter.",
"Our main results are as follows: First, if the Kondo temperature is defined by $T_{\\rm K}= \\frac{N(N+2)}{3\\pi \\chi ^{\\rm imp}} =\\frac{4 S (S+1)}{3\\pi \\chi ^{\\rm imp}} \\; ,$ where $\\chi ^{\\rm imp}$ is the static impurity susceptibility at zero temperature, the Fermi-liquid coefficients are given by $c_{B} = \\frac{(N+2)^2}{9} , \\quad c_T = \\pi ^2 \\frac{4N +5}{9} , \\quad c_\\varepsilon = \\frac{2N+7}{6} .",
"\\quad \\phantom{.",
"}$ For general $N$ , the formula for $c_T$ has first been found by Yoshimori,[21] while those for $c_B$ and $c_\\varepsilon $ are new (though not difficult to obtain).",
"Second, our numerical results for $N=1, 2$ and 3 are found to agree with the predictions of Eq.",
"(REF ) to within 5%."
],
[
"Fermi-liquid theory",
"In this section, we analytically calculate the Fermi-liquid coefficients $c_B$ , $c_T$ and $c_\\varepsilon $ for general $N$ .",
"With the benefit of hindsight, we selectively combine various elements of the work on FLT of Nozières,[13] NB,[15] AL[26] and PG[29].",
"Detailed justifications for the underlying assumptions are given by these authors in their original publications and hence will not be repeated here.",
"Instead, our goal is to assemble their ideas in such a way that the route to the desired results is short and sweet.",
"We begin by summarizing Nozières' ideas for expressing the $T$ matrix in terms of scattering phase shifts and expanding the latter in terms of phenomenological Fermi-liquid parameters.",
"Next, we recount AL's insight that this expansion can be reproduced systematically by doing perturbation theory in the leading irrelevant operator of the model's zero-temperature fixed point.",
"Then we adopt PG's strategy of performing the expansion in a quasiparticle basis in which the contant part of the phase shift has already been taken into account, which considerably simplifies the calculation.",
"Our own calculation is presented using notation analogous to that of PG, while taking care to highlight the extra terms that arise for $N>1$ .",
"It turns out that their extra contributions can be found with very little extra effort."
],
[
"Phase shift and $T$ matrix",
"Since the ground state of the fully screened Kondo model is a spin singlet, a low-energy quasiparticle scattering off the impurity experiences strong elastic scattering as if the impurity were nonmagnetic.",
"Moreover, it also experiences a weak local interaction if some energy $(\\ll T_{\\rm K})$ is available to weakly excite the singlet, causing some inelastic scattering.",
"Since the singlet binding energy is $T_{\\rm K}$ , the strength of this local interaction is proportional to $1/T_{\\rm K}$ .",
"Nozières[13] realized that this combination of strong elastic scattering and a weak local interaction can naturally be treated in terms of scattering phase shifts.",
"The phase shift of a quasiparticle with quantum numbers $m\\sigma $ and excitation energy $\\varepsilon $ relative to the Fermi energy can be written as $\\delta _{m\\sigma } (\\varepsilon ) = \\delta _{m\\sigma }^0 +\\tilde{\\delta }_{m \\sigma } (\\varepsilon ) \\; ,\\quad \\delta _{m\\sigma }^0 = \\pi /2 \\; .$ Here $\\delta _{m\\sigma }^0$ is the phase shift for $\\varepsilon = B=T=0$ ; it has the maximum possible value for scattering off a non-magnetic impurity, namely $\\pi /2$ .",
"Finite-energy corrections arising from weak excitations of the singlet are described by $\\tilde{\\delta }_{m \\sigma } (\\varepsilon )$ , which is proportional to $1/T_{\\rm K}$ .",
"If inelastic scattering is weak, unitarity of the $S$ matrix can be exploited[13] to write the $T$ matrix in the following form (we use the notation PG[29]; for a detailed analysis, see AL's discussion[26] of the terms arising from their Figs.",
"6 and 7): $1 - 2 \\pi \\nu i {m \\sigma }(\\varepsilon ) & = & e^{2 i \\delta _{m\\sigma }(\\varepsilon )}\\bigl [1 - 2 \\pi \\nu i \\tilde{\\mathcal {T}}^{\\rm in}_{m\\sigma } (\\varepsilon )\\bigr ]\\; .",
"\\qquad $ Here $\\tilde{\\mathcal {T}}^{\\rm in}$ accounts for weak inelastic two-body scattering processes, and is proportional to $1/T_{\\rm K}^2$ .",
"It is to be calculated in a basis of quasiparticle states in which the phase shift $\\delta ^0_{m\\sigma }$ has already been accounted for.",
"(Here and below, tildes will be used on quantities defined with respect to the new basis if they differ from corresponding ones in the original basis.)",
"Expanding Eq.",
"(REF ) in the small (real) number $\\tilde{\\delta }_{m\\sigma }(\\varepsilon )$ and recalling that $e^{2 i \\delta ^0_{m \\sigma }} = -1$ , one finds that the imaginary part of the $T$ matrix, which determines the spectral function, can be expressed as $- \\pi \\nu \\textrm {Im} {m\\sigma } ( \\varepsilon )= 1 - \\bigl [ \\tilde{\\delta }^2_{m \\sigma }(\\varepsilon ) - \\pi \\nu \\textrm {Im} \\tilde{\\mathcal {T}}^{\\rm in}_{m\\sigma } (\\varepsilon ) \\bigr ] \\; , \\qquad \\phantom{.",
"}$ to order $1/T_{\\rm K}^2$ .",
"Comparing this to Eq.",
"(REF ), we conclude that knowing $\\tilde{\\delta }$ to order $1/T_{\\rm K}$ and $ \\tilde{\\mathcal {T}}^{\\rm in}$ to order $1/T_{\\rm K}^2$ suffices to fully determine the Fermi-liquid coefficients $c_B$ , $c_T$ and $c_\\varepsilon $ .",
"Now, a systematic calculation of $\\tilde{\\delta }$ and $\\tilde{\\mathcal {T}}^{\\rm in}$ requires a detailed theory for the strong-coupling fixed point, which became available only with the work of AL in the early 1990s.",
"Nevertheless, Nozières succeeded in treating the case $N=1$ already in 1974,[13] using a phenomenological expansion of $\\tilde{\\delta }_{m\\sigma } (\\varepsilon )$ in powers of $(\\varepsilon -\\varepsilon ^{\\rm Z}_\\sigma )/T_{\\rm K}$ [$\\varepsilon ^{\\rm Z}_\\sigma $ represents the Zeeman energy of quasiparticles in a magnetic field, see Eq.",
"(REF ) below] and $\\delta \\bar{n}_{m^{\\prime }\\sigma ^{\\prime }} = n_{m^{\\prime }\\sigma ^{\\prime }} - n^0_{m^{\\prime }\\sigma ^{\\prime }}$ , the deviation of the total quasiparticle number $n_{m^{\\prime }\\sigma ^{\\prime }}$ from its ground-state value.",
"The prefactors in this expansion have the status of phenomenological Fermi-liquid parameters.",
"Using various ingenious heuristic arguments, he was able to show that all these parameters, and also $\\tilde{\\mathcal {T}}^{\\rm in}$ , are related to each other and can be expressed in terms of a single energy scale, namely the Kondo temperature.",
"Moreover, by choosing the prefactor of $\\varepsilon $ in this expansion to be $1/T_{\\rm K}$ , he suggested a definition of the Kondo temperature that also fixes its numerical prefactor.",
"(Our paper adopts this definition, too.)",
"In 1980, NB generalized this strategy [15] to general $N$ , finding an expansion of the form $\\nonumber \\tilde{\\delta }_{m\\sigma } (\\varepsilon )& = & \\alpha (\\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma ) - 3 \\psi \\delta \\bar{n}_{m, - \\sigma }\\\\& & {} + \\psi \\sum _{m^{\\prime } \\ne m}(\\delta \\bar{n}_{m^{\\prime } \\sigma } - \\delta \\bar{n}_{m^{\\prime }, - \\sigma } )\\; ,$ where $\\alpha $ and $\\psi $ are phenomenological Fermi-liquid parameters related by $\\alpha = 3 \\psi \\nu =1/T_{\\rm K}$ .",
"[NB's initial version of Eq.",
"(REF ) [their Eq.",
"(34)] does not contain the Zeeman contribution $\\varepsilon ^{\\rm Z}_\\sigma $ , but the latter is implicit in their subsequent treatment of the Zeeman field before their Eq. (37).]",
"In the following subsections, we show how NB's expansion for $\\tilde{\\delta }$ can be derived systematically.",
"AL[26] and PG[29] have shown how to do this for $N=1$ ; we will generalize their discussion to arbirtrary $N$ ."
],
[
"Leading irrelevant operator",
"AL showed[26] that NB's heuristic results can be derived in a systematic fashion by doing perturbation theory in the leading irrelevant operator of the model's zero-temperature fixed point.",
"As perturbation, they took the operator with the lowest scaling dimension satisfying the requirements of being (i) local, (ii) independent of the impurity spin operator $\\vec{S}$ , since the latter is fully screened, (iii) SU(2)-spin-invariant, (iv) and independent of the local charge density, just as the Kondo interaction.",
"The operator sastifying these criteria has the form[25] $H_\\lambda = - \\lambda :\\!",
"\\vec{J} (0) \\cdot \\vec{J} (0)\\!",
": \\; ,$ where $\\vec{J}(0)$ is the quasiparticle spin density at the impurity site, and $: \\!",
"\\dots \\!",
":$ denotes the point-splitting regularization procedure (see Appendix).",
"In Appendix D of Ref.",
"AL93, AL showed in great detail how NB's phase shifts can be computed using Eq.",
"(REF ), for the single-channel case of $N=1$ .",
"They did not devote as much attention to the case of general $N$ , though the needed generalizations are clearly implied in their work.",
"We here present the corresponding calculation in some detail, following the notational conventions of PG, which differ from those of AL in some regards (see Appendix).",
"The main difference is that PG formulate the perturbation expansion in a new basis of quasiparticle states, in which the phase shift $\\delta ^0_{m\\sigma }$ has already been accounted for, which somewhat simplifies the discussion.",
"(We remark that PG chose $\\delta ^0_{m\\sigma } = \\sigma \\pi /2$ rather than $\\pi /2$ as used by NB and us, but the extra $\\sigma $ has no consequences for the ensuing arguments.)",
"The quasiparticle Hamiltonian describing the vicinity of the strong-coupling fixed point (fp) has the form $H_{\\rm fp}= H_{\\rm fp,0} + H_\\lambda \\; ,$ where $H_{\\rm fp,0} = \\sum _{m\\sigma k} (\\xi _k + \\varepsilon ^{\\rm Z}_\\sigma ) \\!",
": \\!\\psi ^\\dagger _{km\\sigma } \\psi _{km\\sigma } \\!",
": \\; , \\quad \\varepsilon ^{\\rm Z}_\\sigma = - \\frac{\\sigma B}{2} \\qquad \\phantom{.",
"}$ describes free quasiparticles in a magnetic field $B$ , with Zeeman energy $\\varepsilon ^{\\rm Z}_\\sigma $ .",
"Note that although the Zeeman term in the bare Hamiltonian () is local, it is global in Eq.",
"(REF ), because the effective quasiparticle Hamiltonian $H_{\\rm fp}$ contains no local spin.",
"Using standard point-splitting techniques, which we review in pedagogical detail in the Appendix, the leading irrelevant operator (REF ) can be written as $H_\\lambda = H_1+ H_2+ H_3$ , with $H_1& = & - \\frac{1}{2 \\pi \\nu T_{\\rm K}}\\sum _{m\\sigma kk^{\\prime }} (\\xi _k + \\xi _{k^{\\prime }}): \\!",
"\\psi ^\\dagger _{km\\sigma } \\psi _{k^{\\prime }m\\sigma } \\!",
": \\; , \\quad \\phantom{.",
"}\\\\H_2&= & \\frac{1}{\\pi \\nu ^2 T_{\\rm K}} \\sum _m:\\!",
"\\rho _{m\\uparrow } \\rho _{m \\downarrow }\\!",
": \\; ,\\\\H_3& = & - \\frac{2}{3 \\pi \\nu ^2 T_{\\rm K}}\\sum _{m \\ne m^{\\prime }} : \\!",
"\\vec{j}_m \\cdot \\vec{j}_{m^{\\prime }} \\!",
": \\; ,$ where $\\rho _{m\\sigma } & = &\\sum _{kk^{\\prime }\\sigma } \\psi ^\\dagger _{km\\sigma } \\psi _{k^{\\prime }m\\sigma } \\; ,\\\\\\vec{j}_m &=& \\frac{1}{2}\\sum _{kk^{\\prime }\\sigma \\sigma ^{\\prime }} \\psi ^\\dagger _{km\\sigma } \\vec{\\tau }_{\\sigma \\sigma ^{\\prime }}\\psi _{k^{\\prime }m\\sigma ^{\\prime }} \\; .",
"\\quad \\phantom{.",
"}$ Here we have expressed the coupling constant $\\lambda $ in terms of the inverse Kondo temperature using [cf.",
"Eq.",
"(REF )] $\\lambda = \\frac{8 \\pi (\\hbar v_{\\rm F})^2}{3T_{\\rm K}} \\; ,$ with the numerical proportionality factor chosen such that $T_{\\rm K}$ agrees with definition of the Kondo temperature used by NB and PG, as discussed below.",
"Importantly, the point-splitting procedure fixes the relative prefactors arising in $H_1$ , $H_2$ and $H_3$ (whereas NB's approach requires heuristic arguments to fix them).",
"Our notation for $H_1$ and $H_2$ coincides with that used by PG.",
"$H_3$ contains all new contributions that enter additionally for $N>1$ .",
"Figure REF gives a diagrammtic depiction of all three terms.",
"Figure: (Color online) (a)-(c) Vertices associated withH 1 H_1, H 2 H_2 and H 3 H_3, respectively.",
"(d)-(f) Nonzero second-order contributions tothe quasiparticle self-energy, Σ ˜ mσ R \\tilde{\\Sigma }^{R}_{m\\sigma },involving H 1 2 H_1^2, H 2 2 H_2^2 and H 3 2 H_3^2, respectively.The contributions involving H 1 H 2 H_1H_2, H 1 H 3 H_1H_3and H 2 H 3 H_2H_3 all vanish, the former two due to the odd power ofenergy in the two-leg vertex."
],
[
"First order terms",
"Our first goal is to recover NB's expansion of the phase shift $\\tilde{\\delta }$ to leading order in $\\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma $ and $\\delta \\bar{n}$ .",
"Following PG, this can be done by calculating $\\tilde{\\delta }$ perturbatively to first order order in $1/T_{\\rm K}$ , in the new basis of quasiparticle states that already incorporate the phase shift $\\delta ^0$ .",
"To order $1/T_{\\rm K}$ , no inelastic scattering occurs, and $\\tilde{\\delta }$ is related to the elastic $T$ matrix by $e^{2 i \\tilde{\\delta }_{m\\sigma }(\\varepsilon )} =1 - 2 \\pi \\nu i \\tilde{\\mathcal {T}}^{\\rm el}_{m \\sigma } (\\varepsilon ) \\; .$ The elastic $T$ matrix, in turn, equals the real part of the quasiparticle self-energy, $\\tilde{\\mathcal {T}}^{\\rm el}_{m \\sigma } (\\varepsilon ) =\\textrm {Re} \\tilde{\\Sigma }^{R}_{m\\sigma } (\\varepsilon )$ .",
"(Actually, to order $1/T_{\\rm K}$ , the self-energy is purely real.)",
"By expanding Eq.",
"(REF ) for small $\\tilde{\\delta }$ , the phase shift is thus seen to be given by the real part of the self-energy: $\\tilde{\\delta }_{m\\sigma } (\\varepsilon ) \\simeq - \\pi \\nu \\textrm {Re} \\tilde{\\Sigma }^{R}_{m \\sigma } (\\varepsilon ) \\; .$ Now, as pointed out already by Nozières in 1974,[13] a first-order perturbation calculation of the self-energy is equivalent to treating interaction terms in the mean-field (MF) approximation.",
"They then take the form $H_2^{\\rm MF}\\!",
"&= & \\frac{1}{\\pi \\nu ^2 T_{\\rm K}} \\sum _{m\\sigma }:\\!",
"\\rho _{m\\sigma } \\!",
"\\!",
": \\delta \\bar{n}_{m , - \\sigma }\\\\H_3^{\\rm MF}\\!",
"& = & - \\frac{1}{3 \\pi \\nu ^2 T_{\\rm K}} \\!\\!\\sum _\\sigma \\!\\!",
"\\!",
"\\sum _{m \\ne m^{\\prime }} \\!\\!",
"\\!",
"\\!",
": \\!",
"\\rho _{m\\sigma } \\!",
":\\!",
"(\\delta \\bar{n}_{m^{\\prime } \\sigma } - \\delta \\bar{n}_{m^{\\prime }, - \\sigma } ) , \\qquad \\phantom{.",
"}$ where $\\delta \\bar{n}_{m\\sigma } = \\langle :\\!",
"\\rho _{m\\sigma } \\!",
": \\rangle $ , the quasiparticle number relative to the $B=0$ ground state, is given by $\\delta \\bar{n}_{m\\sigma } = - \\nu \\varepsilon ^{\\rm Z}_\\sigma = \\sigma \\nu B/2 \\; \\; .$ The mean-field version of the leading irrelevant operator thus has the form $H_\\lambda ^{\\rm MF}& = &\\sum _{m\\sigma kk^{\\prime }} h_{m\\sigma } (\\xi _k , \\xi _{k^{\\prime }}): \\!",
"\\psi ^\\dagger _{km\\sigma } \\psi _{k^{\\prime }m\\sigma } \\!",
": \\; , \\qquad \\phantom{.",
"}\\\\h_{m\\sigma } (\\xi _k , \\xi _{k^{\\prime }}) & = &\\frac{1}{\\pi \\nu T_{\\rm K}} \\Biggl [ - \\frac{1}{2}(\\xi _k + \\xi _{k^{\\prime }})+ \\frac{\\delta \\bar{n}_{m, - \\sigma }}{\\nu } \\Biggr .",
"\\\\& &\\nonumber \\quad \\qquad \\Biggl .",
"- \\sum _{m^{\\prime } \\ne m}\\frac{ \\delta \\bar{n}_{m^{\\prime } \\sigma } - \\delta \\bar{n}_{m^{\\prime }, -\\sigma }}{3 \\nu }\\Biggr ] \\; .$ For such a single-particle perturbation, the self-energy can be directly read off from $h_{m\\sigma }$ using $\\tilde{\\Sigma }^{R}_{m\\sigma }(\\varepsilon ) = h_{m\\sigma }(\\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma , \\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma ) \\; ,$ because $k$ sums of the type $\\sum _k 1/(\\varepsilon - \\xi _k -\\varepsilon ^{\\rm Z}_\\sigma + i0^+)$ yield residues involving $\\xi _{k} = \\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma $ .",
"Using Eq.",
"(REF ) in Eq.",
"(REF ) for the phase shift, we find $\\tilde{\\delta }_{m\\sigma }(\\varepsilon ) & = &\\frac{1}{T_{\\rm K}} \\Biggl [\\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma - \\frac{\\delta \\bar{n}_{m, - \\sigma }}{\\nu }\\Biggr .",
"\\\\& &\\nonumber \\quad \\qquad \\Biggl .+ \\sum _{m^{\\prime } \\ne m}\\frac{ \\delta \\bar{n}_{m^{\\prime } \\sigma } - \\delta \\bar{n}_{m^{\\prime }, -\\sigma }}{3\\nu } \\Biggr ] \\; .$ This fully agrees with the expansion (REF ) of NB if we make the identification $1/T_{\\rm K}= \\alpha = 3 \\psi \\nu $ , thus confirming the validity of NB's heuristic arguments.",
"Note that the coefficient of $\\varepsilon - \\varepsilon ^{\\rm Z}_\\sigma $ in Eq.",
"(REF ) comes out as $1/T_{\\rm K}$ , in agreement with the conventions of NB and PG, as intended by our choice of numerical prefactor in Eq.",
"(REF ).",
"As consistency check, let us review how NB used Eq.",
"(REF ) to calculate the Wilson ratio.",
"First, Eq.",
"(REF ) implies an impurity-induced change in the density of states per spin and channel of $\\nu ^{\\rm imp}_{m\\sigma }(\\varepsilon ) = \\frac{1}{\\pi } \\partial _\\varepsilon \\delta _{m\\sigma }(\\varepsilon )$ .",
"This yields a corresponding impurity-induced change in the specific heat, $C^{\\rm imp}$ .",
"At zero field (where $\\varepsilon ^{\\rm Z}_\\sigma $ and $\\delta \\bar{n}_{m\\sigma }$ vanish), the change relative to the bulk is given by $\\frac{C^{\\rm imp}}{C} = \\frac{2 N \\nu ^{\\rm imp}_{m\\sigma } (0) }{2N \\nu }= \\frac{1}{\\pi \\nu T_{\\rm K}} \\; .$ Second, the Friedel sum rule for the impurity-induced change in local charge in channel $m$ for spin $\\sigma $ at $T=0$ gives $N^{\\rm imp}_{m\\sigma } & = & \\frac{1}{\\pi } \\delta _{m\\sigma }(0) =\\frac{1}{2} + \\frac{1}{\\pi } \\tilde{\\delta }_{m\\sigma }(0) \\; ,$ and Eq.",
"(REF ), together with Eq.",
"(REF ) for $\\delta \\bar{n}_{m \\sigma }$ , leads to $\\tilde{\\delta }_{m\\sigma }(0)& = & \\frac{\\sigma B}{T_{\\rm K}} \\left[ \\frac{1}{2} + \\frac{1}{2} +\\frac{N-1}{3} \\right]= \\frac{\\sigma B (N+2)}{3 T_{\\rm K}} \\; .",
"\\qquad \\phantom{.",
"}$ The linear response of the impurity-induced magnetization, $M^{\\rm imp}=\\frac{1}{2}\\sum _m(N^{\\rm imp}_{m\\uparrow } - N^{\\rm imp}_{m\\downarrow })$ , then gives the impurity contribution to the spin susceptibility as $\\chi ^{\\rm imp}= \\frac{M^{\\rm imp}}{B} =\\frac{N(N+2)}{3\\pi T_{\\rm K}} =\\frac{4 S(S+1)}{3\\pi T_{\\rm K}} \\; .$ (For all expressions involving $\\chi ^{\\rm imp}$ here and below, the limit $B\\rightarrow 0$ is implied.)",
"The corresponding bulk contribution is $\\chi = \\nu N /2$ .",
"Thus, the Wilson ratio is found to be $R = \\frac{\\chi ^{\\rm imp}/\\chi }{C^{\\rm imp}/C}= \\frac{2(N+2)}{3} = \\frac{4(S+1)}{3} \\; ,$ in agreement with more elaborate calculations by Yoshimori[21] and by Mihály and Zawadowski.",
"[22] Note that Eq.",
"(REF ) relates Nozières' definition of the Kondo temperature to an observable quantity, $\\chi ^{\\rm imp}$ , that can be calculated numerically.",
"We used this as a precise way of defining $T_{\\rm K}$ in our numerical work.",
"(Subtleties involved in calculating $\\chi ^{\\rm imp}$ are discussed in Sec.",
"REF .)",
"Note that up to a prefactor, Eq.",
"(REF ) for $\\chi ^{\\rm imp}$ has the form $\\chi ^{\\rm free}(T_{\\rm K})$ , where $\\chi ^{\\rm free}(T) = S(S+1)/(3 T)$ is the static susceptibility of a free spin $S$ at temperature $T$ .",
"We are now in a position to extract our first Fermi-liquid coefficient, $c_B$ .",
"For this, it suffices to know the spectral function $A$ in Eq.",
"(REF ) to quadratic order in $B$ , at $\\varepsilon =T=0$ , where $\\tilde{\\mathcal {T}}^{\\rm in}=0$ .",
"Inserting the corresponding expression (REF ) for $\\tilde{\\delta }_{m\\sigma }(0)$ into Eq.",
"(REF ) for $\\textrm {Im} , we find\\begin{eqnarray}A_{m\\sigma }(0,0,B) = \\frac{1}{\\nu \\pi ^2}\\left[ 1 - \\frac{(N+2)^2}{9} \\frac{B^2}{T_{\\rm K}^2} \\right] \\; .\\end{eqnarray}Comparing this to Eq.\\,(\\ref {eq:Aexpand}), we read off $ cB = (N+2)2/9$.$ Note that if the definition (REF ) of $T_{\\rm K}$ in terms of $\\chi ^{\\rm imp}$ is taken as given, $c_B$ can actually be derived on the back of an envelope: for a fully screened Kondo model, the impurity-induced spin susceptibility gets equal contributions from all $N$ channels, $\\chi ^{\\rm imp}= N\\chi ^{\\rm imp}_{m}$ , and the Friedel sum rule relates the contribution from each channel to phase shifts, $\\chi ^{\\rm imp}_{m} = M^{\\rm imp}_m/B =[\\tilde{\\delta }_{m\\uparrow }(0) - \\tilde{\\delta }_{m\\downarrow }(0)]/(2 \\pi B)$ , implying $\\tilde{\\delta }_{m\\sigma } (0)= \\sigma (\\pi \\chi ^{\\rm imp}/N)B$ .",
"Using this in Eq.",
"(REF ) yields $A_{m\\sigma }(0,0,B) = \\frac{1}{\\nu \\pi ^2}\\Bigl [ 1 - (\\pi \\chi ^{\\rm imp}/N)^2B^2 \\Bigr ] \\; ,$ which is equivalent to Eq.",
"() if Eq.",
"(REF ) holds."
],
[
"Second order terms",
"We next discuss inelastic scattering for $B=0$ , but at finite temperature.",
"To order $1/T_{\\rm K}^2$ , inelastic scattering is described by the imaginary part of the quasiparticle self-energy arising from the second-order contributions of $H_1$ , $H_2$ and $H_3$ , shown in diagrams (d)-(f) of Fig.",
"REF , respectively.",
"These diagrams give ${\\rm Im} \\tilde{\\Sigma }^{R,1}_{m\\sigma }(\\varepsilon )& = & - \\frac{\\varepsilon ^2}{\\pi \\nu T_{\\rm K}^2} \\; ,\\\\{\\rm Im} \\tilde{\\Sigma }^{R,2}_{m\\sigma }(\\varepsilon )& = & - \\frac{\\varepsilon ^2 + \\pi ^2 T^2}{2 \\pi \\nu T_{\\rm K}^2} \\; ,\\\\{\\rm Im} \\tilde{\\Sigma }^{R,3}_{m\\sigma }(\\varepsilon )& = & \\frac{2}{3} (N-1) \\, {\\rm Im} \\tilde{\\Sigma }^{R,2}_{m\\sigma }(\\varepsilon ) \\; .$ The first two can also be found in the discussion of PG, whose strategy we follow here.",
"(They also appear, in slightly different guise, in the discussion of AL[26].)",
"The third is proportional to the second, and the factor $2/3$ originates from $(2/3)^22 s(s+1) $ with $s=1/2$ , since the relative prefactor between $H_3$ and $H_2$ brings in two powers of $2/3$ , and the algebra of Pauli matrices yields a factor $2s(s+1)$ .",
"Now, the term called $\\tilde{\\mathcal {T}}^{\\rm in}$ in Eq.",
"(REF ) by definition describes the contribution of the two-body terms $H_2$ and $H_3$ to inelastic scattering: $\\textrm {Im} \\tilde{\\mathcal {T}}^{\\rm in}_{m\\sigma }(\\varepsilon ) = \\textrm {Im} \\Bigl [\\tilde{\\Sigma }^{R,2}_{m\\sigma }(\\varepsilon ) +\\tilde{\\Sigma }^{R,3}_{m\\sigma }(\\varepsilon ) \\Bigr ] \\; .$ The contribution ${\\rm Im} \\tilde{\\Sigma }^{R,1}$ from $H_1$ is not included in $\\textrm {Im} \\tilde{\\mathcal {T}}^{\\rm in}$ here, since it actually equals $-\\tilde{\\delta }^2/ \\pi \\nu $ , and hence is already contained in the factor $e^{2 i \\tilde{\\delta }}$ in Eq.",
"(REF ).",
"Indeed, in Eq.",
"(REF ) for the imaginary part of the $T$ matrix in the original basis, the $\\tilde{\\delta }^2$ term equals $- \\pi \\nu {\\rm Im}\\tilde{\\Sigma }^{R,1}$ .",
"Collecting all ingredients, Eq.",
"(REF ) gives $\\nonumber {A_{m\\sigma }(\\varepsilon ,T,0) }\\\\& = & \\frac{1}{\\nu \\pi ^2} \\left[1 - \\frac{\\varepsilon ^2}{T_{\\rm K}^2} -\\frac{\\varepsilon ^2 + \\pi ^2 T^2}{2 T_{\\rm K}^2} \\Bigl (1 + \\frac{2}{3}(N-1)\\Bigr ) \\right]\\nonumber \\\\& = &\\frac{1}{\\nu \\pi ^2} \\left[1 - \\frac{(2N+7)\\varepsilon ^2 + (2N+1)\\pi ^2 T^2}{6 T_{\\rm K}^2}\\right] \\; .\\qquad \\phantom{.",
"}$ For $N=1$ , the second term reduces to the familiar form $- (3\\varepsilon ^2 + \\pi ^2 T^2)/(2 T_{\\rm K}^2)$ found by AL[26] and GP[29].",
"Comparing Eqs.",
"() and (REF ) and (REF ) we read off $c_\\varepsilon = (2N+7)/6$ and $c^{\\prime }_T = \\pi ^2 (2N+1)/6$ , implying $c_T = \\pi ^2 (4N+5)/9$ .",
"In this section, we describe our NRG work.",
"We had set ourselves the goal of achieving an accuracy of better than 5% for the Fermi-liquid coefficients.",
"To achieve this, two ingredients were essential.",
"First, exploiting non-Abelian symmetries; and second, defining the Kondo temperature with due care.",
"The latter is a matter of some subtlety [34] because the wide-band limit assumed in analytical work does not apply in numerical calculations.",
"We begin below by giving the Lehmann representation for the desired spectral function.",
"We then discuss the non-Abelian symmetries used in our NRG calculations and explain how the Kondo temperature was extracted numerically.",
"Finally, we present our numerical results."
],
[
"NRG details",
"To numerically calculate the $T$ matrix of Eq.",
"(REF ), we use equations of motion[37], [38] to express it as ${\\cal T}_{m\\sigma }(\\varepsilon ) &= & J_{\\rm K}\\langle S_z \\rangle +\\langle \\langle O_{m\\sigma }; O^{\\dag }_{m\\sigma } \\rangle \\rangle ,\\\\O_{m\\sigma } & \\equiv & [\\Psi _{m\\sigma }(0),H_{\\rm loc}] =J_{\\rm K}\\sum _{\\sigma ^{\\prime }}{\\vec{S}} \\cdot \\frac{{\\vec{\\tau }}_{\\sigma \\sigma ^{\\prime }}}{2}\\Psi _{m\\sigma ^{\\prime }}(0) .",
"\\qquad \\phantom{.",
"}$ Here $\\langle \\langle \\,\\cdot \\,; \\cdot \\,\\rangle \\rangle $ denotes a retarded correlation function, and $\\Psi _{m\\sigma }(0)=\\frac{1}{\\sqrt{N_{\\rm disc}}}\\sum _k c_{km\\sigma }$ , where $N_{\\rm disc}$ is the number of discrete levels in the band (and hence proportional to the system size).",
"The spectral function is then calculated in its Lehmann-representation, ${A_{m\\sigma }(\\varepsilon ,T,B) =}\\nonumber \\\\& \\quad \\sum _{a,b}\\frac{e^{-\\beta E_a}+e^{-\\beta E_b}}{Z}\\vert \\langle a \\vert O_{m\\sigma } \\vert b \\rangle \\vert ^2 \\delta (\\varepsilon - E_{ab}),$ with $E_{ab}=E_b-E_a$ , using the full density matrix (FDM) approach of NRG.",
"[9], [40], [41], [42] For our numerical work, we take the conduction band energies to lie within the interval $\\xi _k \\in [-D,D]$ , with Fermi energy at 0 and half-bandwidth $D=1$ , and take the density of states per spin, channel and unit length to be constant, as $1/2D$ .",
"(It is related to the extensive density of states used in Sec.",
"by $\\nu = N_{\\rm disc}/2D$ .)",
"For the calculations used to determine the Fermi-liquid parameters, we use exchange coupling $\\nu J_{\\rm K}=0.1$ , so that the Kondo temperature $T_{\\rm K}/D \\propto \\exp [-1/(\\nu J_{\\rm K})]$ has the same order of magnitude for $N=1, 2$ and 3, namely $\\lesssim 10^{-4}$ .",
"Following standard NRG protocol,[7], [8], [10] the conduction band is discretized logarithmically with discretization parameter $\\Lambda $ , mapped onto a Wilson chain, and diagonalized iteratively.",
"NRG truncation at each iteration step is controlled by either specifying the number of kept states per shell, $N_{\\rm K}$ , or the truncation energy, $E_{\\rm tr}$ (in rescaled units, as defined in Ref.",
"Weichselbaum2011), corresponding to the highest kept energy per shell.",
"Spectral data are averaged over $N_z$ different, interleaving logarithmic discretization meshes.",
"[44] The values for NRG-specific parameters used here are given in legends in the figures below.",
"For the fully screened $N$ -channel Kondo model, the dimension of the local Hilbert space of each supersite of the Wilson chain is $4^N$ .",
"Since this increases exponentially with the number of channels, it is essential, specifically so for $N=3$ , to reduce computational costs by exploiting non-Abelian symmetries [11] to combine degenerate states into multiplets.",
"Several large symmetries are available[19]: For $B=0$ , the model has SU(2)$\\times $ U(1)$\\times $ SU$(N)$ spin-charge-channel symmetry.",
"If the bands desribed by $H_0$ are particle-hole symmetric, as assumed here, the model also has a SU(2)$\\times $ [SU(2)]${}^N$ spin-(charge)${}^N$ symmetry, involving SU(2) mixing of particles and holes in each of the $N$ channels.",
"The U(1)$\\times $ SU$(N)$ and [SU(2)]${}^N$ symmetries are not mutually compatible (their generators do not all commute), however, implying that both are subgroups of a larger symmetry group, the symplectic Sp$(2N)$ .",
"Thus the full symmetry of the model for $B=0$ is SU(2)$\\times $ Sp$(2N)$ .",
"For $B\\ne 0$ it is U(1)$\\times $ Sp$(2N)$ , since a finite magnetic field breaks the SU(2) spin symmetry to the Abelian U(1) $S_z$ symmetry.",
"When the model's full symmetry is exploited, the multiplet spaces encountered in NRG calculations exhibit no more degeneracies in energy at all.",
"Using only Abelian symmetries turned out to be clearly insufficient to obtain well converged numerical data for $N=3$ , despite having a relatively large $\\Lambda $ .",
"This, however, is required for accurate Fermi-liquid coefficients with errors below a few percent.",
"For numerically converged data, therefore, it was essential to use non-Abelian symmetries.",
"For our $B=0$ calculations, it turned out to be sufficient to use SU(2)$\\times $ U(1)$\\times $ SU$(N)$ symmetry for calculating $c_T$ , but the full SU(2)$\\times $ Sp$(2N)$ symmetry was needed for calculating $c_\\varepsilon $ .",
"Likewise, for our $B\\ne 0$ calculations of $c_B$ , we needed to use the full U(1)$\\times $ Sp$(2N)$ symmetry.",
"Doing so led to an enormous reduction in memory requirements, the more so the larger the rank of the symmetry group [Sp$(2N)$ has rank $N$ , and SU$(N)$ has rank $N-1$ ].",
"For $N=3$ , for example, we kept $\\lesssim 13\\,500$ multiplets for SU(2)$\\times $ U(1)$\\times $ SU$(3)$ or $\\lesssim 3\\,357$ multiplets for SU(2)$\\times $ Sp$(6)$ during NRG truncation, which, in effect, amounts to keeping $\\lesssim 980\\,000$ individual states.",
"[11]"
],
[
"Definition of $T_{\\rm K}$",
"The Fermi-liquid theory of Sec.",
"implicitly assumes that the model is considered in the so-called scaling limit, in which the ratio of Kondo temperature to bandwidth vanishes, $T_{\\rm K}/D\\rightarrow 0$ .",
"In this limit, physical quantities such as $\\rho (T,B)/\\rho (0,0)$ are universal scaling functions, which depend on their arguments only in the combinations $B/T_{\\rm K}$ and $T/T_{\\rm K}$ .",
"Since the shape of such a scaling function, say $\\rho (0,B)/\\rho (0,0)$ plotted versus $B/T_{\\rm K}$ , is universal, i.e.",
"independent of the bare parameters (coupling $J_{\\rm K}$ and bandwidth $D$ ) used to calculate it, curves generated by different combinations of bare parameters can all be made to collapse onto each other by suitably adjusting the parameter $T_{\\rm K}$ for each.",
"In the same sense the Fermi-liquid parameters $c_B$ , $c_T$ and $c_\\varepsilon $ , being Taylor-coefficients of universal curves, are universal, too.",
"One common way to achieve a scaling collapse, popular particularly in experimental studies, is to identify the Kondo temperature with the field $B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}$ or temperature $T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}$ at which the impurity contribution to the resisitivity has decreased to half its unitary value, $\\rho (0,B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}})= \\rho (0,0)/2 \\; ,\\quad \\rho (T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}},0) = \\rho (0,0)/2 \\; .",
"\\qquad \\phantom{.",
"}$ However, this is approach is not suitable for the purpose of extracting Fermi-liquid coefficients, for which $T_{\\rm K}$ has to be defined in terms of (analytically accessible) low-energy properties characteristic of the strong-coupling fixed point.",
"In Sec.",
"we have therefore adopted Nozières' definition of $T_{\\rm K}$ in terms of the leading energy dependence of the phase shift $\\tilde{\\delta }_{m\\sigma }^0$ [Eq.",
"(REF )], implying that it can be expressed in terms of $\\chi ^{\\rm imp}$ , of the local static spin susceptibility at zero temperature [Eq.",
"(REF )].",
"In the scaling limit, this definition of $T_{\\rm K}$ matches $B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}$ or $T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}$ up to prefactors, i.e.",
"$B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}/T_{\\rm K}$ and $T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}/T_{\\rm K}$ are universal, $N$ -dependent numerical constants, independent of the model's bare parameters.",
"In numerical work, however, the scaling limit is never fully realized, since the bandwidth is always finite.",
"It may thus happen that a scaling collapse expected analytically is not found when the corresponding curves are calculated numerically.",
"For example, if the Kondo temperature is defined, as seems natural, in terms of a purely local susceptibility, $\\chi ^{\\rm loc}$ , involving only the response of the local spin to a local field, $\\frac{4 S(S+1)}{3\\pi T^{\\rm loc}_{\\rm K}}\\equiv \\chi ^{\\rm loc}\\equiv \\frac{d}{d B}\\langle S_z\\rangle \\vert _{B=0} \\; ,\\;$ then curves expected to show a scaling collapse actually do not collapse onto each other, as pointed out recently in Ref.",
"Hanl2013a (see Figs.",
"2(d)-(f) there).",
"That paper also showed how to remedy this problem: the static spin susceptibility used to calculate $T_{\\rm K}$ has to be defined more carefully, and two slightly different definitions have to be used, depending on the context.",
"The first option is needed when studying zero-temperature (i.e.",
"ground state) properties as a function of some external parameter, such as the field dependence of the resisitivity (needed for $c_B$ ).",
"In this case, a corresponding susceptibility defined in terms of the response of the system's total spin to a local field should be used: $\\frac{4 S(S+1)}{3\\pi T^{\\rm FS}_{\\rm K}}\\equiv \\chi ^{\\rm FS}\\equiv \\frac{d}{d B}\\langle S_z^{\\rm tot}\\rangle \\vert _{B=0}\\; .$ The superscript FS stands for “Friedel sum rule”, to highlight the fact that using this rule to calculate the linear response of $\\langle S_z^{\\rm tot}\\rangle $ to a local field directly leads to relation (REF ) between $\\chi ^{\\rm imp}$ and $T_{\\rm K}$ .",
"The second option is needed when studying dynamical or thermal quantities that depend on the system's many-body excitations for given fixed external parameters (e.g.",
"fixed $B=0$ ), such as the temperature-dependence of the resistivity (needed for $c_T$ ), or the curvature of the Kondo resonance (needed for $c_\\varepsilon $ ).",
"In this case, one should use $\\frac{4 S(S+1)}{3\\pi T^{\\rm sc}_{\\rm K}}& \\equiv &\\chi ^{\\rm sc}\\equiv 2\\chi ^{\\rm FS}- \\chi ^{\\rm loc}\\; .$ The superscript sc stands for “scaling”, to indicate that this definition of the Kondo temperature ensures[34] a scaling collapse of dynamical or thermal properties.",
"Figure REF demonstrates that a scaling collapse is indeed found when the field- or temperature-dependent resistivity, plotted versus $B/T^{\\rm FS}_{\\rm K}$ or $T/T^{\\rm sc}_{\\rm K}$ , respectively, is calculated for two different values of $J_{\\rm K}$ (solid and dashed lines, respectively).",
"Note that this works equally well for $N=1, 2$ and 3.",
"(For $N=1$ , such scaling collapses had already been shown in Ref. Hanl2013a.)",
"We remark that the three Kondo temperatures defined in Eqs.",
"(REF )-(REF ) differ quite significantly from each other for the Kondo Hamiltonian of Eq.",
"(), with differences as large as 12%, 31% and 55% for $N=1$ , 2 and 3, respectively, for the parameters used in Fig.",
"REF .",
"This indicates that although we have chosen bare paramters for which $T_{\\rm K}/D$ is smaller than $ 10^{-4}$ , we have still not reached the scaling limit [in which the definitions Eq.",
"(REF )-(REF ) of the Kondo temperature should all coincide numerically[34]].",
"We have checked that the differences between $T^{\\rm loc}_{\\rm K}$ , $T^{\\rm FS}_{\\rm K}$ and $T^{\\rm sc}_{\\rm K}$ decrease when $\\nu J_{\\rm K}$ is reduced in an attempt to get closer to the scaling limit, but estimate that truly reaching that limit would require $\\nu J_{\\rm K}< 0.01$ for the Kondo model, implying $T_{\\rm K}/D < 10^{-45}$ .",
"Thus, reaching the scaling limit by brute force is numerically unfeasible.",
"Therefore, using $T^{\\rm FS}_{\\rm K}$ and $T^{\\rm sc}_{\\rm K}$ rather than $T^{\\rm loc}_{\\rm K}$ is absolutely essential for obtaining scaling collapses.",
"It is similarly essential for an accurate determination of the Fermi-liquid parameters.",
"Correspondingly, for the results discussed below, we have used $T^{\\rm FS}_{\\rm K}$ as definition of the Kondo temperature when extracting $c_B$ , and $T^{\\rm sc}_{\\rm K}$ when extracting $c_T$ and $c_\\varepsilon $ .",
"Figure: (Color online) Scaling collapse of (a) the resistivity atzero temperature as function of field, and (b) at zero field asfunction of temperature, calculated for two different values of thebare coupling, νJ K \\nu J_{\\rm K} (dashed or solid), and for N=1,2N=1, 2 and3.",
"For each NN, the dashed and solid curves overlap so well thatthey are almost indistinguishable.",
"The insets compare the energyscales B 1/2 B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}} and T 1/2 T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}} at which the resistivity hasdecreased to half its unitary value [cf. Eq.",
"()],to the scales T K FS T^{\\rm FS}_{\\rm K} and T K sc T^{\\rm sc}_{\\rm K} [cf. Eqs.",
"() and()], respectively.",
"The shown ratios areuniversal numbers of order unity, but not necessarily veryclose to 1, with a significant dependence on NN: B 1/2 /T K FS =1.22,1.31,1.60B_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}/T^{\\rm FS}_{\\rm K}=1.22, 1.31, 1.60 and T 1/2 /T K sc =0.82,1.02,1.36T_{\\scriptscriptstyle {1 \\!",
"/ \\!",
"2}}/T^{\\rm sc}_{\\rm K}= 0.82, 1.02, 1.36 for N=1N=1,2 and 3, respectively.",
"The legend in the lower left of panel (b)specifies the NRG parameters used for both panels.Figure: (Color online) (a) Resistivity as function of magnetic fieldat T=0T=0, (b) resistivity as function of temperature atB=0B=0, and (c) the weighted spectral function A ¯(τ){\\bar{A}}({\\tau })[cf. Eq.",
"()] at T=B=0T=B=0, all shown for N=1,2,3N=1,2,3.",
"Eachpanel contains NRG data (heavy solid lines), the quadratic term froma fourth order polynomial fit (heavy dashed lines) and thecorresponding predictions from FLT of Eq.",
"() forthe quadratic term (light solid lines).",
"Left and rightvertical dotted lines in matching colors indicate the lower andupper borders of the fitting range used for each NN.The boxed legends specify the NRG parameters used here."
],
[
"Using unbroadened discrete data only",
"When one is interested in spectral properties, one typically has to broaden the discrete data.",
"For the determination of the Fermi-liquid coefficients, however, where high numerical accuracy is required, it is desirable to avoid standard broadening.",
"For the calculation of $c_T$ and $c_B$ this can be achieved[9] by directly inserting the Lehmann sum over $\\delta $ functions for the spectral function $A_{m\\sigma }(\\varepsilon ,T,B)$ [Eq.",
"(REF )] into the energy integral for $\\rho (T,B)$ [Eq.",
"(REF )], resulting in a sum over discrete data points that produces a smooth curve.",
"The curve is smooth because Eq.",
"(REF ) in effect thermally broadens the $\\delta $ peaks in the Lehmann representation.",
"This is true even in the limit $T \\rightarrow 0$ , because in NRG calculations it is realized by taking $T$ nonzero, but much smaller than all other energy scales.",
"For the determination of $c_{\\varepsilon }$ , in contrast, one faces the problem that $A_{m\\sigma }(\\varepsilon ,0,0)$ is represented not as an integral of a sum over discrete $\\delta $ functions, but directly in terms of the latter.",
"To avoid having to broaden these by hand, it is desirable to find a way to extract $c_{\\varepsilon }$ from an expression involving an integral over the discrete spectral data, as for $c_B$ and $c_T$ .",
"This can be achieved as follows.",
"First, note that $c_{\\varepsilon }$ is, by definition, a coefficient in the general Taylor expansion of the normalized spectral function $A^{\\rm norm}(\\varepsilon ) \\equiv A_{m\\sigma }(\\varepsilon ,0,0)/A_{m\\sigma }(0,0,0)$ for small frequencies, $A^{\\rm norm}(\\varepsilon )=\\sum _{n=0}^{\\infty } a_n (\\varepsilon /T_{\\rm K})^n, \\qquad c_\\varepsilon = a_2 \\; .$ Due to particle-hole symmetry, $a_n=0$ for all $n$ odd, and by definition $a_0=1$ .",
"To determine $a_2$ from an integral over discrete data, we consider a weighted average of $A^{\\rm norm}(\\varepsilon )$ over $\\varepsilon $ , ${\\bar{A}}(\\tau ) \\equiv \\int d\\varepsilon A^{\\rm norm}(\\varepsilon )P_{\\tau }(\\varepsilon ),$ where $P_{\\tau }(\\varepsilon )$ is a symmetric weighting function of width $\\tau $ and weight 1, and moments defined by $\\int d\\varepsilon (\\varepsilon /{\\tau })^n P_{\\tau }(\\varepsilon ) \\equiv p_n$ for integer $n \\ge 0$ (with $p_0 = 1$ ).",
"Here we use $P_{\\tau }(\\varepsilon )=\\frac{1}{4\\tau }\\frac{1}{\\cosh ^2\\left( \\varepsilon /2\\tau \\right)}=-\\frac{\\partial f(\\varepsilon , \\tau )}{\\partial \\varepsilon } ,$ but other choices are possible, too (e.g., a Gaussian peak).",
"Clearly, the leading ${\\tau }$ dependence of ${\\bar{A}}({\\tau })$ for small ${\\tau }$ reflects the leading ${\\varepsilon }$ dependence of $A^{\\rm norm}(\\varepsilon )$ and allows for an accurate determination of $a_2$ .",
"Indeed, using Eqs.",
"(REF )-(REF ), we obtain a power-series expansion for ${\\bar{A}}({\\tau })$ of the form ${\\bar{A}}({\\tau })=\\sum _{n} a_n p_n {(\\tau /T_{\\rm K})}^n$ .",
"Thus, by fitting ${\\bar{A}}^{\\rm fit}({\\tau })=\\sum _n f_n {\\tau }^n$ to the NRG data for ${\\bar{A}}({\\tau })$ , one can determine the desired coefficients in (REF ) using $a_n=T_{\\rm K}^n f_n/p_n$ .",
"In particular, the Fermi-liquid coefficient of present interest is given by $c_{\\varepsilon }=a_2=T_{\\rm K}^2 f_2/p_2$ ."
],
[
"Extraction of Fermi-liquid coefficients",
"Figs.",
"REF (a)-(c) show our NRG data (heavy solid lines) for the resistivity plotted versus $B/T^{\\rm FS}_{\\rm K}$ at zero temperature or plotted versus $T/T^{\\rm sc}_{\\rm K}$ at zero field, and for the weighted spectral function plotted versus $\\tau /T^{\\rm sc}_{\\rm K}$ , respectively.",
"We determined the Fermi-liquid coefficients $c_B$ , $c_T$ and $c_{\\varepsilon }$ from the quadratic terms of fourth-order polynomial fits to these curves.",
"Including the fourth-order term allows the fitting range to be extended towards somewhat larger values of the argument, thus increasing the accuracy of the fit.",
"For each solid curve, the quadratic term from the fit is shown by heavy dashed lines; these are found to agree well with the corresponding predictions from FLT, shown by light lines of matching colors.",
"The level of agreement is quite remarkable, given the rather limited range in which the behavior is purely quadratic: with increasing argument, quartic contributions become increasingly important, as reflected by the growing deviations between dashed and solid lines; and at very small values of the argument ($\\lesssim 0.02$ ), the NRG data become unreliable due to known NRG artefacts.",
"Numerical values for the extracted Fermi-liquid coefficients are given in Table REF ; they agree with those predicted analytically to within $\\le 5\\%$ .",
"This can be considered excellent agreement, especially for the numerically very challenging case of $N=3$ .",
"Table: Numerically extracted values of c B c_B, c T c_T andc ε c_{\\varepsilon }, given here relative to the correspondingpredictions from FLT of Eq. ().",
"The deviationsbetween NRG and FLT values are ≤5%\\le 5\\% in all cases.",
"Tonumerically determine these coefficients, we used the quadraticcoefficient of a fourth-order polynomial fit to the corresponding NRGdata.",
"Error bars were estimated by comparing the quartic fitsto polynomial fits of different higher orders."
],
[
"Conclusions",
"Our two main results can be summarized as follows.",
"First, we have presented a compact derivation of three Fermi-liquid coefficients for the fully-screened $N$ -channel Kondo model, by generalizing well-established calculations for $N=1$ to general $N$ .",
"The corresponding calculations, building on ideas of Nozières, Affleck and Ludwig, and Pustilnik and Glazman, are elementary.",
"We hope that our way of presenting them emphasizes this fact, and perhaps paves the way for similar calculations in less trivial quantum impurity problems that also show Fermi-liquid behavior, such as the asymmetric single-impurity Anderson Hamiltonian, or the 0.7-anomaly in quantum point contacts.",
"[45] Second, we have established a benchmark for the quality of NRG results for the fully screened $N$ -channel Kondo model, by showing that it is possible to numerically calculate equilibrium Fermi-liquid coefficients with an accuracy of better than 5% for $N=1$ , 2 and 3.",
"To achieve numerical results of this quality, two technical ingredients were essential, both of which became available only recently: first, exploiting larger-rank non-Abelian symmetries in the numerics;[11], [12] and second, carefully defining the Kondo temperature[34] in such a way that numerically-calculated universal scaling curves are indeed universal, in the sense of showing a proper scaling collapse, despite the fact that the scaling limit $T_{\\rm K}/D \\rightarrow 0$ is typically not achieved in numerical work."
],
[
"Acknowledgements",
"We acknowledge helpful discussions with K. Kikoin, C. Mora, A. Ludwig and G. Zarànd.",
"We are grateful to D. Schuricht for drawing our attention to Ref.",
"H10, and for sending us a preprint of Ref.",
"Horig2014 prior to its submission.",
"The latter work, which we received in the final stages of this work, also uses $H_\\lambda $ of Eq.",
"(REF ) as starting point for calculating Fermi-liquid coefficients for the $N$ -channel Kondo model, and its result for $c_{T}$ is consistent with our own.",
"We gratefully acknowledge financial support from the DFG (WE4819/1-1 for A.W., and SFB-TR12, SFB-631 and the Cluster of Excellence Nanosystems Initiative Munich vor J.v.D., M.H., and A.W.)",
"*"
],
[
"This appendix offers a pedagogical derivation of the Hamiltonian $H_\\lambda $ given in Eq.",
"(REF ) of the main text using the point-splitting regularization strategy, following AL (Appendix D of AL93).",
"Its main purpose is to show how the relation $\\alpha = 3 \\psi \\nu = 1/T_{\\rm K}$ between Fermi-liquid parameters that NB had found by intuitive arguments[15] follows simply and naturally from point splitting.",
"For a detailed discussion of the point-splitting strategy, see Refs. Affleck1986,L92,vonDelft1995.",
"According to AL, the leading irrelevant operator for the fully screened $N$ -channel Kondo model has the form $H_{\\lambda }=-\\lambda : \\!",
"\\vec{J}(0)\\cdot \\vec{J}(0) \\!",
": .$ Here $\\vec{J}(x) =\\sum _{m=1}^{N} : \\!",
"\\vec{J}_m (x) \\!",
":$ is the total (point-split) spin density from all channels at position $x$ (the impurity or dot sits at $x= 0$ ), and $\\vec{J}_m(x)=\\frac{1}{2}\\sum _{\\sigma \\sigma ^{\\prime }}\\Psi ^{\\dagger }_{m\\sigma }(x){\\vec{\\tau }}_{\\sigma \\sigma ^{\\prime }} \\Psi _{m\\sigma ^{\\prime }}(x)$ is the corresponding (non-point-split) spin density for channel $m$ .",
"Here $:...:$ denotes point splitting, $:\\!",
"A(x) B(x)\\!",
": \\equiv \\lim _{\\eta \\rightarrow 0}\\Bigl [ A(x+\\eta ) B(x) - [0.6ex]{}{A}{(x+\\eta )}{B}A(x+\\eta )B(x)\\Bigr ] , \\qquad \\phantom{.",
"}$ a field-theoretic scheme for regularizing products of operators at the same point by subtracting their ground state expecation value, $[0.6ex]{}{A}{}{B}AB =\\langle A B\\rangle $ .",
"(In most cases, point splitting is equivalent to normal ordering.)",
"For present purposes, we follow AL[26] and take $\\Psi _{m\\sigma }(x) = \\frac{1}{\\sqrt{L}} \\sum _{k} e^{- i k x} \\psi _{km\\sigma }$ to be free fermion fields with linear dispersion ($\\xi _k = k \\hbar v_{\\rm F}$ ) in a box of length $L\\rightarrow \\infty $ (with $k\\in 2 \\pi n / L$ , $n \\in \\mathbb {Z}$ ), with normalization $\\lbrace \\psi _{km\\sigma }, \\psi ^\\dagger _{k^{\\prime }m^{\\prime }\\sigma ^{\\prime }}\\rbrace =\\delta _{kk^{\\prime }}\\delta _{mm^{\\prime }}\\delta _{\\sigma \\sigma ^{\\prime }}$ and free ground state correlators $\\langle \\Psi ^\\dagger _{m\\sigma }(x)\\Psi _{m^{\\prime }\\sigma ^{\\prime }}(0)\\rangle =\\langle \\Psi _{m\\sigma }(x)\\Psi ^\\dagger _{m^{\\prime }\\sigma ^{\\prime }}(0)\\rangle =\\frac{\\delta _{mm^{\\prime }}\\delta _{\\sigma \\sigma ^{\\prime }}}{2 \\pi i x} \\; .$ Note that we follow PG in our choice of field normalization, which differs from that used by AL[26] by $\\Psi _{\\rm here} = \\psi _{\\rm AL} /\\sqrt{2 \\pi }$ .",
"Consequently, our coupling constant is related to theirs by $\\lambda _{\\rm here} =(2 \\pi )^2 \\lambda _{\\rm AL}$ .",
"In the definition of $H_\\lambda $ , point splitting is needed because the product of two spin densities, $\\vec{J}(x+\\eta )\\cdot \\vec{J}(x)$ , diverges with decreasing seperation $\\eta $ between their arguments.",
"To make this explicit, we use Wick's theorem, $: \\!",
"\\!",
"AB \\!",
": : \\!",
"CD \\!",
": \\, = \\,: \\!\\!",
"AB CD \\!",
": \\!",
"+ \\!",
": \\!",
"\\!",
"[0.6ex]{A}{B}{}{C} AB CD \\!",
": \\!",
"+ \\!",
": \\!",
"\\!",
"[0.6ex]{}{A}{BC}{D} AB CD \\!",
": \\!",
"+ \\!",
": \\!",
"\\!",
"[0.6ex]{A}{B}{}{C} [1ex]{}{A}{BC}{D} AB CD \\!",
": ,\\nonumber $ to rewrite the product of spin densities as follows: $\\vec{J}(x+\\eta )\\cdot \\vec{J}(x) & = &\\frac{1}{4}\\sum _{m\\sigma \\sigma ^{\\prime }}\\sum _{m^{\\prime } \\bar{\\sigma }\\bar{\\sigma }^{\\prime }}:\\!\\Psi ^{\\dagger }_{m\\sigma }(x+\\eta ){\\vec{\\tau }}_{\\sigma \\sigma ^{\\prime }}\\Psi _{m\\sigma ^{\\prime }}(x+\\eta )\\!",
": \\, : \\!\\Psi ^{\\dagger }_{m^{\\prime }\\bar{\\sigma }}(x){\\vec{\\tau }}_{\\bar{\\sigma }\\bar{\\sigma }^{\\prime }}\\Psi _{m^{\\prime }\\bar{\\sigma }^{\\prime }}(x)\\!",
":\\\\ \\nonumber & = & \\frac{1}{4}\\sum _{m\\sigma \\sigma ^{\\prime }}\\sum _{m^{\\prime } \\bar{\\sigma }\\bar{\\sigma }^{\\prime }}{\\vec{\\tau }}_{\\sigma \\sigma ^{\\prime }}\\cdot {\\vec{\\tau }}_{\\bar{\\sigma }\\bar{\\sigma }^{\\prime }}\\Biggl [:\\!",
"\\Psi ^\\dagger _{m\\sigma }(x+\\eta )\\Psi _{m\\sigma ^{\\prime }}(x+\\eta )\\Psi ^\\dagger _{m^{\\prime }\\bar{\\sigma }}(x)\\Psi _{m^{\\prime }\\bar{\\sigma }^{\\prime }}(x) \\!",
": \\Biggr .\\\\& & \\Biggl .+\\frac{\\delta _{mm^{\\prime }}}{2 \\pi i \\eta }\\Bigl (\\delta _{\\sigma ^{\\prime }\\bar{\\sigma }}:\\!",
"\\Psi ^\\dagger _{m\\sigma }(x+\\eta )\\Psi _{m\\bar{\\sigma }^{\\prime }}(x)\\!",
":{} + {} \\delta _{\\sigma \\bar{\\sigma }^{\\prime }}:\\!",
"\\Psi _{m\\sigma ^{\\prime }}(x+\\eta )\\Psi ^\\dagger _{m\\bar{\\sigma }}(x)\\!",
":\\Bigr )+ \\frac{\\delta _{\\sigma \\bar{\\sigma }^{\\prime }}\\delta _{\\sigma ^{\\prime }\\bar{\\sigma }}\\delta _{mm^{\\prime }}}{(2 \\pi i \\eta )^2} \\Biggr ] \\; .",
"\\qquad \\phantom{.",
"}$ The point-splitting prescription in Eq.",
"(REF ) subtracts off the $1/\\eta ^2$ divergence of the last term of Eq. ().",
"The contributions of the second and first terms to $H_\\lambda $ can be organized as $H_{\\lambda } = H_1+ H_{\\rm int}$ , describing single-particle elastic scattering and two-particle interactions, respectively.",
"Taking $x=0$ and $\\eta \\rightarrow 0$ , we find: $H_1& = &- \\frac{\\lambda }{8 \\pi i } \\lim _{\\eta \\rightarrow 0} \\sum _{m \\sigma \\sigma ^{\\prime }}: \\!",
"\\frac{1}{\\eta } \\Bigl [ \\Psi ^\\dagger _{m\\sigma }(\\eta ) \\vec{\\tau }^2_{\\sigma \\sigma ^{\\prime }}\\Psi _{m\\sigma ^{\\prime }}(0)- \\!",
"\\Psi ^\\dagger _{m\\sigma ^{\\prime }}(0) \\vec{\\tau }^2_{\\sigma ^{\\prime } \\sigma }\\Psi _{m\\sigma }(\\eta ) \\Bigr ] \\!",
": \\\\& = & - \\frac{3 \\lambda }{8 \\pi i} \\lim _{\\eta \\rightarrow 0} \\sum _{m \\sigma }: \\!",
"\\left[ \\frac{1}{\\eta } \\Bigl (\\Psi ^\\dagger _{m\\sigma }(\\eta ) -\\Psi ^\\dagger _{m\\sigma }(0) \\Bigr ) \\Psi _{m\\sigma }(0)- \\Psi ^\\dagger _{m \\sigma }(0)\\frac{1}{\\eta } \\Bigl ( \\Psi _{m\\sigma }(\\eta ) -\\Psi _{m\\sigma }(0) \\Bigr ) \\right] \\!",
": \\; \\\\& = &- \\frac{ 3\\lambda }{8 \\pi i } \\sum _{m\\sigma }: \\!",
"\\Bigl [\\bigl ( \\partial _x \\Psi ^{\\dagger }_{m\\sigma } \\bigr ) (0) \\Psi _{m\\sigma } (0)- \\Psi ^{\\dagger }_{m\\sigma } (0) \\bigl (\\partial _x \\Psi _{m\\sigma })(0) \\Bigr ] \\!",
":,$ $H_{\\rm int}=-\\lambda \\sum _{m m^{\\prime }}: \\!",
"\\vec{J}_m (0)\\cdot \\vec{J}_{m^{\\prime }} (0) \\!",
": .$ To obtain Eq.",
"(), we used $\\vec{\\tau }^2_{\\sigma \\sigma ^{\\prime }}=3\\delta _{\\sigma \\sigma ^{\\prime }}$ and subtracted and added :$\\Psi ^\\dagger _{m\\sigma }(0) \\Psi _{m\\sigma }(0)$ : inside the square brackets.",
"Now pass to the momentum representation, using Eq.",
"(REF ) and the shorthand notations (following PG[29]) $\\rho _{m\\sigma } (0) & = \\frac{1}{L} \\rho _{m\\sigma }, & \\rho _{m\\sigma }& = \\sum _{kk^{\\prime }} \\psi ^\\dagger _{km\\sigma } \\psi _{k^{\\prime }m\\sigma } \\; ,\\\\\\vec{J}_m (0) & = \\frac{1}{L} \\vec{j}_m , & \\vec{j}_m & = \\frac{1}{2}\\sum _{kk^{\\prime }\\sigma \\sigma ^{\\prime }} \\psi ^\\dagger _{km\\sigma } \\vec{\\tau }_{\\sigma \\sigma ^{\\prime }} \\psi _{k^{\\prime }m\\sigma ^{\\prime }} \\; ,$ for the conduction electron channel-$m$ charge and spin densities at the impurity.",
"This gives $H_1& = & - \\frac{\\alpha _1}{2 \\pi \\nu }\\sum _{m\\sigma kk^{\\prime }} (\\xi _k + \\xi _{k^{\\prime }}): \\!",
"\\psi ^\\dagger _{km\\sigma } \\psi _{k^{\\prime }m\\sigma } \\!",
": \\; , \\quad \\phantom{.",
"}\\\\H_{\\rm int}& = & - \\frac{2\\phi _1}{3 \\pi \\nu ^2}\\sum _{mm^{\\prime }} : \\!",
"\\vec{j}_m \\cdot \\vec{j}_{m^{\\prime }} \\!",
": \\; .$ Here $\\nu = L/(2 \\pi \\hbar v_{\\rm F})$ is the extensive 1D density of states per spin and channel, and the prefactors were expressed in terms of the constants $\\alpha _1 = \\phi _1 = \\frac{3 \\lambda }{8 \\pi (\\hbar v_{\\rm F})^2} = \\frac{1}{T_{\\rm K}}\\; .$ (This notation is consistent with that of Ref.",
"Horig2014, where $H_\\lambda $ served starting point for calculating Fermi-liquid corrections, too.)",
"Checking dimensions, with $[H_\\lambda ]$ =$\\mathcal {E}$ and $[\\Psi _{m\\sigma }]$ =$1/\\sqrt{\\mathcal {L}}$ ($\\mathcal {E}$ stands for energy, $\\mathcal {L}$ for length), we see that $[\\lambda ]$ =$\\mathcal {EL}^2$ .",
"Since $[\\nu ]$ =$1/\\mathcal {E}$ , $[\\hbar v_{\\rm F}]$ =$\\mathcal {EL}$ , we have $[\\alpha _1]= [\\phi _1] = 1/\\mathcal {E}$ , thus, $\\alpha _1$ and $\\phi _1$ have dimensions of inverse energy.",
"In the main text, they are identified with $1/T_{\\rm K}$ ; in fact, the numerical prefactor in Eq.",
"(REF ) is purposefully chosen such that the leading term in the expansion (REF ) of the phase shift $\\tilde{\\delta }_{m\\sigma }(\\varepsilon )$ turns out to take the form $\\varepsilon /T_{\\rm K}$ .",
"To elucidate how the case $N>1$ differs from $N = 1$ , we write $H_{\\rm int}= H_2+ H_3$ in the main text, with $H_2$ and $H_3$ given in Eqs.",
"() and (), respectively, where $H_3$ occurs only for $N>1$ ."
]
] | 1403.0497 |
[
[
"Normal forms for CR singular codimension two Levi-flat submanifolds"
],
[
"Abstract Real-analytic Levi-flat codimension two CR singular submanifolds are a natural generalization to ${\\mathbb{C}}^m$, $m > 2$, of Bishop surfaces in ${\\mathbb{C}}^2$.",
"Such submanifolds for example arise as zero sets of mixed-holomorphic equations with one variable antiholomorphic.",
"We classify the codimension two Levi-flat CR singular quadrics, and we notice that new types of submanifolds arise in dimension 3 or greater.",
"In fact, the nondegenerate submanifolds, i.e.",
"higher order purturbations of $z_m=\\bar{z}_1z_2+\\bar{z}_1^2$, have no analogue in dimension 2.",
"We prove that the Levi-foliation extends through the singularity in the real-analytic nondegenerate case.",
"Furthermore, we prove that the quadric is a (convergent) normal form for a natural large class of such submanifolds, and we compute its automorphism group.",
"In general, we find a formal normal form in ${\\mathbb{C}}^3$ in the nondegenerate case that shows infinitely many formal invariants."
