diff --git a/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/2301.11979v1.pdf.txt b/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/2301.11979v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..05d633a0f10ad1ec6f500c5cf26e6eae85d854e3 --- /dev/null +++ b/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/2301.11979v1.pdf.txt @@ -0,0 +1,681 @@ +arXiv:2301.11979v1 [cond-mat.str-el] 27 Jan 2023 +DFT, L(S)DA, LDA+U, LDA+DMFT..., whether we do approach +to a proper description of optical response for strongly correlated +systems? +A.S. Moskvin1 +1Ural Federal University, Ekaterinburg, 620083 Russia +Аннотация +I present a critical overview of so-called "ab initio"DFT (density fuctional theory) based +calculation schemes for the description of the electronic structure, energy spectrum, and optical +response for strongly correlated 3d oxides, in particular, crystal-field and charge transfer transitions +as compared with an "old" cluster model that does generalize crystal-field and ligand-field theory. +As a most instructive illustration of validity of numerous calculation techniques I address the +prototypical 3d insulator NiO predicted to be a metal in frames of a standard LDA (local density +approximation) band theory. +1 + +INTRODUCTION +The electronic states in strongly correlated 3d oxides manifest both significant localization +and dispersional features. One strategy to deal with this dilemma is to restrict oneself to +small many-electron clusters embedded to a whole crystal, then creating model effective +lattice Hamiltonians whose spectra may reasonably well represent the energy and dispersion +of the important excitations of the full problem. Despite some shortcomings the method did +provide a clear physical picture of the complex electronic structure and the energy spectrum, +as well as the possibility of a quantitative modeling. +However, last decades the condensed matter community faced an expanding flurry of +papers with the so called ab initio calculations of electronic structure and physical properties +for strongly correlated systems such as 3d compounds based on density functional theory +(DFT). The modern formulation of the DFT originated in the work of Hohenberg and +Kohn [1], on which based the other classic work in this field by Kohn and Sham [2]. The Kohn- +Sham equation, has become a basic mathematical model of much of present-day methods for +treating electrons in atoms, molecules, condensed matter, solid surfaces, nanomaterials, and +man-made structures [3]. Of the top three most cited physicists in the period 1980-2010, the +first (Perdew: 65 757 citations) and third (Becke: 62 581 citations) were density-functional +theorists [4]. +However, DFT still remains, in some sense, ill-defined: many of DFT statements were +ill-posed or not rigorously proved. Most widely used DFT computational schemes start +with a "metallic-like"approaches making use of approximate energy functionals, firstly LDA +(local density approximation) scheme, which are constructed as expansions around the +homogeneous electron gas limit and fail quite dramatically in capturing the properties of +strongly correlated systems. The LDA+U and LDA+DMFT (DMFT, dynamical mean- +field theory) [5] methods are believed to correct the inaccuracies of approximate DFT +exchange correlation functionals. The main idea of these computational approaches consists +in a selective description of the strongly correlated electronic states, typically, localized +d or f orbitals, using the Hubbard model, while all the other states continue to be +treated at the level of standard approximate DFT functionals. At present the LDA+U +and LDA+DMFT methods are addressed to be most powerful tools for the investigation +of strongly correlated electronic systems, however, these preserve many shortcomings of +2 + +the DFT-LDA approach. Despite many examples of a seemingly good agreement with +experimental data (photoemission and inverse-photoemission spectra, magnetic moments,...) +claimed by the DFT community, both the questionable starting point and many unsolved +and unsoluble problems give rise to serious doubts in quantitative and even qualitative +predictions made within the DFT based techniques. In a certain sense the cluster based +calculations seem to provide a better description of the overall electronic structure of +insulating 3doxides and its optical response than the DFT based band structure calculations, +mainly due to a clear physics and a better account for correlation effects (see, e.g., +Refs. [6, 7]). +The paper is organized as follows. In Sec.II we do present a short critical overview of the +DFT and the DFT based technique with a focus on the NiO oxide. Sec.III is devoted to a +short overview of the cluster model approaches to a proper semiquantitative description of +the optical response in strongly correlated 3d oxides with a focus on the NiO oxide. A short +summary is made in Sec.IV. +SHORT OVERVIEW OF THE DFT BASED TECHNIQUE +Hohenberg-Kohn-Sham DFT +Density functional theory finds its roots in the approach which Thomas and Fermi +elaborated shortly after the creation of quantum mechanics [8, 9]. The Thomas-Fermi theory +of atoms may be interpreted as a semiclassical approximation, where the energy of a system +is written as a functional of the one-particle density. +Justifying earlier attempts directed at generalizing the Thomas-Fermi theory, Hohenberg +and Kohn [1] in 1964 advanced a theorem: "For any system of interacting particles in an +external potential v(r), the external potential is uniquely determined (except for a constant) +by the ground state density n0(r) which states that the exact ground-state energy is a +functional of the exact ground-state one-particle density. Unfortunately, it does not tell +how to construct this functional, i.e., it is an existence theorem for the energy-density +functional. This explains the fact of why so much effort has been dedicated to the task of +obtaining approximate functionals for the description of the ground-state properties of many- +particle systems. Contrary to wavefunction theory, where the objective is to approximate +3 + +the wavefunction, in DFT we choose to make approximations for the functional. +However, DFT still remains, in some sense, ill-defined: many of the DFT statements +were ill-posed or not rigorously proved. Indeed, the HK theorem is the constellation of two +statements: (i) the mathematically rigorous HK lemma, which demonstrates that the same +ground state density cannot correspond to two different potentials of an external field, and +(ii) the hypothesis of the existence of the universal density functional. However, the HK +lemma cannot provide justification of the universal density functional for fermions [10]. In +other words, each external field determines a unique density, and each density determines +a unique external field on the basis of the HK lemma. However, the rule for the last +correspondence can be nonuniversal, as the rule in general depends on the concrete form of +the density. The existence of this nonuniversality violates the HK theorem, although the HK +lemma is believed to be undoubtedly correct [10]. +Furthermore, there are more serious critics. Sarry and Sarry [11] claim that the proof of +the HK theorem is not correct. The authors do emphasize that for a strict many-particle +calculation only the direct mapping: external potential ⇒ ground state wave function ⇒ +electron density +v(r) ⇒ Ψ0(r) ⇒ ρ0(r) +is justified while the inverse mapping +ρ0(r) ⇒ Ψ0(r) ⇒ v(r) +claimed by the HK theorem can be validated only for single-particle self-consistent +calculations. +The DFT exploits the one-to-one correspondence between the single-particle electron +density and an external potential acting upon the system and relies on the existence of +a universal functional F[ρ(r)] which can be minimized in order to find the ground state +energy. However,the correspondence theorem establishes the existence of the functional only +in principle, and provides no unique practical recipe for its construction. The construction +of the functional F[ρ(r)] in the HK-DFT is equivalent to the problem of finding the N- +representability conditions of the reduced density matrix of order two [3, 12], the problem +whose solution has not been found until now. Generally speaking the functional F[ρ(r)] +must be N-dependent, namely, F[N, ρ(r)]. Another important aspect, closely related to N- +representability, is the variational character that either exact or approximate functionals +4 + +F[N, ρ(r)] must have in order to guarantee that the energy remains an upper bound to the +exact value. +The Kohn-Sham (KS) theory goes further in reducing the problem of calculating ground +state properties of a many-electron system in a local external single-particle potential to +solving Hartree-like one-electron KS equations. Within the framework of the HKS-DFT, +the many-body problem of interacting electrons in a static external potential is cast into +a tractable problem of non-interacting electrons moving in an effective potential. The +latter includes the external potential and the effects of the Coulomb interactions between +the electrons, i.e. the Hartree term, describing the electron-electron repulsion, and the +exchange and correlation (XC) interactions, which includes all the many-body interactions. +Modeling the XC interactions is the main difficulty of DFT. In practical calculations, the +XC contribution is approximated, and the results are only as good as the approximation +used. Actually, in HKS-DFT there exist hundreds of XC-approximations for vKS +xc (r) [3]. The +existence of so many approximations, with so little guidance, makes it ever more difficult +for non-specialists to separate the silver from the dross [13]. It is worth noting here that all +the approximate functionals do not comply with the variational principle. +The leading approximation for density functional construction is the so called local density +approximation (LDA), which is based upon exact exchange energy for a uniform electron gas +and only requires the density at each point in space. So the LDA taken from assuming that +the electron density for an atom, molecule, or solid is similarly homogeneous. But molecules +in LDA are typically overbound by about 1 eV/bond, and in the late 1980s the so-called +generalized gradient approximations (GGAs) using both the density and its gradient at +each point in space were elaborated whose accuracy seemed to be acceptable in chemical +calculations [13]. All the GGAs functionals, by definition, are corrections to the LDA, they +all revert to the uniform electron gas at zero density gradient. It should be noted that the +local nature of the standard approximations implies an exponential decay of the inter-site +interaction, in other words, the description of weak interactions such as long-range van der +Waals interactions is well beyond any conventional DFT method [13]. +The DFT calculations are quite different from the usual quantum mechanical methods +where better accuracy depends on computational resources and not on limitations stemming +from the method itself. The Hartree-Fock (HF) results cannot be reproduced within the +framework of Kohn-Sham (KS) theory because the single-particle densities of finite systems +5 + +obtained within the HF calculations are not v-representable, i.e., do not correspond to any +ground state of a N non-interacting electron systems in a local external potential [14]. For +this reason, the KS theory, which finds a minimum on a different subset of all densities, can +overestimate the ground state energy, as compared to the HF result. +In addition to the lack of compliance with N-representability conditions and difficulties +in extending the application of the first HK theorem to finite subspaces, there are still other +problems that beset DFT. They have to do with how to properly include symmetry (i.e., +properties of all operators commuting with the Hamiltonian of a given system). For instance, +translational symmetry in crystalline solids should be applied only to a full many-electron +function rather than to one-electron KS orbitals! +Currently, the KS-DFT is about occupied orbitals only and is far from giving a consistent +and quantitatively accurate description of open-shell spin systems, as the currently available +approximate functionals show unsystematic errors in the (inaccurate) prediction of energies, +geometries, and molecular properties. +Strictly speaking, the DFT is designed for description of ground rather than excited states +with no good scheme for excitations. Because an excited-state density does not uniquely +determine the potential, there is no general analog of HK for excited states. The standard +functionals are inaccurate both for on-site crystal field and for charge transfer excitations [13]. +The DFT based approaches cannot provide the correct atomic limit and the term and +multiplet structure, which is crucial for description of the optical response for 3dcompounds. +Although there are efforts to obtain correct results for spectroscopic properties depending +on spin and orbital density this problem remains as an open one in DFT research. Clearly, +all these difficulties stem from unsolved foundational problems in DFT and are related to +fractional charges and to fractional spins. Thus, these basic unsolved issues in the HKS-DFT +point toward the need for a basic understanding of foundational issues. +In other words, given these background problems, the DFT based models should be +addressed as semi-empirical approximate ones rather than ab initio theories. M. Levy +introduced in 2010 the term DFA to define density functional approximation instead of +DFT, which is believed to quite appropriately describe contemporary DFT [3]. In chemistry, +it is traditional to refer to standard approaches as ab initio, while DFT is regarded as +empirical. Because solid-state calculations are more demanding, for many decades DFT was +the only possible approach. Thus, DFT calculations are referred to as ab initio in solid-state +6 + +physics and materials science [13]. Proceeding with a fixed approximate functional, the DFT +is called "first principles in the sense that the user only chooses the atoms, and the computer +predicts (correctly or not) all properties of the molecule or solid. +LSDA +Basic drawback of the spin-polarized approaches is that these start with a local density +functional in the form (see, e.g. Ref.15) +v(r) = v0[n(r)] + ∆v[n(r), m(r)](ˆσ · m(r) +|m(r)|) , +where n(r), m(r) are the electron and spin magnetic density, respectively, ˆσ is the Pauli +matrix, that is these imply presence of a large fictious local one-electron spin-magnetic field ∝ +(v↑−v↓), where v↑,↓ are the on-site LSDA spin-up and spin-down potentials. Magnitude of the +field is considered to be governed by the intra-atomic Hund exchange, while its orientation +does by the effective molecular, or inter-atomic exchange fields. Despite the supposedly +spin nature of the field it produces an unphysically giant spin-dependent rearrangement of +the charge density that cannot be reproduced within any conventional technique operating +with spin Hamiltonians. Furthermore, a direct link with the orientation of the field makes +the effect of the spin configuration onto the charge distribution to be unphysically large. +However, magnetic long-range order has no significant influence on the redistribution of +the charge density. The DFT-LSDA community needed many years to understand such a +physically clear point. +In general, the LSDA method to handle a spin degree of freedom is absolutely +incompatible with a conventional approach based on the spin Hamiltonian concept. There +are some intractable problems with a match making between the conventional formalism +of a spin Hamiltonian and LSDA approach to the exchange and exchange-relativistic +effects. Visibly plausible numerical results for different exchange and exchange-relativistic +parameters reported in many LSDA investigations (see, e.g., Refs. [16]) evidence only a +potential capacity of the LSDA based models for semiquantitative estimations, rather than +for reliable quantitative data. It is worth noting that for all of these "advantageous"instances +the matter concerns the handling of certain classical N´eel-like spin configurations (ferro-, +antiferro-, spiral,...) and search for a compatibility with a mapping made with a conventional +7 + +quantum spin Hamiltonian. It’s quite another matter when one addresses the search of the +charge density redistribution induced by a spin configuration as, for instance, in multiferroics. +In such a case the straightforward application of the LSDA scheme can lead to an unphysical +overestimation of the effects or even to qualitatively incorrect results due to an unphysically +strong effect of a breaking of spatial symmetry induced by a spin configuration (see, e.g. +Refs. [17] and references therein). +Going beyond LSDA:LDA+U, LDA+DMFT, LDA+U+V +It is commonly accepted now that the standard DFT-LDA(GGA) approach is insufficient +to describe the electronic structure of the Mott insulators. +Apparent weaknesses of the DFT approach were exposed especially after the discovery +in 1986 of the copper-oxide superconductors, as it failed to yield the fact that the parent +compound La2CuO4 is an antiferromagnetic insulator. This difficult period for the DFT- +LDA method as many decided was partially ended in the early and mid 1990s especially +when an orbital dependent Hubbard-type U was incorporated in the exchange correlation +functional of the localized 3delectrons within the LDA+U method, while the other electrons +are still described at the LDA level [5]. +Attempts to go beyond LSDA are based on the self-interaction-corrected density +functional theory SIC-DFT, the LDA+U method, and the GW approximation [5]. These +methods represent corrections of the single-particle Kohn-Sham potential in one way or +another and lead to substantial improvements over the LSDA results for the values of the +energy gap and local moment. Within the SIC-DFT and LDA+U methods the occupied and +unoccupied states are split by the Coulomb interaction U, whereas within the LSDA this +splitting is caused by the Stoner parameter J, which is typically one order of magnitude +smaller than U. Therefore, compared with the LSDA, the novel methods capture more +correctly the physics of transition-metal oxides and improve the results for the energy gap +and local moment significantly. +An important drawback of the LDA+U method is that it requires U as a starting +parameter. Even though several schemes for the determination of U exist, it is almost always +chosen such that it reproduces the experimental value of a specific property of the electronic +structure, most often the band gap. Usually the LDA+U calculations imply account of +8 + +the on-site d-d correlations with Udd parameter and do neglect the ligand p-p correlations +though Udd parameter is only twice as large as Upp in oxides [6, 7]. The predictive power +of the novel methods crucially relies on a reliable assessment of the interactions, however, +the value of the interaction parameters, such as Udd, Upp, depends on the choice of the +downfolded model, namely, the orbitals treated in the model as well as the basis functions +employed, as the screened interaction is determined by the various screening processes that +are not considered in the model. Therefore a careful analysis is needed to make a proper +model and choose appropriate parameters. By fitting, one usually finds higher accuracy for +systems similar to those fitted, but usually greater inaccuracies far away. +All efforts to account for the correlations beyond LDA encounter an insoluble problem +of double counting (DC) of interaction terms which had just included into Kohn-Sham +single-particle potential. A well defined analytical expression for the DC potential cannot +be formulated in the context of LDA+U or other technique going beyond LDA [18]. How to +choose the DC correction potential in a manner that is both physically sound and consistent +is unknown. Thus, one has to resort to numerical criteria to fix the value of the DC correction. +However, there is currently no universal and unambiguous expression for DC correction, +and different formulations are used for different classes of materials. Moreover, different +methods for fixing the double counting can drive the result from Mott insulating to almost +metallic [18, 19]. +The LDA+DMFT approach combines band structure theory within the DFT-LDA with +many-body theory as provided by dynamical mean-field theory (DMFT) [5]. Within DMFT, +a lattice model is mapped onto an effective impurity problem embedded in a medium which +has to be determined self-consistently, e.g., by quantum Monte-Carlo (QMC) simulations. +This mapping becomes exact in the limit of infinite dimensions. +The LDA+U and LDA+DMFT methods are believed to correct the inaccuracies of +approximate DFT exchange correlation functionals. The main idea of the both computational +approaches consists in a selective description of the strongly correlated electronic states, +typically, localized d or f orbitals, using the Hubbard model, while all the other states +continue to be treated at the level of standard approximate DFT functionals. At present +the LDA+U and LDA+DMFT methods are addressed to be most powerful tools for +the investigation of strongly correlated electronic systems, however, these preserve many +shortcomings of the basic DFT-LDA approach. +9 + +Current theoretical studies of electronic correlations in transition metal oxides typically +only account for the local repulsion between d-electrons even if oxygen ligand p-states are +an explicit part of the effective Hamiltonian. Interatomic correlations such as Vpd between +d- and (ligand) p-electrons, as well as the on-site and inter-site interaction between p- +electrons (Upp and Vpp), are usually neglected. Strictly speaking, LDA+DMFT scheme +should incorporate both Upp, Vpp, Vpd and Vdd interactions [20]. To this end we need a +proper procedure for their calculation, however, this makes the double counting problem +significantly more sophisticated. +NiO as a main TMO system for so-called ab initio studies +An ongoing challenge during the last 60 years has been the development of a theoretical +model that could offer an accurate description of both the electric and magnetic phenomena +observed in NiO. Nickel oxide is one of the prototypical compounds that has highlighted the +importance of correlation effects in transition metal oxides (TMO). However, despite several +decades of studies there is still no literature consensus on the detailed electronic structure +of NiO. Although exhibiting a partially filled 3dband and predicted by simple band theory +to be a good conductor, NiO has a relatively large band gap (about 4 eV) that cannot be +accounted for in the LDA calculations. +NiO has long been viewed as a prototype "Mott insulator" [21] with the gap formed +by intersite cation-cation d-d charge transfer (CT) transitions, however, this view was +later replaced by that of a "CT insulator"with the gap formed by anion-cation p-d CT +transitions [22]. +Strictly speaking, the DFT is designed for description of ground rather than excited +states. Nevertheless research activity in the condensed matter DFT community is focused +on the single-particle excitation properties of the TMOs, in particular, the photoemission +spectra and energy gap. +The XPS combined with bremsstrahlung-isochromat spectroscopy (BIS) shows a gap +between the top of the valence band and the bottom of the conducting band of 4.3 eV for +NiO [23]. Namely this value appears to be in the focus of the so-called ab initio DFT-LDA +based calculations for NiO. However, the later studies [24] have shown that the exact value of +this conductivity gap is subject to the band position chosen to define the highest valence and +10 + +lowest conducting levels, obtaining values that range from 3.20 to 5.67 eV (!). Experimental +data, in particular, oxygen x-ray emission (XES) and absorption (XAS) spectra [25] point +to strong matrix element effects, that makes reliable estimates of the energy gap to be very +ambiguous adventure. +The standard DFT-LDA band theory predicts NiO to be a metal. LSDA [26] predicts +NiO to be an insulator (with severe underestimated gap of 0.3 eV) only in antiferromagnetic +state (!?). The later GW [27] and LDA+U [28] calculations yielded the larger gap of 3.7 eV. +First LDA+DMFT calculation performed by Ren et al. [29] yielded the value of 4.3 eV. The +authors claimed: "The overall agreement between the calculated single-particle spectrum and +the experimental data is surprisingly good". However, they do neglect the matrix element +effect, p-d covalency, Upp, Vpd, and Vdd, that de facto does invalidate their conclusion. Part +of these effects, in particular, p − d covalency was taken into account later [30], but with a +severe reinterpretation of the DOS. Again, the authors claim: "...we were able to provide +a full description of the valence-band spectrum and, in particular, of the distribution of +spectral weight between the lower Hubbard band and the resonant peak at the top of the +valence band. However, to this day the LDA+DMFT results for NiO strongly depend on +the choice of the DC correction potential driving the result from Mott insulating to metallic +state [18, 19]. +It is rather surprising how little attention has been paid to the DFT based calculations +of the TMO optical properties. Lets turn to a very recent paper by Roedl and Bechstedt [31] +on NiO and other TMOs, whose approach is typical for DFT community. The authors +calculated the dielectric function ǫ(ω) for NiO within the DFT-GGA+U+∆ technique and +claim:"The experimental data agree very well with the calculated curves" (!?). However, +this seeming agreement is a result of a simple fitting when the two model parameters U and +∆ are determined such (U = 3.0, ∆ = 2.0 eV) that the best possible agreement concerning the +positions and intensities of the characteristic peaks in the experimental spectra is obtained. +In addition, the authors arrive at absolutely unphysical conclusion: "The optical absorption +of NiO is dominated by intra-atomic t2g → eg transitions" (!?). +Nekrasov et al. [19] realized the DMFT calculation of the optical conductivity for NiO. +Just another correlation parameter was chosen: U = 8 eV. The authors claim a general +agreement both with optical and the X-ray experiments. In the calculations, they found +that the main contribution to optical conductivity is due to intra-orbital optical transitions. +11 + +Inter-orbital optical transitions give less than 5% of the optical conductivity intensity in +the frequency range used in the calculations. However, as usual they did neglect a number +of important on-site and inter-site correlation parameters and all the effects due to optical +matrix elements that does invalidate their conclusion. Furthermore, the DFT-LDA based +schemes do not provide the correct atomic limit and the term and multiplet structure. Hence +these cannot correctly describe both the d-d crystal field and p-d and d-d charge transfer +transitions. However, some authors [32] suppose that in future this problem probably can be +solved within the LDA+DMFT. +Surveying these and other literature data we can argue that the conventional DFT based +technique cannot provide a proper description of the optical response for strongly correlated +3dcompounds. As up till now, in future the optical properties of the Mott or charge transfer +insulators will be considered within the framework of cluster approaches initially based on +quantum-chemical calculations. +CLUSTER MODEL IN NIO +Cluster model approach does generalize and advance crystal-field and ligand-field theory. +The method provides a clear physical picture of the complex electronic structure and +the energy spectrum, as well as the possibility of a quantitative modelling. In a certain +sense the cluster calculations might provide a better description of the overall electronic +structure of insulating 3doxides than the band structure calculations, mainly due to a better +account for correlation effects, electron-lattice coupling, and relatively weak interactions +such as spin-orbital and exchange coupling. Cluster models have proven themselves to be +reliable working models for strongly correlated systems such as transition-metal and rare- +earth compounds. These have a long and distinguished history of application in optical and +electron spectroscopy, magnetism, and magnetic resonance. The author with colleagues has +successfully demonstrated great potential of the cluster model for description of the p-d +and d-d charge transfer transitions and their contribution to optical and magneto-optical +response in 3doxides such as ferrites, manganites, cuprates, and nickelates [33]. +Cluster models do widely use the symmetry for atomic orbitals, point group symmetry, +and advanced technique such as Racah algebra and its modifications for point group +symmetry [34]. From the other hand the cluster model is an actual proving-ground for various +12 + +calculation technique from simple quantum chemical MO-LCAO (molecular orbital-linear- +combination-of-atomic-orbitals) method to a more elaborate LDA+MLFT (MLFT, multiplet +ligand-field theory) [35] approach. +Cluster models traditionally combined quantum chemical MO-LCAO calculations [34] +based on atomic Hartree-Fock orbitals with making use parameters fitted to experiments. +Several authors obtained model parameters by performing an LDA calculation for the cluster +and using its Kohn-Sham MOs. First comprehensive description of the electronic structure +of the NiO6 cluster was performed by Fujimori and Minami [36]. Effective transfer and +overlap integrals were evaluated from LCAO parameters of NiO found by Mattheiss [37] +by fitting their APW energy-band results. The localized approach has been shown to +successfully explain the photoemission, optical-absorption, and isochromat spectra of NiO. +Recently, Haverkort et al. [35] suggested a sort of generalization of conventional ligand- +field model with the DFT-based calculations within a so-called "ab initio"LDA+MLFT +technique. They start by performing a DFT calculation for the proper, infinite crystal +using a modern DFT code which employs an accurate density functional and basis set +[e.g., linear augmented plane waves (LAPWs)]. From the (self-consistent) DFT crystal +potential they then calculate a set of Wannier functions suitable as the single-particle basis +for the cluster calculation. The authors compared the theory with experimental spectra +(XAS, nonresonant IXS, photoemission spectroscopy) for SrTiO3, MnO, and NiO and found +overall satisfactory agreement, indicating that their ligand-field parameters are correct to +better than 10%. However, as in Ref. [36] the authors have been forced to treat on-site +correlation parameter Udd and orbitally averaged (spherical) ∆pd parameter as adjustable +ones. Comparing the novel LDA+MFLT technique with that of Fujimori and Minami [36] +one should note very similar level of their quantitative conclusions. Despite the involvement +of powerful calculation techniques the numerical results of the both approaches seem to +be more like semiquantitative ones. In such a situation we should transfer the center of +gravity of the cluster approaches more and more to elaboration of physically sound and clear +semiquantitative models that are maximally take into account all the symmetry requirements +on one hand and refer to experiment on the other. +Hereafter, we do present a most recent and most comprehensive such a cluster model +approach to the description of the p-d and d-d CT transitions in NiO [38] that nicely +illustrates great potential of the model that does combine simple physically clear ligand- +13 + +field analysis, its semiquantitative predictions with a regular appeal to experimental data. +We believe that such an approach should precede and accompany any detailed numerical +calculation providing its physical validation. +Starting with an octahedral NiO6 complex with the point symmetry group Oh we deal +with five Ni 3dand eighteen oxygen O 2p atomic orbitals forming both the hybrid Ni 3d-O +2p bonding and antibonding eg and t2g molecular orbitals (MO), and the purely oxygen +nonbonding a1g(σ), t1g(π), t1u(σ), t1u(π), t2u(π) orbitals. The nonbonding t1u(σ) and t1u(π) +orbitals with the same symmetry are hybridized due to the oxygen-oxygen O 2pπ - O +2pπ transfer. The relative energy position of different nonbonding oxygen orbitals is of +primary importance for the spectroscopy of the oxygen-3d-metal charge transfer. This is +firstly determined by the bare energy separation ∆ǫ2pπσ = ǫ2pπ − ǫ2pσ between O 2pπ and O +2pσ electrons. Since the O 2pσ orbital points towards the two neighboring positive 3d ions, +an electron in this orbital has its energy lowered by the Madelung potential as compared +with the O 2pπ orbitals, which are oriented perpendicular to the respective 3d-O-3d axes. +Thus, the Coulomb arguments favor the positive sign of the π − σ separation ǫpπ − ǫpσ +whose numerical value can be easily estimated in the frames of the well-known point charge +model, and appears to be of the order of 1.0 eV. In a first approximation, all the γ(π) states +t1g(π), t1u(π), t2u(π) have the same energy. However, the O 2pπ-O 2pπ transfer and overlap +yield the energy correction to the bare energies with the largest value and a positive sign for +the t1g(π) state. The energy of the t1u(π) state drops due to a hybridization with the cation +4pt1u(π) state. +The ground state of NiO610− cluster, or nominally Ni2+ ion corresponds to t6 +2ge2 +g +configuration with the Hund 3A2g(F) ground term. Typically for the octahedral MeO6 +clusters [33] the nonbonding t1g(π) oxygen orbital has the highest energy and forms the first +electron removal oxygen state while the other nonbonding oxygen π-orbitals, t2u(π), t1u(π), +and the σ-orbital t1u(σ) have a lower energy with the energy separation ∼ 1 eV inbetween +(see Fig. 1). +The p-d CT transition in NiO10− +6 +center is related to the transfer of O 2p electron to the +partially filled 3deg-subshell with the formation on the Ni-site of the (t6 +2ge3 +g) configuration of +nominal Ni+ ion isoelectronic to the well-known Jahn-Teller Cu2+ ion. Yet actually instead +of a single p-d CT transition we arrive at a series of O 2pγ→ Ni 3deg CT transitions +forming a complex p-d CT band. It should be noted that each single electron γ→eg p-d +14 + +Рис. 1: (Color online) Spectra of the intersite d-d, p-d CT transitions and on-site crystal field d-d +transitions in NiO. Strong dipole-allowed σ−σ d-d and p-d CT transitions are shown by thick solid +uparrows; weak dipole-allowed π − σ p-d transitions by thin solid uparrows; weak dipole-forbidden +low-energy transitions by thin dashed uparrows, respectively. Dashed downarrows point to different +electron-hole relaxation channels, dotted downarrows point to photoluminescence (PL) transitions, +I1,2 are doublet of very narrow lines associated with the recombination of the d-d CT exciton. +The spectrum of the crystal field d-d transitions is reproduced from Ref. [45]. The right hand side +reproduces a fragment of the RIXS spectra for NiO [41]. +CT transition starting with the oxygen γ-orbital gives rise to several many-electron CT +states. For γ=t1,2 these are the singlet and triplet terms 1,3T1, 1,3T2 for the configurations +t6 +2ge3 +gt1,2, where t1,2 denotes the oxygen hole. The complex p-d CT band starts with the +dipole-forbidden t1g(π)→eg, or 3A2g→1,3T1g, 1,3T2g transitions, then includes two formally +dipole-allowed the so-called π→σ p-d CT transitions, the weak t2u(π)→eg, and relatively +strong t1u(π)→eg CT transitions, respectively, each giving rise to 3A2g→3T2u transitions. +15 + +Finally the main p-d CT band is ended by the strongest dipole-allowed σ→σ t1u(σ)→ +eg (3A2g→3T2u) CT transition. The above estimates predict the separation between the +partial p-d bands to be ∼ 1 eV. Thus, if the most intensive CT band with a maximum +around 7 eV observed in the RIXS spectra [39–41] to attribute to the strongest dipole- +allowed O 2pt1u(σ)→Ni 3deg CT transition then one should expect the low-energy p-d +CT counterparts with the maxima around 4, 5, and 6 eV respectively, which are related +to the dipole-forbidden t1g(π)→eg, the weak dipole-allowed t2u(π)→eg, and relatively strong +dipole-allowed t1u(π)→eg CT transitions, respectively (see Fig. 1). It is worth noting that +the π→σ p-d CT t1u(π)−eg transition borrows a portion of the intensity from the strongest +dipole-allowed σ→σ t1u(σ)→eg CT transition because the t1u(π) and t1u(σ) states of the +same symmetry are partly hybridized due to the p-p covalency and overlap. +Thus, the overall width of the p-d CT bands with the final t6 +2ge3 +g configuration occupies +a spectral range from ∼ 4 up to ∼ 7 eV. The left hand side of Fig. 1 summarizes the +main semiquantitative results of the cluster model predictions for the energy and relative +intensities of the p-d CT transitions. Interestingly this assignment finds a strong support in +the reflectance (4.9, 6.1, and 7.2 eV for the allowed p-d CT transitions) spectra of NiO [42]. A +rather strong p(π)-d CT band peaked at 6.3 eV is clearly visible in the absorption spectra of +MgO:Ni [43]. The electroreflectance spectra [44] which detect the dipole-forbidden transitions +clearly point to a low-energy forbidden transition peaked near 3.7 eV missed in the reflectance +and absorption spectra [42, 43, 45], which thus defines a p-d character of the optical CT gap +and can be related to the onset transition for the whole complex p-d CT band. It should +be noted that a peak near 3.8 eV has been also observed in the nonlinear absorption spectra +of NiO [46]. At variance with the bulk NiO a clearly visible intensive CT peak near 3.6- +3.7 eV has been observed in the absorption spectra of NiO nanoparticles [47]. This strongly +supports the conclusion that the 3.7 eV band is related to the bulk-forbidden CT transition +which becomes the partially allowed one in the nanocrystalline state [38]. It is worth noting +that the hole-type photoconductivity threshold in bulk NiO has been observed also at this +"magic" energy 3.7 eV [48], that is the t1g(π)→eg p-d CT transition is believed to produce +itinerant holes. Indeed, the p-d CT transitions in NiO6 cluster are of so-called "anti-Jahn- +Teller" type, that is these are transitions from orbitally nondegenerate state to the final p-d +CT state state formed by two orbitally degenerate states that points to strong electron-lattice +effects in excited state. The final Ni1+ 3d9(t6 +2ge3 +g) configuration is isoelectronic to Cu2+ ion in +16 + +cubic crystal field and presents a well-known textbook example of a Jahn-Teller center that +implies a strong trend to the localization, while a photo-generated hole can move more or +less itinerantly in the O 2p valence band determining the hole-like photoconductivity [48]. It +should be noted that any oxygen π-holes have a larger effective mass than the σ-holes, that +results in a different role of the p(π)-d and p(σ)-d CT transitions both in photoconductivity +and, probably, the luminescence stimulation. +A spectral feature near 6 eV, clearly visible in the NiO photoluminescence excitation +(PLE) spectra [38] can be certainly attributed to a rather strong p(π)-d (t1u(π) → eg) CT +transition while the spectral feature near 5 eV to a weaker p(π)-d (t2u(π) → eg) CT transition. +Interestingly the strongest p(σ)-d (t1u(σ) → eg) CT transition at ∼ 7 eV is actually inactive +in the PLE spectra, most likely, due to a dominating nonradiative relaxation channel for the +oxygen t1u(σ) holes. +However, the p-d CT model cannot explain the main low-energy spectral feature, clearly +visible in the PLE spectra near 4 eV [38], thus pointing to manifestation of another CT-type +mechanism. Indeed, along with the p-d CT transitions an important contribution to the +optical response of the strongly correlated 3doxides can be related to the strong dipole- +allowed d-d CT, or Mott transitions [33]. In NiO one expects a strong d-d CT transition +related to the σ − σ-type eg − eg charge transfer t6 +2ge2 +g + t6 +2ge2 +g→ t6 +2ge3 +g + t6 +2ge1 +g between nnn +Ni sites with the creation of electron NiO611− and hole NiO69− centers (nominally Ni+ and +Ni3+ ions, respectively) thus forming a bound electron-hole dimer, or d-d CT exciton. +The strong dipole-allowed Franck-Condon d(eg)-d(eg) CT transition in NiO manifests +itself as a strong spectral feature near 4.5 eV clearly visible in the absorption of thin +NiO films [49], RIXS spectra [39, 41], the reflectance spectra (4.3 eV) [42]. Such a strong +absorption near 4.5 eV is beyond the predictions of the p-d CT model and indeed is lacking +in the absorption spectra of MgO:Ni [43]. It should be noticed that, unlike all the above +mentioned structureless spectra, the nonlinear absorption spectra [46] of NiO films do reveal +an anticipated "fine" structure of the d-d CT exciton with the two narrow peaks at 4.075 +and 4.33 eV preceding a strong absorption above 4.575 eV. Interestingly the separation 0.2- +0.3 eV between the peaks is typical for the exchange induced splittings in NiO (see, e.g., +the "0.24 eV" optical feature [45]). Accordingly, the 4.1 eV peak in the PLE spectra can be +unambiguously assigned to the d-d CT transition [38]. +The charge, spin, and orbital degeneracy of the final state of this unique double anti- +17 + +Jahn-Teller transition 3A2g + 3A2g→2Eg + 2Eg results in a complex band observed at 4.2-4.5 +eV [38]. The exchange tunnel reaction Ni++Ni3+↔Ni3++Ni+ due to a two-electron transfer +gives rise to the two symmetric (S- and P-) excitons having s- and p-symmetry, respectively, +with the energy separation δ0 = 2|t| and δ1 = +2 +3|t| for the spin singlet and spin triplet +excitons, where t is the two-electron transfer integral whose magnitude is of the order of the +Ni2+-Ni2+ exchange integral: t ≈ Innn. Interestingly the P-exciton is dipole-allowed while +the S-exciton is dipole-forbidden. The anti-Jahn-Teller d-d CT exciton is prone to be self- +trapped in the lattice due to the electron-hole attraction and a particularly strong double +Jahn-Teller effect for both the electron and hole centers. Recombination transitions for such +excitons produce a bulk luminescence with puzzling well isolated doublet of very narrow +lines with close energies near 3.3 eV [38] that corresponds to a reasonable Stokes shift of 1 +eV. To the best of our knowledge it is the first observation of the self-trapping for the d-d +CT excitons. +Thus, we see that a simple cluster model is able to provide a semiquantitative description +of a large body of experimental spectroscopic data, including subtle effects beyond the reach +of any "ab initio"DFT technique. We have shown that the prototype 3doxide NiO, similar +to perovskite manganites RMnO3 or parent cuprates such as La2CuO4 [33], should rather +be sorted neither into the CT insulator nor the Mott-Hubbard insulator in the Zaanen- +Sawatzky-Allen scheme [22]. +SUMMARY +There are still a lot of people who think the Hohenberg-Kohn-Sham DFT within the +LDA has provided a very successful ab initio framework to successfully tackle the problem +of the electronic structure of materials. However, both the starting point and realizations +of the DFT approach have raised serious questions. The HK "theorem"of the existence of +a mythical universal density functional that can resolve everything looks like a way into +Neverland, the DFT heaven is probably unattainable. Various DFAs, density functional +approximations, local or nonlocal, will never be exact. Users are willing to pay this price +for simplicity, efficacy, and speed, combined with useful (but not yet chemical or physical) +accuracy [4, 13]. +The most popular DFA fail for the most interesting systems, such as strongly correlated +18 + +oxides. The standard approximations over-delocalize the d-electrons, leading to highly +incorrect descriptions. Some practical schemes, in particular, DMFT can correct some of +these difficulties, but none has yet become a universal tool of known performance for such +systems [13]. +Any comprehensive physically valid description of the electron and optical spectra for +strongly correlated systems, as we suggest, should combine simple physically clear cluster +ligand-field analysis with a numerical calculation technique such as LDA+MLFT [35], and +a regular appeal to experimental data. +The research was supported by the Ministry of Education and Science of the Russian +Federation, project FEUZ-2020-0054. +[1] H. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). +[2] W. Kohn and J.L. Sham, Phys. Rev. 140, A1133 (1965). +[3] E.S. Kryachko, E.V. Ludena, Physics Reports 544 (2014) 123239. +[4] A.D. Becke, J. Chem. Phys. 140, 18A301 (2014). +[5] G. Kotliar, S.Y. Savrasov, K. Haule, V.S. Oudovenko, O. Parcollet, and C.A. Marianetti, Rev. +Mod. Phys. 78, 865 (2006); V.I. Anisimov and Yu.A. Izyumov, Electronic Structure of Strongly +Correlated Materials (Springer Verlag, Berlin, 2010). +[6] H. Eskes, L.H. Tjeng, and G.A. Sawatzky, Phys. Rev. B 41, 288 (1990). +[7] J. Ghijsen, L.H. Tjeng, J. van Elp, H. Eskes, J. Westerink, G.A. Sawatzky, and M.T. Czyzyk, +Phys. Rev. B 38, 11322 (1988). +[8] L.H. Thomas, Proc. Cambr. Phil. Soc. 23, 542 (1926). +[9] E. Fermi, Rend. Ac. Lincei 6, 602 (1927). +[10] V.B. Bobrov, S.A. Trigger, JETP 143 , 729 (2013). +[11] A.M. Sarry, M.F. Sarry, Phys. Solid State, 54(6), 1315 (2012). +[12] Eduardo V. Ludena, J. Mol. Structure (Theochem) 709 25 (2004). +[13] K. Burke, J. Chem. Phys. 136, 150901 (2012). +[14] M.Ya. Amusia, A.Z. Msezane, V.R. Shaginyan, D. Sokolovski, Phys. Lett. A330:1-2 (2004), +10-15. +[15] L.M. Sandratskii, Phys. Rev. B 64, 134402 (2001). +19 + +[16] V.V. Mazurenko and V.I. Anisimov, Phys. Rev. B 71, 184434 (2005); M.I. Katsnelson, Y.O. +Kvashnin, V.V. Mazurenko, and A.I. Lichtenstein, Phys. Rev. B 82, 100403(R) (2010). +[17] A.S. Moskvin and S.-L. Drechsler, Phys. Rev. B 78, 024102 (2008); Eur. Phys. J. B 71, 331 +(2009); A.S. Moskvin, Yu.D. Panov, S.-L. Drechsler, Phys. Rev. B 79, 104112 (2009). +[18] M. Karolak, G. Ulm, T. O. Wehling, V. Mazurenko, A. Poteryaev, and A. I. Lichtenstein, J. +Electron Spectrosc. Relat. Phenom. 181, 11 (2010). +[19] I.A. Nekrasov, N.S. Pavlov, and M.V. Sadovskii, JETP Lett. 95(11), 581 (2012); JETP 116(4), +620 (2013). +[20] H.J. Kulik and N. Marzari, J. Chem. Phys. 134, 094103 (2011); P. Hansmann, N. Parragh, A. +Toschi, G. Sangiovanni, and K. Held, arXiv:1312.2757v1, 2013. +[21] B. Brandow, Adv. Phys. 26, 651 (1977); S. H¨ufner, Adv. Phys. 43, 183 (1994). +[22] J. Zaanen, G. A. Sawatzky, and J W. Allen, Phys. Rev. Lett. 55, 418 (1985). +[23] G.A. Sawatzky, J.W. Allen, Phys. Rev. Lett. 53, 2239 (1984). +[24] S. H¨ufner, P. Steiner, I. Sander, F. Reinert, and H. Schmitt, Z. Phys. B - Condensed Matter +86, 207-215 (1992). +[25] E. Z. Kurmaev, R. G. Wilks, A. Moewes, L. D. Finkelstein, S. N. Shamin, and Kuneˇs, PRB +77, 165127 (2008). +[26] K. Terakura, T. Oguchi, A.R. Williams, and J. Kubler, Phys. Rev. B 30, 4734 (1984). +[27] S. Massidda, A. Continenza, M. Posternak, and A. Baldereschi, Phys. Rev. B 55, 13494 (1997). +[28] V.I. Anisimov, I.V. Solovyev, M.A. Korotin, M.T. Czyzyk, and G.A. Sawatzky, Phys. Rev. B +48, 16929 (1993). +[29] X. Ren, I. Leonov, G. Keller, M. Kollar, I. Nekrasov, and D. Vollhardt, Phys. Rev. B 74, +195114 (2006). +[30] J. Kunes, V.I. Anisimov, A.V. Lukoyanov, and D. Vollhardt, Phys.Rev. B 75, 165115 (2007). +[31] C. Roedl and F. Bechstedt, Phys. Rev. B 86, 235122 (2012). +[32] M.I. Katsnelson, A.I. Lichtenstein, J. Phys. Cond. Matter. 22, 382201 (2010). +[33] A. S. Moskvin, A. V. Zenkov, E. I. Yuryeva, V. A. Gubanov, Physica B, 168,186 (1991); A. S. +Moskvin, A. V. Zenkov, E. A. Ganshina, G. S. Krinchik, and M. M. Nishanova, J. Phys. Chem. +Solids, 54, 101 (1993); A.V. Zenkov, A.S. Moskvin, J.Phys.Cond. Matter 14 6957 (2002); A. S. +Moskvin, Phys. Rev. B 65, 205113 (2002); A.S. Moskvin, R. Neudert, M. Knupfer, J. Fink, +and R. Hayn, Phys. Rev. B 65, 180512(R) (2002); A.S. Moskvin, J. M´alek, M. Knupfer, R. +20 + +Neudert, J. Fink, R. Hayn, S.-L. Drechsler, N. Motoyama, H. Eisaki, and S. Uchida, Phys. +Rev. Lett. 91, 037001 (2003); R. V. Pisarev, A. S. Moskvin, A. M. Kalashnikova, A. A. Bush, +and Th. Rasing, Phys. Rev. B 74, 132509 (2006); A. S. Moskvin, R. V. Pisarev, Phys. Rev. +B 77, 060102(R) (2008); A.S. Moskvin, R.V. Pisarev, Low Temp. Phys. 36 613 (2010); A. S. +Moskvin, A. A. Makhnev, L. V. Nomerovannaya, N. N. Loshkareva, and A. M. Balbashov, Phys. +Rev. B 82, 035106 (2010); A. S. Moskvin, Optics and Spectroscopy, 111, 403 (2011). +[34] S. Sugano, Y. Tanabe and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals +(Academic, New York, 1970). +[35] M.W. Haverkort, M. Zwierzycki, and O.K. Andersen, Phys. Rev. B 85, 165113 (2012). +[36] A. Fujimori and F. Minami, Phys. Rev. B 30, 957 (1984). +[37] L.F. Mattheiss, Phys. Rev. B 5, 290 (1972). +[38] V.I. Sokolov, V.A. Pustovarov, V.N. Churmanov et al., JETP Letters, 95, 528 (2012); Phys. +Rev. B 86, 115128 (2012); V.I. Sokolov, V.A. Pustovarov, V.N. Churmanov et al., IOP Conf. +Series: Materials Science and Engineering 38, 012007 (2012). +[39] N. Hiraoka, H. Okamura, H. Ishii, I. Jarrige, K.D. Tsuei, and Y.Q. Cai, Eur. Phys. J. B 70, +157 (2009). +[40] M. Magnuson, S.M. Butorin, A. Agui, and J. Nordgren, J. Phys.: Condens. Matter 14, 3669 +(2002). +[41] L.-C. Duda, T. Schmitt, M. Magnuson, J. Forsberg, A. Olsson, J. Nordgren, K. Okada and A. +Kotani, Phys. Rev. Lett. 96, 067402 (2006); B.C. Larson, Wei Ku, J.Z. Tischler, Chi-Cheng +Lee, O.D. Restrepo, A.G. Eguiluz, P. Zschack, and K.D. Finkelstein, Phys. Rev. Lett. 99, +026401 (2007). +[42] R.J. Powell and W.E. Spicer, Phys. Rev. B 2, 2182 (1970). +[43] K.W. Blazey, Physica 89B, 47 (1977). +[44] J.L. McNatt, Phys. Rev. Lett. 23, 915 (1969); R. Glosser, W.C. Walker, Solid State Commun, +9, 1599 (1971). +[45] R. Newman and R.M. Chrenko, Phys. Rev. 114, 1507 (1959). +[46] S.I. Shablaev, and R.V. Pisarev, Physics of the Solid State 45, 1742 (2003). +[47] V.V. Volkov, Z.L. Wang, and B.S. Zou, Chemical Phys. Lett. 337, 117 (2001). +[48] Ja.M. Ksendzov and I.A. Drabkin, Sov. Phys. Solid State, 7, 1519 (1965). +[49] C.E. Rossi and W. Paul, J. Phys. Chem. Solids 30, 2295 (1969). +21 + diff --git a/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/load_file.txt b/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e18e826e936f167b3e4ff46706444f716a4c0fe7 --- /dev/null +++ b/-9FLT4oBgHgl3EQfDS7k/content/tmp_files/load_file.txt @@ -0,0 +1,864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf,len=863 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='11979v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='str-el] 27 Jan 2023 DFT, L(S)DA, LDA+U, LDA+DMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', whether we do approach to a proper description of optical response for strongly correlated systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin1 1Ural Federal University, Ekaterinburg, 620083 Russia Аннотация I present a critical overview of so-called "ab initio"DFT (density fuctional theory) based calculation schemes for the description of the electronic structure, energy spectrum, and optical response for strongly correlated 3d oxides, in particular, crystal-field and charge transfer transitions as compared with an "old" cluster model that does generalize crystal-field and ligand-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' As a most instructive illustration of validity of numerous calculation techniques I address the prototypical 3d insulator NiO predicted to be a metal in frames of a standard LDA (local density approximation) band theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 1 INTRODUCTION The electronic states in strongly correlated 3d oxides manifest both significant localization and dispersional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' One strategy to deal with this dilemma is to restrict oneself to small many-electron clusters embedded to a whole crystal, then creating model effective lattice Hamiltonians whose spectra may reasonably well represent the energy and dispersion of the important excitations of the full problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Despite some shortcomings the method did provide a clear physical picture of the complex electronic structure and the energy spectrum, as well as the possibility of a quantitative modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, last decades the condensed matter community faced an expanding flurry of papers with the so called ab initio calculations of electronic structure and physical properties for strongly correlated systems such as 3d compounds based on density functional theory (DFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The modern formulation of the DFT originated in the work of Hohenberg and Kohn [1], on which based the other classic work in this field by Kohn and Sham [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The Kohn- Sham equation, has become a basic mathematical model of much of present-day methods for treating electrons in atoms, molecules, condensed matter, solid surfaces, nanomaterials, and man-made structures [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Of the top three most cited physicists in the period 1980-2010, the first (Perdew: 65 757 citations) and third (Becke: 62 581 citations) were density-functional theorists [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, DFT still remains, in some sense, ill-defined: many of DFT statements were ill-posed or not rigorously proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Most widely used DFT computational schemes start with a "metallic-like"approaches making use of approximate energy functionals, firstly LDA (local density approximation) scheme, which are constructed as expansions around the homogeneous electron gas limit and fail quite dramatically in capturing the properties of strongly correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The LDA+U and LDA+DMFT (DMFT, dynamical mean- field theory) [5] methods are believed to correct the inaccuracies of approximate DFT exchange correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The main idea of these computational approaches consists in a selective description of the strongly correlated electronic states, typically, localized d or f orbitals, using the Hubbard model, while all the other states continue to be treated at the level of standard approximate DFT functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' At present the LDA+U and LDA+DMFT methods are addressed to be most powerful tools for the investigation of strongly correlated electronic systems, however, these preserve many shortcomings of 2 the DFT-LDA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Despite many examples of a seemingly good agreement with experimental data (photoemission and inverse-photoemission spectra, magnetic moments,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=') claimed by the DFT community, both the questionable starting point and many unsolved and unsoluble problems give rise to serious doubts in quantitative and even qualitative predictions made within the DFT based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In a certain sense the cluster based calculations seem to provide a better description of the overall electronic structure of insulating 3doxides and its optical response than the DFT based band structure calculations, mainly due to a clear physics and a better account for correlation effects (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [6, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='II we do present a short critical overview of the DFT and the DFT based technique with a focus on the NiO oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='III is devoted to a short overview of the cluster model approaches to a proper semiquantitative description of the optical response in strongly correlated 3d oxides with a focus on the NiO oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A short summary is made in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' SHORT OVERVIEW OF THE DFT BASED TECHNIQUE Hohenberg-Kohn-Sham DFT Density functional theory finds its roots in the approach which Thomas and Fermi elaborated shortly after the creation of quantum mechanics [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The Thomas-Fermi theory of atoms may be interpreted as a semiclassical approximation, where the energy of a system is written as a functional of the one-particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Justifying earlier attempts directed at generalizing the Thomas-Fermi theory, Hohenberg and Kohn [1] in 1964 advanced a theorem: "For any system of interacting particles in an external potential v(r), the external potential is uniquely determined (except for a constant) by the ground state density n0(r) which states that the exact ground-state energy is a functional of the exact ground-state one-particle density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Unfortunately, it does not tell how to construct this functional, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', it is an existence theorem for the energy-density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' This explains the fact of why so much effort has been dedicated to the task of obtaining approximate functionals for the description of the ground-state properties of many- particle systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Contrary to wavefunction theory, where the objective is to approximate 3 the wavefunction, in DFT we choose to make approximations for the functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, DFT still remains, in some sense, ill-defined: many of the DFT statements were ill-posed or not rigorously proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Indeed, the HK theorem is the constellation of two statements: (i) the mathematically rigorous HK lemma, which demonstrates that the same ground state density cannot correspond to two different potentials of an external field, and (ii) the hypothesis of the existence of the universal density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, the HK lemma cannot provide justification of the universal density functional for fermions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In other words, each external field determines a unique density, and each density determines a unique external field on the basis of the HK lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, the rule for the last correspondence can be nonuniversal, as the rule in general depends on the concrete form of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The existence of this nonuniversality violates the HK theorem, although the HK lemma is believed to be undoubtedly correct [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Furthermore, there are more serious critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sarry and Sarry [11] claim that the proof of the HK theorem is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The authors do emphasize that for a strict many-particle calculation only the direct mapping: external potential ⇒ ground state wave function ⇒ electron density v(r) ⇒ Ψ0(r) ⇒ ρ0(r) is justified while the inverse mapping ρ0(r) ⇒ Ψ0(r) ⇒ v(r) claimed by the HK theorem can be validated only for single-particle self-consistent calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The DFT exploits the one-to-one correspondence between the single-particle electron density and an external potential acting upon the system and relies on the existence of a universal functional F[ρ(r)] which can be minimized in order to find the ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However,the correspondence theorem establishes the existence of the functional only in principle, and provides no unique practical recipe for its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The construction of the functional F[ρ(r)] in the HK-DFT is equivalent to the problem of finding the N- representability conditions of the reduced density matrix of order two [3, 12], the problem whose solution has not been found until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Generally speaking the functional F[ρ(r)] must be N-dependent, namely, F[N, ρ(r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Another important aspect, closely related to N- representability, is the variational character that either exact or approximate functionals 4 F[N, ρ(r)] must have in order to guarantee that the energy remains an upper bound to the exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The Kohn-Sham (KS) theory goes further in reducing the problem of calculating ground state properties of a many-electron system in a local external single-particle potential to solving Hartree-like one-electron KS equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Within the framework of the HKS-DFT, the many-body problem of interacting electrons in a static external potential is cast into a tractable problem of non-interacting electrons moving in an effective potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The latter includes the external potential and the effects of the Coulomb interactions between the electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' the Hartree term, describing the electron-electron repulsion, and the exchange and correlation (XC) interactions, which includes all the many-body interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Modeling the XC interactions is the main difficulty of DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In practical calculations, the XC contribution is approximated, and the results are only as good as the approximation used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Actually, in HKS-DFT there exist hundreds of XC-approximations for vKS xc (r) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The existence of so many approximations, with so little guidance, makes it ever more difficult for non-specialists to separate the silver from the dross [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It is worth noting here that all the approximate functionals do not comply with the variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The leading approximation for density functional construction is the so called local density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas and only requires the density at each point in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' So the LDA taken from assuming that the electron density for an atom, molecule, or solid is similarly homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' But molecules in LDA are typically overbound by about 1 eV/bond, and in the late 1980s the so-called generalized gradient approximations (GGAs) using both the density and its gradient at each point in space were elaborated whose accuracy seemed to be acceptable in chemical calculations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' All the GGAs functionals, by definition, are corrections to the LDA, they all revert to the uniform electron gas at zero density gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It should be noted that the local nature of the standard approximations implies an exponential decay of the inter-site interaction, in other words, the description of weak interactions such as long-range van der Waals interactions is well beyond any conventional DFT method [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The DFT calculations are quite different from the usual quantum mechanical methods where better accuracy depends on computational resources and not on limitations stemming from the method itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The Hartree-Fock (HF) results cannot be reproduced within the framework of Kohn-Sham (KS) theory because the single-particle densities of finite systems 5 obtained within the HF calculations are not v-representable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', do not correspond to any ground state of a N non-interacting electron systems in a local external potential [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' For this reason, the KS theory, which finds a minimum on a different subset of all densities, can overestimate the ground state energy, as compared to the HF result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In addition to the lack of compliance with N-representability conditions and difficulties in extending the application of the first HK theorem to finite subspaces, there are still other problems that beset DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' They have to do with how to properly include symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', properties of all operators commuting with the Hamiltonian of a given system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' For instance, translational symmetry in crystalline solids should be applied only to a full many-electron function rather than to one-electron KS orbitals!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Currently, the KS-DFT is about occupied orbitals only and is far from giving a consistent and quantitatively accurate description of open-shell spin systems, as the currently available approximate functionals show unsystematic errors in the (inaccurate) prediction of energies, geometries, and molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Strictly speaking, the DFT is designed for description of ground rather than excited states with no good scheme for excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Because an excited-state density does not uniquely determine the potential, there is no general analog of HK for excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The standard functionals are inaccurate both for on-site crystal field and for charge transfer excitations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The DFT based approaches cannot provide the correct atomic limit and the term and multiplet structure, which is crucial for description of the optical response for 3dcompounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Although there are efforts to obtain correct results for spectroscopic properties depending on spin and orbital density this problem remains as an open one in DFT research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Clearly, all these difficulties stem from unsolved foundational problems in DFT and are related to fractional charges and to fractional spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, these basic unsolved issues in the HKS-DFT point toward the need for a basic understanding of foundational issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In other words, given these background problems, the DFT based models should be addressed as semi-empirical approximate ones rather than ab initio theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Levy introduced in 2010 the term DFA to define density functional approximation instead of DFT, which is believed to quite appropriately describe contemporary DFT [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In chemistry, it is traditional to refer to standard approaches as ab initio, while DFT is regarded as empirical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Because solid-state calculations are more demanding, for many decades DFT was the only possible approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, DFT calculations are referred to as ab initio in solid-state 6 physics and materials science [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Proceeding with a fixed approximate functional, the DFT is called "first principles in the sense that the user only chooses the atoms, and the computer predicts (correctly or not) all properties of the molecule or solid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' LSDA Basic drawback of the spin-polarized approaches is that these start with a local density functional in the form (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='15) v(r) = v0[n(r)] + ∆v[n(r), m(r)](ˆσ · m(r) |m(r)|) , where n(r), m(r) are the electron and spin magnetic density, respectively, ˆσ is the Pauli matrix, that is these imply presence of a large fictious local one-electron spin-magnetic field ∝ (v↑−v↓), where v↑,↓ are the on-site LSDA spin-up and spin-down potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Magnitude of the field is considered to be governed by the intra-atomic Hund exchange, while its orientation does by the effective molecular, or inter-atomic exchange fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Despite the supposedly spin nature of the field it produces an unphysically giant spin-dependent rearrangement of the charge density that cannot be reproduced within any conventional technique operating with spin Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Furthermore, a direct link with the orientation of the field makes the effect of the spin configuration onto the charge distribution to be unphysically large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, magnetic long-range order has no significant influence on the redistribution of the charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The DFT-LSDA community needed many years to understand such a physically clear point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In general, the LSDA method to handle a spin degree of freedom is absolutely incompatible with a conventional approach based on the spin Hamiltonian concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' There are some intractable problems with a match making between the conventional formalism of a spin Hamiltonian and LSDA approach to the exchange and exchange-relativistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Visibly plausible numerical results for different exchange and exchange-relativistic parameters reported in many LSDA investigations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [16]) evidence only a potential capacity of the LSDA based models for semiquantitative estimations, rather than for reliable quantitative data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It is worth noting that for all of these "advantageous"instances the matter concerns the handling of certain classical N´eel-like spin configurations (ferro-, antiferro-, spiral,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=') and search for a compatibility with a mapping made with a conventional 7 quantum spin Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It’s quite another matter when one addresses the search of the charge density redistribution induced by a spin configuration as, for instance, in multiferroics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In such a case the straightforward application of the LSDA scheme can lead to an unphysical overestimation of the effects or even to qualitatively incorrect results due to an unphysically strong effect of a breaking of spatial symmetry induced by a spin configuration (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [17] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Going beyond LSDA:LDA+U, LDA+DMFT, LDA+U+V It is commonly accepted now that the standard DFT-LDA(GGA) approach is insufficient to describe the electronic structure of the Mott insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Apparent weaknesses of the DFT approach were exposed especially after the discovery in 1986 of the copper-oxide superconductors, as it failed to yield the fact that the parent compound La2CuO4 is an antiferromagnetic insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' This difficult period for the DFT- LDA method as many decided was partially ended in the early and mid 1990s especially when an orbital dependent Hubbard-type U was incorporated in the exchange correlation functional of the localized 3delectrons within the LDA+U method, while the other electrons are still described at the LDA level [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Attempts to go beyond LSDA are based on the self-interaction-corrected density functional theory SIC-DFT, the LDA+U method, and the GW approximation [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' These methods represent corrections of the single-particle Kohn-Sham potential in one way or another and lead to substantial improvements over the LSDA results for the values of the energy gap and local moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Within the SIC-DFT and LDA+U methods the occupied and unoccupied states are split by the Coulomb interaction U, whereas within the LSDA this splitting is caused by the Stoner parameter J, which is typically one order of magnitude smaller than U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Therefore, compared with the LSDA, the novel methods capture more correctly the physics of transition-metal oxides and improve the results for the energy gap and local moment significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' An important drawback of the LDA+U method is that it requires U as a starting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Even though several schemes for the determination of U exist, it is almost always chosen such that it reproduces the experimental value of a specific property of the electronic structure, most often the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Usually the LDA+U calculations imply account of 8 the on-site d-d correlations with Udd parameter and do neglect the ligand p-p correlations though Udd parameter is only twice as large as Upp in oxides [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The predictive power of the novel methods crucially relies on a reliable assessment of the interactions, however, the value of the interaction parameters, such as Udd, Upp, depends on the choice of the downfolded model, namely, the orbitals treated in the model as well as the basis functions employed, as the screened interaction is determined by the various screening processes that are not considered in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Therefore a careful analysis is needed to make a proper model and choose appropriate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' By fitting, one usually finds higher accuracy for systems similar to those fitted, but usually greater inaccuracies far away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' All efforts to account for the correlations beyond LDA encounter an insoluble problem of double counting (DC) of interaction terms which had just included into Kohn-Sham single-particle potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A well defined analytical expression for the DC potential cannot be formulated in the context of LDA+U or other technique going beyond LDA [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' How to choose the DC correction potential in a manner that is both physically sound and consistent is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, one has to resort to numerical criteria to fix the value of the DC correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, there is currently no universal and unambiguous expression for DC correction, and different formulations are used for different classes of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moreover, different methods for fixing the double counting can drive the result from Mott insulating to almost metallic [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The LDA+DMFT approach combines band structure theory within the DFT-LDA with many-body theory as provided by dynamical mean-field theory (DMFT) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Within DMFT, a lattice model is mapped onto an effective impurity problem embedded in a medium which has to be determined self-consistently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', by quantum Monte-Carlo (QMC) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' This mapping becomes exact in the limit of infinite dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The LDA+U and LDA+DMFT methods are believed to correct the inaccuracies of approximate DFT exchange correlation functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The main idea of the both computational approaches consists in a selective description of the strongly correlated electronic states, typically, localized d or f orbitals, using the Hubbard model, while all the other states continue to be treated at the level of standard approximate DFT functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' At present the LDA+U and LDA+DMFT methods are addressed to be most powerful tools for the investigation of strongly correlated electronic systems, however, these preserve many shortcomings of the basic DFT-LDA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 9 Current theoretical studies of electronic correlations in transition metal oxides typically only account for the local repulsion between d-electrons even if oxygen ligand p-states are an explicit part of the effective Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Interatomic correlations such as Vpd between d- and (ligand) p-electrons, as well as the on-site and inter-site interaction between p- electrons (Upp and Vpp), are usually neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Strictly speaking, LDA+DMFT scheme should incorporate both Upp, Vpp, Vpd and Vdd interactions [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' To this end we need a proper procedure for their calculation, however, this makes the double counting problem significantly more sophisticated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' NiO as a main TMO system for so-called ab initio studies An ongoing challenge during the last 60 years has been the development of a theoretical model that could offer an accurate description of both the electric and magnetic phenomena observed in NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nickel oxide is one of the prototypical compounds that has highlighted the importance of correlation effects in transition metal oxides (TMO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, despite several decades of studies there is still no literature consensus on the detailed electronic structure of NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Although exhibiting a partially filled 3dband and predicted by simple band theory to be a good conductor, NiO has a relatively large band gap (about 4 eV) that cannot be accounted for in the LDA calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' NiO has long been viewed as a prototype "Mott insulator" [21] with the gap formed by intersite cation-cation d-d charge transfer (CT) transitions, however, this view was later replaced by that of a "CT insulator"with the gap formed by anion-cation p-d CT transitions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Strictly speaking, the DFT is designed for description of ground rather than excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nevertheless research activity in the condensed matter DFT community is focused on the single-particle excitation properties of the TMOs, in particular, the photoemission spectra and energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The XPS combined with bremsstrahlung-isochromat spectroscopy (BIS) shows a gap between the top of the valence band and the bottom of the conducting band of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV for NiO [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Namely this value appears to be in the focus of the so-called ab initio DFT-LDA based calculations for NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, the later studies [24] have shown that the exact value of this conductivity gap is subject to the band position chosen to define the highest valence and 10 lowest conducting levels, obtaining values that range from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='20 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='67 eV (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Experimental data, in particular, oxygen x-ray emission (XES) and absorption (XAS) spectra [25] point to strong matrix element effects, that makes reliable estimates of the energy gap to be very ambiguous adventure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The standard DFT-LDA band theory predicts NiO to be a metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' LSDA [26] predicts NiO to be an insulator (with severe underestimated gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV) only in antiferromagnetic state (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The later GW [27] and LDA+U [28] calculations yielded the larger gap of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' First LDA+DMFT calculation performed by Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [29] yielded the value of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The authors claimed: "The overall agreement between the calculated single-particle spectrum and the experimental data is surprisingly good".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, they do neglect the matrix element effect, p-d covalency, Upp, Vpd, and Vdd, that de facto does invalidate their conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Part of these effects, in particular, p − d covalency was taken into account later [30], but with a severe reinterpretation of the DOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Again, the authors claim: ".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='we were able to provide a full description of the valence-band spectrum and, in particular, of the distribution of spectral weight between the lower Hubbard band and the resonant peak at the top of the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, to this day the LDA+DMFT results for NiO strongly depend on the choice of the DC correction potential driving the result from Mott insulating to metallic state [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It is rather surprising how little attention has been paid to the DFT based calculations of the TMO optical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lets turn to a very recent paper by Roedl and Bechstedt [31] on NiO and other TMOs, whose approach is typical for DFT community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The authors calculated the dielectric function ǫ(ω) for NiO within the DFT-GGA+U+∆ technique and claim:"The experimental data agree very well with the calculated curves" (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, this seeming agreement is a result of a simple fitting when the two model parameters U and ∆ are determined such (U = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='0, ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='0 eV) that the best possible agreement concerning the positions and intensities of the characteristic peaks in the experimental spectra is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In addition, the authors arrive at absolutely unphysical conclusion: "The optical absorption of NiO is dominated by intra-atomic t2g → eg transitions" (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nekrasov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [19] realized the DMFT calculation of the optical conductivity for NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Just another correlation parameter was chosen: U = 8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The authors claim a general agreement both with optical and the X-ray experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In the calculations, they found that the main contribution to optical conductivity is due to intra-orbital optical transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 11 Inter-orbital optical transitions give less than 5% of the optical conductivity intensity in the frequency range used in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, as usual they did neglect a number of important on-site and inter-site correlation parameters and all the effects due to optical matrix elements that does invalidate their conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Furthermore, the DFT-LDA based schemes do not provide the correct atomic limit and the term and multiplet structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hence these cannot correctly describe both the d-d crystal field and p-d and d-d charge transfer transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, some authors [32] suppose that in future this problem probably can be solved within the LDA+DMFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Surveying these and other literature data we can argue that the conventional DFT based technique cannot provide a proper description of the optical response for strongly correlated 3dcompounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' As up till now, in future the optical properties of the Mott or charge transfer insulators will be considered within the framework of cluster approaches initially based on quantum-chemical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' CLUSTER MODEL IN NIO Cluster model approach does generalize and advance crystal-field and ligand-field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The method provides a clear physical picture of the complex electronic structure and the energy spectrum, as well as the possibility of a quantitative modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In a certain sense the cluster calculations might provide a better description of the overall electronic structure of insulating 3doxides than the band structure calculations, mainly due to a better account for correlation effects, electron-lattice coupling, and relatively weak interactions such as spin-orbital and exchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cluster models have proven themselves to be reliable working models for strongly correlated systems such as transition-metal and rare- earth compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' These have a long and distinguished history of application in optical and electron spectroscopy, magnetism, and magnetic resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The author with colleagues has successfully demonstrated great potential of the cluster model for description of the p-d and d-d charge transfer transitions and their contribution to optical and magneto-optical response in 3doxides such as ferrites, manganites, cuprates, and nickelates [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cluster models do widely use the symmetry for atomic orbitals, point group symmetry, and advanced technique such as Racah algebra and its modifications for point group symmetry [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' From the other hand the cluster model is an actual proving-ground for various 12 calculation technique from simple quantum chemical MO-LCAO (molecular orbital-linear- combination-of-atomic-orbitals) method to a more elaborate LDA+MLFT (MLFT, multiplet ligand-field theory) [35] approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cluster models traditionally combined quantum chemical MO-LCAO calculations [34] based on atomic Hartree-Fock orbitals with making use parameters fitted to experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Several authors obtained model parameters by performing an LDA calculation for the cluster and using its Kohn-Sham MOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' First comprehensive description of the electronic structure of the NiO6 cluster was performed by Fujimori and Minami [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Effective transfer and overlap integrals were evaluated from LCAO parameters of NiO found by Mattheiss [37] by fitting their APW energy-band results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The localized approach has been shown to successfully explain the photoemission, optical-absorption, and isochromat spectra of NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Recently, Haverkort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [35] suggested a sort of generalization of conventional ligand- field model with the DFT-based calculations within a so-called "ab initio"LDA+MLFT technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' They start by performing a DFT calculation for the proper, infinite crystal using a modern DFT code which employs an accurate density functional and basis set [e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', linear augmented plane waves (LAPWs)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' From the (self-consistent) DFT crystal potential they then calculate a set of Wannier functions suitable as the single-particle basis for the cluster calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The authors compared the theory with experimental spectra (XAS, nonresonant IXS, photoemission spectroscopy) for SrTiO3, MnO, and NiO and found overall satisfactory agreement, indicating that their ligand-field parameters are correct to better than 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [36] the authors have been forced to treat on-site correlation parameter Udd and orbitally averaged (spherical) ∆pd parameter as adjustable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Comparing the novel LDA+MFLT technique with that of Fujimori and Minami [36] one should note very similar level of their quantitative conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Despite the involvement of powerful calculation techniques the numerical results of the both approaches seem to be more like semiquantitative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In such a situation we should transfer the center of gravity of the cluster approaches more and more to elaboration of physically sound and clear semiquantitative models that are maximally take into account all the symmetry requirements on one hand and refer to experiment on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hereafter, we do present a most recent and most comprehensive such a cluster model approach to the description of the p-d and d-d CT transitions in NiO [38] that nicely illustrates great potential of the model that does combine simple physically clear ligand- 13 field analysis, its semiquantitative predictions with a regular appeal to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' We believe that such an approach should precede and accompany any detailed numerical calculation providing its physical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Starting with an octahedral NiO6 complex with the point symmetry group Oh we deal with five Ni 3dand eighteen oxygen O 2p atomic orbitals forming both the hybrid Ni 3d-O 2p bonding and antibonding eg and t2g molecular orbitals (MO), and the purely oxygen nonbonding a1g(σ), t1g(π), t1u(σ), t1u(π), t2u(π) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The nonbonding t1u(σ) and t1u(π) orbitals with the same symmetry are hybridized due to the oxygen-oxygen O 2pπ - O 2pπ transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The relative energy position of different nonbonding oxygen orbitals is of primary importance for the spectroscopy of the oxygen-3d-metal charge transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' This is firstly determined by the bare energy separation ∆ǫ2pπσ = ǫ2pπ − ǫ2pσ between O 2pπ and O 2pσ electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Since the O 2pσ orbital points towards the two neighboring positive 3d ions, an electron in this orbital has its energy lowered by the Madelung potential as compared with the O 2pπ orbitals, which are oriented perpendicular to the respective 3d-O-3d axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, the Coulomb arguments favor the positive sign of the π − σ separation ǫpπ − ǫpσ whose numerical value can be easily estimated in the frames of the well-known point charge model, and appears to be of the order of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='0 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In a first approximation, all the γ(π) states t1g(π), t1u(π), t2u(π) have the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, the O 2pπ-O 2pπ transfer and overlap yield the energy correction to the bare energies with the largest value and a positive sign for the t1g(π) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The energy of the t1u(π) state drops due to a hybridization with the cation 4pt1u(π) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The ground state of NiO610− cluster, or nominally Ni2+ ion corresponds to t6 2ge2 g configuration with the Hund 3A2g(F) ground term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Typically for the octahedral MeO6 clusters [33] the nonbonding t1g(π) oxygen orbital has the highest energy and forms the first electron removal oxygen state while the other nonbonding oxygen π-orbitals, t2u(π), t1u(π), and the σ-orbital t1u(σ) have a lower energy with the energy separation ∼ 1 eV inbetween (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The p-d CT transition in NiO10− 6 center is related to the transfer of O 2p electron to the partially filled 3deg-subshell with the formation on the Ni-site of the (t6 2ge3 g) configuration of nominal Ni+ ion isoelectronic to the well-known Jahn-Teller Cu2+ ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Yet actually instead of a single p-d CT transition we arrive at a series of O 2pγ→ Ni 3deg CT transitions forming a complex p-d CT band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It should be noted that each single electron γ→eg p-d 14 Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 1: (Color online) Spectra of the intersite d-d, p-d CT transitions and on-site crystal field d-d transitions in NiO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Strong dipole-allowed σ−σ d-d and p-d CT transitions are shown by thick solid uparrows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' weak dipole-allowed π − σ p-d transitions by thin solid uparrows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' weak dipole-forbidden low-energy transitions by thin dashed uparrows, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Dashed downarrows point to different electron-hole relaxation channels, dotted downarrows point to photoluminescence (PL) transitions, I1,2 are doublet of very narrow lines associated with the recombination of the d-d CT exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The spectrum of the crystal field d-d transitions is reproduced from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The right hand side reproduces a fragment of the RIXS spectra for NiO [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' CT transition starting with the oxygen γ-orbital gives rise to several many-electron CT states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' For γ=t1,2 these are the singlet and triplet terms 1,3T1, 1,3T2 for the configurations t6 2ge3 gt1,2, where t1,2 denotes the oxygen hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The complex p-d CT band starts with the dipole-forbidden t1g(π)→eg, or 3A2g→1,3T1g, 1,3T2g transitions, then includes two formally dipole-allowed the so-called π→σ p-d CT transitions, the weak t2u(π)→eg, and relatively strong t1u(π)→eg CT transitions, respectively, each giving rise to 3A2g→3T2u transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 15 Finally the main p-d CT band is ended by the strongest dipole-allowed σ→σ t1u(σ)→ eg (3A2g→3T2u) CT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The above estimates predict the separation between the partial p-d bands to be ∼ 1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, if the most intensive CT band with a maximum around 7 eV observed in the RIXS spectra [39–41] to attribute to the strongest dipole- allowed O 2pt1u(σ)→Ni 3deg CT transition then one should expect the low-energy p-d CT counterparts with the maxima around 4, 5, and 6 eV respectively, which are related to the dipole-forbidden t1g(π)→eg, the weak dipole-allowed t2u(π)→eg, and relatively strong dipole-allowed t1u(π)→eg CT transitions, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It is worth noting that the π→σ p-d CT t1u(π)−eg transition borrows a portion of the intensity from the strongest dipole-allowed σ→σ t1u(σ)→eg CT transition because the t1u(π) and t1u(σ) states of the same symmetry are partly hybridized due to the p-p covalency and overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, the overall width of the p-d CT bands with the final t6 2ge3 g configuration occupies a spectral range from ∼ 4 up to ∼ 7 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The left hand side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 1 summarizes the main semiquantitative results of the cluster model predictions for the energy and relative intensities of the p-d CT transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Interestingly this assignment finds a strong support in the reflectance (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='9, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='1, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='2 eV for the allowed p-d CT transitions) spectra of NiO [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A rather strong p(π)-d CT band peaked at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV is clearly visible in the absorption spectra of MgO:Ni [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The electroreflectance spectra [44] which detect the dipole-forbidden transitions clearly point to a low-energy forbidden transition peaked near 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='7 eV missed in the reflectance and absorption spectra [42, 43, 45], which thus defines a p-d character of the optical CT gap and can be related to the onset transition for the whole complex p-d CT band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It should be noted that a peak near 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='8 eV has been also observed in the nonlinear absorption spectra of NiO [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' At variance with the bulk NiO a clearly visible intensive CT peak near 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='6- 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='7 eV has been observed in the absorption spectra of NiO nanoparticles [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' This strongly supports the conclusion that the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='7 eV band is related to the bulk-forbidden CT transition which becomes the partially allowed one in the nanocrystalline state [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It is worth noting that the hole-type photoconductivity threshold in bulk NiO has been observed also at this "magic" energy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='7 eV [48], that is the t1g(π)→eg p-d CT transition is believed to produce itinerant holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Indeed, the p-d CT transitions in NiO6 cluster are of so-called "anti-Jahn- Teller" type, that is these are transitions from orbitally nondegenerate state to the final p-d CT state state formed by two orbitally degenerate states that points to strong electron-lattice effects in excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The final Ni1+ 3d9(t6 2ge3 g) configuration is isoelectronic to Cu2+ ion in 16 cubic crystal field and presents a well-known textbook example of a Jahn-Teller center that implies a strong trend to the localization, while a photo-generated hole can move more or less itinerantly in the O 2p valence band determining the hole-like photoconductivity [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It should be noted that any oxygen π-holes have a larger effective mass than the σ-holes, that results in a different role of the p(π)-d and p(σ)-d CT transitions both in photoconductivity and, probably, the luminescence stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A spectral feature near 6 eV, clearly visible in the NiO photoluminescence excitation (PLE) spectra [38] can be certainly attributed to a rather strong p(π)-d (t1u(π) → eg) CT transition while the spectral feature near 5 eV to a weaker p(π)-d (t2u(π) → eg) CT transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Interestingly the strongest p(σ)-d (t1u(σ) → eg) CT transition at ∼ 7 eV is actually inactive in the PLE spectra, most likely, due to a dominating nonradiative relaxation channel for the oxygen t1u(σ) holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, the p-d CT model cannot explain the main low-energy spectral feature, clearly visible in the PLE spectra near 4 eV [38], thus pointing to manifestation of another CT-type mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Indeed, along with the p-d CT transitions an important contribution to the optical response of the strongly correlated 3doxides can be related to the strong dipole- allowed d-d CT, or Mott transitions [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' In NiO one expects a strong d-d CT transition related to the σ − σ-type eg − eg charge transfer t6 2ge2 g + t6 2ge2 g→ t6 2ge3 g + t6 2ge1 g between nnn Ni sites with the creation of electron NiO611− and hole NiO69− centers (nominally Ni+ and Ni3+ ions, respectively) thus forming a bound electron-hole dimer, or d-d CT exciton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The strong dipole-allowed Franck-Condon d(eg)-d(eg) CT transition in NiO manifests itself as a strong spectral feature near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='5 eV clearly visible in the absorption of thin NiO films [49], RIXS spectra [39, 41], the reflectance spectra (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV) [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Such a strong absorption near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='5 eV is beyond the predictions of the p-d CT model and indeed is lacking in the absorption spectra of MgO:Ni [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' It should be noticed that, unlike all the above mentioned structureless spectra, the nonlinear absorption spectra [46] of NiO films do reveal an anticipated "fine" structure of the d-d CT exciton with the two narrow peaks at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='075 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='33 eV preceding a strong absorption above 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='575 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Interestingly the separation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='2- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV between the peaks is typical for the exchange induced splittings in NiO (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', the "0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='24 eV" optical feature [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Accordingly, the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='1 eV peak in the PLE spectra can be unambiguously assigned to the d-d CT transition [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The charge, spin, and orbital degeneracy of the final state of this unique double anti- 17 Jahn-Teller transition 3A2g + 3A2g→2Eg + 2Eg results in a complex band observed at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='5 eV [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The exchange tunnel reaction Ni++Ni3+↔Ni3++Ni+ due to a two-electron transfer gives rise to the two symmetric (S- and P-) excitons having s- and p-symmetry, respectively, with the energy separation δ0 = 2|t| and δ1 = 2 3|t| for the spin singlet and spin triplet excitons, where t is the two-electron transfer integral whose magnitude is of the order of the Ni2+-Ni2+ exchange integral: t ≈ Innn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Interestingly the P-exciton is dipole-allowed while the S-exciton is dipole-forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The anti-Jahn-Teller d-d CT exciton is prone to be self- trapped in the lattice due to the electron-hole attraction and a particularly strong double Jahn-Teller effect for both the electron and hole centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Recombination transitions for such excitons produce a bulk luminescence with puzzling well isolated doublet of very narrow lines with close energies near 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='3 eV [38] that corresponds to a reasonable Stokes shift of 1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' To the best of our knowledge it is the first observation of the self-trapping for the d-d CT excitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thus, we see that a simple cluster model is able to provide a semiquantitative description of a large body of experimental spectroscopic data, including subtle effects beyond the reach of any "ab initio"DFT technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' We have shown that the prototype 3doxide NiO, similar to perovskite manganites RMnO3 or parent cuprates such as La2CuO4 [33], should rather be sorted neither into the CT insulator nor the Mott-Hubbard insulator in the Zaanen- Sawatzky-Allen scheme [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' SUMMARY There are still a lot of people who think the Hohenberg-Kohn-Sham DFT within the LDA has provided a very successful ab initio framework to successfully tackle the problem of the electronic structure of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' However, both the starting point and realizations of the DFT approach have raised serious questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The HK "theorem"of the existence of a mythical universal density functional that can resolve everything looks like a way into Neverland, the DFT heaven is probably unattainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Various DFAs, density functional approximations, local or nonlocal, will never be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Users are willing to pay this price for simplicity, efficacy, and speed, combined with useful (but not yet chemical or physical) accuracy [4, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The most popular DFA fail for the most interesting systems, such as strongly correlated 18 oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The standard approximations over-delocalize the d-electrons, leading to highly incorrect descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Some practical schemes, in particular, DMFT can correct some of these difficulties, but none has yet become a universal tool of known performance for such systems [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Any comprehensive physically valid description of the electron and optical spectra for strongly correlated systems, as we suggest, should combine simple physically clear cluster ligand-field analysis with a numerical calculation technique such as LDA+MLFT [35], and a regular appeal to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' The research was supported by the Ministry of Education and Science of the Russian Federation, project FEUZ-2020-0054.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hohenberg and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kohn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 136, B864 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kohn and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sham, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 140, A1133 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kryachko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ludena, Physics Reports 544 (2014) 123239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Becke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 140, 18A301 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kotliar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Savrasov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Haule, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Oudovenko, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Parcollet, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Marianetti, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 78, 865 (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Anisimov and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Izyumov, Electronic Structure of Strongly Correlated Materials (Springer Verlag, Berlin, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Eskes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Tjeng, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sawatzky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 41, 288 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ghijsen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Tjeng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' van Elp, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Eskes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Westerink, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sawatzky, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Czyzyk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 38, 11322 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Thomas, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cambr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 23, 542 (1926).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Fermi, Rend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lincei 6, 602 (1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Bobrov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Trigger, JETP 143 , 729 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sarry, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sarry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Solid State, 54(6), 1315 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [12] Eduardo V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ludena, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Structure (Theochem) 709 25 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Burke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 136, 150901 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Ya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Amusia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Msezane, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Shaginyan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sokolovski, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A330:1-2 (2004), 10-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sandratskii, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 64, 134402 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 19 [16] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mazurenko and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Anisimov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 71, 184434 (2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Katsnelson, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kvashnin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mazurenko, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lichtenstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 82, 100403(R) (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Drechsler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 78, 024102 (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 71, 331 (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Panov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Drechsler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 79, 104112 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Karolak, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ulm, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Wehling, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mazurenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Poteryaev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lichtenstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Electron Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Relat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phenom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 181, 11 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [19] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nekrasov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pavlov, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sadovskii, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 95(11), 581 (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' JETP 116(4), 620 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kulik and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Marzari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 134, 094103 (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hansmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Parragh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Toschi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sangiovanni, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Held, arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='2757v1, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [21] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Brandow, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 26, 651 (1977);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' H¨ufner, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 43, 183 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [22] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zaanen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sawatzky, and J W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Allen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 55, 418 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sawatzky, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Allen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 53, 2239 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' H¨ufner, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Steiner, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sander, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Reinert, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Schmitt, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B - Condensed Matter 86, 207-215 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kurmaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Wilks, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moewes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Shamin, and Kuneˇs, PRB 77, 165127 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [26] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Terakura, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Oguchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Williams, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kubler, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 30, 4734 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Massidda, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Continenza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Posternak, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Baldereschi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 55, 13494 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Anisimov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Solovyev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Korotin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Czyzyk, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sawatzky, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 48, 16929 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [29] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ren, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Leonov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Keller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kollar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nekrasov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Vollhardt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 74, 195114 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kunes, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Anisimov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lukoyanov, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Vollhardt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 75, 165115 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Roedl and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Bechstedt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 86, 235122 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Katsnelson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lichtenstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 22, 382201 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zenkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Yuryeva, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Gubanov, Physica B, 168,186 (1991);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zenkov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ganshina, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Krinchik, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nishanova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Solids, 54, 101 (1993);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zenkov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Cond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Matter 14 6957 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 65, 205113 (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Neudert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Knupfer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Fink, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hayn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 65, 180512(R) (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M´alek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Knupfer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 20 Neudert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Fink, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hayn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Drechsler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Motoyama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Eisaki, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Uchida, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 91, 037001 (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pisarev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kalashnikova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Bush, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rasing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 74, 132509 (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pisarev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 77, 060102(R) (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pisarev, Low Temp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 36 613 (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Makhnev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nomerovannaya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Loshkareva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Balbashov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 82, 035106 (2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Moskvin, Optics and Spectroscopy, 111, 403 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sugano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Tanabe and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kamimura, Multiplets of Transition-Metal Ions in Crystals (Academic, New York, 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Haverkort, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zwierzycki, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Andersen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 85, 165113 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Fujimori and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Minami, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 30, 957 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Mattheiss, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 5, 290 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [38] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sokolov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pustovarov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Churmanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', JETP Letters, 95, 528 (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 86, 115128 (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Sokolov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pustovarov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Churmanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=', IOP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Series: Materials Science and Engineering 38, 012007 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [39] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Hiraoka, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Okamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ishii, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Jarrige, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Tsuei, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Cai, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 70, 157 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Magnuson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Butorin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Agui, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nordgren, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Matter 14, 3669 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [41] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Duda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Schmitt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Magnuson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Forsberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Olsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Nordgren, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Okada and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Kotani, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 96, 067402 (2006);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Larson, Wei Ku, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Tischler, Chi-Cheng Lee, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Restrepo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Eguiluz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zschack, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Finkelstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 99, 026401 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [42] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Powell and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Spicer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' B 2, 2182 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [43] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Blazey, Physica 89B, 47 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' McNatt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 23, 915 (1969);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Glosser, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Walker, Solid State Commun, 9, 1599 (1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [45] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Newman and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chrenko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 114, 1507 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [46] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Shablaev, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Pisarev, Physics of the Solid State 45, 1742 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [47] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Volkov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Wang, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Zou, Chemical Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 337, 117 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [48] Ja.