],
[
"Introduction",
"Let $M \\subset {n+1}$ be a real submanifold.",
"A fundamental question in CR geometry is to classify $M$ at a point up to local biholomorphic transformations.",
"One approach is to find a normal form for $M$ .",
"A real-analytic hypersurface $M \\subset {n+1}$ is Levi-flat if the Levi-form vanishes identically.",
"Roughly speaking, a Levi-flat submanifold is a family of complex submanifolds.",
"Intuitively, a Levi-flat submanifold is as close to a complex submanifold as possible.",
"In the real-analytic smooth hypersurface case, it is well-known that $M$ can locally be transformed into the real hyperplane given by $\\operatorname{Im}z_1 = 0 .$ We therefore focus on higher codimension case, in particular on codimension 2.",
"A codimension 2 submanifold is again given by a single equation, but in this case a complex valued equation.",
"A new phenomenon that appears in codimension 2 is that $M$ may no longer be a CR submanifold.",
"Let $T_p^cM \\subset T_pM$ be the largest subspace with $J T_p^c M = T_p^c M$ , where $J$ is the complex structure on ${n+1}$ .",
"A submanifold is CR if $\\dim T_p^cM$ is constant.",
"Real submanifolds of dimension $n+1$ in ${n+1}$ with a non-degenerate complex tangent point has been studied extensively after the fundamental work of E. Bishop .",
"In 2, Bishop studied the submanifolds $w = z\\bar{z} + \\gamma (z^2+\\bar{z}^2) + O(3)$ where $\\gamma \\in [0,\\infty ]$ is called the Bishop invariant, with $\\gamma = \\infty $ interpreted as $w = z^2+ \\bar{z}^2 + O(3)$ .",
"One of Bishop's motivations was to study the hull of holomorphy of the real submanifolds by attaching analytic discs.",
"Bishop's work on the family of attached analytic discs has been refined by Kenig-Webster KenigWebster:82, KenigWebster:84, Huang-Krantz , and Huang .",
"The normal form theory for real submanifolds for Bishop surfaces or submanifolds was established by Moser-Webster ; see also Moser , Gong Gong04, Gong94:duke, Gong94:helv, Huang-Yin , and Coffman.",
"We would like to mention that the Moser-Webster normal form does not deal with the case of vanishing Bishop invariant.",
"The formal normal form and its application to holomorphic classification for surfaces with vanishing Bishop invariant was achieved by Huang-Yin by a completely different method.",
"Real submanifolds with complex tangents have been studied in other situations.",
"See for example , where CR singular submanifolds that are images of CR manifolds were studied.",
"Normal forms for the quadratic part of general codimension two CR singular submanifolds in 3 was completely solved by Coffman .",
"Huang and Yin studied the normal form for codimension two CR singular submanifolds of the form $w=\\left|{z} \\right|^2 + O(3)$ .",
"Dolbeault-Tomassini-Zaitsev DTZ, DTZ2 and Huang-Yin studied CR singular submanifolds of codimension two that are boundaries of Levi-flat hypersurfaces.",
"Burcea constructed the formal normal form for codimension 2 CR singular submanifolds approximating a sphere.",
"Coffman found an algebraic normal form for nondegenerate CR singular manifolds in high codimension and one dimensional complex tangent.",
"To motivate our work, we observe that in Bishop's work, the real submanifolds are Levi-flat away from their CR singular sets.",
"Our purpose is to understand such submanifolds in higher dimensional case with codimension being exactly two.",
"Notice that the latter is the smallest codimension for CR singularity to be present in (smooth) submanifolds.",
"Regarding CR singular Levi-flat real codimension 2 submanifolds on ${n+1}$ as a natural generalization of Bishop surfaces to ${n+1}$ , we wish to find their normal forms.",
"For singular Levi-flat hypersurfaces and related work on foliations with singularity, see Bedford:flat,BG:lf,CerveauLinsNeto,Lebl:lfsing,FernandezPerez:gensing,Brunella:lf.",
"Our techniques revolve around the study of the Levi-map (the generalization of the Levi-form to higher codimension submanifolds) of codimension 2 submanifolds.",
"Extending the CR structure through the singular point via Nash blowup and then extending the Levi-map to this blowup has been studied previously by Garrity .",
"A CR submanifold is Levi-flat if the Levi-map vanishes identically.",
"Locally, all CR real-analytic Levi-flat submanifolds of real codimension 2 can be, after holomorphic change of coordinates, written as $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 .$ If a submanifold $M$ is CR singular, denote by $M_{CR}$ the set of points where $M$ is CR.",
"We say $M$ is Levi-flat if $M_{CR}$ is Levi-flat in the usual sense.",
"A Levi-flat CR singular submanifold has no local biholomorphic invariants at the CR points, just as in the case of Bishop surfaces.",
"A real, real-analytic codimension 2 submanifold that is CR singular at the origin can be written in coordinates $(z,w) \\in {n} \\times {n+1}$ as $w = \\rho (z,\\bar{z})$ for $\\rho $ that is $O(2)$ .",
"We will be concerned with submanifolds where the quadratic part in $\\rho $ is nonzero in any holomorphic coordinates.",
"We say that such submanifolds have a nondegenerate complex tangent.",
"For example, the Bishop surfaces in 2 are precisely the CR singular submanifolds with nondegenerate complex tangent.",
"First, let us classify the quadratic parts of CR singular Levi-flats, and in the process completely classify the CR singular Levi-flat quadrics, that is those where $\\rho $ is a quadratic.",
"Theorem 1.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a germ of a real-analytic real codimension 2 submanifold, CR singular at the origin, written in coordinates $(z,w) \\in {n} \\times as\\begin{equation} w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$, where $ A+B 0$ (nondegenerate complextangent).",
"Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).\\begin{enumerate}[(i)]\\item If M is a quadric, then Mis locally biholomorphically equivalent to one and exactly one of thefollowing:\\begin{equation} \\begin{aligned}\\text{(A.1)} \\quad & w = \\bar{z}_1^2 , \\\\\\text{(A.2)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2, \\\\& \\vdots \\\\\\text{(A.$n$)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2 + \\dots + \\bar{z}_{n}^2 , \\\\[10pt]\\text{(B.$\\gamma $)} \\quad & w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 , ~~ \\gamma \\ge 0, \\\\[10pt]\\text{(C.0)} \\quad & w = \\bar{z}_1z_2 , \\\\\\text{(C.1)} \\quad & w = \\bar{z}_1z_2 + \\bar{z}_1^2 .\\end{aligned}\\end{equation}\\item If M is real-analytic, then the quadricw = A(z,\\bar{z}) + B(z,\\bar{z})is Levi-flat, and can be put via a biholomorphic transformation intoexactly one of the forms (\\ref {eq:quadnormalforms}).\\end{enumerate}$ By part (), the quadratic part in () is an invariant of $M$ at a point.",
"We say the type of $M$ at the origin is A.x, B.$\\gamma $ , or C.x depending on the type of the quadratic form.",
"Following Bishop, we call types B.$\\gamma $ and A.1 Bishop-like, we could think of $\\gamma =\\infty $ as A.1.",
"By type being stable we mean that the type does not change at all complex tangents in a neighborhood of the origin under any small (or higher order) perturbations that stay within the class of Levi-flat CR singular submanifolds.",
"As a consequence of the above theorem and because rank is lower semicontinuous, we get that the only types that are stable are A.$n$ and C.1, although A.$n$ are degenerate because the form $A(z,\\bar{z})$ is identically zero.",
"See also Proposition REF .",
"The quadrics A.$k$ for $k \\ge 2$ do not possess a nonsingular foliation extending the Levi-foliation of $M_{CR}$ through the origin.",
"In fact, there is a singular complex subvariety of dimension 1 through the origin contained in $M$ .",
"See § .",
"In the sequel, when we wish to refer to the quadric of certain type we will use the notation $M_{C.1}$ to denote the quadric of type C.1.",
"The quadratic form $A(z,\\bar{z})$ carries the “Levi-map” of the submanifold.",
"Type C.1 is the unique quadric that is stable and has non-zero $A$ .",
"Having non-zero $A$ is also stable in a neighborhood of the origin under any small (or higher order) perturbations.",
"Therefore, we say a type is non-degenerate if it is C.1 and we focus mostly on such submanifolds.",
"First, we show that submanifolds of type C.x possess a nonsingular real-analytic foliation that extends the Levi-foliation, due to the form $A(z,\\bar{z})$ : Theorem 1.2 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a real-analytic Levi-flat CR singular submanifold of type C.1 or C.0, that is, $M$ is given by $w = \\bar{z}_1z_2 + \\bar{z}_1^2 + O(3)\\qquad \\text{or}\\qquad w = \\bar{z}_1z_2 + O(3).$ Then there exists a nonsingular real-analytic foliation defined on $M$ that extends the Levi-foliation on $M_{CR}$ , and consequently, there exists a CR real-analytic mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1} \\rightarrow {n+1}$ such that $F$ is a diffeomorphism onto $F(U) = M \\cap U^{\\prime }$ , for some neighbourhood $U^{\\prime }$ of 0.",
"Here the CR structure on ${\\mathbb {R}}^2\\times {n-1}$ is induced from $2\\times {n-1}$ .",
"As a corollary of this theorem we obtain in § using the results of that the CR singular set of any type C.1 submanifold is a Levi-flat submanifold of dimension $2n-2$ and CR dimension $n-2$ .",
"The Levi-foliation on a type C.x submanifold cannot extend to a whole neighbourhood of $M$ as a nonsingular holomorphic foliation.",
"If it did, we could flatten the foliation and $M$ would be a Cartesian product, in particular Bishop-like.",
"Thus, the study of normal form theory for the special case when the foliation extends to a neighbourhood is reduced to the case of Bishop surfaces, which have been studied extensively.",
"A codimension 2 submanifold in $\\mathbb {C}^m$ can arise from $ f(\\bar{z}^{\\prime },z^{\\prime \\prime })=0$ for a suitable holomorphic function $f$ in $m$ variables.",
"The zero set admits two holomorphic foliations.",
"We are interested in the case where one of foliations has leaves of maximum dimension $m-2$ , while the other has leaves of minimum dimension 0.",
"Therefore, we will assume that $z^{\\prime }=z_1$ and $z^{\\prime \\prime }=(z_2,\\ldots , z_m)$ .",
"Functions holomorphic in some variables and anti-holomorphic in other variables, such as (REF ), are often called mixed-holomorphic or mixed-analytic, and come up often in complex geometry, the simplest example being the standard inner product.",
"An interesting feature of the mixed-holomorphic setting is that the equation can be complexified into $m$ , so the sets share some of the properties of complex varieties.",
"However, they have a different automorphism group if we wish to classify them under biholomorphic transformations.",
"Such mixed-analytic sets are automatically real codimension 2, are Levi-flat or complex, and may have CR singularities.",
"We study their normal form in § .",
"See also Theorem REF below.",
"When a type C.1 CR singular submanifold has a defining equation that does not depend on $\\bar{z}_2, \\ldots , \\bar{z}_n$ we prove that it is automatically Levi-flat, and it is equivalent to $M_{C.1}$ .",
"Theorem 1.3 Let $M \\subset {n+1}$ , $n \\ge 2$ , be a real-analytic submanifold given by $ w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(3)$ .",
"Then $M$ is Levi-flat and at the origin $M$ is locally biholomorphically equivalent to the quadric $M_{C.1}$ submanifold $ w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ The theorem is also true formally; given a formal submanifold of the form (REF ), it is formally equivalent to $M_{C.1}$ .",
"A key idea in the proof of the convergence of the normalizing transformation is that the form $B(\\bar{z},\\bar{z}) = \\bar{z}_1^2$ induces a natural mixed-holomorphic involution on quadric $M_{C.1}$ .",
"This involution also plays a key role in computing the automorphism group of the quadric in Theorem REF .",
"Finally, we also compute the automorphism group for the quadric $M_{C.1}$ , see Theorem REF .",
"In particular we show that the automorphism group is infinite dimensional.",
"Not every type C.1 Levi-flat submanifold is biholomorphically equivalent to the C.1 quadric.",
"We will find a formal normal form for type C.1 Levi-flat submanifolds in 3 that shows infinitely many formal invariants.",
"Let us give a simplified statement.",
"For details see Theorem REF .",
"Theorem 1.4 Let $M$ be a real-analytic Levi-flat type C.1 submanifold in 3.",
"There exists a formal biholomorphic map transforming $M$ into the image of $\\hat{\\varphi }(z,\\bar{z},\\xi )=\\bigl (z+A(z,\\xi , w)w\\eta , \\xi ,w\\bigr )$ with $\\eta =\\bar{z}+\\frac{1}{2}{\\xi }$ and $w=\\bar{z}\\xi +\\bar{z}^2$ .",
"Here $A=0$ , or $A$ satisfies certain normalizing conditions.",
"When $A \\ne 0$ the formal automorphism group preserving the normal form is finite or 1 dimensional.",
"We do not know if the formal normal form above can be achieved by convergent transformations, even if $A=0$ ."
],
[
"Invariants of codimension 2 CR singular submanifolds",
"Before we impose the Levi-flat condition, let us find some invariants of codimension two CR singular submanifolds in ${n+1}$ with CR singularity at 0.",
"Such a submanifold can locally near the origin be put into the form $ w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),$ where $(z,w) \\in {n} \\times and $ A$ and $ B$ are quadratic forms.We think of $ A$ and $ B$ as matrices and $ z$ as a column vector andwrite the forms as$ z*Az$ and $ z* Bz$ respectively.The matrix $ B$ is not unique.Hence we make $ B$ symmetric to makethe choice of the matrix $ B$ canonical.The following proposition is not difficult andwell-known.",
"Since the details are important and will be used later,let us prove this fact.$ Proposition 2.1 A biholomorphic transformation of (REF ) taking the origin to itself and preserving the form of (REF ) takes the matrices $(A,B)$ to $(\\lambda T^* A T, \\lambda T^* B \\overline{T} ) ,$ for $T \\in GL_n($ and $\\lambda \\in *$ .",
"If $(F_1,\\ldots ,F_n,G) = (F,G)$ is the transformation then the linear part of $G$ is $\\lambda ^{-1} w$ and the linear part of $F$ restricted to $z$ is $Tz$ .",
"Let us emphasize that $A$ is an arbitrary complex matrix and $B$ is a symmetric, but not necessarily Hermitian, matrix.",
"Let $(F_1,\\ldots ,F_n,G) = (F,G)$ be a change of coordinates taking $w = \\widetilde{A}(z,\\bar{z}) + \\widetilde{B}(\\bar{z},\\bar{z}) + O(3) =\\rho (z,\\bar{z})$ to $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3) .$ Then $ G\\bigl (z,\\rho (z,\\bar{z})\\bigr ) =A\\Bigl (F\\bigl (z,\\rho (z,\\bar{z})\\bigr ),\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr )\\Bigr )\\\\+B\\Bigl (\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr ),\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr )\\Bigr ) + O(3)$ is true for all $z$ .",
"The right hand side has no linear terms, so the linear terms in $G$ do not depend on $z$ .",
"That is, $G = \\lambda ^{-1} w + O(2)$ , where $\\lambda $ is a nonzero scalar and the negative power is for convenience.",
"Let $T = [ T_1, T_2 ]$ denote the matrix representing the linear terms of $F$ .",
"Here $T_{1}$ is an $n\\times n$ matrix and $T_{2}$ is $n \\times 1$ .",
"Since the linear terms in $G$ do not depend on any $z_j$ , $T_1$ is nonsingular.",
"Then the quadratic terms in (REF ) are $\\lambda ^{-1} \\bigl (\\widetilde{A}(z,\\bar{z}) + \\widetilde{B}(\\bar{z},\\bar{z}) \\bigr )=z^* T_{1}^* A T_{1} z +z^* T_1^* B \\overline{T}_{1} \\bar{z} .$ In other words as matrices, $\\widetilde{A} = \\lambda T_{1}^* A T_{1} \\qquad \\text{and} \\qquad \\widetilde{B} = \\lambda T_1^* B \\overline{T}_1 .",
"\\text{}$ We will need to at times reduce to the 3-dimensional case, and so we need the following lemma.",
"Lemma 2.2 Let $M \\subset {n+1}$ , $n \\ge 3$ , be a real-analytic Levi-flat CR singular submanifold of the form $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3) ,$ where $A$ and $B$ are quadratic.",
"Let $L$ be a nonsingular $(n-2) \\times n$ matrix $L$ .",
"If $A+B$ is not zero on the set $\\lbrace L z = 0 \\rbrace $ , then the submanifold $M_L = M \\cap \\lbrace L z = 0 \\rbrace $ is a Levi-flat CR singular submanifold.",
"Clearly if $M_L$ is not contained in the CR singularity of $M$ , then $M_L$ is a Levi-flat CR singular submanifold.",
"$M_{L^{\\prime }}$ is not contained in the CR singularity of $M$ for a dense open subset of $(n-2) \\times n$ matrices $L^{\\prime }$ .",
"If $M_L$ is a subset of the CR singularity of $M$ , pick a CR point $p$ of $M_L$ then pick a sequence $L_n$ approaching $L$ such that $M_{L_n}$ are not contained in the CR singularity of $M$ .",
"As $A+B$ is not zero on the set $\\lbrace L z = 0 \\rbrace $ , then $M_L$ is not a complex submanifold, and therefore a CR singular submanifold.",
"Then as the Levi-form of $M_{L_n}$ vanishes at all CR points of $M_{L_n}$ , the Levi-form of $M_L$ vanishes at $p$ , so $M_L$ is Levi-flat."
],
[
"Levi-flat quadrics",
"Let us first focus on Levi-flat quadrics.",
"We will prove later that the quadratic part of a Levi-flat submanifold is Levi-flat.",
"Let $M$ be defined in $(z,w) \\in {n} \\times by\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) .\\end{equation}Being Levi-flat hasseveral equivalent formulations.",
"The main idea is that the $ T(1,0) M T(0,1) M$ vector fields are completely integrable at CR points and we obtain a foliationof $ M$ at CR points by complex submanifolds of complex dimension$ n-1$.",
"An equivalent notion is that the Levi-map is identically zero, see\\cite {BER:book}.",
"The Levi-map for a CR submanifold defined by two real equations$ 1 = 2 = 0$ (for $ 1$ and $ 2$ with linearly independent differentials)is the pair of Hermitian forms\\begin{equation}i \\partial \\bar{\\partial } \\rho _1\\quad \\text{and} \\quad i \\partial \\bar{\\partial } \\rho _2 ,\\end{equation}applied to $ T(1,0) M$ vectors.The full quadratic forms$ i 1$ and $ i 2$of course depend on the defining equations themselves and aretherefore extrinsic information.",
"It is important to notethat for the Levi-map we restrict it to$ T(1,0) M$ vectors.We can define these two forms$ i 1$ and $ i 2$even at a CR singular point $ p M$.$ These forms are the complex Hessian matrices of the defining equations.",
"For our quadric $M$ they are the real and imaginary parts of the $(n+1) \\times (n+1)$ complex matrix $\\widetilde{A} =\\begin{bmatrix}A & 0 \\\\0 & 0\\end{bmatrix} ,$ where the variables are ordered as $(z_1,\\ldots ,z_n,w)$ .",
"For $M$ to be Levi-flat, the quadratic form defined by $\\widetilde{A}$ has to be zero when restricted to the $n-1$ dimensional space spanned by $T^{(1,0)}_p M$ for every $p \\in M_{CR}$ .",
"In other words for every $p \\in M_{CR}$ $v^* \\widetilde{A} v = 0, \\qquad \\text{for all $v \\in T^{(1,0)}_pM$}.$ The space $T^{(1,0)}_pM$ is of dimension $n-1$ , and furthermore, the vector $\\frac{\\partial }{\\partial w}$ is not in $T^{(1,0)}_pM$ .",
"Therefore, $z^* A z = 0$ for $z \\in n$ in a subspace of dimension $n-1$ .",
"Before we proceed let us note the following general fact about CR singular Levi-flat submanifolds.",
"Lemma 3.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a Levi-flat connected real-analytic real codimension 2 submanifold, CR singular at the origin.",
"Then there exists a germ of a complex analytic variety of complex dimension $n-1$ through the origin, contained in $M$ .",
"Through each point of $M_{CR}$ there exists a germ of a complex variety of complex dimension $n-1$ contained in $M$ .",
"The set of CR points is dense in $M$ .",
"Take a sequence $p_k$ of CR points converging to the origin and take complex varieties of dimension $n-1$ , $W_k \\subset M$ with $p_k \\in W_k$ .",
"A theorem of Fornæss (see Theorem 6.23 in for a proof using the methods of Diederich and Fornæss ) implies that there exists a variety through $W \\subset M$ with $0 \\in W$ and of complex dimension at least $n-1$ .",
"Let us first concentrate on $n=2$ .",
"When $n=2$ , $T^{(1,0)} M$ is one dimensional at CR points.",
"Write $A =\\begin{bmatrix}a_{11} & a_{12} \\\\a_{21} & a_{22}\\end{bmatrix},\\qquad B =\\begin{bmatrix}b_{11} & b_{12} \\\\b_{12} & b_{22}\\end{bmatrix}.$ Note that $B$ is symmetric.",
"A short computation shows that the vector field can be written as $\\alpha \\frac{\\partial }{\\partial w} +\\beta _1 \\frac{\\partial }{\\partial z_1} +\\beta _2 \\frac{\\partial }{\\partial z_2}=\\alpha \\frac{\\partial }{\\partial w} +\\beta \\cdot \\frac{\\partial }{\\partial z} ,$ where $\\begin{aligned}& \\beta _1 = \\bar{a}_{21}\\bar{z}_1 + \\bar{a}_{22}\\bar{z}_2+ 2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2 , \\\\& \\beta _2 = - \\bar{a}_{11}\\bar{z}_1 - \\bar{a}_{12}\\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2 , \\\\& \\alpha = a_{11} \\bar{z}_1 \\beta _1 + a_{21} \\bar{z}_2 \\beta _1+ a_{12} \\bar{z}_1 \\beta _2 + a_{22} \\bar{z}_2 \\beta _2 .\\end{aligned}$ Note that since the CR singular set is defined by $\\beta _1=\\beta _2=0$ , then $M_{CR}$ is dense in $M$ .",
"Thus we need to check that $\\begin{bmatrix}\\beta ^* & \\bar{\\alpha }\\end{bmatrix}\\begin{bmatrix}A & 0 \\\\0 & 0\\end{bmatrix}\\begin{bmatrix}\\beta \\\\\\alpha \\end{bmatrix}=\\beta ^* A \\beta $ is identically zero for $M$ to be Levi-flat.",
"If $A$ is the zero matrix, then $M$ is automatically Levi-flat.",
"We diagonalize $B$ via $T$ into a diagonal matrix with ones and zeros on the diagonal.",
"We obtain (recall $n=2$ ) the submanifolds: $\\begin{aligned}& w = \\bar{z}_1^2 , \\qquad \\qquad \\text{or}\\\\& w = \\bar{z}_1^2 + \\bar{z}_2^2 .\\end{aligned}$ The first submanifold is of the form $M \\times where $ M 2$ is a Bishopsurface.$ Let us from now on suppose that $A\\ne 0$ .",
"As $M$ is Levi-flat, then through each CR point $p = (z_p,w_p) \\in M_{CR}$ we have a complex submanifold of dimension 1 in $M$ .",
"It is well-known that this submanifold is contained in the Segre variety (see also § ) $w = A(z,\\bar{z}_p) + B(\\bar{z}_p,\\bar{z}_p), \\qquad \\bar{w}_p = \\overline{A}(\\bar{z}_p,z) + \\overline{B}(z,z) .$ By Lemma REF we obtain a complex variety $V \\subset M$ of dimension one through the origin.",
"Suppose without loss of generality that $V$ is irreducible.",
"$V$ has to be contained in the Segre variety at the origin, in particular $w=0$ on $V$ .",
"Therefore, to simplify notation, let us consider $V$ to be subvariety of $\\lbrace w = 0 \\rbrace $ .",
"Denote by $\\overline{V}$ the complex conjugate of $V$ .",
"Then as $V$ is irreducible, then $V \\times \\overline{V}$ is also irreducible (the smooth part of $V$ is connected and so the smooth part of $V \\times \\overline{V}$ is connected, see ).",
"Hence, by complexifying, we have $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for all $z \\in V$ and $\\xi \\in V$ .",
"If $B \\ne 0$ , then setting $z=0$ , we have $B(\\bar{\\xi },\\bar{\\xi }) = 0$ on $V$ .",
"As $B$ is homogeneous and $V$ is irreducible, $V$ is a one dimensional complex line.",
"If $B=0$ , then $A(z,\\bar{\\zeta })=0$ for $z,\\zeta \\in V$ as mentioned above.",
"We consider two cases.",
"Suppose first that every $\\sum _{j=1}^2 a_{ij}\\bar{\\zeta }_j$ is identically zero for all $\\zeta \\in V$ and $i=1$ and $i=2$ .",
"Then $V$ is contained in some complex line $\\sum _{j=1}^2\\bar{a}_{ij}\\zeta _j=0$ .",
"Suppose now that $A(z,\\bar{\\zeta }_*)$ is not identically zero for some $\\zeta _* \\in V$ .",
"Then $V$ is contained in the complex line $A(z,\\bar{\\zeta }_*)=0$ .",
"This shows that $V$ is a complex line.",
"Thus as $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is zero on a one dimensional linear subspace, we make this subspace $\\lbrace z_1 = 0 \\rbrace $ and so each monomial in $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is divisible by either $z_1$ or $\\bar{z}_1$ .",
"Therefore, $A$ and $B$ are matrices of the form $\\begin{bmatrix}* & * \\\\* & 0\\end{bmatrix} ,$ that is $a_{22} = 0$ and $b_{22} = 0$ .",
"To normalize the pair $(A,B)$ , we apply arbitrary invertible transformations $(T,\\lambda ) \\in GL_{n}( \\times *$ as $(A,B) \\mapsto (\\lambda T^* A T,\\lambda T^* B\\overline{T}) .$ Recall that we are assuming that $A\\ne 0$ .",
"If $a_{21} = 0$ or $a_{12} = 0$ , then $A$ is rank one and via a transformation $T$ of the form $z_1^{\\prime }=z_1, \\quad z_2^{\\prime }=z_2+cz_1\\qquad \\text{or} \\qquad z_2^{\\prime }=z_1, \\quad z_1^{\\prime }=z_2+cz_1$ and rescaling by nonzero $\\lambda $ , the matrix $A$ can be put in the form $\\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} ,\\qquad \\text{or} \\qquad \\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} .$ The transformation $T$ and $\\lambda $ must also be applied to $B$ and this could possibly make $b_{22} \\ne 0$ .",
"However, we will show that we actually have $b_{22}=0$ .",
"Thus $B=0$ on $z_1=0$ still holds true.",
"Let us first focus on $A =\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} .$ We apply the $T^{(1,0)}$ vector field we computed above.",
"Only $a_{11}$ is nonzero in $A$ .",
"Therefore $\\beta ^* A \\beta $ , which must be identically zero, is $0 = \\beta ^* A \\beta =\\bar{\\beta }_1 \\beta _1 =\\overline{(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)}(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)\\\\=4(\\left|{b_{12}} \\right|^2z_1\\bar{z}_1 +\\left|{b_{22}} \\right|^2z_2\\bar{z}_2 +b_{12}\\bar{b}_{22}\\bar{z}_1z_2 +\\bar{b}_{12}b_{22}z_1\\bar{z}_2) .$ This polynomial must be identically zero and hence all coefficients must be identically zero.",
"So $b_{12} = 0$ and $b_{22}=0$ .",
"In other words, only $b_{11}$ in $B$ can be nonzero, in which case we make it nonnegative via a diagonal $T$ to obtain the quadric $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 , \\quad \\gamma \\ge 0 .$ Next let us focus on $A =\\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} .$ As above, we compute $\\beta ^* A \\beta $ : $0 = \\beta ^* A \\beta =\\bar{\\beta }_1 \\beta _2 =\\overline{(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)}(- \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)\\\\=- 2b_{12} \\bar{z}_1\\bar{z}_2- 2 b_{22} \\bar{b}_{11} z_1\\bar{z}_2- 4 \\bar{b}_{11}b_{12}z_1\\bar{z}_1 - 4 b_{12}\\bar{b}_{12} \\bar{z}_1 z_2- 2b_{22}\\bar{z}_2^2 -4 b_{22}\\bar{b}_{12} z_2\\bar{z}_2.$ Again, as this polynomial must be identically zero, all coefficients must be zero.",
"Hence $b_{12} = 0$ and $b_{22} = 0$ .",
"Again only $b_{11}$ is left possibly nonzero.",
"Suppose that $b_{11} \\ne 0$ .",
"Then let $s$ be such that $b_{11} \\bar{s}^2 = 1$ , and let $\\bar{t} = \\frac{1}{\\bar{s}}$ .",
"The matrix $T = \\left[ {\\begin{matrix} s & 0 \\\\ 0 & t \\end{matrix}}\\right]$ is such that $T^* A T = A$ and $T^* B \\overline{T} = \\left[ {\\begin{matrix} 1 & 0 \\\\ 0 & 0\\end{matrix}} \\right]$ .",
"If $b_{11} = 0$ , we have $B=0$ .",
"Therefore we have obtained two distinct possibilities for $B$ , and thus the two submanifolds $\\begin{aligned}& w = \\bar{z}_1z_2 , \\qquad \\qquad \\text{or} \\\\& w = \\bar{z}_1z_2 + \\bar{z}_1^2 .\\end{aligned}$ We emphasize that after $A$ is normalized by a transformation of the form (REF ), only one coordinate change is needed to normalize $b_{11}$ and this coordinate change preserves $A$ .",
"Both are required in a reduction proof for higher dimensions.",
"We have handled the rank one case.",
"Next we focus on the rank two case, that is $a_{21} \\ne 0$ and $a_{12} \\ne 0$ (recall $a_{22} = 0$ ).",
"We normalize (rescale) $A$ to have $a_{12} = 1$ and take $A=\\begin{bmatrix}a_{11} & 1 \\\\a_{21} & 0\\end{bmatrix} .$ Again, let us compute $\\beta ^* A \\beta $ .",
"In the computation for the rank 2 case, recall that we have not done any normalization other than rescaling, so we can safely still assume that $b_{22} = 0$ : $0 = \\beta ^* A \\beta =a_{11}\\bar{\\beta }_1 \\beta _1 +\\bar{\\beta }_1 \\beta _2 +a_{21}\\beta _1 \\bar{\\beta }_2\\\\=a_{11}\\overline{(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)}(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)+\\overline{(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)}(- \\bar{a}_{11}\\bar{z}_1 - \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)\\\\+a_{21}\\overline{(- \\bar{a}_{11}\\bar{z}_1 - \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)}(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)\\\\=(-4 \\left|{b_{12}} \\right|^2 - \\left|{a_{21}} \\right|^2) \\bar{z}_1 z_2+ \\text{(other terms)}.$ All coefficients must be zero.",
"So $a_{21}=0$ , and $A$ would not be rank 2.",
"Let us now focus on $n > 2$ .",
"First let us suppose that $A=0$ .",
"Then as before $M$ is automatically Levi-flat and by diagonalizing $B$ we obtain the $n$ distinct submanifolds: $\\begin{aligned}w & = \\bar{z}_1^2 , \\\\w & = \\bar{z}_1^2 + \\bar{z}_2^2 , \\\\& ~\\vdots \\\\w & = \\bar{z}_1^2 + \\bar{z}_2^2 + \\dots + \\bar{z}_{n}^2 .\\end{aligned}$ Thus suppose from now on that $A \\ne 0$ .",
"As before we have an irreducible $n-1$ dimensional variety $V \\subset M$ through the origin, such that $w = 0$ and $A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) = 0$ on $V$ .",
"We wish to show that $A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) = 0$ on an $n-1$ dimensional linear subspace.",
"For any $\\xi \\in V$ we obtain $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for all $z \\in V$ .",
"If $V$ is contained in the kernel of the matrix $A^*$ , then we have that $V$ is a linear subspace of dimension $n-1$ .",
"So suppose that $\\bar{\\xi }$ is not in the kernel of the matrix $A^t$ .",
"Then for a fixed $\\bar{\\xi }$ we obtain a linear equation $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for $z \\in V$ .",
"Therefore, as $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ needs to be zero on an $n-1$ dimensional subspace we can just make this $\\lbrace z_1 = 0 \\rbrace $ and so each monomial is divisible by either $z_1$ or $\\bar{z}_1$ .",
"Therefore, $A$ and $B$ is of the form $ \\begin{bmatrix}* & * & \\cdots & * \\\\* & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\* & 0 & \\cdots & 0\\end{bmatrix} ,$ that is, only first column and first row are nonzero.",
"We normalize $A$ via $(A,B) \\mapsto (\\lambda T^* A T,\\lambda T^* B\\overline{T}) ,$ as before.",
"We use column operations on all but the first column to make all but the first two columns have nonzero elements.",
"Similarly we can do row operations on all but the first two rows and to make all but first three rows nonzero.",
"That is $A$ has the form $\\begin{bmatrix}* & * & 0 & \\cdots & 0 \\\\* & 0 & 0 & \\cdots & 0 \\\\* & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix} .$ By Lemma REF , setting $z_3 = \\cdots = z_n = 0$ we obtain a Levi-flat submanifold where the matrix corresponding to $A$ is the principal $2 \\times 2$ submatrix of $A$ .",
"This submatrix cannot be of rank 2 and hence either $a_{12} = 0$ or $a_{21} = 0$ .",
"If $a_{21} = 0$ and $a_{12} \\ne 0$ , then setting $z_2 = z_3$ , and $z_4 =\\cdots = z_n = 0$ we again must have a rank one matrix and therefore $a_{31} = 0$ .",
"Therefore, if $a_{12} \\ne 0$ then all but $a_{11}$ and $a_{12}$ are zero.",
"If $a_{12} = 0$ , then via a further linear map not involving $z_1$ we can ensure that $a_{31} = 0$ .",
"In particular, $A$ is of rank 1 and can only be nonzero in the principal $2 \\times 2$ submatrix.",
"At this point $B$ is still of the form (REF ).",
"Via a linear change of coordinates in the first two variables, the principal $2 \\times 2$ submatrix of $A$ can be normalized into one of the 2 possible forms $\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} ,\\qquad \\text{or} \\qquad \\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} .$ Recall that $A=0$ was already handled.",
"Via the 2 dimensional computation we obtain that $b_{22} = b_{12} = b_{21} = 0$ .",
"We use a linear map in $z_1$ and $z_2$ to also normalize the principal $2 \\times 2$ matrix of $B$ , so that the submanifold restricted to $(z_1,z_2,w)$ is in one of the normal forms B.$\\gamma $ , C.0, or C.1.",
"Finally we need to show that all entries of $B$ other than $b_{11}$ are zero.",
"As we have done a linear change of coordinates in $z_1$ and $z_2$ , $B$ may not be in the form (REF ), but we know $b_{jk} = 0$ as long as $j > 2$ and $k > 2$ .",
"Now fix $k = 3,\\ldots ,n$ .",
"Restrict to the submanifold given by $z_1 = \\lambda z_2$ for $\\lambda = 1$ or $\\lambda = -1$ , and $z_j = 0$ for all $j=3,\\ldots ,n$ except for $j=k$ .",
"In the variables $(z_2,z_k,w)$ , we obtain a Levi-flat submanifold where the matrix corresponding to $A$ is $\\left[ {\\begin{matrix}\\lambda & 0 \\\\0 & 0\\end{matrix}} \\right]$ .",
"The matrix corresponding to $B$ is $\\begin{bmatrix}b_{11} & b_{1k} + \\lambda b_{2k} \\\\b_{1k} + \\lambda b_{2k} & 0\\end{bmatrix} .$ Via the 2 dimensional calculation we have $b_{1k} + \\lambda b_{2k} = 0$ .",
"As this is true for $\\lambda = 1$ and $\\lambda = -1$ , we get that $b_{1k} = b_{2k} = 0$ .",
"We have proved the following classification result.",
"It is not difficult to see that the submanifolds in the list are biholomorphically inequivalent by Proposition REF .",
"The ranks of $A$ and $B$ are invariants.",
"It is obvious that the $A$ matrix of B.$\\gamma $ and C.x submanifolds are inequivalent.",
"Therefore, it is only necessary to directly check that B.$\\gamma $ are inequivalent for different $\\gamma \\ge 0$ , which is easy.",
"Lemma 3.2 If $M$ defined in $(z,w) \\in {n} \\times , $ n 1$, by\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})\\end{equation}is Levi-flat, then $ M$ is biholomorphic to one and exactly one of thefollowing:\\begin{equation}\\begin{aligned}\\text{(A.1)} \\quad & w = \\bar{z}_1^2 , \\\\\\text{(A.2)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2 , \\\\& \\vdots \\\\\\text{(A.$n$)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2 + \\dots + \\bar{z}_{n}^2 , \\\\[10pt]\\text{(B.$\\gamma $)} \\quad & w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 , ~~ \\gamma \\ge 0 , \\\\[10pt]\\text{(C.0)} \\quad & w = \\bar{z}_1z_2 , \\\\\\text{(C.1)} \\quad & w = \\bar{z}_1z_2 + \\bar{z}_1^2 .\\end{aligned}\\end{equation}$ The normalizing transformation used above is linear.",
"Lemma 3.3 If $M$ defined by $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3)$ is Levi-flat at all points where $M$ is CR, then the quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is also Levi-flat.",
"Write $M$ as $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + r(z,\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Let $A$ be the matrix giving the quadratic form $A(z,\\bar{z})$ as before.",
"The Levi-map is given by taking the $n\\times n$ matrix $L = L(p) =A+\\begin{bmatrix}\\frac{\\partial ^2 r}{ \\partial z_j \\partial \\bar{z}_k}\\end{bmatrix}_{j,k}$ and applying it to vectors in $\\pi (T^{(1,0)} M)$ , where $\\pi $ is the projection onto the $\\lbrace w = 0 \\rbrace $ plane.",
"That is we parametrize $M$ by the $\\lbrace w = 0 \\rbrace $ plane, and work there as before.",
"Let $\\begin{aligned}& a_j = - \\overline{A}_{z_j} - \\overline{B}_{z_j} - \\bar{r}_{z_j} , \\\\& b = \\overline{A}_{z_1} + \\overline{B}_{z_1} + \\bar{r}_{z_1} , \\\\& c = a_j (A_{z_1} + B_{z_1} + r_{z_1}) + b (A_{z_j} + B_{z_j} + r_{z_j}) .\\end{aligned}$ Then for $j=2,\\ldots ,n$ , we write the $T^{(1,0)}$ vector fields as $X_j =a_j\\frac{\\partial }{\\partial z_1}+b\\frac{\\partial }{\\partial z_j}+c\\frac{\\partial }{\\partial w} .$ Hence $a_j \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial z_j}$ are the vector fields in $\\pi (T^{(1,0)} M)$ .",
"Notice that $a_j$ , $b$ , and $c$ vanish at the origin, and furthermore that if we take the linear terms of $a_j$ , $b$ , and the quadratic terms in $c$ , that is $\\begin{aligned}& \\widetilde{a}_j = - \\overline{A}_{z_j} - \\overline{B}_{z_j} , \\\\& \\widetilde{b} = \\overline{A}_{z_1} + \\overline{B}_{z_1} , \\\\& \\widetilde{c} = \\widetilde{a}_j (A_{z_1} + B_{z_1}) + \\widetilde{b} (A_{z_j} + B_{z_j}) ,\\end{aligned}$ then away from the CR singular set of the quadric $\\widetilde{X}_j =\\widetilde{a}_j\\frac{\\partial }{\\partial z_1}+\\widetilde{b}\\frac{\\partial }{\\partial z_j}+\\widetilde{c}\\frac{\\partial }{\\partial w}$ span the $T^{(1,0)}$ vector fields on the quadric $w = A(z,\\bar{z}) +B(\\bar{z},\\bar{z})$ .",
"Since $M$ is Levi-flat, then we have that $\\pi _*(X_j)^* ~L~ \\pi _*(X_j) = 0 .$ The terms linear in $z$ and $\\bar{z}$ respectively in the expression $\\pi _*(X_j)^* ~L~ \\pi _*(X_j)$ are precisely $\\pi _*(\\widetilde{X}_j)^* ~A~ \\pi _*(\\widetilde{X}_j) .$ As this expression is identically zero, the quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is Levi-flat."
],
[
"Quadratic Levi-flat submanifolds and their Segre varieties",
"A very useful invariant in CR geometry is the Segre variety.",
"Suppose that a real-analytic variety $X \\subset N$ is defined by $\\rho (z,\\bar{z}) = 0 ,$ where $\\rho $ is a real-analytic real vector-valued with $p \\in X$ .",
"Suppose that $\\rho $ converges on some polydisc $\\Delta $ centered at $p$ .",
"We complexify and treat $z$ and $\\bar{z}$ as independent variables, and the power series of $\\rho $ at $(p,\\bar{p})$ converges on $\\Delta \\times \\Delta $ .",
"The Segre variety at $p$ is then defined as the variety $Q_p = \\lbrace z \\in \\Delta : \\rho (z,\\bar{p}) = 0 \\rbrace .$ Of course the variety depends on the defining equation itself and the polydisc $\\Delta $ .",
"For $\\rho $ it is useful to take the defining equation or equations that generate the ideal of the complexified $X$ in $N \\times N$ at $p$ .",
"If $\\rho $ is polynomial we take $\\Delta = N$ .",
"It is well-known that any irreducible complex variety that lies in $X$ and goes through the point $p$ also lies in $Q_p$ .",
"In case of Levi-flat submanifolds we generally get equality as germs.",
"For example, for the CR Levi-flat submanifold $M$ given by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 ,$ the Segre variety $Q_0$ through the origin is precisely $\\lbrace z_1 = z_2 =0\\rbrace $ , which happens to be the unique complex variety in $M$ through the origin.",
"Let us take the Levi-flat quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) .$ As we want to take the generating equations in the complexified space we also need the conjugate $\\bar{w} = \\bar{A}(\\bar{z},z) + \\bar{B}(z,z) .$ The Segre variety is then given by $w = 0, \\qquad \\bar{B}(z,z) = 0 .$ Through any CR singular point of a real-analytic Levi-flat $M$ there is a complex variety of dimension $n-1$ that is the limit of the leaves of the Levi-foliation of $M_{CR}$ , via Lemma REF .",
"Let us take all possible such limits, and call their union $Q^{\\prime }_p$ .",
"Notice that there could be other complex varieties in $M$ through $p$ of dimension $n-1$ .",
"Note that $Q^{\\prime }_p \\subset Q_p$ .",
"Let us write down and classify the Segre varieties for all the quadric Levi-flat submanifolds in ${n+1}$ : Table: NO_CAPTIONThe submanifold C.0 also contains the complex variety $\\lbrace w = 0, z_2 = 0 \\rbrace $ , but this variety is transversal to the leaves of the foliation, and so cannot be in $Q^{\\prime }_0$ Notice that in the cases A.$k$ for all $k$ , B.$\\gamma $ for $\\gamma > 0$ , and C.1, the variety $Q_0$ actually gives the complex variety $Q^{\\prime }_0$ contained in $M$ through the origin.",
"In these cases, the variety is nonsingular only in the set theoretic sense.",
"Scheme-theoretically the variety is always at least a double line or double hyperplane in general."
],
[
"The CR singularity of Levi-flats quadrics",
"Let us study the set of CR singularities for Levi-flat quadrics.",
"The following proposition is well-known.",
"Proposition 5.1 Let $M \\subset {n+1}$ be given by $w = \\rho (z,\\bar{z})$ where $\\rho $ is $O(2)$ , and $M$ is not a complex submanifold.",
"Then the set $S$ of CR singularities of $M$ is given by $S = \\lbrace (z,w) : \\bar{\\partial } \\rho = 0, w = \\rho (z,\\bar{z}) \\rbrace .$ In codimension 2, a real submanifold is either CR singular, complex, or generic.",
"A submanifold is generic if $\\bar{\\partial }$ of all the defining equations are pointwise linearly independent (see ).",
"As $M$ is not complex, to find the set of CR singularities, we find the set of points where $M$ is not generic.",
"We need both defining equations for $M$ , $w = \\rho (z,\\bar{z}), \\qquad \\text{and} \\qquad \\bar{w} = \\rho (z,\\bar{z}) .$ As the second equation always produces a $d\\bar{w}$ while the first does not, the only way that the two can be linearly dependent is for the $\\bar{\\partial }$ of the first equation to be zero.",
"In other words $\\bar{\\partial } \\rho = 0$ .",
"Let us compute and classify the CR singular sets for the CR singular Levi-flat quadrics.",
"Table: NO_CAPTIONBy Levi-flat we mean that $S$ is a Levi-flat CR submanifold in $\\lbrace w = 0 \\rbrace $ .",
"There is a conjecture that a real subvariety that is Levi-flat at CR points has a stratification by Levi-flat CR submanifolds.",
"This computation gives further evidence of this conjecture."
],
[
"Levi-foliations and images of generic Levi-flats",
"A CR Levi-flat submanifold $M \\subset n$ of codimension 2 has a certain canonical foliation defined on it with complex analytic leaves of real codimension 2 in $M$ .",
"The submanifold $M$ is locally equivalent to ${\\mathbb {R}}^2 \\times {n-2}$ , defined by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 .$ The leaves of the foliation are the submanifolds given by fixing $z_1$ and $z_2$ at a real constant.",
"By foliation we always mean the standard nonsingular foliation as locally comes up in the implicit function theorem.",
"This foliation on $M$ is called the Levi-foliation.",
"It is obvious that the Levi-foliation on $M$ extends to a neighbourhood of $M$ as a nonsingular holomorphic foliation.",
"The same is not true in general for CR singular submanifolds.",
"We say that a smooth holomorphic foliation ${\\mathcal {L}}$ defined in a neighborhood of $M$ is an extension of the Levi-foliation of $M_{CR}$ , if ${\\mathcal {L}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"We also say that a smooth real-analytic foliation $\\widetilde{{\\mathcal {L}}}$ on $M$ is an extension of the Levi-foliation on $M_{CR}$ if $\\widetilde{{\\mathcal {L}}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"In our situation (real-analytic), $M_{CR}$ is a dense and open subset of $M$ .",
"This implies that the leaves of ${\\mathcal {L}}$ and $\\widetilde{{\\mathcal {L}}}$ through a CR singular point are complex analytic submanifolds contained in $M$ .",
"The latter could lead to an obvious obstruction to extension.",
"First let us see what happens if the foliation of $M_{CR}$ is the restriction of a nonsingular holomorphic foliation of a whole neighbourhood of $M$ .",
"The Bishop-like quadrics, that is A.1 and B.$\\gamma $ in ${n+1}$ , have a Levi-foliation that extends as a holomorphic foliation to all of ${n+1}$ .",
"That is because these submanifolds are of the form $ N \\times {n-1} .$ For submanifolds of the form (REF ) we can find normal forms using the well-developed theory of Bishop surfaces in 2.",
"Proposition 6.1 Suppose $M \\subset {n+1}$ is a real-analytic Levi-flat CR singular submanifold where the Levi-foliation on $M_{CR}$ extends near $p \\in M$ to a nonsingular holomorphic foliation of a neighbourhood of $p$ in ${n+1}$ .",
"Then at $p$ , $M$ is locally biholomorphically equivalent to a submanifold of the form $ N \\times {n-1}$ where $N \\subset 2$ is a CR singular submanifold of real dimension 2.",
"Therefore if $M$ has a nondegenerate complex tangent, then it is Bishop-like, that is of type A.1 or B.$\\gamma $ .",
"Furthermore, two submanifolds of the form (REF ) are locally biholomorphically (resp.",
"formally) equivalent if and only if the corresponding $N$ s are locally biholomorphically (resp.",
"formally) equivalent in 2.",
"We flatten the holomorphic foliation near $p$ so that in some polydisc $\\Delta $ , the leaves of the foliation are given by $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ for $q \\in 2$ .",
"Let us suppose that $M$ is closed in $\\Delta $ .",
"At any CR point of $M$ , the leaf of the Levi-foliation agrees with the leaf of the holomorphic foliation and therefore the leaf that lies in $M$ agrees with a leaf of the form $\\lbrace q \\rbrace \\times {n-1}$ as a germ and so $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta \\subset M$ .",
"As $M_{CR}$ is dense in $M$ , then $M$ is a union of sets of the form $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ and the first part follows.",
"It is classical that every Bishop surface (2 dimensional real submanifold of 2 with a nondegenerate complex tangent) is equivalent to a submanifold whose quadratic part is of the form A.1 or B.$\\gamma $ .",
"Finally, the proof that two submanifolds of the form (REF ) are equivalent if and only if the $N$ s are equivalent is straightforward.",
"Not every Bishop-like submanifold is a cross product as above.",
"In fact the Bishop invariant may well change from point to point.",
"See § .",
"In such cases the foliation does not extend to a nonsingular holomorphic foliation of a neighbourhood.",
"Let us now focus on extending the Levi-foliation to $M$ , and not to a neighbourhood of $M$ .",
"Let us prove a useful proposition about recognizing certain CR singular Levi-flats from the form of the defining equation.",
"That is if the $r$ in the equation does not depend on $\\bar{z}_2$ through $\\bar{z}_n$ .",
"Proposition 6.2 Suppose near the origin $M \\subset {n+1}$ is given by $w = r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(2)$ and $\\frac{\\partial r}{\\bar{z}_1} \\lnot \\equiv 0$ .",
"Then $M$ is a CR singular Levi-flat submanifold and the Levi-foliation of $M_{CR}$ extends through the origin to a real-analytic foliation on $M$ .",
"Furthermore, there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"Near 0, $M$ is the image of a CR mapping that is a diffeomorphism onto its image of the standard CR Levi-flat.",
"The proposition also holds in two dimensions ($n=1$ ), although in this case it is somewhat trivial.",
"As in , let us define the mapping $F$ by $(x,y,\\xi ) \\mapsto \\bigl (x+iy, \\quad \\xi , \\quad r(x+iy,x-iy, \\xi ) \\bigr ) ,$ where $\\xi = (\\xi _2,\\ldots ,\\xi _n) \\in {n-1}$ .",
"Near points where $M$ is CR, this mapping is a CR diffeomorphism and hence $M$ must be Levi-flat.",
"Furthermore, since $F$ is a diffeomorphism, it takes the Levi-foliation on ${\\mathbb {R}}^2 \\times {n-1}$ to a foliation on $M$ near 0.",
"In fact, we make the following conclusion.",
"Lemma 6.3 Let $M \\subset {n+1}$ be a CR singular real-analytic Levi-flat submanifold of codimension 2 through the origin.",
"Then $M$ is a CR singular Levi-flat submanifold whose Levi-foliation of $M_{CR}$ extends through the origin to a nonsingular real-analytic foliation on $M$ if and only if there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"One direction is easy and was used above.",
"For the other direction, suppose that we have a foliation extending the Levi-foliation through the origin.",
"Let us consider $M_{CR}$ an abstract CR manifold.",
"That is a manifold $M_{CR}$ together with the bundle $T^{(0,1)} M_{CR}\\subset T M_{CR}$ .",
"The extended foliation on $M$ gives a real-analytic subbundle ${\\mathcal {W}}\\subset T M$ .",
"Since we are extending the Levi-foliation, when $p \\in M_{CR}$ , then ${\\mathcal {W}}_p = T_p^c M$ , where $T_p^c M = J(T_p^c M)$ is the complex tangent space and $J$ is the complex structure on ${n+1}$ .",
"Since $M_{CR}$ is dense in $M$ , then $J{\\mathcal {W}}={\\mathcal {W}}$ on $M$ .",
"Define the real-analytic subbundle ${\\mathcal {V}}\\subset T M$ as ${\\mathcal {V}}_p = \\lbrace X + iJ(X) : X \\in {\\mathcal {W}}_p \\rbrace .$ At CR points ${\\mathcal {V}}_p = T_p^{(0,1)} M$ (see for example page 8).",
"Then we can find vector fields $X^1,\\ldots ,X^{n-1}$ in ${\\mathcal {W}}$ such that $X^1,J(X^1),X^2,J(X^2),\\ldots ,X^{n-1},J(X^{n-1})$ is a basis of ${\\mathcal {W}}$ near the origin.",
"Then the basis for ${\\mathcal {V}}$ is given by $X^1+iJ(X^1),X^2+iJ(X^2),\\ldots ,X^{n-1}+iJ(X^{n-1}).$ As the subbundle is integrable, we obtain that $(M,{\\mathcal {V}})$ gives an abstract CR manifold, which at CR points agrees with $M_{CR}$ .",
"This manifold is Levi-flat as it is Levi-flat on a dense open set.",
"As it is real-analytic it is embeddable and hence there exists a real-analytic CR diffeomorphism from a neighbourhood of ${\\mathbb {R}}^2 \\times {n-1}$ to a neighbourhood of 0 in $M$ (as an abstract CR manifold).",
"This is our mapping $F$ .",
"The quadrics A.$k$ , $k \\ge 2$ , defined by $w = \\bar{z}_1^2 + \\cdots + \\bar{z}_k^2 ,$ contain the singular variety defined by $w = 0$ , $z_1^2 + \\cdots + z_k^2 =0$ , and hence the Levi-foliation cannot extend to a nonsingular foliation of the submanifold.",
"The quadric A.1 does admit a holomorphic foliation, but other type A.1 submanifolds do not in general.",
"For example, the submanifold $w = \\bar{z}_1^2 + \\bar{z}_2^3$ is of type A.1 and the unique complex variety through the origin is $0 = z_1^2 + z_2^3$ , which is singular.",
"Therefore the foliation cannot extend to $M$ ."
],
[
"Extending the Levi-foliation of C.x type submanifolds",
"Let us prove Theorem REF , that is, let us start with a type C.0 or C.1 submanifold and show that the Levi-foliation must extend real-analytically to all of $M$ .",
"Equivalently, let us show that the real analytic bundle $T^{(1,0)}M_{CR}$ extends to a real analytic subbundle of $TM$ .",
"Taking real parts we obtain an involutive subbundle of $TM$ extending $T^cM_{CR} = \\operatorname{Re}(T^{(1,0)} M_{CR})$ .",
"Let $M$ be the submanifold given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $\\epsilon = 0,1$ .",
"Let us treat the $z$ variables as the parameters on $M$ .",
"Let $\\pi $ be the projection onto the $\\lbrace w=0 \\rbrace $ plane, which is tangent to $M$ at 0 as a real $2n$ -dimensional hyperplane.",
"We will look at all the vectorfields on this plane $\\lbrace w=0 \\rbrace $ .",
"All vectors in $\\pi (T^{(1,0)} M)$ can be written in terms of $\\frac{\\partial }{\\partial z_j}$ for $j=1,\\ldots ,n$ .",
"The Levi-map is given by taking the $n\\times n$ matrix $L = L(p) =\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}+\\begin{bmatrix}\\frac{\\partial ^2 r}{ \\partial z_j \\partial \\bar{z}_k}\\end{bmatrix}_{j,k}(p)$ to vectors $v \\in \\pi (T^{(1,0)} M)$ ($\\pi $ is the projection) as $v^* L v$ .",
"The excess term in $L$ vanishes at 0.",
"Notice that for $p \\in M_{CR}$ , $\\pi (T^{(1,0)}_p M)$ is $n-1$ dimensional.",
"As $M$ is Levi-flat, then $v^* L v$ vanishes for $v \\in \\pi (T^{(1,0)}_p M)$ .",
"Write the vector $v = (v_1,\\ldots ,v_n)^t$ .",
"The zero set of the function $(z,v) \\in n \\times n \\overset{\\varphi }{\\mapsto } v^* L(z,\\bar{z}) v$ is a variety $V$ of real codimension 2 at the origin of $\\mathbf {C}^n\\times \\mathbf {C}^n$ because of the form of $L$ .",
"That is, at $z=0$ , the only vectors $v$ such that $v^*Lv = 0$ are those where $v_1 = 0$ or $v_2 = 0$ .",
"So the codimension is at least 2.",
"And we know that $v^*Lv$ vanishes for vectors in $\\pi (T^{(1,0)}_p M)$ for $p \\in M$ near 0, which is real codimension 2 at each $z$ corresponding to a CR point.",
"Therefore, $V \\cap (\\pi (M_{CR}) \\times n )$ has a connected component that is equal to a connected component of the real-analytic subbundle $\\pi (T^{(1,0)} M_{CR})$ .",
"We will verify that the latter is connected.",
"We show below that this subbundle extends past the CR singularity.",
"The key point is to show that the restriction of $\\pi \\bigl (T^{(1,0)}(M_{CR})\\bigr )$ extends to a smooth real-analytic submanifold of $T^{(1,0)} n$ .",
"Write $\\varphi (z,v) = v_1\\bar{v}_2 + \\sum a_{jk}(z) v_j\\bar{v}_k$ where $a_{jk}(0) = 0$ .",
"By Proposition REF , $\\pi (M\\setminus M_{CR})$ is contained in $z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1}=0.$ Thus $M_{CR}$ is connected.",
"Assume that $v\\cdot \\frac{\\partial }{\\partial z}\\in T^{(1,0)}_pM$ at a CR point $p$ .",
"Then $(z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1})\\overline{v}_1+\\sum _{j>1} r_{\\overline{z}_j}\\overline{v}_j=0.$ When $p$ is in the open set $U_\\delta \\subset \\pi (M_{CR})$ defined by $\\left|{z_2+2\\epsilon \\overline{z}_1} \\right|>\\left|{z} \\right|/2$ and $0<\\left|{z} \\right|<\\delta $ , $v$ is contained in $V_C\\colon \\left|{v_1} \\right|\\le \\left|{v} \\right|/C.$ When $\\delta $ is sufficiently small, $\\varphi (z,v)=0$ admits a unique solution $v_1=f(z,v_3,\\dots , v_n), \\quad v_2=1$ by imposing $v\\in V_C$ .",
"Note that $f$ is given by convergent power series.",
"For $\\left|{z} \\right|<\\delta $ , define $w_j=\\bigl (w_{j1}(z),\\dots , w_{jn}(z)\\bigr )\\in V_C, \\quad j=2,\\dots , n$ such that $\\varphi (z,w_j(z))=0$ and $ w_{j2}=1, \\quad w_{jk}=\\delta _{jk},\\quad j\\ge 2, k>2.$ To see why we can do so, fix $p\\in U_\\delta $ .",
"First we can find a vector $w_2$ in $E_p=\\pi (T_p^{(1,0)}M_{CR})$ such that $v_2=1$ .",
"Otherwise, $E_p\\subset V_C$ cannnot have dimension $n-1$ .",
"Let $E_p^{\\prime }$ be the vector subspace of $E_p$ with $v_2=0$ .",
"Then $E_p^{\\prime }$ has rank $n-2$ and remains in the cone $V_C$ .",
"Then $E_p^{\\prime }$ has an element $w_2$ with $v_2$ component being 1.",
"Repeating this, we find $w_2,\\dots , w_n$ in $E_p$ such that the $v_j$ component of $w_i$ is 0 for $2<j<i$ .",
"Using linear combinations, we find a unique basis $\\lbrace w_2,\\dots , w_n \\rbrace $ of $E_p$ that satisfies condition (REF ).",
"Assume that $C$ is sufficiently large.",
"By the above uniqueness assertion on $\\varphi (z,v)=0$ , we conclude that when $p\\in U_\\delta $ , $\\lbrace w_{2}(p),\\dots , w_n(p)\\rbrace $ is a base of $\\pi (T^{(1,0)}_pM_{CR})$ .",
"Also it is real analytic at $p=0$ .",
"Define $\\omega _j(z)=w_{j}(z)\\cdot \\frac{\\partial }{\\partial z}, \\quad \\left|{z} \\right|<\\delta .$ We lift the functions $\\omega _j$ via $\\pi $ to a subbundle of $TM$ , let us call these $\\widetilde{\\omega }_j$ .",
"Then consider the vector fields $w^*_j = 2 \\operatorname{Re}\\widetilde{\\omega }_j = \\widetilde{\\omega }_j + \\overline{\\widetilde{\\omega }_j}$ and $w^*_{n+j}=\\operatorname{Im}\\widetilde{\\omega }_j$ for $j=2,\\dots , n$ .",
"Above CR points over $U_\\delta $ , $\\tilde{w}_j$ is in $T M_{CR}\\otimes and so tangentto $ M$.",
"We thus obtain a $ 2n-2$ dimensional real analytic subbundle of $ TM$that agrees with the real analytic realsubbundle of $ TMCR$ induced by the Levi-foliation above $ U$.",
"Since $ MCR$and the subbunldes are real analytic and $ MCR$ is connected, they agree over $ MCR$.$ The real analytic distribution spanned by $\\lbrace \\omega ^*_i\\rbrace $ has constant rank ($2n-2$ ) everywhere and is involutive on an open subset of $M_{CR}$ and hence everywhere."
],
[
"CR singular set of type C.x submanifolds",
"Let $M \\subset {n+1}$ be a codimension two Levi-flat CR singular submanifold that is an image of ${\\mathbb {R}}^2 \\times {n-1}$ via a real-analytic CR map, and let $S \\subset M$ be the CR singular set of $M$ .",
"In it was proved that near a generic point of $S$ exactly one of the following is true: $S$ is Levi-flat submanifold of dimension $2n-2$ and CR dimension $n-2$ .",
"$S$ is a complex submanifold of complex dimension $n-1$ (real dimension $2n-2$ ).",
"$S$ is Levi-flat submanifold of dimension $2n-1$ and CR dimension $n-1$ .",
"We only have the above classification for a generic point of $S$ , and $S$ need not be a CR submanifold everywhere.",
"See for examples.",
"If $M$ is a Levi-flat CR singular submanifold and the Levi-foliation of $M_{CR}$ extends to $M$ , then by Lemma REF at a generic point $S$ has to be of one of the above types.",
"A corollary of Theorem REF is the following result.",
"Corollary 8.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a real-analytic Levi-flat CR singular type C.1 or type C.0 submanifold.",
"Let $S \\subset M$ denote the CR singular set.",
"Then near the origin $S$ is a submanifold of dimension $2n-2$ , and at a generic point, $S$ is either CR Levi-flat of dimension $2n-2$ (CR dimension $n-2$ ) or a complex submanifold of complex dimension $n-1$ .",
"Furthermore, if $M$ is of type C.1, then at the origin $S$ is a CR Levi-flat submanifold of dimension $2n-2$ (CR dimension $n-2$ ).",
"Let us take $M$ to be given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $r$ is $O(3)$ and $\\epsilon = 0$ or $\\epsilon = 1$ .",
"By Proposition REF the CR singular set is exactly where $z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0, \\qquad \\text{and} \\qquad r_{\\bar{z}_j}(z,\\bar{z}) = 0 \\quad \\text{for all $j=2,\\ldots ,n$}.$ By considering the real and imaginary parts of the first equation and applying the implicit function theorem the set $\\widetilde{S} = \\lbrace z : z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) =0 \\rbrace $ is a real submanifold of real dimension $2n-2$ (real codimension 2 in $M$ ).",
"Now $S \\subset \\widetilde{S}$ , but as we saw above $S$ is of dimension at least $2n-2$ .",
"Therefore $S = \\widetilde{S}$ near the origin.",
"The conclusion of the first part then follows from the classification above.",
"The stronger conclusion for C.1 submanifolds follows by noticing that when $\\epsilon = 1$ , the submanifold $z_2 + 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0$ is CR and not complex at the origin."
],
[
"Mixed-holomorphic submanifolds",
"Let us study sets in $m$ defined by $ f(\\bar{z}_1,z_2,\\ldots ,z_m) = 0 ,$ for a single holomorphic function $f$ of $m$ variables.",
"Such sets have much in common with complex varieties, since they are in fact complex varieties when $\\bar{z}_1$ is treated as a complex variable.",
"The distinction is that the automorphism group is different since we are interested in automorphisms that are holomorphic not mixed-holomorphic.",
"Proposition 9.1 If $M \\subset m$ is a submanifold with a defining equation of the form (REF ), where $f$ is a holomorphic function that is not identically zero, then $M$ is a real codimension 2 set and $M$ is either a complex submanifold or a Levi-flat submanifold, possibly CR singular.",
"Furthermore, if $M$ is CR singular at $p \\in M$ , and has a nondegenerate complex tangent at $p$ , then $M$ has type A.$k$ , C.0, or C.1 at $p$ .",
"Since the zero set of $f$ is a complex variety in the $(\\bar{z}_1,z_2,\\ldots ,z_m)$ space, we get automatically that it is real codimension 2.",
"We also have that as it is a submanifold, then it can be written as a graph of one variable over the rest.",
"Let $m = n+1$ for convenience and suppose that $M \\subset {n+1}$ is a submanifold through the origin.",
"By factorization for a germ of holomorphic function and by the smoothness assumption on $M$ we may assume that $df(0) \\ne 0$ .",
"Call the variables $(z_1,\\ldots ,z_n,w)$ and write $M$ as a graph.",
"One possibility is that we write $M$ as $\\bar{w} = \\rho (z_1,\\ldots ,z_n),$ where $\\rho (0) = 0$ and $\\rho $ has no linear terms.",
"$M$ is complex if $\\rho \\equiv 0$ .",
"Otherwise $M$ is CR singular and we rewrite it as $w = \\bar{\\rho }(\\bar{z}_1,\\ldots ,\\bar{z}_n).$ We notice that the matrix representing the Levi-map must be identically zero, so we must get Levi-flat.",
"If there are any quadratic terms we obtain a type A.$k$ submanifold.",
"Alternatively $M$ can be written as $w = \\rho (\\bar{z}_1,z_2,\\ldots ,z_n),$ with $\\rho (0) = 0$ .",
"If $\\rho $ does not depend on $\\bar{z}_1$ then $M$ is complex.",
"Assume that $\\rho $ depends on $\\bar{z}_1$ .",
"If $\\rho $ has linear terms in $\\bar{z}_1$ , then $M$ is CR.",
"Otherwise it is a CR singular submanifold, and near non-CR singular points it is a generic codimension 2 submanifold.",
"The CR singular set of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\overline{z}_1}=0$ .",
"Suppose that $M$ is CR singular.",
"That $M$ is Levi-flat follows from Proposition REF .",
"We can therefore normalize the quadratic term, after linear terms in $z_2,\\ldots ,z_n$ are absorbed into $w$ .",
"If not all quadratic terms are zero, then we notice that we must have an A.$k$ , C.0, or C.1 type submanifold.",
"Let us now study normal forms for such sets in 2 and $m$ , $m \\ge 3$ .",
"First in two variables we can easily completely answer the question.",
"This result is surely well-known and classical.",
"Proposition 9.2 If $M \\subset 2$ is a submanifold with a defining equation of the form (REF ), then it is locally biholomorphically equivalent to a submanifold in coordinates $(z,w) \\in 2$ of the form $w = \\bar{z}^d$ for $d=0,1,2,3,\\ldots $ where $d$ is a local biholomorphic invariant of $M$ .",
"If $d=0$ , $M$ is complex, if $d=1$ it is a CR totally-real submanifold, and if $d \\ge 2$ then $M$ is CR singular.",
"Write the submanifold as a graph of one variable over the other.",
"Without loss of generality and after possibly taking a conjugate of the equation, we have $w = f(\\bar{z})$ for some holomorphic function $f$ .",
"Assume $f(0) = 0$ .",
"If $f$ is identically zero, then $d=0$ and we are finished.",
"If $f$ is not identically zero, then it is locally biholomorphic to a positive power of the variable.",
"We apply a holomorphic change of coordinates in $z$ , and the rest follows easily.",
"In three or more variables, if $M \\subset {n+1}$ , $n \\ge 2$ , is a submanifold through the origin, then if the quadratic part is nonzero we have seen above that it can be a type A.$k$ , C.0, or C.1 submanifold.",
"If the submanifold is the nondegenerate type C.1 submanifold, then we will show in the next section that $M$ is biholomorphically equivalent to the quadric $M_{C.1}$ .",
"Before we move to C.1, let us quickly consider the mixed-holomorphic submanifolds of type A.$n$ .",
"The submanifolds of type A.$n$ in ${n+1}$ can in some sense be considered nondegenerate when talking about mixed-holomorphic submanifolds.",
"Proposition 9.3 If $M \\subset {n+1}$ is a submanifold of type A.$n$ at the origin of the form $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 + r(\\bar{z})$ where $r \\in O(3)$ .",
"Then $M$ is locally near the origin biholomorphically equivalent to the A.$n$ quadric $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 .$ The complex Morse lemma (see e.g.",
"Proposition 3.15 in ) states that there is a local change of coordinates near the origin in just the $z$ variables such that $z_1^2+ \\cdots + z_n^2 + \\bar{r}(z)$ is equivalent to $z_1^2+ \\cdots + z_n^2$ .",
"It is not difficult to see that the normal form for mixed-holomorphic submanifolds in ${n+1}$ of type A.$k$ , $k < n$ , is equivalent to a local normal form for a holomorphic function in $n$ variables.",
"Therefore for example the submanifold $w = \\bar{z}_1^2 +\\bar{z}_2^3$ is of type A.1 and is not equivalent to any quadric."