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Ksendzov and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Drabkin, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Solid State, 7, 1519 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' [49] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Rossi and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Paul, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' Solids 30, 2295 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfDS7k/content/2301.11979v1.pdf'} diff --git a/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/2301.12160v1.pdf.txt b/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/2301.12160v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9121a01f516c3cfd9c198930118b7fbb398596ce --- /dev/null +++ b/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/2301.12160v1.pdf.txt @@ -0,0 +1,708 @@ +arXiv:2301.12160v1 [physics.optics] 28 Jan 2023 +Polarization-independent second-order photonic topological corner states +Linlin Lei,1 Shuyuan Xiao,2, 3 Wenxing Liu,1 Qinghua Liao,1, ∗ Lingjuan He,1 and Tianbao Yu1, † +1School of Physics and Materials Science, +Nanchang University, Nanchang 330031, China +2Institute for Advanced Study, Nanchang University, Nanchang 330031, China +3Jiangxi Key Laboratory for Microscale Interdisciplinary Study, +Nanchang University, Nanchang 330031, China +1 + +Abstract +Recently, much attention has been paid to second-order photonic topological insulators (SPTIs), because +of their support for highly localized corner states with excellent robustness. SPTIs have been implemented +in either transverse magnetic (TM) or transverse electric (TE) polarizations in two-dimensional (2D) pho- +tonic crystals (PCs), and the resultant topological corner states are polarization-dependent, which limits +their application in polarization-independent optics. However, to achieve polarization-independent corner +states is not easy, since they are usually in-gap and the exact location in the topological bandgap is not +known in advance. Here, we report on a SPTI based on a 2D square-lattice PC made of an elliptic metama- +terial, and whether the bandgap is topological or trivial depends on the choice of the unit cell. It is found that +locations of topological bandgaps of TM and TE polarizations in the frequency spectrum can be indepen- +dently controlled by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥, respectively, and more +importantly, the location of in-gap corner states can also be separately manipulated by them. From this, we +achieve topological corner states for both TM and TE polarizations with the same frequency in the PC by +adjusting ε⊥ and ε∥, and their robustness against disorders and defects are numerically demonstrated. The +proposed SPTI provides a potential application scenario for polarization-independent topological photonic +devices. +I. +INTRODUCTION +Recently, the concept of higher-order topological insulators (HOTIs) has been extended from +electronic waves into classic waves[1–12]. It has been shown that HOTIs do not obey the usual +bulk-edge correspondence but comply with the bulk-edge-corner correspondence[13–15]. For +instance, a two-dimensional (2D) second-order topological insulator possesses one-dimensional +(1D) gapped edge states and zero-dimensional (0D) in-gap corner states. In addition to the charac- +teristics of strong field localization and small mode volume, 0D corner states also show excellent +robustness against fabrication flaws[16–18]. On this basis, they have enormous application value +in the topological cavity[18, 19], lasing[20, 21], non-linear optics[22, 23], and sensing[24]. How- +ever, for photonic crystals (PCs), the two kinds of polarization, transverse magnetic (TM) and +transverse electric (TE) modes, are usually studied in a separate way. One reason is either of the +∗ lqhua@ncu.edu.cn +† yutianbao@ncu.edu.cn +2 + +two modes can be excited independently, each with its own band structure, and the other is that +forming a common band gap (CBG) is not easy, especially the topological one. Past researches +have shown the polarization-independent optics is potentially useful in polarization-independent +waveguides relying full bandgaps[25], enhanced nonlinear optical effects[26], and polarization +division multiplexing[27]. Topologically protected polarization-independent optics would give +them additional resistance to perturbation. It is worth noting that dual-polarization second-order +photonic topological states have been proposed by Chen et al. recently, based on a topologically +optimized geometric structure within a square-lattice[28]. However, eigenfrequencies of topolog- +ical states for the two polarizations are not the same, despite they have a common topological +bandgap. +In this paper, a 2D second-order photonic topological insulator (SPTI) is proposed, of which the +topological states are polarization-independent. The square-lattice PC having a fishnet structure is +made by an elliptic metamaterial. The permittivity is anisotropic and nevertheless, the geometry +structure is rather simple compared with the previously proposed topologically optimized struc- +ture. That the CBG is either trivial or topological depends on the choice of the unit cell (UC) for +both TM and TE modes. The proposed SPTI can host topological edge states and corner states +for the two modes at the same time. Our results show polarization-independent topological cor- +ner states based a SPTI is not guaranteed by a common topological bandgap. However, we find +that locations of bandgaps and corner states in the frequency spectrum can be manipulated inde- +pendently by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥ for TM and TE modes, +respectively, which gives an effective way to achieve overlapped corner states for the two modes. +On this basis, corner states independent of polarization can be realized by choosing appropriate +ε⊥ and ε∥. Numerical simulations further show the corner states are topologically protected, with +strong robustness to disorders and defects. Our work shows potential applications in polarization- +independent topological photonic devices. +II. +STRUCTURE DESIGN AND BAND TOPOLOGY +For PCs, it is well known that TM bandgaps are favored in dielectric rods, while TE bandgaps +prefer dielectric veins[29]. From this, the proposed square-lattice PC is constructed by thin dielec- +tric veins with dielectric rods located at lattice sites, as shown in Fig. 1(a). a is the lattice constant, +and the circle radius r and vein width d are 0.3a and 0.18a, respectively. The dielectric material is +3 + +anisotropic, an elliptic metamaterial with the permittivity ε = (ε∥, ε∥, ε⊥) = (16.9, 16.9, 10). Gen- +erally, topological corner states lie in a topological bandgap[13], and hence a topological CBG +of TM and TE polarizations is the prerequisite for polarization-independent topological corner +states. The choice of the elliptic metamaterial is based on the consideration that bandgap locations +of TM and TE polarizations in the frequency spectrum can be manipulated independently by ε⊥ +and ε∥, respectively. In practical, we can use the multilayer model to construct the anisotropic +permittivity[30]. The multilayer consists of two alternative dielectrics with high and low permit- +tivity, and it is placed horizontally in the x-y plane. According to the formulisms (16) and (17) +proposed in ref[31], the PC slab with permittivity (16.9,16.9,10) can be approximately built by +the high dielectric with the permittivity of 17.67 and the air layer when the filling ratio of high +dielectric is 0.954. Herein, the calculation of band structures and numerical simulations are based +on the finite element method using the commercial software COMSOL Multiphysics. +FIG. 1. (a) Fishnet PC and two kinds of unit cells (UC), UC1 and UC2. (b) Band structures of TM and +TE modes, denoted by red and blue dot-lines, respectively. Even and odd parities of UC1 (UC2) at high +symmetric points are indicated by plus and minus symbols, colored in red and blue for TM and TE modes, +respectively. (c) Ez field patterns of the two TM bands at the X point for UC1 and UC2. (d) Hz field patterns +of the two TE bands at the X point for UC1 and UC2. +4 + +(a) +(b) +0.4 +TM +TE +Frequency(c/a) +0.3 ++ ++(-) +UC1 +(+) +0.2 ++(-) +-(+) +0.1 +0.0 +X(Y) +M +(c) +(d) +TM +TE +UC1 +UC1 +UC2 +UC2 +1st +2nd +1st +2nd ++1 +-1 ++1 +-1 +Ez +HzBased on the common square lattice, two kinds of unit cells (UCs), UC1 and UC2, are selected +in Fig. 1(a). Note that the two UCs are consistent with each other after shifting the center of one +of the UCs by half of the period along x and y directions. Therefore, they share the same band +structure as plotted in Fig. 1(b), with red and blue dot-lines denoting the TM and TE modes, re- +spectively. One can find that there is a CBG indicated by the gray region lying between the first +and second bands of TM modes. However, for the two UCs, the CBG possesses different topo- +logical behaviors characterized by the 2D Zak phase [see Appendix A], which has the following +form[32–34]: +θZak +j += +� +dkxdkyTr[ ˆA j(kx, ky)], +(1) +where j = x or y, and the Berry connection ˆA j = i⟨u(k)|∇kj|u(k)⟩ with u(k) being the periodic part +of the Bloch function. The 2D Zak phase can also be understood by the 2D bulk polarization via +θZak +j += 2πP j with +P j = 1 +2( +� +n +qn +j mod 2), +(−1)qn +j = η(X j) +η(Γ) +(2) +where P j is determined by the parity η associated with π rotation at Γ and X(Y) points and the +summation is over all the occupied bands below the bandgap. Here, Px is equal to Py, namely, +Px = Py, due to the C4 symmetry[35, 36]. Eigenfield patterns at the X point of the two bands +for TM and TE modes are shown in Figs. 1(c) and 1(d), respectively, with the monopole an even +parity and dipole an odd parity. As can be seen, the parities of the two bands at the X point have an +inversion between UC1 and UC2 for both the two modes, whereas the parities at the Γ point stay +the same. Moreover, parities of the same UC at the X point are opposite for TM and TE modes, +which gives the same UC distinct topological properties for the two modes. Concretely, for TM +modes, the distinct parties of UC1 at the X and Γ points give the 2D bulk polarization (Px, Py) a +value of (0, 0) and the 2D Zak phase (θZak +x , θZak +y ) a value of (0, 0), while the same parity of UC2 at +the X and T points makes (Px, Py) = (1 +2, 1 +2) and (θZak +x , θZak +y ) = (π, π). The opposite is true for the TE +modes. As a result, the bandgap of UC1 is trivial and of UC2 is topological for TM modes, and it +is reversed for TE modes. +5 + +FIG. 2. Projected band structures of (a) TM and (b) TE modes, with edge modes colored in red and blue, +respectively. Eigenfields at kx = 0 show the edge modes can be well confined at the interface between +UC1s and UC2s for both TM and TE modes. (c) Dependence of bandgaps and eigenfrequencies of one of +the corner states on ε⊥ and ε∥ for the two modes. The area shaded in light red indicates TM bandgaps, +while the area shaded in light blue indicates TE bandgaps. The red and blue lines denote one of the corner +states of TM and TE modes, respectively. (d) TM corner states (colored in red) and TE corner states +(colored in blue) under any combination of ε⊥ and ε∥ in the same parameter range of (c). The yellow +intersecting line denotes the combinations that have overlapped corner states. The yellow points on the +intersecting line is two of the combinations, and their anisotropic permittivity (ε∥, ε∥, ε⊥) are (16.9,16.9,10) +and (16.375,16.375,9.7), respectively. The green points are the two points that share the same anisotropic +permittivity (16.7,16.7,10.4) but have different eigenfrequencies. +III. +POLARIZATION-INDEPENDENT TOPOLOGICAL CORNER STATES +The topological distinction between UC1 and UC2 ensures the existence of topological edge +states[37–40]. To show this, we construct a supercell composed of five UC1s and five UC2s along +the y direction, and projected band structures are shown in Figs. 2(a) and 2(b) for TM and TE +modes, respectively. In the calculation, periodic boundary conditions are applied to the x direction. +6 + +a +0.3 +(b) +0.3 +Frequency(c/a) +0.2 +0.2 +0.1 +0.1 ++1 +TM +TE +0.0 +0.0 +0 +1 +-1 +0 +k(π/a) +k,=0 +kx(元/a) +k,=0 +(c) +16.4 16.6 16.8 17.0 17.2 174 +(d) +TMcornerstates +0.28 +0.266 +TEcornerstates +0.264 +Frequency(c/a) +0.27 +0.262 +(e) +0.260 +Frequency(c/ +0.26 +0.258 +0.256 +0.25 +0.254 +TM bandgap +（16.375,9.7) +TM corner state +0.252 +0.24 +TE bandgap +(16.9,10) +TE corner state +(16.7,10.4) +3 +9.4 +9.6 +9.8 +10.010.210.410.6 +81FIG. 3. (a) Schematic of the finite-size box-shaped PC, with 15×15 UC1s surrounded by 6-layer UC2s. (b) +Eigenfrequencies of the box-shaped PC. TM and TE modes are denoted by pentagons and circles, with their +corner states colored in red and blue, respectively. Edge modes are shown as cyan. (c) Eigenfields of the +overlapped edge and corner modes. +FIG. 4. (a) Box-shaped PC with four disorders (red dots) around four corners of the internal PC composed +UC1s. The enlarged view shows one of the four disorders, with 10% decrease in radius and 0.1a deviation +from the lattice site along x and y directions. Eigenfields of four corner modes of (b) TM and (c) TE modes, +under the influence of the disorders. +As can be seen, there is one in-gap edge state for both the two modes, which does not occupy the +7 + +(a) +(b +EC1 +EC2 +0.25895(c/a) +0.25897(c/a) +TEC3 +TEC4 +0.25876(c/a +0.25897(c/a) +0.25897(c/a) +O +Ez +Hz(a) +(b) +0.28 +TM bulk +TE bulk +6-layer Uc2s +TM edge +TEedge +0.27 +★ +TMcorner +TE corner +requency(c/a) +0.26 +0.25883(c/a) +L5x15UC1s +0.25 +★ +0.24 +0.23 +0 +10 +20 +30 +40 +50 +60 +Solution number +(c) +TMC2 ++1 +5209(c/a +0.25883 +5883 +-1 +TEC1 +TEC2 +TEC3 +TEC4 ++1 +OHz +0.25883(c/a) +-1 +0.24836(c/a) +0.25881(c/a) +0.25883(c/a) +0.25883(c/a)FIG. 5. (a) Box-shaped PC with defects produced by removing five UC1s in the center and four UC2s near +the edge of the PC. Eigenfields of four corner modes of (b) TM and (c) TE modes, under the influence of +the defects. +whole bulk bandgap and canbe confined at the interface between UC1s and UC2s. Since therer is a +C4 symmetry for the PC, we can define a corner topological index: Qc = 1 +4([X1] + 2[M1] + 3[M2]), +where [Πp] = #Πp − #Γp and #Πp is defined as the number of bands below the bandgap with +rotation eigenvalues Πn +p = e[2πi(p−1)/n] for p=1, 2, 3, 4. For the nontrivial TM and TE cases, they +both have [X1] = −1, [M1] = −1, [M2] = 0. Therefore, the corner topological index is Qc = 1 +4 +for both the two modes, indicating 1 +4 fractionalized corner states at each of the four corners[40]. It +is noteworthy that the existence of polarization-independent corner states is not guaranteed by the +CBG. In Fig. 2(c), we change ε⊥ and ε∥ in the certain range near (16.9, 16.9, 10) to solely adjust +the positions of supercell bandgaps in the frequency spectrum for TM and TE modes, respectively. +Specifically, for the TM band gap, we increase ε⊥ from 9.4 to 10.6 and keep ε∥ at any value, while +for the TE band gap, we increase ε∥ from 16.3 to 17.5 and keep ε⊥ at an arbitrary value. As can +be seen, the positions of the two bandgaps descend as the corresponding permittivity increases, +and the TM bandgap (light red area) is completely embedded in the TE bandgap (light blue area), +forming the CBG. We also calculate the eigenfrequencies of TM (red line) and TE (blue line) +corner states from the finite-size box-shaped PC shown in Fig. 3(a), and find that they are in +the CBG and the variation trend of the corner states with the permittivity is the same as that of +the bandgaps. Since the two kinds of polarized corner states are independent of each other, in +order to search for the overlapped ones, we plot their eigenfrequencies under any combination of +ε⊥ and ε∥ in Fig. 2(d). It can be observed that corner states of the two modes do not coincide +with each other except on the yellow intersecting line. The yellow points on the intersecting +8 + +(a) +(b) +EC1 +EC2 +0.25881(c/a) +0.25883(c/a) +TEC3 +TEC4 +0.25883(c/a) +0.25883(c/a) ++1 +E7 +Hzline are two of the combinations that have the overlapped corner states, and the corresponding +anisotropic permittivities (ε∥, ε∥, ε⊥) are (16.9,16.9,10) and (16.375,16.375,9.7). As a contrast, +green points are the two points that share the same anisotropic permittivity (16.7,16.7,10.4) but +have different eigenfrequencies. Therefore, the anisotropic permittivity provides an additional +freedom to manipulate the location of corner states of the two modes, making the corner states +either polarization-independent or polarization-separable [see Appendix B]. +To verify the existence of the polarization-independent corner states, a box-shaped PC of finite +size is constructed, which is composed of 15 × 15 UC1s surrounded by six-layer UC2s, as shown +in Fig. 3(a). The calculated eigenfrequencies of TM and TE modes based on the anisotropic per- +mittivity (16.9,16.9,10) are shown in Fig. 3(b). As can be seen, both of them show gapped edge +modes and four in-gap corner modes. Red and blue dotted lines go through the overlapped cor- +ner and edge states, respectively. In Fig. 3(c), eigenfields of these topological states indicate that +the edge modes can be well confined along the whole interface between UC1s and UC2s, while +the corner states are highly localized at the corners of the internal PC formed by the UC1s. Re- +markably, topological corner states for the two modes do share the same eigenfrequencies, and the +common eigenfrequency of the corner states is 0.25883(c/a). This is different from the previously +reported dual-polarization topological corner states, which possess the topological CBG, but their +eigenfrequencies are not overlapped at all[28]. +FIG. 6. (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.375,16.375,9.7), +showing overlapped corner states of TM and TE modes. Pentagons and circles denote TM and TE modes, +and their corner states are colored in red and blue, respectively. (b). Eigenfields of the corner states of TM +modes. (c) Eigenfields of the corner states of TE modes. +The polarization-independent photonic corner states are topologically protected due to their +9 + +(b) +(c) +2 +0.28 +M +TEC1 +TEC2 +★ +TM bulk/edge +★ +TM corner +TE bulk/edge +a +0.27 +TEcorner +0.26247(c/a) +0.26244(c/a) +0.26247(c/a) +0.26247(c/a) +★食鱼食★★★★ +★★★★★ +FMC3 +FMC4 +TEC3 +TEC4 +0.25 +0 +2 +4 +6 +8 +10 +12 +14 +0.26252(c/a) +0.26256(c/a) +0.26247(c/a) +0.26247(c/a) +Solution number ++1 ++1 +0 +-1 +Ez +HzFIG. 7. (a) Box-shaped PC with a disorder (red dot) located at the left bottom corner of the internal PC +composed UC1s. The enlarged view shows the single disorder, with 10% decrease in radius and 0.1a +deviation from the lattice site along x and y directions. Eigenfields of four corner modes of (b) TM and (c) +TE modes, under the influence of the disorder. +FIG. 8. (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.7,16.7,10.4), +which shows corner states of TM and TE modes are not overlapped. Pentagons and circles denote TM and +TE modes, and their corner states are colored in red and blue, respectively. (b). Eigenfields of the corner +states of TM modes. (c) Eigenfields of the corner states of TE modes. +topology origin[41, 42]. To verify this, we introduce four disorders marked by red dots around +the four corners into the instead perfect PC, as shown in Fig. 4(a). The enlarged view in Fig. 4(a) +exhibits the single disorder, with 10% decrease in radius and 0.1a deviation from the lattice site +along x and y directions. Eigenfields of the corner states of TM and TE modes are shown in +Figs. 4(b) and 4(c), respectively, from which we can see that the corner states still exist with +negligible offsets of the eigenfrequencies. Beyond that, defects, produced by removing five UC1s +10 + +(b) +(c) +0.28 +TEC1 +TEC2 +TM bulk/edge +★ +TM corner +TEbulk/edge +0.27 +TE corner +0.25416(c/a) +025411c3 +0.26017(c/a) +0.26020(c/a) +TMC3 +TEC3 +M4 +TEC4 +★ +0.25 +0 +2 +4 +6 +8 +10 +12 +14 +0.25416(c/a +0.25423(c/ +0.26020(c/a) +0.26020(c/a +Solution number ++1 ++1 +O +-1 +Ez +Hz(a) +(b) +(c) +TMC1 +TEC1 +TEC2 +0.26225(c/a) +0.26245(c/a) +0.26244(c/a) +0.26247(c/a) +TMC3 +TMC4 +TEC3 +TEC4 +0.26249(c/a) +0.26255(c/a) +0.26247(c/a) +0.26427(c/a) ++1 +O1 ++1 +0 +-1 +Ez +Hzand four UC2s in the center and near the edge of the PC respectively, are also introduced, as shown +in Fig. 5(a). As can be seen in Figs. 5(b) and 5(c), since the defects are far away from the corners, +the eigenfrequencies of the corner states for the TM and TE modes remain unchanged, although +the defects have a more destructive effect on the PC structure[43]. +IV. +CONCLUSION +In summary, a polarization-independent SPTI is achieved, based on a 2D square-lattice PC. The +dielectric is an elliptic metamaterial, and the geometric structure is rather simple nevertheless. By +selecting appropriate geometric parameters and anisotropic permittivity, a CBG is can be obtained +for TM and TE modes. That the CBG of a certain UC is either trivial or topological depends on the +polarization modes. Topological corner states of TM and TE modes can coexist in the CBG, but +only the combinations of in-plane permittivity ε∥ and out-of-plane permittivity ε⊥ that lie on the +intersecting line in the eigenfrequency-permittivity space can make them overlapped. Numerical +simulations further show they have strong robustness to disorders and defects. The proposed +scheme can also be extended to corner states induced by the quadrupole topological phase in +square-lattices, pseudo-spin and valley-spin degrees of freedom. Our work would pave the way +toward designing high-performance polarization-independent topological photonic devices, such +as the polarization-independent topological laser and coupled cavity-waveguide. +ACKNOWLEDGMENTS +The work was jointly supported by the National Natural Science Foundation of China (12064025, +12264028) and Natural Science Foundation of Jiangxi Province (20212ACB202006) and Major +Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJ22012). +Appendix A: Tight-binding model +The tight-binding model gives the topological phase transition between the UC1 and UC2 a +well description, in which one can take the dielectric rods as lattice sites for TM modes while the +air holes act the part for TE modes. The Hamiltonain has the following form, +11 + +H = − +� +ij +tijc† +i cj, +(A1) +where tij is the hopping amplitude between the nearest lattice sites and c† +i (cj) is the creation (an- +nihilation) operator. As there is only one lattice cite in UCs, the one band below the photonic +bandgap can be expressed as +E = −t0(eikx + e−ikx + eiky + e−iky) = −2t0(cos kx + cos ky). +(A2) +Look at TM modes first, for UC1, the lattice site choosed as the inversion center is at the center +of the UC1, and the inversion operator is I = 1. Hence, parities at Γ and X points are the same. +For UC2, lattice sites are at the four corners and the inversion operator I = e±i(kx+ky) hinges on +which lattice site is referenced. Thus, the parity is +1 at the Γ = (0, 0) point, while it is -1 at the +X = (π, 0) point[34]. +For TE modes, lattice sites of UC1 choosed as the inversion center are at the four corners, since +the air holes instead of the dielectric rods act the role of lattice sites. For UC2, the lattice site +choosed as the inversion center is at the center of UC2. As a consequence, parities at Γ and X +points are oppostie for UC1, while they are the same for the UC2. The results are consistence with +parities showed in Fig. 1(b). +Appendix B: Switch between polarization-independent and polarization-separable corner states +Here, we would like to show another anisotropic permittivity lying on the intersecting line that +can achieve polarization-independent topological corner states. The anisotropic permittivity is +(16.375, 16.375, 9.7), as indicated in the Fig. 2(e). Fig. 6(a) shows the calculated eigenfrequencies, +from which we can see that the corner states of the two modes can be overlapped under this +permittivity. Eigenfrequencies and eigenfields of the corner states are shown in Figs. 6(b) and +6(c), and one can see the overlapped eigenfrequency is 0.26247(c/a). In Fig. 7, if we introduce a +single disorder into the box-shaped PC, the corner states still survive with litte frequency shit, but +monopole and quadrupole of TM modes no longer exist due to the broken of the C4 symmetry of +the box-shaped PC. +Noting that if the anisotropic permittivity is off the intersecting line, polarization-independent +corner states will be changed into polarization-separable corner states. As shown in the Fig. 2(e), +12 + +if the anisotropic permittivity is (16.7,16.7,10.4), eigenfrequencies of the corner states of the two +modes are apart from each other. In detail, we plot the eigenfrequencies in Fig. 8(a), and one can +find that none of the four corner states of the two modes are the same, and the maximum frequency +difference between the two modes is 0.00606(c/a) as shown in Figs. 8(b) and 8(c). +[1] C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, Nature 555, 346 (2018). +[2] H. R. Xue, Y. H. Yang, F. Gao, Y. D. Chong, and B. L. Zhang, Nature Materials 18, 108 (2019). +[3] S. Y. Huo, H. B. Huang, L. Y. Feng, and J. J. Chen, Applied Physics Express 12, 094003 (2019). +[4] Y. T. Yang, +Z. Y. Jia, +Y. J. Wu, +R. C. Xiao, +Z. H. Hang, +H. Jiang, and X. Xie, +Science Bulletin 65, 531 (2020). +[5] Z. Z. Yang, X. Li, Y. Y. Peng, X. Y. Zou, and J. C. Cheng, Phys. Rev. Lett. 125, 255502 (2020). +[6] H. M. Mu, B. Liu, T. Y. Hu, and Z. F. Wang, Nano Letters 22, 1122 (2022). +[7] S. G. Marc, V. Peri, R. S¨usstrunk, R. B. Osama, L. Tom, L. G. Villanueva, and D. H. Sebastian, Nature +555, 342 (2018). +[8] X. Ni, M. Weiner, A. Al`u, and A. B. Khanikaev, Nature Materials 18, 113 (2019). +[9] L. He, Z. Addison, E. J. Mele, and B. Zhen, Nature Communications 11, 3119 (2020). +[10] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Science 357, 61 (2017). +[11] S. W. An, T. Liu, H. Y. Fan, H. Gao, Z. M. Gu, S. J. Liang, S. B. Huang, Y. Zheng, Y. F. Chen, +L. Cheng, and J. Zhu, International Journal of Mechanical Sciences 224, 107337 (2022). +[12] B. Y. Xie, H. F. Wang, H. X. Wang, X. Y. Zhu, J. H. Jiang, M. H. Lu, and Y. F. Chen, +Phys. Rev. B 98, 205147 (2018). +[13] Z. G. Chen, C. Q. Xu, J. R. Al, J. Mei, and Y. Wu, Phys. Rev. B 100, 075120 (2019). +[14] M. Li, D. Zhirihin, M. Gorlach, X. Ni, D. Filonov, A. Slobozhanyuk, A. Al`u, and A. B. Khanikaev, +Nature Photonics 14, 89 (2020). +[15] X. Zhang, H.-X. Wang, Z.-K. Lin, Y. Tian, B. Xie, M.-H. Lu, Y.-F. Chen, and J.-H. Jiang, +Nature Physics 15, 582 (2019). +[16] A. Q. Shi, B. Yan, R. Ge, J. L. Xie, Y. C. Peng, H. Li, W. E. I. Sha, and J. J. Liu, +Opt. Lett. 46, 1089 (2021). +[17] X. +D. +Chen, +W. +M. +Deng, +F. +L. +Shi, +F. +L. +Zhao, +M. +Chen, +and +J. +W. +Dong, +Phys. Rev. Lett. 122, 233902 (2019). +13 + +[18] O. Yasutomo, L. Feng, K. Ryota, W. Katsuyuki, W. Katsunori, A. Yasuhiko, and I. Satoshi, +Optica 6, 786 (2019). +[19] X. Xie, W. Zhang, X. He, S. Wu, J. Dang, K. Peng, F. Song, L. Yang, H. Ni, Z. Niu, C. Wang, K. Jin, +X. Zhang, and X. Xu, Laser & Photonics Reviews 14, 1900425 (2020). +[20] H. R. Kim, M. S. Hwang, D. Smirnova, K. Y. Jeong, Y. Kivshar, and H. G. Park, Nature Communica- +tions 11, 5758 (2020). +[21] H. Zhong, Y. V. Kartashov, A. Szameit, Y. Li, C. Liu, and Y. Zhang, APL Photonics 6, 040802 (2021). +[22] Z. Hu, D. Bongiovanni, D. Juki´c, E. Jajti´c, S. Xia, D. Song, J. Xu, R. Morandotti, H. Buljan, and +Z. Chen, Light: Science & Applications 10, 164 (2021). +[23] S. S. Kruk, W. Gao, D.-Y. Choi, T. Zentgraf, S. Zhang, and Y. Kivshar, Nano Letters 21, 4592 (2021). +[24] M. H. Chao, Q. S. Liu, W. J. Zhang, L. Y. Zhuang, and G. F. Song, Opt. Express 30, 29258 (2022). +[25] H. Wu, D. S. Citrin, L. Y. Jiang, and X. Y. Li, IEEE Photonics Technology Letters 27, 840 (2015). +[26] Y. N. Zhang, W. M. Murray, B. B. Lan, and L. Marko, Opt. Lett. 34, 2694 (2009). +[27] D. X. Dai and J. E. Bowers, Nanophotonics 3, 283 (2014). +[28] Y. F. Chen, F. Meng, Z. H. Lan, B. H. Jia, and X. D. Huang, Phys. Rev. Applied 15, 034053 (2021). +[29] Y. F. Chau, F. L. Wu, Z. H. Jiang, and H. Y. Li, Opt. Express 19, 4862 (2011). +[30] J. R. Wang, X. D. Chen, F. L. Zhao, and J. W. Dong, Scientific Reports 6, 22739 (2016). +[31] D. L. X. Z. Z. L. Lorenzo Ferrari, Chihhui Wu, Progress in Quantum Electronics 40, 1 (2015). +[32] Y. F. Chen, Z. H. Lan, and J. Zhu, Phys. Rev. Applied 17, 054003 (2022). +[33] F. Liu and K. Wakabayashi, Phys. Rev. Lett. 118, 076803 (2017). +[34] Y. F. Chen, Z. H. Lan, and J. Zhu, Nanophotonics 11, 1345 (2022). +[35] X. T. He, M. Y. Li, H. Y. Qiu, W. S. Ruan, L. D. Zhou, L. Liu, X. D. Chen, W. J. Chen, F. L. Zhao, +and J. W. Dong, Photon. Res. 9, 1423 (2021). +[36] Z. J. Zhang, J. B. Yang, T. Du, and X. P. Jiang, Photon. Res. 10, 855 (2022). +[37] L. Liu, Y. K. Wang, F. X. Zheng, and T. Sang, Opt. Lett. 47, 2634 (2022). +[38] H. +Zhong, +Y. +V. +Kartashov, +A. +Szameit, +Y. +D. +Li, +C. +L. +Liu, +and +Y. +Q. +Zhang, +APL Photonics 6, 040802 (2021). +[39] G. C. Ma, M. Xiao, and C. T. Chan, Nature Reviews Physics 1, 281 (2019). +[40] B. Y. Xie, G. X. Su, H. F. Wang, H. Su, X. P. Shen, P. Zhan, M. H. Lu, Z. L. Wang, and Y. F. Chen, +Phys. Rev. Lett. 122, 233903 (2019). +[41] M. Proctor, P. A. Huidobro, B. Bradlyn, M. B. de Paz, M. G. Vergniory, D. Bercioux, and A. Garc´ıa- +14 + +Etxarri, Phys. Rev. Research 2, 042038 (2020). +[42] S. Wu, B. Jiang, Y. Liu, and J.-H. Jiang, Photon. Res. 9, 668 (2021). +[43] Z. W. Zhang, H. Y. Long, C. Liu, C. Shao, Y. Cheng, X. J. Liu, and J. Christensen, +Advanced Materials 31, 1904682 (2019). +15 + +This figure "fig_1.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_2.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_3.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_4.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_5.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_6.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_7.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + +This figure "fig_8.png" is available in "png"� format from: +http://arxiv.org/ps/2301.12160v1 + diff --git a/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/load_file.txt b/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..685f1f70b9941b96b97d187ddcac7a39946ed7a4 --- /dev/null +++ b/-9FLT4oBgHgl3EQfvi-U/content/tmp_files/load_file.txt @@ -0,0 +1,845 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf,len=844 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='optics] 28 Jan 2023 Polarization-independent second-order photonic topological corner states Linlin Lei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 Shuyuan Xiao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3 Wenxing Liu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 Qinghua Liao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' ∗ Lingjuan He,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 and Tianbao Yu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' † 1School of Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang 330031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' China 2Institute for Advanced Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang 330031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' China 3Jiangxi Key Laboratory for Microscale Interdisciplinary Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Nanchang 330031,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' China 1 Abstract Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' much attention has been paid to second-order photonic topological insulators (SPTIs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' because of their support for highly localized corner states with excellent robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' SPTIs have been implemented in either transverse magnetic (TM) or transverse electric (TE) polarizations in two-dimensional (2D) pho- tonic crystals (PCs), and the resultant topological corner states are polarization-dependent, which limits their application in polarization-independent optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' However, to achieve polarization-independent corner states is not easy, since they are usually in-gap and the exact location in the topological bandgap is not known in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Here, we report on a SPTI based on a 2D square-lattice PC made of an elliptic metama- terial, and whether the bandgap is topological or trivial depends on the choice of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' It is found that locations of topological bandgaps of TM and TE polarizations in the frequency spectrum can be indepen- dently controlled by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥, respectively, and more importantly, the location of in-gap corner states can also be separately manipulated by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' From this, we achieve topological corner states for both TM and TE polarizations with the same frequency in the PC by adjusting ε⊥ and ε∥, and their robustness against disorders and defects are numerically demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The proposed SPTI provides a potential application scenario for polarization-independent topological photonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' INTRODUCTION Recently, the concept of higher-order topological insulators (HOTIs) has been extended from electronic waves into classic waves[1–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' It has been shown that HOTIs do not obey the usual bulk-edge correspondence but comply with the bulk-edge-corner correspondence[13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' For instance, a two-dimensional (2D) second-order topological insulator possesses one-dimensional (1D) gapped edge states and zero-dimensional (0D) in-gap corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In addition to the charac- teristics of strong field localization and small mode volume, 0D corner states also show excellent robustness against fabrication flaws[16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' On this basis, they have enormous application value in the topological cavity[18, 19], lasing[20, 21], non-linear optics[22, 23], and sensing[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' How- ever, for photonic crystals (PCs), the two kinds of polarization, transverse magnetic (TM) and transverse electric (TE) modes, are usually studied in a separate way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' One reason is either of the ∗ lqhua@ncu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='cn † yutianbao@ncu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='cn 2 two modes can be excited independently, each with its own band structure, and the other is that forming a common band gap (CBG) is not easy, especially the topological one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Past researches have shown the polarization-independent optics is potentially useful in polarization-independent waveguides relying full bandgaps[25], enhanced nonlinear optical effects[26], and polarization division multiplexing[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Topologically protected polarization-independent optics would give them additional resistance to perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' It is worth noting that dual-polarization second-order photonic topological states have been proposed by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' recently, based on a topologically optimized geometric structure within a square-lattice[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' However, eigenfrequencies of topolog- ical states for the two polarizations are not the same, despite they have a common topological bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In this paper, a 2D second-order photonic topological insulator (SPTI) is proposed, of which the topological states are polarization-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The square-lattice PC having a fishnet structure is made by an elliptic metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The permittivity is anisotropic and nevertheless, the geometry structure is rather simple compared with the previously proposed topologically optimized struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' That the CBG is either trivial or topological depends on the choice of the unit cell (UC) for both TM and TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The proposed SPTI can host topological edge states and corner states for the two modes at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Our results show polarization-independent topological cor- ner states based a SPTI is not guaranteed by a common topological bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' However, we find that locations of bandgaps and corner states in the frequency spectrum can be manipulated inde- pendently by the out-of-plane permittivity ε⊥ and in-plane permittivity ε∥ for TM and TE modes, respectively, which gives an effective way to achieve overlapped corner states for the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' On this basis, corner states independent of polarization can be realized by choosing appropriate ε⊥ and ε∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Numerical simulations further show the corner states are topologically protected, with strong robustness to disorders and defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Our work shows potential applications in polarization- independent topological photonic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' STRUCTURE DESIGN AND BAND TOPOLOGY For PCs, it is well known that TM bandgaps are favored in dielectric rods, while TE bandgaps prefer dielectric veins[29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' From this, the proposed square-lattice PC is constructed by thin dielec- tric veins with dielectric rods located at lattice sites, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' a is the lattice constant, and the circle radius r and vein width d are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='3a and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='18a, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The dielectric material is 3 anisotropic, an elliptic metamaterial with the permittivity ε = (ε∥, ε∥, ε⊥) = (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gen- erally, topological corner states lie in a topological bandgap[13], and hence a topological CBG of TM and TE polarizations is the prerequisite for polarization-independent topological corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The choice of the elliptic metamaterial is based on the consideration that bandgap locations of TM and TE polarizations in the frequency spectrum can be manipulated independently by ε⊥ and ε∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In practical, we can use the multilayer model to construct the anisotropic permittivity[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The multilayer consists of two alternative dielectrics with high and low permit- tivity, and it is placed horizontally in the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' According to the formulisms (16) and (17) proposed in ref[31], the PC slab with permittivity (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,10) can be approximately built by the high dielectric with the permittivity of 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='67 and the air layer when the filling ratio of high dielectric is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Herein, the calculation of band structures and numerical simulations are based on the finite element method using the commercial software COMSOL Multiphysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Fishnet PC and two kinds of unit cells (UC), UC1 and UC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (b) Band structures of TM and TE modes, denoted by red and blue dot-lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Even and odd parities of UC1 (UC2) at high symmetric points are indicated by plus and minus symbols, colored in red and blue for TM and TE modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (c) Ez field patterns of the two TM bands at the X point for UC1 and UC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (d) Hz field patterns of the two TE bands at the X point for UC1 and UC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 4 (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4 TM TE Frequency(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='3 + +(-) UC1 (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='2 +(-) (+) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='0 X(Y) M (c) (d) TM TE UC1 UC1 UC2 UC2 1st 2nd 1st 2nd +1 1 +1 1 Ez HzBased on the common square lattice, two kinds of unit cells (UCs), UC1 and UC2, are selected in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Note that the two UCs are consistent with each other after shifting the center of one of the UCs by half of the period along x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Therefore, they share the same band structure as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1(b), with red and blue dot-lines denoting the TM and TE modes, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' One can find that there is a CBG indicated by the gray region lying between the first and second bands of TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' However, for the two UCs, the CBG possesses different topo- logical behaviors characterized by the 2D Zak phase [see Appendix A], which has the following form[32–34]: θZak j = � dkxdkyTr[ ˆA j(kx, ky)], (1) where j = x or y, and the Berry connection ˆA j = i⟨u(k)|∇kj|u(k)⟩ with u(k) being the periodic part of the Bloch function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The 2D Zak phase can also be understood by the 2D bulk polarization via θZak j = 2πP j with P j = 1 2( � n qn j mod 2), (−1)qn j = η(X j) η(Γ) (2) where P j is determined by the parity η associated with π rotation at Γ and X(Y) points and the summation is over all the occupied bands below the bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Here, Px is equal to Py, namely, Px = Py, due to the C4 symmetry[35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfield patterns at the X point of the two bands for TM and TE modes are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1(c) and 1(d), respectively, with the monopole an even parity and dipole an odd parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As can be seen, the parities of the two bands at the X point have an inversion between UC1 and UC2 for both the two modes, whereas the parities at the Γ point stay the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Moreover, parities of the same UC at the X point are opposite for TM and TE modes, which gives the same UC distinct topological properties for the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Concretely, for TM modes, the distinct parties of UC1 at the X and Γ points give the 2D bulk polarization (Px, Py) a value of (0, 0) and the 2D Zak phase (θZak x , θZak y ) a value of (0, 0), while the same parity of UC2 at the X and T points makes (Px, Py) = (1 2, 1 2) and (θZak x , θZak y ) = (π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The opposite is true for the TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As a result, the bandgap of UC1 is trivial and of UC2 is topological for TM modes, and it is reversed for TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Projected band structures of (a) TM and (b) TE modes, with edge modes colored in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields at kx = 0 show the edge modes can be well confined at the interface between UC1s and UC2s for both TM and TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (c) Dependence of bandgaps and eigenfrequencies of one of the corner states on ε⊥ and ε∥ for the two modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The area shaded in light red indicates TM bandgaps, while the area shaded in light blue indicates TE bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The red and blue lines denote one of the corner states of TM and TE modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (d) TM corner states (colored in red) and TE corner states (colored in blue) under any combination of ε⊥ and ε∥ in the same parameter range of (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The yellow intersecting line denotes the combinations that have overlapped corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The yellow points on the intersecting line is two of the combinations, and their anisotropic permittivity (ε∥, ε∥, ε⊥) are (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,10) and (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The green points are the two points that share the same anisotropic permittivity (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4) but have different eigenfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' POLARIZATION-INDEPENDENT TOPOLOGICAL CORNER STATES The topological distinction between UC1 and UC2 ensures the existence of topological edge states[37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' To show this, we construct a supercell composed of five UC1s and five UC2s along the y direction, and projected band structures are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2(a) and 2(b) for TM and TE modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In the calculation, periodic boundary conditions are applied to the x direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 6 a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='3 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='3 Frequency(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1 +1 TM TE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='0 0 1 1 0 k(π/a) k,=0 kx(元/a) k,=0 (c) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='2 174 (d) TMcornerstates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='266 TEcornerstates 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='264 Frequency(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='262 (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='260 Frequency(c/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='254 TM bandgap （16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7) TM corner state 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='24 TE bandgap (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,10) TE corner state (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4) 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='6 81FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Schematic of the finite-size box-shaped PC, with 15×15 UC1s surrounded by 6-layer UC2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (b) Eigenfrequencies of the box-shaped PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' TM and TE modes are denoted by pentagons and circles, with their corner states colored in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Edge modes are shown as cyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (c) Eigenfields of the overlapped edge and corner modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Box-shaped PC with four disorders (red dots) around four corners of the internal PC composed UC1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The enlarged view shows one of the four disorders, with 10% decrease in radius and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1a deviation from the lattice site along x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of four corner modes of (b) TM and (c) TE modes, under the influence of the disorders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As can be seen, there is one in-gap edge state for both the two modes, which does not occupy the 7 (a) (b EC1 EC2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25895(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25897(c/a) TEC3 TEC4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25876(c/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25897(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25897(c/a) O Ez Hz(a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='28 TM bulk TE bulk 6-layer Uc2s TM edge TEedge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='27 ★ TMcorner TE corner requency(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) L5x15UC1s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25 ★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='23 0 10 20 30 40 50 60 Solution number (c) TMC2 +1 5209(c/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883 5883 1 TEC1 TEC2 TEC3 TEC4 +1 OHz 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='24836(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25881(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a)FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Box-shaped PC with defects produced by removing five UC1s in the center and four UC2s near the edge of the PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of four corner modes of (b) TM and (c) TE modes, under the influence of the defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' whole bulk bandgap and canbe confined at the interface between UC1s and UC2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Since therer is a C4 symmetry for the PC, we can define a corner topological index: Qc = 1 4([X1] + 2[M1] + 3[M2]), where [Πp] = #Πp − #Γp and #Πp is defined as the number of bands below the bandgap with rotation eigenvalues Πn p = e[2πi(p−1)/n] for p=1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' For the nontrivial TM and TE cases, they both have [X1] = −1, [M1] = −1, [M2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Therefore, the corner topological index is Qc = 1 4 for both the two modes, indicating 1 4 fractionalized corner states at each of the four corners[40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' It is noteworthy that the existence of polarization-independent corner states is not guaranteed by the CBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2(c), we change ε⊥ and ε∥ in the certain range near (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9, 10) to solely adjust the positions of supercell bandgaps in the frequency spectrum for TM and TE modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Specifically, for the TM band gap, we increase ε⊥ from 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4 to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='6 and keep ε∥ at any value, while for the TE band gap, we increase ε∥ from 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='3 to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='5 and keep ε⊥ at an arbitrary value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As can be seen, the positions of the two bandgaps descend as the corresponding permittivity increases, and the TM bandgap (light red area) is completely embedded in the TE bandgap (light blue area), forming the CBG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' We also calculate the eigenfrequencies of TM (red line) and TE (blue line) corner states from the finite-size box-shaped PC shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3(a), and find that they are in the CBG and the variation trend of the corner states with the permittivity is the same as that of the bandgaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Since the two kinds of polarized corner states are independent of each other, in order to search for the overlapped ones, we plot their eigenfrequencies under any combination of ε⊥ and ε∥ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' It can be observed that corner states of the two modes do not coincide with each other except on the yellow intersecting line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The yellow points on the intersecting 8 (a) (b) EC1 EC2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25881(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) TEC3 TEC4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a) +1 E7 Hzline are two of the combinations that have the overlapped corner states, and the corresponding anisotropic permittivities (ε∥, ε∥, ε⊥) are (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,10) and (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As a contrast, green points are the two points that share the same anisotropic permittivity (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4) but have different eigenfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Therefore, the anisotropic permittivity provides an additional freedom to manipulate the location of corner states of the two modes, making the corner states either polarization-independent or polarization-separable [see Appendix B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' To verify the existence of the polarization-independent corner states, a box-shaped PC of finite size is constructed, which is composed of 15 × 15 UC1s surrounded by six-layer UC2s, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The calculated eigenfrequencies of TM and TE modes based on the anisotropic per- mittivity (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='9,10) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As can be seen, both of them show gapped edge modes and four in-gap corner modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Red and blue dotted lines go through the overlapped cor- ner and edge states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 3(c), eigenfields of these topological states indicate that the edge modes can be well confined along the whole interface between UC1s and UC2s, while the corner states are highly localized at the corners of the internal PC formed by the UC1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Re- markably, topological corner states for the two modes do share the same eigenfrequencies, and the common eigenfrequency of the corner states is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25883(c/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' This is different from the previously reported dual-polarization topological corner states, which possess the topological CBG, but their eigenfrequencies are not overlapped at all[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375,9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7), showing overlapped corner states of TM and TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Pentagons and circles denote TM and TE modes, and their corner states are colored in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of the corner states of TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (c) Eigenfields of the corner states of TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The polarization-independent photonic corner states are topologically protected due to their 9 (b) (c) 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='28 M TEC1 TEC2 ★ TM bulk/edge ★ TM corner TE bulk/edge a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='27 TEcorner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26244(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) ★食鱼食★★★★ ★★★★★ FMC3 FMC4 TEC3 TEC4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25 0 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26252(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26256(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) Solution number +1 +1 0 1 Ez HzFIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Box-shaped PC with a disorder (red dot) located at the left bottom corner of the internal PC composed UC1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The enlarged view shows the single disorder, with 10% decrease in radius and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1a deviation from the lattice site along x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of four corner modes of (b) TM and (c) TE modes, under the influence of the disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (a) Eigenfrequencies of the box-shaped PC with an anisotropic permittivity of (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4), which shows corner states of TM and TE modes are not overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Pentagons and circles denote TM and TE modes, and their corner states are colored in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of the corner states of TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (c) Eigenfields of the corner states of TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' topology origin[41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' To verify this, we introduce four disorders marked by red dots around the four corners into the instead perfect PC, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The enlarged view in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 4(a) exhibits the single disorder, with 10% decrease in radius and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='1a deviation from the lattice site along x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfields of the corner states of TM and TE modes are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 4(b) and 4(c), respectively, from which we can see that the corner states still exist with negligible offsets of the eigenfrequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Beyond that, defects, produced by removing five UC1s 10 (b) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='28 TEC1 TEC2 TM bulk/edge ★ TM corner TEbulk/edge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='27 TE corner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25416(c/a) 025411c3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26017(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26020(c/a) TMC3 TEC3 M4 TEC4 ★ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25 0 2 4 6 8 10 12 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25416(c/a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='25423(c/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26020(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26020(c/a Solution number +1 +1 O 1 Ez Hz(a) (b) (c) TMC1 TEC1 TEC2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26225(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26245(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26244(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) TMC3 TMC4 TEC3 TEC4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26249(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26255(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26427(c/a) +1 O1 +1 0 1 Ez Hzand four UC2s in the center and near the edge of the PC respectively, are also introduced, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As can be seen in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 5(b) and 5(c), since the defects are far away from the corners, the eigenfrequencies of the corner states for the TM and TE modes remain unchanged, although the defects have a more destructive effect on the PC structure[43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' CONCLUSION In summary, a polarization-independent SPTI is achieved, based on a 2D square-lattice PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The dielectric is an elliptic metamaterial, and the geometric structure is rather simple nevertheless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' By selecting appropriate geometric parameters and anisotropic permittivity, a CBG is can be obtained for TM and TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' That the CBG of a certain UC is either trivial or topological depends on the polarization modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Topological corner states of TM and TE modes can coexist in the CBG, but only the combinations of in-plane permittivity ε∥ and out-of-plane permittivity ε⊥ that lie on the intersecting line in the eigenfrequency-permittivity space can make them overlapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Numerical simulations further show they have strong robustness to disorders and defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The proposed scheme can also be extended to corner states induced by the quadrupole topological phase in square-lattices, pseudo-spin and valley-spin degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Our work would pave the way toward designing high-performance polarization-independent topological photonic devices, such as the polarization-independent topological laser and coupled cavity-waveguide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work was jointly supported by the National Natural Science Foundation of China (12064025, 12264028) and Natural Science Foundation of Jiangxi Province (20212ACB202006) and Major Discipline Academic and Technical Leaders Training Program of Jiangxi Province (20204BCJ22012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Appendix A: Tight-binding model The tight-binding model gives the topological phase transition between the UC1 and UC2 a well description, in which one can take the dielectric rods as lattice sites for TM modes while the air holes act the part for TE modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The Hamiltonain has the following form, 11 H = − � ij tijc† i cj, (A1) where tij is the hopping amplitude between the nearest lattice sites and c† i (cj) is the creation (an- nihilation) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As there is only one lattice cite in UCs, the one band below the photonic bandgap can be expressed as E = −t0(eikx + e−ikx + eiky + e−iky) = −2t0(cos kx + cos ky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' (A2) Look at TM modes first, for UC1, the lattice site choosed as the inversion center is at the center of the UC1, and the inversion operator is I = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hence, parities at Γ and X points are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' For UC2, lattice sites are at the four corners and the inversion operator I = e±i(kx+ky) hinges on which lattice site is referenced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Thus, the parity is +1 at the Γ = (0, 0) point, while it is -1 at the X = (π, 0) point[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' For TE modes, lattice sites of UC1 choosed as the inversion center are at the four corners, since the air holes instead of the dielectric rods act the role of lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' For UC2, the lattice site choosed as the inversion center is at the center of UC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As a consequence, parities at Γ and X points are oppostie for UC1, while they are the same for the UC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The results are consistence with parities showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Appendix B: Switch between polarization-independent and polarization-separable corner states Here, we would like to show another anisotropic permittivity lying on the intersecting line that can achieve polarization-independent topological corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' The anisotropic permittivity is (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375, 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='375, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7), as indicated in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 6(a) shows the calculated eigenfrequencies, from which we can see that the corner states of the two modes can be overlapped under this permittivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Eigenfrequencies and eigenfields of the corner states are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 6(b) and 6(c), and one can see the overlapped eigenfrequency is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='26247(c/a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 7, if we introduce a single disorder into the box-shaped PC, the corner states still survive with litte frequency shit, but monopole and quadrupole of TM modes no longer exist due to the broken of the C4 symmetry of the box-shaped PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Noting that if the anisotropic permittivity is off the intersecting line, polarization-independent corner states will be changed into polarization-separable corner states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' As shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 2(e), 12 if the anisotropic permittivity is (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='7,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='4), eigenfrequencies of the corner states of the two modes are apart from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' In detail, we plot the eigenfrequencies in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 8(a), and one can find that none of the four corner states of the two modes are the same, and the maximum frequency difference between the two modes is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='00606(c/a) as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 8(b) and 8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Peterson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Benalcazar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hughes, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bahl, Nature 555, 346 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xue, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chong, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, Nature Materials 18, 108 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Huo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Feng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Applied Physics Express 12, 094003 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xiao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, Science Bulletin 65, 531 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [5] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zou, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Cheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 125, 255502 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Mu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hu, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, Nano Letters 22, 1122 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Marc, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Peri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S¨usstrunk, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Osama, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Tom, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Villanueva, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Sebastian, Nature 555, 342 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Weiner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Al`u, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Khanikaev, Nature Materials 18, 113 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Addison, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Mele, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhen, Nature Communications 11, 3119 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Benalcazar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bernevig, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hughes, Science 357, 61 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' An, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Cheng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhu, International Journal of Mechanical Sciences 224, 107337 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B 98, 205147 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [13] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Al, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Mei, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B 100, 075120 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhirihin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gorlach, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Filonov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Slobozhanyuk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Al`u, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Khanikaev, Nature Photonics 14, 89 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [15] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Tian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, Nature Physics 15, 582 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Shi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Peng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Sha, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 46, 1089 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [17] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Deng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Shi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Dong, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 122, 233902 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 13 [18] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yasutomo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Feng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ryota, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Katsuyuki, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Katsunori, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yasuhiko, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Satoshi, Optica 6, 786 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [19] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' He, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Dang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Peng, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Song, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ni, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Niu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jin, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xu, Laser & Photonics Reviews 14, 1900425 