],
[
"Formal normal form for certain C.1 type submanifolds I",
"In this section we prove the formal normal form in Theorem REF .",
"That is, we prove that if $M \\subset {n+1}$ is defined by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(3)$ , then $M$ is Levi-flat and formally equivalent to $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ That $M$ is Levi-flat follows from Proposition REF .",
"Lemma 10.1 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ formal power series then $M$ is formally equivalent to $M_{C.1}$ given by $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ In fact, the normalizing transformation can be of the form $(z,w) = (z_1,\\ldots ,z_n,w)\\mapsto \\bigl (z_1, \\quad f(z,w), \\quad z_3, \\quad \\ldots , \\quad z_n, \\quad g(z,w)\\bigr ) ,$ where $f$ and $g$ are formal power series.",
"Suppose that the normalization was done to degree $d-1$ , then suppose that $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) +r_2(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) ,$ where $r_1$ is degree $d$ homogeneous and $r_2$ is $O(d+1)$ .",
"Write $r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) =\\sum _{j=0}^k \\sum _{\\left|{\\alpha } \\right|+j = d} c_{j,\\alpha }\\bar{z}_1^j z^\\alpha ,$ where $k$ is the highest power of $\\bar{z}_1$ in $r_1$ , and $\\alpha $ is a multiindex.",
"If $k$ is even, then use the transformation that replaces $w$ with $w + \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{k/2} z^\\alpha .$ Let us look at the degree $d$ terms in $(\\bar{z}_1 z_2 + \\bar{z}_1^2)+ \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha }{(\\bar{z}_1 z_2 + \\bar{z}_1^2)}^{k/2} z^\\alpha =\\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ We need not include $r_2$ as the terms are all degree $d+1$ or more.",
"After cancelling out the new terms on the left, we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"Next suppose that $k$ is odd.",
"We use the transformation that replaces $z_2$ with $z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha .$ Let us look at the degree $d$ terms in $\\bar{z}_1 z_2 + \\bar{z}_1^2=\\bar{z}_1 \\left(z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha \\right)+ \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ Again we need not include $r_2$ as the terms are all degree $d+1$ or more, and we need not add the new terms to $z_2$ in the argument list for $r_1$ since all those terms would be of higher degree.",
"Again we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"The procedure above does not change the form of the submanifold, but it lowers the degree of $\\bar{z}_1$ by one.",
"Since we can assume that all terms in $r_1$ depend on $\\bar{z}_1$ , we are finished with degree $d$ terms after $k$ iterations of the above procedure."
],
[
"Convergence of normalization for certain C.1 type submanifolds",
"A key point in the computation below is the following natural involution for the quadric $M_{C.1}$ .",
"Notice that the map $(z_1,z_2,\\ldots ,z_n,w) \\mapsto (-\\bar{z}_2-z_1, \\quad z_2, \\quad \\ldots ,\\quad z_n, \\quad w)$ takes $M_{C.1}$ to itself.",
"The involution simply replaces the $\\bar{z}_1$ in the equation with $-z_2-\\bar{z}_1$ .",
"The way this involution is defined is by noticing that the equation $w = \\bar{z}_1 z_2 + \\bar{z}_1^2$ has generically two solutions for $\\bar{z}_1$ keeping $z_2$ and $w$ fixed.",
"In the same way we could define an involution on all type C.1 submanifolds of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1,z_2,\\ldots ,z_n)$ , although we will not require this construction.",
"We prove convergence via the following well-known lemma: Lemma 11.1 Let $m_1, \\ldots , m_N$ be positive integers.",
"Suppose $T(z)$ is a formal power series in $z \\in N$ .",
"Suppose $T(t^{m_1}v_1,\\ldots , t^{m_N}v_N)$ is a convergent power series in $t \\in forall $ v N$.",
"Then $ T$ is convergent.$ The proof is a standard application of the Baire category theorem and the Cauchy inequality.",
"See (Theorem 5.5.30, p. 153) where all $m_j$ are 1.",
"For $m_j > 1$ we first change variables by setting $v_j = w_j^{m_j}$ and apply the lemma with $m_j=1$ .",
"The following lemma finishes the proof of Theorem REF .",
"By absorbing any holomorphic terms into $w$ , we assume that $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"In Lemma REF we have also constructed a formal transformation that only changed the $z_2$ and $w$ coordinates, so it is enough to prove convergence in this case.",
"Key points of this proof are that the right hand side of the defining equation for $M_{C.1}$ is homogeneous, and that we have a natural involution on $M_{C.1}$ .",
"Lemma 11.2 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ and convergent, and $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"Suppose that two formal power series $f(z,w)$ and $g(z,w)$ satisfy $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{z}_1 f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ \\bar{z}_1^2 + r(z_1,\\bar{z}_1,f(z,\\bar{z}_1z_2 +\\bar{z}_1^2),z_3,\\ldots ,z_n) .$ Then $f$ and $g$ are convergent.",
"The equation (REF ) is true formally, treating $z_1$ and $\\bar{z}_1$ as independent variables.",
"Notice that (REF ) has one equation for 2 unknown functions.",
"We now use the involution on $M_{C.1}$ to create a system that we can solve uniquely.",
"We replace $\\bar{z}_1$ with $-z_2-\\bar{z}_1$ .",
"We leave $z_1$ untouched (treating as an independent variable).",
"We obtain an identity in formal power series: $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = (-z_2-\\bar{z}_1) f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ (-z_2-\\bar{z}_1)^2 \\\\+ r(z_1,(-z_2-\\bar{z}_1),f(z,\\bar{z}_1z_2 + \\bar{z}_1^2),z_3,\\ldots ,z_n) .$ The formal series $\\xi = f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ and $\\omega = g(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ are solutions of the system $\\omega & = \\bar{z}_1 \\xi + \\bar{z}_1^2+ r(z_1,\\bar{z}_1,\\xi ,z_3,\\ldots ,z_n) , \\\\\\omega & = (-z_2-\\bar{z}_1) \\xi + (-z_2-\\bar{z}_1)^2+ r(z_1,(-z_2-\\bar{z}_1),\\xi ,z_3,\\ldots ,z_n) .$ We next replace $z_j$ with $t z_j$ and $\\bar{z}_1$ with $t \\bar{z}_1$ for $t \\in .",
"Because $ z1z2 + z12$ is homogeneous ofdegree 2, we obtain that for every $ (z1,z1,z2,...,zn) n+1$ the formal series in $ t$ given by$ (t) = f(tz,t2(z1z2 + z12))$,$ (t) = g(tz,t2(z1z2 + z12))$are solutions of the system{\\begin{@align}{1}{-1}\\omega & = t \\bar{z}_1 \\xi + t^2 \\bar{z}_1^2+ r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) , \\\\\\omega & = t (-z_2-\\bar{z}_1) \\xi + t^2 (-z_2-\\bar{z}_1)^2+ r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n) .\\end{@align}}We eliminate $$ to obtain an equation for $$:\\begin{equation}t (2 \\bar{z}_1 + z_2) ( \\xi - t z_2)=\\\\r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) .\\end{equation}We now treat $$ as a variable and we have a holomorphic (convergent)equation.",
"The right hand size must be divisible by$ t (2 z1 + z2)$: It is divisible by $ t$ since$ r$ was divisible by $ z1$.",
"It is also divisible by$ 2 z1 + z2$ as setting $ z2 = -2 z1$ makes the righthand side vanish.",
"Therefore,\\begin{equation}\\xi - t z_2=\\frac{r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n)}{t (2 \\bar{z}_1 + z_2)} ,\\end{equation}where the right hand side is a holomorphic function (that is, a convergentpower series) in $ z1,z1,z2,...,zn,t,$.For any fixed $ z1,z1,z2,...,zn$, we solve for $$ in terms of $ t$via the implicit function theorem,and we obtain that $$ is a holomorphicfunction of $ t$.",
"The power series of $$ is given by$ (t) = f(tz,t2(z1z2 + z12))$.$ Let $v \\in {n+1}$ be any nonzero vector.",
"Via a proper choice of $z_1,\\bar{z}_1,z_2,\\ldots ,z_n$ (still treating $\\bar{z}_1$ and $z_1$ as independent variables) we write $v =(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ .",
"We apply the above argument to $\\xi (t) = f(tv_1,\\ldots , tv_n, t^2v_{n+1})$ , and $\\xi (t)$ converges as a series in $t$ .",
"As we get convergence for every $v \\in {n+1}$ we obtain that $f$ converges by Lemma REF .",
"Once $f$ converges, then via () we obtain that $g(tv_1,\\ldots , tv_n, t^2v_{n+1})$ converges as a series in $t$ for all $v$ , and hence $g$ converges."
],
[
"Automorphism group of the C.1 quadric",
"With the normal form achieved in previous sections, let us study the automorphism group of the C.1 quadric in this section.",
"We will again use the mixed-holomorphic involution that is obtained from the quadric.",
"We study the local automorphism group at the origin.",
"That is the set of germs at the origin of biholomorphic transformations taking $M$ to $M$ and fixing the origin.",
"First we look at the linear parts of automorphisms.",
"We already know that the linear term of the last component only depends on $w$ .",
"For $M_{C.1}$ we can say more about the first two components.",
"Proposition 12.1 Let $(F,G) = (F_1,\\ldots ,F_n,G)$ be a formal invertible or biholomorphic automorphism of $M_{C.1} \\subset {n+1}$ , that is the submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Then $F_1(z,w) = a z_1 + \\alpha w + O(2)$ , $F_2(z,w) = \\bar{a} z_2 + \\beta w + O(2)$ , and $G(z,w) = \\bar{a}^2 w + O(2)$ , where $a \\ne 0$ .",
"Let $a = (a_1,\\ldots ,a_n)$ and $b = (b_1,\\ldots ,b_n)$ be such that $F_1(z,w) = a \\cdot z + \\alpha w + O(2)$ and $F_2(z,w) = b \\cdot z + \\beta w + O(2)$ .",
"Then from Proposition REF we have $\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} a \\\\ b \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1 b_2 = 1$ , and $\\bar{a}_j b_k = 0$ for all $(j,k) \\ne (1,2)$ .",
"Similarly $\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} \\bar{a} \\\\ \\bar{b} \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1^2 = 1$ , and $\\bar{a}_j \\bar{a}_k = 0$ for all $(j,k)\\ne (1,1)$ .",
"Putting these two together we obtain that $a_j = 0$ for all $j \\ne 1$ , and as $a_1 \\ne 0$ we get $b_j = 0$ for all $j \\ne 2$ .",
"As $\\lambda $ is the reciprocal of the coefficient of $w$ in $G$ , we are finished.",
"Lemma 12.2 Let $M_{C.1} \\subset 3$ be given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Suppose that a local biholomorphism (resp.",
"formal automorphism) $(F_1,F_2,G)$ transforms $M_{C.1}$ into $M_{C.1}$ .",
"Then $F_1$ depends only on $z_1$ , and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"Let us define a $(1,0)$ tangent vector field on $M$ by $Z=\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} .$ Write $F = (F_1,F_2,G)$ .",
"$F$ must take $Z$ into a multiple of itself when restricted to $M_{C.1}$ .",
"That is on $M_{C.1}$ we have $& \\frac{\\partial F_1}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_1}{\\partial w}= 0 ,\\\\& \\frac{\\partial F_2}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_2}{\\partial w}= \\lambda ,\\\\& \\frac{\\partial G}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial G}{\\partial w}= \\lambda \\overline{F_1}(\\bar{z},\\bar{w}) ,$ for some function $\\lambda $ .",
"Let us take the first equation and plug in the defining equation for $M_1$ : $ \\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+\\bar{z}_1\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This equation is true for all $z \\in 2$ , and so we may treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We have an involution on $M_{C.1}$ that takes $\\bar{z}_1$ to $-z_2-\\bar{z}_1$ .",
"Therefore we also have $\\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+(-z_2-\\bar{z}_1)\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This means that $\\frac{\\partial F_1}{\\partial w}$ and therefore $\\frac{\\partial F_1}{\\partial z_2}$ must be identically zero.",
"That is, $F_1$ only depends on $z_1$ .",
"We have that the following must hold for all $z$ : $G(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)F_2(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+{\\left(\\overline{F_1}(\\bar{z}_1) \\right)}^2 .$ Again we treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We differentiate with respect to $z_1$ : $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ We plug in the involution again to obtain $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(-z_2-\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ Therefore as $F_1$ is not identically zero, then as before both $\\frac{\\partial F_2}{\\partial z_1}$ and $\\frac{\\partial G}{\\partial z_1}$ must be identically zero.",
"Lemma 12.3 Take $M_{C.1} \\subset 3$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and let $(F_1,F_2,G)$ be a local automorphism at the origin.",
"Then $F_1$ uniquely determines $F_2$ and $G$ .",
"Furthermore, given any invertible function of one variable $F_1$ with $F_1(0) = 0$ , there exist unique $F_2$ and $G$ that complete an automorphism and they are determined by $ \\begin{aligned}F_2(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = \\bar{F}_1(\\bar{z}_1)+\\bar{F}_1(-\\bar{z}_1-z_2),\\\\G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = -\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2).\\end{aligned}$ We should note that the lemma also works formally.",
"Given any formal $F_1$ , there exist unique formal $F_2$ and $G$ satisfying the above property.",
"By Lemma REF , $F_1$ depends only on $z_1$ and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"We write the automorphism as a composition of the two mappings $\\bigl (F_1(z_1),z_2,w\\bigr )$ and $\\bigl (z_1,F_2(z_2,w),G(z_2,w)\\bigr )$ .",
"We plug the transformation into the defining equation for $M_{C.1}$ .",
"$ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(\\bar{z}_1)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2 .$ We use the involution $(z_1,z_2) \\mapsto (-\\bar{z}_1-z_2,z_2)$ which preserves $M_{C.1}$ and obtain a second equation $ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(-\\bar{z}_1-z_2)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2 .$ We eliminate $G$ and solve for $F_2$ : $ F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= \\frac{{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2-{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2}{\\bar{F}_1(\\bar{z}_1)-\\bar{F}_1(-\\bar{z}_1-z_2)}=\\bar{F}_1(\\bar{z}_1)+ \\bar{F}_1(-\\bar{z}_1-z_2) .$ Next we note that trivially, $F_2$ is unique if it exists: its difference vanishes on $M_{C.1}$ .",
"If we suppose that $F_1$ is convergent, then just as before, substituting $z_2$ with $tz_2$ and $\\bar{z}_1$ with $t\\bar{z}_1$ , we are restricting to curves $(tz_2,t^2w)$ for all $(z_2,w)$ .",
"The series is convergent in $t$ for every fixed $z_2$ and $w$ .",
"Therefore if $F_2$ exists and $F_1$ is convergent, then $F_2$ is convergent by Lemma REF .",
"Now we need to show the existence of the formal solution $F_2$ .",
"Notice that the right-hand side of (REF ) is invariant under the involution.",
"It suffices to show that any power series in $\\bar{z_1}, z_2$ that is invariant under the involution is a formal power series in $z_2$ and $\\bar{z}_1z_2+\\bar{z}_1^2$ .",
"Let us treat $\\xi =\\bar{z}_1$ as an independent variable.",
"The original involution becomes a holomorphic involution in $\\xi ,z_2$ : $\\tau \\colon \\xi \\rightarrow -\\xi -z_2, \\qquad z_2\\rightarrow z_2.$ By a theorem of Noether we obtain a set of generators for the ring of invariants can be obtained by applying the averaging operation $R(f) = \\frac{1}{2} ( f + f \\circ \\tau )$ to all monomials in $\\xi $ and $z_2$ of degree 2 or less.",
"By direct calculation it is not difficult to see that $\\xi ,\\xi z_2+\\xi ^2$ generate the ring of invariants.",
"Therefore any invariant power series in $z_2,\\xi $ is a power series in $\\xi ,\\xi z_2+\\xi ^2$ .",
"This shows the existence of $F_2$ .",
"The existence of $G$ follows the same.",
"The equation for $G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)=-\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2)$ is obtained by plugging in the equation for $F_2$ .",
"Its existence, uniqueness, and convergence in case $F_1$ converges, follows exactly the same as for $F_2$ .",
"Theorem 12.4 If $M \\subset {n+1}$ , $n \\ge 2$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and $(F_1,F_2,\\ldots ,F_n,G)$ is a local automorphism at the origin, then $F_1$ depends only on $z_1$ , $F_2$ and $G$ depend only on $z_2$ and $w$ , and $F_1$ completely determines $F_2$ and $G$ via (REF ).",
"The mapping $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin.",
"Furthermore, given any invertible function $F_1$ of one variable with $F_1(0) = 0$ , and arbitrary holomorphic functions $F_3,\\ldots ,F_n$ with $F_j(0) = 0$ , and such that $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin, then there exist unique $F_2$ and $G$ that complete an automorphism.",
"Let $(F_1,\\ldots ,F_n,G)$ be an automorphism.",
"Then we have $G(z_1,\\ldots ,z_n,w) =\\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w})F_2(z_1,\\ldots ,z_n,w) +{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w}) \\bigr )}^2 .$ Proposition REF says that the linear terms in $G$ only depend on $w$ , the linear terms of $F_1$ depend only on $z_1$ and $w$ and the linear terms of $F_2$ only depend on $z_2$ and $w$ .",
"Let us embed $M_{C.1} \\subset 3$ into $M$ via setting $z_3 = \\alpha _3 z_2$ , $\\ldots $ , $z_n = \\alpha _n z_2$ , for arbitrary $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Then we obtain $ G(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) = \\\\\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w})F_2(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w}) \\bigr )}^2 .$ By noting what the linear terms are, we notice that the above is the equation for an automorphism of $M_{C.1}$ .",
"Therefore by Lemma REF we have $\\frac{\\partial F_1}{\\partial w} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial F_2}{\\partial z_1} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial G}{\\partial z_1} = 0 ,$ as that is true for all $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Plugging in the defining equation for $M_{C.1}$ we obtain an equation that holds for all $z$ and we can treat $z$ and $\\bar{z}$ independently.",
"We plug in $z = 0$ to obtain $0 =\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0)F_2(0,\\ldots ,0,\\bar{z}_1^2) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0) \\bigr )}^2 .$ Differentiating with respect to $\\bar{\\alpha }_j$ we obtain $\\frac{\\partial F_1}{\\partial z_j} = 0$ , for $j=3,\\ldots ,n$ .",
"We set $\\bar{\\alpha }_j = 0$ in the equation, differentiate with respect to $\\bar{z}_2$ and obtain that $\\frac{\\partial F_1}{\\partial z_2} = 0$ .",
"In other words $F_1$ is a function of $z_1$ only.",
"We rewrite (REF ) by writing $F_1$ as a function of $z_1$ only and $F_2$ and $G$ as functions of $z_2,\\ldots ,z_n,w$ , and we plug in $w = \\bar{z}_1z_2 + \\bar{z}_1^2$ to obtain $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\\\\\overline{F_1}(\\bar{z}_1)F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) +{\\bigl ( \\overline{F_1}(\\bar{z}_1) \\bigr )}^2 .$ By Lemma REF , we know that $F_1$ now uniquely determines $F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ and $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ .",
"These two functions therefore do not depend on $\\alpha _3,\\ldots ,\\alpha _n$ , and in turn $F_2$ and $G$ do not depend on $z_3,\\ldots ,z_n$ as claimed.",
"Furthermore $F_1$ does uniquely determine $F_2$ and $G$ .",
"Finally since the mapping is a biholomorphism, and from what we know about the linear parts of $F_1$ , $F_2$ , and $G$ , it is clear that $(z_1,z_2,F_3,\\ldots ,F_n)$ is rank $n$ .",
"The other direction follows by applying Lemma REF .",
"We start with $F_1$ , determine $F_2$ and $G$ as in 3 dimensions.",
"Then adding $F_3,\\ldots ,F_n$ and the rank condition guarantees an automorphism."
],
[
"Normal form for certain C.1 type submanifolds II",
"The goal of this section is to find the normal form for Levi-flat submanifolds $M \\subset {n+1}$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + \\operatorname{Re}f(z) ,$ for a holomorphic $f(z)$ of order $O(3)$ .",
"Since $f(z)$ can be absorbed into $w$ via a holomorphic transformation, the goal is really to prove the following theorem.",
"Theorem 13.1 Let $M \\subset {n+1}$ be a real-analytic Levi-flat given by $ w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Then $M$ can be put into the $M_{C.1}$ normal form $ w = \\bar{z}_1z_2 + \\bar{z}_1^2 ,$ by a convergent normalizing transformation.",
"Furthermore, if $r$ is a polynomial and the coefficient of $\\bar{z}_1^3$ in $r$ is zero, then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"In Theorem REF , we have already shown that a submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ is necessarily Levi-flat and has the normal form $M_{C.1}$ .",
"The first part of Theorem REF will follow once we prove: Lemma 13.2 If $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z})$ where $r$ is $O(3)$ and $M$ is Levi-flat, then $r$ depends only on $\\bar{z}_1$ .",
"First let us assume that $n=2$ .",
"For $p \\in M_{CR}$ , $T^{(1,0)}_p M$ is one dimensional.",
"The Levi-map is the matrix $L =\\begin{bmatrix}0 & 1 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0\\end{bmatrix}$ applied to the $T^{(1,0)} M$ vectors.",
"As $M$ is Levi-flat, then the Levi-map has to vanish.",
"The only vectors $v$ for which $v^* L v = 0$ , are the ones without $\\frac{\\partial }{\\partial z_1}$ component or $\\frac{\\partial }{\\partial z_2}$ component.",
"That is vectors of the form $a \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial w},\\qquad \\text{or} \\qquad a \\frac{\\partial }{\\partial z_2} + b \\frac{\\partial }{\\partial w}.$ We apply these vectors to the defining equation and its conjugate and we obtain in the first case the equations $b = 0, \\qquad a \\left( \\bar{z}_2 + 2z_1 + \\frac{\\partial \\bar{r}}{\\partial z_1} \\right) = 0 .$ This cannot be satisfied identically on $M$ since this is supposed to be true for all $z$ , but $a$ cannot be identically zero and the second factor in the second equation has only one nonholomorphic term, which is $\\bar{z}_2$ .",
"Let us try the second form and we obtain the equations $b = a \\bar{z}_1 , \\qquad a \\left( \\frac{\\partial \\bar{r}}{\\partial z_2} \\right) = 0 .$ Again $a$ cannot be identically zero, and hence the second factor of the second equation $\\frac{\\partial \\bar{r}}{\\partial z_2}$ must be identically zero, which is possible only if $r$ depends only on $\\bar{z}_1$ .",
"Finally, it is possible to pick $b=\\bar{z}_1$ and $a=1$ , to obtain a $T^{(1,0)}$ vector field $\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} ,$ and therefore these submanifolds are necessarily Levi-flat.",
"Next suppose that $n > 2$ .",
"Notice that replacing $z_k$ with $\\lambda _k \\xi $ for $k \\ge 2$ and then fixing $\\lambda _k$ for $k \\ge 2$ , we get $w = \\bar{z}_1 \\lambda _2 \\xi + \\bar{z}_1^2 + r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi }) .$ By Lemma REF , we obtain a Levi-flat submanifold in $(z_1,\\xi ,w) \\in 3$ , and hence can apply the above reasoning to obtain that $r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi })$ does not depend on $\\bar{\\xi }$ .",
"As this was true for any $\\lambda _k$ 's, we have that $r$ can only depend on $\\bar{z}_1$ .",
"It is left to prove the claim about the polynomial normalizing transformation.",
"Lemma 13.3 Suppose $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ where $r$ is a polynomial that vanishes to fourth order.",
"Then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"We will take a transformation of the form $(z_1,z_2,w) \\mapsto \\bigl (z_1,z_2+f(z_2,w),w+g(z_2,w) \\bigr ) .$ We are therefore trying to find polynomial $f$ and $g$ that satisfy $\\bar{z}_1z_2 + \\bar{z}_1^2+g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)=\\bar{z}_1 \\bigl (z_2 +f(z_2,\\bar{z}_1z_2 +\\bar{z}_1^2)\\bigr ) + \\bar{z}_1^2 + r(\\bar{z}_1) .$ If we simplify we obtain $g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)-\\bar{z}_1 f(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= r(\\bar{z}_1) .$ Consider the involution $S\\colon (\\bar{z}_1,z_2)\\rightarrow (-\\bar{z}_1-z_2,z_2)$ .",
"Its invariant polynomials $u(\\bar{z}_1,z_2)$ are precisely the polynomials in $z_2,z_2\\bar{z}_1+\\bar{z}_1^2$ .",
"The polynomial $r(\\bar{z}_1)$ can be uniquely written as $r^+(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)+\\Bigl (\\bar{z}_1+\\frac{z_2}{2}\\Bigr )r^-(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)$ in two polynomials $r^\\pm $ .",
"Taking $f=-r^-$ and $g=r^++\\frac{z_2}{2}r^-$ , we find the desired solutions."
],
[
"Normal form for general type C.1 submanifolds",
"In this section we show that generically a Levi-flat type C.1 submanifold is not formally equivalent to the quadric $M_{C.1}$ submanifold.",
"In fact, we find a formal normal form that shows infinitely many invariants.",
"There are obviously infinitely many invariants if we do not impose the Levi-flat condition.",
"The trick therefore is, how to impose the Levi-flat condition and still obtain a formal normal form.",
"Let $M \\subset 3$ be a real-analytic Levi-flat type C.1 submanifold through the origin.",
"We know that $M$ is an image of ${\\mathbb {R}}^2 \\times under a real-analytic CR map that is a diffeomorphism onto itstarget; see Theorem~\\ref {thm:folextendsCxtype}.After a linear change of coordinates we assume thatthe mapping is\\begin{equation}\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (& x+iy + a(x,y,\\xi ), \\\\& \\xi + b(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) ,\\end{split}\\end{equation}where $ a$, $ b$ are $ O(2)$ and $ r$ is $ O(3)$.",
"As the mapping is a CR mapping and a local diffeomorphism, then given anysuch $ a$, $ b$, and $ r$, the image isnecessarily Levi-flat at CR points.",
"Therefore the set of all thesemappings gives us all type C.1 Levi-flat submanifolds.$ We precompose with an automorphism of ${\\mathbb {R}}^2 \\times to make $ b = 0$.We cannot similarly remove $ a$ as anyautomorphism must have real valued first two components (the new $ x$ and thenew $ y$), and hence thosecomponents can only depend on $ x$ and $ y$ and not on $$.",
"So if $ a$depends on $$, we cannot remove it by precomposing.$ Next we notice that we can treat $M$ as an abstract CR manifold.",
"Suppose we have two equivalent submanifolds $M_1$ and $M_2$ , with $F$ being the biholomorphic map taking $M_1$ to $M_2$ .",
"If $M_j$ is the image of a map $\\varphi _j$ , then note that $\\varphi _2^{-1}$ is CR on ${(M_2)}_{CR}$ .",
"Therefore, $G = \\varphi _2^{-1} \\circ F \\circ \\varphi _1$ is CR on ${(F \\circ \\varphi _1)}^{-1}\\bigl ({(M_2)}_{CR}\\bigr )$ , which is dense in a neighbourhood of the origin of ${\\mathbb {R}}^2 \\times (theCR singularity of $ M2$ is a thin set, and we pull it back by tworeal-analytic diffeomorphisms).",
"A real-analytic diffeomorphism thatis CR on a dense set is a CR mapping.",
"The same argumentworks for the inverse of $ G$,and therefore we have a CR diffeomorphism of $ R2 .",
"The conclusion we make is the following proposition.",
"Proposition 14.1 If $M_j \\subset 3$ , $j=1,2$ are given by the maps $\\varphi _j$ $\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times {\\varphi _j}{\\mapsto }\\bigl (& x+iy + a_j(x,y,\\xi ), \\\\& \\xi + b_j(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r_j(x,y,\\xi )\\bigr ) ,\\end{split}$ and $M_1$ and $M_2$ are locally biholomorphically (resp.",
"formally) equivalent at 0, then there exists local biholomorphisms (resp.",
"formal equivalences) $F$ and $G$ at 0, with $F(M_1) = M_2$ , $G({\\mathbb {R}}^2 \\times = {\\mathbb {R}}^2 \\times as germs(resp.\\ formally) and\\begin{equation}\\varphi _2 = F \\circ \\varphi _1 \\circ G .\\end{equation}$ In other words, the proposition states that if we find a normal form for the mapping we find a normal form for the submanifolds.",
"Let us prove that the proposition also works formally.",
"We have to prove that $G$ restricted to ${\\mathbb {R}}^2 \\times is CR, that is,$ G = 0$.",
"Let us consider\\begin{equation}\\varphi _2 \\circ G = F \\circ \\varphi _1 .\\end{equation}The right hand side does not depend on $$ and thus the left handside does not either.",
"Write $ G = (G1,G2,G3)$.",
"Let us write $ b = b2$and $ r = r2$for simplicity.",
"Taking derivative of $ 2 G$ with respectto $$ we get:\\begin{equation}\\begin{aligned}& G^1_{\\bar{\\xi }} +i G^2_{\\bar{\\xi }} +a_x(G) G^1_{\\bar{\\xi }} +a_y(G) G^2_{\\bar{\\xi }} +a_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\& G^3_{\\bar{\\xi }} +b_x(G) G^1_{\\bar{\\xi }} +b_y(G) G^2_{\\bar{\\xi }} +b_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\&(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}) G^3+(G^1 - i G^2) G^3_{\\bar{\\xi }} +2 (G^1 - i G^2)(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }})\\\\& \\qquad +r_x(G) G^1_{\\bar{\\xi }} +r_y(G) G^2_{\\bar{\\xi }} +r_\\xi (G) G^3_{\\bar{\\xi }} = 0 .\\end{aligned}\\end{equation}Suppose that the homogeneous parts of $ Gj$ are zero for alldegrees up to degree $ d-1$.",
"If we look at the degree $ d$ homogeneous partsof the first two equations above we immediately note that it must be that$ G1 + i G2 = 0$ and$ G3 = 0$ in degree $ d$.",
"We then look at the degree $ d+1$part of the third equation.",
"Recall that $ []d$ is the degree $ d$part of an expression.",
"We get\\begin{equation}{[G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}]}_{d}{[G^3 + 2 G^1 - i 2 G^2]}_{1} = 0 .\\end{equation}As $ G$ is an automorphism we cannot have the linear terms be linearlydependent and hence$ G1 = G2 = 0$ in degree $ d$.",
"We finishby induction on $ d$.$ Using the proposition we can restate the result of Theorem REF using the parametrization.",
"Corollary 14.2 A real-analytic Levi-flat type C.1 submanifold $M \\subset 3$ is biholomorphically equivalent to the quadric $M_{C.1}$ if and only if the mapping giving $M$ is equivalent to a mapping of the form $(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (x+iy,\\quad \\xi ,\\quad (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) .$ That is, $M$ is equivalent to $M_{C.1}$ if and only if we can get rid of the $a(x,y,\\xi )$ via pre and post composing with automorphisms.",
"The proof of the corollary follows as a submanifold that is realized by this map must be of the form $w = \\bar{z}_1z_2 + \\bar{z}_1^2 + \\rho (z_1,\\bar{z}_1,z_2)$ and we apply Theorem REF .",
"We have seen that the involution $\\tau $ on $M$ , in particular when $M$ is the quadric, is useful to compute the automorphism group and to construct Levi-flat submanifolds of type $C.1$ .",
"We will also need to deal with power series in $z,\\bar{z}, \\xi $ .",
"Thus we extend $\\tau $ , which is originally defined on 2, as follows $\\sigma (z,\\bar{z},\\xi )=(z,-\\bar{z}-\\xi ,\\xi ).$ Here $z,\\bar{z},\\xi $ are treated as independent variables.",
"Note that $z,\\xi ,w=\\bar{z}\\xi +\\bar{z}^2$ are invariant by $\\sigma $ , while $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ is skew invariant by $\\sigma $ .",
"A power series in $z,\\bar{z},\\xi $ that is invariant by $\\sigma $ is precisely a power series in $z,\\xi ,w$ .",
"In general, a power series $u$ in $z,\\bar{z},\\xi $ admits a unique decomposition $u(z,\\bar{z},\\xi )=u^+(z,\\xi ,w)+\\eta u^-(z,\\xi ,w).$ First we introduce degree for power series $u(z,\\bar{z},\\xi )$ and weights for power series $v(z,\\xi ,w)$ .",
"As usual we assign degree $i+j+k$ to the monomial $z^i\\bar{z}^j\\xi ^k$ .",
"We assign weight $i+j+2k$ to the monomial $z^i\\xi ^jw^k$ .",
"For simplicity, we will call them weight in both situations.",
"Let us also denote $[u]_d(z,\\bar{z},\\xi )=\\sum _{i+j+k=d}u_{ijk}z^i\\bar{z}^j\\xi ^k, \\quad [v]_d(z,\\xi ,w)=\\sum _{i+j+2k=d}v_{ijk}z^i\\xi ^jw^k.$ Set $[u]_{i}^j=[u]_i+\\cdots +[u]_j$ and $[v]_i^j=[v]_i+\\cdots +[v]_j$ for $i\\le j$ .",
"Theorem 14.3 Let $M$ be a real-analytic Levi-flat type C.1 submanifold in 3.",
"There exists a formal biholomorphic map transforming $M$ into the image of $\\hat{\\varphi }(z,\\bar{z},\\xi )=\\bigl (z+A(z,\\xi , w)w\\eta , \\xi ,w\\bigr )$ with $\\eta =\\bar{z}+\\frac{1}{2}{\\xi }$ and $w=\\bar{z}\\xi +\\bar{z}^2$ .",
"Suppose further that $A\\lnot \\equiv 0$ .",
"Fix $i_*,j_*,k_*$ such that $j_*$ is the largest integer satisfying $A_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Then we can achieve $A_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ Furthermore, the power series $A$ is uniquely determined up to the transformation $A(z,\\xi ,w)\\rightarrow \\bar{c}^{3}A(cz,\\bar{c}\\xi ,\\bar{c}^2w), \\quad c\\in \\lbrace 0\\rbrace .$ In the above normal form with $A\\lnot \\equiv 0$ , the group of formal biholomorphisms that preserve the normal form consists of dilations $(z,\\xi ,w)\\rightarrow (\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)$ satisfying $\\bar{\\nu }^{3}A(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)=A(z,\\xi ,w)$ .",
"It will be convenient to write the CR diffeomorphism $G$ of ${\\mathbb {R}}^2\\times as $ (G1,G2)$ where $ G1$ is complex-valued and depends on $ z,z$,while $ G2$ depends on $ z,z,$.",
"Let $ M$ be the image of a mapping $$ defined by\\begin{equation}\\begin{aligned}(z,\\bar{z},\\xi ) \\overset{\\varphi }{\\mapsto }\\bigl (& z + a(z,\\bar{z},\\xi ),\\\\& \\xi ,\\\\& \\bar{z} \\xi + {\\bar{z}}^2 + r(z,\\bar{z},\\xi )\\bigr )\\end{aligned}\\end{equation}with $ a=O(2), r=O(3)$.",
"We want to find a formal biholomorphic map $ F$ of $ 3$and a formal CR diffeomorphism $ G$ of $ R2 such that $F\\hat{\\varphi }G^{-1}=\\varphi $ with $\\hat{\\varphi }$ in the normal form.",
"To simplify the computation, we will first achieve a preliminary normal form where $r=0$ and the function $a$ is skew-invariant by $\\sigma $ .",
"For the preliminary normal form we will only apply $F, G$ that are tangent to the identity.",
"We will then use the general $F, G$ to obtain the final normal form.",
"Let us assume that $F, G$ are tangent to the identity.",
"Let $M=F\\bigl (\\hat{\\varphi }({\\mathbb {R}}^2\\times \\bigr )$ where $\\hat{\\varphi }$ is determined by $\\hat{a}, \\hat{r}$ .",
"We write $F=I+(f_1,f_2,f_3), \\quad G=I+(g_1,g_2).$ The $\\xi $ components in $\\varphi G=F\\hat{\\varphi }$ give us $g_2(z,\\bar{z},\\xi )=f_2\\bigl (z+\\hat{a}(z,\\bar{z},\\xi ), \\xi , \\bar{z}\\xi +\\bar{z}^2+\\hat{r}(z,\\bar{z},\\xi )\\bigr ).$ Thus, we are allowed to define $g_2$ by the above identity for any choice of $f_2=O(2)$ .",
"Eliminating $g_2$ in other components of $\\varphi G=F\\hat{\\varphi }$ , we obtain $f_1\\circ \\hat{\\varphi }-g_1&= a\\circ G-\\hat{a},\\\\f_3\\circ \\hat{\\varphi }-\\bar{z}f_2\\circ \\hat{\\varphi }&=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2,$ where $\\tilde{g}_1(z,\\bar{z})=\\bar{g}_1(\\bar{z},z)$ and $(a\\circ G)(z,\\bar{z},\\xi ):=a\\bigl (G_1(z,\\bar{z}),\\bar{G}_1(\\bar{z},z),G_2(z,\\bar{z},\\xi )\\bigr ).$ Each power series $r(z,\\bar{z},\\xi )$ admits a unique decomposition $r(z,\\bar{z},\\xi )=r^+(z,\\xi ,w)+\\eta r^-(z,\\xi ,w),$ where both $r^\\pm $ are invariant by $\\sigma $ .",
"Note that $r(z,\\bar{z},\\xi )$ is a power series in $z,\\xi $ and $w$ , if and only if it is invariant by $\\sigma $ , i.e.",
"if $r^-=0$ .",
"We write $r^+={wt}\\, (k), \\quad \\text{or}\\quad {wt} \\, (r^+)\\ge k,$ if $r^+_{abc}=0$ for $a+b+2c<k$ .",
"Define $r^-=wt (k)$ analogously and write $\\eta r^-={wt}\\,(k)$ if $r^-={wt}\\,(k-1)$ .",
"We write $r={wt}\\,(k)$ if $(r^+,\\eta r^-)={wt}\\,(k)$ .",
"Note that $r=O(k)\\Rightarrow r={wt}\\,(k); \\quad wt \\, (rs)\\ge wt\\, (r)+wt\\,(s).$ The power series in $z,\\bar{z}$ play a special role in describing normal forms.",
"Let us define $T^\\pm $ via $u(z,\\bar{z})=(T^+u)(z,\\xi ,w)+(T^-u)(z,\\xi ,w)\\eta .$ Let $S^+_k$ (resp.",
"$S^-_k$ ) be spanned by monomials in $z,\\bar{z},\\xi $ which have weight $k$ and are invariant (resp.",
"skew-invariant) by $\\sigma $ .",
"Then the range of $\\eta T^-$ in $S_k^-$ is a linear subspace $R_k$ .",
"We decompose $S_k^-=R_k\\oplus (S_k^-\\ominus R_k).$ The decomposition is of course not unique.",
"We will take $S_k^-\\ominus R_k=\\bigoplus _{a+b+2c=k-1, c>0} z̏^a\\xi ^bw^c\\eta .$ Here we have used $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ , $\\eta ^2=w+\\frac{1}{4}\\xi ^2$ , and $T^+u(z,\\xi , w)=\\sum _{i,j\\ge 0}\\sum _{0\\le \\alpha \\le j/2} u_{ij} \\binom{j}{2\\alpha }z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha },\\\\T^-u(z,\\xi ,w)=\\sum _{i\\ge 0,j>0}\\sum _{0\\le \\alpha <j/2} u_{ij}\\binom{j}{2\\alpha +1}z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha -1}.$ In particular, we have $T^-u(z,\\xi ,0)=\\sum _{i\\ge 0,j>0}(-1)^{j-1}u_{ij} z^i \\xi ^{j-1}.$ This shows that $T^-u(z,\\xi ,0)=\\frac{1}{-\\xi }\\bigl (u(z,-\\xi )-u(z,0)\\bigr ).$ We are ready to show that under the condition that $g_1(z,\\bar{z})$ has no pure holomorphic terms, there exists a unique $(F,G)$ which is tangent to the identity such that $\\hat{r}=0$ and $\\hat{a}\\in \\mathcal {N}:=\\bigoplus \\mathcal {N}_k, \\quad \\mathcal {N}_k:=S_k^-\\ominus R_k.$ We start with terms of weight 2 in (REF )-() to get $[f_1]_2-[g_1]_2=[a]_2-\\eta [\\hat{a}^-]_1,\\\\[f_3]_2=0.$ Note that $f_j^-=0$ .",
"The first identity implies that $[f_1]_2-[T^+g_1]_2=[a^+]_2, \\quad [T^-g_1]_1=[\\hat{a}^-]_1-[a^-]_1.$ The first equation is solvable with kernel defined by $[f_1]_k-[T^+g_1]_k=0 $ for $k=2$ .",
"This shows that $[g_1]_2$ is still arbitrary and we use it to achieve $\\eta [\\hat{a}^-]_1\\in S_2^-\\ominus R_2=\\lbrace 0\\rbrace .$ Then the kernel space is defined by (REF ) and $[g_1(z,\\bar{z})-g_1(z,0)]_k=0$ with $k=2$ .",
"In particular, under the restriction $[g_1(z,0)]_k=0,$ for $k=2$ , we have achieved $\\hat{a}^-\\in \\mathcal {N}_2$ by unique $[f_1]_2, [g_1]_2, [f_2]_1, [f_3]_2$ .",
"By induction, we verify that if (REF ) holds for all $k$ , we determine uniquely $[f_1]_k, [g_1]_k$ by normalizing $[\\hat{a}]_k\\in \\mathcal {N}_k$ .",
"We then determine $[f_2]_k, [f_3]_{k+1}$ uniquely to normalize $[\\hat{r}]_{k+1}=0$ .",
"For the details, let us find formula for the solutions.",
"We rewrite (REF ) as $T^-g_1=-(a\\circ G-\\hat{a}-f_1\\circ \\hat{\\varphi })^-,\\\\( f_1\\circ \\hat{\\varphi })^+=(a\\circ G-\\hat{a})^++T^+g_1.$ Using (REF ), we can solve $(-1)^{j-1}g_{1,ij}=-({(a\\circ G)}^-)_{i(j-1)0}, \\quad j\\ge 1, \\quad i+j=k.$ Then we have $(\\hat{a}^-)_{ij0}=0, \\quad i+j=k-1; \\\\(\\hat{a}^-)_{ij m}=((a\\circ G-f_1\\circ \\hat{\\varphi }+g_1)^-)_{ijm}, \\quad m\\ge 1, i+j+m=k-1.$ Note that $- [g_1]_k(z,-\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,\\bar{z},0)$ .",
"We obtain $[g_1]_k(z,\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,-\\bar{z},0).$ Having determined $[g_1]_k$ , we take $[ f_1]_k=[(a\\circ G-\\hat{a}+g_1)^+]_k.$ We then solve () by taking $[ f_2]_k=[E^-]_k, \\quad [f_3]_{k+1}=[(E-\\frac{1}{2}\\xi f_2)^+]_{k+1},\\\\E:=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2.$ We have achieved the preliminary normalization.",
"Assume now that $\\varphi (z,\\bar{z},\\xi )=(z+a^-(z,\\xi ,w)\\eta , \\xi ,w), \\quad \\hat{\\varphi }(z,\\bar{z},\\xi )=(z+\\hat{a}^-(z,\\xi ,w)\\eta , \\xi ,w)$ are in the preliminary normal form, i.e.",
"$w|a^-(z,\\xi ,w), \\quad w|\\hat{a}^-(z,\\xi ,w).$ Let us assume that $a^-(z,\\xi ,w)=wt (s), \\quad [a^-]_s\\lnot \\equiv 0; \\quad \\hat{a}^-(z,\\xi ,w)=wt(s).$ We assume that $\\varphi G=F\\hat{\\varphi }$ with $F(z,\\xi ,w)=I+(f_1,f_2,f_3),\\\\G(z,\\bar{z},\\xi )=(z+g_1(z,\\bar{z}), \\xi +g_2(z,\\bar{z},\\xi )).$ Here $f_i,g_j$ start with terms of weight and order at least 2.",
"In particular, we have $f_i=wt(N), \\quad g_i=wt (N),\\quad i=1,2; \\quad f_3=wt(N^{\\prime }); \\quad N^{\\prime }\\ge N\\ge 2.$ Set $(P,Q,R):=\\varphi G$ .",
"Using $N\\ge 2$ , $s\\ge 2$ , and the Taylor theorem, we obtain $P&=z+g_1(z,\\bar{z})+a^-(z,\\xi ,w)\\eta +a^-(z,\\xi ,w)(\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi ))\\\\&\\quad +\\eta \\nabla a^-(z,\\xi ,w)\\cdot \\Bigl (g_1(z,\\bar{z}), g_2(z,\\bar{z},\\xi ),(\\xi +2\\bar{z})\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )\\Bigr )\\\\&\\quad +wt(s+N+1),\\\\Q&=\\xi +g_2(z,\\bar{z},\\xi ),\\\\R&=w+(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N).$ We also have $(P,Q,R)=F\\hat{\\varphi }$ .",
"Thus $P&=z+\\hat{a}^-(z,\\xi ,w)\\eta +f_1(z,\\xi ,w)+\\partial _zf_1(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\Q&=\\xi +f_2(z,\\xi ,w)+\\partial _zf_2(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\R&=w+f_3(z,\\xi ,w)+\\partial _zf_3(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N^{\\prime }+s+1).$ We will use the above 6 identities for $P,Q,R$ in two ways.",
"First we use their lower order terms to get $f_1(z,\\xi ,w)=g_1(z,\\bar{z})+( a^-(z,\\xi ,w)-\\hat{a}^-(z,\\xi ,w))\\eta +wt(N+s),\\\\\\quad f_2(z,\\xi ,w)=g_2(z,\\bar{z},\\xi )+wt(N+s), \\\\ f_3(z,\\xi ,w)=(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N)+wt(N^{\\prime }+s).$ Hence, we can take $N^{\\prime }=N+1$ .",
"By (REF ) and the preliminary normalization, we first know that $\\hat{a}=a+wt(N+s-1), \\\\f_1(z,\\xi ,w)=b(z)+wt (N+s), \\quad g_1(z,\\bar{z})=b(z)+wt (N+s).",
"$ We compose () by $\\sigma $ and then take the difference of the two equations to get $f_2(z,\\xi ,w)=-\\bar{b}(\\bar{z})-\\bar{b}( -\\bar{z}-\\xi )+wt(2N-1)+wt(N+s), \\\\f_3(z,\\xi ,w)=-\\bar{z}\\bar{b}(-\\bar{z}-\\xi )+(\\bar{z}+\\xi )\\bar{b}(\\bar{z})+wt(2N)+wt(N+s+1).$ Here we have used $N^{\\prime }=N+1$ .",
"Let $b(z)=b_Nz^N+wt(N+1)$ .",
"Therefore, we have $g_2(z,\\bar{z},\\xi )=-\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N)+wt(N+1),\\\\\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi )=\\eta \\bar{b}_N\\sum \\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}+wt(N+1),\\\\(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )=\\bar{b}_N(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})w+wt(N+2).$ Next, we use the two formulae for $P$ and (REF ) to get the identity in higher weight: $\\hat{a}^-&=a^-+g_1^-+Lb_N+wt(N+s), \\quad f_1-g_1^+=wt (N+s+1).$ Here we have used $f_1^-=0$ and $Lb_N(z,\\xi ,w)&:=-Nb_Nz^{N-1} [a^-]_s(z,\\xi ,w)-[a^-]_s(z,\\xi ,w)\\bar{b}_N\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}\\nonumber \\\\&\\quad +\\nabla [a^-]_s\\cdot \\Bigl (b_Nz^N, -\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N),\\bar{b}_Nw(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})\\Bigr ).$ Recall that $w|a^-$ and $w|\\hat{a}^-$ .",
"We also have that $w|Lb_N(z,\\xi ,w)$ and $Lb_N$ is homogenous in weighted variables and of weight $N+s-1$ .",
"This shows that $[g_1^-(z,\\xi ,0)]_{N+s-1}=0$ .",
"By (REF ), we get $[g_1(z,\\bar{z})]_{N+s}=[g_1(z,0)]_{N+s}, \\quad [\\hat{a}^-]_{s+N-1}=[ a^-]_{s+N-1}+Lb_N.$ Let us make some observations.",
"First, $Lb_N$ depends only on $b_N$ and it does not depend on coefficients of $b(z)$ of degree larger than $N$ .",
"We observe that the first identity says that all coefficients of $[g_1]_{N+s}$ must be zero, except that the coefficient $g_{1,(N+s)0}$ is arbitrary.",
"On the other hand $Lb_N$ , which has weight $N+s-1$ , depends only on $g_{1,N0}$ , while $N+s-1>N$ .",
"Let us assume for the moment that we have $Lb_N\\ne 0$ for all $b_N\\ne 0$ .",
"We will then choose a suitable complement subspace ${\\mathcal {N}}^*_{N+s-1}$ in the space of weighted homogenous polynomials in $z,\\xi ,w$ of weight $N+s-1$ for $Lb_N$ .",
"Then $\\hat{a}^-\\in w\\sum _{N>1}{\\mathcal {N}}^*_{N+s-1}$ will be the required normal form.",
"The normal form will be obtained by the following procedures: Assume that $\\varphi $ is not formally equivalent to the quadratic mapping in the preliminary normalization.",
"We first achieve the preliminary normal form by a mapping $F^0=I+(f_1^0,f_2^0,f_3^0)$ and $G^0=I+(g_1^0,g_2^0)$ which are tangent to the identity.",
"We can make $F^0,G^0$ to be unique by requiring $f^1_1(z,0)=0$ .",
"Then $a$ is normalized such that $\\hat{a}=\\hat{a}^-\\eta $ with $[\\hat{a}^-]_s$ being non-zero homogenous part of the lowest weight.",
"We may assume that $[a]_{s+1}=[\\hat{a}]_{s+1}$ .",
"Inductively, we choose $f^1_{1,N00}$ ($N=2, 3, \\ldots $ ) to achieve $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ .",
"In this step for a given $N$ , we determine mappings $F^1=I+(f_1^1,f_2^1,f_3^1)$ and $G^1=I+(g_1^1,g_2^1)$ by requiring that $f_1^1(z,\\xi ,w)$ contains only one term $\\xi ^N$ , while $f_1^1,f_2^1,g_1^1,g_2^1$ have weight at most $N$ and $f_3^1$ has weight at most $N+1$ .",
"In the process, we also show that $[f_1^1(z,\\xi ,w)]_2^{N+s}$ depends only on $z$ , if we do not want to impose the restriction on $f_1^1$ .",
"Moreover, the coefficient of $\\xi ^{N+s-1}$ of $f_1^1$ can still be arbitrarily chosen without changing the normalization achieved for $[\\hat{a}^-]_{N+s-1}$ via $[f_1^1]_{N}$ .",
"However, by achieving $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ via $F^1,G^1$ , we may destroy the preliminary normalization achieved via $F_0,G_0$ .",
"We will then restore the preliminary normalization via $F^2=I+(f_1^2,f_2^2,f_3^2), G^2=I+(g_1^2,g_2^2)$ satisfying $g^2_1(z,0)=0$ .",
"This amounts to determining $g_1^2=g_1$ and $f_1^2=f_1$ via (REF ) and () for which the terms of weight at most $N+s$ have been determined by (REF ), and then $f_2^2=f_2,f_3^2=f_3,g_2^2=g_2$ are determined by (REF )-() and (), respectively.",
"This allows us to repeat the procedure to achieve the normalization in any higher weight.",
"We will then remove the restriction that the normalizing mappings must be tangent to the identity.",
"This will alter the normal form only by suitable linear dilations.",
"Suppose that $b_N\\ne 0$ .",
"Let us verify that $Lb_N\\ne 0.$ We will also identify one of non-zero coefficients to describe the normalizing condition on $\\hat{a}$ .",
"We write the two invariant polynomials $\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^jw^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^jw^k.$ If we plug in $w=\\bar{z}^2+\\bar{z}\\xi $ we obtain a polynomial identity in the variables $z,\\bar{z},\\xi $ .",
"$\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k.$ If we set $\\bar{z} = z = 0$ , we obtain that $\\lambda _N = \\lambda ^{\\prime }_N = {(-1)}^N .$ Recall that $j_*$ is the largest integer such that $(a^-)_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Since $w|[a^-]_s$ , then $k_*>0$ .",
"We obtain $(Lb_N)_{i_*(j_*+N-1)k_*}=(a^-)_{i_*j_*k_*}\\bar{b}_N(-\\lambda _{N-1}^{\\prime }-j_*\\lambda _{N-1}+k_*\\lambda _N)\\ne 0.$ Therefore, we can achieve $(\\hat{a}^-)_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ This determines uniquely all $b_2, b_3, \\ldots .$ We now remove the restriction that $F$ and $G$ are tangent to the identity.",
"Suppose that both $\\varphi $ and $\\hat{\\varphi }$ are in the normal form.",
"Suppose that $F\\varphi =\\hat{\\varphi }G$ .",
"Then looking at the quadratic terms, we know that the linear parts $F,G$ must be dilations.",
"In fact, the linear part of $F$ must be the linear automorphism of the quadric.",
"Thus the linear parts of $F$ and $G$ have the forms $G^{\\prime }\\colon (z,\\xi )=(\\nu z,\\bar{\\nu }\\xi ), \\quad F^{\\prime }(z,\\xi ,w)=(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w).$ Then $(F^{\\prime })^{-1}\\hat{\\varphi }G^{\\prime }$ is still in the normal form.",
"Since $(F^{\\prime })^{-1}F$ is holomorphic and $(G^{\\prime })^{-1}G$ is CR, by the uniqueness of the normalization, we know that $F^{\\prime }=F$ and $G^{\\prime }=G$ .",
"Therefore, $F$ and $G$ change the normal form $a^-$ as follows $a^-(z,\\xi ,w)= \\bar{\\nu }\\hat{a}^-(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w), \\quad \\nu \\in \\lbrace 0\\rbrace .$ When $[\\hat{a}^-]_s=[a^-]_s\\ne 0$ , we see that $|\\nu |=1$ .",
"Therefore, the formal automorphism group is discrete or one-dimensional.",
"In , Coffman used an analogous method of even/odd function decomposition to obtain a quadratic normal form for non Levi-flat real analytic $m$ -submanifolds in $n$ with an CR singularity satisfying certain non-degeneracy conditions, provided $\\frac{3(n+1)}{2} \\le m<n$ .",
"He was able to achieve the convergent normalization by a rapid iteration method.",
"Using the above decomposition of invariant and skew-invariant functions of the involution $\\sigma $ , one might achieve a convergent solution for approximate equations when $M$ is formaly equivalent to the quadric.",
"However, when the iteration is employed, each new CR mapping $\\hat{\\varphi }$ might only be defined on a domain that is proportional to that of the previous $\\varphi $ in a constant factor.",
"This is significantly different from the situations of Moser and Coffman , , where rapid iteration methods are applicable.",
"Therefore, even if $M$ is formally equivalent to the quadric, we do not know if they are holomorphically equivalent.",
"Instability of Bishop-like submanifolds Let us now discuss stability of Levi-flat submanifolds under small perturbations that keep the submanifolds Levi-flat, in particular we discuss which quadratic invariants are stable when moving from point to point on the submanifold.",
"The only stable submanifolds are A.$n$ and C.1.",
"The Bishop-like submanifolds (or even just the Bishop invariant) are not stable under perturbation, which we show by constructing examples.",
"Proposition 15.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a connected real-analytic real codimension 2 submanifold that has a non-degenerate CR singular at the origin.",
"$M$ can be written in coordinates $(z,w) \\in {n} \\times as\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$.In a neighborhood of the origin all complex tangentsof $ M$ are non-degenerate, while ranks of $ A,B$ are upper semicontinuous.Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).The CR singular set of $ M$ that is not of type B.$ 12$ at the origin is areal analytic subset of $ M$ of codimension at least $ 2$, while the CRsingular set of $ M$ that is of type B.$ 12$ the origin has codimension atleast $ 1$.",
"A.$ n$ has an isolated CR singular point at the origin and sodoes C.1 in $ 3$.Let $ S0 M$ be the set of CR singular points.There is a neighborhood $ U$ of the origin such that for $ S=S0U$we have the following.\\begin{enumerate}[(i)]\\item If M is of type A.k for k \\ge 2 at the origin, then it is of type A.j at each pointof S for somej \\ge k.\\item If M is of type C.1 at the origin, then it is of type C.1 on S.If M is of type C.0 at the origin, then it is of type C.0 or C.1 on S.\\item There exists an M that is of type B.\\gamma at one point and ofC.1 at CR singular points arbitrarily near.",
"Similarly there exists an Mof type A.1 at p \\in M that is either of type C.1, or B.\\gamma , atpoints arbitrarily near p. There alsoexists an M of type B.\\gamma at every point but where \\gamma varies from point to point.\\end{enumerate}$ First we show that the rank of $A$ and the rank of $B$ are lower semicontinuous on $S_0$ , without imposing Levi-flatness condition.",
"Similarly the real dimension of the range of $A(z,\\bar{z})$ is lower semicontinuous on $S_0$ .",
"Write $M$ as $w = \\rho (z,\\bar{z}) ,$ where $\\rho $ vanishes to second order at 0.",
"If we move to a different point of $S_0$ via an affine map $(z,w) \\mapsto (Z+z_0,W+w_0)$ .",
"Then we have $W+w_0 = \\rho (Z+z_0,\\bar{Z}+\\bar{z}_0) .$ We compute the Taylor coefficients $W =\\frac{\\partial \\rho }{\\partial z} (z_0,\\bar{z}_0) \\cdot Z +\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z} + \\\\+Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^t\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial z} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] \\bar{Z} +O(3) .$ The holomorphic terms can be absorbed into $W$ .",
"If $\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z}$ is nonzero, then this complex defining function has a linear term in $W$ and linear term in $\\bar{Z}$ and the submanifold is CR at this point.",
"Therefore the set of complex tangents of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\bar{z}} =0$ and each complex tangent point is non-degenerate.",
"At a complex tangent point at the origin, $A$ is given by $\\left[ \\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ and $B$ is given by $\\frac{1}{2} \\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ .",
"In particular these matrices change continuously as we move along $S$ .",
"We first conclude that all CR singular points of $M$ in a neighborhood of the origin are non-degenerate.",
"Further holomorphic transformations act on $A$ and $B$ using Proposition REF .",
"Therefore the ranks of $A$ and $B$ as well as the real dimension of the range of $A(z,\\bar{z})$ are lower semicontinuous on $S_0$ as claimed.",
"Furthermore as $M$ is real-analytic, the points where the rank drops lie on a real-analytic subvariety of $S_0$ , or in other words a thin set.",
"Let $U$ be a small enough neighbourhood of the origin so that $S = S_0 \\cap U$ is connected.",
"Imposing the condition that $M$ is Levi-flat, we apply Theorem REF .",
"By a simple computation, unless $M$ is of type B.$\\frac{1}{2}$ , the set of complex tangents of $M$ has codimension at least 2; and A.$n$ has isolated CR singular point and so does C.1 in 3.",
"The item () follows as A.$k$ are the only types where the rank of $B$ is greater than 1, and the theorem says $M$ must be one of these types.",
"For () note that since $A$ is of rank 1 when $M$ as C.$x$ at a point, $M$ cannot be of type A.$k$ nearby.",
"If $M$ is of type C.1 at a point then the range of $A$ must be of real dimension 2 in a neighbourhood, and hence on this neighbourhood $M$ cannot be of type B.$\\gamma $ .",
"The examples proving item () are given below.",
"Example 15.2 Define $M$ via $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 + \\bar{z}_1z_2z_3 .$ It is Levi-flat by Proposition REF .",
"At the origin $M$ is a type B.$\\gamma $ , but at a point where $z_1 = z_2 = 0$ but $z_3 \\ne 0$ , the submanifold is CR singular and it is of type C.1.",
"Example 15.3 Similarly if we define $M$ via $w = \\bar{z}_1^2 + \\bar{z}_1z_2 z_3 ,$ we obtain a CR singular Levi-flat $M$ that is A.1 at the origin, but C.1 at nearby CR singular points.",
"Example 15.4 If we define $M$ via $w = \\gamma \\bar{z}_1^2 + \\left|{z_1} \\right|^2 z_2 ,$ then $M$ is a CR singular Levi-flat type A.1 submanifold at the origin, but type B.$\\gamma $ at points where $z_1 = 0$ but $z_2 \\ne 0$ .",
"Example 15.5 The Bishop invariant can vary from point to point.",
"Define $M$ via $w = \\left|{z_1} \\right|^2 + \\bar{z}_1^2 \\bigl (\\gamma _1 (1-z_2) + \\gamma _2 z_2 \\bigr ) ,$ where $\\gamma _1 , \\gamma _2 \\ge 0$ .",
"It is not hard to see that $M$ is Levi-flat.",
"Again it is an image of $2 \\times {\\mathbb {R}}^2$ in a similar way as above.",
"At the origin, the submanifold is Bishop-like with Bishop invariant $\\gamma _1$ .",
"When $z_1=0$ and $z_2 = 1$ , the Bishop invariant is $\\gamma _2$ .",
"In fact when $z_1=0$ , the Bishop invariant at that point is $\\left|{\\gamma _1 (1-z_2) + \\gamma _2z_2} \\right| .$ Proposition REF says that this submanifold possesses a real-analytic foliation extending the Levi-foliation through the singular points.",
"Proposition REF says that if a foliation on $M$ extends to a (nonsingular) holomorphic foliation, then the submanifold would be a simple product of a Bishop submanifold and $.",
"Therefore,if $ 1 = 2$ then the Levi-foliation on $ M$cannot extend to a holomorphic foliation of a neighbourhood of $ M$.$ Bishop65article author=Bishop, Errett, title=Differentiable manifolds in complex Euclidean space, journal=Duke Math.",
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],
[
"Invariants of codimension 2 CR singular submanifolds",
"Before we impose the Levi-flat condition, let us find some invariants of codimension two CR singular submanifolds in ${n+1}$ with CR singularity at 0.",
"Such a submanifold can locally near the origin be put into the form $ w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),$ where $(z,w) \\in {n} \\times and $ A$ and $ B$ are quadratic forms.We think of $ A$ and $ B$ as matrices and $ z$ as a column vector andwrite the forms as$ z*Az$ and $ z* Bz$ respectively.The matrix $ B$ is not unique.Hence we make $ B$ symmetric to makethe choice of the matrix $ B$ canonical.The following proposition is not difficult andwell-known.",
"Since the details are important and will be used later,let us prove this fact.$ Proposition 2.1 A biholomorphic transformation of (REF ) taking the origin to itself and preserving the form of (REF ) takes the matrices $(A,B)$ to $(\\lambda T^* A T, \\lambda T^* B \\overline{T} ) ,$ for $T \\in GL_n($ and $\\lambda \\in *$ .",
"If $(F_1,\\ldots ,F_n,G) = (F,G)$ is the transformation then the linear part of $G$ is $\\lambda ^{-1} w$ and the linear part of $F$ restricted to $z$ is $Tz$ .",
"Let us emphasize that $A$ is an arbitrary complex matrix and $B$ is a symmetric, but not necessarily Hermitian, matrix.",
"Let $(F_1,\\ldots ,F_n,G) = (F,G)$ be a change of coordinates taking $w = \\widetilde{A}(z,\\bar{z}) + \\widetilde{B}(\\bar{z},\\bar{z}) + O(3) =\\rho (z,\\bar{z})$ to $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3) .$ Then $ G\\bigl (z,\\rho (z,\\bar{z})\\bigr ) =A\\Bigl (F\\bigl (z,\\rho (z,\\bar{z})\\bigr ),\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr )\\Bigr )\\\\+B\\Bigl (\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr ),\\bar{F}\\bigl (\\bar{z},\\bar{\\rho }(\\bar{z},z)\\bigr )\\Bigr ) + O(3)$ is true for all $z$ .",
"The right hand side has no linear terms, so the linear terms in $G$ do not depend on $z$ .",
"That is, $G = \\lambda ^{-1} w + O(2)$ , where $\\lambda $ is a nonzero scalar and the negative power is for convenience.",
"Let $T = [ T_1, T_2 ]$ denote the matrix representing the linear terms of $F$ .",
"Here $T_{1}$ is an $n\\times n$ matrix and $T_{2}$ is $n \\times 1$ .",
"Since the linear terms in $G$ do not depend on any $z_j$ , $T_1$ is nonsingular.",
"Then the quadratic terms in (REF ) are $\\lambda ^{-1} \\bigl (\\widetilde{A}(z,\\bar{z}) + \\widetilde{B}(\\bar{z},\\bar{z}) \\bigr )=z^* T_{1}^* A T_{1} z +z^* T_1^* B \\overline{T}_{1} \\bar{z} .$ In other words as matrices, $\\widetilde{A} = \\lambda T_{1}^* A T_{1} \\qquad \\text{and} \\qquad \\widetilde{B} = \\lambda T_1^* B \\overline{T}_1 .",
"\\text{}$ We will need to at times reduce to the 3-dimensional case, and so we need the following lemma.",
"Lemma 2.2 Let $M \\subset {n+1}$ , $n \\ge 3$ , be a real-analytic Levi-flat CR singular submanifold of the form $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3) ,$ where $A$ and $B$ are quadratic.",
"Let $L$ be a nonsingular $(n-2) \\times n$ matrix $L$ .",
"If $A+B$ is not zero on the set $\\lbrace L z = 0 \\rbrace $ , then the submanifold $M_L = M \\cap \\lbrace L z = 0 \\rbrace $ is a Levi-flat CR singular submanifold.",
"Clearly if $M_L$ is not contained in the CR singularity of $M$ , then $M_L$ is a Levi-flat CR singular submanifold.",
"$M_{L^{\\prime }}$ is not contained in the CR singularity of $M$ for a dense open subset of $(n-2) \\times n$ matrices $L^{\\prime }$ .",
"If $M_L$ is a subset of the CR singularity of $M$ , pick a CR point $p$ of $M_L$ then pick a sequence $L_n$ approaching $L$ such that $M_{L_n}$ are not contained in the CR singularity of $M$ .",
"As $A+B$ is not zero on the set $\\lbrace L z = 0 \\rbrace $ , then $M_L$ is not a complex submanifold, and therefore a CR singular submanifold.",
"Then as the Levi-form of $M_{L_n}$ vanishes at all CR points of $M_{L_n}$ , the Levi-form of $M_L$ vanishes at $p$ , so $M_L$ is Levi-flat."
],
[
"Levi-flat quadrics",
"Let us first focus on Levi-flat quadrics.",
"We will prove later that the quadratic part of a Levi-flat submanifold is Levi-flat.",
"Let $M$ be defined in $(z,w) \\in {n} \\times by\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) .\\end{equation}Being Levi-flat hasseveral equivalent formulations.",
"The main idea is that the $ T(1,0) M T(0,1) M$ vector fields are completely integrable at CR points and we obtain a foliationof $ M$ at CR points by complex submanifolds of complex dimension$ n-1$.",
"An equivalent notion is that the Levi-map is identically zero, see\\cite {BER:book}.",
"The Levi-map for a CR submanifold defined by two real equations$ 1 = 2 = 0$ (for $ 1$ and $ 2$ with linearly independent differentials)is the pair of Hermitian forms\\begin{equation}i \\partial \\bar{\\partial } \\rho _1\\quad \\text{and} \\quad i \\partial \\bar{\\partial } \\rho _2 ,\\end{equation}applied to $ T(1,0) M$ vectors.The full quadratic forms$ i 1$ and $ i 2$of course depend on the defining equations themselves and aretherefore extrinsic information.",
"It is important to notethat for the Levi-map we restrict it to$ T(1,0) M$ vectors.We can define these two forms$ i 1$ and $ i 2$even at a CR singular point $ p M$.$ These forms are the complex Hessian matrices of the defining equations.",
"For our quadric $M$ they are the real and imaginary parts of the $(n+1) \\times (n+1)$ complex matrix $\\widetilde{A} =\\begin{bmatrix}A & 0 \\\\0 & 0\\end{bmatrix} ,$ where the variables are ordered as $(z_1,\\ldots ,z_n,w)$ .",
"For $M$ to be Levi-flat, the quadratic form defined by $\\widetilde{A}$ has to be zero when restricted to the $n-1$ dimensional space spanned by $T^{(1,0)}_p M$ for every $p \\in M_{CR}$ .",
"In other words for every $p \\in M_{CR}$ $v^* \\widetilde{A} v = 0, \\qquad \\text{for all $v \\in T^{(1,0)}_pM$}.$ The space $T^{(1,0)}_pM$ is of dimension $n-1$ , and furthermore, the vector $\\frac{\\partial }{\\partial w}$ is not in $T^{(1,0)}_pM$ .",
"Therefore, $z^* A z = 0$ for $z \\in n$ in a subspace of dimension $n-1$ .",
"Before we proceed let us note the following general fact about CR singular Levi-flat submanifolds.",
"Lemma 3.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a Levi-flat connected real-analytic real codimension 2 submanifold, CR singular at the origin.",
"Then there exists a germ of a complex analytic variety of complex dimension $n-1$ through the origin, contained in $M$ .",
"Through each point of $M_{CR}$ there exists a germ of a complex variety of complex dimension $n-1$ contained in $M$ .",
"The set of CR points is dense in $M$ .",
"Take a sequence $p_k$ of CR points converging to the origin and take complex varieties of dimension $n-1$ , $W_k \\subset M$ with $p_k \\in W_k$ .",
"A theorem of Fornæss (see Theorem 6.23 in for a proof using the methods of Diederich and Fornæss ) implies that there exists a variety through $W \\subset M$ with $0 \\in W$ and of complex dimension at least $n-1$ .",
"Let us first concentrate on $n=2$ .",
"When $n=2$ , $T^{(1,0)} M$ is one dimensional at CR points.",
"Write $A =\\begin{bmatrix}a_{11} & a_{12} \\\\a_{21} & a_{22}\\end{bmatrix},\\qquad B =\\begin{bmatrix}b_{11} & b_{12} \\\\b_{12} & b_{22}\\end{bmatrix}.$ Note that $B$ is symmetric.",
"A short computation shows that the vector field can be written as $\\alpha \\frac{\\partial }{\\partial w} +\\beta _1 \\frac{\\partial }{\\partial z_1} +\\beta _2 \\frac{\\partial }{\\partial z_2}=\\alpha \\frac{\\partial }{\\partial w} +\\beta \\cdot \\frac{\\partial }{\\partial z} ,$ where $\\begin{aligned}& \\beta _1 = \\bar{a}_{21}\\bar{z}_1 + \\bar{a}_{22}\\bar{z}_2+ 2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2 , \\\\& \\beta _2 = - \\bar{a}_{11}\\bar{z}_1 - \\bar{a}_{12}\\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2 , \\\\& \\alpha = a_{11} \\bar{z}_1 \\beta _1 + a_{21} \\bar{z}_2 \\beta _1+ a_{12} \\bar{z}_1 \\beta _2 + a_{22} \\bar{z}_2 \\beta _2 .\\end{aligned}$ Note that since the CR singular set is defined by $\\beta _1=\\beta _2=0$ , then $M_{CR}$ is dense in $M$ .",
"Thus we need to check that $\\begin{bmatrix}\\beta ^* & \\bar{\\alpha }\\end{bmatrix}\\begin{bmatrix}A & 0 \\\\0 & 0\\end{bmatrix}\\begin{bmatrix}\\beta \\\\\\alpha \\end{bmatrix}=\\beta ^* A \\beta $ is identically zero for $M$ to be Levi-flat.",
"If $A$ is the zero matrix, then $M$ is automatically Levi-flat.",
"We diagonalize $B$ via $T$ into a diagonal matrix with ones and zeros on the diagonal.",
"We obtain (recall $n=2$ ) the submanifolds: $\\begin{aligned}& w = \\bar{z}_1^2 , \\qquad \\qquad \\text{or}\\\\& w = \\bar{z}_1^2 + \\bar{z}_2^2 .\\end{aligned}$ The first submanifold is of the form $M \\times where $ M 2$ is a Bishopsurface.$ Let us from now on suppose that $A\\ne 0$ .",
"As $M$ is Levi-flat, then through each CR point $p = (z_p,w_p) \\in M_{CR}$ we have a complex submanifold of dimension 1 in $M$ .",
"It is well-known that this submanifold is contained in the Segre variety (see also § ) $w = A(z,\\bar{z}_p) + B(\\bar{z}_p,\\bar{z}_p), \\qquad \\bar{w}_p = \\overline{A}(\\bar{z}_p,z) + \\overline{B}(z,z) .$ By Lemma REF we obtain a complex variety $V \\subset M$ of dimension one through the origin.",
"Suppose without loss of generality that $V$ is irreducible.",
"$V$ has to be contained in the Segre variety at the origin, in particular $w=0$ on $V$ .",
"Therefore, to simplify notation, let us consider $V$ to be subvariety of $\\lbrace w = 0 \\rbrace $ .",
"Denote by $\\overline{V}$ the complex conjugate of $V$ .",
"Then as $V$ is irreducible, then $V \\times \\overline{V}$ is also irreducible (the smooth part of $V$ is connected and so the smooth part of $V \\times \\overline{V}$ is connected, see ).",
"Hence, by complexifying, we have $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for all $z \\in V$ and $\\xi \\in V$ .",
"If $B \\ne 0$ , then setting $z=0$ , we have $B(\\bar{\\xi },\\bar{\\xi }) = 0$ on $V$ .",
"As $B$ is homogeneous and $V$ is irreducible, $V$ is a one dimensional complex line.",
"If $B=0$ , then $A(z,\\bar{\\zeta })=0$ for $z,\\zeta \\in V$ as mentioned above.",
"We consider two cases.",
"Suppose first that every $\\sum _{j=1}^2 a_{ij}\\bar{\\zeta }_j$ is identically zero for all $\\zeta \\in V$ and $i=1$ and $i=2$ .",
"Then $V$ is contained in some complex line $\\sum _{j=1}^2\\bar{a}_{ij}\\zeta _j=0$ .",
"Suppose now that $A(z,\\bar{\\zeta }_*)$ is not identically zero for some $\\zeta _* \\in V$ .",
"Then $V$ is contained in the complex line $A(z,\\bar{\\zeta }_*)=0$ .",
"This shows that $V$ is a complex line.",
"Thus as $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is zero on a one dimensional linear subspace, we make this subspace $\\lbrace z_1 = 0 \\rbrace $ and so each monomial in $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is divisible by either $z_1$ or $\\bar{z}_1$ .",
"Therefore, $A$ and $B$ are matrices of the form $\\begin{bmatrix}* & * \\\\* & 0\\end{bmatrix} ,$ that is $a_{22} = 0$ and $b_{22} = 0$ .",
"To normalize the pair $(A,B)$ , we apply arbitrary invertible transformations $(T,\\lambda ) \\in GL_{n}( \\times *$ as $(A,B) \\mapsto (\\lambda T^* A T,\\lambda T^* B\\overline{T}) .$ Recall that we are assuming that $A\\ne 0$ .",
"If $a_{21} = 0$ or $a_{12} = 0$ , then $A$ is rank one and via a transformation $T$ of the form $z_1^{\\prime }=z_1, \\quad z_2^{\\prime }=z_2+cz_1\\qquad \\text{or} \\qquad z_2^{\\prime }=z_1, \\quad z_1^{\\prime }=z_2+cz_1$ and rescaling by nonzero $\\lambda $ , the matrix $A$ can be put in the form $\\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} ,\\qquad \\text{or} \\qquad \\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} .$ The transformation $T$ and $\\lambda $ must also be applied to $B$ and this could possibly make $b_{22} \\ne 0$ .",
"However, we will show that we actually have $b_{22}=0$ .",
"Thus $B=0$ on $z_1=0$ still holds true.",
"Let us first focus on $A =\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} .$ We apply the $T^{(1,0)}$ vector field we computed above.",
"Only $a_{11}$ is nonzero in $A$ .",
"Therefore $\\beta ^* A \\beta $ , which must be identically zero, is $0 = \\beta ^* A \\beta =\\bar{\\beta }_1 \\beta _1 =\\overline{(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)}(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)\\\\=4(\\left|{b_{12}} \\right|^2z_1\\bar{z}_1 +\\left|{b_{22}} \\right|^2z_2\\bar{z}_2 +b_{12}\\bar{b}_{22}\\bar{z}_1z_2 +\\bar{b}_{12}b_{22}z_1\\bar{z}_2) .$ This polynomial must be identically zero and hence all coefficients must be identically zero.",
"So $b_{12} = 0$ and $b_{22}=0$ .",
"In other words, only $b_{11}$ in $B$ can be nonzero, in which case we make it nonnegative via a diagonal $T$ to obtain the quadric $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 , \\quad \\gamma \\ge 0 .$ Next let us focus on $A =\\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} .$ As above, we compute $\\beta ^* A \\beta $ : $0 = \\beta ^* A \\beta =\\bar{\\beta }_1 \\beta _2 =\\overline{(2 \\bar{b}_{12}z_1 + 2 \\bar{b}_{22} z_2)}(- \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)\\\\=- 2b_{12} \\bar{z}_1\\bar{z}_2- 2 b_{22} \\bar{b}_{11} z_1\\bar{z}_2- 4 \\bar{b}_{11}b_{12}z_1\\bar{z}_1 - 4 b_{12}\\bar{b}_{12} \\bar{z}_1 z_2- 2b_{22}\\bar{z}_2^2 -4 b_{22}\\bar{b}_{12} z_2\\bar{z}_2.$ Again, as this polynomial must be identically zero, all coefficients must be zero.",
"Hence $b_{12} = 0$ and $b_{22} = 0$ .",
"Again only $b_{11}$ is left possibly nonzero.",
"Suppose that $b_{11} \\ne 0$ .",
"Then let $s$ be such that $b_{11} \\bar{s}^2 = 1$ , and let $\\bar{t} = \\frac{1}{\\bar{s}}$ .",
"The matrix $T = \\left[ {\\begin{matrix} s & 0 \\\\ 0 & t \\end{matrix}}\\right]$ is such that $T^* A T = A$ and $T^* B \\overline{T} = \\left[ {\\begin{matrix} 1 & 0 \\\\ 0 & 0\\end{matrix}} \\right]$ .",
"If $b_{11} = 0$ , we have $B=0$ .",
"Therefore we have obtained two distinct possibilities for $B$ , and thus the two submanifolds $\\begin{aligned}& w = \\bar{z}_1z_2 , \\qquad \\qquad \\text{or} \\\\& w = \\bar{z}_1z_2 + \\bar{z}_1^2 .\\end{aligned}$ We emphasize that after $A$ is normalized by a transformation of the form (REF ), only one coordinate change is needed to normalize $b_{11}$ and this coordinate change preserves $A$ .",
"Both are required in a reduction proof for higher dimensions.",
"We have handled the rank one case.",
"Next we focus on the rank two case, that is $a_{21} \\ne 0$ and $a_{12} \\ne 0$ (recall $a_{22} = 0$ ).",
"We normalize (rescale) $A$ to have $a_{12} = 1$ and take $A=\\begin{bmatrix}a_{11} & 1 \\\\a_{21} & 0\\end{bmatrix} .$ Again, let us compute $\\beta ^* A \\beta $ .",
"In the computation for the rank 2 case, recall that we have not done any normalization other than rescaling, so we can safely still assume that $b_{22} = 0$ : $0 = \\beta ^* A \\beta =a_{11}\\bar{\\beta }_1 \\beta _1 +\\bar{\\beta }_1 \\beta _2 +a_{21}\\beta _1 \\bar{\\beta }_2\\\\=a_{11}\\overline{(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)}(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)+\\overline{(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)}(- \\bar{a}_{11}\\bar{z}_1 - \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)\\\\+a_{21}\\overline{(- \\bar{a}_{11}\\bar{z}_1 - \\bar{z}_2- 2 \\bar{b}_{11}z_1 - 2 \\bar{b}_{12} z_2)}(\\bar{a}_{21}\\bar{z}_1+ 2 \\bar{b}_{12}z_1)\\\\=(-4 \\left|{b_{12}} \\right|^2 - \\left|{a_{21}} \\right|^2) \\bar{z}_1 z_2+ \\text{(other terms)}.$ All coefficients must be zero.",
"So $a_{21}=0$ , and $A$ would not be rank 2.",
"Let us now focus on $n > 2$ .",
"First let us suppose that $A=0$ .",
"Then as before $M$ is automatically Levi-flat and by diagonalizing $B$ we obtain the $n$ distinct submanifolds: $\\begin{aligned}w & = \\bar{z}_1^2 , \\\\w & = \\bar{z}_1^2 + \\bar{z}_2^2 , \\\\& ~\\vdots \\\\w & = \\bar{z}_1^2 + \\bar{z}_2^2 + \\dots + \\bar{z}_{n}^2 .\\end{aligned}$ Thus suppose from now on that $A \\ne 0$ .",
"As before we have an irreducible $n-1$ dimensional variety $V \\subset M$ through the origin, such that $w = 0$ and $A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) = 0$ on $V$ .",
"We wish to show that $A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) = 0$ on an $n-1$ dimensional linear subspace.",
"For any $\\xi \\in V$ we obtain $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for all $z \\in V$ .",
"If $V$ is contained in the kernel of the matrix $A^*$ , then we have that $V$ is a linear subspace of dimension $n-1$ .",
"So suppose that $\\bar{\\xi }$ is not in the kernel of the matrix $A^t$ .",
"Then for a fixed $\\bar{\\xi }$ we obtain a linear equation $A(z,\\bar{\\xi }) + B(\\bar{\\xi },\\bar{\\xi }) = 0$ for $z \\in V$ .",
"Therefore, as $A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ needs to be zero on an $n-1$ dimensional subspace we can just make this $\\lbrace z_1 = 0 \\rbrace $ and so each monomial is divisible by either $z_1$ or $\\bar{z}_1$ .",
"Therefore, $A$ and $B$ is of the form $ \\begin{bmatrix}* & * & \\cdots & * \\\\* & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\* & 0 & \\cdots & 0\\end{bmatrix} ,$ that is, only first column and first row are nonzero.",
"We normalize $A$ via $(A,B) \\mapsto (\\lambda T^* A T,\\lambda T^* B\\overline{T}) ,$ as before.",
"We use column operations on all but the first column to make all but the first two columns have nonzero elements.",
"Similarly we can do row operations on all but the first two rows and to make all but first three rows nonzero.",
"That is $A$ has the form $\\begin{bmatrix}* & * & 0 & \\cdots & 0 \\\\* & 0 & 0 & \\cdots & 0 \\\\* & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix} .$ By Lemma REF , setting $z_3 = \\cdots = z_n = 0$ we obtain a Levi-flat submanifold where the matrix corresponding to $A$ is the principal $2 \\times 2$ submatrix of $A$ .",
"This submatrix cannot be of rank 2 and hence either $a_{12} = 0$ or $a_{21} = 0$ .",
"If $a_{21} = 0$ and $a_{12} \\ne 0$ , then setting $z_2 = z_3$ , and $z_4 =\\cdots = z_n = 0$ we again must have a rank one matrix and therefore $a_{31} = 0$ .",
"Therefore, if $a_{12} \\ne 0$ then all but $a_{11}$ and $a_{12}$ are zero.",
"If $a_{12} = 0$ , then via a further linear map not involving $z_1$ we can ensure that $a_{31} = 0$ .",
"In particular, $A$ is of rank 1 and can only be nonzero in the principal $2 \\times 2$ submatrix.",
"At this point $B$ is still of the form (REF ).",
"Via a linear change of coordinates in the first two variables, the principal $2 \\times 2$ submatrix of $A$ can be normalized into one of the 2 possible forms $\\begin{bmatrix}1 & 0 \\\\0 & 0\\end{bmatrix} ,\\qquad \\text{or} \\qquad \\begin{bmatrix}0 & 1 \\\\0 & 0\\end{bmatrix} .$ Recall that $A=0$ was already handled.",
"Via the 2 dimensional computation we obtain that $b_{22} = b_{12} = b_{21} = 0$ .",
"We use a linear map in $z_1$ and $z_2$ to also normalize the principal $2 \\times 2$ matrix of $B$ , so that the submanifold restricted to $(z_1,z_2,w)$ is in one of the normal forms B.$\\gamma $ , C.0, or C.1.",
"Finally we need to show that all entries of $B$ other than $b_{11}$ are zero.",
"As we have done a linear change of coordinates in $z_1$ and $z_2$ , $B$ may not be in the form (REF ), but we know $b_{jk} = 0$ as long as $j > 2$ and $k > 2$ .",
"Now fix $k = 3,\\ldots ,n$ .",
"Restrict to the submanifold given by $z_1 = \\lambda z_2$ for $\\lambda = 1$ or $\\lambda = -1$ , and $z_j = 0$ for all $j=3,\\ldots ,n$ except for $j=k$ .",
"In the variables $(z_2,z_k,w)$ , we obtain a Levi-flat submanifold where the matrix corresponding to $A$ is $\\left[ {\\begin{matrix}\\lambda & 0 \\\\0 & 0\\end{matrix}} \\right]$ .",
"The matrix corresponding to $B$ is $\\begin{bmatrix}b_{11} & b_{1k} + \\lambda b_{2k} \\\\b_{1k} + \\lambda b_{2k} & 0\\end{bmatrix} .$ Via the 2 dimensional calculation we have $b_{1k} + \\lambda b_{2k} = 0$ .",
"As this is true for $\\lambda = 1$ and $\\lambda = -1$ , we get that $b_{1k} = b_{2k} = 0$ .",
"We have proved the following classification result.",
"It is not difficult to see that the submanifolds in the list are biholomorphically inequivalent by Proposition REF .",
"The ranks of $A$ and $B$ are invariants.",
"It is obvious that the $A$ matrix of B.$\\gamma $ and C.x submanifolds are inequivalent.",
"Therefore, it is only necessary to directly check that B.$\\gamma $ are inequivalent for different $\\gamma \\ge 0$ , which is easy.",
"Lemma 3.2 If $M$ defined in $(z,w) \\in {n} \\times , $ n 1$, by\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})\\end{equation}is Levi-flat, then $ M$ is biholomorphic to one and exactly one of thefollowing:\\begin{equation}\\begin{aligned}\\text{(A.1)} \\quad & w = \\bar{z}_1^2 , \\\\\\text{(A.2)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2 , \\\\& \\vdots \\\\\\text{(A.$n$)} \\quad & w = \\bar{z}_1^2 + \\bar{z}_2^2 + \\dots + \\bar{z}_{n}^2 , \\\\[10pt]\\text{(B.$\\gamma $)} \\quad & w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 , ~~ \\gamma \\ge 0 , \\\\[10pt]\\text{(C.0)} \\quad & w = \\bar{z}_1z_2 , \\\\\\text{(C.1)} \\quad & w = \\bar{z}_1z_2 + \\bar{z}_1^2 .\\end{aligned}\\end{equation}$ The normalizing transformation used above is linear.",
"Lemma 3.3 If $M$ defined by $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3)$ is Levi-flat at all points where $M$ is CR, then the quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is also Levi-flat.",
"Write $M$ as $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + r(z,\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Let $A$ be the matrix giving the quadratic form $A(z,\\bar{z})$ as before.",
"The Levi-map is given by taking the $n\\times n$ matrix $L = L(p) =A+\\begin{bmatrix}\\frac{\\partial ^2 r}{ \\partial z_j \\partial \\bar{z}_k}\\end{bmatrix}_{j,k}$ and applying it to vectors in $\\pi (T^{(1,0)} M)$ , where $\\pi $ is the projection onto the $\\lbrace w = 0 \\rbrace $ plane.",
"That is we parametrize $M$ by the $\\lbrace w = 0 \\rbrace $ plane, and work there as before.",
"Let $\\begin{aligned}& a_j = - \\overline{A}_{z_j} - \\overline{B}_{z_j} - \\bar{r}_{z_j} , \\\\& b = \\overline{A}_{z_1} + \\overline{B}_{z_1} + \\bar{r}_{z_1} , \\\\& c = a_j (A_{z_1} + B_{z_1} + r_{z_1}) + b (A_{z_j} + B_{z_j} + r_{z_j}) .\\end{aligned}$ Then for $j=2,\\ldots ,n$ , we write the $T^{(1,0)}$ vector fields as $X_j =a_j\\frac{\\partial }{\\partial z_1}+b\\frac{\\partial }{\\partial z_j}+c\\frac{\\partial }{\\partial w} .$ Hence $a_j \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial z_j}$ are the vector fields in $\\pi (T^{(1,0)} M)$ .",
"Notice that $a_j$ , $b$ , and $c$ vanish at the origin, and furthermore that if we take the linear terms of $a_j$ , $b$ , and the quadratic terms in $c$ , that is $\\begin{aligned}& \\widetilde{a}_j = - \\overline{A}_{z_j} - \\overline{B}_{z_j} , \\\\& \\widetilde{b} = \\overline{A}_{z_1} + \\overline{B}_{z_1} , \\\\& \\widetilde{c} = \\widetilde{a}_j (A_{z_1} + B_{z_1}) + \\widetilde{b} (A_{z_j} + B_{z_j}) ,\\end{aligned}$ then away from the CR singular set of the quadric $\\widetilde{X}_j =\\widetilde{a}_j\\frac{\\partial }{\\partial z_1}+\\widetilde{b}\\frac{\\partial }{\\partial z_j}+\\widetilde{c}\\frac{\\partial }{\\partial w}$ span the $T^{(1,0)}$ vector fields on the quadric $w = A(z,\\bar{z}) +B(\\bar{z},\\bar{z})$ .",
"Since $M$ is Levi-flat, then we have that $\\pi _*(X_j)^* ~L~ \\pi _*(X_j) = 0 .$ The terms linear in $z$ and $\\bar{z}$ respectively in the expression $\\pi _*(X_j)^* ~L~ \\pi _*(X_j)$ are precisely $\\pi _*(\\widetilde{X}_j)^* ~A~ \\pi _*(\\widetilde{X}_j) .$ As this expression is identically zero, the quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z})$ is Levi-flat."
],
[
"Quadratic Levi-flat submanifolds and their Segre varieties",
"A very useful invariant in CR geometry is the Segre variety.",
"Suppose that a real-analytic variety $X \\subset N$ is defined by $\\rho (z,\\bar{z}) = 0 ,$ where $\\rho $ is a real-analytic real vector-valued with $p \\in X$ .",
"Suppose that $\\rho $ converges on some polydisc $\\Delta $ centered at $p$ .",
"We complexify and treat $z$ and $\\bar{z}$ as independent variables, and the power series of $\\rho $ at $(p,\\bar{p})$ converges on $\\Delta \\times \\Delta $ .",
"The Segre variety at $p$ is then defined as the variety $Q_p = \\lbrace z \\in \\Delta : \\rho (z,\\bar{p}) = 0 \\rbrace .$ Of course the variety depends on the defining equation itself and the polydisc $\\Delta $ .",
"For $\\rho $ it is useful to take the defining equation or equations that generate the ideal of the complexified $X$ in $N \\times N$ at $p$ .",
"If $\\rho $ is polynomial we take $\\Delta = N$ .",
"It is well-known that any irreducible complex variety that lies in $X$ and goes through the point $p$ also lies in $Q_p$ .",
"In case of Levi-flat submanifolds we generally get equality as germs.",
"For example, for the CR Levi-flat submanifold $M$ given by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 ,$ the Segre variety $Q_0$ through the origin is precisely $\\lbrace z_1 = z_2 =0\\rbrace $ , which happens to be the unique complex variety in $M$ through the origin.",
"Let us take the Levi-flat quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) .$ As we want to take the generating equations in the complexified space we also need the conjugate $\\bar{w} = \\bar{A}(\\bar{z},z) + \\bar{B}(z,z) .$ The Segre variety is then given by $w = 0, \\qquad \\bar{B}(z,z) = 0 .$ Through any CR singular point of a real-analytic Levi-flat $M$ there is a complex variety of dimension $n-1$ that is the limit of the leaves of the Levi-foliation of $M_{CR}$ , via Lemma REF .",
"Let us take all possible such limits, and call their union $Q^{\\prime }_p$ .",
"Notice that there could be other complex varieties in $M$ through $p$ of dimension $n-1$ .",
"Note that $Q^{\\prime }_p \\subset Q_p$ .",
"Let us write down and classify the Segre varieties for all the quadric Levi-flat submanifolds in ${n+1}$ : Table: NO_CAPTIONThe submanifold C.0 also contains the complex variety $\\lbrace w = 0, z_2 = 0 \\rbrace $ , but this variety is transversal to the leaves of the foliation, and so cannot be in $Q^{\\prime }_0$ Notice that in the cases A.$k$ for all $k$ , B.$\\gamma $ for $\\gamma > 0$ , and C.1, the variety $Q_0$ actually gives the complex variety $Q^{\\prime }_0$ contained in $M$ through the origin.",
"In these cases, the variety is nonsingular only in the set theoretic sense.",
"Scheme-theoretically the variety is always at least a double line or double hyperplane in general."
],
[
"The CR singularity of Levi-flats quadrics",
"Let us study the set of CR singularities for Levi-flat quadrics.",
"The following proposition is well-known.",
"Proposition 5.1 Let $M \\subset {n+1}$ be given by $w = \\rho (z,\\bar{z})$ where $\\rho $ is $O(2)$ , and $M$ is not a complex submanifold.",
"Then the set $S$ of CR singularities of $M$ is given by $S = \\lbrace (z,w) : \\bar{\\partial } \\rho = 0, w = \\rho (z,\\bar{z}) \\rbrace .$ In codimension 2, a real submanifold is either CR singular, complex, or generic.",
"A submanifold is generic if $\\bar{\\partial }$ of all the defining equations are pointwise linearly independent (see ).",
"As $M$ is not complex, to find the set of CR singularities, we find the set of points where $M$ is not generic.",
"We need both defining equations for $M$ , $w = \\rho (z,\\bar{z}), \\qquad \\text{and} \\qquad \\bar{w} = \\rho (z,\\bar{z}) .$ As the second equation always produces a $d\\bar{w}$ while the first does not, the only way that the two can be linearly dependent is for the $\\bar{\\partial }$ of the first equation to be zero.",
"In other words $\\bar{\\partial } \\rho = 0$ .",
"Let us compute and classify the CR singular sets for the CR singular Levi-flat quadrics.",
"Table: NO_CAPTIONBy Levi-flat we mean that $S$ is a Levi-flat CR submanifold in $\\lbrace w = 0 \\rbrace $ .",
"There is a conjecture that a real subvariety that is Levi-flat at CR points has a stratification by Levi-flat CR submanifolds.",
"This computation gives further evidence of this conjecture."
],
[
"Levi-foliations and images of generic Levi-flats",
"A CR Levi-flat submanifold $M \\subset n$ of codimension 2 has a certain canonical foliation defined on it with complex analytic leaves of real codimension 2 in $M$ .",
"The submanifold $M$ is locally equivalent to ${\\mathbb {R}}^2 \\times {n-2}$ , defined by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 .$ The leaves of the foliation are the submanifolds given by fixing $z_1$ and $z_2$ at a real constant.",
"By foliation we always mean the standard nonsingular foliation as locally comes up in the implicit function theorem.",
"This foliation on $M$ is called the Levi-foliation.",
"It is obvious that the Levi-foliation on $M$ extends to a neighbourhood of $M$ as a nonsingular holomorphic foliation.",
"The same is not true in general for CR singular submanifolds.",
"We say that a smooth holomorphic foliation ${\\mathcal {L}}$ defined in a neighborhood of $M$ is an extension of the Levi-foliation of $M_{CR}$ , if ${\\mathcal {L}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"We also say that a smooth real-analytic foliation $\\widetilde{{\\mathcal {L}}}$ on $M$ is an extension of the Levi-foliation on $M_{CR}$ if $\\widetilde{{\\mathcal {L}}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"In our situation (real-analytic), $M_{CR}$ is a dense and open subset of $M$ .",
"This implies that the leaves of ${\\mathcal {L}}$ and $\\widetilde{{\\mathcal {L}}}$ through a CR singular point are complex analytic submanifolds contained in $M$ .",
"The latter could lead to an obvious obstruction to extension.",
"First let us see what happens if the foliation of $M_{CR}$ is the restriction of a nonsingular holomorphic foliation of a whole neighbourhood of $M$ .",
"The Bishop-like quadrics, that is A.1 and B.$\\gamma $ in ${n+1}$ , have a Levi-foliation that extends as a holomorphic foliation to all of ${n+1}$ .",
"That is because these submanifolds are of the form $ N \\times {n-1} .$ For submanifolds of the form (REF ) we can find normal forms using the well-developed theory of Bishop surfaces in 2.",
"Proposition 6.1 Suppose $M \\subset {n+1}$ is a real-analytic Levi-flat CR singular submanifold where the Levi-foliation on $M_{CR}$ extends near $p \\in M$ to a nonsingular holomorphic foliation of a neighbourhood of $p$ in ${n+1}$ .",
"Then at $p$ , $M$ is locally biholomorphically equivalent to a submanifold of the form $ N \\times {n-1}$ where $N \\subset 2$ is a CR singular submanifold of real dimension 2.",
"Therefore if $M$ has a nondegenerate complex tangent, then it is Bishop-like, that is of type A.1 or B.$\\gamma $ .",
"Furthermore, two submanifolds of the form (REF ) are locally biholomorphically (resp.",
"formally) equivalent if and only if the corresponding $N$ s are locally biholomorphically (resp.",
"formally) equivalent in 2.",
"We flatten the holomorphic foliation near $p$ so that in some polydisc $\\Delta $ , the leaves of the foliation are given by $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ for $q \\in 2$ .",
"Let us suppose that $M$ is closed in $\\Delta $ .",
"At any CR point of $M$ , the leaf of the Levi-foliation agrees with the leaf of the holomorphic foliation and therefore the leaf that lies in $M$ agrees with a leaf of the form $\\lbrace q \\rbrace \\times {n-1}$ as a germ and so $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta \\subset M$ .",
"As $M_{CR}$ is dense in $M$ , then $M$ is a union of sets of the form $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ and the first part follows.",
"It is classical that every Bishop surface (2 dimensional real submanifold of 2 with a nondegenerate complex tangent) is equivalent to a submanifold whose quadratic part is of the form A.1 or B.$\\gamma $ .",
"Finally, the proof that two submanifolds of the form (REF ) are equivalent if and only if the $N$ s are equivalent is straightforward.",
"Not every Bishop-like submanifold is a cross product as above.",
"In fact the Bishop invariant may well change from point to point.",
"See § .",
"In such cases the foliation does not extend to a nonsingular holomorphic foliation of a neighbourhood.",
"Let us now focus on extending the Levi-foliation to $M$ , and not to a neighbourhood of $M$ .",
"Let us prove a useful proposition about recognizing certain CR singular Levi-flats from the form of the defining equation.",
"That is if the $r$ in the equation does not depend on $\\bar{z}_2$ through $\\bar{z}_n$ .",
"Proposition 6.2 Suppose near the origin $M \\subset {n+1}$ is given by $w = r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(2)$ and $\\frac{\\partial r}{\\bar{z}_1} \\lnot \\equiv 0$ .",
"Then $M$ is a CR singular Levi-flat submanifold and the Levi-foliation of $M_{CR}$ extends through the origin to a real-analytic foliation on $M$ .",
"Furthermore, there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"Near 0, $M$ is the image of a CR mapping that is a diffeomorphism onto its image of the standard CR Levi-flat.",
"The proposition also holds in two dimensions ($n=1$ ), although in this case it is somewhat trivial.",
"As in , let us define the mapping $F$ by $(x,y,\\xi ) \\mapsto \\bigl (x+iy, \\quad \\xi , \\quad r(x+iy,x-iy, \\xi ) \\bigr ) ,$ where $\\xi = (\\xi _2,\\ldots ,\\xi _n) \\in {n-1}$ .",
"Near points where $M$ is CR, this mapping is a CR diffeomorphism and hence $M$ must be Levi-flat.",
"Furthermore, since $F$ is a diffeomorphism, it takes the Levi-foliation on ${\\mathbb {R}}^2 \\times {n-1}$ to a foliation on $M$ near 0.",
"In fact, we make the following conclusion.",
"Lemma 6.3 Let $M \\subset {n+1}$ be a CR singular real-analytic Levi-flat submanifold of codimension 2 through the origin.",
"Then $M$ is a CR singular Levi-flat submanifold whose Levi-foliation of $M_{CR}$ extends through the origin to a nonsingular real-analytic foliation on $M$ if and only if there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"One direction is easy and was used above.",
"For the other direction, suppose that we have a foliation extending the Levi-foliation through the origin.",
"Let us consider $M_{CR}$ an abstract CR manifold.",
"That is a manifold $M_{CR}$ together with the bundle $T^{(0,1)} M_{CR}\\subset T M_{CR}$ .",
"The extended foliation on $M$ gives a real-analytic subbundle ${\\mathcal {W}}\\subset T M$ .",
"Since we are extending the Levi-foliation, when $p \\in M_{CR}$ , then ${\\mathcal {W}}_p = T_p^c M$ , where $T_p^c M = J(T_p^c M)$ is the complex tangent space and $J$ is the complex structure on ${n+1}$ .",
"Since $M_{CR}$ is dense in $M$ , then $J{\\mathcal {W}}={\\mathcal {W}}$ on $M$ .",
"Define the real-analytic subbundle ${\\mathcal {V}}\\subset T M$ as ${\\mathcal {V}}_p = \\lbrace X + iJ(X) : X \\in {\\mathcal {W}}_p \\rbrace .$ At CR points ${\\mathcal {V}}_p = T_p^{(0,1)} M$ (see for example page 8).",
"Then we can find vector fields $X^1,\\ldots ,X^{n-1}$ in ${\\mathcal {W}}$ such that $X^1,J(X^1),X^2,J(X^2),\\ldots ,X^{n-1},J(X^{n-1})$ is a basis of ${\\mathcal {W}}$ near the origin.",
"Then the basis for ${\\mathcal {V}}$ is given by $X^1+iJ(X^1),X^2+iJ(X^2),\\ldots ,X^{n-1}+iJ(X^{n-1}).$ As the subbundle is integrable, we obtain that $(M,{\\mathcal {V}})$ gives an abstract CR manifold, which at CR points agrees with $M_{CR}$ .",
"This manifold is Levi-flat as it is Levi-flat on a dense open set.",
"As it is real-analytic it is embeddable and hence there exists a real-analytic CR diffeomorphism from a neighbourhood of ${\\mathbb {R}}^2 \\times {n-1}$ to a neighbourhood of 0 in $M$ (as an abstract CR manifold).",
"This is our mapping $F$ .",
"The quadrics A.$k$ , $k \\ge 2$ , defined by $w = \\bar{z}_1^2 + \\cdots + \\bar{z}_k^2 ,$ contain the singular variety defined by $w = 0$ , $z_1^2 + \\cdots + z_k^2 =0$ , and hence the Levi-foliation cannot extend to a nonsingular foliation of the submanifold.",
"The quadric A.1 does admit a holomorphic foliation, but other type A.1 submanifolds do not in general.",
"For example, the submanifold $w = \\bar{z}_1^2 + \\bar{z}_2^3$ is of type A.1 and the unique complex variety through the origin is $0 = z_1^2 + z_2^3$ , which is singular.",
"Therefore the foliation cannot extend to $M$ ."
],
[
"Extending the Levi-foliation of C.x type submanifolds",
"Let us prove Theorem REF , that is, let us start with a type C.0 or C.1 submanifold and show that the Levi-foliation must extend real-analytically to all of $M$ .",
"Equivalently, let us show that the real analytic bundle $T^{(1,0)}M_{CR}$ extends to a real analytic subbundle of $TM$ .",
"Taking real parts we obtain an involutive subbundle of $TM$ extending $T^cM_{CR} = \\operatorname{Re}(T^{(1,0)} M_{CR})$ .",
"Let $M$ be the submanifold given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $\\epsilon = 0,1$ .",
"Let us treat the $z$ variables as the parameters on $M$ .",
"Let $\\pi $ be the projection onto the $\\lbrace w=0 \\rbrace $ plane, which is tangent to $M$ at 0 as a real $2n$ -dimensional hyperplane.",
"We will look at all the vectorfields on this plane $\\lbrace w=0 \\rbrace $ .",
"All vectors in $\\pi (T^{(1,0)} M)$ can be written in terms of $\\frac{\\partial }{\\partial z_j}$ for $j=1,\\ldots ,n$ .",
"The Levi-map is given by taking the $n\\times n$ matrix $L = L(p) =\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}+\\begin{bmatrix}\\frac{\\partial ^2 r}{ \\partial z_j \\partial \\bar{z}_k}\\end{bmatrix}_{j,k}(p)$ to vectors $v \\in \\pi (T^{(1,0)} M)$ ($\\pi $ is the projection) as $v^* L v$ .",
"The excess term in $L$ vanishes at 0.",
"Notice that for $p \\in M_{CR}$ , $\\pi (T^{(1,0)}_p M)$ is $n-1$ dimensional.",
"As $M$ is Levi-flat, then $v^* L v$ vanishes for $v \\in \\pi (T^{(1,0)}_p M)$ .",
"Write the vector $v = (v_1,\\ldots ,v_n)^t$ .",
"The zero set of the function $(z,v) \\in n \\times n \\overset{\\varphi }{\\mapsto } v^* L(z,\\bar{z}) v$ is a variety $V$ of real codimension 2 at the origin of $\\mathbf {C}^n\\times \\mathbf {C}^n$ because of the form of $L$ .",
"That is, at $z=0$ , the only vectors $v$ such that $v^*Lv = 0$ are those where $v_1 = 0$ or $v_2 = 0$ .",
"So the codimension is at least 2.",
"And we know that $v^*Lv$ vanishes for vectors in $\\pi (T^{(1,0)}_p M)$ for $p \\in M$ near 0, which is real codimension 2 at each $z$ corresponding to a CR point.",
"Therefore, $V \\cap (\\pi (M_{CR}) \\times n )$ has a connected component that is equal to a connected component of the real-analytic subbundle $\\pi (T^{(1,0)} M_{CR})$ .",
"We will verify that the latter is connected.",
"We show below that this subbundle extends past the CR singularity.",
"The key point is to show that the restriction of $\\pi \\bigl (T^{(1,0)}(M_{CR})\\bigr )$ extends to a smooth real-analytic submanifold of $T^{(1,0)} n$ .",
"Write $\\varphi (z,v) = v_1\\bar{v}_2 + \\sum a_{jk}(z) v_j\\bar{v}_k$ where $a_{jk}(0) = 0$ .",
"By Proposition REF , $\\pi (M\\setminus M_{CR})$ is contained in $z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1}=0.$ Thus $M_{CR}$ is connected.",
"Assume that $v\\cdot \\frac{\\partial }{\\partial z}\\in T^{(1,0)}_pM$ at a CR point $p$ .",
"Then $(z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1})\\overline{v}_1+\\sum _{j>1} r_{\\overline{z}_j}\\overline{v}_j=0.$ When $p$ is in the open set $U_\\delta \\subset \\pi (M_{CR})$ defined by $\\left|{z_2+2\\epsilon \\overline{z}_1} \\right|>\\left|{z} \\right|/2$ and $0<\\left|{z} \\right|<\\delta $ , $v$ is contained in $V_C\\colon \\left|{v_1} \\right|\\le \\left|{v} \\right|/C.$ When $\\delta $ is sufficiently small, $\\varphi (z,v)=0$ admits a unique solution $v_1=f(z,v_3,\\dots , v_n), \\quad v_2=1$ by imposing $v\\in V_C$ .",
"Note that $f$ is given by convergent power series.",
"For $\\left|{z} \\right|<\\delta $ , define $w_j=\\bigl (w_{j1}(z),\\dots , w_{jn}(z)\\bigr )\\in V_C, \\quad j=2,\\dots , n$ such that $\\varphi (z,w_j(z))=0$ and $ w_{j2}=1, \\quad w_{jk}=\\delta _{jk},\\quad j\\ge 2, k>2.$ To see why we can do so, fix $p\\in U_\\delta $ .",
"First we can find a vector $w_2$ in $E_p=\\pi (T_p^{(1,0)}M_{CR})$ such that $v_2=1$ .",
"Otherwise, $E_p\\subset V_C$ cannnot have dimension $n-1$ .",
"Let $E_p^{\\prime }$ be the vector subspace of $E_p$ with $v_2=0$ .",
"Then $E_p^{\\prime }$ has rank $n-2$ and remains in the cone $V_C$ .",
"Then $E_p^{\\prime }$ has an element $w_2$ with $v_2$ component being 1.",
"Repeating this, we find $w_2,\\dots , w_n$ in $E_p$ such that the $v_j$ component of $w_i$ is 0 for $2<j<i$ .",
"Using linear combinations, we find a unique basis $\\lbrace w_2,\\dots , w_n \\rbrace $ of $E_p$ that satisfies condition (REF ).",
"Assume that $C$ is sufficiently large.",
"By the above uniqueness assertion on $\\varphi (z,v)=0$ , we conclude that when $p\\in U_\\delta $ , $\\lbrace w_{2}(p),\\dots , w_n(p)\\rbrace $ is a base of $\\pi (T^{(1,0)}_pM_{CR})$ .",
"Also it is real analytic at $p=0$ .",
"Define $\\omega _j(z)=w_{j}(z)\\cdot \\frac{\\partial }{\\partial z}, \\quad \\left|{z} \\right|<\\delta .$ We lift the functions $\\omega _j$ via $\\pi $ to a subbundle of $TM$ , let us call these $\\widetilde{\\omega }_j$ .",
"Then consider the vector fields $w^*_j = 2 \\operatorname{Re}\\widetilde{\\omega }_j = \\widetilde{\\omega }_j + \\overline{\\widetilde{\\omega }_j}$ and $w^*_{n+j}=\\operatorname{Im}\\widetilde{\\omega }_j$ for $j=2,\\dots , n$ .",
"Above CR points over $U_\\delta $ , $\\tilde{w}_j$ is in $T M_{CR}\\otimes and so tangentto $ M$.",
"We thus obtain a $ 2n-2$ dimensional real analytic subbundle of $ TM$that agrees with the real analytic realsubbundle of $ TMCR$ induced by the Levi-foliation above $ U$.",
"Since $ MCR$and the subbunldes are real analytic and $ MCR$ is connected, they agree over $ MCR$.$ The real analytic distribution spanned by $\\lbrace \\omega ^*_i\\rbrace $ has constant rank ($2n-2$ ) everywhere and is involutive on an open subset of $M_{CR}$ and hence everywhere."
],
[
"CR singular set of type C.x submanifolds",
"Let $M \\subset {n+1}$ be a codimension two Levi-flat CR singular submanifold that is an image of ${\\mathbb {R}}^2 \\times {n-1}$ via a real-analytic CR map, and let $S \\subset M$ be the CR singular set of $M$ .",
"In it was proved that near a generic point of $S$ exactly one of the following is true: $S$ is Levi-flat submanifold of dimension $2n-2$ and CR dimension $n-2$ .",
"$S$ is a complex submanifold of complex dimension $n-1$ (real dimension $2n-2$ ).",
"$S$ is Levi-flat submanifold of dimension $2n-1$ and CR dimension $n-1$ .",
"We only have the above classification for a generic point of $S$ , and $S$ need not be a CR submanifold everywhere.",
"See for examples.",
"If $M$ is a Levi-flat CR singular submanifold and the Levi-foliation of $M_{CR}$ extends to $M$ , then by Lemma REF at a generic point $S$ has to be of one of the above types.",
"A corollary of Theorem REF is the following result.",
"Corollary 8.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a real-analytic Levi-flat CR singular type C.1 or type C.0 submanifold.",
"Let $S \\subset M$ denote the CR singular set.",
"Then near the origin $S$ is a submanifold of dimension $2n-2$ , and at a generic point, $S$ is either CR Levi-flat of dimension $2n-2$ (CR dimension $n-2$ ) or a complex submanifold of complex dimension $n-1$ .",
"Furthermore, if $M$ is of type C.1, then at the origin $S$ is a CR Levi-flat submanifold of dimension $2n-2$ (CR dimension $n-2$ ).",
"Let us take $M$ to be given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $r$ is $O(3)$ and $\\epsilon = 0$ or $\\epsilon = 1$ .",
"By Proposition REF the CR singular set is exactly where $z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0, \\qquad \\text{and} \\qquad r_{\\bar{z}_j}(z,\\bar{z}) = 0 \\quad \\text{for all $j=2,\\ldots ,n$}.$ By considering the real and imaginary parts of the first equation and applying the implicit function theorem the set $\\widetilde{S} = \\lbrace z : z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) =0 \\rbrace $ is a real submanifold of real dimension $2n-2$ (real codimension 2 in $M$ ).",
"Now $S \\subset \\widetilde{S}$ , but as we saw above $S$ is of dimension at least $2n-2$ .",
"Therefore $S = \\widetilde{S}$ near the origin.",
"The conclusion of the first part then follows from the classification above.",
"The stronger conclusion for C.1 submanifolds follows by noticing that when $\\epsilon = 1$ , the submanifold $z_2 + 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0$ is CR and not complex at the origin."
],
[
"Mixed-holomorphic submanifolds",
"Let us study sets in $m$ defined by $ f(\\bar{z}_1,z_2,\\ldots ,z_m) = 0 ,$ for a single holomorphic function $f$ of $m$ variables.",
"Such sets have much in common with complex varieties, since they are in fact complex varieties when $\\bar{z}_1$ is treated as a complex variable.",
"The distinction is that the automorphism group is different since we are interested in automorphisms that are holomorphic not mixed-holomorphic.",
"Proposition 9.1 If $M \\subset m$ is a submanifold with a defining equation of the form (REF ), where $f$ is a holomorphic function that is not identically zero, then $M$ is a real codimension 2 set and $M$ is either a complex submanifold or a Levi-flat submanifold, possibly CR singular.",
"Furthermore, if $M$ is CR singular at $p \\in M$ , and has a nondegenerate complex tangent at $p$ , then $M$ has type A.$k$ , C.0, or C.1 at $p$ .",
"Since the zero set of $f$ is a complex variety in the $(\\bar{z}_1,z_2,\\ldots ,z_m)$ space, we get automatically that it is real codimension 2.",
"We also have that as it is a submanifold, then it can be written as a graph of one variable over the rest.",
"Let $m = n+1$ for convenience and suppose that $M \\subset {n+1}$ is a submanifold through the origin.",
"By factorization for a germ of holomorphic function and by the smoothness assumption on $M$ we may assume that $df(0) \\ne 0$ .",
"Call the variables $(z_1,\\ldots ,z_n,w)$ and write $M$ as a graph.",
"One possibility is that we write $M$ as $\\bar{w} = \\rho (z_1,\\ldots ,z_n),$ where $\\rho (0) = 0$ and $\\rho $ has no linear terms.",
"$M$ is complex if $\\rho \\equiv 0$ .",
"Otherwise $M$ is CR singular and we rewrite it as $w = \\bar{\\rho }(\\bar{z}_1,\\ldots ,\\bar{z}_n).$ We notice that the matrix representing the Levi-map must be identically zero, so we must get Levi-flat.",
"If there are any quadratic terms we obtain a type A.$k$ submanifold.",
"Alternatively $M$ can be written as $w = \\rho (\\bar{z}_1,z_2,\\ldots ,z_n),$ with $\\rho (0) = 0$ .",
"If $\\rho $ does not depend on $\\bar{z}_1$ then $M$ is complex.",
"Assume that $\\rho $ depends on $\\bar{z}_1$ .",
"If $\\rho $ has linear terms in $\\bar{z}_1$ , then $M$ is CR.",
"Otherwise it is a CR singular submanifold, and near non-CR singular points it is a generic codimension 2 submanifold.",
"The CR singular set of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\overline{z}_1}=0$ .",
"Suppose that $M$ is CR singular.",
"That $M$ is Levi-flat follows from Proposition REF .",
"We can therefore normalize the quadratic term, after linear terms in $z_2,\\ldots ,z_n$ are absorbed into $w$ .",
"If not all quadratic terms are zero, then we notice that we must have an A.$k$ , C.0, or C.1 type submanifold.",
"Let us now study normal forms for such sets in 2 and $m$ , $m \\ge 3$ .",
"First in two variables we can easily completely answer the question.",
"This result is surely well-known and classical.",
"Proposition 9.2 If $M \\subset 2$ is a submanifold with a defining equation of the form (REF ), then it is locally biholomorphically equivalent to a submanifold in coordinates $(z,w) \\in 2$ of the form $w = \\bar{z}^d$ for $d=0,1,2,3,\\ldots $ where $d$ is a local biholomorphic invariant of $M$ .",
"If $d=0$ , $M$ is complex, if $d=1$ it is a CR totally-real submanifold, and if $d \\ge 2$ then $M$ is CR singular.",
"Write the submanifold as a graph of one variable over the other.",
"Without loss of generality and after possibly taking a conjugate of the equation, we have $w = f(\\bar{z})$ for some holomorphic function $f$ .",
"Assume $f(0) = 0$ .",
"If $f$ is identically zero, then $d=0$ and we are finished.",
"If $f$ is not identically zero, then it is locally biholomorphic to a positive power of the variable.",
"We apply a holomorphic change of coordinates in $z$ , and the rest follows easily.",
"In three or more variables, if $M \\subset {n+1}$ , $n \\ge 2$ , is a submanifold through the origin, then if the quadratic part is nonzero we have seen above that it can be a type A.$k$ , C.0, or C.1 submanifold.",
"If the submanifold is the nondegenerate type C.1 submanifold, then we will show in the next section that $M$ is biholomorphically equivalent to the quadric $M_{C.1}$ .",
"Before we move to C.1, let us quickly consider the mixed-holomorphic submanifolds of type A.$n$ .",
"The submanifolds of type A.$n$ in ${n+1}$ can in some sense be considered nondegenerate when talking about mixed-holomorphic submanifolds.",
"Proposition 9.3 If $M \\subset {n+1}$ is a submanifold of type A.$n$ at the origin of the form $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 + r(\\bar{z})$ where $r \\in O(3)$ .",
"Then $M$ is locally near the origin biholomorphically equivalent to the A.$n$ quadric $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 .$ The complex Morse lemma (see e.g.",
"Proposition 3.15 in ) states that there is a local change of coordinates near the origin in just the $z$ variables such that $z_1^2+ \\cdots + z_n^2 + \\bar{r}(z)$ is equivalent to $z_1^2+ \\cdots + z_n^2$ .",
"It is not difficult to see that the normal form for mixed-holomorphic submanifolds in ${n+1}$ of type A.$k$ , $k < n$ , is equivalent to a local normal form for a holomorphic function in $n$ variables.",
"Therefore for example the submanifold $w = \\bar{z}_1^2 +\\bar{z}_2^3$ is of type A.1 and is not equivalent to any quadric."
],
[
"Formal normal form for certain C.1 type submanifolds I",
"In this section we prove the formal normal form in Theorem REF .",
"That is, we prove that if $M \\subset {n+1}$ is defined by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(3)$ , then $M$ is Levi-flat and formally equivalent to $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ That $M$ is Levi-flat follows from Proposition REF .",
"Lemma 10.1 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ formal power series then $M$ is formally equivalent to $M_{C.1}$ given by $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ In fact, the normalizing transformation can be of the form $(z,w) = (z_1,\\ldots ,z_n,w)\\mapsto \\bigl (z_1, \\quad f(z,w), \\quad z_3, \\quad \\ldots , \\quad z_n, \\quad g(z,w)\\bigr ) ,$ where $f$ and $g$ are formal power series.",
"Suppose that the normalization was done to degree $d-1$ , then suppose that $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) +r_2(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) ,$ where $r_1$ is degree $d$ homogeneous and $r_2$ is $O(d+1)$ .",
"Write $r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) =\\sum _{j=0}^k \\sum _{\\left|{\\alpha } \\right|+j = d} c_{j,\\alpha }\\bar{z}_1^j z^\\alpha ,$ where $k$ is the highest power of $\\bar{z}_1$ in $r_1$ , and $\\alpha $ is a multiindex.",
"If $k$ is even, then use the transformation that replaces $w$ with $w + \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{k/2} z^\\alpha .$ Let us look at the degree $d$ terms in $(\\bar{z}_1 z_2 + \\bar{z}_1^2)+ \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha }{(\\bar{z}_1 z_2 + \\bar{z}_1^2)}^{k/2} z^\\alpha =\\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ We need not include $r_2$ as the terms are all degree $d+1$ or more.",
"After cancelling out the new terms on the left, we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"Next suppose that $k$ is odd.",
"We use the transformation that replaces $z_2$ with $z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha .$ Let us look at the degree $d$ terms in $\\bar{z}_1 z_2 + \\bar{z}_1^2=\\bar{z}_1 \\left(z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha \\right)+ \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ Again we need not include $r_2$ as the terms are all degree $d+1$ or more, and we need not add the new terms to $z_2$ in the argument list for $r_1$ since all those terms would be of higher degree.",
"Again we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"The procedure above does not change the form of the submanifold, but it lowers the degree of $\\bar{z}_1$ by one.",
"Since we can assume that all terms in $r_1$ depend on $\\bar{z}_1$ , we are finished with degree $d$ terms after $k$ iterations of the above procedure."
],
[
"Convergence of normalization for certain C.1 type submanifolds",
"A key point in the computation below is the following natural involution for the quadric $M_{C.1}$ .",
"Notice that the map $(z_1,z_2,\\ldots ,z_n,w) \\mapsto (-\\bar{z}_2-z_1, \\quad z_2, \\quad \\ldots ,\\quad z_n, \\quad w)$ takes $M_{C.1}$ to itself.",
"The involution simply replaces the $\\bar{z}_1$ in the equation with $-z_2-\\bar{z}_1$ .",
"The way this involution is defined is by noticing that the equation $w = \\bar{z}_1 z_2 + \\bar{z}_1^2$ has generically two solutions for $\\bar{z}_1$ keeping $z_2$ and $w$ fixed.",
"In the same way we could define an involution on all type C.1 submanifolds of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1,z_2,\\ldots ,z_n)$ , although we will not require this construction.",
"We prove convergence via the following well-known lemma: Lemma 11.1 Let $m_1, \\ldots , m_N$ be positive integers.",
"Suppose $T(z)$ is a formal power series in $z \\in N$ .",
"Suppose $T(t^{m_1}v_1,\\ldots , t^{m_N}v_N)$ is a convergent power series in $t \\in forall $ v N$.",
"Then $ T$ is convergent.$ The proof is a standard application of the Baire category theorem and the Cauchy inequality.",
"See (Theorem 5.5.30, p. 153) where all $m_j$ are 1.",
"For $m_j > 1$ we first change variables by setting $v_j = w_j^{m_j}$ and apply the lemma with $m_j=1$ .",
"The following lemma finishes the proof of Theorem REF .",
"By absorbing any holomorphic terms into $w$ , we assume that $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"In Lemma REF we have also constructed a formal transformation that only changed the $z_2$ and $w$ coordinates, so it is enough to prove convergence in this case.",
"Key points of this proof are that the right hand side of the defining equation for $M_{C.1}$ is homogeneous, and that we have a natural involution on $M_{C.1}$ .",
"Lemma 11.2 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ and convergent, and $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"Suppose that two formal power series $f(z,w)$ and $g(z,w)$ satisfy $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{z}_1 f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ \\bar{z}_1^2 + r(z_1,\\bar{z}_1,f(z,\\bar{z}_1z_2 +\\bar{z}_1^2),z_3,\\ldots ,z_n) .$ Then $f$ and $g$ are convergent.",
"The equation (REF ) is true formally, treating $z_1$ and $\\bar{z}_1$ as independent variables.",
"Notice that (REF ) has one equation for 2 unknown functions.",
"We now use the involution on $M_{C.1}$ to create a system that we can solve uniquely.",
"We replace $\\bar{z}_1$ with $-z_2-\\bar{z}_1$ .",
"We leave $z_1$ untouched (treating as an independent variable).",
"We obtain an identity in formal power series: $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = (-z_2-\\bar{z}_1) f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ (-z_2-\\bar{z}_1)^2 \\\\+ r(z_1,(-z_2-\\bar{z}_1),f(z,\\bar{z}_1z_2 + \\bar{z}_1^2),z_3,\\ldots ,z_n) .$ The formal series $\\xi = f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ and $\\omega = g(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ are solutions of the system $\\omega & = \\bar{z}_1 \\xi + \\bar{z}_1^2+ r(z_1,\\bar{z}_1,\\xi ,z_3,\\ldots ,z_n) , \\\\\\omega & = (-z_2-\\bar{z}_1) \\xi + (-z_2-\\bar{z}_1)^2+ r(z_1,(-z_2-\\bar{z}_1),\\xi ,z_3,\\ldots ,z_n) .$ We next replace $z_j$ with $t z_j$ and $\\bar{z}_1$ with $t \\bar{z}_1$ for $t \\in .",
"Because $ z1z2 + z12$ is homogeneous ofdegree 2, we obtain that for every $ (z1,z1,z2,...,zn) n+1$ the formal series in $ t$ given by$ (t) = f(tz,t2(z1z2 + z12))$,$ (t) = g(tz,t2(z1z2 + z12))$are solutions of the system{\\begin{@align}{1}{-1}\\omega & = t \\bar{z}_1 \\xi + t^2 \\bar{z}_1^2+ r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) , \\\\\\omega & = t (-z_2-\\bar{z}_1) \\xi + t^2 (-z_2-\\bar{z}_1)^2+ r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n) .\\end{@align}}We eliminate $$ to obtain an equation for $$:\\begin{equation}t (2 \\bar{z}_1 + z_2) ( \\xi - t z_2)=\\\\r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) .\\end{equation}We now treat $$ as a variable and we have a holomorphic (convergent)equation.",
"The right hand size must be divisible by$ t (2 z1 + z2)$: It is divisible by $ t$ since$ r$ was divisible by $ z1$.",
"It is also divisible by$ 2 z1 + z2$ as setting $ z2 = -2 z1$ makes the righthand side vanish.",
"Therefore,\\begin{equation}\\xi - t z_2=\\frac{r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n)}{t (2 \\bar{z}_1 + z_2)} ,\\end{equation}where the right hand side is a holomorphic function (that is, a convergentpower series) in $ z1,z1,z2,...,zn,t,$.For any fixed $ z1,z1,z2,...,zn$, we solve for $$ in terms of $ t$via the implicit function theorem,and we obtain that $$ is a holomorphicfunction of $ t$.",
"The power series of $$ is given by$ (t) = f(tz,t2(z1z2 + z12))$.$ Let $v \\in {n+1}$ be any nonzero vector.",
"Via a proper choice of $z_1,\\bar{z}_1,z_2,\\ldots ,z_n$ (still treating $\\bar{z}_1$ and $z_1$ as independent variables) we write $v =(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ .",
"We apply the above argument to $\\xi (t) = f(tv_1,\\ldots , tv_n, t^2v_{n+1})$ , and $\\xi (t)$ converges as a series in $t$ .",
"As we get convergence for every $v \\in {n+1}$ we obtain that $f$ converges by Lemma REF .",
"Once $f$ converges, then via () we obtain that $g(tv_1,\\ldots , tv_n, t^2v_{n+1})$ converges as a series in $t$ for all $v$ , and hence $g$ converges."