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hwang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Smirnova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jeong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kivshar, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Park, Nature Communica- tions 11, 5758 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kartashov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Szameit, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, APL Photonics 6, 040802 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [22] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Hu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bongiovanni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Juki´c, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jajti´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Song, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Morandotti, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Buljan, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Light: Science & Applications 10, 164 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kruk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Gao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Choi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zentgraf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kivshar, Nano Letters 21, 4592 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhuang, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Song, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Express 30, 29258 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [25] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Citrin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, IEEE Photonics Technology Letters 27, 840 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [26] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Murray, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lan, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Marko, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 34, 2694 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Dai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bowers, Nanophotonics 3, 283 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Meng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jia, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Huang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Applied 15, 034053 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chau, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Express 19, 4862 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Dong, Scientific Reports 6, 22739 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [31] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lorenzo Ferrari, Chihhui Wu, Progress in Quantum Electronics 40, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Applied 17, 054003 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [33] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wakabayashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 118, 076803 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [34] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhu, Nanophotonics 11, 1345 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [35] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' He, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Qiu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ruan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Dong, Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 9, 1423 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [36] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Du, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 10, 855 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zheng, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Sang, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 47, 2634 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhong, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Kartashov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Szameit, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, APL Photonics 6, 040802 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [39] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Ma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xiao, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chan, Nature Reviews Physics 1, 281 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [40] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Xie, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Su, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Su, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Shen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 122, 233903 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [41] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Proctor, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Huidobro, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bradlyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' de Paz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Vergniory, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Bercioux, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Garc´ıa- 14 Etxarri, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Research 2, 042038 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [42] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Wu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Jiang, Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 9, 668 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' [43] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Long, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Shao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Liu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' Christensen, Advanced Materials 31, 1904682 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content=' 15 This figure "fig_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1 This figure "fig_8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='png" is available in "png"� format from: http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='org/ps/2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} +page_content='12160v1' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-9FLT4oBgHgl3EQfvi-U/content/2301.12160v1.pdf'} diff --git a/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf b/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..52e1d277a4864cc96ccefbd29ea5d24fd70cea0d --- /dev/null +++ b/-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c8cf0a8bfda9f4ff0476d4691e51d43897843e51d480f024d5a5f34270d5d60 +size 7071049 diff --git a/.gitattributes b/.gitattributes index 42ac6de76ee62c452ca2065687dc073813a58e9d..b6a62510ca4a6ec1fbe4d2d643752ee23ca6a539 100644 --- a/.gitattributes +++ b/.gitattributes @@ -260,3 +260,12 @@ hNE4T4oBgHgl3EQfrQ2w/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex QNAzT4oBgHgl3EQfIvuz/content/2301.01068v1.pdf filter=lfs diff=lfs merge=lfs -text ptAzT4oBgHgl3EQf5v4s/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 1tAzT4oBgHgl3EQf8_4M/content/2301.01911v1.pdf filter=lfs diff=lfs merge=lfs -text +-NAzT4oBgHgl3EQfSvt8/content/2301.01237v1.pdf filter=lfs diff=lfs merge=lfs -text +pNE0T4oBgHgl3EQf9AKq/content/2301.02796v1.pdf filter=lfs diff=lfs merge=lfs -text +BtFAT4oBgHgl3EQfsR7g/content/2301.08658v1.pdf filter=lfs diff=lfs merge=lfs -text +7dE4T4oBgHgl3EQfcww5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +7dE4T4oBgHgl3EQfcww5/content/2301.05085v1.pdf filter=lfs diff=lfs merge=lfs -text +MtE0T4oBgHgl3EQfjQEE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +qdFRT4oBgHgl3EQfejci/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +MtE0T4oBgHgl3EQfjQEE/content/2301.02455v1.pdf filter=lfs diff=lfs merge=lfs -text +6tAyT4oBgHgl3EQfpvgE/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text diff --git a/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/2301.03503v1.pdf.txt b/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/2301.03503v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0415f74aa7b364d4daa1e1c48cbba8cf35b67a31 --- /dev/null +++ b/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/2301.03503v1.pdf.txt @@ -0,0 +1,1261 @@ +arXiv:2301.03503v1 [hep-th] 9 Jan 2023 +IPARCOS-23-001 +Embedding Unimodular Gravity in String Theory +Luis J. Garay∗ +Departamento de F´ısica Te´orica and IPARCOS, +Universidad Complutense de Madrid, 28040 Madrid, Spain +Gerardo Garc´ıa-Moreno† +Instituto de Astrof´ısica de Andaluc´ıa (IAA-CSIC), +Glorieta de la Astronom´ıa, 18008 Granada, Spain +Abstract +Unimodular Gravity is a theory displaying Weyl rescalings of the metric and transverse (volume- +preserving) diffeomorphisms as gauge symmetries, as opposed to the full set of diffeomorphisms +displayed by General Relativity. Recently, we presented a systematic comparison of both theories, +concluding that both of them are equivalent in everything but the behaviour of the cosmological +constant under radiative corrections. A careful study of how Unimodular Gravity can be embedded +in the string theory framework has not been provided yet and was not analyzed there in detail. +In this article, we provide such an explicit analysis, filling the gap in the literature. We restrict +ourselves to the unoriented bosonic string theory in critical dimension for the sake of simplicity, +although we argue that no differences are expected for other string theories. Our conclusions are +that both a Diff and a WTDiff invariance principle are equally valid for describing the massless +excitations of the string spectrum. +∗ luisj.garay@ucm.es +† ggarcia@iaa.es +1 + +CONTENTS +I. Introduction +2 +II. Unimodular Gravity and General Relativity: Matching global degrees of freedom +5 +III. String perturbation theory in trivial backgrounds +10 +IV. Strings in general backgrounds +16 +A. Determination of the Weyl anomaly +17 +B. Including string-loop corrections +21 +C. EFTs for the theory +23 +V. Conclusions +25 +Acknowledgments +26 +References +27 +I. +INTRODUCTION +Unimodular gravity (UG) is a theory which is so similar to General Relativity (GR) that +one may wonder to what extent both of them are equivalent. Recently we presented a sys- +tematic comparison of both theories in all the regimes and situations in which a potential +difference might appear, which was still lacking [1]. We concluded that for all of the possible +regimes analyzed there, both theories are equivalent except for the behaviour of the cosmo- +logical constant. Whereas the cosmological constant is radiatively stable in UG [2] (it is +simply an integration constant of the equations of motion), in GR it is radiatively unstable. +In this way, if one uses technical naturalness in the sense introduced by ’t Hooft [1, 3, 4] as +a guiding principle toward building theories, UG theories are much more desirable than GR +theories since the cosmological constant is technically natural. +There are mainly three arguments used to argue that the low-energy limit of string theory +is given by the effective field theory (EFT) consisting of GR coupled to some other fields. +First of all, when one analyzes the massless spectrum (leaving aside the tachyon field) of +bosonic string theory propagating on top of flat spacetime one finds that for oriented strings +2 + +it contains a graviton, a Kalb-Ramond field, and a dilaton; and for unoriented strings +it contains only a graviton and a dilaton. +In principle, for computing observables only +involving massless states, one expects that one can write down an effective action which +simply involves fields that account for these massless excitations, i.e., a graviton-field hµν, +(possibly) a Kalb-Ramond field Bµν, and a dilaton field Φ. +As usual, the fundamental +observable considered is the S-matrix. +Now, we come to the arguments used to argue that GR “emerges naturally” as the low- +energy description of such degrees of freedom. First of all, it has been argued that the only +self-consistent way of coupling the graviton (massless spin-2 representation of the Poincar´e +group) to itself is through GR. In that way, having a massless spin-2 field in the spectrum, +one necessarily guesses that the non-linear structure of the theory needs to be GR up to +potential higher-derivative corrections arising in the EFT. However, we argued [1, 5] that +the self-coupling of UG gravitons (those displaying linerarized WTDiff gauge-invariance) to +themselves also gives rise to the full non-linear UG in a consistent way, although the coupling +of the graviton to itself is through the traceless part of the energy-momentum tensor, instead +of the full one. Hence, this first argument does not allow one to discern whether UG or GR +is preferred from the string point of view since one is as legitimate as the other. +The second argument comes from the analysis of string scattering amplitudes, which was +already revisited in [1]. One can compute within string perturbation theory the scattering +amplitudes for graviton asymptotic states. The result is that, to the lowest order in α′ and +at string tree level, one obtains the same scattering amplitudes obtained in GR. The point is +that UG scattering amplitudes are exactly the same as the GR scattering amplitudes [6, 7]. +In that sense, GR is not preferred over UG from the point of view of scattering amplitudes +either, as it was concluded in [1]. +The final argument comes from analyzing perturbatively in α′ the non-linear sigma model +that arises from coupling the string degrees of freedom to an arbitrary background metric (or +conformal structure), Kalb-Ramond field, and dilaton field generated by the string degrees +of freedom themselves. For such a model, the Weyl symmetry of the worldsheet, which is +potentially anomalous, needs to be handled carefully. However, although in flat spacetime +and zero background fields it simply constrains the dimension of spacetime to be 26 (critical +dimension), in this case constraints also appear for the spacetime fields entering the sigma +model construction. Such constraints arise from imposing a cancellation of the Weyl anomaly +3 + +to make it a sensible theory. +The equations that arise are basically Einstein equations, +although interpreted as β-functionals. Both GR and UG give rise to Einstein equations, +hence from this point of view we show that it is possible to write both a GR and UG-like +EFT for the massless degrees of freedom of the string. Moreover, both actions are also +consistent with the previous argument since they reproduce all the scattering amplitudes +involving massless states of the string. The only difference that seems to appear, is that, +whereas in the GR EFT the cosmological constant is a coupling constant that needs to be +set to zero, in the UG EFT it is an integration constant that needs to be set to zero. In +other words, UG contains the space of theories which is GR with all possible values of the +cosmological constant within a single theory. +The α′-expansion on its own points toward a zero cosmological constant. However, once +we include string loop corrections, the situation changes. +We will revisit the Fischler- +Susskind approach [8–10]towards including the lowest order string loop correction in the +picture. In this way, an arbitrary cosmological constant is generated through the string- +loop corrections in the EFT. In this way, the EFT that we need to write down within the +GR EFT to include the string-loop corrections contains an arbitrary cosmological constant, +which is exactly what happens with the UG EFT, although in the former case it is a coupling +constant whereas in the latter it is an integration constant. In this way, we conclude that +both the UG and the GR EFTs can account for the low-energy description of massless string +states with the only difference arising in the nature of the cosmological constant. +It is worth remarking that this analysis gives further evidence for UG as a sensible classical +theory of gravitation according to the criteria invoked by Weinberg in [11]. According to +Weinberg, one of the key aspects that needs to be addressed to regard UG as a reasonable +classic theory of gravitation is to understand whether it can be obtained as a low energy +limit of a quantum theory of gravitation. By embedding UG within the framework of string +theory, here we answer here in the affirmative. +The remain of this article is structured as follows. In Section II we introduce the frame- +work of UG, making special emphasis on the existence of a priviliged background volume +form and the existence of an additional global degree of freedom with respect to GR. Then, +we introduce a modification of GR in which a new global degree of freedom, precisely the +cosmological constant is assigned to a (D+1)-form field, to make clear the difference between +UG and the standard formulation of GR. In Sec. III we review the basics of the quantiza- +4 + +tion of strings in flat spacetimes and explain why UG and GR are both valid as the low +energy description of string theory from the point of view of scattering amplitudes involving +massless particles. In Sec. IV we move on to analyze strings in general backgrounds. In +Subsec. IV A we rederive the consistency conditions (Weyl anomaly cancellation) from the +perturbative α′ expansion of the sigma model. Some of the details of the computation that +are well explained in the literature and not relevant for our purposes are skipped and we +refer the reader to the literature at those points. In Subsec. IV B we introduce the Susskind- +Fischler approach for cancelling some of the divergences arising from string loops, with the +divergences of the sigma model on the trivial genus worldsheet. The main novelty that this +mechanism introduces is a cosmological-constant-like term in the β functions. We close this +section by analyzing in Subsec. IV C how these consistency conditions can be derived from +an effective action once they are interpreted as equations of motion for the background fields. +We emphasize the consistency of this approach when computing scattering amplitudes in- +volving the massless excitations. we close this section. In Sec. V we summarize the results +and draw the conclusions that can be taken from our analysis. We also point interesting +future lines of work that seem promising in virtue of our analysis presented here. +Notation and conventions: Our convention for the signature of the metric is (−, +, ..., +) +for the (D+1)-dimensional target space metric and (−, +) for the worldsheet metric. Tensor +objects will be represented by bold symbols, whereas their components in a given basis will +be written with the same (not bold) symbol and indices, e.g., the Minkowski metric η +will be represented in components as ηµν. We will use Greek letters for spacetime indices +(µ, ν, ...) whereas we will reserve lower case latin indices (a, b, ...) for the worldsheet indices. +Curvature quantities like the Riemann tensor are defined following Misner-Thorne-Wheeler’s +conventions [12] and we will specify explicitly the metric it depends on, e.g. Rα +βγδ(g). We +also represent the (D + 1)-dimensional Newton’s constant as κ2 = 16πG. +II. +UNIMODULAR GRAVITY AND GENERAL RELATIVITY: MATCHING +GLOBAL DEGREES OF FREEDOM +It is well accepted that metric theories of gravity, those in which the fundamental object +describing the gravitational field at a given point is a metric, are suitable for describing +gravitational experiments to great accuracy [13]. The metric at a given point of the spacetime +5 + +is completely specified by the lightcone at that point up to a conformal factor. Although +the conformal structure of the spacetime is allowed to fluctuate both in UG and GR, the +difference arises in the conformal factor. Whereas in UG the conformal factor is fixed to be +a fiducial (non-dynamical) volume form that we represent as ω = +1 +(D+1)!ω(x)dx0 ∧ ... ∧ dxD +and hence it does not have any dynamics, in GR it is also dynamical like the lightcone itself. +Naively, one could conclude that this reduction in the number of independent components +of the metric may lead to a reduction of the independent degrees of freedom of the theory. +However, it reduces the gauge symmetries of the theory to only transverse diffeomorphisms +(those preserving the background volume form) and hence it is not surprising that the theory +displays the same number of local degrees of freedom as GR does. Actually, it displays +an additional global degree of freedom associated with the cosmological constant. In this +section we will introduce the basic formulation of UG, emphasizing the presence of this new +additional global degree of freedom. Furthermore, we will present a formulation of GR closer +in spirit to UG, since the cosmological constant appears as a combination of an arbitrary +integration constant and the renormalized cosmological constant entering the action and we +still have the invariance under the full set of diffeomorphisms. +Let us begin with the standard formulation of UG. UG is a theory in which the group of +gauge transformations is WTDiff (Weyl rescalings of the metric and Transverse Diffeomor- +phisms) instead of the whole group of Diffs (Diffeomorphisms), see [1] for further details. In +order to define such a theory, we need to use the non-dynamical volume form that we have +already introduced ω. It is useful to introduce the Weyl-invariant auxiliary metric +˜gµν = gµν +�ω2 +|g| +� +1 +D+1 +. +(1) +In this way, every curvature scalar built from the auxiliary metric ˜gµν inherits the invariance +under Weyl rescalings and is also invariant under transverse-diffeomorphism transformations +by construction. The simplest action principle that one can think for an UG-like theory is +the UG version of the Einstein-Hilbert action: +SUG = +1 +2κ2 +� +dD+1xωR (g) . +(2) +We can also add a coupling to some matter fields which need to couple to the auxiliary metric, +i.e., the matter action will be of the form Sm (˜g, Φ), so that it remains Weyl-invariant (note +that the matter fields are not affected by Weyl transformations). The equations of motion +6 + +of this theory are the traceless Einstein equations: +Rµν(˜g) − +1 +D + 1R(˜g)˜gµν = κ2 +� +Tµν(˜g) − +1 +D + 1T(˜g)˜gµν +� +. +(3) +Upon using the Bianchi identities, they become Einstein equations with the cosmological +constant entering as an integration constant [1] +Rµν(˜g) − 1 +2R(˜g)˜gµν + Λ˜gµν = κ2Tµν(˜g), +(4) +provided that ˜∇µT µν (˜g) = 0. +It is clear that the Weyl invariance is trivial in the sense that its gauge fixing is trivial, +we simply need to fix the volume form given by the determinant of the metric +� +|g| to be +the background volume form. Actually, this can be done also at the level of the action. The +result is still a local action for the metric which does not contain any mention to the Weyl +symmetry. In that sense, the resulting action is the most minimalistic action that one can +conceive for a metric field. If one tried to make a gauge fixing of the remaining degrees of +freedom, one would end up with a non-local action for the actual physical degrees of freedom +encoded in the field gµν. +In this way, it seems clear that both theories display the same number of local degrees +of freedom of GR, except for the cosmological constant that we will analyze now. To put +it in other words, leaving aside the cosmological constant, from the point of view of initial +conditions, the same amount of initial data are needed to specify a solution to the equations. +The cosmological constant in this case appears with a difference, it is an additional global +degree of freedom. The simplest way to see this is from the point of view of such constant +being an integration constant. This means that it is a constant that parametrizes the space +of solutions, which is separate from the initial data required in GR. In that sense, it is a +constant to be fixed by initial conditions which makes the space of solutions of UG bigger +than the GR space of solutions, precisely by this cosmological constant as an integration +constant. This analysis can be made much more precise by making a Hamiltonian analysis +of the theory, as it has been done in [14], reaching the same conclusions. +We have concluded that UG is equivalent to GR, up to a global degree of freedom which +is precisely playing the role of the cosmological constant. To make it more explicit, we will +introduce now an additional field in GR that accounts for this global degree of freedom, to +sharpen the difference. We need to introduce a (D +1)-form field which is the differential of +7 + +a D-form [15, 16]. Explicitly, we want to introduce a D + 1 form F which is the differential +of a D-form A. In components, this reads: +Fµ0...µD = ∇[µ0Aµµ1...µD ]. +(5) +We can write down the action principle which is the Einstein-Hilbert action with an arbitrary +cosmological constant and a Maxwell-like term for F , namely: +S = +1 +2κ2 +� +dD+1x√−g +� +−2Λ + R(g) − +K +(D + 1)!Fµ0...µDF µ0...µD +� +, +(6) +where K is simply a coupling constant which can be both positive or negative. The equations +of motion for the F -field are +∇µ0F µ0...µD = 0. +(7) +In a (D + 1)-dimensional manifold, a completely antisymmetric volume form like F needs +to be proportional to the ǫ pseudotensor. Hence, the equations of motion simply fixed the +proportionality function to be a constant, i.e. +Fµ0...µD = c√−gǫµ0...µD. +(8) +From the point of view of the initial value problem, this constant c is precisely a global degree +of freedom that needs to be fixed in terms of initial conditions. From that point of view, it +is akin to the cosmological constant in UG, since it is completely fixed in terms of the initial +conditions. We can sharpen the analogy by examining how does this constant c enter the +equations of motion for the metric. The energy-momentum tensor once we evaluate the F +form on shell, behaves exactly as a cosmological constant [15, 16]. Assuming the existence of +additional matter fields, the equations of motion for the gravitational field take the following +form: +Rµν(g) − 1 +2R(g)gµν + Λeffgµν = κ2Tµν(g), +(9) +where the constant Λeff is expressed in terms of the action as +Λeff = Λ + NDKc2, +(10) +with ND an irrelevant numerical factor depending on the spacetime dimension. In this way, +the cosmological constant entering the equations of motion for the metric are a combination +of an initial condition c and the cosmological constant Λ entering the action. +8 + +From a purely classical point of view, we have presented a theory akin to GR, exhibiting +the whole set of diffeomorphisms as gauge symmetries and containing an additional global +degree of freedom encoded in a (D + 1)-form. The equations of motion for this volume form +enforce that it is proportional to the Levi-Civita pseudotensor, with the proportionality +constant been called here c. The constant of proportionality enters the equations of motion +for the metric as an effective cosmological constant. In this way, it plays a similar role to +the one played by the global degree of freedom of UG. Independently of the value that we +assign to the cosmological constant entering the action Λ, the resulting effective cosmological +constant entering Einstein equations Λeff is given by a combination of Λ and c. In terms of +the initial conditions, it is possible to adjust c in order to make Λeff take any desired value. +This formulation of GR with the additional (D + 1) form field is equivalent to UG, in the +sense that it displays the same amount of degrees of freedom, both local and global, and the +global degree of freedom plays the role of a cosmological constant. +At the quantum level, both formulations seem to be different from the point of view of +radiative corrections. The reason behind this mismatch is that, whereas in UG the cos- +mological constant does not receive any radiative corrections and this makes it technically +natural [1, 2]1, in this formulation of GR, the cosmological constant in the action Λ does re- +ceive radiative corrections, and hence it is not technically natural. However, the cosmological +constant relevant for the dynamics is the effective one Λeff that combines the renormalized +Λ with the initial value constant c. It is possible to obtain any value for the cosmological +constant Λeff independently of the potentially huge radiative corrections that Λ may receive. +The equivalence once quantum corrections are included into the picture is unclear. Whether +this formulation is then completely equivalent to UG at the semiclassical level is something +that deserves a separate and detailed study on its own. +Our point here was mainly to introduce a formulation within the GR setup that is close +to the UG version, so that both theories can be compared easily. We have made explicit +the difference existing in the global degrees of freedom of UG and GR (UG contains the +whole space of GR with arbitrary values of the cosmological constant coupling). This only +difference in the two theories, will be also the only difference appearing from the point of +1 We note that technical naturalness is a definition that only applies to coupling constants appearing in +the action. In that sense it is not completely legitimate to say that in UG the cosmological constant is +technically natural since it is not a coupling constant. However, making an abuse of language we find it +convenient to say that it is technically natural. +9 + +view of regarding UG as the low energy EFT for massless string states. +III. +STRING PERTURBATION THEORY IN TRIVIAL BACKGROUNDS +This section contains a review of the quantization of strings in a flat background as well +as the computation of string scattering amplitudes for gravitons from string theory. This is +well-known material that can be found in any textbook [17, 18]. Also we think that a reader +unfamiliarized with string theory might find here a quick introduction to the arguments +presented in the literature leading to the conclusion that GR is the EFT describing the +excitation in massless degrees of freedom. We find convenient to make such introduction +here to expand the discussion presented in [1] about how the scattering amplitudes can be +equivalently obtained from a GR and a UG-like EFT. +The starting point of our discussion of perturbative string theory is the action describing +relativistic strings propagating in flat spacetime. For relativistic free particles it is natural to +consider the action to be the proper time of the particle trajectory i.e., the embedding of the +worldline in the target space. In the same way, for strings it is natural to consider the area +swept out by the worldsheet to replace the proper time of the particle trajectory. For that +purpose, let us introduce a coordinate system in the worldsheet, a pair σa (a = 0, 1) which +correspond to the time coordinate σ0 ∈ (−∞, ∞) and a spatial coordinate σ1. Furthermore, +we will restrict our attention to closed strings (those giving rise to graviton excitations) in +which the points at σ1 and σ1 + 2π are identified. If we endow the (D + 1) dimensional flat +spacetime with coordinates Xµ, we look for an action such that the area density swept by +the string is expressed in terms of derivatives of the embedding Xµ(τ, σ). We notice that +the induced metric on the worldsheet is given by +hab = ηµν∂aXµ∂bXν. +(11) +If we take the action to be the area swept out by the string, we write down the Nambu-Goto +action as +SNG[X] = − +1 +2πα′ +� +d2σ +√ +−h. +(12) +The constant α′ represents the string tension, i.e., the energy density per unit length. Al- +though this action is perfectly reasonable classically, from the point of view of quantization +10 + +is problematic. This is because it is not quadratic in its variables: we have a square root +appearing explicitly in the action. +To circumvent this problem, one can work with the +Polyakov action, which is given by +SP[X, γ] = − +1 +4πα′ +� +d2σ√−γγab∂aXµ∂bXνηµν. +(13) +In this action, there is an additional configuration variable γab which is a metric in the +worldsheet. Now, this action is clearly quadratic in the Xµ variables over which we will +path-integrate to quantize the theory. To see the equivalence among these two actions, we +can compute the equations of motion for the γab variable. Actually, following the standard +conventions, we can define a two-dimensional energy-momentum tensor as the variation of +the Polyakov action with respect to the worldsheet metric, i.e. γab: +Tab = − +1 +√−γ +δSP +δγab = +1 +4πα′ +� +∂aXµ∂bXν − 1 +2γabγad∂cXµ∂dXν +� +ηµν. +(14) +The Polyakov action does not contain any derivatives of the metric γab, and hence the +equations of motion for the metric can be regarded as a constraint Tab = 0 (as a consequence, +strictly speaking it is not a dynamical variable). Actually, this constraint can be used to +solve γab in terms of the Xµ variables. When we plug the result back into the Polyakov +action, we find the Nambu-Goto action we began with. +It is worth pausing at this point and discussing the continuous symmetries of the theory: +• Poincar´e invariance. This is a global symmetry on the worldsheet +Xµ → Λµ +νXν + cµ. +(15) +• Reparametrization invariance or diffeomorphism invariance in the worldsheet σa → +˜σa(σ). Whereas the Xµ fields transform as worldsheet scalars, γab transforms as a +two-index covariant tensor: +Xµ(σ) → Xµ(˜σ) = Xµ(σ), +(16) +γab(σ) → ˜γab(˜σ) = ∂σc +∂˜σa +∂σd +∂˜σb γcd(σ). +(17) +• Weyl invariance of the worldsheet metric γab. This transformation leaves invariant the +Xµ coordinates and the metric gets a local rescaling +Xµ(σ) → Xµ(σ), +(18) +γab(σ) → e2φ(σ)γab(σ). +(19) +11 + +We can distinguish now between oriented and unoriented strings. The former have a well +defined transformation law under the parity transformation σ1 → 2π − σ1. We will focus on +the unoriented strings for the sake of simplicity. +Not all the symmetries that we have introduced are directly preserved through the process +of quantization. Actually, the Weyl symmetry is anomalous, as it is well known. However, +in this case the Weyl symmetry is a gauge symmetry that we must insist on preserving +at the quantum level to remove unphysical states. We will further discuss this point later +when we deal with strings in general backgrounds. For the time being, let us focus on the +quantization of the theory through a path-integral procedure. +Let us illustrate the quantization of the theory through a path-integral procedure as +well as the spectrum that the theory displays. +Let us define the generating functional +following the usual Faddeev-Popov procedure. First of all, we would write down the action +in Euclidean space, in order to make the quantization procedure sensible. We write down +the generating functional as +Z = +1 +V (gauge) +� +DγDXe−SP [X,γ], +(20) +where V (gauge) represents the volume of the gauge group. We recall that we have the Weyl +rescalings of the metric and diffeomorphisms as gauge symmetries of our theory. Hence, +we need to avoid counting more than once physical configurations and that is the reason +for taking the quotient by the volume of the gauge group. As usual, we will introduce a +Faddeev-Popov determinant ∆F P[γ] to take this volume into account. +The integral over the gauge orbits cancels with the volume of the gauge group and we +reach the expression for the generating functional which is +Z[γ] = +� +DX∆F P[γ]e−SP [X,γ]. +(21) +Choosing a convenient normalization for the action, we can rewrite the Faddeev-Popov +determinant as +∆F P[γ] = +� +DbDce−Sg[b,c], +(22) +where b and c are ghosts Grassman-values that anticommute and +Sg = 1 +2π +� +d2σ√γbab∇acb. +(23) +12 + +At this point, we have reduced the evaluation of the path integral for the bosonic string +theory to the evaluation of the path integral: +Z = +� +DbDcDXe−SP [γ,X]−Sg[γ,b,c], +(24) +which is the CFT of D + 1 scalar fields (the Xµ) and the bc-ghost system [17, 19]. If the +theory is going to preserve the Weyl invariance, we need the theory to have a total zero +central charge. This is precisely the consistency condition that we mentioned would appear. +Weyl invariance means that the trace of the two-dimensional energy momentum tensor needs +to vanish. In two-dimensions, the trace of the energy-momentum tensor is determined by +the central charge and the trace anomaly +⟨T a +a⟩ = − c +12R [γ] . +(25) +The system of the Xµ-scalars and the bc-ghost system is linear, and hence the total central +charge is the sum of the central charges of the two systems independently: +c = cg + cX. +(26) +The bc-ghost system [17] has a central charge cg = −26 while each scalar field gives a +contribution of 1 to the central charge cX = D + 1. Ensuring Weyl-invariance means that +we need the spacetime dimension to be 26. This is the well-known way in which the critical +dimension of bosonic string theory emerges. +Now that we have ensured how to preserve the gauge invariance at the quantum level in +order to make the theory consistent, it is time to talk about the spectrum of the strings. +Our point is simply to illustrate that the spectrum of the closed unoriented bosonic string +contains a tachyon, a dilaton, and a graviton. Hence, for this purpose, we can skip the +detailed BRST analysis and focus only on the states generated by the X-fields which are +the “physical fields”. +In order to characterize the spectrum, the simplest way to do it is to use the so called +state-operator map for CFTs [20, 21], in which states are replaced by operator insertions +that generate them by acting in a neighbourhood of the vacuum. For this purpose, it is +first easier to use complex coordinates σ → (z, ¯z) on the worldsheet. Furthermore, we now +need the operators to be gauge invariant. The diffeomorphism invariance can be ensured +by integrating local operators O(z, ¯z) over the worldsheet, i.e. constructing operators of the +13 + +form +V = +� +d2zO(z, ¯z), +(27) +with V standing for vertex operators. Weyl invariance is ensured by choosing the operators +O to transform adequately under Weyl rescalings, i.e., having a suitable weight. The measure +of integration, d2z has a conformal weight (−1, −1) under such rescalings. Hence, O needs +to be a primary operator of the CFT with weight (+1, +1) to compensate it. +The kind of operators that give rise to the lowest energy states of the string are eip·X +and Pµν∂Xµ∂Xνeip·X, with p a given momentum that we endow the string with and Pµν +the polarization tensor [17, 18]. +The operator eip·X gives rise to the tachyon, since we +need to impose that −p2 = −4/α′ < 0 for the operator to be Weyl invariant. The operator +Pµν∂Xµ∂Xνeip·X corresponds to the dilaton (pure trace part of Pµν) and the symmetric part +of Pµν gives rise to the graviton, since p2 = 0 (massless condition) and pµPµν = 0 (transverse +condition) needs to be imposed to ensure the Weyl invariance. The antisymmetric part does +not appear for unoriented strings since it corresponds to the Kalb-Ramond excitation [17]. +Up to this point, we have analyzed the spectrum of the closed unoriented bosonic string +theory and found that the massless states correspond to the dilaton and the graviton. The +Polyakov action per se does not give rise to interactions. We will now make a small digression +on how interactions among the massless states arise in string theory. There is a term that we +can add to the Polyakov action which is an Einstein-Hilbert term that is purely topological +in two-dimensions +Sint = λ +4π +� +d2σ√γR(γ) = 2λ(1 − g), +(28) +being g the genus of the worldsheet and λ a coupling constant which we assume to be small +in order to do perturbation theory. Hence, if we add this term to the string action, we will +get +Z = +� +topologies +� +DXDγe−SP −Sint = +∞ +� +g=0 +e−2λ(1−g) +� +DXDγe−SP . +(29) +If we call eλ = gs, as it is common, this gives a good expansion as long as gs ≪ 1. The whole +series is known to be a divergent series as the standard perturbative series in QFT [22]. In +addition to this problem, there is a harder problem which is the finiteness of each of the +terms in the series, i.e., the path integral over the different geometries. For a fixed topology, +14 + +the path integral with the Polyakov action requires to compute a sum over the moduli +of conformally inequivalent surfaces. +In general, for higher loop orders (i.e. +non-trivial +topologies) this requires to perform an integral over a moduli space that is not obviously +convergent, although some results in the literature point toward its finiteness [23]. +Now it comes to the point of computing some observables. The observable to compute +in string theory is the string S-matrix. This means, we plug some “in” state of the free +string spectrum and compute the probability amplitude of generating another “out” state of +free string spectrum. These states are generated by introducing their corresponding vertex +operators. +For our purposes of analyzing how GR or UG might emerge from string theory, we are +interested in computing the scattering amplitude involving m gravitons with momenta pi and +polarization tensors ei which we represent as A(m)(p1, e1; p2, e2; ...pm, em). This is computed +as a suitable path integral for the Polyakov action SP that schematically reads [17, 18] +A(m)(1h1, 2h2, ..., mhm) = 1 +g2s +1 +Vgauge +� +DXDg e−SP[X,g] +m +� +i=1 +Vi(pi, hi), +(30) +where Vi represents the vertex operator associated with a graviton insertion with a given +spin and momentum. To begin with, we particularize the amplitude for three gravitons and +we find +A(p1, e1; p2, e2; p3, e3) = igs(α′)6 +2 +(2π)26δ26 (p1 + p2 + p3) e1µνe2αβe3γδT µαγT νβδ, +(31) +where +T µαγ = pµ +23ηαγ + pα +31ηγµ + pγ +12ηµα + α′ +8 pµ +23pα +31pγ +12, +(32) +pµ +ij = pµ +i − pµ +j . The terms of order O(α′) in T µαγ contribute as O(p4) to the amplitude. +If we focus just on the lowest order terms O(p2), this amplitude is equivalent to the ones +computed at tree level from the Einstein-Hilbert action upon the identification κ = gs(α′)6. +The same agreement is found with amplitudes involving an arbitrary number of gravitons: +if we neglect the higher-order contribution from the string amplitude, they agree with those +computed from the Einstein-Hilbert action [17, 18], with the same identification of κ and +the string constants. +As it has been already discussed in the literature [6, 7], the tree-level scattering amplitudes +of gravitons computed in GR and UG are identical. +Hence, from the point of view of +15 + +scattering amplitudes, string theory does not point toward GR in a univocal way: both UG +and GR are equivalent from a low-energy effective field theory point of view. This result +was already advanced in [1] and we have reproduced here the analysis in more detail for the +sake of completeness. We will come back to this analysis later, when we introduce the low +energy EFTs for the massless degrees of freedom of the string: both the UG and the GR-like +actions. +IV. +STRINGS IN GENERAL BACKGROUNDS +Up to now, we have only considered strings propagating in flat spacetime. However, the +spectrum of the strings contains some excitations which typically interact among themselves +and could lead to the generation of a non-trivial background. In particular, it contains a +graviton and, necessarily, gravitons need to interact gravitationally. At low energies, all the +excitations that matter are the massless ones. In the same way a laser is a coherent state of +photons, we expect that a coherent state of gravitons might look like a curved background +and a string propagating on top of it needs to be described appropiately. The same comment +applies to the dilaton field. As such, we can write down the most general renormalizable +action including those fields, which is the following non-linear σ-model +S[X, γ] = SP[X, γ] + SD[X, γ] = − +1 +4πα′ +� +d2σ√−γ +� +γabGµν(X)∂aXµ∂bXν + α′R (γ) Φ(X) +� +, +(33) +where Gµν(X) represents a metric (graviton excitations), Φ(X) represents the dilaton back- +ground field, and R[γ] represents the Ricci-scalar of the two-dimensional metric. This term +breaks explicitly the Weyl invariance in the worldsheet. This term is of a higher dimension +than the Weyl-invariant terms, and it does not require to be normalized with a dimensionful +constant. In virtue of the expansion in α′ that we will perform, we will cancel the tree-level +contribution to the anomaly of this last term with the one-loop contribution of the classically +Weyl-invariant terms. The result of this procedure is a reasonable effective field theory for +the massless degrees of freedom of the string. +There are two missing terms that still give rise to a renormalizable theory. The first of +these terms is the coupling to the Kalb-Ramond field. However, if we focus on unoriented +strings, we can skip it since the divergences of the rest of the terms do not require this term +16 + +to be renormalized. In case we deal with oriented strings, this term gives a contribution to +the conformal anomaly [17]. +The additional term that we can add to the action corresponds to a coupling to the +background tachyon field T(X) +ST = 1 +4π +� +d2σ√−γT (X) . +(34) +In principle this term is needed to cancel some of the quadratic divergences arising from +vacuum to vacuum diagrams. However, if we use a renormalization scheme such that those +divergences are absent (for example, dimensional regularization), we can safely skip those +terms. Hence, we will work with a renormalization scheme fullfilling this property. Fur- +thermore, it is worth mentioning that supersymmetry in the worldsheet ensures that those +quadratic divergences are absent in superstrings due to the characteristic cancellation among +fermionic and bosonic degrees of freedom, with independence of the renormalization scheme. +A. +Determination of the Weyl anomaly +Anomalies always appear when there are two symmetries that the theory displays at the +classical level, but it is not possible to quantize such theory preserving both of them. This +means, there is a trade-off between the two symmetries and it is only possible to preserve +one of them in the process. For example, the chiral anomaly is a trade-off between the vector +and axial currents for massless fermion fields. If we use a regularization procedure which +automatically preserves one of those currents, then straightforwardly the other current will +be anomalous. +In the case of the chiral anomaly, it is standard to use a regularization +scheme that preserves gauge invariance and hence yields to the conservation of the vector +current, leading to an anomalous axial current. In the case of Weyl invariance for strings, we +are using a regularization scheme that preserves diffeomorphism invariance, while the Weyl +symmetry becomes potentially anomalous. We need to ensure that the non-linear sigma +model is chosen in such a way that it gives rise to a Weyl-invariant theory. In a language +closer to particle physics, this means that we need to choose our theory in such a way that +we cancel the potential gauge anomalies, which in this case corresponds to choosing the +background fields in such a way that the theory is not Weyl-anomalous. +In the case of +the Standard Model, since it corresponds to a chiral gauge theory, arbitrary matter fields +17 + +would lead to an anomalous theory. However, the matter content is such that the potential +anomaly is absent. This is precisely what we have done in the previous section to fix the +target space dimension to be 26; otherwise, the Weyl-symmetry becomes anomalous. In +this case, we expect constraints also on the background fields entering the non-linear sigma +models, i.e., constraints that the Gµν(X) and the Φ(X) fields need to obey. +We want now to write down the most general form that the Weyl anomaly can display. +Following D’Hoker [24], it is possible to show that the structure of the anomaly for unoriented +strings in a curved background needs to be of the form +⟨T a +a ⟩ = βG +µν(X)∂aXµ∂bXνγab + βΦ (X) R (γ) + βV +µ (X)gabD∗ +a∂bXµ, +(35) +where D∗ +a represents the covariant derivative on the product space of the cotangent space +of the worldsheet and the tangent space of the target space, and it can be explicitly written +down as +D∗ +a∂bXµ = ∂a∂bXµ − Γc +ab∂cXµ + Γµ +νρ∂aXν∂bXρ, +(36) +where Γc +ab are the Christoffel symbols of the metric γab and Γµ +νρ represent the Christoffel +symbols of the metric Gµν. The last term in the Weyl anomaly, βV can be removed through +a transformation on the Xµ fields, since we are always able to perform a local transformation +on the Xµ fields at the same time that we perform a Weyl-rescaling of the metric. This +leaves only two independent β functionals: βG and βΦ2. +Hence we need to determine the β functionals obtained from the action (33). We want +to study perturbatively this action order by order in the α′ expansion, which is done by +assuming that the background fields Gµν(X), Φ(X) vary smoothly with respect to the scale +α′. It is conventional to do the computations in the background field formalism. In this for- +malism, we decompose the fields Xµ in a background part Xµ +0 and its quantum fluctuations +Y µ +Xµ (σ) = Xµ +0 (σ) + Y µ (σ) , +(37) +where the integration is now performed with respect to the quantum fluctuations instead of +Xµ. We define the effective action Γ[X0, g] following [25] as +e−Γ[X0,g] = +� +DY e +− +� +S(X0,Y )−S(X0)−� d2σY µ(σ) δS +δXµ +0 +� +, +(38) +2 For oriented strings there will be another β-functional associated with the Kalb-Ramond field. +18 + +which is the generating functional of the Feynman diagrams relevant for the computation of +the β-functionals. +At this point, it is better to pause and mention a crucial step in the computations. The +coordinate difference does not transform in a covariant way under changes of coordinates. +Hence, in order to obtain results that are manifestly covariant, it is better to do the com- +putation in variables that are manifestly covariant at intermediate steps. This can be done +as follows. Imagine that the coordinates Xµ +0 correspond to a given point p0 and the coor- +dinates Xµ = Xµ +0 + Y µ to a point p. If both points are close enough, there exists only one +geodesic with respect to Gµν connecting both of them. Hence, we can replace the coordinate +difference Y µ which characterizes the point p by the tangent vector tµ of the geodesic at the +point p0, which transforms covariantly under changes of coordinates. Hence, it is better to +use this vector as the integration variable in the path integral. +In fact, we can use this tangent vector tµ to perform a covariant Taylor expansion based +on Xµ +0 of an arbitrary tensor living in the target manifold. To put it explicitly, any tensor +Tµ1...µn(X) can be expanded as +Tµ1...µn(X0 + t) = +∞ +� +k=0 +T (k) +µ1...µnν1...νk(X0)tν1 . . . tνk, +(39) +where each of the terms T (k) +µ1...µnν1...νk is a combination of covariant derivatives of the tensor +Tµ1...µn and contractions with curvature tensors evaluated at X0. This expansion can be +achieved with the help of the normal coordinate expansion although we emphasize that it +remains valid in an arbitrary coordinate system since it is a tensor expression. We are inter- +ested in the expansion of the tensors Gµν, Φ(X) (the latter is a trivial tensor, i.e. a scalar), +and the object ∂a (Xµ +0 + Y µ). These expansions can be obtained after a straightforward +computation, see [25] for details: +Gµν(X) = Gµν(X0) + 1 +3Rµρσνtρtσ + ..., +(40) +Φ(X) = Φ(X0) + ∇µΦ(X0)tµ + 1 +2∇µ∇νΦ(X0)tµtν + ..., +(41) +∂a (Xµ +0 + Y µ) = ∂aXµ +0 + ∇atµ + 1 +3Rµ +νρσ∂aXσ +0 tνtρ + ..., +(42) +where Rµ +νρσ represents the Riemann tensor associated with Gµν. +We are not ready to perform the diagrammatic computation yet. There is a problem +arising from the fact that the term that gives us the propagator for the quantum fields over +19 + +which we integrate, tµ, contains an arbitrary metric in front of it, i.e. we need to invert +a term that looks like Gµν(X0)∇atµ∇btν. The way to deal with this problem and obtain a +simple propagator is to introduce a vielbein eA +µ(X0) which fulfills the property +eA +µ(X0)eB +ν(X0)ηAB = Gµν(X0), +(43) +with ηAB a Lorentzian metric. In this way, we can rewrite all the vector expressions in the +non-holonomic basis eA +µ and get a trivial propagator for the tA = eA +µtµ fields. This comes +with a subtlety, because now the derivatives ∇a involve the spin-connection of the spacetime +ω AB +µ +; for example, +∇atA = ∂atA + ω AB +µ +∂aXµ +0 tCηBC. +(44) +Obtaining a trivial propagator means breaking the SO(D, 1) invariance that the theory +displays, but since we are working in a formalism that is explicitly gauge covariant, we +automatically know that there will always be contributions in the diagrammatic expansion +that make the theory explicitly gauge covariant in intermediate steps. Up to this point, +collecting all the information, we have performed the following expansion for the Polyakov +piece of the action: +SP = SP[X0] + +1 +2πα′ +� +d2σ√γγabGµν(X0)∂aXµ +0 ∇btν +(45) ++ +1 +4πα′ +� +d2σ√γγab � +ηAB∇atA∇btB� +(46) ++ +1 +3πα′ +� +d2σ√γγabRµABC∂aXµ +0 tAtB∇btC +(47) ++ +1 +12πα′ +� +d2σ√γγabRABCDtBtC∇a∇atA∇btD, +(48) +and for the dilaton part we have the trivial structure: +SD[X0 + t] =SD[X0] − 1 +8π +� +d2σ√γ∇AΦ(X0)tA +(49) +− +1 +16π +� +d2σ√γ∇A∇BΦ(X0)tAtB + ... . +(50) +We recall that we can safely impose the equations of motion for the classical fields and safely +drop the linear terms. This is tantamount to a legitimate field redefinition. +Now we can determine the trace anomaly, see Eq. (35) from the effective action introduced +above. The computation requires to go to the next higher order in loops in the dilaton field, +20 + +since the piece of the action for the dilaton field α′ comes with an additional α′ with respect +to the other field. The computation is rather lengthy and hence we do not reproduce it +here [25]. We simply write down the result as +βG +µν = Rµν (G) − ∇µ∇νΦ + O(α′), +(51) +βΦ = D − 26 +6 ++ α′ � +−R (G) + 2∇2Φ + (∇Φ)2� ++ O +� +α′2� +. +(52) +A comment is in order now. If we are dealing with a flat worldsheet, the vanishing of βG +is enough to ensure the Weyl invariance at the quantum level, as long as we are working in +D = 26 dimensions, the critical dimension (see Eq. (35). Hence, in principle, we expect that +the same applies to non-flat worldsheets, i.e. that the condition βΦ = 0 is not independent +of βG = 0. Actually, we have a non-trivial constraint coming from the Bianchi identity +∇µ +� +Rµν (G) − 1 +2R (G) Gµν +� += 0. +(53) +This ensures that we have to the computed order the following identity whenever βG +µν = 0 +∇µβG +µν = ∇νβΦ = 0, +(54) +as can be seen by direct calculation. This implies that βΦ is a constant as long as βG = 0. +By continuity, this automatically implies at this level that βΦ = 0 for D = 26 [25]. From +now on we will restrict ourselves to work in D = 26 and make a comment on strings on +non-critical dimension later. +B. +Including string-loop corrections +At this point, we have only focused on the zeroth-order in the gs-expansion. Although +it is clear that string loops should modify the results, it is not completely clear how those +corrections must be included. One of the most accepted proposals is the Fischler-Susskind +approach [8–10]. The idea behind such mechanism is that string loop divergences can be +absorbed through a renormalization of the background fields in the non-linear sigma models. +Let us illustrate this explicitly for unoriented closed bosonic strings. For the purpose of this +section, it is simpler to work with a a sharp cut-off as regularization scheme. +The divergences in string loops appear when we have to sum over conformally inequivalent +surfaces of a fixed topology (i.e. genus). For a fixed but arbitrary topology (i.e. we focus +21 + +here on non-trivial topologies), this sum is an integral over a finite-dimensional parameter +space, the so-called Teichm¨uller space [17, 18]. +These integrals are divergent, but these +divergences arise from handles that shrink to zero size. These divergences are equivalent to +the divergences coming from inserting a local operator on the trivial-genus worldsheet. In a +flat spacetime, the divergence appearing for the torus topology can be eliminated through +the insertion of an operator log Λ +2π γabηµν∂aXµ∂bXν, with a suitable coefficient. Here Λ is a +suitable cut-off in the Teichm¨uller space. +If we move to a curved geometry Gµν with a non-trivial zero mode of the dilaton field Φ, +we need to substitute the metric Gµν and include a relative factor e−Φ to account for the +dependence of the path integral on the topology of the surface. We recall that the asymptotic +value of the dilaton field λ = ⟨Φ⟩ is identified with the string coupling constant gs = eλ +through an exponential relation, as it can be seen by comparison of the actions in Eq. (33) +and Eq. (28) [17, 18]. Explicitly for the first non-trivial order (torus topology) we have the +following divergences: +δSloop = log Λ +2π +� +d2σ√−γγabe−ΦGµν(X)∂aXµ∂bXν. +(55) +The e−Φ factor ensures that, when evaluated on the trivial topology on the worldsheet, it +captures the divergences in the torus. If the dilaton field displays a non-trivial background +profile Φ(X), not only a zero mode λ, we expect that replacing Φ with Φ(X) would lead to +a first term in an α′ expansion of the term. This term modifies the β-functional (we will +refer from now on to those β-functionals modified due to the the presence of string loop +corrections as ˜β) associated with the metric through the addition of a term δβG +µν to the +functional βG +µν above +˜βG +µν = βG +µν + δβG +µν, +(56) +which looks like a cosmological constant term, i.e. +δβG +µν = Ce−ΦGµν, +(57) +where C is an arbitrary constant that arises in the renormalization procedure. On equal +footing, an additional contribution to the dilaton, which we call δβΦ will also appear, al- +though it is hard to evaluate explicitly. +Instead, it is easier to obtain it by applying a +consistency argument [8–10]. As we have argued above, in principle the vanishing of the +22 + +modified ˜βΦ-functional through string loop corrections is not independent of the vanishing +of the ˜βG +µν functional. As we have seen, in the CFT computation, it being constant is pre- +cisely a consequence of the vanishing of the remaining β-functionals. By this consistency +condition, it is possible to derive an equation for the ˜βΦ-function. +Taking the divergence of the ˜βG +µν and simplifying it through Bianchi identities and using +also the vanishing of ˜βG +µν itself, we find: +∇µ ˜βG +µν = ∇ν +�1 +2R (G) − ∇2Φ − 1 +2 (∇Φ)2 +� +(58) +This leads us to the following ˜βΦ functional for the dilaton field: +˜ +βΦ = α′ +� +−R (G) + 2∇2Φ + 1 +2 (∇Φ)2 +� +, +(59) +which knowing that is a constant, can be safely chosen to be equal to zero. In case that we +were dealing with strings in non-critical dimension, an additional D − 26/6 factor should be +included arising from the bc-ghost system contribution to the Weyl-anomaly at the string +tree level. Notice that we have introduced α′ as a dimensionful parameter. Once we have +reached this point, it is better to pause and recapitulate what we have done until now. We +began analyzing the α′-expansion of the sigma model describing the propagation of strings in +arbitrary backgrounds. We determined the β-functionals of the Weyl anomaly to the lowest +order. Then we jumped into the problem of including string-loop corrections that should +clearly modify the constraints that the background fields should obey. For the purpose of +including such corrections, we noticed that the divergences arising from the string loops can +be absorbed into a renormalization of the background fields Gµν and Φ. Hence, up to this +point we have found a set of equations that these background fields need to obey for the +consistent propagation of the strings. +C. +EFTs for the theory +The consistency equations that we found arising from the Weyl anomaly cancellation and +the cancellation of the divergences from string loop corrections resemble a lot the equations +of motion of a given field theory for Gµ(X) and Φ(X): +˜βG +µν = Rµν (G) − ∇µ∇νΦ + Ce−ΦGµν + O(α′), +(60) +˜βΦ = D − 26 +6 ++ α′ � +−R (G) + 2∇2Φ + (∇Φ)2� ++ O +� +α′2� +. +(61) +23 + +Setting C = 0 corresponds to omitting the string loop corrections. The natural question is +then whether it is possible to obtain an effective action whose dynamics correctly reproduce +these equations. In addition, such effective action needs to correctly account for the scatter- +ing amplitudes involving only massless excitations of the string (to the lowest order in the +α′ expansion) in order to be a sensible action. There are (at least) two effective actions that +fullfill these criteria: match the scattering amplitudes involving gravitons and dilatons and +their equations of motion give rise to the β-functionals. These two actions correspond to a +GR-like EFT and a UG-like EFT. The GR-like EFT can be given as: +SGR +eff = +1 +2κ2 +� +dD+1X +√ +−GeΦ +� +−(D − 26) +6α′ +− 2Ce−Φ + R (G) + (∇Φ)2 +� ++ O(α′). +(62) +From this action principle it is straightforward to obtain the β-functionals as +˜βΦ = −2κ2 e−Φ +√ +−G +δSGR +eff +δΦ , +(63) +˜βG +µν = 2κ2 e−Φ +√ +−G +�δSGR +eff +δGµν + 1 +2Gµν +δSGR +eff +δΦ +� +. +(64) +Furthermore, it is possible to perform a field redefinition to map this action to the Einstein +Frame [18]. +Following [1] we know that it is also possible to write down an action principle which +reproduces the same equations of motion that Eq. (62) displays, with the cosmological con- +stant C entering as an integration constant instead of a coupling constant. To be concrete, +we can write down the following action principle: +SUG +eff = +1 +2κ2 +� +dD+1XωeΦ +� +−(D − 26) +6α′ ++ R( ˜ +G) + ( ˜∇Φ)2 +� ++ O(α′). +(65) +If we compute the variation with respect to Gµν we obtain the traceless version of the +equations obtained from Eq. (62). Explicitly, if we define +δSUG +eff +δGµν += Kµν (G) − 1 +2K (G) , +(66) +for the variation of SUG +eff we obtain the following: +δSUG +eff +δGµν = Kµν( ˜ +G) − +1 +D + 1K( ˜ +G) ˜Gµν = 0. +(67) +with K( ˜ +G) = ˜GµνKµν( ˜ +G). Upon taking the divergence and using the generalized Bianchi +identities for the corresponding tensor K entering the equations (see [1] for further details) +24 + +we find: +Eµν = Kµν( ˜ +G) − 1 +2K( ˜ +G) ˜Gµν + C ˜Gµν = 0. +(68) +Again, a suitable combination of these equations with the equation obtained from the equa- +tion of motion for Φ we find: +˜βΦ = −2κ2e−Φ +ω +δSUG +eff +δΦ , +(69) +˜βG +µν = 2κ2e−Φ +ω +� +Eµν + 1 +2 +˜Gµν +δSeff +δΦ +� +, +(70) +confirming our claim that the Unimodular Gravity action (65) reproduces the β-functionals. +Notice that this effective action does not only reproduce the β-functionals but it also repro- +duces all of the scattering amplitudes involving massless excitations of the string (graviton +and dilaton asymptotic states), as derived following the procedure sketched in the previous +section. In that sense, both actions reproduce the desired properties and hence none of them +is preferred over the other one from the perspective of using them as EFTs for the massless +modes of the string. +V. +CONCLUSIONS +We have analyzed the embedding of UG in string theory from the point of view of the +consistent quantization of the strings in an arbitrary background. Furthermore, we have +followed the proposal by Susskind and Fischler towards cancelling divergences arising from +string loops with suitable counterterms in the non-linear sigma model. Our analysis here +does not unveil any preference for UG or GR as a low energy description of string theory. +This ties up the loose ends that were not analyzed in [1], regarding the embedding of UG +in string theory. To put it explicitly: both UG and GR are equally valid as low energy +descriptions of the massless modes of string theory and none of them seems to be preferred +over the other one. +Regarding future directions of work, we recall that our analysis here has focused on +bosonic string theory. At first sight, the extension to superstring theory seems straightfor- +ward although subtleties may arise in a careful study. Previous considerations of supergrav- +ity in a UG-like context suggest that some of the vacua may spontenously break SUSY and +hence both theories may develop a potential inequivalence at the quantum level [26–28]. +25 + +Although there is no analysis of the global degrees of freedom in such contexts, it should be +mentioned that it seems possible that a careful implementation of SUSY in that contexts +requires also from a fermionic global degree of freedom, which is the responsible for the +apparent SUSY-breaking presented there. +A second direction of work that is worthwhile exploring is that of non-perturbative defini- +tions of string theory and its interplay with UG. For instance, the gauge/gravity correspon- +dence (also called usually AdS/CFT) [29–31] and matrix models [32], among them probably +we could highlight the BFSS matrix model [33]. In such contexts, we have not explored +whether it is easy or not to accomodate a UG principle instead of a GR principle. +ACKNOWLEDGMENTS +The authors would like to thank Carlos Barcel´o and Ra´ul Carballo-Rubio for collab- +oration in early stages of this project and invaluable discussions during the preparation +of the manuscript. +We would also like to thank Tom´as Ort´ın for helpful conversations. +Financial support was provided by the Spanish Government through the projects PID2020- +118159GB-C43, PID2020-118159GB-C44, and by the Junta de Andaluc´ıa through the +project FQM219. GGM acknowledges financial support from the grant CEX2021-001131-S +funded by MCIN/AEI/10.13039/501100011033. GGM is funded by the Spanish Government +fellowship FPU20/01684. +26 + +[1] R. Carballo-Rubio, L. J. Garay, and G. Garc´ıa-Moreno, (2022), arXiv:2207.08499 [gr-qc]. +[2] R. Carballo-Rubio, Phys. Rev. D 91, 124071 (2015), arXiv:1502.05278 [gr-qc]. +[3] G. ’t Hooft, NATO Sci. Ser. B 59, 135 (1980). +[4] C. P. Burgess, Introduction to Effective Field Theory (Cambridge University Press, 2020). +[5] A. Delhom, G. Garc´ıa-Moreno, M. Hohmann, A. Jim´enez-Cano, and T. S. Koivisto, (2022), +arXiv:2211.13056 [gr-qc]. +[6] E. ´Alvarez, S. Gonzalez-Martin, +and C. P. Mart´ın, Eur. Phys. J. C 76, 554 (2016), +arXiv:1605.02667 [hep-th]. +[7] R. Carballo-Rubio, F. Di Filippo, and N. Moynihan, JCAP 10, 030 (2019), arXiv:1811.08192 +[hep-th]. +[8] W. Fischler and L. Susskind, Phys. Lett. B 171, 383 (1986). +[9] W. Fischler and L. Susskind, Phys. Lett. B 173, 262 (1986). +[10] W. Fischler, I. R. Klebanov, and L. Susskind, Nucl. Phys. B 306, 271 (1988). +[11] S. Weinberg, Rev. Mod. Phys. 61, 1 (1989). +[12] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, +1973). +[13] C. M. Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, +2018). +[14] M. Henneaux and C. Teitelboim, Physics Letters B 222, 195 (1989). +[15] M. Henneaux and C. Teitelboim, Phys. Lett. B 143, 415 (1984). +[16] E. Mottola, JHEP 11, 037 (2022), arXiv:2205.04703 [hep-th]. +[17] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge Mono- +graphs on Mathematical Physics (Cambridge University Press, 2007). +[18] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory. Vol. 1: Introduction, Cam- +bridge Monographs on Mathematical Physics (1988). +[19] Y. Nakayama, Phys. Rept. 569, 1 (2015), arXiv:1302.0884 [hep-th]. +[20] J. D. Qualls, (2015), arXiv:1511.04074 [hep-th]. +[21] P. Di Francesco, P. Mathieu, and D. Senechal, Conformal Field Theory, Graduate Texts in +Contemporary Physics (Springer-Verlag, New York, 1997). +27 + +[22] D. J. Gross and V. Periwal, Phys. Rev. Lett. 60, 2105 (1988). +[23] S. Mandelstam, Phys. Lett. B 277, 82 (1992). +[24] P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, +and E. Witten, eds., Quantum fields and strings: A course for mathematicians. Vol. 1, 2 +(1999). +[25] C. G. Callan, Jr. and L. Thorlacius, in Theoretical Advanced Study Institute in Elementary +Particle Physics: Particles, Strings and Supernovae (TASI 88) (1989). +[26] J. Anero, C. P. Martin, and R. Santos-Garcia, JHEP 01, 145 (2020), arXiv:1911.04160 [hep- +-th]. +[27] J. Anero, C. P. Martin, +and R. Santos-Garcia, JCAP 03, 006 (2020), arXiv:2001.05365 +[hep-th]. +[28] S. Bansal, S. Nagy, A. Padilla, and I. Zavala, JHEP 01, 146 (2021), arXiv:2010.13758 [hep-th]. +[29] J. M. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998), arXiv:hep-th/9711200. +[30] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri, +and Y. Oz, Phys. Rept. 323, 183 +(2000), arXiv:hep-th/9905111. +[31] J. Polchinski, in Theoretical Advanced Study Institute in Elementary Particle Physics: String +theory and its Applications: From meV to the Planck Scale (2010) pp. 3–46, arXiv:1010.6134 +[hep-th]. +[32] P. H. Ginsparg and G. W. Moore, in Theoretical Advanced Study Institute (TASI 92): From +Black Holes and Strings to Particles (1993) pp. 277–469, arXiv:hep-th/9304011. +[33] T. Banks, W. Fischler, S. H. Shenker, +and L. Susskind, Phys. Rev. D 55, 5112 (1997), +arXiv:hep-th/9610043. +28 + diff --git a/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/load_file.txt b/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b2c064bbaee690d48828dc675b8c4df0455d738d --- /dev/null +++ b/0NE1T4oBgHgl3EQf4wUX/content/tmp_files/load_file.txt @@ -0,0 +1,695 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf,len=694 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='03503v1 [hep-th] 9 Jan 2023 IPARCOS-23-001 Embedding Unimodular Gravity in String Theory Luis J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Garay∗ Departamento de F´ısica Te´orica and IPARCOS, Universidad Complutense de Madrid, 28040 Madrid, Spain Gerardo Garc´ıa-Moreno† Instituto de Astrof´ısica de Andaluc´ıa (IAA-CSIC), Glorieta de la Astronom´ıa, 18008 Granada, Spain Abstract Unimodular Gravity is a theory displaying Weyl rescalings of the metric and transverse (volume- preserving) diffeomorphisms as gauge symmetries, as opposed to the full set of diffeomorphisms displayed by General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Recently, we presented a systematic comparison of both theories, concluding that both of them are equivalent in everything but the behaviour of the cosmological constant under radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' A careful study of how Unimodular Gravity can be embedded in the string theory framework has not been provided yet and was not analyzed there in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this article, we provide such an explicit analysis, filling the gap in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We restrict ourselves to the unoriented bosonic string theory in critical dimension for the sake of simplicity, although we argue that no differences are expected for other string theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Our conclusions are that both a Diff and a WTDiff invariance principle are equally valid for describing the massless excitations of the string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' ∗ luisj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='garay@ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='es † ggarcia@iaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='es 1 CONTENTS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Introduction 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Unimodular Gravity and General Relativity: Matching global degrees of freedom 5 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' String perturbation theory in trivial backgrounds 10 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Strings in general backgrounds 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Determination of the Weyl anomaly 17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Including string-loop corrections 21 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' EFTs for the theory 23 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Conclusions 25 Acknowledgments 26 References 27 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' INTRODUCTION Unimodular gravity (UG) is a theory which is so similar to General Relativity (GR) that one may wonder to what extent both of them are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Recently we presented a sys- tematic comparison of both theories in all the regimes and situations in which a potential difference might appear, which was still lacking [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We concluded that for all of the possible regimes analyzed there, both theories are equivalent except for the behaviour of the cosmo- logical constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Whereas the cosmological constant is radiatively stable in UG [2] (it is simply an integration constant of the equations of motion), in GR it is radiatively unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, if one uses technical naturalness in the sense introduced by ’t Hooft [1, 3, 4] as a guiding principle toward building theories, UG theories are much more desirable than GR theories since the cosmological constant is technically natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' There are mainly three arguments used to argue that the low-energy limit of string theory is given by the effective field theory (EFT) consisting of GR coupled to some other fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' First of all, when one analyzes the massless spectrum (leaving aside the tachyon field) of bosonic string theory propagating on top of flat spacetime one finds that for oriented strings 2 it contains a graviton, a Kalb-Ramond field, and a dilaton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' and for unoriented strings it contains only a graviton and a dilaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In principle, for computing observables only involving massless states, one expects that one can write down an effective action which simply involves fields that account for these massless excitations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', a graviton-field hµν, (possibly) a Kalb-Ramond field Bµν, and a dilaton field Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As usual, the fundamental observable considered is the S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Now, we come to the arguments used to argue that GR “emerges naturally” as the low- energy description of such degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' First of all, it has been argued that the only self-consistent way of coupling the graviton (massless spin-2 representation of the Poincar´e group) to itself is through GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that way, having a massless spin-2 field in the spectrum, one necessarily guesses that the non-linear structure of the theory needs to be GR up to potential higher-derivative corrections arising in the EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, we argued [1, 5] that the self-coupling of UG gravitons (those displaying linerarized WTDiff gauge-invariance) to themselves also gives rise to the full non-linear UG in a consistent way, although the coupling of the graviton to itself is through the traceless part of the energy-momentum tensor, instead of the full one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, this first argument does not allow one to discern whether UG or GR is preferred from the string point of view since one is as legitimate as the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The second argument comes from the analysis of string scattering amplitudes, which was already revisited in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' One can compute within string perturbation theory the scattering amplitudes for graviton asymptotic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The result is that, to the lowest order in α′ and at string tree level, one obtains the same scattering amplitudes obtained in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The point is that UG scattering amplitudes are exactly the same as the GR scattering amplitudes [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that sense, GR is not preferred over UG from the point of view of scattering amplitudes either, as it was concluded in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The final argument comes from analyzing perturbatively in α′ the non-linear sigma model that arises from coupling the string degrees of freedom to an arbitrary background metric (or conformal structure), Kalb-Ramond field, and dilaton field generated by the string degrees of freedom themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For such a model, the Weyl symmetry of the worldsheet, which is potentially anomalous, needs to be handled carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, although in flat spacetime and zero background fields it simply constrains the dimension of spacetime to be 26 (critical dimension), in this case constraints also appear for the spacetime fields entering the sigma model construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Such constraints arise from imposing a cancellation of the Weyl anomaly 3 to make it a sensible theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The equations that arise are basically Einstein equations, although interpreted as β-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Both GR and UG give rise to Einstein equations, hence from this point of view we show that it is possible to write both a GR and UG-like EFT for the massless degrees of freedom of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Moreover, both actions are also consistent with the previous argument since they reproduce all the scattering amplitudes involving massless states of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The only difference that seems to appear, is that, whereas in the GR EFT the cosmological constant is a coupling constant that needs to be set to zero, in the UG EFT it is an integration constant that needs to be set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In other words, UG contains the space of theories which is GR with all possible values of the cosmological constant within a single theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The α′-expansion on its own points toward a zero cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, once we include string loop corrections, the situation changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will revisit the Fischler- Susskind approach [8–10]towards including the lowest order string loop correction in the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, an arbitrary cosmological constant is generated through the string- loop corrections in the EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, the EFT that we need to write down within the GR EFT to include the string-loop corrections contains an arbitrary cosmological constant, which is exactly what happens with the UG EFT, although in the former case it is a coupling constant whereas in the latter it is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, we conclude that both the UG and the GR EFTs can account for the low-energy description of massless string states with the only difference arising in the nature of the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is worth remarking that this analysis gives further evidence for UG as a sensible classical theory of gravitation according to the criteria invoked by Weinberg in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' According to Weinberg, one of the key aspects that needs to be addressed to regard UG as a reasonable classic theory of gravitation is to understand whether it can be obtained as a low energy limit of a quantum theory of gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' By embedding UG within the framework of string theory, here we answer here in the affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The remain of this article is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Section II we introduce the frame- work of UG, making special emphasis on the existence of a priviliged background volume form and the existence of an additional global degree of freedom with respect to GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Then, we introduce a modification of GR in which a new global degree of freedom, precisely the cosmological constant is assigned to a (D+1)-form field, to make clear the difference between UG and the standard formulation of GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' III we review the basics of the quantiza- 4 tion of strings in flat spacetimes and explain why UG and GR are both valid as the low energy description of string theory from the point of view of scattering amplitudes involving massless particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' IV we move on to analyze strings in general backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' IV A we rederive the consistency conditions (Weyl anomaly cancellation) from the perturbative α′ expansion of the sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Some of the details of the computation that are well explained in the literature and not relevant for our purposes are skipped and we refer the reader to the literature at those points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' IV B we introduce the Susskind- Fischler approach for cancelling some of the divergences arising from string loops, with the divergences of the sigma model on the trivial genus worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The main novelty that this mechanism introduces is a cosmological-constant-like term in the β functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We close this section by analyzing in Subsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' IV C how these consistency conditions can be derived from an effective action once they are interpreted as equations of motion for the background fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We emphasize the consistency of this approach when computing scattering amplitudes in- volving the massless excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' we close this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' V we summarize the results and draw the conclusions that can be taken from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We also point interesting future lines of work that seem promising in virtue of our analysis presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Notation and conventions: Our convention for the signature of the metric is (−, +, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', +) for the (D+1)-dimensional target space metric and (−, +) for the worldsheet metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Tensor objects will be represented by bold symbols, whereas their components in a given basis will be written with the same (not bold) symbol and indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', the Minkowski metric η will be represented in components as ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will use Greek letters for spacetime indices (µ, ν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=') whereas we will reserve lower case latin indices (a, b, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=') for the worldsheet indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Curvature quantities like the Riemann tensor are defined following Misner-Thorne-Wheeler’s conventions [12] and we will specify explicitly the metric it depends on, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rα βγδ(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We also represent the (D + 1)-dimensional Newton’s constant as κ2 = 16πG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' UNIMODULAR GRAVITY AND GENERAL RELATIVITY: MATCHING GLOBAL DEGREES OF FREEDOM It is well accepted that metric theories of gravity, those in which the fundamental object describing the gravitational field at a given point is a metric, are suitable for describing gravitational experiments to great accuracy [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The metric at a given point of the spacetime 5 is completely specified by the lightcone at that point up to a conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Although the conformal structure of the spacetime is allowed to fluctuate both in UG and GR, the difference arises in the conformal factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Whereas in UG the conformal factor is fixed to be a fiducial (non-dynamical) volume form that we represent as ω = 1 (D+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='ω(x)dx0 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' ∧ dxD and hence it does not have any dynamics, in GR it is also dynamical like the lightcone itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Naively, one could conclude that this reduction in the number of independent components of the metric may lead to a reduction of the independent degrees of freedom of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, it reduces the gauge symmetries of the theory to only transverse diffeomorphisms (those preserving the background volume form) and hence it is not surprising that the theory displays the same number of local degrees of freedom as GR does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, it displays an additional global degree of freedom associated with the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this section we will introduce the basic formulation of UG, emphasizing the presence of this new additional global degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Furthermore, we will present a formulation of GR closer in spirit to UG, since the cosmological constant appears as a combination of an arbitrary integration constant and the renormalized cosmological constant entering the action and we still have the invariance under the full set of diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Let us begin with the standard formulation of UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' UG is a theory in which the group of gauge transformations is WTDiff (Weyl rescalings of the metric and Transverse Diffeomor- phisms) instead of the whole group of Diffs (Diffeomorphisms), see [1] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In order to define such a theory, we need to use the non-dynamical volume form that we have already introduced ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is useful to introduce the Weyl-invariant auxiliary metric ˜gµν = gµν �ω2 |g| � 1 D+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (1) In this way, every curvature scalar built from the auxiliary metric ˜gµν inherits the invariance under Weyl rescalings and is also invariant under transverse-diffeomorphism transformations by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The simplest action principle that one can think for an UG-like theory is the UG version of the Einstein-Hilbert action: SUG = 1 2κ2 � dD+1xωR (g) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (2) We can also add a coupling to some matter fields which need to couple to the auxiliary metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', the matter action will be of the form Sm (˜g, Φ), so that it remains Weyl-invariant (note that the matter fields are not affected by Weyl transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The equations of motion 6 of this theory are the traceless Einstein equations: Rµν(˜g) − 1 D + 1R(˜g)˜gµν = κ2 � Tµν(˜g) − 1 D + 1T(˜g)˜gµν � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (3) Upon using the Bianchi identities, they become Einstein equations with the cosmological constant entering as an integration constant [1] Rµν(˜g) − 1 2R(˜g)˜gµν + Λ˜gµν = κ2Tµν(˜g), (4) provided that ˜∇µT µν (˜g) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is clear that the Weyl invariance is trivial in the sense that its gauge fixing is trivial, we simply need to fix the volume form given by the determinant of the metric � |g| to be the background volume form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, this can be done also at the level of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The result is still a local action for the metric which does not contain any mention to the Weyl symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that sense, the resulting action is the most minimalistic action that one can conceive for a metric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If one tried to make a gauge fixing of the remaining degrees of freedom, one would end up with a non-local action for the actual physical degrees of freedom encoded in the field gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, it seems clear that both theories display the same number of local degrees of freedom of GR, except for the cosmological constant that we will analyze now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To put it in other words, leaving aside the cosmological constant, from the point of view of initial conditions, the same amount of initial data are needed to specify a solution to the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The cosmological constant in this case appears with a difference, it is an additional global degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The simplest way to see this is from the point of view of such constant being an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This means that it is a constant that parametrizes the space of solutions, which is separate from the initial data required in GR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that sense, it is a constant to be fixed by initial conditions which makes the space of solutions of UG bigger than the GR space of solutions, precisely by this cosmological constant as an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This analysis can be made much more precise by making a Hamiltonian analysis of the theory, as it has been done in [14], reaching the same conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We have concluded that UG is equivalent to GR, up to a global degree of freedom which is precisely playing the role of the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To make it more explicit, we will introduce now an additional field in GR that accounts for this global degree of freedom, to sharpen the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We need to introduce a (D +1)-form field which is the differential of 7 a D-form [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Explicitly, we want to introduce a D + 1 form F which is the differential of a D-form A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In components, this reads: Fµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD = ∇[µ0Aµµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (5) We can write down the action principle which is the Einstein-Hilbert action with an arbitrary cosmological constant and a Maxwell-like term for F , namely: S = 1 2κ2 � dD+1x√−g � −2Λ + R(g) − K (D + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='Fµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µDF µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD � , (6) where K is simply a coupling constant which can be both positive or negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The equations of motion for the F -field are ∇µ0F µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (7) In a (D + 1)-dimensional manifold, a completely antisymmetric volume form like F needs to be proportional to the ǫ pseudotensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, the equations of motion simply fixed the proportionality function to be a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD = c√−gǫµ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (8) From the point of view of the initial value problem, this constant c is precisely a global degree of freedom that needs to be fixed in terms of initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' From that point of view, it is akin to the cosmological constant in UG, since it is completely fixed in terms of the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We can sharpen the analogy by examining how does this constant c enter the equations of motion for the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The energy-momentum tensor once we evaluate the F form on shell, behaves exactly as a cosmological constant [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Assuming the existence of additional matter fields, the equations of motion for the gravitational field take the following form: Rµν(g) − 1 2R(g)gµν + Λeffgµν = κ2Tµν(g), (9) where the constant Λeff is expressed in terms of the action as Λeff = Λ + NDKc2, (10) with ND an irrelevant numerical factor depending on the spacetime dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, the cosmological constant entering the equations of motion for the metric are a combination of an initial condition c and the cosmological constant Λ entering the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 8 From a purely classical point of view, we have presented a theory akin to GR, exhibiting the whole set of diffeomorphisms as gauge symmetries and containing an additional global degree of freedom encoded in a (D + 1)-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The equations of motion for this volume form enforce that it is proportional to the Levi-Civita pseudotensor, with the proportionality constant been called here c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The constant of proportionality enters the equations of motion for the metric as an effective cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, it plays a similar role to the one played by the global degree of freedom of UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Independently of the value that we assign to the cosmological constant entering the action Λ, the resulting effective cosmological constant entering Einstein equations Λeff is given by a combination of Λ and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In terms of the initial conditions, it is possible to adjust c in order to make Λeff take any desired value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This formulation of GR with the additional (D + 1) form field is equivalent to UG, in the sense that it displays the same amount of degrees of freedom, both local and global, and the global degree of freedom plays the role of a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' At the quantum level, both formulations seem to be different from the point of view of radiative corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The reason behind this mismatch is that, whereas in UG the cos- mological constant does not receive any radiative corrections and this makes it technically natural [1, 2]1, in this formulation of GR, the cosmological constant in the action Λ does re- ceive radiative corrections, and hence it is not technically natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, the cosmological constant relevant for the dynamics is the effective one Λeff that combines the renormalized Λ with the initial value constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is possible to obtain any value for the cosmological constant Λeff independently of the potentially huge radiative corrections that Λ may receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The equivalence once quantum corrections are included into the picture is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Whether this formulation is then completely equivalent to UG at the semiclassical level is something that deserves a separate and detailed study on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Our point here was mainly to introduce a formulation within the GR setup that is close to the UG version, so that both theories can be compared easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We have made explicit the difference existing in the global degrees of freedom of UG and GR (UG contains the whole space of GR with arbitrary values of the cosmological constant coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This only difference in the two theories, will be also the only difference appearing from the point of 1 We note that technical naturalness is a definition that only applies to coupling constants appearing in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that sense it is not completely legitimate to say that in UG the cosmological constant is technically natural since it is not a coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, making an abuse of language we find it convenient to say that it is technically natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 9 view of regarding UG as the low energy EFT for massless string states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' STRING PERTURBATION THEORY IN TRIVIAL BACKGROUNDS This section contains a review of the quantization of strings in a flat background as well as the computation of string scattering amplitudes for gravitons from string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is well-known material that can be found in any textbook [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Also we think that a reader unfamiliarized with string theory might find here a quick introduction to the arguments presented in the literature leading to the conclusion that GR is the EFT describing the excitation in massless degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We find convenient to make such introduction here to expand the discussion presented in [1] about how the scattering amplitudes can be equivalently obtained from a GR and a UG-like EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The starting point of our discussion of perturbative string theory is the action describing relativistic strings propagating in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For relativistic free particles it is natural to consider the action to be the proper time of the particle trajectory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', the embedding of the worldline in the target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In the same way, for strings it is natural to consider the area swept out by the worldsheet to replace the proper time of the particle trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For that purpose, let us introduce a coordinate system in the worldsheet, a pair σa (a = 0, 1) which correspond to the time coordinate σ0 ∈ (−∞, ∞) and a spatial coordinate σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Furthermore, we will restrict our attention to closed strings (those giving rise to graviton excitations) in which the points at σ1 and σ1 + 2π are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If we endow the (D + 1) dimensional flat spacetime with coordinates Xµ, we look for an action such that the area density swept by the string is expressed in terms of derivatives of the embedding Xµ(τ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We notice that the induced metric on the worldsheet is given by hab = ηµν∂aXµ∂bXν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (11) If we take the action to be the area swept out by the string, we write down the Nambu-Goto action as SNG[X] = − 1 2πα′ � d2σ √ −h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (12) The constant α′ represents the string tension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', the energy density per unit length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Al- though this action is perfectly reasonable classically, from the point of view of quantization 10 is problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is because it is not quadratic in its variables: we have a square root appearing explicitly in the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To circumvent this problem, one can work with the Polyakov action, which is given by SP[X, γ] = − 1 4πα′ � d2σ√−γγab∂aXµ∂bXνηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (13) In this action, there is an additional configuration variable γab which is a metric in the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Now, this action is clearly quadratic in the Xµ variables over which we will path-integrate to quantize the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To see the equivalence among these two actions, we can compute the equations of motion for the γab variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, following the standard conventions, we can define a two-dimensional energy-momentum tensor as the variation of the Polyakov action with respect to the worldsheet metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' γab: Tab = − 1 √−γ δSP δγab = 1 4πα′ � ∂aXµ∂bXν − 1 2γabγad∂cXµ∂dXν � ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (14) The Polyakov action does not contain any derivatives of the metric γab, and hence the equations of motion for the metric can be regarded as a constraint Tab = 0 (as a consequence, strictly speaking it is not a dynamical variable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, this constraint can be used to solve γab in terms of the Xµ variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' When we plug the result back into the Polyakov action, we find the Nambu-Goto action we began with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is worth pausing at this point and discussing the continuous symmetries of the theory: Poincar´e invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is a global symmetry on the worldsheet Xµ → Λµ νXν + cµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (15) Reparametrization invariance or diffeomorphism invariance in the worldsheet σa → ˜σa(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Whereas the Xµ fields transform as worldsheet scalars, γab transforms as a two-index covariant tensor: Xµ(σ) → Xµ(˜σ) = Xµ(σ), (16) γab(σ) → ˜γab(˜σ) = ∂σc ∂˜σa ∂σd ∂˜σb γcd(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (17) Weyl invariance of the worldsheet metric γab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This transformation leaves invariant the Xµ coordinates and the metric gets a local rescaling Xµ(σ) → Xµ(σ), (18) γab(σ) → e2φ(σ)γab(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (19) 11 We can distinguish now between oriented and unoriented strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The former have a well defined transformation law under the parity transformation σ1 → 2π − σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will focus on the unoriented strings for the sake of simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Not all the symmetries that we have introduced are directly preserved through the process of quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, the Weyl symmetry is anomalous, as it is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, in this case the Weyl symmetry is a gauge symmetry that we must insist on preserving at the quantum level to remove unphysical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will further discuss this point later when we deal with strings in general backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For the time being, let us focus on the quantization of the theory through a path-integral procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Let us illustrate the quantization of the theory through a path-integral procedure as well as the spectrum that the theory displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Let us define the generating functional following the usual Faddeev-Popov procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' First of all, we would write down the action in Euclidean space, in order to make the quantization procedure sensible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We write down the generating functional as Z = 1 V (gauge) � DγDXe−SP [X,γ], (20) where V (gauge) represents the volume of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We recall that we have the Weyl rescalings of the metric and diffeomorphisms as gauge symmetries of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, we need to avoid counting more than once physical configurations and that is the reason for taking the quotient by the volume of the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As usual, we will introduce a Faddeev-Popov determinant ∆F P[γ] to take this volume into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The integral over the gauge orbits cancels with the volume of the gauge group and we reach the expression for the generating functional which is Z[γ] = � DX∆F P[γ]e−SP [X,γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (21) Choosing a convenient normalization for the action, we can rewrite the Faddeev-Popov determinant as ∆F P[γ] = � DbDce−Sg[b,c], (22) where b and c are ghosts Grassman-values that anticommute and Sg = 1 2π � d2σ√γbab∇acb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (23) 12 At this point, we have reduced the evaluation of the path integral for the bosonic string theory to the evaluation of the path integral: Z = � DbDcDXe−SP [γ,X]−Sg[γ,b,c], (24) which is the CFT of D + 1 scalar fields (the Xµ) and the bc-ghost system [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If the theory is going to preserve the Weyl invariance, we need the theory to have a total zero central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is precisely the consistency condition that we mentioned would appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Weyl invariance means that the trace of the two-dimensional energy momentum tensor needs to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In two-dimensions, the trace of the energy-momentum tensor is determined by the central charge and the trace anomaly ⟨T a a⟩ = − c 12R [γ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (25) The system of the Xµ-scalars and the bc-ghost system is linear, and hence the total central charge is the sum of the central charges of the two systems independently: c = cg + cX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (26) The bc-ghost system [17] has a central charge cg = −26 while each scalar field gives a contribution of 1 to the central charge cX = D + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Ensuring Weyl-invariance means that we need the spacetime dimension to be 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is the well-known way in which the critical dimension of bosonic string theory emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Now that we have ensured how to preserve the gauge invariance at the quantum level in order to make the theory consistent, it is time to talk about the spectrum of the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Our point is simply to illustrate that the spectrum of the closed unoriented bosonic string contains a tachyon, a dilaton, and a graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, for this purpose, we can skip the detailed BRST analysis and focus only on the states generated by the X-fields which are the “physical fields”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In order to characterize the spectrum, the simplest way to do it is to use the so called state-operator map for CFTs [20, 21], in which states are replaced by operator insertions that generate them by acting in a neighbourhood of the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For this purpose, it is first easier to use complex coordinates σ → (z, ¯z) on the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Furthermore, we now need the operators to be gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The diffeomorphism invariance can be ensured by integrating local operators O(z, ¯z) over the worldsheet, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' constructing operators of the 13 form V = � d2zO(z, ¯z), (27) with V standing for vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Weyl invariance is ensured by choosing the operators O to transform adequately under Weyl rescalings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', having a suitable weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The measure of integration, d2z has a conformal weight (−1, −1) under such rescalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, O needs to be a primary operator of the CFT with weight (+1, +1) to compensate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The kind of operators that give rise to the lowest energy states of the string are eip·X and Pµν∂Xµ∂Xνeip·X, with p a given momentum that we endow the string with and Pµν the polarization tensor [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The operator eip·X gives rise to the tachyon, since we need to impose that −p2 = −4/α′ < 0 for the operator to be Weyl invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The operator Pµν∂Xµ∂Xνeip·X corresponds to the dilaton (pure trace part of Pµν) and the symmetric part of Pµν gives rise to the graviton, since p2 = 0 (massless condition) and pµPµν = 0 (transverse condition) needs to be imposed to ensure the Weyl invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The antisymmetric part does not appear for unoriented strings since it corresponds to the Kalb-Ramond excitation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Up to this point, we have analyzed the spectrum of the closed unoriented bosonic string theory and found that the massless states correspond to the dilaton and the graviton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The Polyakov action per se does not give rise to interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will now make a small digression on how interactions among the massless states arise in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' There is a term that we can add to the Polyakov action which is an Einstein-Hilbert term that is purely topological in two-dimensions Sint = λ 4π � d2σ√γR(γ) = 2λ(1 − g), (28) being g the genus of the worldsheet and λ a coupling constant which we assume to be small in order to do perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, if we add this term to the string action, we will get Z = � topologies � DXDγe−SP −Sint = ∞ � g=0 e−2λ(1−g) � DXDγe−SP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (29) If we call eλ = gs, as it is common, this gives a good expansion as long as gs ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The whole series is known to be a divergent series as the standard perturbative series in QFT [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In addition to this problem, there is a harder problem which is the finiteness of each of the terms in the series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', the path integral over the different geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For a fixed topology, 14 the path integral with the Polyakov action requires to compute a sum over the moduli of conformally inequivalent surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In general, for higher loop orders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' non-trivial topologies) this requires to perform an integral over a moduli space that is not obviously convergent, although some results in the literature point toward its finiteness [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Now it comes to the point of computing some observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The observable to compute in string theory is the string S-matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This means, we plug some “in” state of the free string spectrum and compute the probability amplitude of generating another “out” state of free string spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' These states are generated by introducing their corresponding vertex operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For our purposes of analyzing how GR or UG might emerge from string theory, we are interested in computing the scattering amplitude involving m gravitons with momenta pi and polarization tensors ei which we represent as A(m)(p1, e1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' p2, e2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='pm, em).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is computed as a suitable path integral for the Polyakov action SP that schematically reads [17, 18] A(m)(1h1, 2h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', mhm) = 1 g2s 1 Vgauge � DXDg e−SP[X,g] m � i=1 Vi(pi, hi), (30) where Vi represents the vertex operator associated with a graviton insertion with a given spin and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To begin with, we particularize the amplitude for three gravitons and we find A(p1, e1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' p2, e2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' p3, e3) = igs(α′)6 2 (2π)26δ26 (p1 + p2 + p3) e1µνe2αβe3γδT µαγT νβδ, (31) where T µαγ = pµ 23ηαγ + pα 31ηγµ + pγ 12ηµα + α′ 8 pµ 23pα 31pγ 12, (32) pµ ij = pµ i − pµ j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The terms of order O(α′) in T µαγ contribute as O(p4) to the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If we focus just on the lowest order terms O(p2), this amplitude is equivalent to the ones computed at tree level from the Einstein-Hilbert action upon the identification κ = gs(α′)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The same agreement is found with amplitudes involving an arbitrary number of gravitons: if we neglect the higher-order contribution from the string amplitude, they agree with those computed from the Einstein-Hilbert action [17, 18], with the same identification of κ and the string constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As it has been already discussed in the literature [6, 7], the tree-level scattering amplitudes of gravitons computed in GR and UG are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, from the point of view of 15 scattering amplitudes, string theory does not point toward GR in a univocal way: both UG and GR are equivalent from a low-energy effective field theory point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This result was already advanced in [1] and we have reproduced here the analysis in more detail for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We will come back to this analysis later, when we introduce the low energy EFTs for the massless degrees of freedom of the string: both the UG and the GR-like actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' STRINGS IN GENERAL BACKGROUNDS Up to now, we have only considered strings propagating in flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, the spectrum of the strings contains some excitations which typically interact among themselves and could lead to the generation of a non-trivial background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In particular, it contains a graviton and, necessarily, gravitons need to interact gravitationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' At low energies, all the excitations that matter are the massless ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In the same way a laser is a coherent state of photons, we expect that a coherent state of gravitons might look like a curved background and a string propagating on top of it needs to be described appropiately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The same comment applies to the dilaton field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As such, we can write down the most general renormalizable action including those fields, which is the following non-linear σ-model S[X, γ] = SP[X, γ] + SD[X, γ] = − 1 4πα′ � d2σ√−γ � γabGµν(X)∂aXµ∂bXν + α′R (γ) Φ(X) � , (33) where Gµν(X) represents a metric (graviton excitations), Φ(X) represents the dilaton back- ground field, and R[γ] represents the Ricci-scalar of the two-dimensional metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This term breaks explicitly the Weyl invariance in the worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This term is of a higher dimension than the Weyl-invariant terms, and it does not require to be normalized with a dimensionful constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In virtue of the expansion in α′ that we will perform, we will cancel the tree-level contribution to the anomaly of this last term with the one-loop contribution of the classically Weyl-invariant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The result of this procedure is a reasonable effective field theory for the massless degrees of freedom of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' There are two missing terms that still give rise to a renormalizable theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The first of these terms is the coupling to the Kalb-Ramond field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, if we focus on unoriented strings, we can skip it since the divergences of the rest of the terms do not require this term 16 to be renormalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In case we deal with oriented strings, this term gives a contribution to the conformal anomaly [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The additional term that we can add to the action corresponds to a coupling to the background tachyon field T(X) ST = 1 4π � d2σ√−γT (X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (34) In principle this term is needed to cancel some of the quadratic divergences arising from vacuum to vacuum diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, if we use a renormalization scheme such that those divergences are absent (for example, dimensional regularization), we can safely skip those terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, we will work with a renormalization scheme fullfilling this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fur- thermore, it is worth mentioning that supersymmetry in the worldsheet ensures that those quadratic divergences are absent in superstrings due to the characteristic cancellation among fermionic and bosonic degrees of freedom, with independence of the renormalization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Determination of the Weyl anomaly Anomalies always appear when there are two symmetries that the theory displays at the classical level, but it is not possible to quantize such theory preserving both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This means, there is a trade-off between the two symmetries and it is only possible to preserve one of them in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For example, the chiral anomaly is a trade-off between the vector and axial currents for massless fermion fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If we use a regularization procedure which automatically preserves one of those currents, then straightforwardly the other current will be anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In the case of the chiral anomaly, it is standard to use a regularization scheme that preserves gauge invariance and hence yields to the conservation of the vector current, leading to an anomalous axial current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In the case of Weyl invariance for strings, we are using a regularization scheme that preserves diffeomorphism invariance, while the Weyl symmetry becomes potentially anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We need to ensure that the non-linear sigma model is chosen in such a way that it gives rise to a Weyl-invariant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In a language closer to particle physics, this means that we need to choose our theory in such a way that we cancel the potential gauge anomalies, which in this case corresponds to choosing the background fields in such a way that the theory is not Weyl-anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In the case of the Standard Model, since it corresponds to a chiral gauge theory, arbitrary matter fields 17 would lead to an anomalous theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' However, the matter content is such that the potential anomaly is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is precisely what we have done in the previous section to fix the target space dimension to be 26;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' otherwise, the Weyl-symmetry becomes anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this case, we expect constraints also on the background fields entering the non-linear sigma models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', constraints that the Gµν(X) and the Φ(X) fields need to obey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We want now to write down the most general form that the Weyl anomaly can display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Following D’Hoker [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' it is possible to show that the structure of the anomaly for unoriented strings in a curved background needs to be of the form ⟨T a a ⟩ = βG µν(X)∂aXµ∂bXνγab + βΦ (X) R (γ) + βV µ (X)gabD∗ a∂bXµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (35) where D∗ a represents the covariant derivative on the product space of the cotangent space of the worldsheet and the tangent space of the target space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' and it can be explicitly written down as D∗ a∂bXµ = ∂a∂bXµ − Γc ab∂cXµ + Γµ νρ∂aXν∂bXρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (36) where Γc ab are the Christoffel symbols of the metric γab and Γµ νρ represent the Christoffel symbols of the metric Gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The last term in the Weyl anomaly, βV can be removed through a transformation on the Xµ fields, since we are always able to perform a local transformation on the Xµ fields at the same time that we perform a Weyl-rescaling of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This leaves only two independent β functionals: βG and βΦ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence we need to determine the β functionals obtained from the action (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We want to study perturbatively this action order by order in the α′ expansion, which is done by assuming that the background fields Gµν(X), Φ(X) vary smoothly with respect to the scale α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' It is conventional to do the computations in the background field formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this for- malism, we decompose the fields Xµ in a background part Xµ 0 and its quantum fluctuations Y µ Xµ (σ) = Xµ 0 (σ) + Y µ (σ) , (37) where the integration is now performed with respect to the quantum fluctuations instead of Xµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We define the effective action Γ[X0, g] following [25] as e−Γ[X0,g] = � DY e − � S(X0,Y )−S(X0)−� d2σY µ(σ) δS δXµ 0 � , (38) 2 For oriented strings there will be another β-functional associated with the Kalb-Ramond field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 18 which is the generating functional of the Feynman diagrams relevant for the computation of the β-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' At this point, it is better to pause and mention a crucial step in the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The coordinate difference does not transform in a covariant way under changes of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, in order to obtain results that are manifestly covariant, it is better to do the com- putation in variables that are manifestly covariant at intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This can be done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Imagine that the coordinates Xµ 0 correspond to a given point p0 and the coor- dinates Xµ = Xµ 0 + Y µ to a point p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If both points are close enough, there exists only one geodesic with respect to Gµν connecting both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, we can replace the coordinate difference Y µ which characterizes the point p by the tangent vector tµ of the geodesic at the point p0, which transforms covariantly under changes of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, it is better to use this vector as the integration variable in the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In fact, we can use this tangent vector tµ to perform a covariant Taylor expansion based on Xµ 0 of an arbitrary tensor living in the target manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To put it explicitly, any tensor Tµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µn(X) can be expanded as Tµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µn(X0 + t) = ∞ � k=0 T (k) µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µnν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='νk(X0)tν1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' tνk, (39) where each of the terms T (k) µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µnν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='νk is a combination of covariant derivatives of the tensor Tµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='µn and contractions with curvature tensors evaluated at X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This expansion can be achieved with the help of the normal coordinate expansion although we emphasize that it remains valid in an arbitrary coordinate system since it is a tensor expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We are inter- ested in the expansion of the tensors Gµν, Φ(X) (the latter is a trivial tensor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' a scalar), and the object ∂a (Xµ 0 + Y µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' These expansions can be obtained after a straightforward computation, see [25] for details: Gµν(X) = Gµν(X0) + 1 3Rµρσνtρtσ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', (40) Φ(X) = Φ(X0) + ∇µΦ(X0)tµ + 1 2∇µ∇νΦ(X0)tµtν + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', (41) ∂a (Xµ 0 + Y µ) = ∂aXµ 0 + ∇atµ + 1 3Rµ νρσ∂aXσ 0 tνtρ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', (42) where Rµ νρσ represents the Riemann tensor associated with Gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We are not ready to perform the diagrammatic computation yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' There is a problem arising from the fact that the term that gives us the propagator for the quantum fields over 19 which we integrate, tµ, contains an arbitrary metric in front of it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' we need to invert a term that looks like Gµν(X0)∇atµ∇btν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The way to deal with this problem and obtain a simple propagator is to introduce a vielbein eA µ(X0) which fulfills the property eA µ(X0)eB ν(X0)ηAB = Gµν(X0), (43) with ηAB a Lorentzian metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In this way, we can rewrite all the vector expressions in the non-holonomic basis eA µ and get a trivial propagator for the tA = eA µtµ fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This comes with a subtlety, because now the derivatives ∇a involve the spin-connection of the spacetime ω AB µ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' for example, ∇atA = ∂atA + ω AB µ ∂aXµ 0 tCηBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (44) Obtaining a trivial propagator means breaking the SO(D, 1) invariance that the theory displays, but since we are working in a formalism that is explicitly gauge covariant, we automatically know that there will always be contributions in the diagrammatic expansion that make the theory explicitly gauge covariant in intermediate steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Up to this point, collecting all the information, we have performed the following expansion for the Polyakov piece of the action: SP = SP[X0] + 1 2πα′ � d2σ√γγabGµν(X0)∂aXµ 0 ∇btν (45) + 1 4πα′ � d2σ√γγab � ηAB∇atA∇btB� (46) + 1 3πα′ � d2σ√γγabRµABC∂aXµ 0 tAtB∇btC (47) + 1 12πα′ � d2σ√γγabRABCDtBtC∇a∇atA∇btD, (48) and for the dilaton part we have the trivial structure: SD[X0 + t] =SD[X0] − 1 8π � d2σ√γ∇AΦ(X0)tA (49) − 1 16π � d2σ√γ∇A∇BΦ(X0)tAtB + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (50) We recall that we can safely impose the equations of motion for the classical fields and safely drop the linear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This is tantamount to a legitimate field redefinition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Now we can determine the trace anomaly, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (35) from the effective action introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The computation requires to go to the next higher order in loops in the dilaton field, 20 since the piece of the action for the dilaton field α′ comes with an additional α′ with respect to the other field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The computation is rather lengthy and hence we do not reproduce it here [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We simply write down the result as βG µν = Rµν (G) − ∇µ∇νΦ + O(α′), (51) βΦ = D − 26 6 + α′ � −R (G) + 2∇2Φ + (∇Φ)2� + O � α′2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (52) A comment is in order now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If we are dealing with a flat worldsheet, the vanishing of βG is enough to ensure the Weyl invariance at the quantum level, as long as we are working in D = 26 dimensions, the critical dimension (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, in principle, we expect that the same applies to non-flat worldsheets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' that the condition βΦ = 0 is not independent of