],
[
"Automorphism group of the C.1 quadric",
"With the normal form achieved in previous sections, let us study the automorphism group of the C.1 quadric in this section.",
"We will again use the mixed-holomorphic involution that is obtained from the quadric.",
"We study the local automorphism group at the origin.",
"That is the set of germs at the origin of biholomorphic transformations taking $M$ to $M$ and fixing the origin.",
"First we look at the linear parts of automorphisms.",
"We already know that the linear term of the last component only depends on $w$ .",
"For $M_{C.1}$ we can say more about the first two components.",
"Proposition 12.1 Let $(F,G) = (F_1,\\ldots ,F_n,G)$ be a formal invertible or biholomorphic automorphism of $M_{C.1} \\subset {n+1}$ , that is the submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Then $F_1(z,w) = a z_1 + \\alpha w + O(2)$ , $F_2(z,w) = \\bar{a} z_2 + \\beta w + O(2)$ , and $G(z,w) = \\bar{a}^2 w + O(2)$ , where $a \\ne 0$ .",
"Let $a = (a_1,\\ldots ,a_n)$ and $b = (b_1,\\ldots ,b_n)$ be such that $F_1(z,w) = a \\cdot z + \\alpha w + O(2)$ and $F_2(z,w) = b \\cdot z + \\beta w + O(2)$ .",
"Then from Proposition REF we have $\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} a \\\\ b \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1 b_2 = 1$ , and $\\bar{a}_j b_k = 0$ for all $(j,k) \\ne (1,2)$ .",
"Similarly $\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} \\bar{a} \\\\ \\bar{b} \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1^2 = 1$ , and $\\bar{a}_j \\bar{a}_k = 0$ for all $(j,k)\\ne (1,1)$ .",
"Putting these two together we obtain that $a_j = 0$ for all $j \\ne 1$ , and as $a_1 \\ne 0$ we get $b_j = 0$ for all $j \\ne 2$ .",
"As $\\lambda $ is the reciprocal of the coefficient of $w$ in $G$ , we are finished.",
"Lemma 12.2 Let $M_{C.1} \\subset 3$ be given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Suppose that a local biholomorphism (resp.",
"formal automorphism) $(F_1,F_2,G)$ transforms $M_{C.1}$ into $M_{C.1}$ .",
"Then $F_1$ depends only on $z_1$ , and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"Let us define a $(1,0)$ tangent vector field on $M$ by $Z=\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} .$ Write $F = (F_1,F_2,G)$ .",
"$F$ must take $Z$ into a multiple of itself when restricted to $M_{C.1}$ .",
"That is on $M_{C.1}$ we have $& \\frac{\\partial F_1}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_1}{\\partial w}= 0 ,\\\\& \\frac{\\partial F_2}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_2}{\\partial w}= \\lambda ,\\\\& \\frac{\\partial G}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial G}{\\partial w}= \\lambda \\overline{F_1}(\\bar{z},\\bar{w}) ,$ for some function $\\lambda $ .",
"Let us take the first equation and plug in the defining equation for $M_1$ : $ \\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+\\bar{z}_1\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This equation is true for all $z \\in 2$ , and so we may treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We have an involution on $M_{C.1}$ that takes $\\bar{z}_1$ to $-z_2-\\bar{z}_1$ .",
"Therefore we also have $\\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+(-z_2-\\bar{z}_1)\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This means that $\\frac{\\partial F_1}{\\partial w}$ and therefore $\\frac{\\partial F_1}{\\partial z_2}$ must be identically zero.",
"That is, $F_1$ only depends on $z_1$ .",
"We have that the following must hold for all $z$ : $G(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)F_2(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+{\\left(\\overline{F_1}(\\bar{z}_1) \\right)}^2 .$ Again we treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We differentiate with respect to $z_1$ : $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ We plug in the involution again to obtain $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(-z_2-\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ Therefore as $F_1$ is not identically zero, then as before both $\\frac{\\partial F_2}{\\partial z_1}$ and $\\frac{\\partial G}{\\partial z_1}$ must be identically zero.",
"Lemma 12.3 Take $M_{C.1} \\subset 3$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and let $(F_1,F_2,G)$ be a local automorphism at the origin.",
"Then $F_1$ uniquely determines $F_2$ and $G$ .",
"Furthermore, given any invertible function of one variable $F_1$ with $F_1(0) = 0$ , there exist unique $F_2$ and $G$ that complete an automorphism and they are determined by $ \\begin{aligned}F_2(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = \\bar{F}_1(\\bar{z}_1)+\\bar{F}_1(-\\bar{z}_1-z_2),\\\\G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = -\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2).\\end{aligned}$ We should note that the lemma also works formally.",
"Given any formal $F_1$ , there exist unique formal $F_2$ and $G$ satisfying the above property.",
"By Lemma REF , $F_1$ depends only on $z_1$ and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"We write the automorphism as a composition of the two mappings $\\bigl (F_1(z_1),z_2,w\\bigr )$ and $\\bigl (z_1,F_2(z_2,w),G(z_2,w)\\bigr )$ .",
"We plug the transformation into the defining equation for $M_{C.1}$ .",
"$ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(\\bar{z}_1)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2 .$ We use the involution $(z_1,z_2) \\mapsto (-\\bar{z}_1-z_2,z_2)$ which preserves $M_{C.1}$ and obtain a second equation $ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(-\\bar{z}_1-z_2)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2 .$ We eliminate $G$ and solve for $F_2$ : $ F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= \\frac{{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2-{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2}{\\bar{F}_1(\\bar{z}_1)-\\bar{F}_1(-\\bar{z}_1-z_2)}=\\bar{F}_1(\\bar{z}_1)+ \\bar{F}_1(-\\bar{z}_1-z_2) .$ Next we note that trivially, $F_2$ is unique if it exists: its difference vanishes on $M_{C.1}$ .",
"If we suppose that $F_1$ is convergent, then just as before, substituting $z_2$ with $tz_2$ and $\\bar{z}_1$ with $t\\bar{z}_1$ , we are restricting to curves $(tz_2,t^2w)$ for all $(z_2,w)$ .",
"The series is convergent in $t$ for every fixed $z_2$ and $w$ .",
"Therefore if $F_2$ exists and $F_1$ is convergent, then $F_2$ is convergent by Lemma REF .",
"Now we need to show the existence of the formal solution $F_2$ .",
"Notice that the right-hand side of (REF ) is invariant under the involution.",
"It suffices to show that any power series in $\\bar{z_1}, z_2$ that is invariant under the involution is a formal power series in $z_2$ and $\\bar{z}_1z_2+\\bar{z}_1^2$ .",
"Let us treat $\\xi =\\bar{z}_1$ as an independent variable.",
"The original involution becomes a holomorphic involution in $\\xi ,z_2$ : $\\tau \\colon \\xi \\rightarrow -\\xi -z_2, \\qquad z_2\\rightarrow z_2.$ By a theorem of Noether we obtain a set of generators for the ring of invariants can be obtained by applying the averaging operation $R(f) = \\frac{1}{2} ( f + f \\circ \\tau )$ to all monomials in $\\xi $ and $z_2$ of degree 2 or less.",
"By direct calculation it is not difficult to see that $\\xi ,\\xi z_2+\\xi ^2$ generate the ring of invariants.",
"Therefore any invariant power series in $z_2,\\xi $ is a power series in $\\xi ,\\xi z_2+\\xi ^2$ .",
"This shows the existence of $F_2$ .",
"The existence of $G$ follows the same.",
"The equation for $G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)=-\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2)$ is obtained by plugging in the equation for $F_2$ .",
"Its existence, uniqueness, and convergence in case $F_1$ converges, follows exactly the same as for $F_2$ .",
"Theorem 12.4 If $M \\subset {n+1}$ , $n \\ge 2$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and $(F_1,F_2,\\ldots ,F_n,G)$ is a local automorphism at the origin, then $F_1$ depends only on $z_1$ , $F_2$ and $G$ depend only on $z_2$ and $w$ , and $F_1$ completely determines $F_2$ and $G$ via (REF ).",
"The mapping $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin.",
"Furthermore, given any invertible function $F_1$ of one variable with $F_1(0) = 0$ , and arbitrary holomorphic functions $F_3,\\ldots ,F_n$ with $F_j(0) = 0$ , and such that $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin, then there exist unique $F_2$ and $G$ that complete an automorphism.",
"Let $(F_1,\\ldots ,F_n,G)$ be an automorphism.",
"Then we have $G(z_1,\\ldots ,z_n,w) =\\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w})F_2(z_1,\\ldots ,z_n,w) +{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w}) \\bigr )}^2 .$ Proposition REF says that the linear terms in $G$ only depend on $w$ , the linear terms of $F_1$ depend only on $z_1$ and $w$ and the linear terms of $F_2$ only depend on $z_2$ and $w$ .",
"Let us embed $M_{C.1} \\subset 3$ into $M$ via setting $z_3 = \\alpha _3 z_2$ , $\\ldots $ , $z_n = \\alpha _n z_2$ , for arbitrary $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Then we obtain $ G(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) = \\\\\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w})F_2(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w}) \\bigr )}^2 .$ By noting what the linear terms are, we notice that the above is the equation for an automorphism of $M_{C.1}$ .",
"Therefore by Lemma REF we have $\\frac{\\partial F_1}{\\partial w} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial F_2}{\\partial z_1} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial G}{\\partial z_1} = 0 ,$ as that is true for all $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Plugging in the defining equation for $M_{C.1}$ we obtain an equation that holds for all $z$ and we can treat $z$ and $\\bar{z}$ independently.",
"We plug in $z = 0$ to obtain $0 =\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0)F_2(0,\\ldots ,0,\\bar{z}_1^2) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0) \\bigr )}^2 .$ Differentiating with respect to $\\bar{\\alpha }_j$ we obtain $\\frac{\\partial F_1}{\\partial z_j} = 0$ , for $j=3,\\ldots ,n$ .",
"We set $\\bar{\\alpha }_j = 0$ in the equation, differentiate with respect to $\\bar{z}_2$ and obtain that $\\frac{\\partial F_1}{\\partial z_2} = 0$ .",
"In other words $F_1$ is a function of $z_1$ only.",
"We rewrite (REF ) by writing $F_1$ as a function of $z_1$ only and $F_2$ and $G$ as functions of $z_2,\\ldots ,z_n,w$ , and we plug in $w = \\bar{z}_1z_2 + \\bar{z}_1^2$ to obtain $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\\\\\overline{F_1}(\\bar{z}_1)F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) +{\\bigl ( \\overline{F_1}(\\bar{z}_1) \\bigr )}^2 .$ By Lemma REF , we know that $F_1$ now uniquely determines $F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ and $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ .",
"These two functions therefore do not depend on $\\alpha _3,\\ldots ,\\alpha _n$ , and in turn $F_2$ and $G$ do not depend on $z_3,\\ldots ,z_n$ as claimed.",
"Furthermore $F_1$ does uniquely determine $F_2$ and $G$ .",
"Finally since the mapping is a biholomorphism, and from what we know about the linear parts of $F_1$ , $F_2$ , and $G$ , it is clear that $(z_1,z_2,F_3,\\ldots ,F_n)$ is rank $n$ .",
"The other direction follows by applying Lemma REF .",
"We start with $F_1$ , determine $F_2$ and $G$ as in 3 dimensions.",
"Then adding $F_3,\\ldots ,F_n$ and the rank condition guarantees an automorphism."
],
[
"Normal form for certain C.1 type submanifolds II",
"The goal of this section is to find the normal form for Levi-flat submanifolds $M \\subset {n+1}$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + \\operatorname{Re}f(z) ,$ for a holomorphic $f(z)$ of order $O(3)$ .",
"Since $f(z)$ can be absorbed into $w$ via a holomorphic transformation, the goal is really to prove the following theorem.",
"Theorem 13.1 Let $M \\subset {n+1}$ be a real-analytic Levi-flat given by $ w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Then $M$ can be put into the $M_{C.1}$ normal form $ w = \\bar{z}_1z_2 + \\bar{z}_1^2 ,$ by a convergent normalizing transformation.",
"Furthermore, if $r$ is a polynomial and the coefficient of $\\bar{z}_1^3$ in $r$ is zero, then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"In Theorem REF , we have already shown that a submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ is necessarily Levi-flat and has the normal form $M_{C.1}$ .",
"The first part of Theorem REF will follow once we prove: Lemma 13.2 If $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z})$ where $r$ is $O(3)$ and $M$ is Levi-flat, then $r$ depends only on $\\bar{z}_1$ .",
"First let us assume that $n=2$ .",
"For $p \\in M_{CR}$ , $T^{(1,0)}_p M$ is one dimensional.",
"The Levi-map is the matrix $L =\\begin{bmatrix}0 & 1 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0\\end{bmatrix}$ applied to the $T^{(1,0)} M$ vectors.",
"As $M$ is Levi-flat, then the Levi-map has to vanish.",
"The only vectors $v$ for which $v^* L v = 0$ , are the ones without $\\frac{\\partial }{\\partial z_1}$ component or $\\frac{\\partial }{\\partial z_2}$ component.",
"That is vectors of the form $a \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial w},\\qquad \\text{or} \\qquad a \\frac{\\partial }{\\partial z_2} + b \\frac{\\partial }{\\partial w}.$ We apply these vectors to the defining equation and its conjugate and we obtain in the first case the equations $b = 0, \\qquad a \\left( \\bar{z}_2 + 2z_1 + \\frac{\\partial \\bar{r}}{\\partial z_1} \\right) = 0 .$ This cannot be satisfied identically on $M$ since this is supposed to be true for all $z$ , but $a$ cannot be identically zero and the second factor in the second equation has only one nonholomorphic term, which is $\\bar{z}_2$ .",
"Let us try the second form and we obtain the equations $b = a \\bar{z}_1 , \\qquad a \\left( \\frac{\\partial \\bar{r}}{\\partial z_2} \\right) = 0 .$ Again $a$ cannot be identically zero, and hence the second factor of the second equation $\\frac{\\partial \\bar{r}}{\\partial z_2}$ must be identically zero, which is possible only if $r$ depends only on $\\bar{z}_1$ .",
"Finally, it is possible to pick $b=\\bar{z}_1$ and $a=1$ , to obtain a $T^{(1,0)}$ vector field $\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} ,$ and therefore these submanifolds are necessarily Levi-flat.",
"Next suppose that $n > 2$ .",
"Notice that replacing $z_k$ with $\\lambda _k \\xi $ for $k \\ge 2$ and then fixing $\\lambda _k$ for $k \\ge 2$ , we get $w = \\bar{z}_1 \\lambda _2 \\xi + \\bar{z}_1^2 + r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi }) .$ By Lemma REF , we obtain a Levi-flat submanifold in $(z_1,\\xi ,w) \\in 3$ , and hence can apply the above reasoning to obtain that $r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi })$ does not depend on $\\bar{\\xi }$ .",
"As this was true for any $\\lambda _k$ 's, we have that $r$ can only depend on $\\bar{z}_1$ .",
"It is left to prove the claim about the polynomial normalizing transformation.",
"Lemma 13.3 Suppose $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ where $r$ is a polynomial that vanishes to fourth order.",
"Then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"We will take a transformation of the form $(z_1,z_2,w) \\mapsto \\bigl (z_1,z_2+f(z_2,w),w+g(z_2,w) \\bigr ) .$ We are therefore trying to find polynomial $f$ and $g$ that satisfy $\\bar{z}_1z_2 + \\bar{z}_1^2+g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)=\\bar{z}_1 \\bigl (z_2 +f(z_2,\\bar{z}_1z_2 +\\bar{z}_1^2)\\bigr ) + \\bar{z}_1^2 + r(\\bar{z}_1) .$ If we simplify we obtain $g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)-\\bar{z}_1 f(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= r(\\bar{z}_1) .$ Consider the involution $S\\colon (\\bar{z}_1,z_2)\\rightarrow (-\\bar{z}_1-z_2,z_2)$ .",
"Its invariant polynomials $u(\\bar{z}_1,z_2)$ are precisely the polynomials in $z_2,z_2\\bar{z}_1+\\bar{z}_1^2$ .",
"The polynomial $r(\\bar{z}_1)$ can be uniquely written as $r^+(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)+\\Bigl (\\bar{z}_1+\\frac{z_2}{2}\\Bigr )r^-(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)$ in two polynomials $r^\\pm $ .",
"Taking $f=-r^-$ and $g=r^++\\frac{z_2}{2}r^-$ , we find the desired solutions."
],
[
"Normal form for general type C.1 submanifolds",
"In this section we show that generically a Levi-flat type C.1 submanifold is not formally equivalent to the quadric $M_{C.1}$ submanifold.",
"In fact, we find a formal normal form that shows infinitely many invariants.",
"There are obviously infinitely many invariants if we do not impose the Levi-flat condition.",
"The trick therefore is, how to impose the Levi-flat condition and still obtain a formal normal form.",
"Let $M \\subset 3$ be a real-analytic Levi-flat type C.1 submanifold through the origin.",
"We know that $M$ is an image of ${\\mathbb {R}}^2 \\times under a real-analytic CR map that is a diffeomorphism onto itstarget; see Theorem~\\ref {thm:folextendsCxtype}.After a linear change of coordinates we assume thatthe mapping is\\begin{equation}\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (& x+iy + a(x,y,\\xi ), \\\\& \\xi + b(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) ,\\end{split}\\end{equation}where $ a$, $ b$ are $ O(2)$ and $ r$ is $ O(3)$.",
"As the mapping is a CR mapping and a local diffeomorphism, then given anysuch $ a$, $ b$, and $ r$, the image isnecessarily Levi-flat at CR points.",
"Therefore the set of all thesemappings gives us all type C.1 Levi-flat submanifolds.$ We precompose with an automorphism of ${\\mathbb {R}}^2 \\times to make $ b = 0$.We cannot similarly remove $ a$ as anyautomorphism must have real valued first two components (the new $ x$ and thenew $ y$), and hence thosecomponents can only depend on $ x$ and $ y$ and not on $$.",
"So if $ a$depends on $$, we cannot remove it by precomposing.$ Next we notice that we can treat $M$ as an abstract CR manifold.",
"Suppose we have two equivalent submanifolds $M_1$ and $M_2$ , with $F$ being the biholomorphic map taking $M_1$ to $M_2$ .",
"If $M_j$ is the image of a map $\\varphi _j$ , then note that $\\varphi _2^{-1}$ is CR on ${(M_2)}_{CR}$ .",
"Therefore, $G = \\varphi _2^{-1} \\circ F \\circ \\varphi _1$ is CR on ${(F \\circ \\varphi _1)}^{-1}\\bigl ({(M_2)}_{CR}\\bigr )$ , which is dense in a neighbourhood of the origin of ${\\mathbb {R}}^2 \\times (theCR singularity of $ M2$ is a thin set, and we pull it back by tworeal-analytic diffeomorphisms).",
"A real-analytic diffeomorphism thatis CR on a dense set is a CR mapping.",
"The same argumentworks for the inverse of $ G$,and therefore we have a CR diffeomorphism of $ R2 .",
"The conclusion we make is the following proposition.",
"Proposition 14.1 If $M_j \\subset 3$ , $j=1,2$ are given by the maps $\\varphi _j$ $\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times {\\varphi _j}{\\mapsto }\\bigl (& x+iy + a_j(x,y,\\xi ), \\\\& \\xi + b_j(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r_j(x,y,\\xi )\\bigr ) ,\\end{split}$ and $M_1$ and $M_2$ are locally biholomorphically (resp.",
"formally) equivalent at 0, then there exists local biholomorphisms (resp.",
"formal equivalences) $F$ and $G$ at 0, with $F(M_1) = M_2$ , $G({\\mathbb {R}}^2 \\times = {\\mathbb {R}}^2 \\times as germs(resp.\\ formally) and\\begin{equation}\\varphi _2 = F \\circ \\varphi _1 \\circ G .\\end{equation}$ In other words, the proposition states that if we find a normal form for the mapping we find a normal form for the submanifolds.",
"Let us prove that the proposition also works formally.",
"We have to prove that $G$ restricted to ${\\mathbb {R}}^2 \\times is CR, that is,$ G = 0$.",
"Let us consider\\begin{equation}\\varphi _2 \\circ G = F \\circ \\varphi _1 .\\end{equation}The right hand side does not depend on $$ and thus the left handside does not either.",
"Write $ G = (G1,G2,G3)$.",
"Let us write $ b = b2$and $ r = r2$for simplicity.",
"Taking derivative of $ 2 G$ with respectto $$ we get:\\begin{equation}\\begin{aligned}& G^1_{\\bar{\\xi }} +i G^2_{\\bar{\\xi }} +a_x(G) G^1_{\\bar{\\xi }} +a_y(G) G^2_{\\bar{\\xi }} +a_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\& G^3_{\\bar{\\xi }} +b_x(G) G^1_{\\bar{\\xi }} +b_y(G) G^2_{\\bar{\\xi }} +b_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\&(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}) G^3+(G^1 - i G^2) G^3_{\\bar{\\xi }} +2 (G^1 - i G^2)(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }})\\\\& \\qquad +r_x(G) G^1_{\\bar{\\xi }} +r_y(G) G^2_{\\bar{\\xi }} +r_\\xi (G) G^3_{\\bar{\\xi }} = 0 .\\end{aligned}\\end{equation}Suppose that the homogeneous parts of $ Gj$ are zero for alldegrees up to degree $ d-1$.",
"If we look at the degree $ d$ homogeneous partsof the first two equations above we immediately note that it must be that$ G1 + i G2 = 0$ and$ G3 = 0$ in degree $ d$.",
"We then look at the degree $ d+1$part of the third equation.",
"Recall that $ []d$ is the degree $ d$part of an expression.",
"We get\\begin{equation}{[G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}]}_{d}{[G^3 + 2 G^1 - i 2 G^2]}_{1} = 0 .\\end{equation}As $ G$ is an automorphism we cannot have the linear terms be linearlydependent and hence$ G1 = G2 = 0$ in degree $ d$.",
"We finishby induction on $ d$.$ Using the proposition we can restate the result of Theorem REF using the parametrization.",
"Corollary 14.2 A real-analytic Levi-flat type C.1 submanifold $M \\subset 3$ is biholomorphically equivalent to the quadric $M_{C.1}$ if and only if the mapping giving $M$ is equivalent to a mapping of the form $(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (x+iy,\\quad \\xi ,\\quad (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) .$ That is, $M$ is equivalent to $M_{C.1}$ if and only if we can get rid of the $a(x,y,\\xi )$ via pre and post composing with automorphisms.",
"The proof of the corollary follows as a submanifold that is realized by this map must be of the form $w = \\bar{z}_1z_2 + \\bar{z}_1^2 + \\rho (z_1,\\bar{z}_1,z_2)$ and we apply Theorem REF .",
"We have seen that the involution $\\tau $ on $M$ , in particular when $M$ is the quadric, is useful to compute the automorphism group and to construct Levi-flat submanifolds of type $C.1$ .",
"We will also need to deal with power series in $z,\\bar{z}, \\xi $ .",
"Thus we extend $\\tau $ , which is originally defined on 2, as follows $\\sigma (z,\\bar{z},\\xi )=(z,-\\bar{z}-\\xi ,\\xi ).$ Here $z,\\bar{z},\\xi $ are treated as independent variables.",
"Note that $z,\\xi ,w=\\bar{z}\\xi +\\bar{z}^2$ are invariant by $\\sigma $ , while $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ is skew invariant by $\\sigma $ .",
"A power series in $z,\\bar{z},\\xi $ that is invariant by $\\sigma $ is precisely a power series in $z,\\xi ,w$ .",
"In general, a power series $u$ in $z,\\bar{z},\\xi $ admits a unique decomposition $u(z,\\bar{z},\\xi )=u^+(z,\\xi ,w)+\\eta u^-(z,\\xi ,w).$ First we introduce degree for power series $u(z,\\bar{z},\\xi )$ and weights for power series $v(z,\\xi ,w)$ .",
"As usual we assign degree $i+j+k$ to the monomial $z^i\\bar{z}^j\\xi ^k$ .",
"We assign weight $i+j+2k$ to the monomial $z^i\\xi ^jw^k$ .",
"For simplicity, we will call them weight in both situations.",
"Let us also denote $[u]_d(z,\\bar{z},\\xi )=\\sum _{i+j+k=d}u_{ijk}z^i\\bar{z}^j\\xi ^k, \\quad [v]_d(z,\\xi ,w)=\\sum _{i+j+2k=d}v_{ijk}z^i\\xi ^jw^k.$ Set $[u]_{i}^j=[u]_i+\\cdots +[u]_j$ and $[v]_i^j=[v]_i+\\cdots +[v]_j$ for $i\\le j$ .",
"Theorem 14.3 Let $M$ be a real-analytic Levi-flat type C.1 submanifold in 3.",
"There exists a formal biholomorphic map transforming $M$ into the image of $\\hat{\\varphi }(z,\\bar{z},\\xi )=\\bigl (z+A(z,\\xi , w)w\\eta , \\xi ,w\\bigr )$ with $\\eta =\\bar{z}+\\frac{1}{2}{\\xi }$ and $w=\\bar{z}\\xi +\\bar{z}^2$ .",
"Suppose further that $A\\lnot \\equiv 0$ .",
"Fix $i_*,j_*,k_*$ such that $j_*$ is the largest integer satisfying $A_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Then we can achieve $A_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ Furthermore, the power series $A$ is uniquely determined up to the transformation $A(z,\\xi ,w)\\rightarrow \\bar{c}^{3}A(cz,\\bar{c}\\xi ,\\bar{c}^2w), \\quad c\\in \\lbrace 0\\rbrace .$ In the above normal form with $A\\lnot \\equiv 0$ , the group of formal biholomorphisms that preserve the normal form consists of dilations $(z,\\xi ,w)\\rightarrow (\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)$ satisfying $\\bar{\\nu }^{3}A(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)=A(z,\\xi ,w)$ .",
"It will be convenient to write the CR diffeomorphism $G$ of ${\\mathbb {R}}^2\\times as $ (G1,G2)$ where $ G1$ is complex-valued and depends on $ z,z$,while $ G2$ depends on $ z,z,$.",
"Let $ M$ be the image of a mapping $$ defined by\\begin{equation}\\begin{aligned}(z,\\bar{z},\\xi ) \\overset{\\varphi }{\\mapsto }\\bigl (& z + a(z,\\bar{z},\\xi ),\\\\& \\xi ,\\\\& \\bar{z} \\xi + {\\bar{z}}^2 + r(z,\\bar{z},\\xi )\\bigr )\\end{aligned}\\end{equation}with $ a=O(2), r=O(3)$.",
"We want to find a formal biholomorphic map $ F$ of $ 3$and a formal CR diffeomorphism $ G$ of $ R2 such that $F\\hat{\\varphi }G^{-1}=\\varphi $ with $\\hat{\\varphi }$ in the normal form.",
"To simplify the computation, we will first achieve a preliminary normal form where $r=0$ and the function $a$ is skew-invariant by $\\sigma $ .",
"For the preliminary normal form we will only apply $F, G$ that are tangent to the identity.",
"We will then use the general $F, G$ to obtain the final normal form.",
"Let us assume that $F, G$ are tangent to the identity.",
"Let $M=F\\bigl (\\hat{\\varphi }({\\mathbb {R}}^2\\times \\bigr )$ where $\\hat{\\varphi }$ is determined by $\\hat{a}, \\hat{r}$ .",
"We write $F=I+(f_1,f_2,f_3), \\quad G=I+(g_1,g_2).$ The $\\xi $ components in $\\varphi G=F\\hat{\\varphi }$ give us $g_2(z,\\bar{z},\\xi )=f_2\\bigl (z+\\hat{a}(z,\\bar{z},\\xi ), \\xi , \\bar{z}\\xi +\\bar{z}^2+\\hat{r}(z,\\bar{z},\\xi )\\bigr ).$ Thus, we are allowed to define $g_2$ by the above identity for any choice of $f_2=O(2)$ .",
"Eliminating $g_2$ in other components of $\\varphi G=F\\hat{\\varphi }$ , we obtain $f_1\\circ \\hat{\\varphi }-g_1&= a\\circ G-\\hat{a},\\\\f_3\\circ \\hat{\\varphi }-\\bar{z}f_2\\circ \\hat{\\varphi }&=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2,$ where $\\tilde{g}_1(z,\\bar{z})=\\bar{g}_1(\\bar{z},z)$ and $(a\\circ G)(z,\\bar{z},\\xi ):=a\\bigl (G_1(z,\\bar{z}),\\bar{G}_1(\\bar{z},z),G_2(z,\\bar{z},\\xi )\\bigr ).$ Each power series $r(z,\\bar{z},\\xi )$ admits a unique decomposition $r(z,\\bar{z},\\xi )=r^+(z,\\xi ,w)+\\eta r^-(z,\\xi ,w),$ where both $r^\\pm $ are invariant by $\\sigma $ .",
"Note that $r(z,\\bar{z},\\xi )$ is a power series in $z,\\xi $ and $w$ , if and only if it is invariant by $\\sigma $ , i.e.",
"if $r^-=0$ .",
"We write $r^+={wt}\\, (k), \\quad \\text{or}\\quad {wt} \\, (r^+)\\ge k,$ if $r^+_{abc}=0$ for $a+b+2c<k$ .",
"Define $r^-=wt (k)$ analogously and write $\\eta r^-={wt}\\,(k)$ if $r^-={wt}\\,(k-1)$ .",
"We write $r={wt}\\,(k)$ if $(r^+,\\eta r^-)={wt}\\,(k)$ .",
"Note that $r=O(k)\\Rightarrow r={wt}\\,(k); \\quad wt \\, (rs)\\ge wt\\, (r)+wt\\,(s).$ The power series in $z,\\bar{z}$ play a special role in describing normal forms.",
"Let us define $T^\\pm $ via $u(z,\\bar{z})=(T^+u)(z,\\xi ,w)+(T^-u)(z,\\xi ,w)\\eta .$ Let $S^+_k$ (resp.",
"$S^-_k$ ) be spanned by monomials in $z,\\bar{z},\\xi $ which have weight $k$ and are invariant (resp.",
"skew-invariant) by $\\sigma $ .",
"Then the range of $\\eta T^-$ in $S_k^-$ is a linear subspace $R_k$ .",
"We decompose $S_k^-=R_k\\oplus (S_k^-\\ominus R_k).$ The decomposition is of course not unique.",
"We will take $S_k^-\\ominus R_k=\\bigoplus _{a+b+2c=k-1, c>0} z̏^a\\xi ^bw^c\\eta .$ Here we have used $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ , $\\eta ^2=w+\\frac{1}{4}\\xi ^2$ , and $T^+u(z,\\xi , w)=\\sum _{i,j\\ge 0}\\sum _{0\\le \\alpha \\le j/2} u_{ij} \\binom{j}{2\\alpha }z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha },\\\\T^-u(z,\\xi ,w)=\\sum _{i\\ge 0,j>0}\\sum _{0\\le \\alpha <j/2} u_{ij}\\binom{j}{2\\alpha +1}z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha -1}.$ In particular, we have $T^-u(z,\\xi ,0)=\\sum _{i\\ge 0,j>0}(-1)^{j-1}u_{ij} z^i \\xi ^{j-1}.$ This shows that $T^-u(z,\\xi ,0)=\\frac{1}{-\\xi }\\bigl (u(z,-\\xi )-u(z,0)\\bigr ).$ We are ready to show that under the condition that $g_1(z,\\bar{z})$ has no pure holomorphic terms, there exists a unique $(F,G)$ which is tangent to the identity such that $\\hat{r}=0$ and $\\hat{a}\\in \\mathcal {N}:=\\bigoplus \\mathcal {N}_k, \\quad \\mathcal {N}_k:=S_k^-\\ominus R_k.$ We start with terms of weight 2 in (REF )-() to get $[f_1]_2-[g_1]_2=[a]_2-\\eta [\\hat{a}^-]_1,\\\\[f_3]_2=0.$ Note that $f_j^-=0$ .",
"The first identity implies that $[f_1]_2-[T^+g_1]_2=[a^+]_2, \\quad [T^-g_1]_1=[\\hat{a}^-]_1-[a^-]_1.$ The first equation is solvable with kernel defined by $[f_1]_k-[T^+g_1]_k=0 $ for $k=2$ .",
"This shows that $[g_1]_2$ is still arbitrary and we use it to achieve $\\eta [\\hat{a}^-]_1\\in S_2^-\\ominus R_2=\\lbrace 0\\rbrace .$ Then the kernel space is defined by (REF ) and $[g_1(z,\\bar{z})-g_1(z,0)]_k=0$ with $k=2$ .",
"In particular, under the restriction $[g_1(z,0)]_k=0,$ for $k=2$ , we have achieved $\\hat{a}^-\\in \\mathcal {N}_2$ by unique $[f_1]_2, [g_1]_2, [f_2]_1, [f_3]_2$ .",
"By induction, we verify that if (REF ) holds for all $k$ , we determine uniquely $[f_1]_k, [g_1]_k$ by normalizing $[\\hat{a}]_k\\in \\mathcal {N}_k$ .",
"We then determine $[f_2]_k, [f_3]_{k+1}$ uniquely to normalize $[\\hat{r}]_{k+1}=0$ .",
"For the details, let us find formula for the solutions.",
"We rewrite (REF ) as $T^-g_1=-(a\\circ G-\\hat{a}-f_1\\circ \\hat{\\varphi })^-,\\\\( f_1\\circ \\hat{\\varphi })^+=(a\\circ G-\\hat{a})^++T^+g_1.$ Using (REF ), we can solve $(-1)^{j-1}g_{1,ij}=-({(a\\circ G)}^-)_{i(j-1)0}, \\quad j\\ge 1, \\quad i+j=k.$ Then we have $(\\hat{a}^-)_{ij0}=0, \\quad i+j=k-1; \\\\(\\hat{a}^-)_{ij m}=((a\\circ G-f_1\\circ \\hat{\\varphi }+g_1)^-)_{ijm}, \\quad m\\ge 1, i+j+m=k-1.$ Note that $- [g_1]_k(z,-\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,\\bar{z},0)$ .",
"We obtain $[g_1]_k(z,\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,-\\bar{z},0).$ Having determined $[g_1]_k$ , we take $[ f_1]_k=[(a\\circ G-\\hat{a}+g_1)^+]_k.$ We then solve () by taking $[ f_2]_k=[E^-]_k, \\quad [f_3]_{k+1}=[(E-\\frac{1}{2}\\xi f_2)^+]_{k+1},\\\\E:=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2.$ We have achieved the preliminary normalization.",
"Assume now that $\\varphi (z,\\bar{z},\\xi )=(z+a^-(z,\\xi ,w)\\eta , \\xi ,w), \\quad \\hat{\\varphi }(z,\\bar{z},\\xi )=(z+\\hat{a}^-(z,\\xi ,w)\\eta , \\xi ,w)$ are in the preliminary normal form, i.e.",
"$w|a^-(z,\\xi ,w), \\quad w|\\hat{a}^-(z,\\xi ,w).$ Let us assume that $a^-(z,\\xi ,w)=wt (s), \\quad [a^-]_s\\lnot \\equiv 0; \\quad \\hat{a}^-(z,\\xi ,w)=wt(s).$ We assume that $\\varphi G=F\\hat{\\varphi }$ with $F(z,\\xi ,w)=I+(f_1,f_2,f_3),\\\\G(z,\\bar{z},\\xi )=(z+g_1(z,\\bar{z}), \\xi +g_2(z,\\bar{z},\\xi )).$ Here $f_i,g_j$ start with terms of weight and order at least 2.",
"In particular, we have $f_i=wt(N), \\quad g_i=wt (N),\\quad i=1,2; \\quad f_3=wt(N^{\\prime }); \\quad N^{\\prime }\\ge N\\ge 2.$ Set $(P,Q,R):=\\varphi G$ .",
"Using $N\\ge 2$ , $s\\ge 2$ , and the Taylor theorem, we obtain $P&=z+g_1(z,\\bar{z})+a^-(z,\\xi ,w)\\eta +a^-(z,\\xi ,w)(\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi ))\\\\&\\quad +\\eta \\nabla a^-(z,\\xi ,w)\\cdot \\Bigl (g_1(z,\\bar{z}), g_2(z,\\bar{z},\\xi ),(\\xi +2\\bar{z})\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )\\Bigr )\\\\&\\quad +wt(s+N+1),\\\\Q&=\\xi +g_2(z,\\bar{z},\\xi ),\\\\R&=w+(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N).$ We also have $(P,Q,R)=F\\hat{\\varphi }$ .",
"Thus $P&=z+\\hat{a}^-(z,\\xi ,w)\\eta +f_1(z,\\xi ,w)+\\partial _zf_1(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\Q&=\\xi +f_2(z,\\xi ,w)+\\partial _zf_2(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\R&=w+f_3(z,\\xi ,w)+\\partial _zf_3(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N^{\\prime }+s+1).$ We will use the above 6 identities for $P,Q,R$ in two ways.",
"First we use their lower order terms to get $f_1(z,\\xi ,w)=g_1(z,\\bar{z})+( a^-(z,\\xi ,w)-\\hat{a}^-(z,\\xi ,w))\\eta +wt(N+s),\\\\\\quad f_2(z,\\xi ,w)=g_2(z,\\bar{z},\\xi )+wt(N+s), \\\\ f_3(z,\\xi ,w)=(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N)+wt(N^{\\prime }+s).$ Hence, we can take $N^{\\prime }=N+1$ .",
"By (REF ) and the preliminary normalization, we first know that $\\hat{a}=a+wt(N+s-1), \\\\f_1(z,\\xi ,w)=b(z)+wt (N+s), \\quad g_1(z,\\bar{z})=b(z)+wt (N+s).",
"$ We compose () by $\\sigma $ and then take the difference of the two equations to get $f_2(z,\\xi ,w)=-\\bar{b}(\\bar{z})-\\bar{b}( -\\bar{z}-\\xi )+wt(2N-1)+wt(N+s), \\\\f_3(z,\\xi ,w)=-\\bar{z}\\bar{b}(-\\bar{z}-\\xi )+(\\bar{z}+\\xi )\\bar{b}(\\bar{z})+wt(2N)+wt(N+s+1).$ Here we have used $N^{\\prime }=N+1$ .",
"Let $b(z)=b_Nz^N+wt(N+1)$ .",
"Therefore, we have $g_2(z,\\bar{z},\\xi )=-\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N)+wt(N+1),\\\\\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi )=\\eta \\bar{b}_N\\sum \\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}+wt(N+1),\\\\(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )=\\bar{b}_N(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})w+wt(N+2).$ Next, we use the two formulae for $P$ and (REF ) to get the identity in higher weight: $\\hat{a}^-&=a^-+g_1^-+Lb_N+wt(N+s), \\quad f_1-g_1^+=wt (N+s+1).$ Here we have used $f_1^-=0$ and $Lb_N(z,\\xi ,w)&:=-Nb_Nz^{N-1} [a^-]_s(z,\\xi ,w)-[a^-]_s(z,\\xi ,w)\\bar{b}_N\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}\\nonumber \\\\&\\quad +\\nabla [a^-]_s\\cdot \\Bigl (b_Nz^N, -\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N),\\bar{b}_Nw(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})\\Bigr ).$ Recall that $w|a^-$ and $w|\\hat{a}^-$ .",
"We also have that $w|Lb_N(z,\\xi ,w)$ and $Lb_N$ is homogenous in weighted variables and of weight $N+s-1$ .",
"This shows that $[g_1^-(z,\\xi ,0)]_{N+s-1}=0$ .",
"By (REF ), we get $[g_1(z,\\bar{z})]_{N+s}=[g_1(z,0)]_{N+s}, \\quad [\\hat{a}^-]_{s+N-1}=[ a^-]_{s+N-1}+Lb_N.$ Let us make some observations.",
"First, $Lb_N$ depends only on $b_N$ and it does not depend on coefficients of $b(z)$ of degree larger than $N$ .",
"We observe that the first identity says that all coefficients of $[g_1]_{N+s}$ must be zero, except that the coefficient $g_{1,(N+s)0}$ is arbitrary.",
"On the other hand $Lb_N$ , which has weight $N+s-1$ , depends only on $g_{1,N0}$ , while $N+s-1>N$ .",
"Let us assume for the moment that we have $Lb_N\\ne 0$ for all $b_N\\ne 0$ .",
"We will then choose a suitable complement subspace ${\\mathcal {N}}^*_{N+s-1}$ in the space of weighted homogenous polynomials in $z,\\xi ,w$ of weight $N+s-1$ for $Lb_N$ .",
"Then $\\hat{a}^-\\in w\\sum _{N>1}{\\mathcal {N}}^*_{N+s-1}$ will be the required normal form.",
"The normal form will be obtained by the following procedures: Assume that $\\varphi $ is not formally equivalent to the quadratic mapping in the preliminary normalization.",
"We first achieve the preliminary normal form by a mapping $F^0=I+(f_1^0,f_2^0,f_3^0)$ and $G^0=I+(g_1^0,g_2^0)$ which are tangent to the identity.",
"We can make $F^0,G^0$ to be unique by requiring $f^1_1(z,0)=0$ .",
"Then $a$ is normalized such that $\\hat{a}=\\hat{a}^-\\eta $ with $[\\hat{a}^-]_s$ being non-zero homogenous part of the lowest weight.",
"We may assume that $[a]_{s+1}=[\\hat{a}]_{s+1}$ .",
"Inductively, we choose $f^1_{1,N00}$ ($N=2, 3, \\ldots $ ) to achieve $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ .",
"In this step for a given $N$ , we determine mappings $F^1=I+(f_1^1,f_2^1,f_3^1)$ and $G^1=I+(g_1^1,g_2^1)$ by requiring that $f_1^1(z,\\xi ,w)$ contains only one term $\\xi ^N$ , while $f_1^1,f_2^1,g_1^1,g_2^1$ have weight at most $N$ and $f_3^1$ has weight at most $N+1$ .",
"In the process, we also show that $[f_1^1(z,\\xi ,w)]_2^{N+s}$ depends only on $z$ , if we do not want to impose the restriction on $f_1^1$ .",
"Moreover, the coefficient of $\\xi ^{N+s-1}$ of $f_1^1$ can still be arbitrarily chosen without changing the normalization achieved for $[\\hat{a}^-]_{N+s-1}$ via $[f_1^1]_{N}$ .",
"However, by achieving $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ via $F^1,G^1$ , we may destroy the preliminary normalization achieved via $F_0,G_0$ .",
"We will then restore the preliminary normalization via $F^2=I+(f_1^2,f_2^2,f_3^2), G^2=I+(g_1^2,g_2^2)$ satisfying $g^2_1(z,0)=0$ .",
"This amounts to determining $g_1^2=g_1$ and $f_1^2=f_1$ via (REF ) and () for which the terms of weight at most $N+s$ have been determined by (REF ), and then $f_2^2=f_2,f_3^2=f_3,g_2^2=g_2$ are determined by (REF )-() and (), respectively.",
"This allows us to repeat the procedure to achieve the normalization in any higher weight.",
"We will then remove the restriction that the normalizing mappings must be tangent to the identity.",
"This will alter the normal form only by suitable linear dilations.",
"Suppose that $b_N\\ne 0$ .",
"Let us verify that $Lb_N\\ne 0.$ We will also identify one of non-zero coefficients to describe the normalizing condition on $\\hat{a}$ .",
"We write the two invariant polynomials $\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^jw^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^jw^k.$ If we plug in $w=\\bar{z}^2+\\bar{z}\\xi $ we obtain a polynomial identity in the variables $z,\\bar{z},\\xi $ .",
"$\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k.$ If we set $\\bar{z} = z = 0$ , we obtain that $\\lambda _N = \\lambda ^{\\prime }_N = {(-1)}^N .$ Recall that $j_*$ is the largest integer such that $(a^-)_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Since $w|[a^-]_s$ , then $k_*>0$ .",
"We obtain $(Lb_N)_{i_*(j_*+N-1)k_*}=(a^-)_{i_*j_*k_*}\\bar{b}_N(-\\lambda _{N-1}^{\\prime }-j_*\\lambda _{N-1}+k_*\\lambda _N)\\ne 0.$ Therefore, we can achieve $(\\hat{a}^-)_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ This determines uniquely all $b_2, b_3, \\ldots .$ We now remove the restriction that $F$ and $G$ are tangent to the identity.",
"Suppose that both $\\varphi $ and $\\hat{\\varphi }$ are in the normal form.",
"Suppose that $F\\varphi =\\hat{\\varphi }G$ .",
"Then looking at the quadratic terms, we know that the linear parts $F,G$ must be dilations.",
"In fact, the linear part of $F$ must be the linear automorphism of the quadric.",
"Thus the linear parts of $F$ and $G$ have the forms $G^{\\prime }\\colon (z,\\xi )=(\\nu z,\\bar{\\nu }\\xi ), \\quad F^{\\prime }(z,\\xi ,w)=(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w).$ Then $(F^{\\prime })^{-1}\\hat{\\varphi }G^{\\prime }$ is still in the normal form.",
"Since $(F^{\\prime })^{-1}F$ is holomorphic and $(G^{\\prime })^{-1}G$ is CR, by the uniqueness of the normalization, we know that $F^{\\prime }=F$ and $G^{\\prime }=G$ .",
"Therefore, $F$ and $G$ change the normal form $a^-$ as follows $a^-(z,\\xi ,w)= \\bar{\\nu }\\hat{a}^-(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w), \\quad \\nu \\in \\lbrace 0\\rbrace .$ When $[\\hat{a}^-]_s=[a^-]_s\\ne 0$ , we see that $|\\nu |=1$ .",
"Therefore, the formal automorphism group is discrete or one-dimensional.",
"In , Coffman used an analogous method of even/odd function decomposition to obtain a quadratic normal form for non Levi-flat real analytic $m$ -submanifolds in $n$ with an CR singularity satisfying certain non-degeneracy conditions, provided $\\frac{3(n+1)}{2} \\le m<n$ .",
"He was able to achieve the convergent normalization by a rapid iteration method.",
"Using the above decomposition of invariant and skew-invariant functions of the involution $\\sigma $ , one might achieve a convergent solution for approximate equations when $M$ is formaly equivalent to the quadric.",
"However, when the iteration is employed, each new CR mapping $\\hat{\\varphi }$ might only be defined on a domain that is proportional to that of the previous $\\varphi $ in a constant factor.",
"This is significantly different from the situations of Moser and Coffman , , where rapid iteration methods are applicable.",
"Therefore, even if $M$ is formally equivalent to the quadric, we do not know if they are holomorphically equivalent.",
"Instability of Bishop-like submanifolds Let us now discuss stability of Levi-flat submanifolds under small perturbations that keep the submanifolds Levi-flat, in particular we discuss which quadratic invariants are stable when moving from point to point on the submanifold.",
"The only stable submanifolds are A.$n$ and C.1.",
"The Bishop-like submanifolds (or even just the Bishop invariant) are not stable under perturbation, which we show by constructing examples.",
"Proposition 15.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a connected real-analytic real codimension 2 submanifold that has a non-degenerate CR singular at the origin.",
"$M$ can be written in coordinates $(z,w) \\in {n} \\times as\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$.In a neighborhood of the origin all complex tangentsof $ M$ are non-degenerate, while ranks of $ A,B$ are upper semicontinuous.Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).The CR singular set of $ M$ that is not of type B.$ 12$ at the origin is areal analytic subset of $ M$ of codimension at least $ 2$, while the CRsingular set of $ M$ that is of type B.$ 12$ the origin has codimension atleast $ 1$.",
"A.$ n$ has an isolated CR singular point at the origin and sodoes C.1 in $ 3$.Let $ S0 M$ be the set of CR singular points.There is a neighborhood $ U$ of the origin such that for $ S=S0U$we have the following.\\begin{enumerate}[(i)]\\item If M is of type A.k for k \\ge 2 at the origin, then it is of type A.j at each pointof S for somej \\ge k.\\item If M is of type C.1 at the origin, then it is of type C.1 on S.If M is of type C.0 at the origin, then it is of type C.0 or C.1 on S.\\item There exists an M that is of type B.\\gamma at one point and ofC.1 at CR singular points arbitrarily near.",
"Similarly there exists an Mof type A.1 at p \\in M that is either of type C.1, or B.\\gamma , atpoints arbitrarily near p. There alsoexists an M of type B.\\gamma at every point but where \\gamma varies from point to point.\\end{enumerate}$ First we show that the rank of $A$ and the rank of $B$ are lower semicontinuous on $S_0$ , without imposing Levi-flatness condition.",
"Similarly the real dimension of the range of $A(z,\\bar{z})$ is lower semicontinuous on $S_0$ .",
"Write $M$ as $w = \\rho (z,\\bar{z}) ,$ where $\\rho $ vanishes to second order at 0.",
"If we move to a different point of $S_0$ via an affine map $(z,w) \\mapsto (Z+z_0,W+w_0)$ .",
"Then we have $W+w_0 = \\rho (Z+z_0,\\bar{Z}+\\bar{z}_0) .$ We compute the Taylor coefficients $W =\\frac{\\partial \\rho }{\\partial z} (z_0,\\bar{z}_0) \\cdot Z +\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z} + \\\\+Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^t\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial z} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] \\bar{Z} +O(3) .$ The holomorphic terms can be absorbed into $W$ .",
"If $\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z}$ is nonzero, then this complex defining function has a linear term in $W$ and linear term in $\\bar{Z}$ and the submanifold is CR at this point.",
"Therefore the set of complex tangents of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\bar{z}} =0$ and each complex tangent point is non-degenerate.",
"At a complex tangent point at the origin, $A$ is given by $\\left[ \\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ and $B$ is given by $\\frac{1}{2} \\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ .",
"In particular these matrices change continuously as we move along $S$ .",
"We first conclude that all CR singular points of $M$ in a neighborhood of the origin are non-degenerate.",
"Further holomorphic transformations act on $A$ and $B$ using Proposition REF .",
"Therefore the ranks of $A$ and $B$ as well as the real dimension of the range of $A(z,\\bar{z})$ are lower semicontinuous on $S_0$ as claimed.",
"Furthermore as $M$ is real-analytic, the points where the rank drops lie on a real-analytic subvariety of $S_0$ , or in other words a thin set.",
"Let $U$ be a small enough neighbourhood of the origin so that $S = S_0 \\cap U$ is connected.",
"Imposing the condition that $M$ is Levi-flat, we apply Theorem REF .",
"By a simple computation, unless $M$ is of type B.$\\frac{1}{2}$ , the set of complex tangents of $M$ has codimension at least 2; and A.$n$ has isolated CR singular point and so does C.1 in 3.",
"The item () follows as A.$k$ are the only types where the rank of $B$ is greater than 1, and the theorem says $M$ must be one of these types.",
"For () note that since $A$ is of rank 1 when $M$ as C.$x$ at a point, $M$ cannot be of type A.$k$ nearby.",
"If $M$ is of type C.1 at a point then the range of $A$ must be of real dimension 2 in a neighbourhood, and hence on this neighbourhood $M$ cannot be of type B.$\\gamma $ .",
"The examples proving item () are given below.",
"Example 15.2 Define $M$ via $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 + \\bar{z}_1z_2z_3 .$ It is Levi-flat by Proposition REF .",
"At the origin $M$ is a type B.$\\gamma $ , but at a point where $z_1 = z_2 = 0$ but $z_3 \\ne 0$ , the submanifold is CR singular and it is of type C.1.",
"Example 15.3 Similarly if we define $M$ via $w = \\bar{z}_1^2 + \\bar{z}_1z_2 z_3 ,$ we obtain a CR singular Levi-flat $M$ that is A.1 at the origin, but C.1 at nearby CR singular points.",
"Example 15.4 If we define $M$ via $w = \\gamma \\bar{z}_1^2 + \\left|{z_1} \\right|^2 z_2 ,$ then $M$ is a CR singular Levi-flat type A.1 submanifold at the origin, but type B.$\\gamma $ at points where $z_1 = 0$ but $z_2 \\ne 0$ .",
"Example 15.5 The Bishop invariant can vary from point to point.",
"Define $M$ via $w = \\left|{z_1} \\right|^2 + \\bar{z}_1^2 \\bigl (\\gamma _1 (1-z_2) + \\gamma _2 z_2 \\bigr ) ,$ where $\\gamma _1 , \\gamma _2 \\ge 0$ .",
"It is not hard to see that $M$ is Levi-flat.",
"Again it is an image of $2 \\times {\\mathbb {R}}^2$ in a similar way as above.",
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"Proposition REF says that if a foliation on $M$ extends to a (nonsingular) holomorphic foliation, then the submanifold would be a simple product of a Bishop submanifold and $.",
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],
[
"Quadratic Levi-flat submanifolds and their Segre varieties",
"A very useful invariant in CR geometry is the Segre variety.",
"Suppose that a real-analytic variety $X \\subset N$ is defined by $\\rho (z,\\bar{z}) = 0 ,$ where $\\rho $ is a real-analytic real vector-valued with $p \\in X$ .",
"Suppose that $\\rho $ converges on some polydisc $\\Delta $ centered at $p$ .",
"We complexify and treat $z$ and $\\bar{z}$ as independent variables, and the power series of $\\rho $ at $(p,\\bar{p})$ converges on $\\Delta \\times \\Delta $ .",
"The Segre variety at $p$ is then defined as the variety $Q_p = \\lbrace z \\in \\Delta : \\rho (z,\\bar{p}) = 0 \\rbrace .$ Of course the variety depends on the defining equation itself and the polydisc $\\Delta $ .",
"For $\\rho $ it is useful to take the defining equation or equations that generate the ideal of the complexified $X$ in $N \\times N$ at $p$ .",
"If $\\rho $ is polynomial we take $\\Delta = N$ .",
"It is well-known that any irreducible complex variety that lies in $X$ and goes through the point $p$ also lies in $Q_p$ .",
"In case of Levi-flat submanifolds we generally get equality as germs.",
"For example, for the CR Levi-flat submanifold $M$ given by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 ,$ the Segre variety $Q_0$ through the origin is precisely $\\lbrace z_1 = z_2 =0\\rbrace $ , which happens to be the unique complex variety in $M$ through the origin.",
"Let us take the Levi-flat quadric $w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) .$ As we want to take the generating equations in the complexified space we also need the conjugate $\\bar{w} = \\bar{A}(\\bar{z},z) + \\bar{B}(z,z) .$ The Segre variety is then given by $w = 0, \\qquad \\bar{B}(z,z) = 0 .$ Through any CR singular point of a real-analytic Levi-flat $M$ there is a complex variety of dimension $n-1$ that is the limit of the leaves of the Levi-foliation of $M_{CR}$ , via Lemma REF .",
"Let us take all possible such limits, and call their union $Q^{\\prime }_p$ .",
"Notice that there could be other complex varieties in $M$ through $p$ of dimension $n-1$ .",
"Note that $Q^{\\prime }_p \\subset Q_p$ .",
"Let us write down and classify the Segre varieties for all the quadric Levi-flat submanifolds in ${n+1}$ : Table: NO_CAPTIONThe submanifold C.0 also contains the complex variety $\\lbrace w = 0, z_2 = 0 \\rbrace $ , but this variety is transversal to the leaves of the foliation, and so cannot be in $Q^{\\prime }_0$ Notice that in the cases A.$k$ for all $k$ , B.$\\gamma $ for $\\gamma > 0$ , and C.1, the variety $Q_0$ actually gives the complex variety $Q^{\\prime }_0$ contained in $M$ through the origin.",
"In these cases, the variety is nonsingular only in the set theoretic sense.",
"Scheme-theoretically the variety is always at least a double line or double hyperplane in general."
],
[
"The CR singularity of Levi-flats quadrics",
"Let us study the set of CR singularities for Levi-flat quadrics.",
"The following proposition is well-known.",
"Proposition 5.1 Let $M \\subset {n+1}$ be given by $w = \\rho (z,\\bar{z})$ where $\\rho $ is $O(2)$ , and $M$ is not a complex submanifold.",
"Then the set $S$ of CR singularities of $M$ is given by $S = \\lbrace (z,w) : \\bar{\\partial } \\rho = 0, w = \\rho (z,\\bar{z}) \\rbrace .$ In codimension 2, a real submanifold is either CR singular, complex, or generic.",
"A submanifold is generic if $\\bar{\\partial }$ of all the defining equations are pointwise linearly independent (see ).",
"As $M$ is not complex, to find the set of CR singularities, we find the set of points where $M$ is not generic.",
"We need both defining equations for $M$ , $w = \\rho (z,\\bar{z}), \\qquad \\text{and} \\qquad \\bar{w} = \\rho (z,\\bar{z}) .$ As the second equation always produces a $d\\bar{w}$ while the first does not, the only way that the two can be linearly dependent is for the $\\bar{\\partial }$ of the first equation to be zero.",
"In other words $\\bar{\\partial } \\rho = 0$ .",
"Let us compute and classify the CR singular sets for the CR singular Levi-flat quadrics.",
"Table: NO_CAPTIONBy Levi-flat we mean that $S$ is a Levi-flat CR submanifold in $\\lbrace w = 0 \\rbrace $ .",
"There is a conjecture that a real subvariety that is Levi-flat at CR points has a stratification by Levi-flat CR submanifolds.",
"This computation gives further evidence of this conjecture."
],
[
"Levi-foliations and images of generic Levi-flats",
"A CR Levi-flat submanifold $M \\subset n$ of codimension 2 has a certain canonical foliation defined on it with complex analytic leaves of real codimension 2 in $M$ .",
"The submanifold $M$ is locally equivalent to ${\\mathbb {R}}^2 \\times {n-2}$ , defined by $\\operatorname{Im}z_1 = 0, \\qquad \\operatorname{Im}z_2 = 0 .$ The leaves of the foliation are the submanifolds given by fixing $z_1$ and $z_2$ at a real constant.",
"By foliation we always mean the standard nonsingular foliation as locally comes up in the implicit function theorem.",
"This foliation on $M$ is called the Levi-foliation.",
"It is obvious that the Levi-foliation on $M$ extends to a neighbourhood of $M$ as a nonsingular holomorphic foliation.",
"The same is not true in general for CR singular submanifolds.",
"We say that a smooth holomorphic foliation ${\\mathcal {L}}$ defined in a neighborhood of $M$ is an extension of the Levi-foliation of $M_{CR}$ , if ${\\mathcal {L}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"We also say that a smooth real-analytic foliation $\\widetilde{{\\mathcal {L}}}$ on $M$ is an extension of the Levi-foliation on $M_{CR}$ if $\\widetilde{{\\mathcal {L}}}$ and the Levi-foliation have the same germs of leaves at each CR point of $M$ .",
"In our situation (real-analytic), $M_{CR}$ is a dense and open subset of $M$ .",
"This implies that the leaves of ${\\mathcal {L}}$ and $\\widetilde{{\\mathcal {L}}}$ through a CR singular point are complex analytic submanifolds contained in $M$ .",
"The latter could lead to an obvious obstruction to extension.",
"First let us see what happens if the foliation of $M_{CR}$ is the restriction of a nonsingular holomorphic foliation of a whole neighbourhood of $M$ .",
"The Bishop-like quadrics, that is A.1 and B.$\\gamma $ in ${n+1}$ , have a Levi-foliation that extends as a holomorphic foliation to all of ${n+1}$ .",
"That is because these submanifolds are of the form $ N \\times {n-1} .$ For submanifolds of the form (REF ) we can find normal forms using the well-developed theory of Bishop surfaces in 2.",
"Proposition 6.1 Suppose $M \\subset {n+1}$ is a real-analytic Levi-flat CR singular submanifold where the Levi-foliation on $M_{CR}$ extends near $p \\in M$ to a nonsingular holomorphic foliation of a neighbourhood of $p$ in ${n+1}$ .",
"Then at $p$ , $M$ is locally biholomorphically equivalent to a submanifold of the form $ N \\times {n-1}$ where $N \\subset 2$ is a CR singular submanifold of real dimension 2.",
"Therefore if $M$ has a nondegenerate complex tangent, then it is Bishop-like, that is of type A.1 or B.$\\gamma $ .",
"Furthermore, two submanifolds of the form (REF ) are locally biholomorphically (resp.",
"formally) equivalent if and only if the corresponding $N$ s are locally biholomorphically (resp.",
"formally) equivalent in 2.",
"We flatten the holomorphic foliation near $p$ so that in some polydisc $\\Delta $ , the leaves of the foliation are given by $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ for $q \\in 2$ .",
"Let us suppose that $M$ is closed in $\\Delta $ .",
"At any CR point of $M$ , the leaf of the Levi-foliation agrees with the leaf of the holomorphic foliation and therefore the leaf that lies in $M$ agrees with a leaf of the form $\\lbrace q \\rbrace \\times {n-1}$ as a germ and so $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta \\subset M$ .",
"As $M_{CR}$ is dense in $M$ , then $M$ is a union of sets of the form $\\lbrace q \\rbrace \\times {n-1} \\cap \\Delta $ and the first part follows.",
"It is classical that every Bishop surface (2 dimensional real submanifold of 2 with a nondegenerate complex tangent) is equivalent to a submanifold whose quadratic part is of the form A.1 or B.$\\gamma $ .",
"Finally, the proof that two submanifolds of the form (REF ) are equivalent if and only if the $N$ s are equivalent is straightforward.",
"Not every Bishop-like submanifold is a cross product as above.",
"In fact the Bishop invariant may well change from point to point.",
"See § .",
"In such cases the foliation does not extend to a nonsingular holomorphic foliation of a neighbourhood.",
"Let us now focus on extending the Levi-foliation to $M$ , and not to a neighbourhood of $M$ .",
"Let us prove a useful proposition about recognizing certain CR singular Levi-flats from the form of the defining equation.",
"That is if the $r$ in the equation does not depend on $\\bar{z}_2$ through $\\bar{z}_n$ .",
"Proposition 6.2 Suppose near the origin $M \\subset {n+1}$ is given by $w = r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(2)$ and $\\frac{\\partial r}{\\bar{z}_1} \\lnot \\equiv 0$ .",
"Then $M$ is a CR singular Levi-flat submanifold and the Levi-foliation of $M_{CR}$ extends through the origin to a real-analytic foliation on $M$ .",
"Furthermore, there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"Near 0, $M$ is the image of a CR mapping that is a diffeomorphism onto its image of the standard CR Levi-flat.",
"The proposition also holds in two dimensions ($n=1$ ), although in this case it is somewhat trivial.",
"As in , let us define the mapping $F$ by $(x,y,\\xi ) \\mapsto \\bigl (x+iy, \\quad \\xi , \\quad r(x+iy,x-iy, \\xi ) \\bigr ) ,$ where $\\xi = (\\xi _2,\\ldots ,\\xi _n) \\in {n-1}$ .",
"Near points where $M$ is CR, this mapping is a CR diffeomorphism and hence $M$ must be Levi-flat.",
"Furthermore, since $F$ is a diffeomorphism, it takes the Levi-foliation on ${\\mathbb {R}}^2 \\times {n-1}$ to a foliation on $M$ near 0.",
"In fact, we make the following conclusion.",
"Lemma 6.3 Let $M \\subset {n+1}$ be a CR singular real-analytic Levi-flat submanifold of codimension 2 through the origin.",
"Then $M$ is a CR singular Levi-flat submanifold whose Levi-foliation of $M_{CR}$ extends through the origin to a nonsingular real-analytic foliation on $M$ if and only if there exists a real-analytic CR mapping $F \\colon U \\subset {\\mathbb {R}}^2 \\times {n-1}\\rightarrow {n+1}$ , $F(0) = 0$ , which is a diffeomorphism onto its image $F(U)\\subset M$ .",
"One direction is easy and was used above.",
"For the other direction, suppose that we have a foliation extending the Levi-foliation through the origin.",
"Let us consider $M_{CR}$ an abstract CR manifold.",
"That is a manifold $M_{CR}$ together with the bundle $T^{(0,1)} M_{CR}\\subset T M_{CR}$ .",
"The extended foliation on $M$ gives a real-analytic subbundle ${\\mathcal {W}}\\subset T M$ .",
"Since we are extending the Levi-foliation, when $p \\in M_{CR}$ , then ${\\mathcal {W}}_p = T_p^c M$ , where $T_p^c M = J(T_p^c M)$ is the complex tangent space and $J$ is the complex structure on ${n+1}$ .",
"Since $M_{CR}$ is dense in $M$ , then $J{\\mathcal {W}}={\\mathcal {W}}$ on $M$ .",
"Define the real-analytic subbundle ${\\mathcal {V}}\\subset T M$ as ${\\mathcal {V}}_p = \\lbrace X + iJ(X) : X \\in {\\mathcal {W}}_p \\rbrace .$ At CR points ${\\mathcal {V}}_p = T_p^{(0,1)} M$ (see for example page 8).",
"Then we can find vector fields $X^1,\\ldots ,X^{n-1}$ in ${\\mathcal {W}}$ such that $X^1,J(X^1),X^2,J(X^2),\\ldots ,X^{n-1},J(X^{n-1})$ is a basis of ${\\mathcal {W}}$ near the origin.",
"Then the basis for ${\\mathcal {V}}$ is given by $X^1+iJ(X^1),X^2+iJ(X^2),\\ldots ,X^{n-1}+iJ(X^{n-1}).$ As the subbundle is integrable, we obtain that $(M,{\\mathcal {V}})$ gives an abstract CR manifold, which at CR points agrees with $M_{CR}$ .",
"This manifold is Levi-flat as it is Levi-flat on a dense open set.",
"As it is real-analytic it is embeddable and hence there exists a real-analytic CR diffeomorphism from a neighbourhood of ${\\mathbb {R}}^2 \\times {n-1}$ to a neighbourhood of 0 in $M$ (as an abstract CR manifold).",
"This is our mapping $F$ .",
"The quadrics A.$k$ , $k \\ge 2$ , defined by $w = \\bar{z}_1^2 + \\cdots + \\bar{z}_k^2 ,$ contain the singular variety defined by $w = 0$ , $z_1^2 + \\cdots + z_k^2 =0$ , and hence the Levi-foliation cannot extend to a nonsingular foliation of the submanifold.",
"The quadric A.1 does admit a holomorphic foliation, but other type A.1 submanifolds do not in general.",
"For example, the submanifold $w = \\bar{z}_1^2 + \\bar{z}_2^3$ is of type A.1 and the unique complex variety through the origin is $0 = z_1^2 + z_2^3$ , which is singular.",
"Therefore the foliation cannot extend to $M$ ."
],
[
"Extending the Levi-foliation of C.x type submanifolds",
"Let us prove Theorem REF , that is, let us start with a type C.0 or C.1 submanifold and show that the Levi-foliation must extend real-analytically to all of $M$ .",
"Equivalently, let us show that the real analytic bundle $T^{(1,0)}M_{CR}$ extends to a real analytic subbundle of $TM$ .",
"Taking real parts we obtain an involutive subbundle of $TM$ extending $T^cM_{CR} = \\operatorname{Re}(T^{(1,0)} M_{CR})$ .",
"Let $M$ be the submanifold given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $\\epsilon = 0,1$ .",
"Let us treat the $z$ variables as the parameters on $M$ .",
"Let $\\pi $ be the projection onto the $\\lbrace w=0 \\rbrace $ plane, which is tangent to $M$ at 0 as a real $2n$ -dimensional hyperplane.",
"We will look at all the vectorfields on this plane $\\lbrace w=0 \\rbrace $ .",
"All vectors in $\\pi (T^{(1,0)} M)$ can be written in terms of $\\frac{\\partial }{\\partial z_j}$ for $j=1,\\ldots ,n$ .",
"The Levi-map is given by taking the $n\\times n$ matrix $L = L(p) =\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}+\\begin{bmatrix}\\frac{\\partial ^2 r}{ \\partial z_j \\partial \\bar{z}_k}\\end{bmatrix}_{j,k}(p)$ to vectors $v \\in \\pi (T^{(1,0)} M)$ ($\\pi $ is the projection) as $v^* L v$ .",
"The excess term in $L$ vanishes at 0.",
"Notice that for $p \\in M_{CR}$ , $\\pi (T^{(1,0)}_p M)$ is $n-1$ dimensional.",
"As $M$ is Levi-flat, then $v^* L v$ vanishes for $v \\in \\pi (T^{(1,0)}_p M)$ .",
"Write the vector $v = (v_1,\\ldots ,v_n)^t$ .",
"The zero set of the function $(z,v) \\in n \\times n \\overset{\\varphi }{\\mapsto } v^* L(z,\\bar{z}) v$ is a variety $V$ of real codimension 2 at the origin of $\\mathbf {C}^n\\times \\mathbf {C}^n$ because of the form of $L$ .",
"That is, at $z=0$ , the only vectors $v$ such that $v^*Lv = 0$ are those where $v_1 = 0$ or $v_2 = 0$ .",
"So the codimension is at least 2.",
"And we know that $v^*Lv$ vanishes for vectors in $\\pi (T^{(1,0)}_p M)$ for $p \\in M$ near 0, which is real codimension 2 at each $z$ corresponding to a CR point.",
"Therefore, $V \\cap (\\pi (M_{CR}) \\times n )$ has a connected component that is equal to a connected component of the real-analytic subbundle $\\pi (T^{(1,0)} M_{CR})$ .",
"We will verify that the latter is connected.",
"We show below that this subbundle extends past the CR singularity.",
"The key point is to show that the restriction of $\\pi \\bigl (T^{(1,0)}(M_{CR})\\bigr )$ extends to a smooth real-analytic submanifold of $T^{(1,0)} n$ .",
"Write $\\varphi (z,v) = v_1\\bar{v}_2 + \\sum a_{jk}(z) v_j\\bar{v}_k$ where $a_{jk}(0) = 0$ .",
"By Proposition REF , $\\pi (M\\setminus M_{CR})$ is contained in $z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1}=0.$ Thus $M_{CR}$ is connected.",
"Assume that $v\\cdot \\frac{\\partial }{\\partial z}\\in T^{(1,0)}_pM$ at a CR point $p$ .",
"Then $(z_2+2\\epsilon \\overline{z}_1+r_{\\overline{z}_1})\\overline{v}_1+\\sum _{j>1} r_{\\overline{z}_j}\\overline{v}_j=0.$ When $p$ is in the open set $U_\\delta \\subset \\pi (M_{CR})$ defined by $\\left|{z_2+2\\epsilon \\overline{z}_1} \\right|>\\left|{z} \\right|/2$ and $0<\\left|{z} \\right|<\\delta $ , $v$ is contained in $V_C\\colon \\left|{v_1} \\right|\\le \\left|{v} \\right|/C.$ When $\\delta $ is sufficiently small, $\\varphi (z,v)=0$ admits a unique solution $v_1=f(z,v_3,\\dots , v_n), \\quad v_2=1$ by imposing $v\\in V_C$ .",
"Note that $f$ is given by convergent power series.",
"For $\\left|{z} \\right|<\\delta $ , define $w_j=\\bigl (w_{j1}(z),\\dots , w_{jn}(z)\\bigr )\\in V_C, \\quad j=2,\\dots , n$ such that $\\varphi (z,w_j(z))=0$ and $ w_{j2}=1, \\quad w_{jk}=\\delta _{jk},\\quad j\\ge 2, k>2.$ To see why we can do so, fix $p\\in U_\\delta $ .",
"First we can find a vector $w_2$ in $E_p=\\pi (T_p^{(1,0)}M_{CR})$ such that $v_2=1$ .",
"Otherwise, $E_p\\subset V_C$ cannnot have dimension $n-1$ .",
"Let $E_p^{\\prime }$ be the vector subspace of $E_p$ with $v_2=0$ .",
"Then $E_p^{\\prime }$ has rank $n-2$ and remains in the cone $V_C$ .",
"Then $E_p^{\\prime }$ has an element $w_2$ with $v_2$ component being 1.",
"Repeating this, we find $w_2,\\dots , w_n$ in $E_p$ such that the $v_j$ component of $w_i$ is 0 for $2<j<i$ .",
"Using linear combinations, we find a unique basis $\\lbrace w_2,\\dots , w_n \\rbrace $ of $E_p$ that satisfies condition (REF ).",
"Assume that $C$ is sufficiently large.",
"By the above uniqueness assertion on $\\varphi (z,v)=0$ , we conclude that when $p\\in U_\\delta $ , $\\lbrace w_{2}(p),\\dots , w_n(p)\\rbrace $ is a base of $\\pi (T^{(1,0)}_pM_{CR})$ .",
"Also it is real analytic at $p=0$ .",
"Define $\\omega _j(z)=w_{j}(z)\\cdot \\frac{\\partial }{\\partial z}, \\quad \\left|{z} \\right|<\\delta .$ We lift the functions $\\omega _j$ via $\\pi $ to a subbundle of $TM$ , let us call these $\\widetilde{\\omega }_j$ .",
"Then consider the vector fields $w^*_j = 2 \\operatorname{Re}\\widetilde{\\omega }_j = \\widetilde{\\omega }_j + \\overline{\\widetilde{\\omega }_j}$ and $w^*_{n+j}=\\operatorname{Im}\\widetilde{\\omega }_j$ for $j=2,\\dots , n$ .",
"Above CR points over $U_\\delta $ , $\\tilde{w}_j$ is in $T M_{CR}\\otimes and so tangentto $ M$.",
"We thus obtain a $ 2n-2$ dimensional real analytic subbundle of $ TM$that agrees with the real analytic realsubbundle of $ TMCR$ induced by the Levi-foliation above $ U$.",
"Since $ MCR$and the subbunldes are real analytic and $ MCR$ is connected, they agree over $ MCR$.$ The real analytic distribution spanned by $\\lbrace \\omega ^*_i\\rbrace $ has constant rank ($2n-2$ ) everywhere and is involutive on an open subset of $M_{CR}$ and hence everywhere."
],
[
"CR singular set of type C.x submanifolds",
"Let $M \\subset {n+1}$ be a codimension two Levi-flat CR singular submanifold that is an image of ${\\mathbb {R}}^2 \\times {n-1}$ via a real-analytic CR map, and let $S \\subset M$ be the CR singular set of $M$ .",
"In it was proved that near a generic point of $S$ exactly one of the following is true: $S$ is Levi-flat submanifold of dimension $2n-2$ and CR dimension $n-2$ .",
"$S$ is a complex submanifold of complex dimension $n-1$ (real dimension $2n-2$ ).",
"$S$ is Levi-flat submanifold of dimension $2n-1$ and CR dimension $n-1$ .",
"We only have the above classification for a generic point of $S$ , and $S$ need not be a CR submanifold everywhere.",
"See for examples.",
"If $M$ is a Levi-flat CR singular submanifold and the Levi-foliation of $M_{CR}$ extends to $M$ , then by Lemma REF at a generic point $S$ has to be of one of the above types.",
"A corollary of Theorem REF is the following result.",
"Corollary 8.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a real-analytic Levi-flat CR singular type C.1 or type C.0 submanifold.",
"Let $S \\subset M$ denote the CR singular set.",
"Then near the origin $S$ is a submanifold of dimension $2n-2$ , and at a generic point, $S$ is either CR Levi-flat of dimension $2n-2$ (CR dimension $n-2$ ) or a complex submanifold of complex dimension $n-1$ .",
"Furthermore, if $M$ is of type C.1, then at the origin $S$ is a CR Levi-flat submanifold of dimension $2n-2$ (CR dimension $n-2$ ).",
"Let us take $M$ to be given by $w = \\bar{z}_1 z_2 + \\epsilon \\bar{z}_1^2 + r(z,\\bar{z})$ where $r$ is $O(3)$ and $\\epsilon = 0$ or $\\epsilon = 1$ .",
"By Proposition REF the CR singular set is exactly where $z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0, \\qquad \\text{and} \\qquad r_{\\bar{z}_j}(z,\\bar{z}) = 0 \\quad \\text{for all $j=2,\\ldots ,n$}.$ By considering the real and imaginary parts of the first equation and applying the implicit function theorem the set $\\widetilde{S} = \\lbrace z : z_2 + \\epsilon 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) =0 \\rbrace $ is a real submanifold of real dimension $2n-2$ (real codimension 2 in $M$ ).",
"Now $S \\subset \\widetilde{S}$ , but as we saw above $S$ is of dimension at least $2n-2$ .",
"Therefore $S = \\widetilde{S}$ near the origin.",
"The conclusion of the first part then follows from the classification above.",
"The stronger conclusion for C.1 submanifolds follows by noticing that when $\\epsilon = 1$ , the submanifold $z_2 + 2 \\bar{z}_1 + r_{\\bar{z}_1}(z,\\bar{z}) = 0$ is CR and not complex at the origin."
],
[
"Mixed-holomorphic submanifolds",
"Let us study sets in $m$ defined by $ f(\\bar{z}_1,z_2,\\ldots ,z_m) = 0 ,$ for a single holomorphic function $f$ of $m$ variables.",
"Such sets have much in common with complex varieties, since they are in fact complex varieties when $\\bar{z}_1$ is treated as a complex variable.",
"The distinction is that the automorphism group is different since we are interested in automorphisms that are holomorphic not mixed-holomorphic.",
"Proposition 9.1 If $M \\subset m$ is a submanifold with a defining equation of the form (REF ), where $f$ is a holomorphic function that is not identically zero, then $M$ is a real codimension 2 set and $M$ is either a complex submanifold or a Levi-flat submanifold, possibly CR singular.",
"Furthermore, if $M$ is CR singular at $p \\in M$ , and has a nondegenerate complex tangent at $p$ , then $M$ has type A.$k$ , C.0, or C.1 at $p$ .",
"Since the zero set of $f$ is a complex variety in the $(\\bar{z}_1,z_2,\\ldots ,z_m)$ space, we get automatically that it is real codimension 2.",
"We also have that as it is a submanifold, then it can be written as a graph of one variable over the rest.",
"Let $m = n+1$ for convenience and suppose that $M \\subset {n+1}$ is a submanifold through the origin.",
"By factorization for a germ of holomorphic function and by the smoothness assumption on $M$ we may assume that $df(0) \\ne 0$ .",
"Call the variables $(z_1,\\ldots ,z_n,w)$ and write $M$ as a graph.",
"One possibility is that we write $M$ as $\\bar{w} = \\rho (z_1,\\ldots ,z_n),$ where $\\rho (0) = 0$ and $\\rho $ has no linear terms.",
"$M$ is complex if $\\rho \\equiv 0$ .",
"Otherwise $M$ is CR singular and we rewrite it as $w = \\bar{\\rho }(\\bar{z}_1,\\ldots ,\\bar{z}_n).$ We notice that the matrix representing the Levi-map must be identically zero, so we must get Levi-flat.",
"If there are any quadratic terms we obtain a type A.$k$ submanifold.",
"Alternatively $M$ can be written as $w = \\rho (\\bar{z}_1,z_2,\\ldots ,z_n),$ with $\\rho (0) = 0$ .",
"If $\\rho $ does not depend on $\\bar{z}_1$ then $M$ is complex.",
"Assume that $\\rho $ depends on $\\bar{z}_1$ .",
"If $\\rho $ has linear terms in $\\bar{z}_1$ , then $M$ is CR.",
"Otherwise it is a CR singular submanifold, and near non-CR singular points it is a generic codimension 2 submanifold.",
"The CR singular set of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\overline{z}_1}=0$ .",
"Suppose that $M$ is CR singular.",
"That $M$ is Levi-flat follows from Proposition REF .",
"We can therefore normalize the quadratic term, after linear terms in $z_2,\\ldots ,z_n$ are absorbed into $w$ .",
"If not all quadratic terms are zero, then we notice that we must have an A.$k$ , C.0, or C.1 type submanifold.",
"Let us now study normal forms for such sets in 2 and $m$ , $m \\ge 3$ .",
"First in two variables we can easily completely answer the question.",
"This result is surely well-known and classical.",
"Proposition 9.2 If $M \\subset 2$ is a submanifold with a defining equation of the form (REF ), then it is locally biholomorphically equivalent to a submanifold in coordinates $(z,w) \\in 2$ of the form $w = \\bar{z}^d$ for $d=0,1,2,3,\\ldots $ where $d$ is a local biholomorphic invariant of $M$ .",
"If $d=0$ , $M$ is complex, if $d=1$ it is a CR totally-real submanifold, and if $d \\ge 2$ then $M$ is CR singular.",
"Write the submanifold as a graph of one variable over the other.",
"Without loss of generality and after possibly taking a conjugate of the equation, we have $w = f(\\bar{z})$ for some holomorphic function $f$ .",
"Assume $f(0) = 0$ .",
"If $f$ is identically zero, then $d=0$ and we are finished.",
"If $f$ is not identically zero, then it is locally biholomorphic to a positive power of the variable.",
"We apply a holomorphic change of coordinates in $z$ , and the rest follows easily.",
"In three or more variables, if $M \\subset {n+1}$ , $n \\ge 2$ , is a submanifold through the origin, then if the quadratic part is nonzero we have seen above that it can be a type A.$k$ , C.0, or C.1 submanifold.",
"If the submanifold is the nondegenerate type C.1 submanifold, then we will show in the next section that $M$ is biholomorphically equivalent to the quadric $M_{C.1}$ .",
"Before we move to C.1, let us quickly consider the mixed-holomorphic submanifolds of type A.$n$ .",
"The submanifolds of type A.$n$ in ${n+1}$ can in some sense be considered nondegenerate when talking about mixed-holomorphic submanifolds.",
"Proposition 9.3 If $M \\subset {n+1}$ is a submanifold of type A.$n$ at the origin of the form $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 + r(\\bar{z})$ where $r \\in O(3)$ .",
"Then $M$ is locally near the origin biholomorphically equivalent to the A.$n$ quadric $w = \\bar{z}_1^2+ \\cdots + \\bar{z}_n^2 .$ The complex Morse lemma (see e.g.",
"Proposition 3.15 in ) states that there is a local change of coordinates near the origin in just the $z$ variables such that $z_1^2+ \\cdots + z_n^2 + \\bar{r}(z)$ is equivalent to $z_1^2+ \\cdots + z_n^2$ .",
"It is not difficult to see that the normal form for mixed-holomorphic submanifolds in ${n+1}$ of type A.$k$ , $k < n$ , is equivalent to a local normal form for a holomorphic function in $n$ variables.",
"Therefore for example the submanifold $w = \\bar{z}_1^2 +\\bar{z}_2^3$ is of type A.1 and is not equivalent to any quadric."
],
[
"Formal normal form for certain C.1 type submanifolds I",
"In this section we prove the formal normal form in Theorem REF .",
"That is, we prove that if $M \\subset {n+1}$ is defined by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n) ,$ where $r$ is $O(3)$ , then $M$ is Levi-flat and formally equivalent to $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ That $M$ is Levi-flat follows from Proposition REF .",
"Lemma 10.1 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ formal power series then $M$ is formally equivalent to $M_{C.1}$ given by $w = \\bar{z}_1z_2 + \\bar{z}_1^2 .$ In fact, the normalizing transformation can be of the form $(z,w) = (z_1,\\ldots ,z_n,w)\\mapsto \\bigl (z_1, \\quad f(z,w), \\quad z_3, \\quad \\ldots , \\quad z_n, \\quad g(z,w)\\bigr ) ,$ where $f$ and $g$ are formal power series.",
"Suppose that the normalization was done to degree $d-1$ , then suppose that $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) +r_2(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) ,$ where $r_1$ is degree $d$ homogeneous and $r_2$ is $O(d+1)$ .",
"Write $r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) =\\sum _{j=0}^k \\sum _{\\left|{\\alpha } \\right|+j = d} c_{j,\\alpha }\\bar{z}_1^j z^\\alpha ,$ where $k$ is the highest power of $\\bar{z}_1$ in $r_1$ , and $\\alpha $ is a multiindex.",
"If $k$ is even, then use the transformation that replaces $w$ with $w + \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{k/2} z^\\alpha .$ Let us look at the degree $d$ terms in $(\\bar{z}_1 z_2 + \\bar{z}_1^2)+ \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha }{(\\bar{z}_1 z_2 + \\bar{z}_1^2)}^{k/2} z^\\alpha =\\bar{z}_1 z_2 + \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ We need not include $r_2$ as the terms are all degree $d+1$ or more.",
"After cancelling out the new terms on the left, we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"Next suppose that $k$ is odd.",
"We use the transformation that replaces $z_2$ with $z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha .$ Let us look at the degree $d$ terms in $\\bar{z}_1 z_2 + \\bar{z}_1^2=\\bar{z}_1 \\left(z_2 - \\sum _{\\left|{\\alpha } \\right|+k = d} c_{j,\\alpha } w^{(k-1)/2} z^\\alpha \\right)+ \\bar{z}_1^2 + r_1(z_1,\\bar{z}_1,z_2,\\ldots ,z_n) .$ Again we need not include $r_2$ as the terms are all degree $d+1$ or more, and we need not add the new terms to $z_2$ in the argument list for $r_1$ since all those terms would be of higher degree.",
"Again we notice that the formal transformation removed all the terms in $r_1$ with a power $\\bar{z}_1^k$ and replaced them with terms that have a smaller power of $\\bar{z}_1$ .",
"The procedure above does not change the form of the submanifold, but it lowers the degree of $\\bar{z}_1$ by one.",
"Since we can assume that all terms in $r_1$ depend on $\\bar{z}_1$ , we are finished with degree $d$ terms after $k$ iterations of the above procedure."
],
[
"Convergence of normalization for certain C.1 type submanifolds",
"A key point in the computation below is the following natural involution for the quadric $M_{C.1}$ .",
"Notice that the map $(z_1,z_2,\\ldots ,z_n,w) \\mapsto (-\\bar{z}_2-z_1, \\quad z_2, \\quad \\ldots ,\\quad z_n, \\quad w)$ takes $M_{C.1}$ to itself.",
"The involution simply replaces the $\\bar{z}_1$ in the equation with $-z_2-\\bar{z}_1$ .",
"The way this involution is defined is by noticing that the equation $w = \\bar{z}_1 z_2 + \\bar{z}_1^2$ has generically two solutions for $\\bar{z}_1$ keeping $z_2$ and $w$ fixed.",
"In the same way we could define an involution on all type C.1 submanifolds of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1,z_2,\\ldots ,z_n)$ , although we will not require this construction.",
"We prove convergence via the following well-known lemma: Lemma 11.1 Let $m_1, \\ldots , m_N$ be positive integers.",
"Suppose $T(z)$ is a formal power series in $z \\in N$ .",
"Suppose $T(t^{m_1}v_1,\\ldots , t^{m_N}v_N)$ is a convergent power series in $t \\in forall $ v N$.",
"Then $ T$ is convergent.$ The proof is a standard application of the Baire category theorem and the Cauchy inequality.",
"See (Theorem 5.5.30, p. 153) where all $m_j$ are 1.",
"For $m_j > 1$ we first change variables by setting $v_j = w_j^{m_j}$ and apply the lemma with $m_j=1$ .",
"The following lemma finishes the proof of Theorem REF .",
"By absorbing any holomorphic terms into $w$ , we assume that $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"In Lemma REF we have also constructed a formal transformation that only changed the $z_2$ and $w$ coordinates, so it is enough to prove convergence in this case.",
"Key points of this proof are that the right hand side of the defining equation for $M_{C.1}$ is homogeneous, and that we have a natural involution on $M_{C.1}$ .",
"Lemma 11.2 If $M \\subset {n+1}$ , $n \\ge 2$ , is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(z_1,\\bar{z}_1,z_2,z_3,\\ldots ,z_n)$ where $r$ is $O(3)$ and convergent, and $r(z_1,0,z_2,\\ldots ,z_n) \\equiv 0$ .",
"Suppose that two formal power series $f(z,w)$ and $g(z,w)$ satisfy $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{z}_1 f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ \\bar{z}_1^2 + r(z_1,\\bar{z}_1,f(z,\\bar{z}_1z_2 +\\bar{z}_1^2),z_3,\\ldots ,z_n) .$ Then $f$ and $g$ are convergent.",
"The equation (REF ) is true formally, treating $z_1$ and $\\bar{z}_1$ as independent variables.",
"Notice that (REF ) has one equation for 2 unknown functions.",
"We now use the involution on $M_{C.1}$ to create a system that we can solve uniquely.",
"We replace $\\bar{z}_1$ with $-z_2-\\bar{z}_1$ .",
"We leave $z_1$ untouched (treating as an independent variable).",
"We obtain an identity in formal power series: $g(z,\\bar{z}_1z_2 + \\bar{z}_1^2) = (-z_2-\\bar{z}_1) f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)+ (-z_2-\\bar{z}_1)^2 \\\\+ r(z_1,(-z_2-\\bar{z}_1),f(z,\\bar{z}_1z_2 + \\bar{z}_1^2),z_3,\\ldots ,z_n) .$ The formal series $\\xi = f(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ and $\\omega = g(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ are solutions of the system $\\omega & = \\bar{z}_1 \\xi + \\bar{z}_1^2+ r(z_1,\\bar{z}_1,\\xi ,z_3,\\ldots ,z_n) , \\\\\\omega & = (-z_2-\\bar{z}_1) \\xi + (-z_2-\\bar{z}_1)^2+ r(z_1,(-z_2-\\bar{z}_1),\\xi ,z_3,\\ldots ,z_n) .$ We next replace $z_j$ with $t z_j$ and $\\bar{z}_1$ with $t \\bar{z}_1$ for $t \\in .",
"Because $ z1z2 + z12$ is homogeneous ofdegree 2, we obtain that for every $ (z1,z1,z2,...,zn) n+1$ the formal series in $ t$ given by$ (t) = f(tz,t2(z1z2 + z12))$,$ (t) = g(tz,t2(z1z2 + z12))$are solutions of the system{\\begin{@align}{1}{-1}\\omega & = t \\bar{z}_1 \\xi + t^2 \\bar{z}_1^2+ r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) , \\\\\\omega & = t (-z_2-\\bar{z}_1) \\xi + t^2 (-z_2-\\bar{z}_1)^2+ r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n) .\\end{@align}}We eliminate $$ to obtain an equation for $$:\\begin{equation}t (2 \\bar{z}_1 + z_2) ( \\xi - t z_2)=\\\\r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n) .\\end{equation}We now treat $$ as a variable and we have a holomorphic (convergent)equation.",
"The right hand size must be divisible by$ t (2 z1 + z2)$: It is divisible by $ t$ since$ r$ was divisible by $ z1$.",
"It is also divisible by$ 2 z1 + z2$ as setting $ z2 = -2 z1$ makes the righthand side vanish.",
"Therefore,\\begin{equation}\\xi - t z_2=\\frac{r(tz_1,t(-z_2-\\bar{z}_1),\\xi ,t z_3,\\ldots , t z_n)- r(tz_1,t\\bar{z}_1,\\xi ,tz_3,\\ldots ,tz_n)}{t (2 \\bar{z}_1 + z_2)} ,\\end{equation}where the right hand side is a holomorphic function (that is, a convergentpower series) in $ z1,z1,z2,...,zn,t,$.For any fixed $ z1,z1,z2,...,zn$, we solve for $$ in terms of $ t$via the implicit function theorem,and we obtain that $$ is a holomorphicfunction of $ t$.",
"The power series of $$ is given by$ (t) = f(tz,t2(z1z2 + z12))$.$ Let $v \\in {n+1}$ be any nonzero vector.",
"Via a proper choice of $z_1,\\bar{z}_1,z_2,\\ldots ,z_n$ (still treating $\\bar{z}_1$ and $z_1$ as independent variables) we write $v =(z,\\bar{z}_1z_2 + \\bar{z}_1^2)$ .",
"We apply the above argument to $\\xi (t) = f(tv_1,\\ldots , tv_n, t^2v_{n+1})$ , and $\\xi (t)$ converges as a series in $t$ .",
"As we get convergence for every $v \\in {n+1}$ we obtain that $f$ converges by Lemma REF .",
"Once $f$ converges, then via () we obtain that $g(tv_1,\\ldots , tv_n, t^2v_{n+1})$ converges as a series in $t$ for all $v$ , and hence $g$ converges."
],
[
"Automorphism group of the C.1 quadric",
"With the normal form achieved in previous sections, let us study the automorphism group of the C.1 quadric in this section.",
"We will again use the mixed-holomorphic involution that is obtained from the quadric.",
"We study the local automorphism group at the origin.",
"That is the set of germs at the origin of biholomorphic transformations taking $M$ to $M$ and fixing the origin.",
"First we look at the linear parts of automorphisms.",
"We already know that the linear term of the last component only depends on $w$ .",
"For $M_{C.1}$ we can say more about the first two components.",
"Proposition 12.1 Let $(F,G) = (F_1,\\ldots ,F_n,G)$ be a formal invertible or biholomorphic automorphism of $M_{C.1} \\subset {n+1}$ , that is the submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Then $F_1(z,w) = a z_1 + \\alpha w + O(2)$ , $F_2(z,w) = \\bar{a} z_2 + \\beta w + O(2)$ , and $G(z,w) = \\bar{a}^2 w + O(2)$ , where $a \\ne 0$ .",
"Let $a = (a_1,\\ldots ,a_n)$ and $b = (b_1,\\ldots ,b_n)$ be such that $F_1(z,w) = a \\cdot z + \\alpha w + O(2)$ and $F_2(z,w) = b \\cdot z + \\beta w + O(2)$ .",
"Then from Proposition REF we have $\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} a \\\\ b \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1 b_2 = 1$ , and $\\bar{a}_j b_k = 0$ for all $(j,k) \\ne (1,2)$ .",
"Similarly $\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} \\bar{a} \\\\ \\bar{b} \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1^2 = 1$ , and $\\bar{a}_j \\bar{a}_k = 0$ for all $(j,k)\\ne (1,1)$ .",
"Putting these two together we obtain that $a_j = 0$ for all $j \\ne 1$ , and as $a_1 \\ne 0$ we get $b_j = 0$ for all $j \\ne 2$ .",
"As $\\lambda $ is the reciprocal of the coefficient of $w$ in $G$ , we are finished.",
"Lemma 12.2 Let $M_{C.1} \\subset 3$ be given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Suppose that a local biholomorphism (resp.",
"formal automorphism) $(F_1,F_2,G)$ transforms $M_{C.1}$ into $M_{C.1}$ .",
"Then $F_1$ depends only on $z_1$ , and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"Let us define a $(1,0)$ tangent vector field on $M$ by $Z=\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} .$ Write $F = (F_1,F_2,G)$ .",
"$F$ must take $Z$ into a multiple of itself when restricted to $M_{C.1}$ .",
"That is on $M_{C.1}$ we have $& \\frac{\\partial F_1}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_1}{\\partial w}= 0 ,\\\\& \\frac{\\partial F_2}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_2}{\\partial w}= \\lambda ,\\\\& \\frac{\\partial G}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial G}{\\partial w}= \\lambda \\overline{F_1}(\\bar{z},\\bar{w}) ,$ for some function $\\lambda $ .",
"Let us take the first equation and plug in the defining equation for $M_1$ : $ \\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+\\bar{z}_1\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This equation is true for all $z \\in 2$ , and so we may treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We have an involution on $M_{C.1}$ that takes $\\bar{z}_1$ to $-z_2-\\bar{z}_1$ .",
"Therefore we also have $\\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+(-z_2-\\bar{z}_1)\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This means that $\\frac{\\partial F_1}{\\partial w}$ and therefore $\\frac{\\partial F_1}{\\partial z_2}$ must be identically zero.",
"That is, $F_1$ only depends on $z_1$ .",
"We have that the following must hold for all $z$ : $G(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)F_2(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+{\\left(\\overline{F_1}(\\bar{z}_1) \\right)}^2 .$ Again we treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We differentiate with respect to $z_1$ : $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ We plug in the involution again to obtain $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(-z_2-\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ Therefore as $F_1$ is not identically zero, then as before both $\\frac{\\partial F_2}{\\partial z_1}$ and $\\frac{\\partial G}{\\partial z_1}$ must be identically zero.",
"Lemma 12.3 Take $M_{C.1} \\subset 3$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and let $(F_1,F_2,G)$ be a local automorphism at the origin.",
"Then $F_1$ uniquely determines $F_2$ and $G$ .",
"Furthermore, given any invertible function of one variable $F_1$ with $F_1(0) = 0$ , there exist unique $F_2$ and $G$ that complete an automorphism and they are determined by $ \\begin{aligned}F_2(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = \\bar{F}_1(\\bar{z}_1)+\\bar{F}_1(-\\bar{z}_1-z_2),\\\\G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = -\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2).\\end{aligned}$ We should note that the lemma also works formally.",
"Given any formal $F_1$ , there exist unique formal $F_2$ and $G$ satisfying the above property.",
"By Lemma REF , $F_1$ depends only on $z_1$ and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"We write the automorphism as a composition of the two mappings $\\bigl (F_1(z_1),z_2,w\\bigr )$ and $\\bigl (z_1,F_2(z_2,w),G(z_2,w)\\bigr )$ .",
"We plug the transformation into the defining equation for $M_{C.1}$ .",
"$ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(\\bar{z}_1)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2 .$ We use the involution $(z_1,z_2) \\mapsto (-\\bar{z}_1-z_2,z_2)$ which preserves $M_{C.1}$ and obtain a second equation $ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(-\\bar{z}_1-z_2)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2 .$ We eliminate $G$ and solve for $F_2$ : $ F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= \\frac{{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2-{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2}{\\bar{F}_1(\\bar{z}_1)-\\bar{F}_1(-\\bar{z}_1-z_2)}=\\bar{F}_1(\\bar{z}_1)+ \\bar{F}_1(-\\bar{z}_1-z_2) .$ Next we note that trivially, $F_2$ is unique if it exists: its difference vanishes on $M_{C.1}$ .",
"If we suppose that $F_1$ is convergent, then just as before, substituting $z_2$ with $tz_2$ and $\\bar{z}_1$ with $t\\bar{z}_1$ , we are restricting to curves $(tz_2,t^2w)$ for all $(z_2,w)$ .",
"The series is convergent in $t$ for every fixed $z_2$ and $w$ .",
"Therefore if $F_2$ exists and $F_1$ is convergent, then $F_2$ is convergent by Lemma REF .",
"Now we need to show the existence of the formal solution $F_2$ .",
"Notice that the right-hand side of (REF ) is invariant under the involution.",
"It suffices to show that any power series in $\\bar{z_1}, z_2$ that is invariant under the involution is a formal power series in $z_2$ and $\\bar{z}_1z_2+\\bar{z}_1^2$ .",
"Let us treat $\\xi =\\bar{z}_1$ as an independent variable.",
"The original involution becomes a holomorphic involution in $\\xi ,z_2$ : $\\tau \\colon \\xi \\rightarrow -\\xi -z_2, \\qquad z_2\\rightarrow z_2.$ By a theorem of Noether we obtain a set of generators for the ring of invariants can be obtained by applying the averaging operation $R(f) = \\frac{1}{2} ( f + f \\circ \\tau )$ to all monomials in $\\xi $ and $z_2$ of degree 2 or less.",
"By direct calculation it is not difficult to see that $\\xi ,\\xi z_2+\\xi ^2$ generate the ring of invariants.",
"Therefore any invariant power series in $z_2,\\xi $ is a power series in $\\xi ,\\xi z_2+\\xi ^2$ .",
"This shows the existence of $F_2$ .",
"The existence of $G$ follows the same.",
"The equation for $G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)=-\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2)$ is obtained by plugging in the equation for $F_2$ .",
"Its existence, uniqueness, and convergence in case $F_1$ converges, follows exactly the same as for $F_2$ .",
"Theorem 12.4 If $M \\subset {n+1}$ , $n \\ge 2$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and $(F_1,F_2,\\ldots ,F_n,G)$ is a local automorphism at the origin, then $F_1$ depends only on $z_1$ , $F_2$ and $G$ depend only on $z_2$ and $w$ , and $F_1$ completely determines $F_2$ and $G$ via (REF ).",
"The mapping $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin.",
"Furthermore, given any invertible function $F_1$ of one variable with $F_1(0) = 0$ , and arbitrary holomorphic functions $F_3,\\ldots ,F_n$ with $F_j(0) = 0$ , and such that $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin, then there exist unique $F_2$ and $G$ that complete an automorphism.",
"Let $(F_1,\\ldots ,F_n,G)$ be an automorphism.",
"Then we have $G(z_1,\\ldots ,z_n,w) =\\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w})F_2(z_1,\\ldots ,z_n,w) +{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w}) \\bigr )}^2 .$ Proposition REF says that the linear terms in $G$ only depend on $w$ , the linear terms of $F_1$ depend only on $z_1$ and $w$ and the linear terms of $F_2$ only depend on $z_2$ and $w$ .",
"Let us embed $M_{C.1} \\subset 3$ into $M$ via setting $z_3 = \\alpha _3 z_2$ , $\\ldots $ , $z_n = \\alpha _n z_2$ , for arbitrary $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Then we obtain $ G(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) = \\\\\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w})F_2(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w}) \\bigr )}^2 .$ By noting what the linear terms are, we notice that the above is the equation for an automorphism of $M_{C.1}$ .",
"Therefore by Lemma REF we have $\\frac{\\partial F_1}{\\partial w} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial F_2}{\\partial z_1} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial G}{\\partial z_1} = 0 ,$ as that is true for all $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Plugging in the defining equation for $M_{C.1}$ we obtain an equation that holds for all $z$ and we can treat $z$ and $\\bar{z}$ independently.",
"We plug in $z = 0$ to obtain $0 =\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0)F_2(0,\\ldots ,0,\\bar{z}_1^2) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0) \\bigr )}^2 .$ Differentiating with respect to $\\bar{\\alpha }_j$ we obtain $\\frac{\\partial F_1}{\\partial z_j} = 0$ , for $j=3,\\ldots ,n$ .",
"We set $\\bar{\\alpha }_j = 0$ in the equation, differentiate with respect to $\\bar{z}_2$ and obtain that $\\frac{\\partial F_1}{\\partial z_2} = 0$ .",
"In other words $F_1$ is a function of $z_1$ only.",
"We rewrite (REF ) by writing $F_1$ as a function of $z_1$ only and $F_2$ and $G$ as functions of $z_2,\\ldots ,z_n,w$ , and we plug in $w = \\bar{z}_1z_2 + \\bar{z}_1^2$ to obtain $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\\\\\overline{F_1}(\\bar{z}_1)F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) +{\\bigl ( \\overline{F_1}(\\bar{z}_1) \\bigr )}^2 .$ By Lemma REF , we know that $F_1$ now uniquely determines $F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ and $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ .",
"These two functions therefore do not depend on $\\alpha _3,\\ldots ,\\alpha _n$ , and in turn $F_2$ and $G$ do not depend on $z_3,\\ldots ,z_n$ as claimed.",
"Furthermore $F_1$ does uniquely determine $F_2$ and $G$ .",
"Finally since the mapping is a biholomorphism, and from what we know about the linear parts of $F_1$ , $F_2$ , and $G$ , it is clear that $(z_1,z_2,F_3,\\ldots ,F_n)$ is rank $n$ .",
"The other direction follows by applying Lemma REF .",
"We start with $F_1$ , determine $F_2$ and $G$ as in 3 dimensions.",
"Then adding $F_3,\\ldots ,F_n$ and the rank condition guarantees an automorphism."
],
[
"Normal form for certain C.1 type submanifolds II",
"The goal of this section is to find the normal form for Levi-flat submanifolds $M \\subset {n+1}$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + \\operatorname{Re}f(z) ,$ for a holomorphic $f(z)$ of order $O(3)$ .",
"Since $f(z)$ can be absorbed into $w$ via a holomorphic transformation, the goal is really to prove the following theorem.",
"Theorem 13.1 Let $M \\subset {n+1}$ be a real-analytic Levi-flat given by $ w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Then $M$ can be put into the $M_{C.1}$ normal form $ w = \\bar{z}_1z_2 + \\bar{z}_1^2 ,$ by a convergent normalizing transformation.",
"Furthermore, if $r$ is a polynomial and the coefficient of $\\bar{z}_1^3$ in $r$ is zero, then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"In Theorem REF , we have already shown that a submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ is necessarily Levi-flat and has the normal form $M_{C.1}$ .",
"The first part of Theorem REF will follow once we prove: Lemma 13.2 If $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z})$ where $r$ is $O(3)$ and $M$ is Levi-flat, then $r$ depends only on $\\bar{z}_1$ .",
"First let us assume that $n=2$ .",
"For $p \\in M_{CR}$ , $T^{(1,0)}_p M$ is one dimensional.",
"The Levi-map is the matrix $L =\\begin{bmatrix}0 & 1 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0\\end{bmatrix}$ applied to the $T^{(1,0)} M$ vectors.",
"As $M$ is Levi-flat, then the Levi-map has to vanish.",
"The only vectors $v$ for which $v^* L v = 0$ , are the ones without $\\frac{\\partial }{\\partial z_1}$ component or $\\frac{\\partial }{\\partial z_2}$ component.",
"That is vectors of the form $a \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial w},\\qquad \\text{or} \\qquad a \\frac{\\partial }{\\partial z_2} + b \\frac{\\partial }{\\partial w}.$ We apply these vectors to the defining equation and its conjugate and we obtain in the first case the equations $b = 0, \\qquad a \\left( \\bar{z}_2 + 2z_1 + \\frac{\\partial \\bar{r}}{\\partial z_1} \\right) = 0 .$ This cannot be satisfied identically on $M$ since this is supposed to be true for all $z$ , but $a$ cannot be identically zero and the second factor in the second equation has only one nonholomorphic term, which is $\\bar{z}_2$ .",
"Let us try the second form and we obtain the equations $b = a \\bar{z}_1 , \\qquad a \\left( \\frac{\\partial \\bar{r}}{\\partial z_2} \\right) = 0 .$ Again $a$ cannot be identically zero, and hence the second factor of the second equation $\\frac{\\partial \\bar{r}}{\\partial z_2}$ must be identically zero, which is possible only if $r$ depends only on $\\bar{z}_1$ .",
"Finally, it is possible to pick $b=\\bar{z}_1$ and $a=1$ , to obtain a $T^{(1,0)}$ vector field $\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} ,$ and therefore these submanifolds are necessarily Levi-flat.",
"Next suppose that $n > 2$ .",
"Notice that replacing $z_k$ with $\\lambda _k \\xi $ for $k \\ge 2$ and then fixing $\\lambda _k$ for $k \\ge 2$ , we get $w = \\bar{z}_1 \\lambda _2 \\xi + \\bar{z}_1^2 + r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi }) .$ By Lemma REF , we obtain a Levi-flat submanifold in $(z_1,\\xi ,w) \\in 3$ , and hence can apply the above reasoning to obtain that $r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi })$ does not depend on $\\bar{\\xi }$ .",
"As this was true for any $\\lambda _k$ 's, we have that $r$ can only depend on $\\bar{z}_1$ .",
"It is left to prove the claim about the polynomial normalizing transformation.",
"Lemma 13.3 Suppose $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ where $r$ is a polynomial that vanishes to fourth order.",
"Then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"We will take a transformation of the form $(z_1,z_2,w) \\mapsto \\bigl (z_1,z_2+f(z_2,w),w+g(z_2,w) \\bigr ) .$ We are therefore trying to find polynomial $f$ and $g$ that satisfy $\\bar{z}_1z_2 + \\bar{z}_1^2+g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)=\\bar{z}_1 \\bigl (z_2 +f(z_2,\\bar{z}_1z_2 +\\bar{z}_1^2)\\bigr ) + \\bar{z}_1^2 + r(\\bar{z}_1) .$ If we simplify we obtain $g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)-\\bar{z}_1 f(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= r(\\bar{z}_1) .$ Consider the involution $S\\colon (\\bar{z}_1,z_2)\\rightarrow (-\\bar{z}_1-z_2,z_2)$ .",
"Its invariant polynomials $u(\\bar{z}_1,z_2)$ are precisely the polynomials in $z_2,z_2\\bar{z}_1+\\bar{z}_1^2$ .",
"The polynomial $r(\\bar{z}_1)$ can be uniquely written as $r^+(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)+\\Bigl (\\bar{z}_1+\\frac{z_2}{2}\\Bigr )r^-(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)$ in two polynomials $r^\\pm $ .",
"Taking $f=-r^-$ and $g=r^++\\frac{z_2}{2}r^-$ , we find the desired solutions."
],
[
"Normal form for general type C.1 submanifolds",
"In this section we show that generically a Levi-flat type C.1 submanifold is not formally equivalent to the quadric $M_{C.1}$ submanifold.",
"In fact, we find a formal normal form that shows infinitely many invariants.",
"There are obviously infinitely many invariants if we do not impose the Levi-flat condition.",
"The trick therefore is, how to impose the Levi-flat condition and still obtain a formal normal form.",
"Let $M \\subset 3$ be a real-analytic Levi-flat type C.1 submanifold through the origin.",
"We know that $M$ is an image of ${\\mathbb {R}}^2 \\times under a real-analytic CR map that is a diffeomorphism onto itstarget; see Theorem~\\ref {thm:folextendsCxtype}.After a linear change of coordinates we assume thatthe mapping is\\begin{equation}\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (& x+iy + a(x,y,\\xi ), \\\\& \\xi + b(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) ,\\end{split}\\end{equation}where $ a$, $ b$ are $ O(2)$ and $ r$ is $ O(3)$.",
"As the mapping is a CR mapping and a local diffeomorphism, then given anysuch $ a$, $ b$, and $ r$, the image isnecessarily Levi-flat at CR points.",
"Therefore the set of all thesemappings gives us all type C.1 Levi-flat submanifolds.$ We precompose with an automorphism of ${\\mathbb {R}}^2 \\times to make $ b = 0$.We cannot similarly remove $ a$ as anyautomorphism must have real valued first two components (the new $ x$ and thenew $ y$), and hence thosecomponents can only depend on $ x$ and $ y$ and not on $$.",
"So if $ a$depends on $$, we cannot remove it by precomposing.$ Next we notice that we can treat $M$ as an abstract CR manifold.",
"Suppose we have two equivalent submanifolds $M_1$ and $M_2$ , with $F$ being the biholomorphic map taking $M_1$ to $M_2$ .",
"If $M_j$ is the image of a map $\\varphi _j$ , then note that $\\varphi _2^{-1}$ is CR on ${(M_2)}_{CR}$ .",
"Therefore, $G = \\varphi _2^{-1} \\circ F \\circ \\varphi _1$ is CR on ${(F \\circ \\varphi _1)}^{-1}\\bigl ({(M_2)}_{CR}\\bigr )$ , which is dense in a neighbourhood of the origin of ${\\mathbb {R}}^2 \\times (theCR singularity of $ M2$ is a thin set, and we pull it back by tworeal-analytic diffeomorphisms).",
"A real-analytic diffeomorphism thatis CR on a dense set is a CR mapping.",
"The same argumentworks for the inverse of $ G$,and therefore we have a CR diffeomorphism of $ R2 .",
"The conclusion we make is the following proposition.",
"Proposition 14.1 If $M_j \\subset 3$ , $j=1,2$ are given by the maps $\\varphi _j$ $\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times {\\varphi _j}{\\mapsto }\\bigl (& x+iy + a_j(x,y,\\xi ), \\\\& \\xi + b_j(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r_j(x,y,\\xi )\\bigr ) ,\\end{split}$ and $M_1$ and $M_2$ are locally biholomorphically (resp.",
"formally) equivalent at 0, then there exists local biholomorphisms (resp.",
"formal equivalences) $F$ and $G$ at 0, with $F(M_1) = M_2$ , $G({\\mathbb {R}}^2 \\times = {\\mathbb {R}}^2 \\times as germs(resp.\\ formally) and\\begin{equation}\\varphi _2 = F \\circ \\varphi _1 \\circ G .\\end{equation}$ In other words, the proposition states that if we find a normal form for the mapping we find a normal form for the submanifolds.",
"Let us prove that the proposition also works formally.",
"We have to prove that $G$ restricted to ${\\mathbb {R}}^2 \\times is CR, that is,$ G = 0$.",
"Let us consider\\begin{equation}\\varphi _2 \\circ G = F \\circ \\varphi _1 .\\end{equation}The right hand side does not depend on $$ and thus the left handside does not either.",
"Write $ G = (G1,G2,G3)$.",
"Let us write $ b = b2$and $ r = r2$for simplicity.",
"Taking derivative of $ 2 G$ with respectto $$ we get:\\begin{equation}\\begin{aligned}& G^1_{\\bar{\\xi }} +i G^2_{\\bar{\\xi }} +a_x(G) G^1_{\\bar{\\xi }} +a_y(G) G^2_{\\bar{\\xi }} +a_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\& G^3_{\\bar{\\xi }} +b_x(G) G^1_{\\bar{\\xi }} +b_y(G) G^2_{\\bar{\\xi }} +b_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\&(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}) G^3+(G^1 - i G^2) G^3_{\\bar{\\xi }} +2 (G^1 - i G^2)(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }})\\\\& \\qquad +r_x(G) G^1_{\\bar{\\xi }} +r_y(G) G^2_{\\bar{\\xi }} +r_\\xi (G) G^3_{\\bar{\\xi }} = 0 .\\end{aligned}\\end{equation}Suppose that the homogeneous parts of $ Gj$ are zero for alldegrees up to degree $ d-1$.",
"If we look at the degree $ d$ homogeneous partsof the first two equations above we immediately note that it must be that$ G1 + i G2 = 0$ and$ G3 = 0$ in degree $ d$.",
"We then look at the degree $ d+1$part of the third equation.",
"Recall that $ []d$ is the degree $ d$part of an expression.",
"We get\\begin{equation}{[G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}]}_{d}{[G^3 + 2 G^1 - i 2 G^2]}_{1} = 0 .\\end{equation}As $ G$ is an automorphism we cannot have the linear terms be linearlydependent and hence$ G1 = G2 = 0$ in degree $ d$.",
"We finishby induction on $ d$.$ Using the proposition we can restate the result of Theorem REF using the parametrization.",
"Corollary 14.2 A real-analytic Levi-flat type C.1 submanifold $M \\subset 3$ is biholomorphically equivalent to the quadric $M_{C.1}$ if and only if the mapping giving $M$ is equivalent to a mapping of the form $(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (x+iy,\\quad \\xi ,\\quad (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) .$ That is, $M$ is equivalent to $M_{C.1}$ if and only if we can get rid of the $a(x,y,\\xi )$ via pre and post composing with automorphisms.",
"The proof of the corollary follows as a submanifold that is realized by this map must be of the form $w = \\bar{z}_1z_2 + \\bar{z}_1^2 + \\rho (z_1,\\bar{z}_1,z_2)$ and we apply Theorem REF .",
"We have seen that the involution $\\tau $ on $M$ , in particular when $M$ is the quadric, is useful to compute the automorphism group and to construct Levi-flat submanifolds of type $C.1$ .",
"We will also need to deal with power series in $z,\\bar{z}, \\xi $ .",
"Thus we extend $\\tau $ , which is originally defined on 2, as follows $\\sigma (z,\\bar{z},\\xi )=(z,-\\bar{z}-\\xi ,\\xi ).$ Here $z,\\bar{z},\\xi $ are treated as independent variables.",
"Note that $z,\\xi ,w=\\bar{z}\\xi +\\bar{z}^2$ are invariant by $\\sigma $ , while $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ is skew invariant by $\\sigma $ .",
"A power series in $z,\\bar{z},\\xi $ that is invariant by $\\sigma $ is precisely a power series in $z,\\xi ,w$ .",
"In general, a power series $u$ in $z,\\bar{z},\\xi $ admits a unique decomposition $u(z,\\bar{z},\\xi )=u^+(z,\\xi ,w)+\\eta u^-(z,\\xi ,w).$ First we introduce degree for power series $u(z,\\bar{z},\\xi )$ and weights for power series $v(z,\\xi ,w)$ .",
"As usual we assign degree $i+j+k$ to the monomial $z^i\\bar{z}^j\\xi ^k$ .",
"We assign weight $i+j+2k$ to the monomial $z^i\\xi ^jw^k$ .",
"For simplicity, we will call them weight in both situations.",
"Let us also denote $[u]_d(z,\\bar{z},\\xi )=\\sum _{i+j+k=d}u_{ijk}z^i\\bar{z}^j\\xi ^k, \\quad [v]_d(z,\\xi ,w)=\\sum _{i+j+2k=d}v_{ijk}z^i\\xi ^jw^k.$ Set $[u]_{i}^j=[u]_i+\\cdots +[u]_j$ and $[v]_i^j=[v]_i+\\cdots +[v]_j$ for $i\\le j$ .",
"Theorem 14.3 Let $M$ be a real-analytic Levi-flat type C.1 submanifold in 3.",
"There exists a formal biholomorphic map transforming $M$ into the image of $\\hat{\\varphi }(z,\\bar{z},\\xi )=\\bigl (z+A(z,\\xi , w)w\\eta , \\xi ,w\\bigr )$ with $\\eta =\\bar{z}+\\frac{1}{2}{\\xi }$ and $w=\\bar{z}\\xi +\\bar{z}^2$ .",
"Suppose further that $A\\lnot \\equiv 0$ .",
"Fix $i_*,j_*,k_*$ such that $j_*$ is the largest integer satisfying $A_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Then we can achieve $A_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ Furthermore, the power series $A$ is uniquely determined up to the transformation $A(z,\\xi ,w)\\rightarrow \\bar{c}^{3}A(cz,\\bar{c}\\xi ,\\bar{c}^2w), \\quad c\\in \\lbrace 0\\rbrace .$ In the above normal form with $A\\lnot \\equiv 0$ , the group of formal biholomorphisms that preserve the normal form consists of dilations $(z,\\xi ,w)\\rightarrow (\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)$ satisfying $\\bar{\\nu }^{3}A(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)=A(z,\\xi ,w)$ .",
"It will be convenient to write the CR diffeomorphism $G$ of ${\\mathbb {R}}^2\\times as $ (G1,G2)$ where $ G1$ is complex-valued and depends on $ z,z$,while $ G2$ depends on $ z,z,$.",
"Let $ M$ be the image of a mapping $$ defined by\\begin{equation}\\begin{aligned}(z,\\bar{z},\\xi ) \\overset{\\varphi }{\\mapsto }\\bigl (& z + a(z,\\bar{z},\\xi ),\\\\& \\xi ,\\\\& \\bar{z} \\xi + {\\bar{z}}^2 + r(z,\\bar{z},\\xi )\\bigr )\\end{aligned}\\end{equation}with $ a=O(2), r=O(3)$.",
"We want to find a formal biholomorphic map $ F$ of $ 3$and a formal CR diffeomorphism $ G$ of $ R2 such that $F\\hat{\\varphi }G^{-1}=\\varphi $ with $\\hat{\\varphi }$ in the normal form.",
"To simplify the computation, we will first achieve a preliminary normal form where $r=0$ and the function $a$ is skew-invariant by $\\sigma $ .",
"For the preliminary normal form we will only apply $F, G$ that are tangent to the identity.",
"We will then use the general $F, G$ to obtain the final normal form.",
"Let us assume that $F, G$ are tangent to the identity.",
"Let $M=F\\bigl (\\hat{\\varphi }({\\mathbb {R}}^2\\times \\bigr )$ where $\\hat{\\varphi }$ is determined by $\\hat{a}, \\hat{r}$ .",
"We write $F=I+(f_1,f_2,f_3), \\quad G=I+(g_1,g_2).$ The $\\xi $ components in $\\varphi G=F\\hat{\\varphi }$ give us $g_2(z,\\bar{z},\\xi )=f_2\\bigl (z+\\hat{a}(z,\\bar{z},\\xi ), \\xi , \\bar{z}\\xi +\\bar{z}^2+\\hat{r}(z,\\bar{z},\\xi )\\bigr ).$ Thus, we are allowed to define $g_2$ by the above identity for any choice of $f_2=O(2)$ .",
"Eliminating $g_2$ in other components of $\\varphi G=F\\hat{\\varphi }$ , we obtain $f_1\\circ \\hat{\\varphi }-g_1&= a\\circ G-\\hat{a},\\\\f_3\\circ \\hat{\\varphi }-\\bar{z}f_2\\circ \\hat{\\varphi }&=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2,$ where $\\tilde{g}_1(z,\\bar{z})=\\bar{g}_1(\\bar{z},z)$ and $(a\\circ G)(z,\\bar{z},\\xi ):=a\\bigl (G_1(z,\\bar{z}),\\bar{G}_1(\\bar{z},z),G_2(z,\\bar{z},\\xi )\\bigr ).$ Each power series $r(z,\\bar{z},\\xi )$ admits a unique decomposition $r(z,\\bar{z},\\xi )=r^+(z,\\xi ,w)+\\eta r^-(z,\\xi ,w),$ where both $r^\\pm $ are invariant by $\\sigma $ .",
"Note that $r(z,\\bar{z},\\xi )$ is a power series in $z,\\xi $ and $w$ , if and only if it is invariant by $\\sigma $ , i.e.",
"if $r^-=0$ .",
"We write $r^+={wt}\\, (k), \\quad \\text{or}\\quad {wt} \\, (r^+)\\ge k,$ if $r^+_{abc}=0$ for $a+b+2c<k$ .",
"Define $r^-=wt (k)$ analogously and write $\\eta r^-={wt}\\,(k)$ if $r^-={wt}\\,(k-1)$ .",
"We write $r={wt}\\,(k)$ if $(r^+,\\eta r^-)={wt}\\,(k)$ .",
"Note that $r=O(k)\\Rightarrow r={wt}\\,(k); \\quad wt \\, (rs)\\ge wt\\, (r)+wt\\,(s).$ The power series in $z,\\bar{z}$ play a special role in describing normal forms.",
"Let us define $T^\\pm $ via $u(z,\\bar{z})=(T^+u)(z,\\xi ,w)+(T^-u)(z,\\xi ,w)\\eta .$ Let $S^+_k$ (resp.",
"$S^-_k$ ) be spanned by monomials in $z,\\bar{z},\\xi $ which have weight $k$ and are invariant (resp.",
"skew-invariant) by $\\sigma $ .",
"Then the range of $\\eta T^-$ in $S_k^-$ is a linear subspace $R_k$ .",
"We decompose $S_k^-=R_k\\oplus (S_k^-\\ominus R_k).$ The decomposition is of course not unique.",
"We will take $S_k^-\\ominus R_k=\\bigoplus _{a+b+2c=k-1, c>0} z̏^a\\xi ^bw^c\\eta .$ Here we have used $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ , $\\eta ^2=w+\\frac{1}{4}\\xi ^2$ , and $T^+u(z,\\xi , w)=\\sum _{i,j\\ge 0}\\sum _{0\\le \\alpha \\le j/2} u_{ij} \\binom{j}{2\\alpha }z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha },\\\\T^-u(z,\\xi ,w)=\\sum _{i\\ge 0,j>0}\\sum _{0\\le \\alpha <j/2} u_{ij}\\binom{j}{2\\alpha +1}z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha -1}.$ In particular, we have $T^-u(z,\\xi ,0)=\\sum _{i\\ge 0,j>0}(-1)^{j-1}u_{ij} z^i \\xi ^{j-1}.$ This shows that $T^-u(z,\\xi ,0)=\\frac{1}{-\\xi }\\bigl (u(z,-\\xi )-u(z,0)\\bigr ).$ We are ready to show that under the condition that $g_1(z,\\bar{z})$ has no pure holomorphic terms, there exists a unique $(F,G)$ which is tangent to the identity such that $\\hat{r}=0$ and $\\hat{a}\\in \\mathcal {N}:=\\bigoplus \\mathcal {N}_k, \\quad \\mathcal {N}_k:=S_k^-\\ominus R_k.$ We start with terms of weight 2 in (REF )-() to get $[f_1]_2-[g_1]_2=[a]_2-\\eta [\\hat{a}^-]_1,\\\\[f_3]_2=0.$ Note that $f_j^-=0$ .",
"The first identity implies that $[f_1]_2-[T^+g_1]_2=[a^+]_2, \\quad [T^-g_1]_1=[\\hat{a}^-]_1-[a^-]_1.$ The first equation is solvable with kernel defined by $[f_1]_k-[T^+g_1]_k=0 $ for $k=2$ .",
"This shows that $[g_1]_2$ is still arbitrary and we use it to achieve $\\eta [\\hat{a}^-]_1\\in S_2^-\\ominus R_2=\\lbrace 0\\rbrace .$ Then the kernel space is defined by (REF ) and $[g_1(z,\\bar{z})-g_1(z,0)]_k=0$ with $k=2$ .",
"In particular, under the restriction $[g_1(z,0)]_k=0,$ for $k=2$ , we have achieved $\\hat{a}^-\\in \\mathcal {N}_2$ by unique $[f_1]_2, [g_1]_2, [f_2]_1, [f_3]_2$ .",
"By induction, we verify that if (REF ) holds for all $k$ , we determine uniquely $[f_1]_k, [g_1]_k$ by normalizing $[\\hat{a}]_k\\in \\mathcal {N}_k$ .",
"We then determine $[f_2]_k, [f_3]_{k+1}$ uniquely to normalize $[\\hat{r}]_{k+1}=0$ .",
"For the details, let us find formula for the solutions.",
"We rewrite (REF ) as $T^-g_1=-(a\\circ G-\\hat{a}-f_1\\circ \\hat{\\varphi })^-,\\\\( f_1\\circ \\hat{\\varphi })^+=(a\\circ G-\\hat{a})^++T^+g_1.$ Using (REF ), we can solve $(-1)^{j-1}g_{1,ij}=-({(a\\circ G)}^-)_{i(j-1)0}, \\quad j\\ge 1, \\quad i+j=k.$ Then we have $(\\hat{a}^-)_{ij0}=0, \\quad i+j=k-1; \\\\(\\hat{a}^-)_{ij m}=((a\\circ G-f_1\\circ \\hat{\\varphi }+g_1)^-)_{ijm}, \\quad m\\ge 1, i+j+m=k-1.$ Note that $- [g_1]_k(z,-\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,\\bar{z},0)$ .",
"We obtain $[g_1]_k(z,\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,-\\bar{z},0).$ Having determined $[g_1]_k$ , we take $[ f_1]_k=[(a\\circ G-\\hat{a}+g_1)^+]_k.$ We then solve () by taking $[ f_2]_k=[E^-]_k, \\quad [f_3]_{k+1}=[(E-\\frac{1}{2}\\xi f_2)^+]_{k+1},\\\\E:=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2.$ We have achieved the preliminary normalization.",
"Assume now that $\\varphi (z,\\bar{z},\\xi )=(z+a^-(z,\\xi ,w)\\eta , \\xi ,w), \\quad \\hat{\\varphi }(z,\\bar{z},\\xi )=(z+\\hat{a}^-(z,\\xi ,w)\\eta , \\xi ,w)$ are in the preliminary normal form, i.e.",
"$w|a^-(z,\\xi ,w), \\quad w|\\hat{a}^-(z,\\xi ,w).$ Let us assume that $a^-(z,\\xi ,w)=wt (s), \\quad [a^-]_s\\lnot \\equiv 0; \\quad \\hat{a}^-(z,\\xi ,w)=wt(s).$ We assume that $\\varphi G=F\\hat{\\varphi }$ with $F(z,\\xi ,w)=I+(f_1,f_2,f_3),\\\\G(z,\\bar{z},\\xi )=(z+g_1(z,\\bar{z}), \\xi +g_2(z,\\bar{z},\\xi )).$ Here $f_i,g_j$ start with terms of weight and order at least 2.",
"In particular, we have $f_i=wt(N), \\quad g_i=wt (N),\\quad i=1,2; \\quad f_3=wt(N^{\\prime }); \\quad N^{\\prime }\\ge N\\ge 2.$ Set $(P,Q,R):=\\varphi G$ .",
"Using $N\\ge 2$ , $s\\ge 2$ , and the Taylor theorem, we obtain $P&=z+g_1(z,\\bar{z})+a^-(z,\\xi ,w)\\eta +a^-(z,\\xi ,w)(\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi ))\\\\&\\quad +\\eta \\nabla a^-(z,\\xi ,w)\\cdot \\Bigl (g_1(z,\\bar{z}), g_2(z,\\bar{z},\\xi ),(\\xi +2\\bar{z})\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )\\Bigr )\\\\&\\quad +wt(s+N+1),\\\\Q&=\\xi +g_2(z,\\bar{z},\\xi ),\\\\R&=w+(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N).$ We also have $(P,Q,R)=F\\hat{\\varphi }$ .",
"Thus $P&=z+\\hat{a}^-(z,\\xi ,w)\\eta +f_1(z,\\xi ,w)+\\partial _zf_1(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\Q&=\\xi +f_2(z,\\xi ,w)+\\partial _zf_2(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\R&=w+f_3(z,\\xi ,w)+\\partial _zf_3(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N^{\\prime }+s+1).$ We will use the above 6 identities for $P,Q,R$ in two ways.",
"First we use their lower order terms to get $f_1(z,\\xi ,w)=g_1(z,\\bar{z})+( a^-(z,\\xi ,w)-\\hat{a}^-(z,\\xi ,w))\\eta +wt(N+s),\\\\\\quad f_2(z,\\xi ,w)=g_2(z,\\bar{z},\\xi )+wt(N+s), \\\\ f_3(z,\\xi ,w)=(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N)+wt(N^{\\prime }+s).$ Hence, we can take $N^{\\prime }=N+1$ .",
"By (REF ) and the preliminary normalization, we first know that $\\hat{a}=a+wt(N+s-1), \\\\f_1(z,\\xi ,w)=b(z)+wt (N+s), \\quad g_1(z,\\bar{z})=b(z)+wt (N+s).",
"$ We compose () by $\\sigma $ and then take the difference of the two equations to get $f_2(z,\\xi ,w)=-\\bar{b}(\\bar{z})-\\bar{b}( -\\bar{z}-\\xi )+wt(2N-1)+wt(N+s), \\\\f_3(z,\\xi ,w)=-\\bar{z}\\bar{b}(-\\bar{z}-\\xi )+(\\bar{z}+\\xi )\\bar{b}(\\bar{z})+wt(2N)+wt(N+s+1).$ Here we have used $N^{\\prime }=N+1$ .",
"Let $b(z)=b_Nz^N+wt(N+1)$ .",
"Therefore, we have $g_2(z,\\bar{z},\\xi )=-\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N)+wt(N+1),\\\\\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi )=\\eta \\bar{b}_N\\sum \\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}+wt(N+1),\\\\(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )=\\bar{b}_N(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})w+wt(N+2).$ Next, we use the two formulae for $P$ and (REF ) to get the identity in higher weight: $\\hat{a}^-&=a^-+g_1^-+Lb_N+wt(N+s), \\quad f_1-g_1^+=wt (N+s+1).$ Here we have used $f_1^-=0$ and $Lb_N(z,\\xi ,w)&:=-Nb_Nz^{N-1} [a^-]_s(z,\\xi ,w)-[a^-]_s(z,\\xi ,w)\\bar{b}_N\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}\\nonumber \\\\&\\quad +\\nabla [a^-]_s\\cdot \\Bigl (b_Nz^N, -\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N),\\bar{b}_Nw(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})\\Bigr ).$ Recall that $w|a^-$ and $w|\\hat{a}^-$ .",
"We also have that $w|Lb_N(z,\\xi ,w)$ and $Lb_N$ is homogenous in weighted variables and of weight $N+s-1$ .",
"This shows that $[g_1^-(z,\\xi ,0)]_{N+s-1}=0$ .",
"By (REF ), we get $[g_1(z,\\bar{z})]_{N+s}=[g_1(z,0)]_{N+s}, \\quad [\\hat{a}^-]_{s+N-1}=[ a^-]_{s+N-1}+Lb_N.$ Let us make some observations.",
"First, $Lb_N$ depends only on $b_N$ and it does not depend on coefficients of $b(z)$ of degree larger than $N$ .",
"We observe that the first identity says that all coefficients of $[g_1]_{N+s}$ must be zero, except that the coefficient $g_{1,(N+s)0}$ is arbitrary.",
"On the other hand $Lb_N$ , which has weight $N+s-1$ , depends only on $g_{1,N0}$ , while $N+s-1>N$ .",
"Let us assume for the moment that we have $Lb_N\\ne 0$ for all $b_N\\ne 0$ .",
"We will then choose a suitable complement subspace ${\\mathcal {N}}^*_{N+s-1}$ in the space of weighted homogenous polynomials in $z,\\xi ,w$ of weight $N+s-1$ for $Lb_N$ .",
"Then $\\hat{a}^-\\in w\\sum _{N>1}{\\mathcal {N}}^*_{N+s-1}$ will be the required normal form.",
"The normal form will be obtained by the following procedures: Assume that $\\varphi $ is not formally equivalent to the quadratic mapping in the preliminary normalization.",
"We first achieve the preliminary normal form by a mapping $F^0=I+(f_1^0,f_2^0,f_3^0)$ and $G^0=I+(g_1^0,g_2^0)$ which are tangent to the identity.",
"We can make $F^0,G^0$ to be unique by requiring $f^1_1(z,0)=0$ .",
"Then $a$ is normalized such that $\\hat{a}=\\hat{a}^-\\eta $ with $[\\hat{a}^-]_s$ being non-zero homogenous part of the lowest weight.",
"We may assume that $[a]_{s+1}=[\\hat{a}]_{s+1}$ .",
"Inductively, we choose $f^1_{1,N00}$ ($N=2, 3, \\ldots $ ) to achieve $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ .",
"In this step for a given $N$ , we determine mappings $F^1=I+(f_1^1,f_2^1,f_3^1)$ and $G^1=I+(g_1^1,g_2^1)$ by requiring that $f_1^1(z,\\xi ,w)$ contains only one term $\\xi ^N$ , while $f_1^1,f_2^1,g_1^1,g_2^1$ have weight at most $N$ and $f_3^1$ has weight at most $N+1$ .",
"In the process, we also show that $[f_1^1(z,\\xi ,w)]_2^{N+s}$ depends only on $z$ , if we do not want to impose the restriction on $f_1^1$ .",
"Moreover, the coefficient of $\\xi ^{N+s-1}$ of $f_1^1$ can still be arbitrarily chosen without changing the normalization achieved for $[\\hat{a}^-]_{N+s-1}$ via $[f_1^1]_{N}$ .",
"However, by achieving $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ via $F^1,G^1$ , we may destroy the preliminary normalization achieved via $F_0,G_0$ .",
"We will then restore the preliminary normalization via $F^2=I+(f_1^2,f_2^2,f_3^2), G^2=I+(g_1^2,g_2^2)$ satisfying $g^2_1(z,0)=0$ .",
"This amounts to determining $g_1^2=g_1$ and $f_1^2=f_1$ via (REF ) and () for which the terms of weight at most $N+s$ have been determined by (REF ), and then $f_2^2=f_2,f_3^2=f_3,g_2^2=g_2$ are determined by (REF )-() and (), respectively.",
"This allows us to repeat the procedure to achieve the normalization in any higher weight.",
"We will then remove the restriction that the normalizing mappings must be tangent to the identity.",
"This will alter the normal form only by suitable linear dilations.",
"Suppose that $b_N\\ne 0$ .",
"Let us verify that $Lb_N\\ne 0.$ We will also identify one of non-zero coefficients to describe the normalizing condition on $\\hat{a}$ .",
"We write the two invariant polynomials $\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^jw^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^jw^k.$ If we plug in $w=\\bar{z}^2+\\bar{z}\\xi $ we obtain a polynomial identity in the variables $z,\\bar{z},\\xi $ .",
"$\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k.$ If we set $\\bar{z} = z = 0$ , we obtain that $\\lambda _N = \\lambda ^{\\prime }_N = {(-1)}^N .$ Recall that $j_*$ is the largest integer such that $(a^-)_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Since $w|[a^-]_s$ , then $k_*>0$ .",
"We obtain $(Lb_N)_{i_*(j_*+N-1)k_*}=(a^-)_{i_*j_*k_*}\\bar{b}_N(-\\lambda _{N-1}^{\\prime }-j_*\\lambda _{N-1}+k_*\\lambda _N)\\ne 0.$ Therefore, we can achieve $(\\hat{a}^-)_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ This determines uniquely all $b_2, b_3, \\ldots .$ We now remove the restriction that $F$ and $G$ are tangent to the identity.",
"Suppose that both $\\varphi $ and $\\hat{\\varphi }$ are in the normal form.",
"Suppose that $F\\varphi =\\hat{\\varphi }G$ .",
"Then looking at the quadratic terms, we know that the linear parts $F,G$ must be dilations.",
"In fact, the linear part of $F$ must be the linear automorphism of the quadric.",
"Thus the linear parts of $F$ and $G$ have the forms $G^{\\prime }\\colon (z,\\xi )=(\\nu z,\\bar{\\nu }\\xi ), \\quad F^{\\prime }(z,\\xi ,w)=(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w).$ Then $(F^{\\prime })^{-1}\\hat{\\varphi }G^{\\prime }$ is still in the normal form.",
"Since $(F^{\\prime })^{-1}F$ is holomorphic and $(G^{\\prime })^{-1}G$ is CR, by the uniqueness of the normalization, we know that $F^{\\prime }=F$ and $G^{\\prime }=G$ .",
"Therefore, $F$ and $G$ change the normal form $a^-$ as follows $a^-(z,\\xi ,w)= \\bar{\\nu }\\hat{a}^-(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w), \\quad \\nu \\in \\lbrace 0\\rbrace .$ When $[\\hat{a}^-]_s=[a^-]_s\\ne 0$ , we see that $|\\nu |=1$ .",
"Therefore, the formal automorphism group is discrete or one-dimensional.",
"In , Coffman used an analogous method of even/odd function decomposition to obtain a quadratic normal form for non Levi-flat real analytic $m$ -submanifolds in $n$ with an CR singularity satisfying certain non-degeneracy conditions, provided $\\frac{3(n+1)}{2} \\le m<n$ .",
"He was able to achieve the convergent normalization by a rapid iteration method.",
"Using the above decomposition of invariant and skew-invariant functions of the involution $\\sigma $ , one might achieve a convergent solution for approximate equations when $M$ is formaly equivalent to the quadric.",
"However, when the iteration is employed, each new CR mapping $\\hat{\\varphi }$ might only be defined on a domain that is proportional to that of the previous $\\varphi $ in a constant factor.",
"This is significantly different from the situations of Moser and Coffman , , where rapid iteration methods are applicable.",
"Therefore, even if $M$ is formally equivalent to the quadric, we do not know if they are holomorphically equivalent.",
"Instability of Bishop-like submanifolds Let us now discuss stability of Levi-flat submanifolds under small perturbations that keep the submanifolds Levi-flat, in particular we discuss which quadratic invariants are stable when moving from point to point on the submanifold.",
"The only stable submanifolds are A.$n$ and C.1.",
"The Bishop-like submanifolds (or even just the Bishop invariant) are not stable under perturbation, which we show by constructing examples.",
"Proposition 15.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a connected real-analytic real codimension 2 submanifold that has a non-degenerate CR singular at the origin.",
"$M$ can be written in coordinates $(z,w) \\in {n} \\times as\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$.In a neighborhood of the origin all complex tangentsof $ M$ are non-degenerate, while ranks of $ A,B$ are upper semicontinuous.Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).The CR singular set of $ M$ that is not of type B.$ 12$ at the origin is areal analytic subset of $ M$ of codimension at least $ 2$, while the CRsingular set of $ M$ that is of type B.$ 12$ the origin has codimension atleast $ 1$.",
"A.$ n$ has an isolated CR singular point at the origin and sodoes C.1 in $ 3$.Let $ S0 M$ be the set of CR singular points.There is a neighborhood $ U$ of the origin such that for $ S=S0U$we have the following.\\begin{enumerate}[(i)]\\item If M is of type A.k for k \\ge 2 at the origin, then it is of type A.j at each pointof S for somej \\ge k.\\item If M is of type C.1 at the origin, then it is of type C.1 on S.If M is of type C.0 at the origin, then it is of type C.0 or C.1 on S.\\item There exists an M that is of type B.\\gamma at one point and ofC.1 at CR singular points arbitrarily near.",
"Similarly there exists an Mof type A.1 at p \\in M that is either of type C.1, or B.\\gamma , atpoints arbitrarily near p. There alsoexists an M of type B.\\gamma at every point but where \\gamma varies from point to point.\\end{enumerate}$ First we show that the rank of $A$ and the rank of $B$ are lower semicontinuous on $S_0$ , without imposing Levi-flatness condition.",
"Similarly the real dimension of the range of $A(z,\\bar{z})$ is lower semicontinuous on $S_0$ .",
"Write $M$ as $w = \\rho (z,\\bar{z}) ,$ where $\\rho $ vanishes to second order at 0.",
"If we move to a different point of $S_0$ via an affine map $(z,w) \\mapsto (Z+z_0,W+w_0)$ .",
"Then we have $W+w_0 = \\rho (Z+z_0,\\bar{Z}+\\bar{z}_0) .$ We compute the Taylor coefficients $W =\\frac{\\partial \\rho }{\\partial z} (z_0,\\bar{z}_0) \\cdot Z +\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z} + \\\\+Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^t\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial z} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] \\bar{Z} +O(3) .$ The holomorphic terms can be absorbed into $W$ .",
"If $\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z}$ is nonzero, then this complex defining function has a linear term in $W$ and linear term in $\\bar{Z}$ and the submanifold is CR at this point.",
"Therefore the set of complex tangents of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\bar{z}} =0$ and each complex tangent point is non-degenerate.",
"At a complex tangent point at the origin, $A$ is given by $\\left[ \\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ and $B$ is given by $\\frac{1}{2} \\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ .",
"In particular these matrices change continuously as we move along $S$ .",
"We first conclude that all CR singular points of $M$ in a neighborhood of the origin are non-degenerate.",
"Further holomorphic transformations act on $A$ and $B$ using Proposition REF .",
"Therefore the ranks of $A$ and $B$ as well as the real dimension of the range of $A(z,\\bar{z})$ are lower semicontinuous on $S_0$ as claimed.",
"Furthermore as $M$ is real-analytic, the points where the rank drops lie on a real-analytic subvariety of $S_0$ , or in other words a thin set.",
"Let $U$ be a small enough neighbourhood of the origin so that $S = S_0 \\cap U$ is connected.",
"Imposing the condition that $M$ is Levi-flat, we apply Theorem REF .",
"By a simple computation, unless $M$ is of type B.$\\frac{1}{2}$ , the set of complex tangents of $M$ has codimension at least 2; and A.$n$ has isolated CR singular point and so does C.1 in 3.",
"The item () follows as A.$k$ are the only types where the rank of $B$ is greater than 1, and the theorem says $M$ must be one of these types.",
"For () note that since $A$ is of rank 1 when $M$ as C.$x$ at a point, $M$ cannot be of type A.$k$ nearby.",
"If $M$ is of type C.1 at a point then the range of $A$ must be of real dimension 2 in a neighbourhood, and hence on this neighbourhood $M$ cannot be of type B.$\\gamma $ .",
"The examples proving item () are given below.",
"Example 15.2 Define $M$ via $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 + \\bar{z}_1z_2z_3 .$ It is Levi-flat by Proposition REF .",
"At the origin $M$ is a type B.$\\gamma $ , but at a point where $z_1 = z_2 = 0$ but $z_3 \\ne 0$ , the submanifold is CR singular and it is of type C.1.",
"Example 15.3 Similarly if we define $M$ via $w = \\bar{z}_1^2 + \\bar{z}_1z_2 z_3 ,$ we obtain a CR singular Levi-flat $M$ that is A.1 at the origin, but C.1 at nearby CR singular points.",
"Example 15.4 If we define $M$ via $w = \\gamma \\bar{z}_1^2 + \\left|{z_1} \\right|^2 z_2 ,$ then $M$ is a CR singular Levi-flat type A.1 submanifold at the origin, but type B.$\\gamma $ at points where $z_1 = 0$ but $z_2 \\ne 0$ .",
"Example 15.5 The Bishop invariant can vary from point to point.",
"Define $M$ via $w = \\left|{z_1} \\right|^2 + \\bar{z}_1^2 \\bigl (\\gamma _1 (1-z_2) + \\gamma _2 z_2 \\bigr ) ,$ where $\\gamma _1 , \\gamma _2 \\ge 0$ .",
"It is not hard to see that $M$ is Levi-flat.",
"Again it is an image of $2 \\times {\\mathbb {R}}^2$ in a similar way as above.",
"At the origin, the submanifold is Bishop-like with Bishop invariant $\\gamma _1$ .",
"When $z_1=0$ and $z_2 = 1$ , the Bishop invariant is $\\gamma _2$ .",
"In fact when $z_1=0$ , the Bishop invariant at that point is $\\left|{\\gamma _1 (1-z_2) + \\gamma _2z_2} \\right| .$ Proposition REF says that this submanifold possesses a real-analytic foliation extending the Levi-foliation through the singular points.",
"Proposition REF says that if a foliation on $M$ extends to a (nonsingular) holomorphic foliation, then the submanifold would be a simple product of a Bishop submanifold and $.",
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],
[
"Automorphism group of the C.1 quadric",
"With the normal form achieved in previous sections, let us study the automorphism group of the C.1 quadric in this section.",
"We will again use the mixed-holomorphic involution that is obtained from the quadric.",
"We study the local automorphism group at the origin.",
"That is the set of germs at the origin of biholomorphic transformations taking $M$ to $M$ and fixing the origin.",
"First we look at the linear parts of automorphisms.",
"We already know that the linear term of the last component only depends on $w$ .",
"For $M_{C.1}$ we can say more about the first two components.",
"Proposition 12.1 Let $(F,G) = (F_1,\\ldots ,F_n,G)$ be a formal invertible or biholomorphic automorphism of $M_{C.1} \\subset {n+1}$ , that is the submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Then $F_1(z,w) = a z_1 + \\alpha w + O(2)$ , $F_2(z,w) = \\bar{a} z_2 + \\beta w + O(2)$ , and $G(z,w) = \\bar{a}^2 w + O(2)$ , where $a \\ne 0$ .",
"Let $a = (a_1,\\ldots ,a_n)$ and $b = (b_1,\\ldots ,b_n)$ be such that $F_1(z,w) = a \\cdot z + \\alpha w + O(2)$ and $F_2(z,w) = b \\cdot z + \\beta w + O(2)$ .",
"Then from Proposition REF we have $\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}0 & 1 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} a \\\\ b \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1 b_2 = 1$ , and $\\bar{a}_j b_k = 0$ for all $(j,k) \\ne (1,2)$ .",
"Similarly $\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}=\\lambda \\begin{bmatrix} a^* & b^* & \\cdots \\end{bmatrix}\\begin{bmatrix}1 & 0 & 0 & \\cdots & 0 \\\\0 & 0 & 0 & \\cdots & 0 \\\\\\vdots & \\vdots &\\vdots & \\ddots & \\vdots \\\\0 & 0 & 0 & \\cdots & 0\\end{bmatrix}\\begin{bmatrix} \\bar{a} \\\\ \\bar{b} \\\\ \\vdots \\end{bmatrix} .$ Therefore $\\lambda \\bar{a}_1^2 = 1$ , and $\\bar{a}_j \\bar{a}_k = 0$ for all $(j,k)\\ne (1,1)$ .",
"Putting these two together we obtain that $a_j = 0$ for all $j \\ne 1$ , and as $a_1 \\ne 0$ we get $b_j = 0$ for all $j \\ne 2$ .",
"As $\\lambda $ is the reciprocal of the coefficient of $w$ in $G$ , we are finished.",
"Lemma 12.2 Let $M_{C.1} \\subset 3$ be given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2.$ Suppose that a local biholomorphism (resp.",
"formal automorphism) $(F_1,F_2,G)$ transforms $M_{C.1}$ into $M_{C.1}$ .",
"Then $F_1$ depends only on $z_1$ , and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"Let us define a $(1,0)$ tangent vector field on $M$ by $Z=\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} .$ Write $F = (F_1,F_2,G)$ .",
"$F$ must take $Z$ into a multiple of itself when restricted to $M_{C.1}$ .",
"That is on $M_{C.1}$ we have $& \\frac{\\partial F_1}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_1}{\\partial w}= 0 ,\\\\& \\frac{\\partial F_2}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial F_2}{\\partial w}= \\lambda ,\\\\& \\frac{\\partial G}{\\partial z_2} + \\bar{z}_1 \\frac{\\partial G}{\\partial w}= \\lambda \\overline{F_1}(\\bar{z},\\bar{w}) ,$ for some function $\\lambda $ .",
"Let us take the first equation and plug in the defining equation for $M_1$ : $ \\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+\\bar{z}_1\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This equation is true for all $z \\in 2$ , and so we may treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We have an involution on $M_{C.1}$ that takes $\\bar{z}_1$ to $-z_2-\\bar{z}_1$ .",
"Therefore we also have $\\frac{\\partial F_1}{\\partial z_2}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+(-z_2-\\bar{z}_1)\\frac{\\partial F_1}{\\partial w}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)= 0 .$ This means that $\\frac{\\partial F_1}{\\partial w}$ and therefore $\\frac{\\partial F_1}{\\partial z_2}$ must be identically zero.",
"That is, $F_1$ only depends on $z_1$ .",
"We have that the following must hold for all $z$ : $G(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)F_2(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2)+{\\left(\\overline{F_1}(\\bar{z}_1) \\right)}^2 .$ Again we treat $z_1$ and $\\bar{z}_1$ as independent variables.",
"We differentiate with respect to $z_1$ : $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ We plug in the involution again to obtain $\\frac{\\partial G}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2 )=\\overline{F_1}(-z_2-\\bar{z}_1)\\frac{\\partial F_2}{\\partial z_1}(z_1,z_2,\\bar{z}_1 z_2 + \\bar{z}_1^2) .$ Therefore as $F_1$ is not identically zero, then as before both $\\frac{\\partial F_2}{\\partial z_1}$ and $\\frac{\\partial G}{\\partial z_1}$ must be identically zero.",
"Lemma 12.3 Take $M_{C.1} \\subset 3$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and let $(F_1,F_2,G)$ be a local automorphism at the origin.",
"Then $F_1$ uniquely determines $F_2$ and $G$ .",
"Furthermore, given any invertible function of one variable $F_1$ with $F_1(0) = 0$ , there exist unique $F_2$ and $G$ that complete an automorphism and they are determined by $ \\begin{aligned}F_2(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = \\bar{F}_1(\\bar{z}_1)+\\bar{F}_1(-\\bar{z}_1-z_2),\\\\G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2) & = -\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2).\\end{aligned}$ We should note that the lemma also works formally.",
"Given any formal $F_1$ , there exist unique formal $F_2$ and $G$ satisfying the above property.",
"By Lemma REF , $F_1$ depends only on $z_1$ and $F_2$ and $G$ depend only on $z_2$ and $w$ .",
"We write the automorphism as a composition of the two mappings $\\bigl (F_1(z_1),z_2,w\\bigr )$ and $\\bigl (z_1,F_2(z_2,w),G(z_2,w)\\bigr )$ .",
"We plug the transformation into the defining equation for $M_{C.1}$ .",
"$ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(\\bar{z}_1)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2 .$ We use the involution $(z_1,z_2) \\mapsto (-\\bar{z}_1-z_2,z_2)$ which preserves $M_{C.1}$ and obtain a second equation $ G(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\bar{F}_1(-\\bar{z}_1-z_2)F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)+{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2 .$ We eliminate $G$ and solve for $F_2$ : $ F_2(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= \\frac{{\\bigl (\\bar{F}_1(-\\bar{z}_1-z_2)\\bigr )}^2-{\\bigl (\\bar{F}_1(\\bar{z}_1)\\bigr )}^2}{\\bar{F}_1(\\bar{z}_1)-\\bar{F}_1(-\\bar{z}_1-z_2)}=\\bar{F}_1(\\bar{z}_1)+ \\bar{F}_1(-\\bar{z}_1-z_2) .$ Next we note that trivially, $F_2$ is unique if it exists: its difference vanishes on $M_{C.1}$ .",
"If we suppose that $F_1$ is convergent, then just as before, substituting $z_2$ with $tz_2$ and $\\bar{z}_1$ with $t\\bar{z}_1$ , we are restricting to curves $(tz_2,t^2w)$ for all $(z_2,w)$ .",
"The series is convergent in $t$ for every fixed $z_2$ and $w$ .",
"Therefore if $F_2$ exists and $F_1$ is convergent, then $F_2$ is convergent by Lemma REF .",
"Now we need to show the existence of the formal solution $F_2$ .",
"Notice that the right-hand side of (REF ) is invariant under the involution.",
"It suffices to show that any power series in $\\bar{z_1}, z_2$ that is invariant under the involution is a formal power series in $z_2$ and $\\bar{z}_1z_2+\\bar{z}_1^2$ .",
"Let us treat $\\xi =\\bar{z}_1$ as an independent variable.",
"The original involution becomes a holomorphic involution in $\\xi ,z_2$ : $\\tau \\colon \\xi \\rightarrow -\\xi -z_2, \\qquad z_2\\rightarrow z_2.$ By a theorem of Noether we obtain a set of generators for the ring of invariants can be obtained by applying the averaging operation $R(f) = \\frac{1}{2} ( f + f \\circ \\tau )$ to all monomials in $\\xi $ and $z_2$ of degree 2 or less.",
"By direct calculation it is not difficult to see that $\\xi ,\\xi z_2+\\xi ^2$ generate the ring of invariants.",
"Therefore any invariant power series in $z_2,\\xi $ is a power series in $\\xi ,\\xi z_2+\\xi ^2$ .",
"This shows the existence of $F_2$ .",
"The existence of $G$ follows the same.",
"The equation for $G(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)=-\\bar{F}_1(\\bar{z}_1)\\bar{F}_1(-\\bar{z}_1-z_2)$ is obtained by plugging in the equation for $F_2$ .",
"Its existence, uniqueness, and convergence in case $F_1$ converges, follows exactly the same as for $F_2$ .",
"Theorem 12.4 If $M \\subset {n+1}$ , $n \\ge 2$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 ,$ and $(F_1,F_2,\\ldots ,F_n,G)$ is a local automorphism at the origin, then $F_1$ depends only on $z_1$ , $F_2$ and $G$ depend only on $z_2$ and $w$ , and $F_1$ completely determines $F_2$ and $G$ via (REF ).",
"The mapping $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin.",
"Furthermore, given any invertible function $F_1$ of one variable with $F_1(0) = 0$ , and arbitrary holomorphic functions $F_3,\\ldots ,F_n$ with $F_j(0) = 0$ , and such that $(z_1,z_2,F_3,\\ldots ,F_n)$ has rank $n$ at the origin, then there exist unique $F_2$ and $G$ that complete an automorphism.",
"Let $(F_1,\\ldots ,F_n,G)$ be an automorphism.",
"Then we have $G(z_1,\\ldots ,z_n,w) =\\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w})F_2(z_1,\\ldots ,z_n,w) +{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\ldots ,\\bar{z}_n,\\bar{w}) \\bigr )}^2 .$ Proposition REF says that the linear terms in $G$ only depend on $w$ , the linear terms of $F_1$ depend only on $z_1$ and $w$ and the linear terms of $F_2$ only depend on $z_2$ and $w$ .",
"Let us embed $M_{C.1} \\subset 3$ into $M$ via setting $z_3 = \\alpha _3 z_2$ , $\\ldots $ , $z_n = \\alpha _n z_2$ , for arbitrary $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Then we obtain $ G(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) = \\\\\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w})F_2(z_1,z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,\\bar{w}) \\bigr )}^2 .$ By noting what the linear terms are, we notice that the above is the equation for an automorphism of $M_{C.1}$ .",
"Therefore by Lemma REF we have $\\frac{\\partial F_1}{\\partial w} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial F_2}{\\partial z_1} = 0 \\qquad \\text{and} \\qquad \\frac{\\partial G}{\\partial z_1} = 0 ,$ as that is true for all $\\alpha _3,\\ldots ,\\alpha _n$ .",
"Plugging in the defining equation for $M_{C.1}$ we obtain an equation that holds for all $z$ and we can treat $z$ and $\\bar{z}$ independently.",
"We plug in $z = 0$ to obtain $0 =\\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0)F_2(0,\\ldots ,0,\\bar{z}_1^2) + \\\\{\\bigl ( \\overline{F_1}(\\bar{z}_1,\\bar{z}_2,\\bar{\\alpha }_3\\bar{z}_2,\\ldots ,\\bar{\\alpha }_n\\bar{z}_2,0) \\bigr )}^2 .$ Differentiating with respect to $\\bar{\\alpha }_j$ we obtain $\\frac{\\partial F_1}{\\partial z_j} = 0$ , for $j=3,\\ldots ,n$ .",
"We set $\\bar{\\alpha }_j = 0$ in the equation, differentiate with respect to $\\bar{z}_2$ and obtain that $\\frac{\\partial F_1}{\\partial z_2} = 0$ .",
"In other words $F_1$ is a function of $z_1$ only.",
"We rewrite (REF ) by writing $F_1$ as a function of $z_1$ only and $F_2$ and $G$ as functions of $z_2,\\ldots ,z_n,w$ , and we plug in $w = \\bar{z}_1z_2 + \\bar{z}_1^2$ to obtain $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) = \\\\\\overline{F_1}(\\bar{z}_1)F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,\\bar{z}_1z_2 + \\bar{z}_1^2) +{\\bigl ( \\overline{F_1}(\\bar{z}_1) \\bigr )}^2 .$ By Lemma REF , we know that $F_1$ now uniquely determines $F_2(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ and $G(z_2,\\alpha _3z_2,\\ldots ,\\alpha _n z_2,w)$ .",
"These two functions therefore do not depend on $\\alpha _3,\\ldots ,\\alpha _n$ , and in turn $F_2$ and $G$ do not depend on $z_3,\\ldots ,z_n$ as claimed.",
"Furthermore $F_1$ does uniquely determine $F_2$ and $G$ .",
"Finally since the mapping is a biholomorphism, and from what we know about the linear parts of $F_1$ , $F_2$ , and $G$ , it is clear that $(z_1,z_2,F_3,\\ldots ,F_n)$ is rank $n$ .",
"The other direction follows by applying Lemma REF .",
"We start with $F_1$ , determine $F_2$ and $G$ as in 3 dimensions.",
"Then adding $F_3,\\ldots ,F_n$ and the rank condition guarantees an automorphism."