βG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Actually, we have a non-trivial constraint coming from the Bianchi identity ∇µ � Rµν (G) − 1 2R (G) Gµν � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (53) This ensures that we have to the computed order the following identity whenever βG µν = 0 ∇µβG µν = ∇νβΦ = 0, (54) as can be seen by direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This implies that βΦ is a constant as long as βG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' By continuity, this automatically implies at this level that βΦ = 0 for D = 26 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' From now on we will restrict ourselves to work in D = 26 and make a comment on strings on non-critical dimension later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Including string-loop corrections At this point, we have only focused on the zeroth-order in the gs-expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Although it is clear that string loops should modify the results, it is not completely clear how those corrections must be included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' One of the most accepted proposals is the Fischler-Susskind approach [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The idea behind such mechanism is that string loop divergences can be absorbed through a renormalization of the background fields in the non-linear sigma models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Let us illustrate this explicitly for unoriented closed bosonic strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For the purpose of this section, it is simpler to work with a a sharp cut-off as regularization scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The divergences in string loops appear when we have to sum over conformally inequivalent surfaces of a fixed topology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' genus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For a fixed but arbitrary topology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' we focus 21 here on non-trivial topologies), this sum is an integral over a finite-dimensional parameter space, the so-called Teichm¨uller space [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' These integrals are divergent, but these divergences arise from handles that shrink to zero size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' These divergences are equivalent to the divergences coming from inserting a local operator on the trivial-genus worldsheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In a flat spacetime, the divergence appearing for the torus topology can be eliminated through the insertion of an operator log Λ 2π γabηµν∂aXµ∂bXν, with a suitable coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Here Λ is a suitable cut-off in the Teichm¨uller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If we move to a curved geometry Gµν with a non-trivial zero mode of the dilaton field Φ, we need to substitute the metric Gµν and include a relative factor e−Φ to account for the dependence of the path integral on the topology of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We recall that the asymptotic value of the dilaton field λ = ⟨Φ⟩ is identified with the string coupling constant gs = eλ through an exponential relation, as it can be seen by comparison of the actions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (33) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (28) [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Explicitly for the first non-trivial order (torus topology) we have the following divergences: δSloop = log Λ 2π � d2σ√−γγabe−ΦGµν(X)∂aXµ∂bXν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (55) The e−Φ factor ensures that, when evaluated on the trivial topology on the worldsheet, it captures the divergences in the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' If the dilaton field displays a non-trivial background profile Φ(X), not only a zero mode λ, we expect that replacing Φ with Φ(X) would lead to a first term in an α′ expansion of the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This term modifies the β-functional (we will refer from now on to those β-functionals modified due to the the presence of string loop corrections as ˜β) associated with the metric through the addition of a term δβG µν to the functional βG µν above ˜βG µν = βG µν + δβG µν, (56) which looks like a cosmological constant term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' δβG µν = Ce−ΦGµν, (57) where C is an arbitrary constant that arises in the renormalization procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' On equal footing, an additional contribution to the dilaton, which we call δβΦ will also appear, al- though it is hard to evaluate explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Instead, it is easier to obtain it by applying a consistency argument [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As we have argued above, in principle the vanishing of the 22 modified ˜βΦ-functional through string loop corrections is not independent of the vanishing of the ˜βG µν functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' As we have seen, in the CFT computation, it being constant is pre- cisely a consequence of the vanishing of the remaining β-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' By this consistency condition, it is possible to derive an equation for the ˜βΦ-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Taking the divergence of the ˜βG µν and simplifying it through Bianchi identities and using also the vanishing of ˜βG µν itself, we find: ∇µ ˜βG µν = ∇ν �1 2R (G) − ∇2Φ − 1 2 (∇Φ)2 � (58) This leads us to the following ˜βΦ functional for the dilaton field: ˜ βΦ = α′ � −R (G) + 2∇2Φ + 1 2 (∇Φ)2 � , (59) which knowing that is a constant, can be safely chosen to be equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In case that we were dealing with strings in non-critical dimension, an additional D − 26/6 factor should be included arising from the bc-ghost system contribution to the Weyl-anomaly at the string tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Notice that we have introduced α′ as a dimensionful parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Once we have reached this point, it is better to pause and recapitulate what we have done until now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We began analyzing the α′-expansion of the sigma model describing the propagation of strings in arbitrary backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We determined the β-functionals of the Weyl anomaly to the lowest order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Then we jumped into the problem of including string-loop corrections that should clearly modify the constraints that the background fields should obey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For the purpose of including such corrections, we noticed that the divergences arising from the string loops can be absorbed into a renormalization of the background fields Gµν and Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hence, up to this point we have found a set of equations that these background fields need to obey for the consistent propagation of the strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' EFTs for the theory The consistency equations that we found arising from the Weyl anomaly cancellation and the cancellation of the divergences from string loop corrections resemble a lot the equations of motion of a given field theory for Gµ(X) and Φ(X): ˜βG µν = Rµν (G) − ∇µ∇νΦ + Ce−ΦGµν + O(α′), (60) ˜βΦ = D − 26 6 + α′ � −R (G) + 2∇2Φ + (∇Φ)2� + O � α′2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (61) 23 Setting C = 0 corresponds to omitting the string loop corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The natural question is then whether it is possible to obtain an effective action whose dynamics correctly reproduce these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In addition, such effective action needs to correctly account for the scatter- ing amplitudes involving only massless excitations of the string (to the lowest order in the α′ expansion) in order to be a sensible action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' There are (at least) two effective actions that fullfill these criteria: match the scattering amplitudes involving gravitons and dilatons and their equations of motion give rise to the β-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' These two actions correspond to a GR-like EFT and a UG-like EFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' The GR-like EFT can be given as: SGR eff = 1 2κ2 � dD+1X √ −GeΦ � −(D − 26) 6α′ − 2Ce−Φ + R (G) + (∇Φ)2 � + O(α′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (62) From this action principle it is straightforward to obtain the β-functionals as ˜βΦ = −2κ2 e−Φ √ −G δSGR eff δΦ , (63) ˜βG µν = 2κ2 e−Φ √ −G �δSGR eff δGµν + 1 2Gµν δSGR eff δΦ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (64) Furthermore, it is possible to perform a field redefinition to map this action to the Einstein Frame [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Following [1] we know that it is also possible to write down an action principle which reproduces the same equations of motion that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (62) displays, with the cosmological con- stant C entering as an integration constant instead of a coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To be concrete, we can write down the following action principle: SUG eff = 1 2κ2 � dD+1XωeΦ � −(D − 26) 6α′ + R( ˜ G) + ( ˜∇Φ)2 � + O(α′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (65) If we compute the variation with respect to Gµν we obtain the traceless version of the equations obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Explicitly, if we define δSUG eff δGµν = Kµν (G) − 1 2K (G) , (66) for the variation of SUG eff we obtain the following: δSUG eff δGµν = Kµν( ˜ G) − 1 D + 1K( ˜ G) ˜Gµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (67) with K( ˜ G) = ˜GµνKµν( ˜ G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Upon taking the divergence and using the generalized Bianchi identities for the corresponding tensor K entering the equations (see [1] for further details) 24 we find: Eµν = Kµν( ˜ G) − 1 2K( ˜ G) ˜Gµν + C ˜Gµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' (68) Again, a suitable combination of these equations with the equation obtained from the equa- tion of motion for Φ we find: ˜βΦ = −2κ2e−Φ ω δSUG eff δΦ , (69) ˜βG µν = 2κ2e−Φ ω � Eµν + 1 2 ˜Gµν δSeff δΦ � , (70) confirming our claim that the Unimodular Gravity action (65) reproduces the β-functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Notice that this effective action does not only reproduce the β-functionals but it also repro- duces all of the scattering amplitudes involving massless excitations of the string (graviton and dilaton asymptotic states), as derived following the procedure sketched in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In that sense, both actions reproduce the desired properties and hence none of them is preferred over the other one from the perspective of using them as EFTs for the massless modes of the string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' CONCLUSIONS We have analyzed the embedding of UG in string theory from the point of view of the consistent quantization of the strings in an arbitrary background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Furthermore, we have followed the proposal by Susskind and Fischler towards cancelling divergences arising from string loops with suitable counterterms in the non-linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Our analysis here does not unveil any preference for UG or GR as a low energy description of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' This ties up the loose ends that were not analyzed in [1], regarding the embedding of UG in string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' To put it explicitly: both UG and GR are equally valid as low energy descriptions of the massless modes of string theory and none of them seems to be preferred over the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Regarding future directions of work, we recall that our analysis here has focused on bosonic string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' At first sight, the extension to superstring theory seems straightfor- ward although subtleties may arise in a careful study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Previous considerations of supergrav- ity in a UG-like context suggest that some of the vacua may spontenously break SUSY and hence both theories may develop a potential inequivalence at the quantum level [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 25 Although there is no analysis of the global degrees of freedom in such contexts, it should be mentioned that it seems possible that a careful implementation of SUSY in that contexts requires also from a fermionic global degree of freedom, which is the responsible for the apparent SUSY-breaking presented there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' A second direction of work that is worthwhile exploring is that of non-perturbative defini- tions of string theory and its interplay with UG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' For instance, the gauge/gravity correspon- dence (also called usually AdS/CFT) [29–31] and matrix models [32], among them probably we could highlight the BFSS matrix model [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' In such contexts, we have not explored whether it is easy or not to accomodate a UG principle instead of a GR principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank Carlos Barcel´o and Ra´ul Carballo-Rubio for collab- oration in early stages of this project and invaluable discussions during the preparation of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' We would also like to thank Tom´as Ort´ın for helpful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Financial support was provided by the Spanish Government through the projects PID2020- 118159GB-C43, PID2020-118159GB-C44, and by the Junta de Andaluc´ıa through the project FQM219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' GGM acknowledges financial support from the grant CEX2021-001131-S funded by MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' GGM is funded by the Spanish Government fellowship FPU20/01684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 26 [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Carballo-Rubio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Garay, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Garc´ıa-Moreno, (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='08499 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Carballo-Rubio, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' D 91, 124071 (2015), arXiv:1502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='05278 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [3] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' ’t Hooft, NATO Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 59, 135 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Burgess, Introduction to Effective Field Theory (Cambridge University Press, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Delhom, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Garc´ıa-Moreno, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Hohmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Jim´enez-Cano, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Koivisto, (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='13056 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' ´Alvarez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Gonzalez-Martin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Mart´ın, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' C 76, 554 (2016), arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='02667 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Carballo-Rubio, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Di Filippo, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Moynihan, JCAP 10, 030 (2019), arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='08192 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fischler and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Susskind, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 171, 383 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [9] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fischler and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Susskind, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 173, 262 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fischler, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Klebanov, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Susskind, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 306, 271 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Weinberg, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 61, 1 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Misner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Thorne, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Wheeler, Gravitation (W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Freeman, San Francisco, 1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Will, Theory and Experiment in Gravitational Physics (Cambridge University Press, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Henneaux and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Teitelboim, Physics Letters B 222, 195 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Henneaux and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Teitelboim, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 143, 415 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Mottola, JHEP 11, 037 (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='04703 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Polchinski, String theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 1: An introduction to the bosonic string, Cambridge Mono- graphs on Mathematical Physics (Cambridge University Press, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [18] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Green, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Schwarz, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Witten, Superstring Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 1: Introduction, Cam- bridge Monographs on Mathematical Physics (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Nakayama, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 569, 1 (2015), arXiv:1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='0884 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Qualls, (2015), arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='04074 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [21] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Di Francesco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Mathieu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics (Springer-Verlag, New York, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 27 [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Gross and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Periwal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 60, 2105 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Mandelstam, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' B 277, 82 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [24] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Deligne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Etingof, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Freed, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Jeffrey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Kazhdan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Morgan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Morrison, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Witten, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=', Quantum fields and strings: A course for mathematicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 1, 2 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Callan, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Thorlacius, in Theoretical Advanced Study Institute in Elementary Particle Physics: Particles, Strings and Supernovae (TASI 88) (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Anero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Martin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Santos-Garcia, JHEP 01, 145 (2020), arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='04160 [hep- th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Anero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Martin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Santos-Garcia, JCAP 03, 006 (2020), arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='05365 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Bansal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Nagy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Padilla, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Zavala, JHEP 01, 146 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='13758 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Maldacena, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 2, 231 (1998), arXiv:hep-th/9711200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [30] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Aharony, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Gubser, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Maldacena, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Ooguri, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Oz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 323, 183 (2000), arXiv:hep-th/9905111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [31] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Polchinski, in Theoretical Advanced Study Institute in Elementary Particle Physics: String theory and its Applications: From meV to the Planck Scale (2010) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 3–46, arXiv:1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content='6134 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Ginsparg and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Moore, in Theoretical Advanced Study Institute (TASI 92): From Black Holes and Strings to Particles (1993) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 277–469, arXiv:hep-th/9304011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' [33] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Banks, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Fischler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Shenker, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Susskind, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' D 55, 5112 (1997), arXiv:hep-th/9610043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} +page_content=' 28' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQf4wUX/content/2301.03503v1.pdf'} diff --git a/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/2301.13603v1.pdf.txt b/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/2301.13603v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ce0e3db9e10f044afce76be0953a42d50127f7c5 --- /dev/null +++ b/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/2301.13603v1.pdf.txt @@ -0,0 +1,1291 @@ +arXiv:2301.13603v1 [math.LO] 31 Jan 2023 +Limits of structures and Total NP Search Problems∗ +Ondřej Ježil +ondrej.jezil@email.cz +Faculty of Mathematics and Physics, Charles University† +Abstract +For a class of finite graphs, we define a limit object relative to some computation- +ally restricted class of functions. The properties of the limit object then reflect how +a computationally restricted viewer “sees” a generic instance from the class. The +construction uses Krajíček’s forcing with random variables [7]. We prove sufficient +conditions for universal and existential sentences to be valid in the limit, provide sev- +eral examples, and prove that such a limit object can then be expanded to a model +of weak arithmetic. We then take the limit of all finite pointed paths to obtain a +model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the +complete problem for PPA) is not. This can be viewed as a logical separation of +the oracle classes of total NP search problems, which in our setting implies standard +nonreducibility of Leaf to OntoWeakPigeon. +1 +Introduction +There exist several logical constructions of limits of classes of finite structures such as +the ultraproduct and the compactness theorem. The latter was used in [2] to prove the +0–1 law for structures over relational vocabularies. +In combinatorics there are also several notions of limits of finite graphs. For example +the dense graph limit defined for a sequence of graphs {Gk}k>0 satisfying the condition +that +t(F, Gn) = |hom(F, G)| +|Gn||F | +, +converges for every fixed connected graph F, where hom(F, G) denotes the set of all graph +homomorphisms from F to G. This provided a framework (see [8]) to restate and find new +proofs for results in extremal graph theory — for instance Goodman’s theorem relating +the number of edges to the number of triangles in a graph. There are other notions of +limits of sequences of graphs, and we refer the interested reader to [10]. Another recent +use of limit objects for the results of extremal combinatorics was by Razborov in [11]. +∗ This work has been supported by Charles University Research Center program No.UNCE/SCI/022. +† Sokolovská 83, Prague, 186 75, The Czech Republic +1 + +In this work, we define a new construction of a limit object. Given a class of finite +graphs G, whose vertex sets are initial segments of N, we can stratify it into the sequence +of sets {Gk}∞ +k=1 as follows +Gk = {G ∈ G; G has {0, . . . , k − 1} as its vertex set}. +Our construction would yield a pseudofinite structure if limk→∞|Gk| = 1, but an +ordinary application of the compactness theorem suffices for that, we therefore generally +care about the case, where limk→∞|Gk| = ∞.1 We call such a sequence of sets of graphs +a wide sequence and the limit object its wide limit. +Let F be a class of functions with some computational restrictions, for example take +F to be the set of functions computed by decision trees of some small depth. We define +the wide limit denoted limF Gn, where n is a technical parameter to be defined later. +The wide limit limF Gn is a Boolean-valued graph2 — its edge relation does not only +permit the truth values 0 and 1 but also many other values from some infinite complete +Boolean algebra B. This algebra is in fact also a σ-algebra with a measure µ on it, +so to any statement formulated as a first order sentence ϕ we can assign a real number +µ([[ϕ]]) ∈ [0, 1] which measures how far is the truth value of ϕ (denoted [[ϕ]]) from the value +0. The key method we use is arithmetical forcing with random variables, developed in [7], +which allows us to construct models of (weak) arithmetical theories and by restricting to +a language of graphs gives us Boolean-valued graphs. In these Boolean-valued graphs, +validity of existential quantifiers corresponds to the ability of F to solve search problems +over the class of graphs we are considering. +Our limit object can be expanded to the original model Krajíček’s method would +otherwise construct. We prove (Theorem 5.8) that the truth values of first order sentences +concerning the object are preserved even when evaluated in the model of arithmetic +relativized to the wide limit (under a mild condition on the family F). +As an application of this construction, we take the limit of all finite paths starting +at the vertex 0 relative to the class of functions computed by oracle trees of subex- +ponential depth and obtain the Boolean-valued graph limFnb ∗PATHn which is an infi- +nite path with only one endpoint. This object is then expanded to a Boolean-valued +model of weak second order arithmetic K(∗PATHn, Fnb, Gnb) in which every instance of +OntoWeakPigeon has a solution. However, the object limFnb ∗PATHn in the model +K(∗PATHn, Fnb, Gnb) is an instance of the PPA-complete problem Leaf which does not +have a solution. This can be seen as a logical analogue of an oracle separation of these +two classes, which is known to hold3. We then show the result implies a separation of +those classes under stronger notion of reducibility. +1The case where the limit tends to some other positive number results in a structure which after +collapsing to a two-valued boolean algebra becomes pseudofinite too. +2Generally, we can do this with any L-structures for some first order language L. The limit object is +then a Boolean-valued L-structure limF Gn. In this work we restrict ourselves to the language of graphs +L = {E} to simplify the presentation. +3OntoWeakPigeon can be reduced to WeakPigeon which is known to be in PPP [6] and it is +known [1] that Leaf cannot be reduced to any problem in PPP. +2 + +There is already an established connection between complexity of search problems and +logic (namely bounded arithmetic, see [4]). The model we construct is not known nor ex- +pected to be a model of any theory which has been considered under these investigations. +However, we show that open induction and open comprehension is valid in this model, +and thus we show these principles along with the principle that OntoWeakPigeon is +total cannot prove that the problem Leaf is total. The way the model is constructed +also implies nonreducibility from Leaf to OntoWeakPigeon for subexponential time +oracle machines. Moreover, one can at least in theory tweak our construction (e.g. by +extending the family Fnb) to obtain a model of a stronger theory. This has been success- +fully done for several models already in [7, Chapter 10, Chapter 14, Chapter 21] using +the switching lemma. +2 +Preliminaries +By graphs we mean structures in a language with a single binary relation denoted E +which is antireflexive and possibly symmetric if the graph in question is undirected. We +will denote any particular graph by ω as it will be used in some sense as a sample of a +discrete probability space. The edge relation of a particular graph ω will be denoted Eω. +In the rest of this section we recall notions needed for Krajíček’s forcing construc- +tion. Fundamental notion we use throughout the work is of nonstandard models of (true) +arithmetic. Let Lall be the language containing the names of all relations and functions +on the natural numbers and let ThLall(N) denote the set of true sentences in this lan- +guage. By classical results of logic there exist Lall-structures in which all sentences from +ThLall(N) are valid but which are not isomorphic to N. These are called nonstandard +models (of ThLall(N)). +All nonstandard models of ThLall(N) (and even much weaker theories) contain an +isomorphic copy of N as an initial segment. Therefore, we can assume that in fact all +models we encounter satisfy N ⊆ M. +After considering a concrete nonstandard model M (of ThLall(N)) we shall call the +elements of M\N nonstandard numbers. These can be intuitively understood as “infinite +natural numbers”. The key feature of those elements is that all functions and relations +from Lall are defined even on nonstandard numbers. This includes functions for coding +sequences and sets by numbers, and therefore we can use notation like a0, . . . , an−1 even +for a nonstandard number n. The notation then means that for each i ∈ M such that +i < n we have an object ai coded by a number in M and that this whole sequence is +coded by some number {ai}n−1 +i=0 ∈ M. For a nonstandard number S ∈ M coding a set +we denote its nonstandard size (cardinality) to be |S|. In the case where we talk about +a binary string x the notation |x| denotes the bit length of x (which is nonstandard if x +is). +In the next section we will fix a nonstandard model M which has the model theoretic +property that it is ℵ1-saturated. There is a self-contained construction of such model +in [7, Appendix]. The only consequence of the ℵ1-saturation we shall use is the following. +Property. Let {ai}∞ +i=0 be a sequence of standard numbers. Then there exists t ∈ M \N +3 + +and a sequence {bi}t +i=0 ∈ M such that for all i ∈ N it holds that ai = bi. We shall call +the sequence of {bi}t +i=0 the nonstandard prolongation of {ai}∞ +i=0. +The language Lall contains symbols for all relations on N. +Since every sequence +of numbers can be defined by some relation it turns out that in our case there is a +unique nonstandard prolongation which matches the definition of the wide sequence (up +to length which can be chosen arbitrarily high). We can therefore allow ourselves to use +nonstandard numbers as indices of any sequences of objects unambiguously. +Any nonstandard model M can be extended to an ordered ring ZM by adding neg- +ative elements. This ring then can be extended to a fraction field QM. We shall call +elements of QM M-rationals. The field QM contains an isomorphic copy of Q as a sub- +structure. We call an element in QM with absolute valued greater than all k +1, k ∈ N, +infinite otherwise we call it finite. We call elements in QM with absolute value smaller +than all 1 +k, k ∈ N infinitesimal. +We will denote the set of finite M-rationals as QM +fin and one can check it forms an +ordered ring. +Lemma (The existence of a standard part). There is a function st : QM +fin → R assigning +to each finite M-rational a real number. st is a ring homomorphism and the kernel of st +is exactly the ideal of infinitesimal numbers. When q is a finite M-rational we call st(q) +its standard part. +We shall use the structure QM analogously to how hyperreal numbers are used in +nonstandard analysis. For more details about nonstandard analysis we recommend [3] +to the interested reader. The following result characterizes convergence of sequences of +rational numbers using QM. +Theorem. Let {ai}∞ +i=0 be a sequence of rational numbers and let r ∈ R. +Then the +following are equivalent. +• limi→∞ ai = r +• For every {bi}t +i=0, t ∈ M\N, which is a nonstandard prolongation of {ai}∞ +i=0, there +is an nonstandard s0 ≤ t, such that for every nonstandard s ≤ s0: st(as) = r. +It is important for forcing with random variables to consider discrete probability +spaces of nonstandard size. We shall always use uniform distribution on the samples +(although this is not necessary for the general construction). Thus, the probability of an +event coded by an element A ∈ M is then just the M-rational number |A|/|S| where S +is the set of samples of such a space. +We conclude this section by restating classical inequalities used in this work using +the nonstandard approach. +Theorem (Bernoulli’s inequlity). Let y ∈ M, x ∈ QM and x ≥ −1, then +(1 + x)y ≥ 1 + yx. +Theorem (Exponential inequality). Let x ∈ M \ N, then +st +�� +1 − 1 +x +�x� +≤ e−1. +4 + +3 +Wide limits +3.1 +The definition +We shall define a wide limit of every sequence of the following form. +Definition 3.1. A sequence of sets of graphs {Gk}∞ +k=1 is called a wide sequence if the +following holds: +• Every graph ω ∈ Gk has the vertex set {0, . . . , k − 1}. +• limk→∞|Gk| = ∞. +By abuse of notation we will simply talk about a wide sequence Gk instead of {Gk}∞ +k=1. +Since a wide limit is a Boolean-valued graph, we need to construct a Boolean algebra in +which the truth evaluation of statements shall take place. +For the construction of the Boolean algebra we will closely follow [7, Chapter 1] albeit +with slight changes. Let us now fix for the rest of this work an ℵ1-saturated model of +ThLall(N) which we will denote M. +Definition 3.2. Let n ∈ M. We define +An = {A ⊆ {0, . . . , n − 1}; A ∈ M}, +in words An is the set of subsets of {0, . . . , n − 1} coded by an element in M. This is +a boolean algebra and to each A ∈ An we assign an M-rational |A|/n which we call its +counting measure. +Even though An is a boolean algebra with a “measure” it is not a σ-algebra. Indeed, +An contains all singletons, but the countable set of those elements in {0, . . . , n − 1} with +only finitely many predecessors is not definable by compactness. However, having infinite +joins and meets at our disposal allows us to interpret quantifiers in the boolean valued +case, so we now want to ‘tweak’ this Boolean algebra. +Definition 3.3. Let I be the ideal of An consisting of elements with infinitesimal count- +ing measure. We define Bn = An/I. Each element in Bn is of the form A/I, where +A ∈ An, and we define µ(A/I) = st(|A|/n). We will denote the maximal element of Bn +by 1 and the minimal element by 0. +One can easily check that µ is well-defined since for all A ∈ I it holds that st(|A|/n) = +0. The measure µ is called the Loeb measure. The following then holds. +Lemma 3.4 ( [7, Lemma 1.2.1]). Bn is a σ-algebra with a real valued measure µ. More- +over, Bn is a complete boolean algebra. +It is important to note that 1 ∈ Bn is the only element of Bn with measure µ(1) = 1 +and similarly 0 ∈ Bn is the only element with measure µ(0) = 0. Also, for B, B′ ∈ Bn +the inequality B ≤ B′ implies µ(B) ≤ µ(B′). +5 + +We now define precisely what we mean by the family of functions F relative to which +we will be taking the wide limit. This is still a part of Krajíček’s construction, we just +modify it to make it compatible with our setup — where we start with a wide sequence. +For every k ∈ N the set Gk is finite and thus can be coded by a number. Therefore, +there is a nonstandard prolongation of this sequence, and we can consider the set coded +by the nonstandard number Gn, which matches the definition of the wide sequence in M. +Definition 3.5. Let {Gk}∞ +k=1 be a wide sequence and let n ∈ M \ N. We say that F is a +family of random variables on Gn if every α ∈ F is a function coded by a number in M +with domain Gn and taking values in M. We say α ∈ F is an F-vertex if for all ω ∈ Gn +it holds that α(ω) ∈ {0, . . . , n − 1}. The set of all F-vertices is denoted U(F). +If the wide sequence {Gk}∞ +k=1 and the number n ∈ M \ N is clear from context we +just say F is a family of random variables. This is for now everything we need to recall +from [7], and we can proceed to define the central object of our work. +Definition 3.6 (The wide limit). Let {Gk}∞ +k=1 be a wide sequence, let n ∈ M \ N and +let F be a family of random variables on Gn. We define the wide limit limF,n{Gk}∞ +k=1 as +a Bn-valued structure in the language consisting of a single binary relation symbol {E} +as follows. The universe of the wide limit is taken as the set of all F-vertices. We now +inductively define the truth values for all {E}-sentences. +• [[α = β]] = {ω ∈ Gn; α(ω) = β(ω)}/I +• [[E(α, β)]] = {ω ∈ Gn; Eω(α(ω), β(ω))}/I +• [[ − ]] commutes with ¬, ∧ and ∨ +• [[(∃x)A(x)]] = � +α∈U(F ) [[A(α)]] +• [[(∀x)A(x)]] = � +α∈U(F ) [[A(α)]] +By abuse of notation we will denote the wide limit limF,n{Gk}∞ +k=1 by limF Gn. To +stress in which boolean valued structure is the truth evaluation [[ − ]] taking place we will +sometimes denote the evaluation C1[[ − ]], C2[[ − ]] for boolean valued structures C1 and C2 +respectively. Furthermore, if C1[[ϕ]] = 1 for some sentence ϕ we say ϕ is valid in C1. +Note that since Gn can be recovered from F as the domain of its elements the wide +limit strictly speaking only depends on F. We keep Gn in the notation to cover the +situation where we have a very general family of functions (e.g. the family of polynomial +functions FPV) which can be applied to every wide sequence. Thus, the notation limF Gn +means that F is restricted to those functions which take elements of Gn as an input even +when F possibly contains other functions too. +The variability of the parameter n may also seem unnecessary and indeed in our +applications it is the case, but generally there are examples of wide sequences where n +directly affects the wide limit. +6 + +Example 3.7. Let Fconst be the family of all constant functions with domain Gn and +range anywhere in M. Let +Gk = +� +{({0, . . . , k − 1}, E); |E| = 2, (0, 1) ∈ E} +k even +{({0, . . . , k − 1}, E); |E| = 1, (0, 1) ̸∈ E} +k odd +then +lim +Fconst Gn[[E(0, 1)]] = +� +1 +n even +0 +n odd. +3.2 +An example of a wide limit relative to shallow decision trees +Now we shall define the first nontrivial family of random variables relative to which we +shall take wide limits of several sequences. The functions in the family will be computed +by shallow decision trees. So the shape of the wide limit reflects what can ‘superloga- +rithmic’ trees witness in the wide sequence with probability arbitrarily close to 1. +Definition 3.8. Let Trud be a family of labeled rooted binary trees in M of the following +form. At each vertex the tree is labeled by an element of {0, . . . , n − 1} × {0, . . . , n − 1} +and the two outgoing edges incident to it are labeled as 0 and 1 respectively. The leaves +are labeled by an element of M. The depth of the tree is bounded by a number of a +form n1/t (rounded to the nearest element of M) for some t ∈ M \ N. +A computation of a T ∈ Trud on some ω ∈ Gn is defined as follows. Start at the root +and interpret each label (i, j) of the vertex as a question whether the pair (i, j) is in +the edge set Eω and follow a path through T reading 1 as a positive answer and 0 as a +negative answer. The label of the leaf visited at the end of the path is the output of T +on ω, denoted T(ω). +We define Frud to be the set of all functions computed by a tree T ∈ Trud. +The simplest wide sequence we shall consider is the following sequence of sets of +undirected graphs with exactly one edge. +Definition 3.9. EDGEk = {({0, . . . , k − 1}, E); |E| = 1} +Since any ω ∈ EDGEk has only 1 edge in all potential k · (k − 1)/2 edges, it is not +likely a shallow tree will find the edge. This is the idea behind the proof of the following +theorem. +Theorem 3.10. +lim +Frud +EDGEn[[(∃x)(∃y)E(x, y)]] = 0 +Proof. Let α, β ∈ U(Frud), we proceed by proving that +[[E(α, β)]] = 0 +which is enough to prove the theorem since even an infinite disjunction of the values 0 +is 0. +7 + +Let α and β be computed by T ∈ Trud and S ∈ Trud respectively. Let the depth of +both T and S be at most n1/t, where t ∈ M \N. Walk down T from the root and always +prolong the path along the edge labeled 0. On this path we have a set of at most n1/t +different pairs of vertices and a label of the leaf lT . +We do the same for S, and we find another set of at most n1/t pairs of vertices and +a label of the leaf lS. lS and lT are then combined to one last pair (lS, lT ). Now we just +need to compute the probability that none of these 2n1/t + 1 pairs of vertices are not in +the edge set Eω. +There are +�n +2 +� +different graphs in EDGEn and +�n−4n1/t−2 +2 +� +graphs which fulfill our +requirements. The probability is thus +�n−4n1/t−2 +2 +� +�n +2 +� += (n − 4n1/t − 2)(n − 4n1/t − 3) +n(n − 1) +≥ (n − 4n1/t − 3)2 +n2 +≥ +� +1 − 4n1/t + 3 +n +�2 +≥ +� +1 − 8n1/t + 6 +n +� +after taking the standard part of the last line we get st(1 − 8n1/t+6 +n +) = 1. Therefore, +µ([[E(α, β)]]) = 0 and [[E(α, β)]] = 0. +3.3 +Sufficient conditions for validity of existential and universal sen- +tences +To analyze wide limits we need ideally to know the values of sentences which describe +properties whose complexity we are interested in. Generally this can be hard, so in this +section we prove sufficient conditions at least for the validity of universal and existential +sentences. +We will start with the simpler condition for the validity of universal sentences. This +is important also because we would like to know that a wide limit of a wide sequence +of graphs (i.e. antireflexive {E}-structures) is also a graph and that a wide limit of a +wide sequence of undirected graphs (directed graphs with E symmetric) is an undirected +graph. All of these properties are expressible as universal sentences. +Theorem 3.11. Let Gk be a wide sequence and let F be any family of random variables. +Let ϕ(x0, . . . , xl−1) be an open {E}-formula and assume that +lim +k→∞ Pr +ω∈Gk +[ω |= (∀x)ϕ(x)] = 1. +Then limF Gn[[(∀x)ϕ(x)]] = 1. +8 + +Proof. By induction in M we have that st(Prω∈Gn[ω |= (∀x)ϕ(x)]) = 1. Therefore, we +have for every tuple of F-vertices α that [[ϕ(α)]] = 1. Now +[[(∀x)ϕ(x)]] = +� +α∈U(F )l +[[ϕ(α)]] += +� +α∈U(F )l +1 += 1. +Corollary 3.12. Let Gk be a wide sequence and F any family of random variables. +• If all ω ∈ Gk, k ∈ N, are directed graphs ({E}-structures satisfying that E is antire- +flexive) then limF Gn is a Boolean-valued {E}-structure in which the antireflexivity +of E is valid (i.e. limF Gn is a Boolean-valued graph). +• If all ω ∈ Gk, k ∈ N, are undirected graphs (directed graphs where E is symmetric) +then limF Gn is an {E}-structure in which both antireflexivity and symmetry of E +is valid. (i.e. limF is a Boolean-valued undirected graph) +Now to give a sufficient condition for the validity of an existential sentence (∃x)ϕ(x) +we use the auxiliary value of density of ϕ(x0, . . . , xl−1) defined as the probability that a +random graph ω ∈ Gk and a random tuple a ∈ {0, . . . , k − 1}l satisfy ω |= ϕ(a) and show +that the limiting density gives a lower bound for the measure of [[(∃x)ϕ(x)]]. +Theorem 3.13. Let Gk be a wide sequence and let F be a family of random variables +which contains all constant functions. Let ϕ(x0, . . . , xl−1) be an open {E}-formula and +let p ∈ [0, 1]. Assume that +lim +k→∞ Pr +ω∈Gk +a +[ω |= ϕ(a)] ≥ p, +where a is sampled uniformly over all elements of {0, . . . , k − 1}l. Then +µ(lim +F Gn[[(∃x)ϕ(x)]]) ≥ p. +In particular if p = 1 then limF Gn[[(∃x)ϕ(x)]] = 1. +Proof. Consider an array C indexed by ω ∈ Gn and a ∈ {0, . . . , n − 1}l such that +Cω,a = +� +1 +ω |= ϕ(a) +0 +otherwise. +By the assumption and induction in M we have that +st +� +1 +nl|Gn| +� +ω∈Gn +� +a +Cω,a +� +≥ p. +9 + +We now claim that there exists a specific b ∈ {0, . . . , n−1}l such that st(Prω∈Gn[ω |= +ϕ(b)]) ≥ p. Assume for contradiction that the claim is false. Then +1 +|Gn|nl +� +ω∈Gn +� +a +Cω,α = 1 +nl +� +a +Pr +ω∈Gn[ω |= ϕ(a)] +≤ Pr +ω∈Gn[ω |= ϕ(a0)], +where we pick a0 such that it maximizes Prω∈Gn[ω |= ϕ(a0)]. +But after taking the +standard part of the inequality we obtain that +st +� +1 +nl|Gn| +� +ω∈Gn +� +a +Cω,a +� +≤ st( Pr +ω∈Gn[ω |= ϕ(a0)]) < p. +Which is a contradiction and so the claim is true. Let γb be a tuple of constant +functions which is at every sample equal to b. We have +[[(∃x)ϕ(x)]] = +� +α∈U(F )l +[[ϕ(α)]] +≥ [[ϕ(γb)]] +and by taking µ of this inequality we finally obtain that µ([[(∃x)ϕ(x)]]) ≥ p. +In the following example we use Theorem 3.13 to show that in the wide limit of graphs +which have exactly one large clique and no other edges the nonexistence of a standard +sized clique cannot be valid relative to any F with all constants. +Example 3.14. Consider the wide sequence +SK1/2 +k += {({0, . . . , k − 1}, E); E has a clique of size ⌊k/2⌋ and no other edges}. +We will check that for an {E}-formula ϕl(x) which states that x forms a clique of size l +we have +lim +k→∞ +Pr +ω∈SK1/2 +k +a +[ω |= ϕl(a)] ≥ (1/2)l. +Notice that we can compute the probability for a fixed a such that ai ̸= aj whenever +i ̸= j, since the ratio of tuples containing some vertex twice is infinitesimal. So we have +Pr +ω∈SK1/2 +k +[ω |= ϕl(a)] = +l−1 +� +i=0 +� +1 − k − ⌊k/2⌋ +k − i +� +≥ +� +1 − k − ⌊k/2⌋ +k − l +�l +≥ +� +1 − +1 +2(1 − l/k) − +1 +k − l +�l +10 + +and since l ∈ N we just take the limit of the inner expression. But one can see that +limk→∞(1 − l/k) = 1 and that limk→∞(1/(k − l)) = 1. +Now by Theorem 3.13 we obtain that for any F that contains all constants we have +lim +F SK1/2 +n [[(∃x)ϕl(x)]] > 0. +The following example demonstrates that Theorem 3.11 cannot be generalized to a +similar hypothesis as Theorem 3.13. +Example 3.15. Let Gk consist of all undirected graphs on the vertex set {0, . . . , k − 1} +with exactly ⌈ k(k−1) +2 log(k)⌉edges. One can see that +lim +k→∞ Pr +ω∈Gk +x,y +[ω |= ¬E(x, y)] = 1, +but in fact limFrud Gn[[(∀x)(∀y)¬E(x, y)]] = 0. +Let t ∈ M \ N such that n1/t is not bounded above by a standard number. Let T be +a tree which queries on all paths a fixed set of n1/t different potential edges. If we prove +that any such set in Gn has to contain at least one edge with probability infinitesimally +close to 1 then we can construct Frud-vertices α and β using T such that [[E(α, β)]] = 1 +by simply taking T and labeling each leaf on a path which finds an edge with one of the +vertices incident to this edge. +Let S be the set of potential edges queried by T and let m = +�n +2 +� +. Now we have +Pr +ω∈Gn[S contains no edge in ω] = +(m − n1/t)!(m − ⌈ m +log n⌉!) +m!(m − ⌈ +m +log m⌉ − n1/t)! += +n1/t−1 +� +i=0 +m − ⌈ m +log n⌉ − i +m − i +≤ +� +1 − +⌈ m +log n⌉ +m +�n1/t +≤ +� +1 − +1 +2 log n +�n1/t +standard part of which is for all k ∈ N bounded above by +st +�� +1 − +1 +2 log n +�k·2 log n� +≤ e−k +which tends to 0 as k → ∞. +11 + +4 +A wide limit of Leaf instances relative to oracle trees +The class of total NP search problems TFNP, first defined in [9], consists of all relations +on binary strings P(x, y) such that: +• (verifiability in polynomial time) There is a polynomial time machine M which, +given x, y, can decide whether P(x, y) holds. +• (totality) There exists a polynomial p and for every x there exists at least one y +satisfying |y| ≤ p(|x|) such that P(x, y) holds. +Two particular problems are relevant for us. +The problem Leaf is formulated as follows. An instance is given by a number k and +a undirected graph ω on the vertex set {0, . . . , 2|k| − 1}, presented by a Boolean circuit +of polynomial size in |k| computing its neighborhood function, such that degω(0) = 1 +and ∀v : degω(v) ≤ 2. The task is then to find some nonzero v with degω(v) = 1. The +corresponding combinatorial principle being the handshaking lemma, which assures the +problem is total. +The problem OntoWeakPigeon is formulated as follows. +An instance is given +by a number k and two functions A : {0, . . . , 2|k| − 1} → {0, . . . , 2|k|−1 − 1} and B : +{0, . . . , 2|k|−1 − 1} → {0, . . . , 2|k| − 1}, each presented by a Boolean circuit of polynomial +size in |k|. The task is then to find some x such that B(A(x)) ̸= x or some y such +that A(B(y)) ̸= y. The corresponding combinatorial principle being the bijective weak +pigeonhole principle, which assures the problem is total. The domain of A is twice as +large as its range, so B and A cannot form a pair of inverse functions between their +respective domains. +So far, we presented what is called ‘type 1’ problem in [1]. We are interested about the +‘type 2’ problems which replace the input function(s) with oracle(s). So in the ‘type 2’ +Leaf problem, the input is a pair (α, x) where α is an oracle which describes the neighbor +function on G with vertex set {0, . . . , 2|x| − 1}. For the ‘type 2’ OntoWeakPigeon +problem, the input is a triple (α, β, x), where α and β are oracles describing the functions +α : {0, . . . , 2|x| − 1} → {0, . . . , 2|x|−1 − 1} and β : {0, . . . , 2|x|−1 − 1} → {0, . . . , 2|x| − 1}. +The associated computational models for the ‘type 1’ problems are Turing machines +and for the ‘type 2’ problems oracle Turing machines. +4.1 +The wide limit and oracle trees +The wide sequence ∗PATHk (pointed paths on k vertices) consists of all undirected graphs +ω on the vertex set {0, 1, . . . , k − 1} which are isomorphic to a path with k − 1 edges +connecting all vertices and degω(0) = 1. Graphs in ∗PATHk are ‘the hardest instances +of Leaf’ so we can expect the wide limit to reflect the complexity of these instances +relative to the family F we choose. +Since each ω ∈ ∗PATHk has only k − 1 edges we can proceed similarly to the proof +of Theorem 3.10 to get the following. +Lemma 4.1. limFrud ∗PATHn[[(∃x)(∃y)E(x, y)]] = 0 +12 + +To get a result which reflects the properties of the wide sequence more faithfully we +will define a new family of random variables on ∗PATHn. +Definition 4.2. We define Tnb as the set of all labeled rooted trees of the following shape: +• Each non-leaf node is labeled by some v ∈ {0, . . . , n − 1}. +• For each {u, w} ⊆ {0, . . . , n − 1} and a node v there is an outgoing edge from v +labeled {u, w} (it can be that u = w). +• Each leaf is labeled by some m ∈ M. +• The depth of the tree is defined as the maximal number of edges in a path from +the root, and we require it is at most n1/t (rounded to the nearest element of M) +for some t ∈ M \ N. +The computation of such a tree in Tnb on ω ∈ ∗PATHn is defined as follows. We +build a path by starting at the root and interpreting every vertex labeled by some v as +a question ‘what are the neighbors of the vertex v?’ and we follow the output edge with +the answer and continue analogously until we find a leaf. The label of the leaf is defined +to be the output of the computation. +We define Fnb to be the set of all functions on ∗PATHn which are computed by some +T ∈ Tnb. +The trees computing the functions in Fnb can be thought of as a protocol describing +the behavior of a machine M communicating with an oracle describing a particular +ω ∈ ∗PATHn. In the study of total NP search problems presented by oracles, we usually +denote the size of the object by some 2|x| where x is an additional input to the problems. +If 2|x| = n then n1/t = 2|x|/t which for t ∈ M \ N corresponds to protocols describing +non-uniform subexponential-time computations. If we prove that no tuple of Fnb-vertices +satisfies some open {E}-formula in limFnb ∗PATH we also prove that subexponential-time +oracle machines cannot solve the corresponding type 2 problem on a non-diminishing +fraction of the inputs. In the rest of this section we proceed to prove that limFnb ∗PATHn +has no vertex with degree 1 other than the vertex 0. +To do so, we will consider computations of trees on samples with different nonstandard +lengths. For the rest of this section we put Gm = ∗PATHm for all m ∈ M, but we can +assume m to be smaller than n. We define T (m) +nb +to be the subset of Tnb consisting of +all the trees that have the vertex labels from {0, . . . , m − 1}. For trees in T (m) +nb +we can +extend the definition of a computation to input graphs from Gm in a straight forward +way. +Definition 4.3. We say a tree T ∈ T (m) +nb +fails on ω ∈ Gm if the output of T on ω has +degree 2. +Definition 4.4. Let m ∈ M, v ∈ {0, . . . , m − 1} and {u, w} ⊆ {0, . . . , m − 1} we define +Gv:{u,w} +m += {ω ∈ Gm; ω |= E(v, u) ∧ E(v, w)}. +13 + +Lemma 4.5. Let m ∈ M and let u, v and w be distinct elements of {1, . . . , m−1}. Then +there are bijections: +Gv:{u,w} +m +∼= Gm−2 × {L, R} +Gv:{u,0} +m +∼= Gm−2 +Gv:{u} +m +∼= Gm−1 +G0:{u} +m +∼= Gm−1 +Proof. For the first case a bijection can be given as follows. +Contract u, v and w to +just one vertex min{u, v, w} and if u is closer to 0 than w pick L otherwise pick R +and relabel the remaining vertices using a function ‘new’ which has a property that if +u′, v′ remain and u′ < v′ as numbers then new(u′) < new(w′) and the range of new is +{0, . . . , m−2}. This can be inverted by first renaming the vertices using new−1 and then +replacing min{u, v, w} by a path (u, v, w) with the orientation given either by L or R. +The second bijection is almost the same, but the orientation is clear since u is always +the neighbor further from 0 since the other neighbor is 0. +The third and fourth bijections are given by just removing one end of the graph and +relabeling. +Definition 4.6. Let m ∈ M and v ∈ {0, . . . , m − 1}. +Let u and w be elements of +{0, . . . , m − 1} \ {v} and let T ∈ T (m) +nb +be a tree with the root labeled v. By Tv:{u,w} we +denote the induced subtree whose root is the vertex neighboring the root of T via the +edge labeled {u, w}. +Lemma 4.7. Let m ∈ M. Let T ∈ T (m) +nb +be a tree with the root labeled v ̸= 0. For +each u and w which are distinct elements of {0, . . . , m − 1} \ {v} there exists a tree +˜Tv:{u,w} ∈ T (m−2) +nb +of the same depth as Tv:{u,w} such that +Pr +ω∈Gm[Tv:{u,w} fails | ω |= E(v, u) ∧ E(v, w)] = +Pr +ω∈Gm−2[ ˜Tv:{u,w} fails]. +If T has the root labeled 0 then there exists a tree ˜T0:{u} ∈ T (m−1) +nb +of the same depth +as T0:{u} such that +Pr +ω∈Gm[T0:{u} fails | ω |= E(0, u)] = +Pr +ω∈Gm−1[ ˜T0:{u} fails]. +Proof. In the case where the root is labeled by v ∈ {1, . . . , m − 1} we can construct +the tree ˜Tv:{u,w} by simply relabeling vertices of Tv:{u,w}. We use the relabeling function +‘new’ from the proof of Lemma 4.5. Now for every ω ∈ Gm there is by the first bijection in +Lemma 4.5 a uniquely determined ω′ ∈ Gm−2. The computation of ˜Tv:{u,w} on ω′ is then +of the same shape as the computation of Tv:{u,w} on ω assuming ω |= E(v, u) ∧ E(v, w). +And ˜Tv:{u,w}(ω′) has the same degree in ω′ as Tv:{u,w}(ω) does in ω. +The case where the root is labeled by 0 is analogous, but we instead use the relabeling +from the fourth bijection in Lemma 4.5. +14 + +Lemma 4.8. Let T ∈ T (m) +nb +of depth d ∈ M and let d ≤ m. Then we have +Pr +ω∈Gm[T fails] ≥ +d +� +i=0 +� +1 − +2 +m − 2i − 2 +� +. +Proof. We proceed by induction on d. The case d = 0 follows from +Pr +ω∈Gm[T fails] ≥ +� +1 − +1 +m − 1 +� +≥ +� +1 − +2 +m − 2 +� +. +Now for the inductive case we assume the lemma holds for d − 1, and prove it for d. If +the root of T is labeled 0 we proceed as follows. For a given T let u0 be the vertex which +minimizes the value Prω∈Gm[T fails | E(0, u0)] which exists by the least number principle +in M. Then by Lemma 4.7 and the induction hypothesis +Pr +ω∈Gm[T fails] ≥ +Pr +ω∈Gm[T fails | E(0, u0)] += +Pr +ω∈Gm−1[ ˜T0:{u0} fails] +≥ +d−1 +� +i=0 +� +1 − +2 +m − 2i − 3 +� +≥ +d +� +i=0 +� +1 − +2 +m − 2i − 2 +� +. +Now for the case where the root of T is labeled by nonzero v. First we note that +Pr +ω∈Gm[v has degree 2 ∧ ¬E(v, 0)] = 1 − +2 +m − 1. +Now we choose distinct u0, w0 such that they minimize +Pr +ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)]. +Then by the Lemma 4.7 and the induction hypothesis we have +Pr +ω∈Gm[T fails] ≥ +� +1 − +2 +m − 1 +� +Pr +ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)] += +� +1 − +2 +m − 1 +� +Pr +ω∈Gm−2[ ˜Tv:{u0,w0} fails] +≥ +� +1 − +2 +m − 1 +� d−1 +� +i=0 +� +1 − +2 +m − 2i − 4 +� +≥ +� +1 − +2 +m − 1 +� +d +� +i=1 +� +1 − +2 +m − 2i − 2 +� +≥ +d +� +i=0 +� +1 − +2 +m − 2i − 2 +� +. +15 + +Lemma 4.9. Let T ∈ Tnb, then st (Prω∈Gn[T fails]) = 1. +Proof. The depth of T is bounded by n1/t for some t ∈ M \ N. We have by Lemma 4.8 +that +Pr +ω∈Gn[T fails] ≥ +n1/t +� +i=0 +� +1 − +2 +n − 2i − 2 +� +(1) +≥ +� +1 − +2(n1/t + 1) +n − 2n1/t − 2 +� +(2) +and the standard part of this lower bound is 1. +Finally, in the next theorem we prove that a formalization of ‘there is a nonzero +vertex of degree 1’ is not valid in limFnb ∗PATHn and in fact its boolean value is 0. +Theorem 4.10. +lim +Fnb ∗PATHn[[(∃v)(∃u)(∀w)(v ̸= 0 ∧ E(v, u) ∧ (E(v, w) → w = u))]] = 0 +Proof. By expanding the left-hand side of the statement we get +� +α∈U(Fnb) +� +β∈U(Fnb) +� +γ∈U(Fnb) +[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]]. +Therefore, it is enough if we prove that for each Fnb-vertices α and β there exists an +Fnb-vertex γ such that +[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] = 0. +For any α, β ∈ U(Fnb) we can append the tree computing β to every leaf of a tree +computing α. This is still a tree in Tnb as its depth is at most twice the maximum of +depths of the original trees. By relabeling the leaves of the resulting tree we can obtain +a tree computing a function +γ(ω) = +� +v +if degω(α(ω)) = 1 and v is the only neighbor of α(ω) +w +if degω(α(ω)) = 2, w is a neighbor of α(ω) and w ̸= β(ω). +This is obviously an Fnb-vertex. Let us assume for contradiction that +[[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] > 0. +By definition this gives us +0 < st +� +Pr +ω∈Gn[α(ω) ̸= 0 ∧ Eω(α(ω), β(ω)) ∧ (E(α(ω), γ(ω)) → γ(ω) = β(ω))] +� +≤ st +� +Pr +ω∈Gn[α(ω) ̸= 0 ∧ degω(α(ω)) = 1] +� +, +but this is in contradiction with Lemma 4.9. +16 + +5 +The expanded model with a Leaf instance without a so- +lution and with total OntoWeakPigeon +As a part of the proof of Theorem 4.10 we proved what can be reformulated as the +statement that oracle instances of Leaf are not in oracle time O(2f(|x|)) with f ∈ o(|x|1/c) +for every c ∈ N even when we just require it to be correct on any nondiminishing ratio +of inputs as |x| grows. In this section we proceed to compare strength of (type 2) NP +search problems not only with oracle FP but also with other NP search problems via +relative consistency of their totality and nontotality. We will show that there is a model +of weak second order arithmetic in which the problem Leaf is not total even though +OntoWeakPigeon is. +5.1 +The structures K(F, G) +We will now recall the construction of second order models of weak arithmetic K(F, G) +defined in [7, Chapter 5]. We will take the liberty to define them as an extension of +the definition of a wide limit to obtain structures K(Gn, F, G) 4 which under the right +conditions result in a structure in some sublanguage of Lall with two sorts: numbers and +bounded sets of numbers which contains the wide limit as an object of the second sort. +Definition 5.1. Let L ⊆ Lall. This determines a language L2 which we get by adding +to L second order variables X, Y, . . . whose intended interpretation are bounded sets and +the equality symbol for second order variables (denoted the same as the first order one). +All second order variables are treated as function symbols and can form terms with the +first order terms as arguments. +We will also use the second order variables as relation symbols, and we define the +atomic formula X(x0, . . . , xk−1) simply to be evaluated as the formula X(x0, . . . , xk−1) ̸= +0. +Now we assume we fix a number n, a wide sequence Gk and a family of random +variables on Gn which all together determine a wide limit limF Gn. +Definition 5.2. We define Mn ⊆ M to be the subset of M consisting of all numbers +bounded above by 2n1/t for some t ∈ M \ N. +Definition 5.3. We define Ln ⊆ Lall to contain all relation symbols from Lall and all +functions from Lall for which their values on any element of Mn is still in Mn. We say +F is Ln-closed if for every function symbol f ∈ Ln we have that f(α0, . . . , αk−1) ∈ F. +Note that Mn is then a substructure of the Ln-reduct of M. +Definition 5.4. We say that G is a family of random functions (on Gn) if every Θ ∈ G +assigns to each ω ∈ Gn a function Θω ∈ Mn. +4This notation is just making some parameters of the construction explicit, the models constructed +can be obtained by the original method without first constructing the wide limit. Our contribution is in +observing that the truth values of first order sentences concerning the wide limit is preserved between +the wide limit and the structure K(Gn, F, G). +17 + +We say G is F-compatible if for every α ∈ F, Θ ∈ G we have that the function Θ(α) +defined as +Θ(α)(ω) = +� +Θω(α(ω)) +if α(ω) ∈ dom(Θω) +0 +otherwise +is in fact in F. +Definition 5.5. Let F be an Ln-closed family of random variables with values in Mn. +Let G be an F-compatible family of random functions. We define K(Gn, F, G) to be +a Bn-valued L2 +n structure with first order sort of the universe F and second order sort +of the universe G. The valuation of formulas is then given by the following inductive +definition: +• [[α = β]] = {ω ∈ Gn; α(ω) = β(ω)}/I, where α, β ∈ F +• [[R(α0, . . . , αk−1)]] = {ω ∈ Gn; ω |= R(α0(ω), . . . , αk−1(ω))}/I, where α0, . . . , αk−1 +are from F and is R a relation symbol in Ln +• [[Θ = Ξ]] = {ω ∈ Gn; Θω = Ξω}/I, where Θ, Ξ ∈ G +• [[(∀x)A(x)]] = � +α∈F [[A(α)]] +• [[(∃x)A(x)]] = � +α∈F [[A(α)]] +• [[(∀X)A(X)]] = � +Θ∈G [[A(Θ)]] +• [[(∃X)A(X)]] = � +Θ∈G [[A(Θ)]]. +5.2 +Preservation of sentences concerning the wide limit +We will now prove (under a mild condition on F) that in a structure K(Gn, F, G) which +represents the wide limit limF Gn by a second order object are the values of all sentences +regarding the object the same as in the wide limit. This lets us construct models in +which an object elementary equivalent to the wide limit might be desired. +Definition 5.6. We say that the edge relation of the wide limit limF Gn is represented +in G by Γ if Γ ∈ G and for all α, β ∈ U(F) we have that +K(Gn, F, G)[[Γ(α, β)]] = lim +F Gn[[E(α, β)]]. +Definition 5.7. We say a family of random variables F has restrictable ranges if for +every α ∈ F and m ∈ Mn there is ˜αm ∈ F such that +˜αm(ω) = +� +α(ω) +α(ω) < m +0 +otherwise. +18 + +Theorem 5.8. Let ϕ be a {E}-sentence. +Let F be Ln-closed and have restrictable +ranges and let G be F-compatible. Let the edge relation of the wide limit limF Gn be +represented in G by Γ. We define ˜ϕ(Γ) to be the L2 +n-sentence obtained by replacing +all the occurrences of the relation symbol E by Γ, keeping the structure of the logical +connectives and replacing all quantifiers (∀x)(. . . ) by (∀x)(x < n → (. . . )) and (∃x)(. . . ) +by (∃x)(x < n ∧ . . . ). +Then we have that for all {E}-sentences that +lim +F Gn[[ϕ]] = K(Gn, F, G)[[ ˜ϕ(Γ)]]. +Proof. We will prove that for all {E}-formulas ϕ(x) and all α ∈ F we have that +lim +F Gn[[ϕ(α)]] = K(Gn, F, G)[[ ˜ϕ(Γ, α)]]. +We proceed by induction on the complexity of the formula. +The case for atomic +formulas is clear and the step for logical connectives also since [[ − ]] commutes with +them. +With the induction step for negation in hand it is now enough to prove the +inductive step for the universal quantifier. +We assume that the statement works for a formula of lower complexity ϕ(y, x). By +the restrictability of ranges in F we get that for all β ∈ F there is ˜βn ∈ U(F) such that +K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]] ≤ K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]]. +Now we have that for all α ∈ U(F) +K(Gn, F, G)[[(∀y) ˜ϕ(Γ, y, α)]] = +� +α∈F +K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]] += +� +˜βn∈U(F ) +K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]] += +� +˜βn∈U(F ) +lim +F Gn[[ϕ(˜βn, α)]] += lim +F Gn[[(∀y)ϕ(y, α)]]. +5.3 +Failure of totality of Leaf +Now we are in a situation that lets us construct a model of weak second order arithmetic +that contains an instance of the problem Leaf without a solution. Consider a suitable +family Gnb in which we can define not only the wide limit itself but instances of some +other search problem. We can then ask: ‘Do all these instances have a solution?’ This +is a way to compare the strength of the different total NP search problems by relative +unprovability results. We will pick the family Gnb such that validity of totality of some +search problem P implies the nonexistence of a suitable reduction from Leaf to P. +19 + +Definition 5.9. Let Gnb be the family of all random functions on ∗PATHn such that for +each Θ ∈ Gnb there exists a tuple (γ0, . . . , γm−1) ∈ M so that γi ∈ Fnb and +Θ(α)(ω) = +� +γα(ω)(ω) +α(ω) < m +0 +otherwise. +In the models Mn we are working with there is a pairing function ⟨i, j⟩ which can +code pairs of numbers by a single number thus we can represent functions of any finite +arity by functions from Gnb. +One can understand the tuples which compute the random functions from Gnb as +tuples of protocols describing the computations of subexponential oracle machines. Such +a tuple defines a function which is at every index of the tuple computed using queries to +some ω ∈ ∗PATHn. If we prove that every instance of a search problem P represented +by such a tuple has a solution in K(∗PATHn, Fnb, Gnb), and we know that Leaf in +K(∗PATHn, Fnb, Gnb) is not total, which implies nonexistence of a subexponential oracle +machine which converts solutions of P to solutions of Leaf even on any standard fraction +of instances from ∗PATHn and thus a nonexistence of a many-one reduction from Leaf +to P as defined in [1]. +Lemma 5.10. +1. Fnb has restrictable ranges +2. Fnb is Ln-closed +3. Gnb is Fnb-compatible +4. Gnb represents the edge relation of limFnb ∗PATHn. +Proof. 1, 2: Here we can proceed simply by relabeling the leaves of the trees computing +the functions from Fnb. +3: Assume that Θ ∈ Gnb is computed by a tuple (γ0, . . . , γm−1). By induction in +M there exists t ∈ M \ N such that ∀i ∈ {0, . . . , m − 1} the depth of γi is at most +n1/t. Therefore, for all α ∈ Fnb we have that Θ(α) has also depth at most n1/t′ for some +t′ ∈ M \ N. Thus, Gnb is Fnb-compatible. +4: Let γ⟨i,j⟩ ∈ Fnb be computed by a tree in Tnb which queries i and outputs 1 +if the neighbor set contains j otherwise it outputs 0. Let Γ be computed by a tuple +(γ⟨i,j⟩)n−1 +i,j=0. Then we have +K(∗PATHn, Fnb, Gnb)[[Γ(α, β)]] = lim +Fnb ∗PATHn[[E(α, β)]]. +Definition 5.11. The L2 +n-formula ϕLeaf(X, Y, m) is defined as the disjunction of the +following formulas +(X(0) ̸= Y (0) ∨ X(0) = 0) +(∃x)((x < m) ∧ (X(x) > m − 1 ∨ Y (x) > m − 1)) +(∃x)((x < m) ∧ ((X(x) = x ∧ Y (x) ̸= x) ∨ (X(x) ̸= x ∧ Y (x) = x))) +(∃x)((x < m) ∧ (Y (X(x)) ̸= x ∧ X(X(x)) ̸= x) ∨ (X(Y (x)) ̸= x ∧ Y (Y (x)) ̸= x))) +(∃x)((0 < x < m) ∧ (X(x) = Y (x) ∧ X(x) ̸= x)), +20 + +this formula formalizes that if X and Y are functions representing the neighbor set of +each x < m as {X(x), Y (x)} \ {x} and 0 has only one neighbor then there has to exist +another y < x which also has only one neighbor. +Theorem 5.12. +K(∗PATHn, F, G)[[(∃X)(∃Y )(∃m)¬ϕLeaf(X, Y, m)]] = 1 +Proof. We can find Θ1, Θ2 ∈ Gnb such that for each v ∈ {0, . . . , n − 1} we have that +{Θ1(v)(ω), Θ2(v)(ω)} is the neighbor set of v on ω ∈ ∗PATHn. (We can just query v and +split the answer between Θ1 and Θ2.) +By Theorem 4.10 we know that limF Gn has one degree 1 vertex and all other vertices +of degree 2 and by Lemma 5.10 we know that it can be represented by some Γ ∈ Gnb. +Furthermore, we can verify that +[[(Γ(α, β)) ≡ (Θ1(α) = β ∨ Θ2(α) = β)]] = 1, +thus Θ1 and Θ2 do not satisfy the last disjunct of ϕLeaf otherwise it would be in contra- +diction with Theorem 4.10. By their construction and the definition of ∗PATHk we have +that (Θ1, Θ2, n) does not satisfy the other disjuncts either. +5.4 +Totality of OntoWeakPigeon +Definition 5.13. The L2 +n formula ϕOntoWeakPigeon(X, Y, m) is defined as the disjunc- +tion of the following formulas +(∃x)((x < 2m) ∧ (X(x) > m − 1)) +(∃y)((y < m) ∧ Y (y) > m − 1)) +(∃x)((x < 2m) ∧ Y (X(x)) ̸= x) +(∃y)((y < m) ∧ X(Y (y)) ̸= y) +it formalizes the bijective weak pigeonhole principle which claims that any pair of func- +tions +X :{0, . . . , 2m − 1} → {0, . . . , m − 1} +Y :{0, . . . , m − 1} → {0, . . . 2m − 1} +is not a pair of inverse bijections. +To prove that ϕOntoWeakPigeon(X, Y, m) is valid in K(∗PATHn, Fnb, Gnb) we will +construct a tree which finds some x such that Yω(Xω(x)) ̸= x or Xω(x) > m − 1 with +probability infinitesimally close to one. +Definition 5.14. Let Θ, Ξ ∈ Gnb, and ζ ∈ Fnb. We say that a tree T ∈ Tnb fails for +(Θ, Ξ, ζ) on ω if +Θω(T(ω)) < ζ(ω) +and +Ξω(Θω(T(ω))) = T(ω). +In words if T does not witness the failure of Ξ being the inverse function to Θ. +21 + +Lemma 5.15. Let Θ, Ξ ∈ Gnb and ζ ∈ Fnb. Then there is a tree T such that +st +� +Pr +ω∈Gn[T fails for (Θ, Ξ, ζ)] +� += 0. +Proof. Without loss of generality we may assume that ζ is actually constant, and its +value is r ∈ Mn which we pick to be the least possible output of ζ on any sample. +Furthermore, let Θ be computed by (θ0, . . . , θ2r−1) and Ξ by (ξ0, . . . , ξr−1). +We construct T by stages and at each stage it will have some potential output. First +we notice that at the beginning stage there is at least one i ∈ {0, . . . , 2r − 1} such that +the probability that θi < r or ξθi = i is at most 1 +2. The tree T0 is thus the constant tree +always outputting i. +Assume Td−1 have been constructed and pick any path p ∈ Td−1. If p did not fail we +leave it as it is otherwise we extend Td−1 along this path and after extending all such +paths this will become the new stage Td. The path p has a leaf with some label i. We +can check whether i fails by first appending the tree θi to this path and then to each +new leaf (labeled with a number < r) appending ξθi, let the leaves which confirm the +nonfailure of i be labeled by i. Now consider a path p′ extending p without determined +output. We claim that there is j ∈ {0, . . . , 2r − 1} such that +Pr +Gn[θj < r ∧ ξθj = j | p′ is compatible with ω] ≤ 1 +2, +where p′ being compatible with ω means that the computation along p′ agrees with the +edge labels which would be chosen according to ω. +To prove the claim we notice that along p′ it was confirmed that already d-many +distinct elements of {0, . . . , 2r − 1} are in bijection with some d-many elements of the +set {0, . . . , r − 1}. Therefore, to fail further there are only at most (r − d)-many other +values j′ in {0, . . . , 2r−1} for which it holds that ξθj′ = j′. By an analogous argument to +the proof of Theorem 3.13 this is enough to show that at least for one of them the claim +holds since r−d +2r ≤ 1 +2. Thus, we let j to be the label of the leaf of p′ which concludes the +construction. +Therefore, by construction for each d ∈ Mn, d < 2r we have +Pr +ω∈Gn[Td fails for (Θ, Ξ, ζ)] ≤ 2−d. +If r is in Mn \ N then we put T = Tt′ for any nonstandard t′ such that the depth of +T is still bounded by some n1/t, where t ∈ Mn \ N. Otherwise, we put T = T2r−1 and +since this tree can go through the whole range of Θ it can never fail. +Theorem 5.16. +K(∗PATHn, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕOntoWeakPigeon(X, Y, m)]] = 1 +Proof. By Lemma 5.15 we can construct for each (Θ, Ξ, ζ) a tree T which computes some +function α which validates the third disjunct of ϕOntoWeakPigeon. +22 + +Theorem 5.17. Let ϕ(x) be an L2 +n-formula with parameters from Fnb and Gnb. Then +for every m ∈ Mn the open comprehension principle +(∃X)(∀y < m)(X(y) ≡ ϕ(y)) +and the open induction principle +¬ϕ(0) ∨ ϕ(m) ∨ (∃x < m)(ϕ(x) ∧ ¬ϕ(x + 1)) +are both valid in K(∗PATH, Fnb, Gnb). +Proof. This can be proven completely analogously to [7, Lemma 20.2.5]. +Compiling the results we have about K(∗PATH, Fnb, Gnb) we get the following. +Corollary 5.18. In the structure K(∗PATH, Fnb, Gnb) the following are valid: +• open induction with parameters from Fnb and Gnb +• open comprehension with parameters from Fnb and Gnb +• every instance of OntoWeakPigeon has a solution +• there is an instance of Leaf which does not have a solution. +Concluding remarks +We have to note that the problem OntoWeakPigeon has not been considered in +the context of oracle NP search problems and the proof of Theorem 5.16 cannot be +adapted to prove that every instance of the stronger WeakPigeon 5 has a solution +in K(∗PATH, Fnb, Gnb) because the presence of the inverse function is essential to the +construction of the witness. +A stronger problem called SourceOrSink is well established in the study of NP +search problems (it is the complete problem for PPAD, see [1]) and can be formulated +as follows: Given a directed graph ω on the vertex set {0, . . . , 2|x| − 1} with the property +that any vertex v has outdegree bounded by 1 and indegree also bounded by 1 and the +indegree of the zero vertex is 0 find a nonzero vertex which is either a source or a sink. +In the type 2 setting the problem is given by a tuple (α, β, x), where x is a binary string +and α and β functions presented by an oracle with domain {0, . . . , 2|x| − 1} computing +the potential successor or predecessor of a given vertex. +It was established in [1] that Leaf is not many-one reducible to SourceOrSink and +therefore this nonreducibility may be reflected in our model K(∗PATH, Fnb, Gnb). The +way SourceOrSink is presented is similar to how OntoWeakPigeon is presented, +and thus a similar strategy could be potentially used to solve the following problem. +Problem. Let ϕSourceOrSink(X, Y, m) be the formula formalizing that (X, Y, m) as an +instance of SourceOrSink has a solution. Is it true that +K(∗PATH, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕSourceOrSink(X, Y, m)]] = 1? +5The problem to witness that α : {0, . . . , 2|x| − 1} → {0, . . . , 2|x|−1 − 1} is not injective. +23 + +Acknowledgement +This work is based on the author’s master’s thesis [5] which was completed under the +supervision of Jan Krajíček. The author also thanks Eitetsu Ken for comments on a draft +of this paper. +References +[1] +Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann +Pitassi. +The relative complexity of np search problems. +In Proceedings of the +Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’95, page +303–314, New York, NY, USA, 1995. Association for Computing Machinery. +[2] +Ronald Fagin. +Probabilities on finite models. +The Journal of Symbolic Logic, +41(1):50–58, 1976. +[3] +Isaac Goldbring. Lecture notes on nonstandard analysis, 2014. Available at https:// +www.math.uci.edu/~isaac/NSA%20notes.pdf (last accessed 9th of January 2023). +[4] +J. Hanika. Search problems and Bounded Arithmetic. PhD thesis, Charles University, +Prague, 2004. +[5] +Ondřej Ježil. Pseudofinite structures and limits. Master’s thesis, Charles University, +Prague, 2022. Also available in Electronic Colloquium on Computational Complex- +ity. +[6] +Emil Jeřábek. Integer factoring and modular square roots. Journal of Computer +and System Sciences, 82(2):380–394, 2016. +[7] +Jan Krajíček. Forcing with random variables and proof complexity, volume 382 of +London Mathematical Society Lecture Note Series. +Cambridge University Press, +2011. +[8] +László Lovász and Balázs Szegedy. Limits of dense graph sequences. Journal of +Combinatorial Theory, Series B, 96(6):933–957, 2006. +[9] +Nimrod Megiddo and Christos H Papadimitriou. On total functions, existence theo- +rems and computational complexity. Theoretical Computer Science, 81(2):317–324, +1991. +[10] Jaroslav Nešetřil and Patrice Ossona de Mendez. +A model theory approach to +structural limits. Commentationes Mathematicae Universitatis Carolinae, 53:581– +603, 11 2012. +[11] Alexander A. Razborov. Flag algebras. The Journal of Symbolic Logic, 72(4):1239– +1282, 2007. +24 + diff --git a/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/load_file.txt b/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..e1fdf5e520348cbf68062f68ca68b29a55c2aeb7 --- /dev/null +++ b/2NFRT4oBgHgl3EQfmzfK/content/tmp_files/load_file.txt @@ -0,0 +1,833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf,len=832 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13603v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='LO] 31 Jan 2023 Limits of structures and Total NP Search Problems∗ Ondřej Ježil ondrej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='jezil@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='cz Faculty of Mathematics and Physics, Charles University† Abstract For a class of finite graphs, we define a limit object relative to some computation- ally restricted class of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The properties of the limit object then reflect how a computationally restricted viewer “sees” a generic instance from the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The construction uses Krajíček’s forcing with random variables [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We prove sufficient conditions for universal and existential sentences to be valid in the limit, provide sev- eral examples, and prove that such a limit object can then be expanded to a model of weak arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We then take the limit of all finite pointed paths to obtain a model of arithmetic where the problem OntoWeakPigeon is total but Leaf (the complete problem for PPA) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This can be viewed as a logical separation of the oracle classes of total NP search problems, which in our setting implies standard nonreducibility of Leaf to OntoWeakPigeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 1 Introduction There exist several logical constructions of limits of classes of finite structures such as the ultraproduct and the compactness theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The latter was used in [2] to prove the 0–1 law for structures over relational vocabularies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In combinatorics there are also several notions of limits of finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For example the dense graph limit defined for a sequence of graphs {Gk}k>0 satisfying the condition that t(F, Gn) = |hom(F, G)| |Gn||F | , converges for every fixed connected graph F, where hom(F, G) denotes the set of all graph homomorphisms from F to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This provided a framework (see [8]) to restate and find new proofs for results in extremal graph theory — for instance Goodman’s theorem relating the number of edges to the number of triangles in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' There are other notions of limits of sequences of graphs, and we refer the interested reader to [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Another recent use of limit objects for the results of extremal combinatorics was by Razborov in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ∗ This work has been supported by Charles University Research Center program No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='UNCE/SCI/022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' † Sokolovská 83, Prague, 186 75, The Czech Republic 1 In this work, we define a new construction of a limit object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Given a class of finite graphs G, whose vertex sets are initial segments of N, we can stratify it into the sequence of sets {Gk}∞ k=1 as follows Gk = {G ∈ G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' G has {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1} as its vertex set}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Our construction would yield a pseudofinite structure if limk→∞|Gk| = 1, but an ordinary application of the compactness theorem suffices for that, we therefore generally care about the case, where limk→∞|Gk| = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1 We call such a sequence of sets of graphs a wide sequence and the limit object its wide limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let F be a class of functions with some computational restrictions, for example take F to be the set of functions computed by decision trees of some small depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define the wide limit denoted limF Gn, where n is a technical parameter to be defined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The wide limit limF Gn is a Boolean-valued graph2 — its edge relation does not only permit the truth values 0 and 1 but also many other values from some infinite complete Boolean algebra B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This algebra is in fact also a σ-algebra with a measure µ on it, so to any statement formulated as a first order sentence ϕ we can assign a real number µ([[ϕ]]) ∈ [0, 1] which measures how far is the truth value of ϕ (denoted [[ϕ]]) from the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The key method we use is arithmetical forcing with random variables, developed in [7], which allows us to construct models of (weak) arithmetical theories and by restricting to a language of graphs gives us Boolean-valued graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In these Boolean-valued graphs, validity of existential quantifiers corresponds to the ability of F to solve search problems over the class of graphs we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Our limit object can be expanded to the original model Krajíček’s method would otherwise construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We prove (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='8) that the truth values of first order sentences concerning the object are preserved even when evaluated in the model of arithmetic relativized to the wide limit (under a mild condition on the family F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' As an application of this construction, we take the limit of all finite paths starting at the vertex 0 relative to the class of functions computed by oracle trees of subex- ponential depth and obtain the Boolean-valued graph limFnb ∗PATHn which is an infi- nite path with only one endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This object is then expanded to a Boolean-valued model of weak second order arithmetic K(∗PATHn, Fnb, Gnb) in which every instance of OntoWeakPigeon has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' However, the object limFnb ∗PATHn in the model K(∗PATHn, Fnb, Gnb) is an instance of the PPA-complete problem Leaf which does not have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This can be seen as a logical analogue of an oracle separation of these two classes, which is known to hold3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We then show the result implies a separation of those classes under stronger notion of reducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 1The case where the limit tends to some other positive number results in a structure which after collapsing to a two-valued boolean algebra becomes pseudofinite too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 2Generally, we can do this with any L-structures for some first order language L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The limit object is then a Boolean-valued L-structure limF Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In this work we restrict ourselves to the language of graphs L = {E} to simplify the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 3OntoWeakPigeon can be reduced to WeakPigeon which is known to be in PPP [6] and it is known [1] that Leaf cannot be reduced to any problem in PPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 2 There is already an established connection between complexity of search problems and logic (namely bounded arithmetic, see [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The model we construct is not known nor ex- pected to be a model of any theory which has been considered under these investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' However, we show that open induction and open comprehension is valid in this model, and thus we show these principles along with the principle that OntoWeakPigeon is total cannot prove that the problem Leaf is total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The way the model is constructed also implies nonreducibility from Leaf to OntoWeakPigeon for subexponential time oracle machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Moreover, one can at least in theory tweak our construction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' by extending the family Fnb) to obtain a model of a stronger theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This has been success- fully done for several models already in [7, Chapter 10, Chapter 14, Chapter 21] using the switching lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 2 Preliminaries By graphs we mean structures in a language with a single binary relation denoted E which is antireflexive and possibly symmetric if the graph in question is undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will denote any particular graph by ω as it will be used in some sense as a sample of a discrete probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The edge relation of a particular graph ω will be denoted Eω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the rest of this section we recall notions needed for Krajíček’s forcing construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Fundamental notion we use throughout the work is of nonstandard models of (true) arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Lall be the language containing the names of all relations and functions on the natural numbers and let ThLall(N) denote the set of true sentences in this lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By classical results of logic there exist Lall-structures in which all sentences from ThLall(N) are valid but which are not isomorphic to N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' These are called nonstandard models (of ThLall(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' All nonstandard models of ThLall(N) (and even much weaker theories) contain an isomorphic copy of N as an initial segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, we can assume that in fact all models we encounter satisfy N ⊆ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' After considering a concrete nonstandard model M (of ThLall(N)) we shall call the elements of M\\N nonstandard numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' These can be intuitively understood as “infinite natural numbers”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The key feature of those elements is that all functions and relations from Lall are defined even on nonstandard numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This includes functions for coding sequences and sets by numbers, and therefore we can use notation like a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , an−1 even for a nonstandard number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The notation then means that for each i ∈ M such that i < n we have an object ai coded by a number in M and that this whole sequence is coded by some number {ai}n−1 i=0 ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For a nonstandard number S ∈ M coding a set we denote its nonstandard size (cardinality) to be |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the case where we talk about a binary string x the notation |x| denotes the bit length of x (which is nonstandard if x is).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the next section we will fix a nonstandard model M which has the model theoretic property that it is ℵ1-saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' There is a self-contained construction of such model in [7, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The only consequence of the ℵ1-saturation we shall use is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let {ai}∞ i=0 be a sequence of standard numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then there exists t ∈ M \\N 3 and a sequence {bi}t i=0 ∈ M such that for all i ∈ N it holds that ai = bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We shall call the sequence of {bi}t i=0 the nonstandard prolongation of {ai}∞ i=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The language Lall contains symbols for all relations on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Since every sequence of numbers can be defined by some relation it turns out that in our case there is a unique nonstandard prolongation which matches the definition of the wide sequence (up to length which can be chosen arbitrarily high).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We can therefore allow ourselves to use nonstandard numbers as indices of any sequences of objects unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Any nonstandard model M can be extended to an ordered ring ZM by adding neg- ative elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This ring then can be extended to a fraction field QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We shall call elements of QM M-rationals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The field QM contains an isomorphic copy of Q as a sub- structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We call an element in QM with absolute valued greater than all k 1, k ∈ N, infinite otherwise we call it finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We call elements in QM with absolute value smaller than all 1 k, k ∈ N infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will denote the set of finite M-rationals as QM fin and one can check it forms an ordered ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lemma (The existence of a standard part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' There is a function st : QM fin → R assigning to each finite M-rational a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' st is a ring homomorphism and the kernel of st is exactly the ideal of infinitesimal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' When q is a finite M-rational we call st(q) its standard part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We shall use the structure QM analogously to how hyperreal numbers are used in nonstandard analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For more details about nonstandard analysis we recommend [3] to the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The following result characterizes convergence of sequences of rational numbers using QM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let {ai}∞ i=0 be a sequence of rational numbers and let r ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then the following are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' limi→∞ ai = r For every {bi}t i=0, t ∈ M\\N, which is a nonstandard prolongation of {ai}∞ i=0, there is an nonstandard s0 ≤ t, such that for every nonstandard s ≤ s0: st(as) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' It is important for forcing with random variables to consider discrete probability spaces of nonstandard size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We shall always use uniform distribution on the samples (although this is not necessary for the general construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Thus, the probability of an event coded by an element A ∈ M is then just the M-rational number |A|/|S| where S is the set of samples of such a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We conclude this section by restating classical inequalities used in this work using the nonstandard approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem (Bernoulli’s inequlity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let y ∈ M, x ∈ QM and x ≥ −1, then (1 + x)y ≥ 1 + yx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem (Exponential inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let x ∈ M \\ N, then st �� 1 − 1 x �x� ≤ e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 4 3 Wide limits 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1 The definition We shall define a wide limit of every sequence of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' A sequence of sets of graphs {Gk}∞ k=1 is called a wide sequence if the following holds: Every graph ω ∈ Gk has the vertex set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' limk→∞|Gk| = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By abuse of notation we will simply talk about a wide sequence Gk instead of {Gk}∞ k=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Since a wide limit is a Boolean-valued graph, we need to construct a Boolean algebra in which the truth evaluation of statements shall take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For the construction of the Boolean algebra we will closely follow [7, Chapter 1] albeit with slight changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let us now fix for the rest of this work an ℵ1-saturated model of ThLall(N) which we will denote M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let n ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define An = {A ⊆ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' A ∈ M}, in words An is the set of subsets of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} coded by an element in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is a boolean algebra and to each A ∈ An we assign an M-rational |A|/n which we call its counting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Even though An is a boolean algebra with a “measure” it is not a σ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Indeed, An contains all singletons, but the countable set of those elements in {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} with only finitely many predecessors is not definable by compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' However, having infinite joins and meets at our disposal allows us to interpret quantifiers in the boolean valued case, so we now want to ‘tweak’ this Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let I be the ideal of An consisting of elements with infinitesimal count- ing measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Bn = An/I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Each element in Bn is of the form A/I, where A ∈ An, and we define µ(A/I) = st(|A|/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will denote the maximal element of Bn by 1 and the minimal element by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' One can easily check that µ is well-defined since for all A ∈ I it holds that st(|A|/n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The measure µ is called the Loeb measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The following then holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='4 ( [7, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Bn is a σ-algebra with a real valued measure µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' More- over, Bn is a complete boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' It is important to note that 1 ∈ Bn is the only element of Bn with measure µ(1) = 1 and similarly 0 ∈ Bn is the only element with measure µ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Also, for B, B′ ∈ Bn the inequality B ≤ B′ implies µ(B) ≤ µ(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5 We now define precisely what we mean by the family of functions F relative to which we will be taking the wide limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is still a part of Krajíček’s construction, we just modify it to make it compatible with our setup — where we start with a wide sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For every k ∈ N the set Gk is finite and thus can be coded by a number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, there is a nonstandard prolongation of this sequence, and we can consider the set coded by the nonstandard number Gn, which matches the definition of the wide sequence in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let {Gk}∞ k=1 be a wide sequence and let n ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say that F is a family of random variables on Gn if every α ∈ F is a function coded by a number in M with domain Gn and taking values in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say α ∈ F is an F-vertex if for all ω ∈ Gn it holds that α(ω) ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The set of all F-vertices is denoted U(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If the wide sequence {Gk}∞ k=1 and the number n ∈ M \\ N is clear from context we just say F is a family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is for now everything we need to recall from [7], and we can proceed to define the central object of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='6 (The wide limit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let {Gk}∞ k=1 be a wide sequence, let n ∈ M \\ N and let F be a family of random variables on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define the wide limit limF,n{Gk}∞ k=1 as a Bn-valued structure in the language consisting of a single binary relation symbol {E} as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The universe of the wide limit is taken as the set of all F-vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We now inductively define the truth values for all {E}-sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [[α = β]] = {ω ∈ Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' α(ω) = β(ω)}/I [[E(α, β)]] = {ω ∈ Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Eω(α(ω), β(ω))}/I [[ − ]] commutes with ¬, ∧ and ∨ [[(∃x)A(x)]] = � α∈U(F ) [[A(α)]] [[(∀x)A(x)]] = � α∈U(F ) [[A(α)]] By abuse of notation we will denote the wide limit limF,n{Gk}∞ k=1 by limF Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' To stress in which boolean valued structure is the truth evaluation [[ − ]] taking place we will sometimes denote the evaluation C1[[ − ]], C2[[ − ]] for boolean valued structures C1 and C2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Furthermore, if C1[[ϕ]] = 1 for some sentence ϕ we say ϕ is valid in C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Note that since Gn can be recovered from F as the domain of its elements the wide limit strictly speaking only depends on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We keep Gn in the notation to cover the situation where we have a very general family of functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' the family of polynomial functions FPV) which can be applied to every wide sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Thus, the notation limF Gn means that F is restricted to those functions which take elements of Gn as an input even when F possibly contains other functions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The variability of the parameter n may also seem unnecessary and indeed in our applications it is the case, but generally there are examples of wide sequences where n directly affects the wide limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 6 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Fconst be the family of all constant functions with domain Gn and range anywhere in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gk = � {({0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}, E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' |E| = 2, (0, 1) ∈ E} k even {({0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}, E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' |E| = 1, (0, 1) ̸∈ E} k odd then lim Fconst Gn[[E(0, 1)]] = � 1 n even 0 n odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2 An example of a wide limit relative to shallow decision trees Now we shall define the first nontrivial family of random variables relative to which we shall take wide limits of several sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The functions in the family will be computed by shallow decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' So the shape of the wide limit reflects what can ‘superloga- rithmic’ trees witness in the wide sequence with probability arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Trud be a family of labeled rooted binary trees in M of the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' At each vertex the tree is labeled by an element of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} × {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} and the two outgoing edges incident to it are labeled as 0 and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The leaves are labeled by an element of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The depth of the tree is bounded by a number of a form n1/t (rounded to the nearest element of M) for some t ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' A computation of a T ∈ Trud on some ω ∈ Gn is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Start at the root and interpret each label (i, j) of the vertex as a question whether the pair (i, j) is in the edge set Eω and follow a path through T reading 1 as a positive answer and 0 as a negative answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The label of the leaf visited at the end of the path is the output of T on ω, denoted T(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Frud to be the set of all functions computed by a tree T ∈ Trud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The simplest wide sequence we shall consider is the following sequence of sets of undirected graphs with exactly one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' EDGEk = {({0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}, E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' |E| = 1} Since any ω ∈ EDGEk has only 1 edge in all potential k · (k − 1)/2 edges, it is not likely a shallow tree will find the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is the idea behind the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' lim Frud EDGEn[[(∃x)(∃y)E(x, y)]] = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let α, β ∈ U(Frud), we proceed by proving that [[E(α, β)]] = 0 which is enough to prove the theorem since even an infinite disjunction of the values 0 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 7 Let α and β be computed by T ∈ Trud and S ∈ Trud respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let the depth of both T and S be at most n1/t, where t ∈ M \\N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Walk down T from the root and always prolong the path along the edge labeled 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' On this path we have a set of at most n1/t different pairs of vertices and a label of the leaf lT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We do the same for S, and we find another set of at most n1/t pairs of vertices and a label of the leaf lS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' lS and lT are then combined to one last pair (lS, lT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now we just need to compute the probability that none of these 2n1/t + 1 pairs of vertices are not in the edge set Eω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' There are �n 2 � different graphs in EDGEn and �n−4n1/t−2 2 � graphs which fulfill our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The probability is thus �n−4n1/t−2 2 � �n 2 � = (n − 4n1/t − 2)(n − 4n1/t − 3) n(n − 1) ≥ (n − 4n1/t − 3)2 n2 ≥ � 1 − 4n1/t + 3 n �2 ≥ � 1 − 8n1/t + 6 n � after taking the standard part of the last line we get st(1 − 8n1/t+6 n ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, µ([[E(α, β)]]) = 0 and [[E(α, β)]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='3 Sufficient conditions for validity of existential and universal sen- tences To analyze wide limits we need ideally to know the values of sentences which describe properties whose complexity we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Generally this can be hard, so in this section we prove sufficient conditions at least for the validity of universal and existential sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will start with the simpler condition for the validity of universal sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is important also because we would like to know that a wide limit of a wide sequence of graphs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' antireflexive {E}-structures) is also a graph and that a wide limit of a wide sequence of undirected graphs (directed graphs with E symmetric) is an undirected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' All of these properties are expressible as universal sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gk be a wide sequence and let F be any family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let ϕ(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , xl−1) be an open {E}-formula and assume that lim k→∞ Pr ω∈Gk [ω |= (∀x)ϕ(x)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then limF Gn[[(∀x)ϕ(x)]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 8 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By induction in M we have that st(Prω∈Gn[ω |= (∀x)ϕ(x)]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, we have for every tuple of F-vertices α that [[ϕ(α)]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now [[(∀x)ϕ(x)]] = � α∈U(F )l [[ϕ(α)]] = � α∈U(F )l 1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gk be a wide sequence and F any family of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If all ω ∈ Gk, k ∈ N, are directed graphs ({E}-structures satisfying that E is antire- flexive) then limF Gn is a Boolean-valued {E}-structure in which the antireflexivity of E is valid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' limF Gn is a Boolean-valued graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If all ω ∈ Gk, k ∈ N, are undirected graphs (directed graphs where E is symmetric) then limF Gn is an {E}-structure in which both antireflexivity and symmetry of E is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' limF is a Boolean-valued undirected graph) Now to give a sufficient condition for the validity of an existential sentence (∃x)ϕ(x) we use the auxiliary value of density of ϕ(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , xl−1) defined as the probability that a random graph ω ∈ Gk and a random tuple a ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}l satisfy ω |= ϕ(a) and show that the limiting density gives a lower bound for the measure of [[(∃x)ϕ(x)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gk be a wide sequence and let F be a family of random variables which contains all constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let ϕ(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , xl−1) be an open {E}-formula and let p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Assume that lim k→∞ Pr ω∈Gk a [ω |= ϕ(a)] ≥ p, where a is sampled uniformly over all elements of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then µ(lim F Gn[[(∃x)ϕ(x)]]) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In particular if p = 1 then limF Gn[[(∃x)ϕ(x)]] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Consider an array C indexed by ω ∈ Gn and a ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1}l such that Cω,a = � 1 ω |= ϕ(a) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By the assumption and induction in M we have that st � 1 nl|Gn| � ω∈Gn � a Cω,a � ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 9 We now claim that there exists a specific b ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n−1}l such that st(Prω∈Gn[ω |= ϕ(b)]) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Assume for contradiction that the claim is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then 1 |Gn|nl � ω∈Gn � a Cω,α = 1 nl � a Pr ω∈Gn[ω |= ϕ(a)] ≤ Pr ω∈Gn[ω |= ϕ(a0)], where we pick a0 such that it maximizes Prω∈Gn[ω |= ϕ(a0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' But after taking the standard part of the inequality we obtain that st � 1 nl|Gn| � ω∈Gn � a Cω,a � ≤ st( Pr ω∈Gn[ω |= ϕ(a0)]) < p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Which is a contradiction and so the claim is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let γb be a tuple of constant functions which is at every sample equal to b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We have [[(∃x)ϕ(x)]] = � α∈U(F )l [[ϕ(α)]] ≥ [[ϕ(γb)]] and by taking µ of this inequality we finally obtain that µ([[(∃x)ϕ(x)]]) ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the following example we use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13 to show that in the wide limit of graphs which have exactly one large clique and no other edges the nonexistence of a standard sized clique cannot be valid relative to any F with all constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Consider the wide sequence SK1/2 k = {({0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1}, E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' E has a clique of size ⌊k/2⌋ and no other edges}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will check that for an {E}-formula ϕl(x) which states that x forms a clique of size l we have lim k→∞ Pr ω∈SK1/2 k a [ω |= ϕl(a)] ≥ (1/2)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Notice that we can compute the probability for a fixed a such that ai ̸= aj whenever i ̸= j, since the ratio of tuples containing some vertex twice is infinitesimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' So we have Pr ω∈SK1/2 k [ω |= ϕl(a)] = l−1 � i=0 � 1 − k − ⌊k/2⌋ k − i � ≥ � 1 − k − ⌊k/2⌋ k − l �l ≥ � 1 − 1 2(1 − l/k) − 1 k − l �l 10 and since l ∈ N we just take the limit of the inner expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' But one can see that limk→∞(1 − l/k) = 1 and that limk→∞(1/(k − l)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13 we obtain that for any F that contains all constants we have lim F SK1/2 n [[(∃x)ϕl(x)]] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The following example demonstrates that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='11 cannot be generalized to a similar hypothesis as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gk consist of all undirected graphs on the vertex set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1} with exactly ⌈ k(k−1) 2 log(k)⌉edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' One can see that lim k→∞ Pr ω∈Gk x,y [ω |= ¬E(x, y)] = 1, but in fact limFrud Gn[[(∀x)(∀y)¬E(x, y)]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let t ∈ M \\ N such that n1/t is not bounded above by a standard number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let T be a tree which queries on all paths a fixed set of n1/t different potential edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If we prove that any such set in Gn has to contain at least one edge with probability infinitesimally close to 1 then we can construct Frud-vertices α and β using T such that [[E(α, β)]] = 1 by simply taking T and labeling each leaf on a path which finds an edge with one of the vertices incident to this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let S be the set of potential edges queried by T and let m = �n 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now we have Pr ω∈Gn[S contains no edge in ω] = (m − n1/t)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' (m − ⌈ m log n⌉!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=') m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' (m − ⌈ m log m⌉ − n1/t)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' = n1/t−1 � i=0 m − ⌈ m log n⌉ − i m − i ≤ � 1 − ⌈ m log n⌉ m �n1/t ≤ � 1 − 1 2 log n �n1/t standard part of which is for all k ∈ N bounded above by st �� 1 − 1 2 log n �k·2 log n� ≤ e−k which tends to 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 11 4 A wide limit of Leaf instances relative to oracle trees The class of total NP search problems TFNP, first defined in [9], consists of all relations on binary strings P(x, y) such that: (verifiability in polynomial time) There is a polynomial time machine M which, given x, y, can decide whether P(x, y) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' (totality) There exists a polynomial p and for every x there exists at least one y satisfying |y| ≤ p(|x|) such that P(x, y) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Two particular problems are relevant for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The problem Leaf is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' An instance is given by a number k and a undirected graph ω on the vertex set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|k| − 1}, presented by a Boolean circuit of polynomial size in |k| computing its neighborhood function, such that degω(0) = 1 and ∀v : degω(v) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The task is then to find some nonzero v with degω(v) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The corresponding combinatorial principle being the handshaking lemma, which assures the problem is total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The problem OntoWeakPigeon is formulated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' An instance is given by a number k and two functions A : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|k| − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|k|−1 − 1} and B : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|k|−1 − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|k| − 1}, each presented by a Boolean circuit of polynomial size in |k|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The task is then to find some x such that B(A(x)) ̸= x or some y such that A(B(y)) ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The corresponding combinatorial principle being the bijective weak pigeonhole principle, which assures the problem is total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The domain of A is twice as large as its range, so B and A cannot form a pair of inverse functions between their respective domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' So far, we presented what is called ‘type 1’ problem in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We are interested about the ‘type 2’ problems which replace the input function(s) with oracle(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' So in the ‘type 2’ Leaf problem, the input is a pair (α, x) where α is an oracle which describes the neighbor function on G with vertex set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For the ‘type 2’ OntoWeakPigeon problem, the input is a triple (α, β, x), where α and β are oracles describing the functions α : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x|−1 − 1} and β : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x|−1 − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The associated computational models for the ‘type 1’ problems are Turing machines and for the ‘type 2’ problems oracle Turing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1 The wide limit and oracle trees The wide sequence ∗PATHk (pointed paths on k vertices) consists of all undirected graphs ω on the vertex set {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , k − 1} which are isomorphic to a path with k − 1 edges connecting all vertices and degω(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Graphs in ∗PATHk are ‘the hardest instances of Leaf’ so we can expect the wide limit to reflect the complexity of these instances relative to the family F we choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Since each ω ∈ ∗PATHk has only k − 1 edges we can proceed similarly to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10 to get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' limFrud ∗PATHn[[(∃x)(∃y)E(x, y)]] = 0 12 To get a result which reflects the properties of the wide sequence more faithfully we will define a new family of random variables on ∗PATHn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Tnb as the set of all labeled rooted trees of the following shape: Each non-leaf node is labeled by some v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For each {u, w} ⊆ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} and a node v there is an outgoing edge from v labeled {u, w} (it can be that u = w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Each leaf is labeled by some m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The depth of the tree is defined as the maximal number of edges in a path from the root, and we require it is at most n1/t (rounded to the nearest element of M) for some t ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The computation of such a tree in Tnb on ω ∈ ∗PATHn is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We build a path by starting at the root and interpreting every vertex labeled by some v as a question ‘what are the neighbors of the vertex v?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' and we follow the output edge with the answer and continue analogously until we find a leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The label of the leaf is defined to be the output of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Fnb to be the set of all functions on ∗PATHn which are computed by some T ∈ Tnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The trees computing the functions in Fnb can be thought of as a protocol describing the behavior of a machine M communicating with an oracle describing a particular ω ∈ ∗PATHn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the study of total NP search problems presented by oracles, we usually denote the size of the object by some 2|x| where x is an additional input to the problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If 2|x| = n then n1/t = 2|x|/t which for t ∈ M \\ N corresponds to protocols describing non-uniform subexponential-time computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If we prove that no tuple of Fnb-vertices satisfies some open {E}-formula in limFnb ∗PATH we also prove that subexponential-time oracle machines cannot solve the corresponding type 2 problem on a non-diminishing fraction of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the rest of this section we proceed to prove that limFnb ∗PATHn has no vertex with degree 1 other than the vertex 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' To do so, we will consider computations of trees on samples with different nonstandard lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For the rest of this section we put Gm = ∗PATHm for all m ∈ M, but we can assume m to be smaller than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define T (m) nb to be the subset of Tnb consisting of all the trees that have the vertex labels from {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For trees in T (m) nb we can extend the definition of a computation to input graphs from Gm in a straight forward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say a tree T ∈ T (m) nb fails on ω ∈ Gm if the output of T on ω has degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let m ∈ M, v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} and {u, w} ⊆ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} we define Gv:{u,w} m = {ω ∈ Gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ω |= E(v, u) ∧ E(v, w)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 13 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let m ∈ M and let u, v and w be distinct elements of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then there are bijections: Gv:{u,w} m ∼= Gm−2 × {L, R} Gv:{u,0} m ∼= Gm−2 Gv:{u} m ∼= Gm−1 G0:{u} m ∼= Gm−1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For the first case a bijection can be given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Contract u, v and w to just one vertex min{u, v, w} and if u is closer to 0 than w pick L otherwise pick R and relabel the remaining vertices using a function ‘new’ which has a property that if u′, v′ remain and u′ < v′ as numbers then new(u′) < new(w′) and the range of new is {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This can be inverted by first renaming the vertices using new−1 and then replacing min{u, v, w} by a path (u, v, w) with the orientation given either by L or R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The second bijection is almost the same, but the orientation is clear since u is always the neighbor further from 0 since the other neighbor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The third and fourth bijections are given by just removing one end of the graph and relabeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let m ∈ M and v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let u and w be elements of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} \\ {v} and let T ∈ T (m) nb be a tree with the root labeled v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By Tv:{u,w} we denote the induced subtree whose root is the vertex neighboring the root of T via the edge labeled {u, w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let T ∈ T (m) nb be a tree with the root labeled v ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For each u and w which are distinct elements of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} \\ {v} there exists a tree ˜Tv:{u,w} ∈ T (m−2) nb of the same depth as Tv:{u,w} such that Pr ω∈Gm[Tv:{u,w} fails | ω |= E(v, u) ∧ E(v, w)] = Pr ω∈Gm−2[ ˜Tv:{u,w} fails].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If T has the root labeled 0 then there exists a tree ˜T0:{u} ∈ T (m−1) nb of the same depth as T0:{u} such that Pr ω∈Gm[T0:{u} fails | ω |= E(0, u)] = Pr ω∈Gm−1[ ˜T0:{u} fails].