],
[
"Normal form for certain C.1 type submanifolds II",
"The goal of this section is to find the normal form for Levi-flat submanifolds $M \\subset {n+1}$ given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + \\operatorname{Re}f(z) ,$ for a holomorphic $f(z)$ of order $O(3)$ .",
"Since $f(z)$ can be absorbed into $w$ via a holomorphic transformation, the goal is really to prove the following theorem.",
"Theorem 13.1 Let $M \\subset {n+1}$ be a real-analytic Levi-flat given by $ w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}) ,$ where $r$ is $O(3)$ .",
"Then $M$ can be put into the $M_{C.1}$ normal form $ w = \\bar{z}_1z_2 + \\bar{z}_1^2 ,$ by a convergent normalizing transformation.",
"Furthermore, if $r$ is a polynomial and the coefficient of $\\bar{z}_1^3$ in $r$ is zero, then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"In Theorem REF , we have already shown that a submanifold of the form $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ is necessarily Levi-flat and has the normal form $M_{C.1}$ .",
"The first part of Theorem REF will follow once we prove: Lemma 13.2 If $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z})$ where $r$ is $O(3)$ and $M$ is Levi-flat, then $r$ depends only on $\\bar{z}_1$ .",
"First let us assume that $n=2$ .",
"For $p \\in M_{CR}$ , $T^{(1,0)}_p M$ is one dimensional.",
"The Levi-map is the matrix $L =\\begin{bmatrix}0 & 1 & 0 \\\\0 & 0 & 0 \\\\0 & 0 & 0\\end{bmatrix}$ applied to the $T^{(1,0)} M$ vectors.",
"As $M$ is Levi-flat, then the Levi-map has to vanish.",
"The only vectors $v$ for which $v^* L v = 0$ , are the ones without $\\frac{\\partial }{\\partial z_1}$ component or $\\frac{\\partial }{\\partial z_2}$ component.",
"That is vectors of the form $a \\frac{\\partial }{\\partial z_1} + b \\frac{\\partial }{\\partial w},\\qquad \\text{or} \\qquad a \\frac{\\partial }{\\partial z_2} + b \\frac{\\partial }{\\partial w}.$ We apply these vectors to the defining equation and its conjugate and we obtain in the first case the equations $b = 0, \\qquad a \\left( \\bar{z}_2 + 2z_1 + \\frac{\\partial \\bar{r}}{\\partial z_1} \\right) = 0 .$ This cannot be satisfied identically on $M$ since this is supposed to be true for all $z$ , but $a$ cannot be identically zero and the second factor in the second equation has only one nonholomorphic term, which is $\\bar{z}_2$ .",
"Let us try the second form and we obtain the equations $b = a \\bar{z}_1 , \\qquad a \\left( \\frac{\\partial \\bar{r}}{\\partial z_2} \\right) = 0 .$ Again $a$ cannot be identically zero, and hence the second factor of the second equation $\\frac{\\partial \\bar{r}}{\\partial z_2}$ must be identically zero, which is possible only if $r$ depends only on $\\bar{z}_1$ .",
"Finally, it is possible to pick $b=\\bar{z}_1$ and $a=1$ , to obtain a $T^{(1,0)}$ vector field $\\frac{\\partial }{\\partial z_2} + \\bar{z}_1 \\frac{\\partial }{\\partial w} ,$ and therefore these submanifolds are necessarily Levi-flat.",
"Next suppose that $n > 2$ .",
"Notice that replacing $z_k$ with $\\lambda _k \\xi $ for $k \\ge 2$ and then fixing $\\lambda _k$ for $k \\ge 2$ , we get $w = \\bar{z}_1 \\lambda _2 \\xi + \\bar{z}_1^2 + r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi }) .$ By Lemma REF , we obtain a Levi-flat submanifold in $(z_1,\\xi ,w) \\in 3$ , and hence can apply the above reasoning to obtain that $r(\\bar{z}_1,\\bar{\\lambda }_2 \\bar{\\xi },\\dots ,\\bar{\\lambda }_n \\bar{\\xi })$ does not depend on $\\bar{\\xi }$ .",
"As this was true for any $\\lambda _k$ 's, we have that $r$ can only depend on $\\bar{z}_1$ .",
"It is left to prove the claim about the polynomial normalizing transformation.",
"Lemma 13.3 Suppose $M \\subset {n+1}$ is given by $w = \\bar{z}_1 z_2 + \\bar{z}_1^2 + r(\\bar{z}_1)$ where $r$ is a polynomial that vanishes to fourth order.",
"Then there exists an invertible polynomial mapping taking $M_{C.1}$ to $M$ .",
"We will take a transformation of the form $(z_1,z_2,w) \\mapsto \\bigl (z_1,z_2+f(z_2,w),w+g(z_2,w) \\bigr ) .$ We are therefore trying to find polynomial $f$ and $g$ that satisfy $\\bar{z}_1z_2 + \\bar{z}_1^2+g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)=\\bar{z}_1 \\bigl (z_2 +f(z_2,\\bar{z}_1z_2 +\\bar{z}_1^2)\\bigr ) + \\bar{z}_1^2 + r(\\bar{z}_1) .$ If we simplify we obtain $g(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)-\\bar{z}_1 f(z_2,\\bar{z}_1z_2 + \\bar{z}_1^2)= r(\\bar{z}_1) .$ Consider the involution $S\\colon (\\bar{z}_1,z_2)\\rightarrow (-\\bar{z}_1-z_2,z_2)$ .",
"Its invariant polynomials $u(\\bar{z}_1,z_2)$ are precisely the polynomials in $z_2,z_2\\bar{z}_1+\\bar{z}_1^2$ .",
"The polynomial $r(\\bar{z}_1)$ can be uniquely written as $r^+(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)+\\Bigl (\\bar{z}_1+\\frac{z_2}{2}\\Bigr )r^-(z_2,\\bar{z}_1z_2+\\bar{z}_1^2)$ in two polynomials $r^\\pm $ .",
"Taking $f=-r^-$ and $g=r^++\\frac{z_2}{2}r^-$ , we find the desired solutions."
],
[
"Normal form for general type C.1 submanifolds",
"In this section we show that generically a Levi-flat type C.1 submanifold is not formally equivalent to the quadric $M_{C.1}$ submanifold.",
"In fact, we find a formal normal form that shows infinitely many invariants.",
"There are obviously infinitely many invariants if we do not impose the Levi-flat condition.",
"The trick therefore is, how to impose the Levi-flat condition and still obtain a formal normal form.",
"Let $M \\subset 3$ be a real-analytic Levi-flat type C.1 submanifold through the origin.",
"We know that $M$ is an image of ${\\mathbb {R}}^2 \\times under a real-analytic CR map that is a diffeomorphism onto itstarget; see Theorem~\\ref {thm:folextendsCxtype}.After a linear change of coordinates we assume thatthe mapping is\\begin{equation}\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (& x+iy + a(x,y,\\xi ), \\\\& \\xi + b(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) ,\\end{split}\\end{equation}where $ a$, $ b$ are $ O(2)$ and $ r$ is $ O(3)$.",
"As the mapping is a CR mapping and a local diffeomorphism, then given anysuch $ a$, $ b$, and $ r$, the image isnecessarily Levi-flat at CR points.",
"Therefore the set of all thesemappings gives us all type C.1 Levi-flat submanifolds.$ We precompose with an automorphism of ${\\mathbb {R}}^2 \\times to make $ b = 0$.We cannot similarly remove $ a$ as anyautomorphism must have real valued first two components (the new $ x$ and thenew $ y$), and hence thosecomponents can only depend on $ x$ and $ y$ and not on $$.",
"So if $ a$depends on $$, we cannot remove it by precomposing.$ Next we notice that we can treat $M$ as an abstract CR manifold.",
"Suppose we have two equivalent submanifolds $M_1$ and $M_2$ , with $F$ being the biholomorphic map taking $M_1$ to $M_2$ .",
"If $M_j$ is the image of a map $\\varphi _j$ , then note that $\\varphi _2^{-1}$ is CR on ${(M_2)}_{CR}$ .",
"Therefore, $G = \\varphi _2^{-1} \\circ F \\circ \\varphi _1$ is CR on ${(F \\circ \\varphi _1)}^{-1}\\bigl ({(M_2)}_{CR}\\bigr )$ , which is dense in a neighbourhood of the origin of ${\\mathbb {R}}^2 \\times (theCR singularity of $ M2$ is a thin set, and we pull it back by tworeal-analytic diffeomorphisms).",
"A real-analytic diffeomorphism thatis CR on a dense set is a CR mapping.",
"The same argumentworks for the inverse of $ G$,and therefore we have a CR diffeomorphism of $ R2 .",
"The conclusion we make is the following proposition.",
"Proposition 14.1 If $M_j \\subset 3$ , $j=1,2$ are given by the maps $\\varphi _j$ $\\begin{split}(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times {\\varphi _j}{\\mapsto }\\bigl (& x+iy + a_j(x,y,\\xi ), \\\\& \\xi + b_j(x,y,\\xi ), \\\\& (x-iy) \\xi + {(x-iy)}^2 + r_j(x,y,\\xi )\\bigr ) ,\\end{split}$ and $M_1$ and $M_2$ are locally biholomorphically (resp.",
"formally) equivalent at 0, then there exists local biholomorphisms (resp.",
"formal equivalences) $F$ and $G$ at 0, with $F(M_1) = M_2$ , $G({\\mathbb {R}}^2 \\times = {\\mathbb {R}}^2 \\times as germs(resp.\\ formally) and\\begin{equation}\\varphi _2 = F \\circ \\varphi _1 \\circ G .\\end{equation}$ In other words, the proposition states that if we find a normal form for the mapping we find a normal form for the submanifolds.",
"Let us prove that the proposition also works formally.",
"We have to prove that $G$ restricted to ${\\mathbb {R}}^2 \\times is CR, that is,$ G = 0$.",
"Let us consider\\begin{equation}\\varphi _2 \\circ G = F \\circ \\varphi _1 .\\end{equation}The right hand side does not depend on $$ and thus the left handside does not either.",
"Write $ G = (G1,G2,G3)$.",
"Let us write $ b = b2$and $ r = r2$for simplicity.",
"Taking derivative of $ 2 G$ with respectto $$ we get:\\begin{equation}\\begin{aligned}& G^1_{\\bar{\\xi }} +i G^2_{\\bar{\\xi }} +a_x(G) G^1_{\\bar{\\xi }} +a_y(G) G^2_{\\bar{\\xi }} +a_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\& G^3_{\\bar{\\xi }} +b_x(G) G^1_{\\bar{\\xi }} +b_y(G) G^2_{\\bar{\\xi }} +b_\\xi (G) G^3_{\\bar{\\xi }} = 0, \\\\&(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}) G^3+(G^1 - i G^2) G^3_{\\bar{\\xi }} +2 (G^1 - i G^2)(G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }})\\\\& \\qquad +r_x(G) G^1_{\\bar{\\xi }} +r_y(G) G^2_{\\bar{\\xi }} +r_\\xi (G) G^3_{\\bar{\\xi }} = 0 .\\end{aligned}\\end{equation}Suppose that the homogeneous parts of $ Gj$ are zero for alldegrees up to degree $ d-1$.",
"If we look at the degree $ d$ homogeneous partsof the first two equations above we immediately note that it must be that$ G1 + i G2 = 0$ and$ G3 = 0$ in degree $ d$.",
"We then look at the degree $ d+1$part of the third equation.",
"Recall that $ []d$ is the degree $ d$part of an expression.",
"We get\\begin{equation}{[G^1_{\\bar{\\xi }} - i G^2_{\\bar{\\xi }}]}_{d}{[G^3 + 2 G^1 - i 2 G^2]}_{1} = 0 .\\end{equation}As $ G$ is an automorphism we cannot have the linear terms be linearlydependent and hence$ G1 = G2 = 0$ in degree $ d$.",
"We finishby induction on $ d$.$ Using the proposition we can restate the result of Theorem REF using the parametrization.",
"Corollary 14.2 A real-analytic Levi-flat type C.1 submanifold $M \\subset 3$ is biholomorphically equivalent to the quadric $M_{C.1}$ if and only if the mapping giving $M$ is equivalent to a mapping of the form $(x,y,\\xi ) \\in {\\mathbb {R}}^2 \\times \\bigl (x+iy,\\quad \\xi ,\\quad (x-iy) \\xi + {(x-iy)}^2 + r(x,y,\\xi )\\bigr ) .$ That is, $M$ is equivalent to $M_{C.1}$ if and only if we can get rid of the $a(x,y,\\xi )$ via pre and post composing with automorphisms.",
"The proof of the corollary follows as a submanifold that is realized by this map must be of the form $w = \\bar{z}_1z_2 + \\bar{z}_1^2 + \\rho (z_1,\\bar{z}_1,z_2)$ and we apply Theorem REF .",
"We have seen that the involution $\\tau $ on $M$ , in particular when $M$ is the quadric, is useful to compute the automorphism group and to construct Levi-flat submanifolds of type $C.1$ .",
"We will also need to deal with power series in $z,\\bar{z}, \\xi $ .",
"Thus we extend $\\tau $ , which is originally defined on 2, as follows $\\sigma (z,\\bar{z},\\xi )=(z,-\\bar{z}-\\xi ,\\xi ).$ Here $z,\\bar{z},\\xi $ are treated as independent variables.",
"Note that $z,\\xi ,w=\\bar{z}\\xi +\\bar{z}^2$ are invariant by $\\sigma $ , while $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ is skew invariant by $\\sigma $ .",
"A power series in $z,\\bar{z},\\xi $ that is invariant by $\\sigma $ is precisely a power series in $z,\\xi ,w$ .",
"In general, a power series $u$ in $z,\\bar{z},\\xi $ admits a unique decomposition $u(z,\\bar{z},\\xi )=u^+(z,\\xi ,w)+\\eta u^-(z,\\xi ,w).$ First we introduce degree for power series $u(z,\\bar{z},\\xi )$ and weights for power series $v(z,\\xi ,w)$ .",
"As usual we assign degree $i+j+k$ to the monomial $z^i\\bar{z}^j\\xi ^k$ .",
"We assign weight $i+j+2k$ to the monomial $z^i\\xi ^jw^k$ .",
"For simplicity, we will call them weight in both situations.",
"Let us also denote $[u]_d(z,\\bar{z},\\xi )=\\sum _{i+j+k=d}u_{ijk}z^i\\bar{z}^j\\xi ^k, \\quad [v]_d(z,\\xi ,w)=\\sum _{i+j+2k=d}v_{ijk}z^i\\xi ^jw^k.$ Set $[u]_{i}^j=[u]_i+\\cdots +[u]_j$ and $[v]_i^j=[v]_i+\\cdots +[v]_j$ for $i\\le j$ .",
"Theorem 14.3 Let $M$ be a real-analytic Levi-flat type C.1 submanifold in 3.",
"There exists a formal biholomorphic map transforming $M$ into the image of $\\hat{\\varphi }(z,\\bar{z},\\xi )=\\bigl (z+A(z,\\xi , w)w\\eta , \\xi ,w\\bigr )$ with $\\eta =\\bar{z}+\\frac{1}{2}{\\xi }$ and $w=\\bar{z}\\xi +\\bar{z}^2$ .",
"Suppose further that $A\\lnot \\equiv 0$ .",
"Fix $i_*,j_*,k_*$ such that $j_*$ is the largest integer satisfying $A_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Then we can achieve $A_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ Furthermore, the power series $A$ is uniquely determined up to the transformation $A(z,\\xi ,w)\\rightarrow \\bar{c}^{3}A(cz,\\bar{c}\\xi ,\\bar{c}^2w), \\quad c\\in \\lbrace 0\\rbrace .$ In the above normal form with $A\\lnot \\equiv 0$ , the group of formal biholomorphisms that preserve the normal form consists of dilations $(z,\\xi ,w)\\rightarrow (\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)$ satisfying $\\bar{\\nu }^{3}A(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w)=A(z,\\xi ,w)$ .",
"It will be convenient to write the CR diffeomorphism $G$ of ${\\mathbb {R}}^2\\times as $ (G1,G2)$ where $ G1$ is complex-valued and depends on $ z,z$,while $ G2$ depends on $ z,z,$.",
"Let $ M$ be the image of a mapping $$ defined by\\begin{equation}\\begin{aligned}(z,\\bar{z},\\xi ) \\overset{\\varphi }{\\mapsto }\\bigl (& z + a(z,\\bar{z},\\xi ),\\\\& \\xi ,\\\\& \\bar{z} \\xi + {\\bar{z}}^2 + r(z,\\bar{z},\\xi )\\bigr )\\end{aligned}\\end{equation}with $ a=O(2), r=O(3)$.",
"We want to find a formal biholomorphic map $ F$ of $ 3$and a formal CR diffeomorphism $ G$ of $ R2 such that $F\\hat{\\varphi }G^{-1}=\\varphi $ with $\\hat{\\varphi }$ in the normal form.",
"To simplify the computation, we will first achieve a preliminary normal form where $r=0$ and the function $a$ is skew-invariant by $\\sigma $ .",
"For the preliminary normal form we will only apply $F, G$ that are tangent to the identity.",
"We will then use the general $F, G$ to obtain the final normal form.",
"Let us assume that $F, G$ are tangent to the identity.",
"Let $M=F\\bigl (\\hat{\\varphi }({\\mathbb {R}}^2\\times \\bigr )$ where $\\hat{\\varphi }$ is determined by $\\hat{a}, \\hat{r}$ .",
"We write $F=I+(f_1,f_2,f_3), \\quad G=I+(g_1,g_2).$ The $\\xi $ components in $\\varphi G=F\\hat{\\varphi }$ give us $g_2(z,\\bar{z},\\xi )=f_2\\bigl (z+\\hat{a}(z,\\bar{z},\\xi ), \\xi , \\bar{z}\\xi +\\bar{z}^2+\\hat{r}(z,\\bar{z},\\xi )\\bigr ).$ Thus, we are allowed to define $g_2$ by the above identity for any choice of $f_2=O(2)$ .",
"Eliminating $g_2$ in other components of $\\varphi G=F\\hat{\\varphi }$ , we obtain $f_1\\circ \\hat{\\varphi }-g_1&= a\\circ G-\\hat{a},\\\\f_3\\circ \\hat{\\varphi }-\\bar{z}f_2\\circ \\hat{\\varphi }&=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2,$ where $\\tilde{g}_1(z,\\bar{z})=\\bar{g}_1(\\bar{z},z)$ and $(a\\circ G)(z,\\bar{z},\\xi ):=a\\bigl (G_1(z,\\bar{z}),\\bar{G}_1(\\bar{z},z),G_2(z,\\bar{z},\\xi )\\bigr ).$ Each power series $r(z,\\bar{z},\\xi )$ admits a unique decomposition $r(z,\\bar{z},\\xi )=r^+(z,\\xi ,w)+\\eta r^-(z,\\xi ,w),$ where both $r^\\pm $ are invariant by $\\sigma $ .",
"Note that $r(z,\\bar{z},\\xi )$ is a power series in $z,\\xi $ and $w$ , if and only if it is invariant by $\\sigma $ , i.e.",
"if $r^-=0$ .",
"We write $r^+={wt}\\, (k), \\quad \\text{or}\\quad {wt} \\, (r^+)\\ge k,$ if $r^+_{abc}=0$ for $a+b+2c<k$ .",
"Define $r^-=wt (k)$ analogously and write $\\eta r^-={wt}\\,(k)$ if $r^-={wt}\\,(k-1)$ .",
"We write $r={wt}\\,(k)$ if $(r^+,\\eta r^-)={wt}\\,(k)$ .",
"Note that $r=O(k)\\Rightarrow r={wt}\\,(k); \\quad wt \\, (rs)\\ge wt\\, (r)+wt\\,(s).$ The power series in $z,\\bar{z}$ play a special role in describing normal forms.",
"Let us define $T^\\pm $ via $u(z,\\bar{z})=(T^+u)(z,\\xi ,w)+(T^-u)(z,\\xi ,w)\\eta .$ Let $S^+_k$ (resp.",
"$S^-_k$ ) be spanned by monomials in $z,\\bar{z},\\xi $ which have weight $k$ and are invariant (resp.",
"skew-invariant) by $\\sigma $ .",
"Then the range of $\\eta T^-$ in $S_k^-$ is a linear subspace $R_k$ .",
"We decompose $S_k^-=R_k\\oplus (S_k^-\\ominus R_k).$ The decomposition is of course not unique.",
"We will take $S_k^-\\ominus R_k=\\bigoplus _{a+b+2c=k-1, c>0} z̏^a\\xi ^bw^c\\eta .$ Here we have used $\\eta =\\bar{z}+\\frac{1}{2}\\xi $ , $\\eta ^2=w+\\frac{1}{4}\\xi ^2$ , and $T^+u(z,\\xi , w)=\\sum _{i,j\\ge 0}\\sum _{0\\le \\alpha \\le j/2} u_{ij} \\binom{j}{2\\alpha }z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha },\\\\T^-u(z,\\xi ,w)=\\sum _{i\\ge 0,j>0}\\sum _{0\\le \\alpha <j/2} u_{ij}\\binom{j}{2\\alpha +1}z^i(w+\\frac{1}{4}\\xi ^2)^\\alpha (-\\frac{1}{2}\\xi )^{j-2\\alpha -1}.$ In particular, we have $T^-u(z,\\xi ,0)=\\sum _{i\\ge 0,j>0}(-1)^{j-1}u_{ij} z^i \\xi ^{j-1}.$ This shows that $T^-u(z,\\xi ,0)=\\frac{1}{-\\xi }\\bigl (u(z,-\\xi )-u(z,0)\\bigr ).$ We are ready to show that under the condition that $g_1(z,\\bar{z})$ has no pure holomorphic terms, there exists a unique $(F,G)$ which is tangent to the identity such that $\\hat{r}=0$ and $\\hat{a}\\in \\mathcal {N}:=\\bigoplus \\mathcal {N}_k, \\quad \\mathcal {N}_k:=S_k^-\\ominus R_k.$ We start with terms of weight 2 in (REF )-() to get $[f_1]_2-[g_1]_2=[a]_2-\\eta [\\hat{a}^-]_1,\\\\[f_3]_2=0.$ Note that $f_j^-=0$ .",
"The first identity implies that $[f_1]_2-[T^+g_1]_2=[a^+]_2, \\quad [T^-g_1]_1=[\\hat{a}^-]_1-[a^-]_1.$ The first equation is solvable with kernel defined by $[f_1]_k-[T^+g_1]_k=0 $ for $k=2$ .",
"This shows that $[g_1]_2$ is still arbitrary and we use it to achieve $\\eta [\\hat{a}^-]_1\\in S_2^-\\ominus R_2=\\lbrace 0\\rbrace .$ Then the kernel space is defined by (REF ) and $[g_1(z,\\bar{z})-g_1(z,0)]_k=0$ with $k=2$ .",
"In particular, under the restriction $[g_1(z,0)]_k=0,$ for $k=2$ , we have achieved $\\hat{a}^-\\in \\mathcal {N}_2$ by unique $[f_1]_2, [g_1]_2, [f_2]_1, [f_3]_2$ .",
"By induction, we verify that if (REF ) holds for all $k$ , we determine uniquely $[f_1]_k, [g_1]_k$ by normalizing $[\\hat{a}]_k\\in \\mathcal {N}_k$ .",
"We then determine $[f_2]_k, [f_3]_{k+1}$ uniquely to normalize $[\\hat{r}]_{k+1}=0$ .",
"For the details, let us find formula for the solutions.",
"We rewrite (REF ) as $T^-g_1=-(a\\circ G-\\hat{a}-f_1\\circ \\hat{\\varphi })^-,\\\\( f_1\\circ \\hat{\\varphi })^+=(a\\circ G-\\hat{a})^++T^+g_1.$ Using (REF ), we can solve $(-1)^{j-1}g_{1,ij}=-({(a\\circ G)}^-)_{i(j-1)0}, \\quad j\\ge 1, \\quad i+j=k.$ Then we have $(\\hat{a}^-)_{ij0}=0, \\quad i+j=k-1; \\\\(\\hat{a}^-)_{ij m}=((a\\circ G-f_1\\circ \\hat{\\varphi }+g_1)^-)_{ijm}, \\quad m\\ge 1, i+j+m=k-1.$ Note that $- [g_1]_k(z,-\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,\\bar{z},0)$ .",
"We obtain $[g_1]_k(z,\\bar{z})=\\bar{z}[(a\\circ G-\\hat{a})^-]_{k-1}(z,-\\bar{z},0).$ Having determined $[g_1]_k$ , we take $[ f_1]_k=[(a\\circ G-\\hat{a}+g_1)^+]_k.$ We then solve () by taking $[ f_2]_k=[E^-]_k, \\quad [f_3]_{k+1}=[(E-\\frac{1}{2}\\xi f_2)^+]_{k+1},\\\\E:=r\\circ G-\\hat{r}+2\\eta \\tilde{g}_1+\\tilde{g}_1f_2\\circ \\hat{\\varphi }+\\tilde{g}_1^2.$ We have achieved the preliminary normalization.",
"Assume now that $\\varphi (z,\\bar{z},\\xi )=(z+a^-(z,\\xi ,w)\\eta , \\xi ,w), \\quad \\hat{\\varphi }(z,\\bar{z},\\xi )=(z+\\hat{a}^-(z,\\xi ,w)\\eta , \\xi ,w)$ are in the preliminary normal form, i.e.",
"$w|a^-(z,\\xi ,w), \\quad w|\\hat{a}^-(z,\\xi ,w).$ Let us assume that $a^-(z,\\xi ,w)=wt (s), \\quad [a^-]_s\\lnot \\equiv 0; \\quad \\hat{a}^-(z,\\xi ,w)=wt(s).$ We assume that $\\varphi G=F\\hat{\\varphi }$ with $F(z,\\xi ,w)=I+(f_1,f_2,f_3),\\\\G(z,\\bar{z},\\xi )=(z+g_1(z,\\bar{z}), \\xi +g_2(z,\\bar{z},\\xi )).$ Here $f_i,g_j$ start with terms of weight and order at least 2.",
"In particular, we have $f_i=wt(N), \\quad g_i=wt (N),\\quad i=1,2; \\quad f_3=wt(N^{\\prime }); \\quad N^{\\prime }\\ge N\\ge 2.$ Set $(P,Q,R):=\\varphi G$ .",
"Using $N\\ge 2$ , $s\\ge 2$ , and the Taylor theorem, we obtain $P&=z+g_1(z,\\bar{z})+a^-(z,\\xi ,w)\\eta +a^-(z,\\xi ,w)(\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi ))\\\\&\\quad +\\eta \\nabla a^-(z,\\xi ,w)\\cdot \\Bigl (g_1(z,\\bar{z}), g_2(z,\\bar{z},\\xi ),(\\xi +2\\bar{z})\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )\\Bigr )\\\\&\\quad +wt(s+N+1),\\\\Q&=\\xi +g_2(z,\\bar{z},\\xi ),\\\\R&=w+(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N).$ We also have $(P,Q,R)=F\\hat{\\varphi }$ .",
"Thus $P&=z+\\hat{a}^-(z,\\xi ,w)\\eta +f_1(z,\\xi ,w)+\\partial _zf_1(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\Q&=\\xi +f_2(z,\\xi ,w)+\\partial _zf_2(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N+s+1),\\\\R&=w+f_3(z,\\xi ,w)+\\partial _zf_3(z,\\xi ,w)\\hat{a}^-(z,\\xi ,w)\\eta +wt(N^{\\prime }+s+1).$ We will use the above 6 identities for $P,Q,R$ in two ways.",
"First we use their lower order terms to get $f_1(z,\\xi ,w)=g_1(z,\\bar{z})+( a^-(z,\\xi ,w)-\\hat{a}^-(z,\\xi ,w))\\eta +wt(N+s),\\\\\\quad f_2(z,\\xi ,w)=g_2(z,\\bar{z},\\xi )+wt(N+s), \\\\ f_3(z,\\xi ,w)=(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )+wt(2N)+wt(N^{\\prime }+s).$ Hence, we can take $N^{\\prime }=N+1$ .",
"By (REF ) and the preliminary normalization, we first know that $\\hat{a}=a+wt(N+s-1), \\\\f_1(z,\\xi ,w)=b(z)+wt (N+s), \\quad g_1(z,\\bar{z})=b(z)+wt (N+s).",
"$ We compose () by $\\sigma $ and then take the difference of the two equations to get $f_2(z,\\xi ,w)=-\\bar{b}(\\bar{z})-\\bar{b}( -\\bar{z}-\\xi )+wt(2N-1)+wt(N+s), \\\\f_3(z,\\xi ,w)=-\\bar{z}\\bar{b}(-\\bar{z}-\\xi )+(\\bar{z}+\\xi )\\bar{b}(\\bar{z})+wt(2N)+wt(N+s+1).$ Here we have used $N^{\\prime }=N+1$ .",
"Let $b(z)=b_Nz^N+wt(N+1)$ .",
"Therefore, we have $g_2(z,\\bar{z},\\xi )=-\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N)+wt(N+1),\\\\\\bar{g}_1(\\bar{z},z)+\\frac{1}{2}g_2(z,\\bar{z},\\xi )=\\eta \\bar{b}_N\\sum \\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}+wt(N+1),\\\\(2\\bar{z}+\\xi )\\bar{g}_1(\\bar{z},z)+\\bar{z}g_2(z,\\bar{z},\\xi )=\\bar{b}_N(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})w+wt(N+2).$ Next, we use the two formulae for $P$ and (REF ) to get the identity in higher weight: $\\hat{a}^-&=a^-+g_1^-+Lb_N+wt(N+s), \\quad f_1-g_1^+=wt (N+s+1).$ Here we have used $f_1^-=0$ and $Lb_N(z,\\xi ,w)&:=-Nb_Nz^{N-1} [a^-]_s(z,\\xi ,w)-[a^-]_s(z,\\xi ,w)\\bar{b}_N\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}\\nonumber \\\\&\\quad +\\nabla [a^-]_s\\cdot \\Bigl (b_Nz^N, -\\bar{b}_N(\\bar{z}^N+(-\\bar{z}-\\xi )^N),\\bar{b}_Nw(\\bar{z}^{N-1}+(-\\bar{z}-\\xi )^{N-1})\\Bigr ).$ Recall that $w|a^-$ and $w|\\hat{a}^-$ .",
"We also have that $w|Lb_N(z,\\xi ,w)$ and $Lb_N$ is homogenous in weighted variables and of weight $N+s-1$ .",
"This shows that $[g_1^-(z,\\xi ,0)]_{N+s-1}=0$ .",
"By (REF ), we get $[g_1(z,\\bar{z})]_{N+s}=[g_1(z,0)]_{N+s}, \\quad [\\hat{a}^-]_{s+N-1}=[ a^-]_{s+N-1}+Lb_N.$ Let us make some observations.",
"First, $Lb_N$ depends only on $b_N$ and it does not depend on coefficients of $b(z)$ of degree larger than $N$ .",
"We observe that the first identity says that all coefficients of $[g_1]_{N+s}$ must be zero, except that the coefficient $g_{1,(N+s)0}$ is arbitrary.",
"On the other hand $Lb_N$ , which has weight $N+s-1$ , depends only on $g_{1,N0}$ , while $N+s-1>N$ .",
"Let us assume for the moment that we have $Lb_N\\ne 0$ for all $b_N\\ne 0$ .",
"We will then choose a suitable complement subspace ${\\mathcal {N}}^*_{N+s-1}$ in the space of weighted homogenous polynomials in $z,\\xi ,w$ of weight $N+s-1$ for $Lb_N$ .",
"Then $\\hat{a}^-\\in w\\sum _{N>1}{\\mathcal {N}}^*_{N+s-1}$ will be the required normal form.",
"The normal form will be obtained by the following procedures: Assume that $\\varphi $ is not formally equivalent to the quadratic mapping in the preliminary normalization.",
"We first achieve the preliminary normal form by a mapping $F^0=I+(f_1^0,f_2^0,f_3^0)$ and $G^0=I+(g_1^0,g_2^0)$ which are tangent to the identity.",
"We can make $F^0,G^0$ to be unique by requiring $f^1_1(z,0)=0$ .",
"Then $a$ is normalized such that $\\hat{a}=\\hat{a}^-\\eta $ with $[\\hat{a}^-]_s$ being non-zero homogenous part of the lowest weight.",
"We may assume that $[a]_{s+1}=[\\hat{a}]_{s+1}$ .",
"Inductively, we choose $f^1_{1,N00}$ ($N=2, 3, \\ldots $ ) to achieve $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ .",
"In this step for a given $N$ , we determine mappings $F^1=I+(f_1^1,f_2^1,f_3^1)$ and $G^1=I+(g_1^1,g_2^1)$ by requiring that $f_1^1(z,\\xi ,w)$ contains only one term $\\xi ^N$ , while $f_1^1,f_2^1,g_1^1,g_2^1$ have weight at most $N$ and $f_3^1$ has weight at most $N+1$ .",
"In the process, we also show that $[f_1^1(z,\\xi ,w)]_2^{N+s}$ depends only on $z$ , if we do not want to impose the restriction on $f_1^1$ .",
"Moreover, the coefficient of $\\xi ^{N+s-1}$ of $f_1^1$ can still be arbitrarily chosen without changing the normalization achieved for $[\\hat{a}^-]_{N+s-1}$ via $[f_1^1]_{N}$ .",
"However, by achieving $[\\hat{a}^-]_{N+s-1}\\in w{\\mathcal {N}}^*_{N+s-1}$ via $F^1,G^1$ , we may destroy the preliminary normalization achieved via $F_0,G_0$ .",
"We will then restore the preliminary normalization via $F^2=I+(f_1^2,f_2^2,f_3^2), G^2=I+(g_1^2,g_2^2)$ satisfying $g^2_1(z,0)=0$ .",
"This amounts to determining $g_1^2=g_1$ and $f_1^2=f_1$ via (REF ) and () for which the terms of weight at most $N+s$ have been determined by (REF ), and then $f_2^2=f_2,f_3^2=f_3,g_2^2=g_2$ are determined by (REF )-() and (), respectively.",
"This allows us to repeat the procedure to achieve the normalization in any higher weight.",
"We will then remove the restriction that the normalizing mappings must be tangent to the identity.",
"This will alter the normal form only by suitable linear dilations.",
"Suppose that $b_N\\ne 0$ .",
"Let us verify that $Lb_N\\ne 0.$ We will also identify one of non-zero coefficients to describe the normalizing condition on $\\hat{a}$ .",
"We write the two invariant polynomials $\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^jw^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^jw^k.$ If we plug in $w=\\bar{z}^2+\\bar{z}\\xi $ we obtain a polynomial identity in the variables $z,\\bar{z},\\xi $ .",
"$\\bar{z}^N+(-\\bar{z}-\\xi )^N=\\lambda _N\\xi ^N+\\sum _{j<N}p_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k,\\\\\\sum _i\\bar{z}^i(-\\bar{z}-\\xi )^{N-1-i}=\\lambda _{N-1}^{\\prime }\\xi ^{N-1}+\\sum _{j<N-1}q_{ijk}z^i\\xi ^j{(\\bar{z}^2+\\bar{z}\\xi )}^k.$ If we set $\\bar{z} = z = 0$ , we obtain that $\\lambda _N = \\lambda ^{\\prime }_N = {(-1)}^N .$ Recall that $j_*$ is the largest integer such that $(a^-)_{i_*j_*k_*}\\ne 0$ and $i_*+j_*+2k_*=s$ .",
"Since $w|[a^-]_s$ , then $k_*>0$ .",
"We obtain $(Lb_N)_{i_*(j_*+N-1)k_*}=(a^-)_{i_*j_*k_*}\\bar{b}_N(-\\lambda _{N-1}^{\\prime }-j_*\\lambda _{N-1}+k_*\\lambda _N)\\ne 0.$ Therefore, we can achieve $(\\hat{a}^-)_{i_*(j_*+n)k_*}=0, \\quad n=1,2,\\ldots .$ This determines uniquely all $b_2, b_3, \\ldots .$ We now remove the restriction that $F$ and $G$ are tangent to the identity.",
"Suppose that both $\\varphi $ and $\\hat{\\varphi }$ are in the normal form.",
"Suppose that $F\\varphi =\\hat{\\varphi }G$ .",
"Then looking at the quadratic terms, we know that the linear parts $F,G$ must be dilations.",
"In fact, the linear part of $F$ must be the linear automorphism of the quadric.",
"Thus the linear parts of $F$ and $G$ have the forms $G^{\\prime }\\colon (z,\\xi )=(\\nu z,\\bar{\\nu }\\xi ), \\quad F^{\\prime }(z,\\xi ,w)=(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w).$ Then $(F^{\\prime })^{-1}\\hat{\\varphi }G^{\\prime }$ is still in the normal form.",
"Since $(F^{\\prime })^{-1}F$ is holomorphic and $(G^{\\prime })^{-1}G$ is CR, by the uniqueness of the normalization, we know that $F^{\\prime }=F$ and $G^{\\prime }=G$ .",
"Therefore, $F$ and $G$ change the normal form $a^-$ as follows $a^-(z,\\xi ,w)= \\bar{\\nu }\\hat{a}^-(\\nu z,\\bar{\\nu }\\xi ,\\bar{\\nu }^2w), \\quad \\nu \\in \\lbrace 0\\rbrace .$ When $[\\hat{a}^-]_s=[a^-]_s\\ne 0$ , we see that $|\\nu |=1$ .",
"Therefore, the formal automorphism group is discrete or one-dimensional.",
"In , Coffman used an analogous method of even/odd function decomposition to obtain a quadratic normal form for non Levi-flat real analytic $m$ -submanifolds in $n$ with an CR singularity satisfying certain non-degeneracy conditions, provided $\\frac{3(n+1)}{2} \\le m<n$ .",
"He was able to achieve the convergent normalization by a rapid iteration method.",
"Using the above decomposition of invariant and skew-invariant functions of the involution $\\sigma $ , one might achieve a convergent solution for approximate equations when $M$ is formaly equivalent to the quadric.",
"However, when the iteration is employed, each new CR mapping $\\hat{\\varphi }$ might only be defined on a domain that is proportional to that of the previous $\\varphi $ in a constant factor.",
"This is significantly different from the situations of Moser and Coffman , , where rapid iteration methods are applicable.",
"Therefore, even if $M$ is formally equivalent to the quadric, we do not know if they are holomorphically equivalent.",
"Instability of Bishop-like submanifolds Let us now discuss stability of Levi-flat submanifolds under small perturbations that keep the submanifolds Levi-flat, in particular we discuss which quadratic invariants are stable when moving from point to point on the submanifold.",
"The only stable submanifolds are A.$n$ and C.1.",
"The Bishop-like submanifolds (or even just the Bishop invariant) are not stable under perturbation, which we show by constructing examples.",
"Proposition 15.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a connected real-analytic real codimension 2 submanifold that has a non-degenerate CR singular at the origin.",
"$M$ can be written in coordinates $(z,w) \\in {n} \\times as\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$.In a neighborhood of the origin all complex tangentsof $ M$ are non-degenerate, while ranks of $ A,B$ are upper semicontinuous.Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).The CR singular set of $ M$ that is not of type B.$ 12$ at the origin is areal analytic subset of $ M$ of codimension at least $ 2$, while the CRsingular set of $ M$ that is of type B.$ 12$ the origin has codimension atleast $ 1$.",
"A.$ n$ has an isolated CR singular point at the origin and sodoes C.1 in $ 3$.Let $ S0 M$ be the set of CR singular points.There is a neighborhood $ U$ of the origin such that for $ S=S0U$we have the following.\\begin{enumerate}[(i)]\\item If M is of type A.k for k \\ge 2 at the origin, then it is of type A.j at each pointof S for somej \\ge k.\\item If M is of type C.1 at the origin, then it is of type C.1 on S.If M is of type C.0 at the origin, then it is of type C.0 or C.1 on S.\\item There exists an M that is of type B.\\gamma at one point and ofC.1 at CR singular points arbitrarily near.",
"Similarly there exists an Mof type A.1 at p \\in M that is either of type C.1, or B.\\gamma , atpoints arbitrarily near p. There alsoexists an M of type B.\\gamma at every point but where \\gamma varies from point to point.\\end{enumerate}$ First we show that the rank of $A$ and the rank of $B$ are lower semicontinuous on $S_0$ , without imposing Levi-flatness condition.",
"Similarly the real dimension of the range of $A(z,\\bar{z})$ is lower semicontinuous on $S_0$ .",
"Write $M$ as $w = \\rho (z,\\bar{z}) ,$ where $\\rho $ vanishes to second order at 0.",
"If we move to a different point of $S_0$ via an affine map $(z,w) \\mapsto (Z+z_0,W+w_0)$ .",
"Then we have $W+w_0 = \\rho (Z+z_0,\\bar{Z}+\\bar{z}_0) .$ We compute the Taylor coefficients $W =\\frac{\\partial \\rho }{\\partial z} (z_0,\\bar{z}_0) \\cdot Z +\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z} + \\\\+Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^t\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial z} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] \\bar{Z} +O(3) .$ The holomorphic terms can be absorbed into $W$ .",
"If $\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z}$ is nonzero, then this complex defining function has a linear term in $W$ and linear term in $\\bar{Z}$ and the submanifold is CR at this point.",
"Therefore the set of complex tangents of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\bar{z}} =0$ and each complex tangent point is non-degenerate.",
"At a complex tangent point at the origin, $A$ is given by $\\left[ \\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ and $B$ is given by $\\frac{1}{2} \\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ .",
"In particular these matrices change continuously as we move along $S$ .",
"We first conclude that all CR singular points of $M$ in a neighborhood of the origin are non-degenerate.",
"Further holomorphic transformations act on $A$ and $B$ using Proposition REF .",
"Therefore the ranks of $A$ and $B$ as well as the real dimension of the range of $A(z,\\bar{z})$ are lower semicontinuous on $S_0$ as claimed.",
"Furthermore as $M$ is real-analytic, the points where the rank drops lie on a real-analytic subvariety of $S_0$ , or in other words a thin set.",
"Let $U$ be a small enough neighbourhood of the origin so that $S = S_0 \\cap U$ is connected.",
"Imposing the condition that $M$ is Levi-flat, we apply Theorem REF .",
"By a simple computation, unless $M$ is of type B.$\\frac{1}{2}$ , the set of complex tangents of $M$ has codimension at least 2; and A.$n$ has isolated CR singular point and so does C.1 in 3.",
"The item () follows as A.$k$ are the only types where the rank of $B$ is greater than 1, and the theorem says $M$ must be one of these types.",
"For () note that since $A$ is of rank 1 when $M$ as C.$x$ at a point, $M$ cannot be of type A.$k$ nearby.",
"If $M$ is of type C.1 at a point then the range of $A$ must be of real dimension 2 in a neighbourhood, and hence on this neighbourhood $M$ cannot be of type B.$\\gamma $ .",
"The examples proving item () are given below.",
"Example 15.2 Define $M$ via $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 + \\bar{z}_1z_2z_3 .$ It is Levi-flat by Proposition REF .",
"At the origin $M$ is a type B.$\\gamma $ , but at a point where $z_1 = z_2 = 0$ but $z_3 \\ne 0$ , the submanifold is CR singular and it is of type C.1.",
"Example 15.3 Similarly if we define $M$ via $w = \\bar{z}_1^2 + \\bar{z}_1z_2 z_3 ,$ we obtain a CR singular Levi-flat $M$ that is A.1 at the origin, but C.1 at nearby CR singular points.",
"Example 15.4 If we define $M$ via $w = \\gamma \\bar{z}_1^2 + \\left|{z_1} \\right|^2 z_2 ,$ then $M$ is a CR singular Levi-flat type A.1 submanifold at the origin, but type B.$\\gamma $ at points where $z_1 = 0$ but $z_2 \\ne 0$ .",
"Example 15.5 The Bishop invariant can vary from point to point.",
"Define $M$ via $w = \\left|{z_1} \\right|^2 + \\bar{z}_1^2 \\bigl (\\gamma _1 (1-z_2) + \\gamma _2 z_2 \\bigr ) ,$ where $\\gamma _1 , \\gamma _2 \\ge 0$ .",
"It is not hard to see that $M$ is Levi-flat.",
"Again it is an image of $2 \\times {\\mathbb {R}}^2$ in a similar way as above.",
"At the origin, the submanifold is Bishop-like with Bishop invariant $\\gamma _1$ .",
"When $z_1=0$ and $z_2 = 1$ , the Bishop invariant is $\\gamma _2$ .",
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"Proposition REF says that if a foliation on $M$ extends to a (nonsingular) holomorphic foliation, then the submanifold would be a simple product of a Bishop submanifold and $.",
"Therefore,if $ 1 = 2$ then the Levi-foliation on $ M$cannot extend to a holomorphic foliation of a neighbourhood of $ M$.$ Bishop65article author=Bishop, Errett, title=Differentiable manifolds in complex Euclidean space, journal=Duke Math.",
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],
[
"Instability of Bishop-like submanifolds",
"Let us now discuss stability of Levi-flat submanifolds under small perturbations that keep the submanifolds Levi-flat, in particular we discuss which quadratic invariants are stable when moving from point to point on the submanifold.",
"The only stable submanifolds are A.$n$ and C.1.",
"The Bishop-like submanifolds (or even just the Bishop invariant) are not stable under perturbation, which we show by constructing examples.",
"Proposition 15.1 Suppose that $M \\subset {n+1}$ , $n \\ge 2$ , is a connected real-analytic real codimension 2 submanifold that has a non-degenerate CR singular at the origin.",
"$M$ can be written in coordinates $(z,w) \\in {n} \\times as\\begin{equation}w = A(z,\\bar{z}) + B(\\bar{z},\\bar{z}) + O(3),\\end{equation}for quadratic $ A$ and $ B$.In a neighborhood of the origin all complex tangentsof $ M$ are non-degenerate, while ranks of $ A,B$ are upper semicontinuous.Suppose that $ M$ is Levi-flat (that is $ MCR$is Levi-flat).The CR singular set of $ M$ that is not of type B.$ 12$ at the origin is areal analytic subset of $ M$ of codimension at least $ 2$, while the CRsingular set of $ M$ that is of type B.$ 12$ the origin has codimension atleast $ 1$.",
"A.$ n$ has an isolated CR singular point at the origin and sodoes C.1 in $ 3$.Let $ S0 M$ be the set of CR singular points.There is a neighborhood $ U$ of the origin such that for $ S=S0U$we have the following.\\begin{enumerate}[(i)]\\item If M is of type A.k for k \\ge 2 at the origin, then it is of type A.j at each pointof S for somej \\ge k.\\item If M is of type C.1 at the origin, then it is of type C.1 on S.If M is of type C.0 at the origin, then it is of type C.0 or C.1 on S.\\item There exists an M that is of type B.\\gamma at one point and ofC.1 at CR singular points arbitrarily near.",
"Similarly there exists an Mof type A.1 at p \\in M that is either of type C.1, or B.\\gamma , atpoints arbitrarily near p. There alsoexists an M of type B.\\gamma at every point but where \\gamma varies from point to point.\\end{enumerate}$ First we show that the rank of $A$ and the rank of $B$ are lower semicontinuous on $S_0$ , without imposing Levi-flatness condition.",
"Similarly the real dimension of the range of $A(z,\\bar{z})$ is lower semicontinuous on $S_0$ .",
"Write $M$ as $w = \\rho (z,\\bar{z}) ,$ where $\\rho $ vanishes to second order at 0.",
"If we move to a different point of $S_0$ via an affine map $(z,w) \\mapsto (Z+z_0,W+w_0)$ .",
"Then we have $W+w_0 = \\rho (Z+z_0,\\bar{Z}+\\bar{z}_0) .$ We compute the Taylor coefficients $W =\\frac{\\partial \\rho }{\\partial z} (z_0,\\bar{z}_0) \\cdot Z +\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z} + \\\\+Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^t\\left[\\frac{\\partial ^2 \\rho }{\\partial z \\partial z} (z_0,\\bar{z}_0)\\right] Z +\\frac{1}{2}Z^*\\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0)\\right] \\bar{Z} +O(3) .$ The holomorphic terms can be absorbed into $W$ .",
"If $\\frac{\\partial \\rho }{\\partial \\bar{z}} (z_0,\\bar{z}_0) \\cdot \\bar{Z}$ is nonzero, then this complex defining function has a linear term in $W$ and linear term in $\\bar{Z}$ and the submanifold is CR at this point.",
"Therefore the set of complex tangents of $M$ is defined by $\\frac{\\partial \\rho }{\\partial \\bar{z}} =0$ and each complex tangent point is non-degenerate.",
"At a complex tangent point at the origin, $A$ is given by $\\left[ \\frac{\\partial ^2 \\rho }{\\partial z \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ and $B$ is given by $\\frac{1}{2} \\left[\\frac{\\partial ^2 \\rho }{\\partial \\bar{z} \\partial \\bar{z}} (z_0,\\bar{z}_0) \\right]$ .",
"In particular these matrices change continuously as we move along $S$ .",
"We first conclude that all CR singular points of $M$ in a neighborhood of the origin are non-degenerate.",
"Further holomorphic transformations act on $A$ and $B$ using Proposition REF .",
"Therefore the ranks of $A$ and $B$ as well as the real dimension of the range of $A(z,\\bar{z})$ are lower semicontinuous on $S_0$ as claimed.",
"Furthermore as $M$ is real-analytic, the points where the rank drops lie on a real-analytic subvariety of $S_0$ , or in other words a thin set.",
"Let $U$ be a small enough neighbourhood of the origin so that $S = S_0 \\cap U$ is connected.",
"Imposing the condition that $M$ is Levi-flat, we apply Theorem REF .",
"By a simple computation, unless $M$ is of type B.$\\frac{1}{2}$ , the set of complex tangents of $M$ has codimension at least 2; and A.$n$ has isolated CR singular point and so does C.1 in 3.",
"The item () follows as A.$k$ are the only types where the rank of $B$ is greater than 1, and the theorem says $M$ must be one of these types.",
"For () note that since $A$ is of rank 1 when $M$ as C.$x$ at a point, $M$ cannot be of type A.$k$ nearby.",
"If $M$ is of type C.1 at a point then the range of $A$ must be of real dimension 2 in a neighbourhood, and hence on this neighbourhood $M$ cannot be of type B.$\\gamma $ .",
"The examples proving item () are given below.",
"Example 15.2 Define $M$ via $w = \\left|{z_1} \\right|^2 + \\gamma \\bar{z}_1^2 + \\bar{z}_1z_2z_3 .$ It is Levi-flat by Proposition REF .",
"At the origin $M$ is a type B.$\\gamma $ , but at a point where $z_1 = z_2 = 0$ but $z_3 \\ne 0$ , the submanifold is CR singular and it is of type C.1.",
"Example 15.3 Similarly if we define $M$ via $w = \\bar{z}_1^2 + \\bar{z}_1z_2 z_3 ,$ we obtain a CR singular Levi-flat $M$ that is A.1 at the origin, but C.1 at nearby CR singular points.",
"Example 15.4 If we define $M$ via $w = \\gamma \\bar{z}_1^2 + \\left|{z_1} \\right|^2 z_2 ,$ then $M$ is a CR singular Levi-flat type A.1 submanifold at the origin, but type B.$\\gamma $ at points where $z_1 = 0$ but $z_2 \\ne 0$ .",
"Example 15.5 The Bishop invariant can vary from point to point.",
"Define $M$ via $w = \\left|{z_1} \\right|^2 + \\bar{z}_1^2 \\bigl (\\gamma _1 (1-z_2) + \\gamma _2 z_2 \\bigr ) ,$ where $\\gamma _1 , \\gamma _2 \\ge 0$ .",
"It is not hard to see that $M$ is Levi-flat.",
"Again it is an image of $2 \\times {\\mathbb {R}}^2$ in a similar way as above.",
"At the origin, the submanifold is Bishop-like with Bishop invariant $\\gamma _1$ .",
"When $z_1=0$ and $z_2 = 1$ , the Bishop invariant is $\\gamma _2$ .",
"In fact when $z_1=0$ , the Bishop invariant at that point is $\\left|{\\gamma _1 (1-z_2) + \\gamma _2z_2} \\right| .$ Proposition REF says that this submanifold possesses a real-analytic foliation extending the Levi-foliation through the singular points.",
"Proposition REF says that if a foliation on $M$ extends to a (nonsingular) holomorphic foliation, then the submanifold would be a simple product of a Bishop submanifold and $.",
"Therefore,if $ 1 = 2$ then the Levi-foliation on $ M$cannot extend to a holomorphic foliation of a neighbourhood of $ M$.$ Bishop65article author=Bishop, Errett, title=Differentiable manifolds in complex Euclidean space, journal=Duke Math.",
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]
] | 1403.0558 |
[
[
"l-adic Monodromy and Shimura curves in positive characteristics"
],
[
"Abstract In this paper, we seek an appropriate definition for a Shimura curve of Hodge type in positive characteristics, i.e.",
"a characterization, in terms of geometry mod p, of curves in positive characteristics which are reduction of Shimura curves over the complex field.",
"Specifically, we study the liftablity of a curve in moduli space of principally polarized abelian varieties over k, char k=p.",
"We show that some conditions on the l-adic monodromy over such a curve imply that this curve can be lifted to a Shimura curve."
],
[
"Introduction",
"This paper is a sequel to our previous paper [15] and [16].",
"All the three paper aim at answering the following question: $\\textit {What is an appropriate definition of Shimura curves in positive characteristics ?",
"}$ The answer is only known for Shimura varieties of PEL type which admit a natural moduli interpretation.",
"In this paper, we consider Shimura curves of Hodge type and give an answer in the generic ordinary case, in terms of $l$ -adic monodromy.",
"In [15], we start with a proper family of abelian varieties in characteristic $p$ and prove if this family admits some special crystalline cycles, then it is a reduction of a Shimura curve of Mumford type.",
"In this paper we present a similar result.",
"We find the conditions on the $l$ -adic monodromy associated to the family, which imply the family is a reduction of a Mumford type family of abelian fourfolds.",
"Let $\\pi :X\\longrightarrow C$ be a family of principally polarized abelian fourfolds over a proper smooth curve $C$ defined over a finite field ${{\\mathbb {F}}}_{q}$ with $q=p^{f}$ , $p>3$ .",
"Let $\\mathcal {E}_{l}$ be the lisse étale $l$ -adic sheaf $R^{1}\\pi _{*}({\\mathbb {Q}}_{l})$ over $C$ .",
"Choose a geometric point $\\bar{\\xi }$ and a closed point $c$ in $C$ , and then $\\mathcal {E}_{l}$ induces a monodromy: $\\rho :\\pi _{1}(C,\\bar{\\xi })\\longrightarrow {\\mathrm {Aut}}(\\mathcal {E}_{l,c})\\cong GL(8,{\\mathbb {Q}}_{l}).$ Let $G_{l}$ be the Zariski closure of $\\rho (\\pi _{1}(C,\\bar{\\xi }))$ in $GL(8,{\\mathbb {Q}}_{l})$ and $G_{l}^{\\mathrm {geom}}$ be that of $\\rho (\\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi }))$ .",
"$1\\longrightarrow \\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi })\\longrightarrow \\pi _{1}(C,\\bar{\\xi })\\longrightarrow \\mbox{Gal}(\\bar{{\\mathbb {F}}}_q/{\\mathbb {F}}_{q})\\longrightarrow 1.$ Then $G_{l}^{\\mathrm {geom}}$ is a normal subgroup of $G_{l}$ .",
"To every closed point $c\\in C$ , it associates a unique (up to conjugation) Frobenius element $F_{c}$ in $\\pi _{1}(C,\\bar{\\xi })$ .",
"Its image in $\\mbox{Gal}(\\bar{{\\mathbb {F}}}_q/{\\mathbb {F}}_{q})$ is the integer $\\deg (\\kappa (c):{\\mathbb {F}}_{q})$ .",
"For notational simplicity, let us define the following representation $\\rho _{0}:SL(2,{\\mathbb {Q}}_{l})^{\\times 3}\\longrightarrow {\\mathrm {Aut}}(\\mathcal {E}_{l,c})$ as the tensor product of three copies of the standard representation of $SL(2,{\\mathbb {Q}}_{l})$ .",
"Fix an embedding ${\\mathbb {Q}}_{l}\\longrightarrow once for all.",
"Our main theorem isas follows.",
"\\begin{thm} If there exists aclosed point c\\in C such that\\begin{enumerate}\\item G_{l}^{\\mathrm {geom},o}\\otimes _{{\\mathbb {Q}}_{l}}{\\rm im}\\:\\rho _{0}\\otimes ,\\item X_{c} is an ordinary abelian variety,\\item In G_{l}, \\rho (F_{c}) generates a maximal torus which is unramifiedover {\\mathbb {Q}}_{p}.\\end{enumerate}Then X\\longrightarrow C is a weak Mumford curve.\\end{thm}If further assume the Higgsfield of $ XC$ is maximal, then there exists a family of polarized abelian fourfolds $ YC'$ such that$ $C^{\\prime }\\longrightarrow C$ is a finite étale covering, $Y\\longrightarrow X^{\\prime }$ is an isogeny between $Y$ and the pullback family of $X$ over $C^{\\prime }$ , $Y\\longrightarrow C^{\\prime }$ is a good reduction of a Mumford curve.",
"For the definition of weak Mumford curve, see Section .",
"Remark 1.1 By REF and Chebotarev density theorem, the Frobenius element over sufficiently many points in $C$ generates the maximal torus in $G_l$ .",
"From [7], we know that the torus generated by the Frobenius $F_{c}$ is defined over ${\\mathbb {Q}}$ .",
"So it makes sense to require the torus is unramified over ${\\mathbb {Q}}_{p}$ , which is equivalent to say that the eigenvalues of the Frobenius are unramified over ${\\mathbb {Q}}_{p}$ .",
"We wonder if (3) can be replaced by a weaker condition, especially under the presence of (1) and (2).",
"More are explained in Remark REF ."
],
[
"Structure of the paper",
"To prove , we reduce it to the main theorem in [15].",
"We need to compute the Frobenius eigenvalues in the $l$ -adic setting and the connection with the crystalline cohomology.",
"In Section , we show some good reductions of Mumford curves satisfy the conditions in so that we are not proving a vacuous theorem.",
"In Section , we compute the dimensions of the Frobenius eigenspaces in $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})$ and $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))$ .",
"$\\begin{aligned}\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})^{F-q^{2}}\\otimes {\\mathbb {Q}}_{q} & =1\\\\\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))^{F}\\otimes {\\mathbb {Q}}_{q} & =4.\\end{aligned}$ In Section , we translate the above result to $p$ -adic case via comparing Lefschetz trace formulas.",
"Therefore the Frobenius eigenspaces in crystalline cohomology spaces have the expected dimensions as in the main theorem of [15].",
"Furthermore, we conclude in Section that $\\Gamma ((C/W(k))_{{\\mathrm {cris}}},{E}nd(\\wedge ^{2}\\mathcal {E}))^{F}\\otimes {\\mathbb {Q}}_{q} \\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras.",
"Then boils down to the main result in [15].",
"Acknowledgements.",
"I am deeply indebted to my thesis advisor Aise Johan de Jong, who introduced me to this subject and encouraged me through this project.",
"Without him I would have given up long time ago.",
"I thank Martin Olsson for useful discussion and feedbacks."
],
[
"Mumford curves",
"In [11], Mumford defines a family of Shimura curves.",
"We briefly recall the construction.",
"Let $F$ be a cubic totally real field and $D$ be a quaternion division algebra over $F$ such that $D\\otimes _{{\\mathbb {Q}}}{\\mathbb {R}}\\cong \\mathbb {H}\\times \\mathbb {H}\\times M_{2}({\\mathbb {R}}),{\\rm {Cor}_{F/{\\mathbb {Q}}}(D)\\cong M_{8}({\\mathbb {Q}}).",
"}$ Here $\\mathbb {H}$ is the quaternion algebra.",
"Let $G=\\lbrace x\\in D^{*}|x\\bar{x}=1\\rbrace $ .",
"Then $G$ is a ${\\mathbb {Q}}$ -simple algebraic group and it is the ${\\mathbb {Q}}$ -form of the ${\\mathbb {R}}$ -algebraic group $SU(2)\\times SU(2)\\times SL(2,{\\mathbb {R}})$ .",
"$h:\\mathbb {S}_{m}({\\mathbb {R}})\\longrightarrow & G({\\mathbb {R}})\\\\e^{i\\theta }\\mapsto & I_{4}\\otimes \\begin{pmatrix}\\cos \\theta & \\sin \\theta \\\\-\\sin \\theta & \\cos \\theta \\end{pmatrix}.$ The pair $(G,h)$ forms a Shimura datum.",
"And it defines Shimura curves, parameterizing abelian fourfolds.",
"We call such curves (with its universal family) Mumford curves.",
"Definition 2.1 A curve in $\\mathcal {A}_{4,1,n}\\otimes is called a \\textit {specialMumford curve} if it is the image of a Mumford curve in $ A4,1,n induced by a universal family.",
"The family $X\\longrightarrow C$ is a weak Mumford curve over $k$ if the image of $C\\longrightarrow \\mathcal {A}_{4,1,n}$ (induced by the family $X/C$ ) is (possible an irreducible component of) a reduction of a special Mumford curve in $\\mathcal {A}_{4,1,n}\\otimes .$ Remark 2.2 The \"weakness\" of the weak Mumford curve, comparing to good reductions, reflects in two aspects: firstly, the image of $C\\longrightarrow \\mathcal {A}_{4,1,n}$ might have singularities.",
"Secondly, the reduction of a special Mumford curve at $k$ might be reducible and that image of $C\\longrightarrow \\mathcal {A}_{4,1,n}$ is just one of the irreducible components.",
"Examples To indicate that is not a vacuous result, we show good reductions of a Mumford curve with an ordinary fiber satisfy the conditions of .",
"Let $f: A\\longrightarrow M$ be the universal family over a Mumford curve defined over $ and $ M$ is defined over the reflex field $ K$.",
"For every$ p$ over which $ K$ splits, $ AM$ admits a smooth and genericallyordinary reduction $ : XC$ over $ p$ (\\cite {Zuo2}).",
"By the definitionof Mumford curve or \\cite [Proposition 2.4]{Xia2}, the image of$$\\rho _{:\\pi _{1}(M)\\longrightarrow R^{1}f_{*}(\\underline{)is SL(2,^{\\times 3}.",
"By Grothendieck specialization theorem of algebraicmonodromy, \\rho _{ factors through\\rho :\\pi _{1}(C,\\bar{\\xi })\\longrightarrow R^{1}f_{*}(\\underline{)with a surjection \\pi _{1}(M)\\longrightarrow \\pi _{1}(\\bar{C},\\bar{\\xi }).",
"Bythe comparison of the l-adic cohomology and de Rham cohomologyR^{1}f_{*}(\\cong R^{1}f_{*}({\\mathbb {Q}}_{l})\\otimes for every l\\ne p,we know conditions (1) and (2) in \\ref {main theorem} hold for such C.}}For condition (3), we choose x to be a CM point on M. SinceA_{x} is simple, there is a degree 8 field L\\subset {\\mathrm {End}}^{o}(A_{x}).We can choose p such that L is unramified over p. Then lookat the reduction \\bar{x} and {\\mathbb {Q}}[F_{\\bar{x}}] is the center of{\\mathrm {End}}(X_{\\bar{x}}).",
"Since L\\subset {\\mathrm {End}}(X_{\\bar{x}}) is the maximalcommutative subalgebra, F\\in L and in particular, F is unramifiedover p.}So with a careful choice of p, the resultant reduction of a Mumfordcurve satisfies all the conditions in \\ref {main theorem}.",
"}$ Frobenius eigenvalues on $\\wedge ^{4}\\mathcal {E}{\\mbox{ or }}{\\mathcal {E}nd(\\wedge ^{2}\\mathcal {E})}$ In this section, we compute the dimension of Frobenius eigenspace at the target spaces and the corresponding eigenvalues.",
"Recall the short exact sequence $1\\longrightarrow \\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi })\\longrightarrow \\pi _{1}(C,\\bar{\\xi })\\longrightarrow {\\rm {Gal}(\\bar{{\\mathbb {F}}}_{q}/{\\mathbb {F}}_{q})\\longrightarrow 1.",
"}$ The Zariski closure $G^{\\mathrm {geom}}_l$ , of $\\rho (\\pi ^{\\mathrm {geom}}_1(C,\\bar{\\xi }))$ is a normal subgroup of $G_l$ .",
"Since $\\mathcal {E}_l$ is pure, it follows from that $G^{\\mathrm {geom}}_l$ is semisimple.",
"The connected component of identity $G^{\\mathrm {geom},o}_l$ is the derived group of $G^o_l$ .",
"Remark 4.1 If we assume $G^{\\mathrm {geom}}_l \\otimes is entirely contained in $ SL(2,3$, then we only need to enlarge the base field $ Fq$ to kill the $ S3$ part.$ For every closed point $c$ of $C$ , the action of $F_{c}$ on $\\mathcal {E}_{l\\, c}\\cong H_{{\\mathrm {et}}}^{1}(X_{\\bar{c}},{\\mathbb {Q}}_{l})$ is semisimple (see [2]).",
"Therefore $F_{c}$ acts on $\\wedge ^{4}\\mathcal {E}_{l,c}$ and ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{l,c})$ semisimply.",
"Firstly let us consider the space $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))\\cong {\\mathrm {End}}(\\wedge ^{2}(\\mathcal {E}_{l,c}))^{\\pi _{1}^{\\mathrm {geom}}(C,\\bar{c})}.$ Since $F_c$ acts on ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{l,c})$ semisimply, we can calculate its eigenvalues over $.$ Let $V$ be the dimension 2 standard representation of $SL(2,$ .",
"Condition (1) of Theorem shows that $\\mathcal {E}_{l.c}\\otimes V^{\\otimes 3} $ is the tensor product of three standard representations of $SL(2,$ .",
"So base change to $,$$H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))\\otimes {\\mathrm {End}}(\\wedge ^{2}(V^{\\otimes 3}))^{SL(2,^{\\times 3}}.$$$ As a representation of $SL(2,^{\\times 3}$ , $\\wedge ^{2}(V^{\\otimes 3})$ decomposes into four distinct irreducible components $\\wedge ^{2}(V^{\\otimes 3})\\cong \\oplus ^4_{i=1}W_{i}.$ There $W_1, W_2, W_3$ are all $S^2 V \\otimes S^2 V$ and $W_4$ is the trivial representation of $SL(2,^{\\times 3}$ .",
"Yet $W_1, W_2, W_3$ are pairwisely non-isomorphic $SL(2, ^{\\times 3}$ representations.",
"Let $p_k$ ($i_k$ , resp.)",
"be the projection from $\\wedge ^{2}(V^{\\otimes 3})$ to $W_k$ (inclusion from $W_k$ to $\\wedge ^{2}(V^{\\otimes 3})$ , resp.).",
"We have ${\\mathrm {End}}(\\wedge ^{2}(V^{\\otimes 3}))^{SL(2,^{\\times 3}}\\cong \\oplus ^4_{k=1} (i_k\\circ \\text{id}_{W_k}\\circ p_k).$ Note each id$_{W_k}$ is invariant under the action of $SL(2,^{\\times 3}$ and the scalar multiplication.",
"Thus id$_{W_k}$ is further invariant under the action of $G_l$ .",
"In particular, the Frobenius $F_c$ fixes each $\\text{id}_{V_{i}}$ .",
"So the invariant space $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))^{F}$ has dimension 4.",
"Secondly, $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})=(\\wedge ^{4}\\mathcal {E}_{l\\,{c}})^{\\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi })}.$ Similarly, base change to $ and it is isomorphic to $ 4(V3)SL(2,3$.One can directly compute by hand, or see the proof of Theorem4.1 in \\cite {Moon} to conclude that this space only has dimension1 which is generated by the polarization.",
"Therefore the corresponding Frobeniuseigenvalues are $ q2$.$ In summary, we have the following results: $\\begin{aligned}\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))^{F} & =4,\\\\\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})^{F-q^{2}} & =1.\\end{aligned}$ Comparison of Lefschetz Trace Formulas In this section, we compare Lefschetz Trace Formulas to obtain a similar result to (REF ) in the case of crystalline cohomology.",
"We firstly consider $\\mathcal {E}_{p}:=R^{1}\\pi _{{\\mathrm {cris}},*}(\\mathcal {O}_{X})$ .",
"Since $\\sigma $ is the identity on ${\\mathbb {F}}_{q}$ , the absolute Frobenius $F$ acts linearly on $\\mathcal {E}_{p,c}$ .",
"Since the local crystalline characteristic polynomial coincides with the $l$ -adic one ([5]) $\\det (1-tF|_{\\mathcal {E}_{p\\, c}})=\\det (1-tF|_{\\mathcal {E}_{l,c}}),$ the eigenvalues of $F$ on $\\mathcal {E}_{l,c}$ and $\\mathcal {E}_{p\\, c}$ are identical.",
"Let $\\mathcal {F}_{l}$ be either $\\wedge ^{4}\\mathcal {E}_{l}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{l})$ .",
"Since $\\mathcal {E}_{l}$ comes from geometry, by Deligne's Weil II, the $l$ -adic relative Lefschetz Trace Formula provides $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{l,c}})=\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}.$ In the $p$ -adic case, we still use $\\mathcal {F}_{p}$ to represent either $\\wedge ^{4}\\mathcal {E}_{p}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{p})$ .",
"Since $\\mathcal {E}_{p}$ is a Dieudonne crystal, $\\mathcal {F}_{p}$ is automatically overconvergent.",
"By a theorem of Etesse and le Stum ([6]), we also have a Lefschetz Trace Formula within crystalline cohomology setting $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{p,c}})=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{{\\mathbb {Z}}_{q}},\\mathcal {F}_{p})})^{(-1)^{i}}.$ Combining with equality (REF ), we have $\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})})^{(-1)^{i}}.$ By Deligne's Weil II [4], the étale cohomology groups $H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})$ is pure of weight $i+j$ where $\\mathcal {F}_{l}$ has weight $j$ .",
"Since $\\mathcal {F}_{p}$ is pointwisely pure, by [6], $H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})$ has the purity which implies, on each side of equality (REF ), there is no cancellation between the numerator and the denominator.",
"All zeros or poles have the expected complex norms.",
"Then we have the following termwise equality from (REF ).",
"$\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})=\\det (1-tF|_{H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})}).$ So the eigenvalues of $F$ on $H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})\\otimes {\\mathbb {Q}}_{q}$ and $H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))\\otimes {\\mathbb {Q}}_{q}$ are identical as on their $l$ -adic counterparts.",
"In particular, $\\begin{aligned}\\dim H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})^{F-q^{2}}\\otimes {\\mathbb {Q}}_{q} & =1,\\\\\\dim _{{\\mathbb {Q}}_{q}}H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))^{F}\\otimes {\\mathbb {Q}}_{q} & =4.\\end{aligned}$ Compute ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}$ In order to apply the main theorem in [15], we need to prove that ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras.",
"Frobenius Torus By [13], ${\\mathbb {Q}}[F]\\cong \\prod K_{i}$ where $K_{i}$ are number fields.",
"The multiplicative group ${\\mathbb {Q}}[F]^{*}$ defines a ${\\mathbb {Q}}$ -torus $T=\\prod _{i}{\\rm {Res}_{K_{i}/{\\mathbb {Q}}}({\\mathbb {G}}_{m}).",
"}$ Viewing $F$ as an element in $G_{l}$ , $T$ can be regarded as the ${\\mathbb {Q}}$ -model of the connected component of 1 in the Zariski closure of the set $\\lbrace \\rho (F)^{n}|n\\in {\\mathbb {Z}}\\rbrace $ in $G_{l}$ (cf.",
"[1]).",
"In particular, $T$ is contained in a maximal torus of $G_{l}$ .",
"By and Chebotarev density theorem for the function field, generic points $c$ on $C$ satisfy that $F_{c}$ generates a maximal torus.",
"For every $c$ , the torus $T$ is defined over $\\mathbb {Q}$ .",
"We say $T$ is unramified over $\\mathbb {Q}_{p}$ if the splitting field of $T$ is unramified over prime $p$ , and equivalently, the eigenvalues of $F_{c}$ are unramified over $p$ .",
"Remark 6.2 Varying the prime $l$ , we obtain a compatible system of $l$ -adic representation as stated in [7].",
"The existence of a point $c$ satisfying (3) in requires that $G_l$ is unramified over ${\\mathbb {Q}}_p$ .",
"On one hand, by [7] and [8], for a subset of primes $l$ of density 1(or even $l$ large enough), $G_l$ is unramified over $Q_l$ .",
"However, most results in the two paper have involved Dirichlet density restriction and hence can not be applied directly to our case.",
"On the other hand, we expect that if $G_l$ is unramified over ${\\mathbb {Q}}_q$ , then there always exists a closed point $c$ satisfying (3) in .",
"We also think generic ordinary property of $X\\longrightarrow C$ should also provide more information on Frobenius eigenvalues.",
"Eigenvalues of $F_c$ on $\\mathcal {E}_{p\\,c}$ Note up to now, we have not used condition (2) and (3) in .",
"Under the condition (2), by REF , we always can find $c$ such that $X_c$ ordinary and $\\rho (F_c)$ a maximal torus.",
"Further with the condition (3), there exists a closed point $c$ which satisfies the following two conditions: $X_{c}$ is ordinary, the Frobenius torus $T$ is a maximal torus in $G_{l}$ .",
"Now we study the eigenvalues of the Frobenius on the fiber over $c$ .",
"Since $X_{c}$ is ordinary, $\\mathcal {E}_{p\\, c}$ is the product of a unit root crystal ${\\mathcal {U}_c}$ and its dual ${\\mathcal {U}^{\\vee }_c}$ .",
"Let $\\lambda _{1},\\cdots ,\\lambda _{4}$ be the eigenvalues of $F_c$ on ${\\mathcal {U}_c}$ .",
"Then on ${\\mathcal {U}^{\\vee }_c}$ , the eigenvalues of $F_c$ are $\\displaystyle \\frac{q}{\\lambda _{i}}$ .",
"Since ${\\mathcal {U}_c}$ is a unit root crystal, $\\lambda _{i}$ are all $p$ -adic units.",
"Since $\\mathcal {E}_{p\\, c}$ has pure weight 1, $\\lambda _{i}$ all have complex norm $q^{\\frac{1}{2}}$ .",
"By REF , the Frobenius $F_c$ also has eigenvalues $\\lambda _1,\\cdots , \\lambda _4, \\displaystyle \\frac{q}{\\lambda _1}, \\cdots , \\displaystyle \\frac{q}{\\lambda _4}$ on $\\mathcal {E}_{l,c}$ .",
"Since the Frobenius torus is the maximal torus, the Frobenius eigenvalues $\\lambda _i$ correspond to the weights in the $SL(2)^{\\times 3}$ representation $V^{\\otimes 3}$ .",
"Let $a, b, c$ be the three highest weights in the three standard representation of $SL(2, $ .",
"Then the eight weights of $V^{\\otimes 3}$ are of the form $\\pm a \\pm b \\pm c$ and they have a configuration as vertices of a cube.",
"In this cube, the four $p$ -adic units $\\lambda _1, \\cdots , \\lambda _4$ lie in the same face.",
"Without loss of generality, we can assume that $\\lambda _1$ corresponds to the highest weight $a+b+c$ and $\\lambda _2, \\lambda _3, \\lambda _4$ correspond to $a+b-c, a+c-b$ and $a-b-c$ .",
"Then the only relation between $\\lambda _1, \\cdots ,\\lambda _4$ is $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"So we have the following lemma.",
"Lemma 6.4 Under the above choice of $c$ , the eigenvalues $\\lambda _i$ have no relations other than those generated by $\\lambda _i\\displaystyle \\frac{q}{\\lambda _i}=q $ and $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"Remark 6.5 Lemma REF also follows from the arguments in [12].",
"Proposition 6.6 ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras.",
"From [15] or basic representation theory of $SL(2)$ , we know the condition REF implies ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\otimes _{{\\mathbb {Q}}_{q}}{\\times 4}$ as algebras.",
"In particular, the algebra ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is commutative.",
"Therefore ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is a product of fields.",
"Note $\\wedge ^{2}\\mathcal {E}_{p}$ has the polarization as a direct summand.",
"So ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}^{\\times 4}_q, {\\mathbb {Q}}_{q}\\times K{\\mbox{ or }}{\\mathbb {Q}}_{q}^{\\times 2}\\times L$ where $K$ is a degree 3 field extension of ${\\mathbb {Q}}_{q}$ and $L$ has degree 2.",
"Comparing with the decomposition over $, there exists $ K K or L L$ such that $ imK $ and $ imL$ are subcrystals in $ 2 Ep$.",
"Further, rank $ imK=27$ and rank $ imL=18$.$ If $K{\\mbox{ or }}L$ is unramified over ${\\mathbb {Q}}_{q}$ , then by enlarging $f$ in $q=p^{f}$ , it becomes a product of copies of ${\\mathbb {Q}}_{q}$ .",
"Therefore we only need to consider the case $K{\\mbox{ or }}L$ ramified over ${\\mathbb {Q}}_{q}$ .",
"Since $p\\ne 2{\\mbox{ or }}3$ , we can assume $L\\cong {\\mathbb {Q}}_{q}(\\sqrt{p})$ and $K\\cong {\\mathbb {Q}}_{q}(\\@root 3 \\of {p})$ and we can choose $\\eta _K=\\@root 3 \\of {p}$ , $\\eta _L=\\sqrt{p}$ .",
"Note $\\mathcal {E}_{p\\, c}\\cong \\mathcal {U}_c\\oplus \\mathcal {U}^\\vee _c$ .",
"Since the eigenvalues have distinct $p$ -adic values, there is no $F_c$ -invariant morphisms between ${\\wedge ^{2}{\\mathcal {U}_{c}}}, \\wedge ^{2}\\mathcal {U}_{c}^{\\vee }$ and $\\mathcal {U}_{c}\\otimes \\mathcal {U}_{c}^{\\vee }$ .",
"Thus we have the decomposition ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F \\longrightarrow {\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})^F \\cong {\\mathrm {End}}({\\wedge ^{2}{\\mathcal {U}_{c}}})\\oplus {\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}^{\\vee })\\oplus {\\mathrm {End}}({\\mathcal {U}_{c}\\otimes {\\mathcal {U}_{c}^{\\vee }).",
"}}}$ The restriction of $F$ to ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})$ is just as $F_c$ .",
"Then by REF , all the eigenvalues of $F$ on $\\wedge ^{2}{\\mathcal {U}_{c}}$ are $\\lambda _{1}\\lambda _{2},\\cdots ,\\lambda _{3}\\lambda _{4}$ and there is no more relations between the eigenvalues of $\\wedge ^{2}{\\mathcal {U}_{c}}$ other than $\\lambda _{4}\\lambda _{1}=\\lambda _{2}\\lambda _{3}$ .",
"So each eigenspace $U_{\\lambda _i\\lambda _j}$ has dimension 1 except for $(1,4){\\mbox{ or }}(2,3)$ .",
"Thereby ${\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}})^F \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathrm {End}}(U_{\\lambda _{i}\\lambda _{j}})\\oplus {\\mathrm {End}}(U_{\\lambda _{1}\\lambda _{3}})\\\\ \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathbb {Q}}_{q}(\\lambda _{i}\\lambda _{j})\\oplus M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4)).$ Since the four eigenvalues $\\lambda _{i}$ are all unramified over ${\\mathbb {Q}}_{q}$ and $L {\\mbox{ or }}K$ is ramified, the image of the composition $L{\\mbox{ or }}K \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {E}_p)^F \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {U}_c)^F$ lies only in ${\\mathrm {End}}(U_{\\lambda _1\\lambda _4})\\cong M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4))$ .",
"Otherwise, it would induce an embedding $L{\\mbox{ or }}K \\hookrightarrow {\\mathbb {Q}}_q(\\lambda _i\\lambda _j)$ .",
"In particular, ${\\eta _K}_{|\\wedge ^2 \\mathcal {U}_c} {\\mbox{ or }}{\\eta _L}_{|\\wedge ^2 \\mathcal {U}_c}$ has only rank 2.",
"Restricted to point $c$ , the image of $\\eta _K$ has dimension at most only 20.",
"Contradiction.",
"For $L$ , we know that ${\\eta _L}_{|\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c}$ is a surjection.",
"Note the eigenvalues of $F_c$ on $\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c$ have the form $\\displaystyle \\frac{q\\lambda _i}{\\lambda _j}$ .",
"Again by REF , among these eigenvalues, $\\displaystyle \\frac{q\\lambda _1}{\\lambda _4}$ has only multiplicity 1.",
"Therefore ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F \\cong {\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )\\oplus \\cdots \\cong {\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4}) \\oplus \\cdots $ as algebras.",
"Since ${\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4})$ is unramified over ${\\mathbb {Q}}_q$ , the image of $L$ in ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F$ excludes ${\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )$ and hence $\\eta _L$ can not be a surjection.",
"The contradiction concludes the proof."
],
[
"Examples",
"To indicate that is not a vacuous result, we show good reductions of a Mumford curve with an ordinary fiber satisfy the conditions of .",
"Let $f: A\\longrightarrow M$ be the universal family over a Mumford curve defined over $ and $ M$ is defined over the reflex field $ K$.",
"For every$ p$ over which $ K$ splits, $ AM$ admits a smooth and genericallyordinary reduction $ : XC$ over $ p$ (\\cite {Zuo2}).",
"By the definitionof Mumford curve or \\cite [Proposition 2.4]{Xia2}, the image of$$\\rho _{:\\pi _{1}(M)\\longrightarrow R^{1}f_{*}(\\underline{)is SL(2,^{\\times 3}.",
"By Grothendieck specialization theorem of algebraicmonodromy, \\rho _{ factors through\\rho :\\pi _{1}(C,\\bar{\\xi })\\longrightarrow R^{1}f_{*}(\\underline{)with a surjection \\pi _{1}(M)\\longrightarrow \\pi _{1}(\\bar{C},\\bar{\\xi }).",
"Bythe comparison of the l-adic cohomology and de Rham cohomologyR^{1}f_{*}(\\cong R^{1}f_{*}({\\mathbb {Q}}_{l})\\otimes for every l\\ne p,we know conditions (1) and (2) in \\ref {main theorem} hold for such C.}}For condition (3), we choose x to be a CM point on M. SinceA_{x} is simple, there is a degree 8 field L\\subset {\\mathrm {End}}^{o}(A_{x}).We can choose p such that L is unramified over p. Then lookat the reduction \\bar{x} and {\\mathbb {Q}}[F_{\\bar{x}}] is the center of{\\mathrm {End}}(X_{\\bar{x}}).",
"Since L\\subset {\\mathrm {End}}(X_{\\bar{x}}) is the maximalcommutative subalgebra, F\\in L and in particular, F is unramifiedover p.}So with a careful choice of p, the resultant reduction of a Mumfordcurve satisfies all the conditions in \\ref {main theorem}.",
"}$"
],
[
"Frobenius eigenvalues on $\\wedge ^{4}\\mathcal {E}{\\mbox{ or }}{\\mathcal {E}nd(\\wedge ^{2}\\mathcal {E})}$",
"In this section, we compute the dimension of Frobenius eigenspace at the target spaces and the corresponding eigenvalues.",
"Recall the short exact sequence $1\\longrightarrow \\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi })\\longrightarrow \\pi _{1}(C,\\bar{\\xi })\\longrightarrow {\\rm {Gal}(\\bar{{\\mathbb {F}}}_{q}/{\\mathbb {F}}_{q})\\longrightarrow 1.",
"}$ The Zariski closure $G^{\\mathrm {geom}}_l$ , of $\\rho (\\pi ^{\\mathrm {geom}}_1(C,\\bar{\\xi }))$ is a normal subgroup of $G_l$ .",
"Since $\\mathcal {E}_l$ is pure, it follows from that $G^{\\mathrm {geom}}_l$ is semisimple.",
"The connected component of identity $G^{\\mathrm {geom},o}_l$ is the derived group of $G^o_l$ .",
"Remark 4.1 If we assume $G^{\\mathrm {geom}}_l \\otimes is entirely contained in $ SL(2,3$, then we only need to enlarge the base field $ Fq$ to kill the $ S3$ part.$ For every closed point $c$ of $C$ , the action of $F_{c}$ on $\\mathcal {E}_{l\\, c}\\cong H_{{\\mathrm {et}}}^{1}(X_{\\bar{c}},{\\mathbb {Q}}_{l})$ is semisimple (see [2]).",
"Therefore $F_{c}$ acts on $\\wedge ^{4}\\mathcal {E}_{l,c}$ and ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{l,c})$ semisimply.",
"Firstly let us consider the space $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))\\cong {\\mathrm {End}}(\\wedge ^{2}(\\mathcal {E}_{l,c}))^{\\pi _{1}^{\\mathrm {geom}}(C,\\bar{c})}.$ Since $F_c$ acts on ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{l,c})$ semisimply, we can calculate its eigenvalues over $.$ Let $V$ be the dimension 2 standard representation of $SL(2,$ .",
"Condition (1) of Theorem shows that $\\mathcal {E}_{l.c}\\otimes V^{\\otimes 3} $ is the tensor product of three standard representations of $SL(2,$ .",
"So base change to $,$$H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))\\otimes {\\mathrm {End}}(\\wedge ^{2}(V^{\\otimes 3}))^{SL(2,^{\\times 3}}.$$$ As a representation of $SL(2,^{\\times 3}$ , $\\wedge ^{2}(V^{\\otimes 3})$ decomposes into four distinct irreducible components $\\wedge ^{2}(V^{\\otimes 3})\\cong \\oplus ^4_{i=1}W_{i}.$ There $W_1, W_2, W_3$ are all $S^2 V \\otimes S^2 V$ and $W_4$ is the trivial representation of $SL(2,^{\\times 3}$ .",
"Yet $W_1, W_2, W_3$ are pairwisely non-isomorphic $SL(2, ^{\\times 3}$ representations.",
"Let $p_k$ ($i_k$ , resp.)",
"be the projection from $\\wedge ^{2}(V^{\\otimes 3})$ to $W_k$ (inclusion from $W_k$ to $\\wedge ^{2}(V^{\\otimes 3})$ , resp.).",
"We have ${\\mathrm {End}}(\\wedge ^{2}(V^{\\otimes 3}))^{SL(2,^{\\times 3}}\\cong \\oplus ^4_{k=1} (i_k\\circ \\text{id}_{W_k}\\circ p_k).$ Note each id$_{W_k}$ is invariant under the action of $SL(2,^{\\times 3}$ and the scalar multiplication.",
"Thus id$_{W_k}$ is further invariant under the action of $G_l$ .",
"In particular, the Frobenius $F_c$ fixes each $\\text{id}_{V_{i}}$ .",
"So the invariant space $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))^{F}$ has dimension 4.",
"Secondly, $H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})=(\\wedge ^{4}\\mathcal {E}_{l\\,{c}})^{\\pi _{1}^{\\mathrm {geom}}(C,\\bar{\\xi })}.$ Similarly, base change to $ and it is isomorphic to $ 4(V3)SL(2,3$.One can directly compute by hand, or see the proof of Theorem4.1 in \\cite {Moon} to conclude that this space only has dimension1 which is generated by the polarization.",
"Therefore the corresponding Frobeniuseigenvalues are $ q2$.$ In summary, we have the following results: $\\begin{aligned}\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},{E} nd(\\wedge ^{2}\\mathcal {E}_{l}))^{F} & =4,\\\\\\dim H_{{\\mathrm {et}}}^{0}(C_{\\bar{{\\mathbb {F}}}_{q}},\\wedge ^{4}\\mathcal {E}_{l})^{F-q^{2}} & =1.\\end{aligned}$"
],
[
"Comparison of Lefschetz Trace Formulas",
"In this section, we compare Lefschetz Trace Formulas to obtain a similar result to (REF ) in the case of crystalline cohomology.",
"We firstly consider $\\mathcal {E}_{p}:=R^{1}\\pi _{{\\mathrm {cris}},*}(\\mathcal {O}_{X})$ .",
"Since $\\sigma $ is the identity on ${\\mathbb {F}}_{q}$ , the absolute Frobenius $F$ acts linearly on $\\mathcal {E}_{p,c}$ .",
"Since the local crystalline characteristic polynomial coincides with the $l$ -adic one ([5]) $\\det (1-tF|_{\\mathcal {E}_{p\\, c}})=\\det (1-tF|_{\\mathcal {E}_{l,c}}),$ the eigenvalues of $F$ on $\\mathcal {E}_{l,c}$ and $\\mathcal {E}_{p\\, c}$ are identical.",
"Let $\\mathcal {F}_{l}$ be either $\\wedge ^{4}\\mathcal {E}_{l}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{l})$ .",
"Since $\\mathcal {E}_{l}$ comes from geometry, by Deligne's Weil II, the $l$ -adic relative Lefschetz Trace Formula provides $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{l,c}})=\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}.$ In the $p$ -adic case, we still use $\\mathcal {F}_{p}$ to represent either $\\wedge ^{4}\\mathcal {E}_{p}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{p})$ .",
"Since $\\mathcal {E}_{p}$ is a Dieudonne crystal, $\\mathcal {F}_{p}$ is automatically overconvergent.",
"By a theorem of Etesse and le Stum ([6]), we also have a Lefschetz Trace Formula within crystalline cohomology setting $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{p,c}})=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{{\\mathbb {Z}}_{q}},\\mathcal {F}_{p})})^{(-1)^{i}}.$ Combining with equality (REF ), we have $\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})})^{(-1)^{i}}.$ By Deligne's Weil II [4], the étale cohomology groups $H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})$ is pure of weight $i+j$ where $\\mathcal {F}_{l}$ has weight $j$ .",
"Since $\\mathcal {F}_{p}$ is pointwisely pure, by [6], $H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})$ has the purity which implies, on each side of equality (REF ), there is no cancellation between the numerator and the denominator.",
"All zeros or poles have the expected complex norms.",
"Then we have the following termwise equality from (REF ).",
"$\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})=\\det (1-tF|_{H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})}).$ So the eigenvalues of $F$ on $H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})\\otimes {\\mathbb {Q}}_{q}$ and $H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))\\otimes {\\mathbb {Q}}_{q}$ are identical as on their $l$ -adic counterparts.",
"In particular, $\\begin{aligned}\\dim H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})^{F-q^{2}}\\otimes {\\mathbb {Q}}_{q} & =1,\\\\\\dim _{{\\mathbb {Q}}_{q}}H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))^{F}\\otimes {\\mathbb {Q}}_{q} & =4.\\end{aligned}$"
],
[
"Compute ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}$",
"In order to apply the main theorem in [15], we need to prove that ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras."
],
[
"Frobenius Torus",
"By [13], ${\\mathbb {Q}}[F]\\cong \\prod K_{i}$ where $K_{i}$ are number fields.",
"The multiplicative group ${\\mathbb {Q}}[F]^{*}$ defines a ${\\mathbb {Q}}$ -torus $T=\\prod _{i}{\\rm {Res}_{K_{i}/{\\mathbb {Q}}}({\\mathbb {G}}_{m}).",
"}$ Viewing $F$ as an element in $G_{l}$ , $T$ can be regarded as the ${\\mathbb {Q}}$ -model of the connected component of 1 in the Zariski closure of the set $\\lbrace \\rho (F)^{n}|n\\in {\\mathbb {Z}}\\rbrace $ in $G_{l}$ (cf.",
"[1]).",
"In particular, $T$ is contained in a maximal torus of $G_{l}$ .",
"By and Chebotarev density theorem for the function field, generic points $c$ on $C$ satisfy that $F_{c}$ generates a maximal torus.",
"For every $c$ , the torus $T$ is defined over $\\mathbb {Q}$ .",
"We say $T$ is unramified over $\\mathbb {Q}_{p}$ if the splitting field of $T$ is unramified over prime $p$ , and equivalently, the eigenvalues of $F_{c}$ are unramified over $p$ .",
"Remark 6.2 Varying the prime $l$ , we obtain a compatible system of $l$ -adic representation as stated in [7].",
"The existence of a point $c$ satisfying (3) in requires that $G_l$ is unramified over ${\\mathbb {Q}}_p$ .",
"On one hand, by [7] and [8], for a subset of primes $l$ of density 1(or even $l$ large enough), $G_l$ is unramified over $Q_l$ .",
"However, most results in the two paper have involved Dirichlet density restriction and hence can not be applied directly to our case.",
"On the other hand, we expect that if $G_l$ is unramified over ${\\mathbb {Q}}_q$ , then there always exists a closed point $c$ satisfying (3) in .",
"We also think generic ordinary property of $X\\longrightarrow C$ should also provide more information on Frobenius eigenvalues."
],
[
"Eigenvalues of $F_c$ on {{formula:9b7265c1-8f72-4262-aea8-ac5d62d5cc1b}}",
"Note up to now, we have not used condition (2) and (3) in .",
"Under the condition (2), by REF , we always can find $c$ such that $X_c$ ordinary and $\\rho (F_c)$ a maximal torus.",
"Further with the condition (3), there exists a closed point $c$ which satisfies the following two conditions: $X_{c}$ is ordinary, the Frobenius torus $T$ is a maximal torus in $G_{l}$ .",
"Now we study the eigenvalues of the Frobenius on the fiber over $c$ .",
"Since $X_{c}$ is ordinary, $\\mathcal {E}_{p\\, c}$ is the product of a unit root crystal ${\\mathcal {U}_c}$ and its dual ${\\mathcal {U}^{\\vee }_c}$ .",
"Let $\\lambda _{1},\\cdots ,\\lambda _{4}$ be the eigenvalues of $F_c$ on ${\\mathcal {U}_c}$ .",
"Then on ${\\mathcal {U}^{\\vee }_c}$ , the eigenvalues of $F_c$ are $\\displaystyle \\frac{q}{\\lambda _{i}}$ .",
"Since ${\\mathcal {U}_c}$ is a unit root crystal, $\\lambda _{i}$ are all $p$ -adic units.",
"Since $\\mathcal {E}_{p\\, c}$ has pure weight 1, $\\lambda _{i}$ all have complex norm $q^{\\frac{1}{2}}$ .",
"By REF , the Frobenius $F_c$ also has eigenvalues $\\lambda _1,\\cdots , \\lambda _4, \\displaystyle \\frac{q}{\\lambda _1}, \\cdots , \\displaystyle \\frac{q}{\\lambda _4}$ on $\\mathcal {E}_{l,c}$ .",
"Since the Frobenius torus is the maximal torus, the Frobenius eigenvalues $\\lambda _i$ correspond to the weights in the $SL(2)^{\\times 3}$ representation $V^{\\otimes 3}$ .",
"Let $a, b, c$ be the three highest weights in the three standard representation of $SL(2, $ .",
"Then the eight weights of $V^{\\otimes 3}$ are of the form $\\pm a \\pm b \\pm c$ and they have a configuration as vertices of a cube.",
"In this cube, the four $p$ -adic units $\\lambda _1, \\cdots , \\lambda _4$ lie in the same face.",
"Without loss of generality, we can assume that $\\lambda _1$ corresponds to the highest weight $a+b+c$ and $\\lambda _2, \\lambda _3, \\lambda _4$ correspond to $a+b-c, a+c-b$ and $a-b-c$ .",
"Then the only relation between $\\lambda _1, \\cdots ,\\lambda _4$ is $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"So we have the following lemma.",
"Lemma 6.4 Under the above choice of $c$ , the eigenvalues $\\lambda _i$ have no relations other than those generated by $\\lambda _i\\displaystyle \\frac{q}{\\lambda _i}=q $ and $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"Remark 6.5 Lemma REF also follows from the arguments in [12].",
"Proposition 6.6 ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras.",
"From [15] or basic representation theory of $SL(2)$ , we know the condition REF implies ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\otimes _{{\\mathbb {Q}}_{q}}{\\times 4}$ as algebras.",
"In particular, the algebra ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is commutative.",
"Therefore ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is a product of fields.",
"Note $\\wedge ^{2}\\mathcal {E}_{p}$ has the polarization as a direct summand.",
"So ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}^{\\times 4}_q, {\\mathbb {Q}}_{q}\\times K{\\mbox{ or }}{\\mathbb {Q}}_{q}^{\\times 2}\\times L$ where $K$ is a degree 3 field extension of ${\\mathbb {Q}}_{q}$ and $L$ has degree 2.",
"Comparing with the decomposition over $, there exists $ K K or L L$ such that $ imK $ and $ imL$ are subcrystals in $ 2 Ep$.",
"Further, rank $ imK=27$ and rank $ imL=18$.$ If $K{\\mbox{ or }}L$ is unramified over ${\\mathbb {Q}}_{q}$ , then by enlarging $f$ in $q=p^{f}$ , it becomes a product of copies of ${\\mathbb {Q}}_{q}$ .",
"Therefore we only need to consider the case $K{\\mbox{ or }}L$ ramified over ${\\mathbb {Q}}_{q}$ .",
"Since $p\\ne 2{\\mbox{ or }}3$ , we can assume $L\\cong {\\mathbb {Q}}_{q}(\\sqrt{p})$ and $K\\cong {\\mathbb {Q}}_{q}(\\@root 3 \\of {p})$ and we can choose $\\eta _K=\\@root 3 \\of {p}$ , $\\eta _L=\\sqrt{p}$ .",
"Note $\\mathcal {E}_{p\\, c}\\cong \\mathcal {U}_c\\oplus \\mathcal {U}^\\vee _c$ .",
"Since the eigenvalues have distinct $p$ -adic values, there is no $F_c$ -invariant morphisms between ${\\wedge ^{2}{\\mathcal {U}_{c}}}, \\wedge ^{2}\\mathcal {U}_{c}^{\\vee }$ and $\\mathcal {U}_{c}\\otimes \\mathcal {U}_{c}^{\\vee }$ .",
"Thus we have the decomposition ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F \\longrightarrow {\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})^F \\cong {\\mathrm {End}}({\\wedge ^{2}{\\mathcal {U}_{c}}})\\oplus {\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}^{\\vee })\\oplus {\\mathrm {End}}({\\mathcal {U}_{c}\\otimes {\\mathcal {U}_{c}^{\\vee }).",
"}}}$ The restriction of $F$ to ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})$ is just as $F_c$ .",
"Then by REF , all the eigenvalues of $F$ on $\\wedge ^{2}{\\mathcal {U}_{c}}$ are $\\lambda _{1}\\lambda _{2},\\cdots ,\\lambda _{3}\\lambda _{4}$ and there is no more relations between the eigenvalues of $\\wedge ^{2}{\\mathcal {U}_{c}}$ other than $\\lambda _{4}\\lambda _{1}=\\lambda _{2}\\lambda _{3}$ .",
"So each eigenspace $U_{\\lambda _i\\lambda _j}$ has dimension 1 except for $(1,4){\\mbox{ or }}(2,3)$ .",
"Thereby ${\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}})^F \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathrm {End}}(U_{\\lambda _{i}\\lambda _{j}})\\oplus {\\mathrm {End}}(U_{\\lambda _{1}\\lambda _{3}})\\\\ \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathbb {Q}}_{q}(\\lambda _{i}\\lambda _{j})\\oplus M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4)).$ Since the four eigenvalues $\\lambda _{i}$ are all unramified over ${\\mathbb {Q}}_{q}$ and $L {\\mbox{ or }}K$ is ramified, the image of the composition $L{\\mbox{ or }}K \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {E}_p)^F \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {U}_c)^F$ lies only in ${\\mathrm {End}}(U_{\\lambda _1\\lambda _4})\\cong M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4))$ .",
"Otherwise, it would induce an embedding $L{\\mbox{ or }}K \\hookrightarrow {\\mathbb {Q}}_q(\\lambda _i\\lambda _j)$ .",
"In particular, ${\\eta _K}_{|\\wedge ^2 \\mathcal {U}_c} {\\mbox{ or }}{\\eta _L}_{|\\wedge ^2 \\mathcal {U}_c}$ has only rank 2.",
"Restricted to point $c$ , the image of $\\eta _K$ has dimension at most only 20.",
"Contradiction.",
"For $L$ , we know that ${\\eta _L}_{|\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c}$ is a surjection.",
"Note the eigenvalues of $F_c$ on $\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c$ have the form $\\displaystyle \\frac{q\\lambda _i}{\\lambda _j}$ .",
"Again by REF , among these eigenvalues, $\\displaystyle \\frac{q\\lambda _1}{\\lambda _4}$ has only multiplicity 1.",
"Therefore ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F \\cong {\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )\\oplus \\cdots \\cong {\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4}) \\oplus \\cdots $ as algebras.",
"Since ${\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4})$ is unramified over ${\\mathbb {Q}}_q$ , the image of $L$ in ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F$ excludes ${\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )$ and hence $\\eta _L$ can not be a surjection.",
"The contradiction concludes the proof."
],
[
"Comparison of Lefschetz Trace Formulas",
"In this section, we compare Lefschetz Trace Formulas to obtain a similar result to (REF ) in the case of crystalline cohomology.",
"We firstly consider $\\mathcal {E}_{p}:=R^{1}\\pi _{{\\mathrm {cris}},*}(\\mathcal {O}_{X})$ .",
"Since $\\sigma $ is the identity on ${\\mathbb {F}}_{q}$ , the absolute Frobenius $F$ acts linearly on $\\mathcal {E}_{p,c}$ .",
"Since the local crystalline characteristic polynomial coincides with the $l$ -adic one ([5]) $\\det (1-tF|_{\\mathcal {E}_{p\\, c}})=\\det (1-tF|_{\\mathcal {E}_{l,c}}),$ the eigenvalues of $F$ on $\\mathcal {E}_{l,c}$ and $\\mathcal {E}_{p\\, c}$ are identical.",
"Let $\\mathcal {F}_{l}$ be either $\\wedge ^{4}\\mathcal {E}_{l}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{l})$ .",
"Since $\\mathcal {E}_{l}$ comes from geometry, by Deligne's Weil II, the $l$ -adic relative Lefschetz Trace Formula provides $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{l,c}})=\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}.$ In the $p$ -adic case, we still use $\\mathcal {F}_{p}$ to represent either $\\wedge ^{4}\\mathcal {E}_{p}{\\mbox{ or }}{E} nd(\\wedge ^{2}\\mathcal {E}_{p})$ .",
"Since $\\mathcal {E}_{p}$ is a Dieudonne crystal, $\\mathcal {F}_{p}$ is automatically overconvergent.",
"By a theorem of Etesse and le Stum ([6]), we also have a Lefschetz Trace Formula within crystalline cohomology setting $\\prod _{c\\in C}\\det (1-tF|_{\\mathcal {F}_{p,c}})=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{{\\mathbb {Z}}_{q}},\\mathcal {F}_{p})})^{(-1)^{i}}.$ Combining with equality (REF ), we have $\\prod _{i}\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})^{(-1)^{i}}=\\prod _{i}\\det (1-tF|_{H^{i}_{\\mathrm {cris}}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})})^{(-1)^{i}}.$ By Deligne's Weil II [4], the étale cohomology groups $H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})$ is pure of weight $i+j$ where $\\mathcal {F}_{l}$ has weight $j$ .",
"Since $\\mathcal {F}_{p}$ is pointwisely pure, by [6], $H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})$ has the purity which implies, on each side of equality (REF ), there is no cancellation between the numerator and the denominator.",
"All zeros or poles have the expected complex norms.",
"Then we have the following termwise equality from (REF ).",
"$\\det (1-tF|_{H_{{\\mathrm {et}}}^{i}(C_{\\bar{{\\mathbb {F}}}_{q}},\\mathcal {F}_{l})})=\\det (1-tF|_{H_{{\\mathrm {cris}}}^{i}(C/{\\mathbb {Z}}_{q},\\mathcal {F}_{p})}).$ So the eigenvalues of $F$ on $H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})\\otimes {\\mathbb {Q}}_{q}$ and $H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))\\otimes {\\mathbb {Q}}_{q}$ are identical as on their $l$ -adic counterparts.",
"In particular, $\\begin{aligned}\\dim H^{0}(C/{\\mathbb {Z}}_{q},\\wedge ^{4}\\mathcal {E}_{p})^{F-q^{2}}\\otimes {\\mathbb {Q}}_{q} & =1,\\\\\\dim _{{\\mathbb {Q}}_{q}}H^{0}(C/{\\mathbb {Z}}_{q},{E} nd(\\wedge ^{2}\\mathcal {E}_{p}))^{F}\\otimes {\\mathbb {Q}}_{q} & =4.\\end{aligned}$"
],
[
"Compute ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}$",
"In order to apply the main theorem in [15], we need to prove that ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras."
],
[
"Frobenius Torus",
"By [13], ${\\mathbb {Q}}[F]\\cong \\prod K_{i}$ where $K_{i}$ are number fields.",
"The multiplicative group ${\\mathbb {Q}}[F]^{*}$ defines a ${\\mathbb {Q}}$ -torus $T=\\prod _{i}{\\rm {Res}_{K_{i}/{\\mathbb {Q}}}({\\mathbb {G}}_{m}).",
"}$ Viewing $F$ as an element in $G_{l}$ , $T$ can be regarded as the ${\\mathbb {Q}}$ -model of the connected component of 1 in the Zariski closure of the set $\\lbrace \\rho (F)^{n}|n\\in {\\mathbb {Z}}\\rbrace $ in $G_{l}$ (cf.",
"[1]).",
"In particular, $T$ is contained in a maximal torus of $G_{l}$ .",
"By and Chebotarev density theorem for the function field, generic points $c$ on $C$ satisfy that $F_{c}$ generates a maximal torus.",
"For every $c$ , the torus $T$ is defined over $\\mathbb {Q}$ .",
"We say $T$ is unramified over $\\mathbb {Q}_{p}$ if the splitting field of $T$ is unramified over prime $p$ , and equivalently, the eigenvalues of $F_{c}$ are unramified over $p$ .",
"Remark 6.2 Varying the prime $l$ , we obtain a compatible system of $l$ -adic representation as stated in [7].",
"The existence of a point $c$ satisfying (3) in requires that $G_l$ is unramified over ${\\mathbb {Q}}_p$ .",
"On one hand, by [7] and [8], for a subset of primes $l$ of density 1(or even $l$ large enough), $G_l$ is unramified over $Q_l$ .",
"However, most results in the two paper have involved Dirichlet density restriction and hence can not be applied directly to our case.",
"On the other hand, we expect that if $G_l$ is unramified over ${\\mathbb {Q}}_q$ , then there always exists a closed point $c$ satisfying (3) in .",
"We also think generic ordinary property of $X\\longrightarrow C$ should also provide more information on Frobenius eigenvalues."
],
[
"Eigenvalues of $F_c$ on {{formula:9b7265c1-8f72-4262-aea8-ac5d62d5cc1b}}",
"Note up to now, we have not used condition (2) and (3) in .",
"Under the condition (2), by REF , we always can find $c$ such that $X_c$ ordinary and $\\rho (F_c)$ a maximal torus.",
"Further with the condition (3), there exists a closed point $c$ which satisfies the following two conditions: $X_{c}$ is ordinary, the Frobenius torus $T$ is a maximal torus in $G_{l}$ .",
"Now we study the eigenvalues of the Frobenius on the fiber over $c$ .",
"Since $X_{c}$ is ordinary, $\\mathcal {E}_{p\\, c}$ is the product of a unit root crystal ${\\mathcal {U}_c}$ and its dual ${\\mathcal {U}^{\\vee }_c}$ .",
"Let $\\lambda _{1},\\cdots ,\\lambda _{4}$ be the eigenvalues of $F_c$ on ${\\mathcal {U}_c}$ .",
"Then on ${\\mathcal {U}^{\\vee }_c}$ , the eigenvalues of $F_c$ are $\\displaystyle \\frac{q}{\\lambda _{i}}$ .",
"Since ${\\mathcal {U}_c}$ is a unit root crystal, $\\lambda _{i}$ are all $p$ -adic units.",
"Since $\\mathcal {E}_{p\\, c}$ has pure weight 1, $\\lambda _{i}$ all have complex norm $q^{\\frac{1}{2}}$ .",
"By REF , the Frobenius $F_c$ also has eigenvalues $\\lambda _1,\\cdots , \\lambda _4, \\displaystyle \\frac{q}{\\lambda _1}, \\cdots , \\displaystyle \\frac{q}{\\lambda _4}$ on $\\mathcal {E}_{l,c}$ .",
"Since the Frobenius torus is the maximal torus, the Frobenius eigenvalues $\\lambda _i$ correspond to the weights in the $SL(2)^{\\times 3}$ representation $V^{\\otimes 3}$ .",
"Let $a, b, c$ be the three highest weights in the three standard representation of $SL(2, $ .",
"Then the eight weights of $V^{\\otimes 3}$ are of the form $\\pm a \\pm b \\pm c$ and they have a configuration as vertices of a cube.",
"In this cube, the four $p$ -adic units $\\lambda _1, \\cdots , \\lambda _4$ lie in the same face.",
"Without loss of generality, we can assume that $\\lambda _1$ corresponds to the highest weight $a+b+c$ and $\\lambda _2, \\lambda _3, \\lambda _4$ correspond to $a+b-c, a+c-b$ and $a-b-c$ .",
"Then the only relation between $\\lambda _1, \\cdots ,\\lambda _4$ is $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"So we have the following lemma.",
"Lemma 6.4 Under the above choice of $c$ , the eigenvalues $\\lambda _i$ have no relations other than those generated by $\\lambda _i\\displaystyle \\frac{q}{\\lambda _i}=q $ and $\\lambda _1\\lambda _4=\\lambda _2\\lambda _3$ .",
"Remark 6.5 Lemma REF also follows from the arguments in [12].",
"Proposition 6.6 ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}_{q}^{\\times 4}$ as algebras.",
"From [15] or basic representation theory of $SL(2)$ , we know the condition REF implies ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\otimes _{{\\mathbb {Q}}_{q}}{\\times 4}$ as algebras.",
"In particular, the algebra ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is commutative.",
"Therefore ${\\mathrm {End}}^{0}(\\wedge ^{2}\\mathcal {E}_{p})^F$ is a product of fields.",
"Note $\\wedge ^{2}\\mathcal {E}_{p}$ has the polarization as a direct summand.",
"So ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^{F}\\cong {\\mathbb {Q}}^{\\times 4}_q, {\\mathbb {Q}}_{q}\\times K{\\mbox{ or }}{\\mathbb {Q}}_{q}^{\\times 2}\\times L$ where $K$ is a degree 3 field extension of ${\\mathbb {Q}}_{q}$ and $L$ has degree 2.",
"Comparing with the decomposition over $, there exists $ K K or L L$ such that $ imK $ and $ imL$ are subcrystals in $ 2 Ep$.",
"Further, rank $ imK=27$ and rank $ imL=18$.$ If $K{\\mbox{ or }}L$ is unramified over ${\\mathbb {Q}}_{q}$ , then by enlarging $f$ in $q=p^{f}$ , it becomes a product of copies of ${\\mathbb {Q}}_{q}$ .",
"Therefore we only need to consider the case $K{\\mbox{ or }}L$ ramified over ${\\mathbb {Q}}_{q}$ .",
"Since $p\\ne 2{\\mbox{ or }}3$ , we can assume $L\\cong {\\mathbb {Q}}_{q}(\\sqrt{p})$ and $K\\cong {\\mathbb {Q}}_{q}(\\@root 3 \\of {p})$ and we can choose $\\eta _K=\\@root 3 \\of {p}$ , $\\eta _L=\\sqrt{p}$ .",
"Note $\\mathcal {E}_{p\\, c}\\cong \\mathcal {U}_c\\oplus \\mathcal {U}^\\vee _c$ .",
"Since the eigenvalues have distinct $p$ -adic values, there is no $F_c$ -invariant morphisms between ${\\wedge ^{2}{\\mathcal {U}_{c}}}, \\wedge ^{2}\\mathcal {U}_{c}^{\\vee }$ and $\\mathcal {U}_{c}\\otimes \\mathcal {U}_{c}^{\\vee }$ .",
"Thus we have the decomposition ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p})^F \\longrightarrow {\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})^F \\cong {\\mathrm {End}}({\\wedge ^{2}{\\mathcal {U}_{c}}})\\oplus {\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}^{\\vee })\\oplus {\\mathrm {End}}({\\mathcal {U}_{c}\\otimes {\\mathcal {U}_{c}^{\\vee }).",
"}}}$ The restriction of $F$ to ${\\mathrm {End}}(\\wedge ^{2}\\mathcal {E}_{p\\, c})$ is just as $F_c$ .",
"Then by REF , all the eigenvalues of $F$ on $\\wedge ^{2}{\\mathcal {U}_{c}}$ are $\\lambda _{1}\\lambda _{2},\\cdots ,\\lambda _{3}\\lambda _{4}$ and there is no more relations between the eigenvalues of $\\wedge ^{2}{\\mathcal {U}_{c}}$ other than $\\lambda _{4}\\lambda _{1}=\\lambda _{2}\\lambda _{3}$ .",
"So each eigenspace $U_{\\lambda _i\\lambda _j}$ has dimension 1 except for $(1,4){\\mbox{ or }}(2,3)$ .",
"Thereby ${\\mathrm {End}}(\\wedge ^{2}{\\mathcal {U}_{c}})^F \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathrm {End}}(U_{\\lambda _{i}\\lambda _{j}})\\oplus {\\mathrm {End}}(U_{\\lambda _{1}\\lambda _{3}})\\\\ \\cong \\oplus _{(i,j)\\ne (1,3),(2,4)}{\\mathbb {Q}}_{q}(\\lambda _{i}\\lambda _{j})\\oplus M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4)).$ Since the four eigenvalues $\\lambda _{i}$ are all unramified over ${\\mathbb {Q}}_{q}$ and $L {\\mbox{ or }}K$ is ramified, the image of the composition $L{\\mbox{ or }}K \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {E}_p)^F \\longrightarrow {\\mathrm {End}}(\\wedge ^2\\mathcal {U}_c)^F$ lies only in ${\\mathrm {End}}(U_{\\lambda _1\\lambda _4})\\cong M_2({\\mathbb {Q}}_q(\\lambda _1\\lambda _4))$ .",
"Otherwise, it would induce an embedding $L{\\mbox{ or }}K \\hookrightarrow {\\mathbb {Q}}_q(\\lambda _i\\lambda _j)$ .",
"In particular, ${\\eta _K}_{|\\wedge ^2 \\mathcal {U}_c} {\\mbox{ or }}{\\eta _L}_{|\\wedge ^2 \\mathcal {U}_c}$ has only rank 2.",
"Restricted to point $c$ , the image of $\\eta _K$ has dimension at most only 20.",
"Contradiction.",
"For $L$ , we know that ${\\eta _L}_{|\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c}$ is a surjection.",
"Note the eigenvalues of $F_c$ on $\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c$ have the form $\\displaystyle \\frac{q\\lambda _i}{\\lambda _j}$ .",
"Again by REF , among these eigenvalues, $\\displaystyle \\frac{q\\lambda _1}{\\lambda _4}$ has only multiplicity 1.",
"Therefore ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F \\cong {\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )\\oplus \\cdots \\cong {\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4}) \\oplus \\cdots $ as algebras.",
"Since ${\\mathbb {Q}}_q(\\frac{q\\lambda _1}{\\lambda _4})$ is unramified over ${\\mathbb {Q}}_q$ , the image of $L$ in ${\\mathrm {End}}(\\mathcal {U}_c \\otimes \\mathcal {U}^\\vee _c)^F$ excludes ${\\mathrm {End}}(\\mathcal {U}_{\\frac{q\\lambda _1}{\\lambda _4}} )$ and hence $\\eta _L$ can not be a surjection.",
"The contradiction concludes the proof."
]
] | 1403.0125 |
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