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the case where the root is labeled by v ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} we can construct the tree ˜Tv:{u,w} by simply relabeling vertices of Tv:{u,w}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We use the relabeling function ‘new’ from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now for every ω ∈ Gm there is by the first bijection in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5 a uniquely determined ω′ ∈ Gm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The computation of ˜Tv:{u,w} on ω′ is then of the same shape as the computation of Tv:{u,w} on ω assuming ω |= E(v, u) ∧ E(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' And ˜Tv:{u,w}(ω′) has the same degree in ω′ as Tv:{u,w}(ω) does in ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The case where the root is labeled by 0 is analogous, but we instead use the relabeling from the fourth bijection in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 14 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let T ∈ T (m) nb of depth d ∈ M and let d ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then we have Pr ω∈Gm[T fails] ≥ d � i=0 � 1 − 2 m − 2i − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We proceed by induction on d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The case d = 0 follows from Pr ω∈Gm[T fails] ≥ � 1 − 1 m − 1 � ≥ � 1 − 2 m − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now for the inductive case we assume the lemma holds for d − 1, and prove it for d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If the root of T is labeled 0 we proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For a given T let u0 be the vertex which minimizes the value Prω∈Gm[T fails | E(0, u0)] which exists by the least number principle in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='7 and the induction hypothesis Pr ω∈Gm[T fails] ≥ Pr ω∈Gm[T fails | E(0, u0)] = Pr ω∈Gm−1[ ˜T0:{u0} fails] ≥ d−1 � i=0 � 1 − 2 m − 2i − 3 � ≥ d � i=0 � 1 − 2 m − 2i − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now for the case where the root of T is labeled by nonzero v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' First we note that Pr ω∈Gm[v has degree 2 ∧ ¬E(v, 0)] = 1 − 2 m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now we choose distinct u0, w0 such that they minimize Pr ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then by the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='7 and the induction hypothesis we have Pr ω∈Gm[T fails] ≥ � 1 − 2 m − 1 � Pr ω∈Gm[Tv:{u0,w0} fails | E(v, u0) ∧ E(v, w0)] = � 1 − 2 m − 1 � Pr ω∈Gm−2[ ˜Tv:{u0,w0} fails] ≥ � 1 − 2 m − 1 � d−1 � i=0 � 1 − 2 m − 2i − 4 � ≥ � 1 − 2 m − 1 � d � i=1 � 1 − 2 m − 2i − 2 � ≥ d � i=0 � 1 − 2 m − 2i − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 15 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let T ∈ Tnb, then st (Prω∈Gn[T fails]) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The depth of T is bounded by n1/t for some t ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We have by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='8 that Pr ω∈Gn[T fails] ≥ n1/t � i=0 � 1 − 2 n − 2i − 2 � (1) ≥ � 1 − 2(n1/t + 1) n − 2n1/t − 2 � (2) and the standard part of this lower bound is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Finally, in the next theorem we prove that a formalization of ‘there is a nonzero vertex of degree 1’ is not valid in limFnb ∗PATHn and in fact its boolean value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' lim Fnb ∗PATHn[[(∃v)(∃u)(∀w)(v ̸= 0 ∧ E(v, u) ∧ (E(v, w) → w = u))]] = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By expanding the left-hand side of the statement we get � α∈U(Fnb) � β∈U(Fnb) � γ∈U(Fnb) [[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, it is enough if we prove that for each Fnb-vertices α and β there exists an Fnb-vertex γ such that [[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' For any α, β ∈ U(Fnb) we can append the tree computing β to every leaf of a tree computing α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is still a tree in Tnb as its depth is at most twice the maximum of depths of the original trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By relabeling the leaves of the resulting tree we can obtain a tree computing a function γ(ω) = � v if degω(α(ω)) = 1 and v is the only neighbor of α(ω) w if degω(α(ω)) = 2, w is a neighbor of α(ω) and w ̸= β(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is obviously an Fnb-vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let us assume for contradiction that [[α ̸= 0 ∧ E(α, β) ∧ (E(α, γ) → γ = β)]] > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By definition this gives us 0 < st � Pr ω∈Gn[α(ω) ̸= 0 ∧ Eω(α(ω), β(ω)) ∧ (E(α(ω), γ(ω)) → γ(ω) = β(ω))] � ≤ st � Pr ω∈Gn[α(ω) ̸= 0 ∧ degω(α(ω)) = 1] � , but this is in contradiction with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 16 5 The expanded model with a Leaf instance without a so- lution and with total OntoWeakPigeon As a part of the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10 we proved what can be reformulated as the statement that oracle instances of Leaf are not in oracle time O(2f(|x|)) with f ∈ o(|x|1/c) for every c ∈ N even when we just require it to be correct on any nondiminishing ratio of inputs as |x| grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In this section we proceed to compare strength of (type 2) NP search problems not only with oracle FP but also with other NP search problems via relative consistency of their totality and nontotality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will show that there is a model of weak second order arithmetic in which the problem Leaf is not total even though OntoWeakPigeon is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1 The structures K(F, G) We will now recall the construction of second order models of weak arithmetic K(F, G) defined in [7, Chapter 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will take the liberty to define them as an extension of the definition of a wide limit to obtain structures K(Gn, F, G) 4 which under the right conditions result in a structure in some sublanguage of Lall with two sorts: numbers and bounded sets of numbers which contains the wide limit as an object of the second sort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let L ⊆ Lall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This determines a language L2 which we get by adding to L second order variables X, Y, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' whose intended interpretation are bounded sets and the equality symbol for second order variables (denoted the same as the first order one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' All second order variables are treated as function symbols and can form terms with the first order terms as arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will also use the second order variables as relation symbols, and we define the atomic formula X(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , xk−1) simply to be evaluated as the formula X(x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , xk−1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now we assume we fix a number n, a wide sequence Gk and a family of random variables on Gn which all together determine a wide limit limF Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Mn ⊆ M to be the subset of M consisting of all numbers bounded above by 2n1/t for some t ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define Ln ⊆ Lall to contain all relation symbols from Lall and all functions from Lall for which their values on any element of Mn is still in Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say F is Ln-closed if for every function symbol f ∈ Ln we have that f(α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , αk−1) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Note that Mn is then a substructure of the Ln-reduct of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say that G is a family of random functions (on Gn) if every Θ ∈ G assigns to each ω ∈ Gn a function Θω ∈ Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 4This notation is just making some parameters of the construction explicit, the models constructed can be obtained by the original method without first constructing the wide limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Our contribution is in observing that the truth values of first order sentences concerning the wide limit is preserved between the wide limit and the structure K(Gn, F, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 17 We say G is F-compatible if for every α ∈ F, Θ ∈ G we have that the function Θ(α) defined as Θ(α)(ω) = � Θω(α(ω)) if α(ω) ∈ dom(Θω) 0 otherwise is in fact in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let F be an Ln-closed family of random variables with values in Mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let G be an F-compatible family of random functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define K(Gn, F, G) to be a Bn-valued L2 n structure with first order sort of the universe F and second order sort of the universe G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The valuation of formulas is then given by the following inductive definition: [[α = β]] = {ω ∈ Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' α(ω) = β(ω)}/I, where α, β ∈ F [[R(α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , αk−1)]] = {ω ∈ Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ω |= R(α0(ω), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , αk−1(ω))}/I, where α0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , αk−1 are from F and is R a relation symbol in Ln [[Θ = Ξ]] = {ω ∈ Gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Θω = Ξω}/I, where Θ, Ξ ∈ G [[(∀x)A(x)]] = � α∈F [[A(α)]] [[(∃x)A(x)]] = � α∈F [[A(α)]] [[(∀X)A(X)]] = � Θ∈G [[A(Θ)]] [[(∃X)A(X)]] = � Θ∈G [[A(Θ)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2 Preservation of sentences concerning the wide limit We will now prove (under a mild condition on F) that in a structure K(Gn, F, G) which represents the wide limit limF Gn by a second order object are the values of all sentences regarding the object the same as in the wide limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This lets us construct models in which an object elementary equivalent to the wide limit might be desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say that the edge relation of the wide limit limF Gn is represented in G by Γ if Γ ∈ G and for all α, β ∈ U(F) we have that K(Gn, F, G)[[Γ(α, β)]] = lim F Gn[[E(α, β)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say a family of random variables F has restrictable ranges if for every α ∈ F and m ∈ Mn there is ˜αm ∈ F such that ˜αm(ω) = � α(ω) α(ω) < m 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 18 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let ϕ be a {E}-sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let F be Ln-closed and have restrictable ranges and let G be F-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let the edge relation of the wide limit limF Gn be represented in G by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We define ˜ϕ(Γ) to be the L2 n-sentence obtained by replacing all the occurrences of the relation symbol E by Γ, keeping the structure of the logical connectives and replacing all quantifiers (∀x)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ) by (∀x)(x < n → (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' )) and (∃x)(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ) by (∃x)(x < n ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then we have that for all {E}-sentences that lim F Gn[[ϕ]] = K(Gn, F, G)[[ ˜ϕ(Γ)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will prove that for all {E}-formulas ϕ(x) and all α ∈ F we have that lim F Gn[[ϕ(α)]] = K(Gn, F, G)[[ ˜ϕ(Γ, α)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We proceed by induction on the complexity of the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The case for atomic formulas is clear and the step for logical connectives also since [[ − ]] commutes with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' With the induction step for negation in hand it is now enough to prove the inductive step for the universal quantifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We assume that the statement works for a formula of lower complexity ϕ(y, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By the restrictability of ranges in F we get that for all β ∈ F there is ˜βn ∈ U(F) such that K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]] ≤ K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now we have that for all α ∈ U(F) K(Gn, F, G)[[(∀y) ˜ϕ(Γ, y, α)]] = � α∈F K(Gn, F, G)[[β < n → ˜ϕ(Γ, β, α)]] = � ˜βn∈U(F ) K(Gn, F, G)[[ ˜ϕ(Γ, ˜βn, α)]] = � ˜βn∈U(F ) lim F Gn[[ϕ(˜βn, α)]] = lim F Gn[[(∀y)ϕ(y, α)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='3 Failure of totality of Leaf Now we are in a situation that lets us construct a model of weak second order arithmetic that contains an instance of the problem Leaf without a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Consider a suitable family Gnb in which we can define not only the wide limit itself but instances of some other search problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We can then ask: ‘Do all these instances have a solution?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This is a way to compare the strength of the different total NP search problems by relative unprovability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We will pick the family Gnb such that validity of totality of some search problem P implies the nonexistence of a suitable reduction from Leaf to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 19 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Gnb be the family of all random functions on ∗PATHn such that for each Θ ∈ Gnb there exists a tuple (γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , γm−1) ∈ M so that γi ∈ Fnb and Θ(α)(ω) = � γα(ω)(ω) α(ω) < m 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the models Mn we are working with there is a pairing function ⟨i, j⟩ which can code pairs of numbers by a single number thus we can represent functions of any finite arity by functions from Gnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' One can understand the tuples which compute the random functions from Gnb as tuples of protocols describing the computations of subexponential oracle machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Such a tuple defines a function which is at every index of the tuple computed using queries to some ω ∈ ∗PATHn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If we prove that every instance of a search problem P represented by such a tuple has a solution in K(∗PATHn, Fnb, Gnb), and we know that Leaf in K(∗PATHn, Fnb, Gnb) is not total, which implies nonexistence of a subexponential oracle machine which converts solutions of P to solutions of Leaf even on any standard fraction of instances from ∗PATHn and thus a nonexistence of a many-one reduction from Leaf to P as defined in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Fnb has restrictable ranges 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Fnb is Ln-closed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Gnb is Fnb-compatible 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Gnb represents the edge relation of limFnb ∗PATHn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 1, 2: Here we can proceed simply by relabeling the leaves of the trees computing the functions from Fnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 3: Assume that Θ ∈ Gnb is computed by a tuple (γ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , γm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By induction in M there exists t ∈ M \\ N such that ∀i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} the depth of γi is at most n1/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, for all α ∈ Fnb we have that Θ(α) has also depth at most n1/t′ for some t′ ∈ M \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Thus, Gnb is Fnb-compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 4: Let γ⟨i,j⟩ ∈ Fnb be computed by a tree in Tnb which queries i and outputs 1 if the neighbor set contains j otherwise it outputs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Γ be computed by a tuple (γ⟨i,j⟩)n−1 i,j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then we have K(∗PATHn, Fnb, Gnb)[[Γ(α, β)]] = lim Fnb ∗PATHn[[E(α, β)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The L2 n-formula ϕLeaf(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' m) is defined as the disjunction of the following formulas (X(0) ̸= Y (0) ∨ X(0) = 0) (∃x)((x < m) ∧ (X(x) > m − 1 ∨ Y (x) > m − 1)) (∃x)((x < m) ∧ ((X(x) = x ∧ Y (x) ̸= x) ∨ (X(x) ̸= x ∧ Y (x) = x))) (∃x)((x < m) ∧ (Y (X(x)) ̸= x ∧ X(X(x)) ̸= x) ∨ (X(Y (x)) ̸= x ∧ Y (Y (x)) ̸= x))) (∃x)((0 < x < m) ∧ (X(x) = Y (x) ∧ X(x) ̸= x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 20 this formula formalizes that if X and Y are functions representing the neighbor set of each x < m as {X(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Y (x)} \\ {x} and 0 has only one neighbor then there has to exist another y < x which also has only one neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' K(∗PATHn, F, G)[[(∃X)(∃Y )(∃m)¬ϕLeaf(X, Y, m)]] = 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We can find Θ1, Θ2 ∈ Gnb such that for each v ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , n − 1} we have that {Θ1(v)(ω), Θ2(v)(ω)} is the neighbor set of v on ω ∈ ∗PATHn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' (We can just query v and split the answer between Θ1 and Θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=') By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10 we know that limF Gn has one degree 1 vertex and all other vertices of degree 2 and by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10 we know that it can be represented by some Γ ∈ Gnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Furthermore, we can verify that [[(Γ(α, β)) ≡ (Θ1(α) = β ∨ Θ2(α) = β)]] = 1, thus Θ1 and Θ2 do not satisfy the last disjunct of ϕLeaf otherwise it would be in contra- diction with Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By their construction and the definition of ∗PATHk we have that (Θ1, Θ2, n) does not satisfy the other disjuncts either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='4 Totality of OntoWeakPigeon Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The L2 n formula ϕOntoWeakPigeon(X, Y, m) is defined as the disjunc- tion of the following formulas (∃x)((x < 2m) ∧ (X(x) > m − 1)) (∃y)((y < m) ∧ Y (y) > m − 1)) (∃x)((x < 2m) ∧ Y (X(x)) ̸= x) (∃y)((y < m) ∧ X(Y (y)) ̸= y) it formalizes the bijective weak pigeonhole principle which claims that any pair of func- tions X :{0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2m − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} Y :{0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , m − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 2m − 1} is not a pair of inverse bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' To prove that ϕOntoWeakPigeon(X, Y, m) is valid in K(∗PATHn, Fnb, Gnb) we will construct a tree which finds some x such that Yω(Xω(x)) ̸= x or Xω(x) > m − 1 with probability infinitesimally close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Θ, Ξ ∈ Gnb, and ζ ∈ Fnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We say that a tree T ∈ Tnb fails for (Θ, Ξ, ζ) on ω if Θω(T(ω)) < ζ(ω) and Ξω(Θω(T(ω))) = T(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In words if T does not witness the failure of Ξ being the inverse function to Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 21 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let Θ, Ξ ∈ Gnb and ζ ∈ Fnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then there is a tree T such that st � Pr ω∈Gn[T fails for (Θ, Ξ, ζ)] � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Without loss of generality we may assume that ζ is actually constant, and its value is r ∈ Mn which we pick to be the least possible output of ζ on any sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Furthermore, let Θ be computed by (θ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , θ2r−1) and Ξ by (ξ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , ξr−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We construct T by stages and at each stage it will have some potential output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' First we notice that at the beginning stage there is at least one i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2r − 1} such that the probability that θi < r or ξθi = i is at most 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The tree T0 is thus the constant tree always outputting i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Assume Td−1 have been constructed and pick any path p ∈ Td−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If p did not fail we leave it as it is otherwise we extend Td−1 along this path and after extending all such paths this will become the new stage Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The path p has a leaf with some label i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We can check whether i fails by first appending the tree θi to this path and then to each new leaf (labeled with a number < r) appending ξθi, let the leaves which confirm the nonfailure of i be labeled by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Now consider a path p′ extending p without determined output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' We claim that there is j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2r − 1} such that Pr Gn[θj < r ∧ ξθj = j | p′ is compatible with ω] ≤ 1 2, where p′ being compatible with ω means that the computation along p′ agrees with the edge labels which would be chosen according to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' To prove the claim we notice that along p′ it was confirmed that already d-many distinct elements of {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2r − 1} are in bijection with some d-many elements of the set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , r − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, to fail further there are only at most (r − d)-many other values j′ in {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2r−1} for which it holds that ξθj′ = j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By an analogous argument to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='13 this is enough to show that at least for one of them the claim holds since r−d 2r ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Thus, we let j to be the label of the leaf of p′ which concludes the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Therefore, by construction for each d ∈ Mn, d < 2r we have Pr ω∈Gn[Td fails for (Θ, Ξ, ζ)] ≤ 2−d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' If r is in Mn \\ N then we put T = Tt′ for any nonstandard t′ such that the depth of T is still bounded by some n1/t, where t ∈ Mn \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Otherwise, we put T = T2r−1 and since this tree can go through the whole range of Θ it can never fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' K(∗PATHn, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕOntoWeakPigeon(X, Y, m)]] = 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='15 we can construct for each (Θ, Ξ, ζ) a tree T which computes some function α which validates the third disjunct of ϕOntoWeakPigeon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 22 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let ϕ(x) be an L2 n-formula with parameters from Fnb and Gnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Then for every m ∈ Mn the open comprehension principle (∃X)(∀y < m)(X(y) ≡ ϕ(y)) and the open induction principle ¬ϕ(0) ∨ ϕ(m) ∨ (∃x < m)(ϕ(x) ∧ ¬ϕ(x + 1)) are both valid in K(∗PATH, Fnb, Gnb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' This can be proven completely analogously to [7, Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Compiling the results we have about K(∗PATH, Fnb, Gnb) we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the structure K(∗PATH, Fnb, Gnb) the following are valid: open induction with parameters from Fnb and Gnb open comprehension with parameters from Fnb and Gnb every instance of OntoWeakPigeon has a solution there is an instance of Leaf which does not have a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Concluding remarks We have to note that the problem OntoWeakPigeon has not been considered in the context of oracle NP search problems and the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='16 cannot be adapted to prove that every instance of the stronger WeakPigeon 5 has a solution in K(∗PATH, Fnb, Gnb) because the presence of the inverse function is essential to the construction of the witness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' A stronger problem called SourceOrSink is well established in the study of NP search problems (it is the complete problem for PPAD, see [1]) and can be formulated as follows: Given a directed graph ω on the vertex set {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1} with the property that any vertex v has outdegree bounded by 1 and indegree also bounded by 1 and the indegree of the zero vertex is 0 find a nonzero vertex which is either a source or a sink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In the type 2 setting the problem is given by a tuple (α, β, x), where x is a binary string and α and β functions presented by an oracle with domain {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1} computing the potential successor or predecessor of a given vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' It was established in [1] that Leaf is not many-one reducible to SourceOrSink and therefore this nonreducibility may be reflected in our model K(∗PATH, Fnb, Gnb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The way SourceOrSink is presented is similar to how OntoWeakPigeon is presented, and thus a similar strategy could be potentially used to solve the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Let ϕSourceOrSink(X, Y, m) be the formula formalizing that (X, Y, m) as an instance of SourceOrSink has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Is it true that K(∗PATH, Fnb, Gnb)[[(∀X)(∀Y )(∀m)ϕSourceOrSink(X, Y, m)]] = 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 5The problem to witness that α : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x| − 1} → {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' , 2|x|−1 − 1} is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 23 Acknowledgement This work is based on the author’s master’s thesis [5] which was completed under the supervision of Jan Krajíček.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The author also thanks Eitetsu Ken for comments on a draft of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' References [1] Paul Beame, Stephen Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The relative complexity of np search problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' In Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, STOC ’95, page 303–314, New York, NY, USA, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [2] Ronald Fagin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Probabilities on finite models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The Journal of Symbolic Logic, 41(1):50–58, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [3] Isaac Goldbring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Lecture notes on nonstandard analysis, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Available at https:// www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='uci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='edu/~isaac/NSA%20notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content='pdf (last accessed 9th of January 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Hanika.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Search problems and Bounded Arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' PhD thesis, Charles University, Prague, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [5] Ondřej Ježil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Pseudofinite structures and limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Master’s thesis, Charles University, Prague, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Also available in Electronic Colloquium on Computational Complex- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [6] Emil Jeřábek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Integer factoring and modular square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Journal of Computer and System Sciences, 82(2):380–394, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [7] Jan Krajíček.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Forcing with random variables and proof complexity, volume 382 of London Mathematical Society Lecture Note Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Cambridge University Press, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [8] László Lovász and Balázs Szegedy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Limits of dense graph sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Journal of Combinatorial Theory, Series B, 96(6):933–957, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [9] Nimrod Megiddo and Christos H Papadimitriou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' On total functions, existence theo- rems and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Theoretical Computer Science, 81(2):317–324, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [10] Jaroslav Nešetřil and Patrice Ossona de Mendez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' A model theory approach to structural limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Commentationes Mathematicae Universitatis Carolinae, 53:581– 603, 11 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' [11] Alexander A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Razborov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' Flag algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' The Journal of Symbolic Logic, 72(4):1239– 1282, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} +page_content=' 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NFRT4oBgHgl3EQfmzfK/content/2301.13603v1.pdf'} diff --git a/39A0T4oBgHgl3EQfNf8t/content/tmp_files/2301.02146v1.pdf.txt b/39A0T4oBgHgl3EQfNf8t/content/tmp_files/2301.02146v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b97dd40477d7fe41ce09cf10ba041fe1ecd2f110 --- /dev/null +++ b/39A0T4oBgHgl3EQfNf8t/content/tmp_files/2301.02146v1.pdf.txt @@ -0,0 +1,3499 @@ +Searching for Lindbladians obeying local conservation laws and showing thermalization +Devashish Tupkary,1, ∗ Abhishek Dhar,2, † Manas Kulkarni,2, ‡ and Archak Purkayastha3, 4, 5, § +1Institute for Quantum Computing and Department of Physics and Astronomy, +University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 +2International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India +3Department of Physics, Indian Institute of Technology, Hyderabad 502285, India +4Centre for complex quantum systems, Aarhus University, Nordre Ringgade 1, 8000 Aarhus C, Denmark +5School of Physics, Trinity College Dublin, College Green, Dublin 2, Ireland +We investigate the possibility of a Markovian quantum master equation (QME) that consistently +describes a finite-dimensional system, a part of which is weakly coupled to a thermal bath. In order +to preserve complete positivity and trace, such a QME must be of Lindblad form. For physical con- +sistency, it should additionally preserve local conservation laws and be able to show thermalization. +First, we show that the microscopically derived Redfield equation (RE) violates complete positivity +unless in extremely special cases. We then prove that imposing complete positivity and demanding +preservation of local conservation laws enforces the Lindblad operators and the lamb-shift Hamil- +tonian to be ‘local’, i.e, to be supported only on the part of the system directly coupled to the +bath. We then cast the problem of finding ‘local’ Lindblad QME which can show thermalization +into a semidefinite program (SDP). We call this the thermalization optimization problem (TOP). +For given system parameters and temperature, the solution of the TOP conclusively shows whether +the desired type of QME is possible up to a given precision. Whenever possible, it also outputs a +form for such a QME. For a XXZ chain of few qubits, fixing a reasonably high precision, we find +that such a QME is impossible over a considerably wide parameter regime when only the first qubit +is coupled to the bath. Remarkably, we find that when the first two qubits are attached to the bath, +such a QME becomes possible over much of the same paramater regime, including a wide range of +temperatures. +I. +INTRODUCTION +A small finite-dimensional quantum system, a part of +which is weakly coupled to a macroscopic thermal bath, +is expected to thermalize to the temperature of the bath. +Describing this dynamics is relevant across various fields +in quantum science and technology, including quantum +information and thermodynamics [1], quantum optics [2], +quantum chemistry [3], engineering [4] and biology [5]. +In absence of coupling to the macroscopic thermal bath, +the dynamics of the density matrix of the system is gov- +erned by the Heisenberg equation of motion. This unitary +evolution is Markovian. +When coupled to the macro- +scopic thermal bath, the dynamics becomes non-unitary, +described by a quantum master equation (QME) [6–8]. +We investigate whether it is possible to have a physically +consistent Markovian QME describing such dynamics. In +order to do so, we are led to introduce the “thermaliza- +tion optimization problem” (TOP). This is a semidefinite +program (SDP), the output of which conclusively shows +whether, for given system parameters and temperature, +such a QME is possible, up to given precision. When- +ever possible, the output also yields one possible form +for such a QME. Whenever impossible, it means that, +for such parameters, the dynamics cannot be described +∗ djtupkary@uwaterloo.ca +† abhishek.dhar@icts.res.in +‡ manas.kulkarni@icts.res.in +§ archak.p@phy.iith.ac.in +FIG. 1. Schematic of the setup we consider: an arbitrary fi- +nite dimensional system described by Hamiltonian HS, a part +of which is weakly coupled to a thermal bath at inverse tem- +perature β. The Hilbert space of the system, HS is divided +into a part HL which directly couples to the bath, and the +remaining part HM. +by any Markovian QME even at weak system-bath cou- +pling, and therefore must have some non-Markovian char- +acter. The SDP can be solved using standard packages +in high-level computing. +We note that, while SDP is +widely used in many branches of quantum information +and communication [9, 10], and also in quantum chem- +istry [11, 12], it has been combined with open quantum +system techniques in only few previous works [13–15], in +very different contexts. +It was shown by Gorini, Kossakowski, Sudarshan, and +arXiv:2301.02146v1 [quant-ph] 5 Jan 2023 + +Hilbert space Hl +Hilbert space HM +Hilbert space Hs = HL HM, Hamiltonian Hs2 +Lindblad (GKSL) [16–18] that any QME that preserves +complete positivity and trace of the density matrix, and +describes Markovian dynamics has to be of the form +∂ρ +∂t = i[ρ, HS + HLS] + D(ρ), +D(ρ) = +d2−1 +� +λ=1 +γλ +� +LλρL† +λ − 1 +2{L† +λLλ, ρ} +� +, +γλ ≥ 0, +(1) +which is commonly called the Lindblad equation. +In +Eq. (1), ρ is the density matrix of the system, d is the +Hilbert space dimension of the system, HS is the sys- +tem Hamiltonian, HLS is the Lamb shift Hamiltonian, +Lλ are the Lindblad operators, γλ are the rates, and +D(ρ) is called the “dissipator” term. The preservation +of complete positivity condition is enforced by demand- +ing γλ ≥ 0. Lindblad equations have been extensively +used in studying both theoretical and experimental se- +tups [6–8, 19–22]. +Given this enormous scope of application, it is of +paramount importance to assess the conditions under +which such a Markovian description emerges from a more +microscopic theory. +The standard way to microscopi- +cally obtain a Markovian QME is to consider the global +Hamiltonian of the system weakly coupled to baths, and +to trace out the baths perturbatively to the leading or- +der. Starting with this microscopic viewpoint, it becomes +clear that only having an equation in the Lindblad form +is not sufficient, there are some additional fundamental +requirements for physical consistency [23]. +In particu- +lar, one must preserve local conservation laws, and if the +steady state is unique, the system is not driven and all +baths have same temperature, the system must thermal- +ize to the temperature of the baths. It would be useful to +have a QME which, by construction, is of Lindblad form +and satisfy these additional requirements. In this paper, +we systematically go about searching for such a QME for +a setup where a part of the system is coupled to a single +bath (see Fig. 1). This is done in three steps, each step +having important consequences: +1. The microscopically derived quantum master equa- +tion to the leading order in system-bath coupling +is the so-called Redfield equation (RE) [6]. +The +RE has been shown to preserve local conservation +laws and be able to show thermalization [23]. Here, +we provide an explicit, model independent proof +that the RE necessarily violates complete positiv- +ity unless the Redfield dissipator happens to act +“locally”, meaning it is identity on the part of the +system that is not directly coupled to the bath. Al- +though the violation of complete positivity by RE +has been previously demonstrated in specific exam- +ples [24–30], we are unaware of any previous work +with such a model independent explicit proof. +2. We then prove that enforcing complete positivity +condition γλ ≥ 0 and preservation of local conser- +vation laws necessarily requires the Lindblad oper- +ators and the Lamb shift Hamiltonian to be “lo- +cal”. That is, they must act only on the part of +the system coupled to the bath, and be identity +on the part of the system that is not connected to +the bath. This directly rules out the possibility of +any so-called ‘global’ Lindblad equation, such as +the eigenbasis Lindblad equation [6, 8], the Uni- +versal Lindblad Equation [31] to be consistent with +local conservation laws. +3. Given the restriction of the dissipator and the +Lamb shift Hamiltonian to be “local’, we devise +a numerical technique using SDP to check conclu- +sively in a case-by-case basis whether such a QME +can show thermalization in a particular situation. +We call this the TOP. We use this method to study +the case of a XXZ chain of few qubits with a part +of it coupled to a bath. If the bath is coupled only +to the first qubit, our method conclusively shows +that over a large regime of system parameters and +temperature, no such QME exists. +However, re- +markably if the bath is coupled to two qubits of +the chain, up to a chosen precision, our method +shows that a Marovian QME respecting all condi- +tions becomes possible over a considerable regime +of parameters, including a wide range of tempera- +tures. Note that the RE for the XXZ chain does +show thermalization and preserve local conserva- +tion laws [23], even when only one qubit is attached +to a bath. But it is not completely positive. +This work is organized as follows. In Sec. II we ex- +plain the setup studied in this work, derive the Redfield +equation for our setup, and show that it will necessarily +violate complete positivity, unless the Redfield dissipa- +tor happens to act “locally”. +In Sec. III we consider +quantum master equations preserving complete positiv- +ity and obeying local conservation laws, and show that +such equations must have a dissipator and Lamb shift +operator that acts only on the part of the system cou- +pled to the bath. In Sec. IV we discuss the possibility of +QMEs respecting complete positivity, local conservation +laws and being able to show thermalization. We intro- +duce the TOP, and use it in the specific case of the few +site XXZ chain with one or two sites attached to bath. +In Sec. V we summarize our results, and discuss future +directions. Certain details are delegated to the appen- +dices. +II. +THE MODEL AND REDFIELD +DESCRIPTION +Our setup is described schematically in Fig. 1. The +full Hamiltonian be given by +H = HS + ϵHSB + HB, +(2) +where HS and HB are the Hamiltonians of the system +and bath respectively, ϵ ≪ 1 is a small dimensionless pa- + +3 +rameter controlling system-bath coupling strength, and +HSB is the system-bath coupling Hamiltonian. At initial +time, the system is considered to be in an arbitrary ini- +tial state ρ(0), while the bath is in a thermal state with +inverse temperature β +ρtot(0) = ρ(0) ⊗ ρB, +ρB = e−βHB +ZB +. +(3) +Starting with this initial state, the whole set-up of the +system and the bath is evolved with the full Hamiltonian +H, and the bath degrees of freedom are traced out to +obtain the state of the system, +ρ(t) = TrB +� +e−iHtρtot(0)eiHt� +, +(4) +where TrB(. . .) denotes trace over bath degrees of free- +dom. +The Eq. (4), by construction, is a completely +positive trace preserving (CPTP) map from ρ(0) to +ρ(t) [6, 7]. +Without any loss of generality, we assume +TrB(HSBρB) = 0, where TrB(. . .) denotes trace over +bath degrees of freedom [6, 7]. +The effective equation +of motion for the system density matrix written to the +leading order in system-bath coupling strength ϵ is the +RE, given by, [6], +∂ρ +∂t =i[ρ(t), HS] ++ϵ2 +� ∞ +0 +dt′ TrB[HSB, [HSB(−t′), ρ(t) ⊗ ρB]], +(5) +where +HSB(t) = ei(HS+HB)tHSBe−i(HS+HB)t +(6) +and ρB is the state of the bath. In complete generality, +we can write the system-bath coupling Hamiltonian as +HSB = +� +l +(SlB† +l + S† +l Bl), +(7) +where Sl and Bl are operators on the system and bath +respectively, and l can be summed over as many indices as +required for HSB. Using Eq.(7) in Eq.(5) and simplifying, +we have +∂ρ +∂t = i[ρ(t), HS] + ϵ2� � +l +� +S† +l , S(1) +l +ρ(t) +� +− +� +S† +l , ρ(t)S(2) +l +� ++ H.c +� +, +(8) +where +S(1) +l += +� +m +� ∞ +0 +dt′ Tr +� +BlB† +m(−t′)ρB +� +Sm(−t′) ++ +� +m +� ∞ +0 +dt′ Tr +� +BlBm(−t′)ρB +� +S† +m(−t′) +S(2) +l += +� +m +� ∞ +0 +dt′ Tr +� +B† +m(−t′)BlρB +� +Sm(−t′) ++ +� +m +� ∞ +0 +dt′ Tr +� +Bm(−t′)BlρB +� +S† +m(−t′). +(9) +Since the actual microscopic evolution is given by a +CPTP map [see Eqs. (3), (4)], one might naively expect +that the evolution obtained from the microscopically de- +rived RE respects complete positivity. However, as we +prove in the next subsection in generality, unless in ex- +tremely special cases, the RE violates complete positivity. +A. +Violation of complete positivity in Redfield +equation +1. +Choosing an operator Basis +As shown in Fig. 1, we consider only a part of the sys- +tem is coupled to the bath. +Let us denote HL as the +Hilbert space of that part of the system that couples to +the bath, and let HM be the Hilbert space of the re- +maining part of the system. In mathematical terms, this +means that any operator OM in the Hilbert space HM +commutes with the system-bath coupling Hamiltonian +HSB, +[OM, HSB] = 0. +(10) +The system Hamiltonian can then be written as +HS = HL + HLM + HM, +(11) +where the Hamiltonian HL is in Hilbert space HL, the +Hamiltonian HM is in Hilbert space HM, and HLM gives +the coupling between the two Hilbert spaces. Note that +we do not consider this coupling to be small. +Let the dimension of HL and HM be dL and dM re- +spectively. Then, one can pick an orthonormal basis of +operators {fi} and {gj} on HL and HM respectively, +where 1 ≤ i ≤ d2 +L and 1 ≤ j ≤ d2 +M, and where or- +thonormality is defined according to the Hilbert Schmidt +inner product given by ⟨A, B⟩ = Tr[A†B]. One can al- +ways choose this basis such that fd2 +L = IL/√dL and +gd2 +M = IM/√dM, where IM and IL are the identity op- +erators on those spaces. Such a basis is required by the +GKSL theorem [6, 7, 16, 18]. Taking the tensor prod- +uct of these two basis, one can obtain an orthonormal +basis {Fk} = {fi} ⊗ {gj} for operators on HS, with +Fd2 +Ld2 +M = IS/ +√ +d, where d = dLdM is the dimension of +the system Hilbert space. Without loss of generality, the +Lindblad equation [Eq. (1)] written in this basis is given +by +∂ρ +∂t = i[ρ, HS + HLS] + +d2−1 +� +α,˜α=1 +Γα˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +� +, +(12) +where complete positivity of ρ is preserved iff Γ is positive +semidefinite [6, 7]. Eq. (12) can be turned into Eq. (1) +by diagonalizing the matrix Γ. +The complete positivity of RE can be checked by tak- +ing the RE to the same form as Eq.(12) and checking if +the corresponding Γ is positive semidefinite. To do so, + +4 +let us relabel the indices so that Fi = fi ⊗ IM/√dM for +1 ≤ i ≤ d2 +L − 1. This allows us to expand the system +operators in Eq. (8) as, +Sl = +d2 +� +α=1 +alαFα, +S† +l = +d2 +� +α=1 +a′ +lαFα, +S(1) +l += +d2 +� +α=1 +blαFα, +S(1)† +l += +d2 +� +α=1 +b′ +lαFα, +S(2) +l += +d2 +� +α=1 +clαFα, +S(2)† +l += +d2 +� +α=1 +c′ +lαFα, +(13) +where alα = a′ +lα = 0, ∀ +d2 +L ≤ α ≤ d2 −1 since Sl and S† +l +are identity on HM. Substituting Eq. (13) into Eq. (8), +we obtain +∂ρ +∂t = i[ρ, HS] − ϵ2 � +l +d2 +� +α,˜α=1 +� +a∗ +lαbl˜α[F † +α, F˜αρ] ++ c′∗ +lαa′ +l˜α[ρF † +α, F˜α] + b∗ +lαal˜α[ρF † +α, F˜α] + a′∗ +lαc′ +l˜α[F † +α, F˜αρ] +� +. +(14) +Using some straightforward algebra (see Appendix A), +Eq. (14) can be simplified to +∂ρ +∂t = i[ρ, HS + HLS] + +d2−1 +� +α,˜α=1 +Γα˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +� +, +(15) +where Γα˜α is a (d2 −1)×(d2 −1) hermitian matrix given +by +Γα˜α = ϵ2 � +l +(a∗ +lαbl˜α + c′∗ +lαa′ +l˜α + b∗ +lαal˜α + a′∗ +lαc′ +l˜α), +(16) +where x∗ in Eq. (16) denotes the complex conjugate of x, +and the expression of HLS that appears in Eq. (15) can +be found in Appendix A. Since Eq. (16) is of the form in +Eq. (12), the condition for preserving complete positivity +of ρ is equivalent to Γ being positive semidefinite. +2. +The dissipator +Let us now look at Γα˜α from Eq. (16). Recall from +Sec. II A 1, that if α ≥ d2 +L, then alα = a′ +lα = 0. Therefore, +Γα˜α = 0 when α, ˜α ≥ d2 +L, and has the following structure: +Γ = +� +Γα,˜α 0, ∀α > d2 +L or (ii) making �Γα˜α = + +5 +0 ∀α < d2 +L. We will see in the next section (Sec. III) that +any equation with (i), will violate local conservation laws. +III. +LINDBLAD DESCRIPTIONS OBEYING +LOCAL CONSERVATION LAWS +A. +Local conservation laws +Let us make precise what we mean by preservation of +local conservation laws. For our setup, since the bath +only acts on HL part of the system, any operator on +HM commutes with the system-bath coupling Hamilto- +nian HSB (see Eq. (10), Fig. 1). So, writing down the +Heisenberg equation of motion with respect to the full +Hamiltonian H [Eq. (2)], and using Eq. (10), we see that +the dynamical equation for the expectation value of any +observable OM on HM is given by +d +dt ⟨IL ⊗ OM⟩ = −i ⟨[IL ⊗ OM, HS]⟩ +(19) +where ⟨X⟩ = Tr[Xρ]. Any effective QME obtained by +integrating out the bath should satisfy this property. We +call QMEs satisfying this property as ones preserving lo- +cal conservation laws. The justification for this name be- +comes clear if we look at an operator in HM that would +remain conserved if there is coupling with HL, i.e, if HLM +in Eq.(11) is zero. One such operator is the Hamiltonian +HM. The dynamical equation for expectation value of +HM gives the energy continuity equation +d +dt ⟨HM⟩ = JL→M, +JL→M = −i ⟨[HM, HLM]⟩ . +(20) +Here, JL→M can be interpreted as the energy current +from the region L to the region M (see Fig. 1). In steady +state, the rate of change of any system operator is zero. +From above equation, this gives, JL→M = 0 in steady +state. Thus, steady state energy current inside the sys- +tem is zero. This is a statement of local conservation of +energy and is one of the fundamental physical require- +ments for a system coupled to a single bath that follows +from the more general requirement Eq. (19). +Importantly, the RE, i.e, Eq. (5), can be shown to sat- +isfy Eq. (19) [23] and thereby preserves local conservation +laws. We can write any QME to leading order in system +bath coupling as +∂ρ +∂t = L0(ρ) + ϵ2L2(ρ), +(21) +where L0(ρ) = i[ρ, HS] and L2(ρ) contains both the dis- +sipator and the Lamb-shift Hamiltonian. Computing the +left hand size of Eq. (19) using Eq. (21), and comparing +with the right hand size of Eq. (19), we obtain [23], +Tr[(IM ⊗ OM)L2(ρ)] = 0. +(22) +This is a necessary condition for satisfying local conser- +vation laws. If we now further restrict the QME to be of +Lindblad form, i.e, of the form Eq. (1), thereby respecting +complete positivity, we obtain the following theorem. +Theorem 2 Any QME of Lindblad form Eq. (1) (thereby +satisfying complete positivity) that also satisfies local con- +servation laws must have the Lindblad operators and the +Lamb-shift Hamiltonian acting only on the part of the +system connected to the bath. That is, Lλ = LL +λ ⊗ IM, +HLS = HL +LS ⊗IM, where LL +λ, HL +LS act only on the Hilbert +space HL which is coupled to the bath by system-bath cou- +pling Hamiltonian HSB, and IM is the identity on the +remaining of the system Hilbert space HM. +In the next subsection, we give the proof of this theorem. +B. +Proof of theorem 2 +We start by writing the most general Lindblad equa- +tion in the basis of Sec. II A 1, +∂ρ +∂t = i[ρ, HS + �HLS] + +d2−1 +� +α,˜α=1 +�Γα˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +� +(23) +where �HLS is some Lamb shift Hamiltonian, and �Γ is a +positive semidefinite matrix. As mentioned before, this +form can be reduced to the form of Eq.(1) by transform- +ing to a basis where matrix �Γ is diagonal. So, it suffices to +work with this form. Since Fi = fi ⊗ IM for 1 ≤ i < d2 +L, +the condition for the Lindblad operators to act only on +HL then translates to the matrix �Γ being of the form +�Γ = +� �Γα,˜α NL, we again see +the non-monotonic behavior. However, in stark contrast +to Fig. 3, we find that over the entire chosen range of β +τopt ≪ δ. Thus, for NL = 2, over a considerable range of +parameters, a QME that simultaneously preserves com- +plete positivity, obeys local conservation laws, and satis- +fies thermalization up to the given precision is possible. +This is a highly non-trivial result. +Previously, local two-qubit Lindblad dissipators have +been used to study energy transport in XXZ-type qubit +chains (for example, Refs. [36, 37]). However, those local +two-qubit Lindblad operators were constructed so as to +thermalize the two qubits only, in absence of coupling to +the rest of the chain. Such Lindblad description is not +guaranteed to thermalize the whole chain to the given +inverse temperature β of the bath [38]. Our result here +shows that it is possible to have a two-qubit local Lind- +blad description that can thermalize the full chain to the +given temperature of the bath to a good approximation. +As mentioned before, CVX also outputs a possible +choice of Γ(L) and H(L) +LS matrices corresponding to τopt. +So, when τopt < δ, we get one possible candidate for the +desired type of QME. For our choice of parameters, we +find that CVX always outputs H(L) +LS = 0, and a non- +trivial value of Γ(L) that would be hard to guess other- +wise. In Table. I, we demonstrate the Γ(L) obtained for +NL = 2, NM = 4, ω(ℓ) +0 += 1, gℓ = 0.1, ∆ℓ = 1, β = 1. The +Γ(L) matrix corresponds to the basis of operators {Fk} +chosen as +Fk = f⌈k/4⌉ ⊗ fk(mod 4) ⊗ IM +(43) +where {fi} = {−σz/ +√ +2, σ−, σ+, I2/ +√ +2}, ⌈k/4⌉ denotes +the nearest integer greater than or equal to k/4, and +k(mod 4) denotes the value of k modulo 4, and k goes +from 1 to 15. We also note that the exact values of Γ(L) +and H(L) +LS computed by CVX may depend on the exact +configuration of the programming environment (such as +internal solvers used by CVX). +For every parameter of the system, there is a differ- +ent τopt, with a corresponding value of Γ(L) and H(L) +LS +given by CVX. If we want to explore a large parameter +space of the system, it seems that we need a different +Γ(L) and H(L) +LS for each parameter point. Surprisingly, +we find that this is not always required. If τopt ≪ δ for +one set of parameters, we can substantially change pa- +rameters of the system far from the qubits attached to +baths, and still obtain a value of τ ≪ δ with the same +value of Γ(L) and H(L) +LS . This is shown in Fig. (6), where +τ is calculated changing various parameters away from +the two qubits coupled with the bath, fixing H(L) +LS = 0 +and Γ(L) to be the same as in Table. I. Over the entire +regime of chosen parameters τ ≪ δ. Note that, in con- +trast to previous plots, this is not be the optimal value of +τ. Nevertheless, if τ ≪ δ, we still get a completely posi- +tive Markovian QME preserving local conservation laws +and showing thermalization up to the chosen precision. +Given Γ(L) and H(L) +LS , it is much easier to just check this +rather than finding the optimal value τopt. +If parameters of the two qubits that are coupled to the +bath are changed, we can no longer use the same Γ(L) +and H(L) +LS . For example, if we choose the same Γ(L) as + +11 +FIG. 6. τ vs g4, for NL = 2, NM = 4, with ω(ℓ) +0 += 1, ∆ℓ = 1, +β = 1 and gℓ = 0.1 for all ℓ unless otherwise mentioned. +τ is computed from Γ(L) and H(L) +LS obtained from CVX for +NL = 2, NM = 4, with ω(ℓ) +0 += 1, ∆ℓ = 1, β = 1 and gℓ = 0.1 +for all ℓ. The modified parameters for the plots are given by (i) +(no parameters changed), (ii) ∆3 = 0.4, ∆4 = 1.2, (iii) ω(3) +0 += +1.5, ω(4) +0 += 1.5, g5 = 0.3, (iv) ω(3) +0 += 1.5, ω(4) +0 += 1.5, g5 = +0.3, ∆4 = 0.5, (v) g3 = 0.3. We find that τ ≪ δ = 10−6 even +if parameters are changed for qubits of the system that are +not coupled to the baths. +in Fig. 6, and change g1 to 0.2 from 0.1, we get τ = +0.0014 ≫ δ. +The above observation suggests that the values of Γ(L) +and H(L) +LS obtained by CVX can be used to define a QME, +independent of the parameters in the bulk of the sys- +tem. This is consistent with underlying picture that each +value of Γ(L) and H(L) +LS corresponds to a different choice of +the bath spectral function and the system-bath coupling +Hamiltonian. If we change any parameter of the qubits +attached to the baths, the change reflects substantially +on the system-bath coupling Hamiltonian, so the value of +τ changes drastically from τopt obtained with original pa- +rameters. If we change any parameter away from the two +qubits, the change reflects much less on the system-bath +coupling Hamiltonian, causing τ to be of the same order +as the original value of τopt. This presents an exciting +prospect for studying the dynamics of the system-bath +setup over a wide range of parameters, including a wide +range of temperatures, with physically consistent Marko- +vian QMEs. Such studies may also be possible for long +chains, since local Markovian dissipation is favourable for +tensor network based numerical techniques. Such dissi- +pation may, also, in principle, be engineered in quantum +computing and quantum simulation platforms, like ion +traps [39, 40], Rydberg atoms [41, 42], superconducting +qubits [43] and quantum dots [44]. +V. +SUMMARY AND OUTLOOK +Searching for a physically consistent Markvian QME +— A physically consistent Markovian QME must satisfy +complete positivity, obey local conservation laws and be +able to show thermalization. In this work, we have sys- +tematically gone about searching for such QMEs. This +is done in three steps, and the result in each step has im- +portant consequences. Especially, we are led to introduce +the TOP problem, which is an optimization problem for +finding a QME with all the above properties up to a given +precision. The TOP opens a completely new avenue in +the study of dissipative quantum systems. +We consider a finite-dimensional undriven system a +part of which is weakly coupled to a thermal bath. The +microscopically derived QME written to leading order +in system-bath coupling is the RE, which is known to +obey local conservation laws and be able to show ther- +malization [23]. +First, we show in generality that the +RE violates complete positivity, unless in extremely spe- +cial cases. Although there are previous works showing +this via specific examples (for instance, [24, 25, 30, 32]), +we are unaware of a model independent proof similar to +ours. Next, we prove that imposing complete positivity +and preservation of local conservation laws enforces the +QME to be of ‘local’ form. That is, the Lindblad op- +erators and the Lamb-shift Hamiltonian must have sup- +port only on the part of the system directly coupled to +the bath, and be identity elsewhere. This rules out the +possibility of any ‘global’ forms of Lindblad equations, +which are usually constructed to show thermalization, +to be consistent with local conservation laws. Then, we +ask if a ‘local’ Lindblad equation can be found which is +able to show thermalization. We find that, the task of +finding such a Lindblad equation can be cast as an op- +timization problem, which we call TOP. Most interest- +ingly, this optimization problem turns out to be a SDP. +For given system and parameters, the SDP can be ef- +ficiently solved using high-level programming packages +like the CVX Matlab package. The output of the TOP +conclusively shows whether the desired type of QME is +possible for the chosen system parameters and tempera- +ture, up to a chosen precision. For numerical example, +we look at the TOP in a XXZ qubit chain of few sites, +fixing a reasonably high precision. When only the first +site is coupled to a bath, we find that, unless in extremes +of temperatures, it is impossible to find a local Lindblad +equation that is capable of showing thermalization up to +the chosen precision. +Discussion in light of various existing forms of QMEs +— Various forms of QMEs have been derived it literature +under various approximations (for example, [31, 35, 45– +54]), along with the standard RE, local and eigenbasis +Lindblad equations [6]. +Although the above example +shows that there is no general form of physically consis- +tent Markovian QME, this does not immediately make +them unusable. +Instead, it turns out that in each of +these forms of QME, some elements of the system den- +sity matrix are given correctly, while the others are not +[23]. So, one needs to be careful in interpreting the re- +sults from them, always keeping in mind their micro- +scopic derivation and approximations. The RE, despite +not being completely positive, is provably more accurate +than all such Lindblad QMEs. To elucidate how this can + +X10-8 +2.5 +2. +1.5 +(i) +(ii) +(ii) +1 +(iv) +中(v) +0.5 +100 +9412 +happen, imagine that, in a given setup, physically, the +population of one energy level, say, ⟨Ej| ρ |Ej⟩, is zero in +steady state. The RE might then give a small negative +value (say, ⟨Ej| ρ |Ej⟩ = −10−3), while any of the Lind- +blad equations will give a larger positive value, which +might be (say, ⟨Ej| ρ |Ej⟩ = 0.1). Either case is a prob- +lem if we want to calculate various kinds of entropies, +as often required in quantum information and thermo- +dynamics. In case of RE, unphysical results can often +be ruled out by checking the scaling with system-bath +coupling [23, 55]. This is often more difficult in Lindblad +QMEs, where approximations are often less controlled. +The state obtained from the recently derived ULE [31], +which been shown to violate local conservation laws [23], +can be corrected to obtain results as accurate as the RE +[48]. This re-instates the local conservation laws, at the +cost of also re-instating the same positivity problem of +the density matrix as in RE. In another recent work, a +general form of QME has been derived [45] which is more +accurate than RE, even though complete positivity of dy- +namics is still not guaranteed. +TOP and (non) Markovianity — In the microscopic +picture, given the temperature of the bath, the QME is +completely defined by the bath spectral functions and +the type of system-bath coupling. The TOP can then be +thought of as varying over all possible bath spectral func- +tions and types of system-bath couplings to find the clos- +est to satisfying thermalization the local Lindblad equa- +tion can be. So, when TOP shows that the desired type +of QME is impossible, it means no matter what type of +bath is attached and how it is coupled to the system, for +the chosen parameters, it is impossible to describe the +dynamics via a completely positive Markovian QME sat- +isfying local conservation laws and showing thermaliza- +tion. The approach to thermal state must then have some +non-Markovian character for such system parameters and +temperature. The output of TOP, τopt, shows non-trivial +dependence on the system parameters and the temper- +ature. This dependence seems to capture how close to +Markovian the dynamics can be for the chosen parame- +ters. +Surprises when two qubits are attached to bath — Sur- +prisingly, we have found that, when first two qubits of +the few-site XXZ chain are attached to a bath, solving +the TOP shows that it is possible to find Lindbladians +obeying local conservation laws and showing thermaliza- +tion up to quite high precision. This holds over a con- +siderable range of parameters, including a wide range of +temperatures. Notably, in this entire parameter regime, +when one qubit was coupled to a bath, such a QME was +impossible. +Whenever the TOP shows a QME respecting all condi- +tions is possible, standard high-level programming pack- +ages used to solve the SDP also outputs one possible form +for such a QME. When two qubits are attached to the +bath, the form of QME so obtained, which respects all the +requirements, is quite non-trivial and would be hard to +guess otherwise. Even more interestingly, we have found +that if we take one such QME obtained for one choice of +system parameters, and change some system parameters +away from two qubits that couple to the bath, the QME +still satisfies all the requirements. This opens several ex- +citing possibilities that we describe below. +Future directions — Our results open the exciting pos- +sibility of studying the dynamics of approach to thermal +state in open quantum many-body systems using phys- +ically consistent Markovian QMEs, over a wide range +of parameters, including a wide range of temperatures. +This is particularly aided by the fact that local Lindblad +equations are favourable for tensor network techniques. +Studying such dynamics at finite temperatures is often +quite challenging otherwise, requiring simulation of non- +Markovian dynamics [56–58]. +The TOP lets us find parameters of the system where +local Lindblad equations can show thermalization. For +two qubits attached to bath, this range of parameters +can be considerably large, as we have seen. It may be +possible to design such local dissipation in quantum com- +puting and quantum simulation platforms like ion traps +[39, 40], Rydberg atoms [41, 42], superconducting qubits +[43] and quantum dots [44]. Especially in ion traps and +Rydberg atom platforms, this offers an interesting way to +controllably prepare finite temperature states of complex +quantum many-body systems in these platforms, which +is presently a technological challenge. Usually, one would +require global Lindblad dissipators to ensure that a ther- +mal state is prepared. This would be hard to design in +quantum simulation platforms if one wants to simulate +complex many-body systems. The possibility of having +local dissipation confined to two qubits offers a much eas- +ier alternative. +Moreover, as we have seen in the example of XXZ qubit +chain, the dependence of the output of the TOP, τopt, on +various parameters of the system already encode rich and +interesting physics. +For complex quantum many-body +systems, one may need more scalable techniques for SDP, +which is itself a direction of research in computer science +[59]. +Using these techniques, the rich behavior of τopt +with various parameters can then be studied. +It is therefore clear that our results, especially the in- +troduction of the TOP, leads to new paradigm within the +fields of quantum information, computation and technol- +ogy. Nevertheless, various questions still remain. One +main question concerns steady-state coherences [55, 60– +62]. When coupled to a thermal bath at any finite cou- +pling, the system density matrix will have coherences in +energy eigenbasis of the system [61, 62]. These coher- +ences can be important in quantum information and ther- +modynamics [63–66] and are given correctly to the lead- +ing order by the RE [23, 55, 62]. However, it is not clear +that the steady-state coherences calculated from physi- +cally consistent Markovian QME obtained via TOP will +be the same as those obtained from RE. Further inves- +tigation is required in this respect, which will be carried +out in future works. +All code used in this work can be found at [67]. + +13 +ACKNOWLEDGEMENTS +MK would like to acknowledge support from the +project 6004-1 of the Indo-French Centre for the Promo- +tion of Advanced Research (IFCPAR), Ramanujan Fel- +lowship (SB/S2/RJN-114/2016), SERB Early Career Re- +search Award (ECR/2018/002085) and SERB Matrics +Grant (MTR/2019/001101) from the Science and En- +gineering Research Board (SERB), Department of Sci- +ence and Technology, Government of India. AD and MK +acknowledge support of the Department of Atomic En- +ergy, Government of India, under Project No. RTI4001. +AP acknowledges funding from the European Research +Council (ERC) under the European Unions Horizon 2020 +research and innovation program (Grant Agreement No. +758403). A.P also acknowledges funding from the Dan- +ish National Research Foundation through the Center of +Excellence “CCQ” (Grant agreement no.: DNRF156). +Appendix A: Casting Eq.(14) to Eq.(15) +In this appendix we show the steps for taking Eq. (14) +to the form of Eq. (15) which is more amenable to study- +ing issues related to conservation of complete positivity. +We start with Eq. (14), which we recall to be +∂ρ +∂t = i[ρ, HS] − ϵ2 � +l +d2 +� +α,˜α=1 +� +a∗ +lαbl˜α[F † +α, F˜αρ] ++ c′∗ +lαa′ +l˜α[ρF † +α, F˜α] + b∗ +lαal˜α[ρF † +α, F˜α] + a′∗ +lαc′ +l˜α[F † +α, F˜αρ] +� +. +(A1) +This can be rewritten as +∂ρ +∂t = i[ρ, HS] + ϵ2 � +l +d2 +� +α,˜α=1 +� +a∗ +lαbl˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +− [F † +αF˜α, ρ] +2 +� ++ c′∗ +lαa′ +l˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 ++ [F † +αF˜α, ρ] +2 +� ++ b∗ +lαal˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 ++ [F † +αF˜α, ρ] +2 +� ++ a′∗ +lαc′ +l˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +− [F † +αF˜α, ρ] +2 +�� +, +(A2) +where, A, B += +AB + BA is the anti-commutator. +Next, +we +note +that +the +summation +�d2 +α,˜α=1 +in +above +equation, +can +be +written +as +�d2 +α,˜α=1 += +� +α=˜α=d2 + �d2−1 +α=1,˜α=d2 + �d2−1 +˜α=1,α=d2 + �d2−1 +α,˜α=1 . +Us- +ing this, and the fact that Fd2 = IS/ +√ +d commutes with +all operators, we combine all commutator terms and +write them as as i[ρ, HS + HLS] to obtain +∂ρ +∂t = i[ρ, HS + HLS] + +d2−1 +� +α,˜α=1 +Γα˜α +� +F˜αρF † +α − {F † +αF˜α, ρ} +2 +� +. +(A3) +Here +HLS = ϵ2 � +l +� +d2 +� +α,˜α=1 +�a∗ +lαbl˜α +2i +− c′∗ +lαa′ +l˜α +2i +− b∗ +lαal˜α +2i ++ a′∗ +lαc′ +l˜α +2i +� +F † +αF˜α ++ +d2−1 +� +α=1 +(a∗ +lαbl,d2 + c′∗ +lαa′ +ld2 + b∗ +lαal,d2 + a′∗ +lαc′ +l,d2) +2i +√ +d +F † +α +− +d2−1 +� +˜α=1 +(a∗ +ld2bl˜α + c′∗ +l,d2a′ +l˜α + b∗ +l,d2al˜α + a′∗ +l,d2c′ +l˜α) +2i +√ +d +F˜α +� +(A4) +and +Γα˜α = ϵ2 � +l +(a∗ +lαbl˜α + c′∗ +lαa′ +l˜α + b∗ +lαal˜α + a′∗ +lαc′ +l˜α), +(A5) +α, ˜α going from 1 to d2 − 1. This is Eq. (15) given in the +main text. +Appendix B: An example of RE violating complete +positivity +In this section, we will present a simple example of the +discussion in Sec. II. Our setup consists of a two-qubit +XXZ qubit chain, where only the first qubit is connected +to the bath modelled by an infinite number of bosonic +modes. +Let H be the Hamiltonian of the full set-up, +given by +H = HS + ϵ HSB + HB, +(B1) +where +HS = ω0 +2 (σ(1) +z ++ σ(2) +z ) +− g(σ(1) +x σ(2) +x ++ σ(1) +y σ(2) +y ++ ∆σ(1) +z σ(2) +z ) +HSB = +∞ +� +r=1 +(κr ˆB† +rσ(1) +− + κ∗ +r ˆBrσ(1) ++ ) +HB = +∞ +� +r=1 +Ωr ˆB† +r ˆBr +(B2) + +14 +where σ(ℓ) +x,y,z denotes the Pauli matrices acting on the ℓth +qubit, σ(ℓ) ++ += (σ(ℓ) +x ++ iσ(ℓ) +y )/2, σ(ℓ) +− += (σ(ℓ) +x +− iσ(ℓ) +y )/2, +ˆBr is bosonic annihilation operator for the rth mode of +the bath. Here, ω0, g, and g∆ represent the magnetic +field, the overall qubit-qubit coupling strength and the +anisotropy respectively. The RE for this setup can be +computed to be [23] +∂ρ +∂t = i[ρ(t), HS] + ϵ2� +[S†, S(1)ρ(t)] − [S†, ρ(t)S(2)] ++ H.c +� +(B3) +with +S† = σ(1) ++ , +S = σ(1) +− +S(1) = +4 +� +j,k=1 +|Ej⟩ ⟨Ej| σ(1) +− |Ek⟩ ⟨Ek| D(j, k), +S(2) = +4 +� +j,k=1 +|Ej⟩ ⟨Ej| σ(1) +− |Ek⟩ ⟨Ek| C(j, k) +, +(B4) +and +C(j, k) = J(Ekj)n(Ekj) +2 +− iP +� ∞ +0 +dω J(ω)n(ω) +ω − Ekj +, +D(j, k) = eβ(Ekj−µℓ)J(Ekj)n(Ekj) +2 +− iP +� ∞ +0 +dω eβ(ω−µ)J(ω)n(ω) +ω − Ekj +, +J(ω) = +∞ +� +k=1 +2π |κk|2 δ(ω − Ωk), +n(ω) = [eβω − 1]−1. +(B5) +In above, J(ω) is called the bath spectral function. Let us +consider bosonic baths described by Ohmic spectral func- +tions with Gaussian cut-offs, J(ω) = ωe−(ω/ωc)2Θ(ω), +where Θ(ω) is the Heaviside step function, and ωc is the +cut-off frequency. The above operators can then be com- +puted numerically. +The next step is to choose the basis fi and gj for oper- +ators on HL and HM. For the general case, one can start +with any set of linearly independent operators that forms +a basis and includes the identity operator, and then ap- +ply the Gram Schmidt orthonormalization procedure to +produce an orthonormal basis that includes the normal- +ized identity operator. For our case, one can easily verify +that the set {−σ(i) +z / +√ +2, σ(i) +− , σ(i) ++ , I(i) +2 / +√ +2} suffices, where +i = 1 for {fi} and i = 2 for {gj}, and I2 is the identity +operator. +The basis for the full system {Fi} can be constructed +from the above basis as described in subsection +II A 1, +and is given by +Fi = fi ⊗ I(2) +2 +2 +(for i = 1, 2, 3) +F3i+j = fi ⊗ gj +(for i = 1, 2, 3, 4 and j = 1, 2, 3) +F16 = I4 +2 +(B6) +Any operator X can be expanded in terms of the above +basis as X = � +α xαFα, where xα = ⟨Fα, X⟩ = Tr(F † +αX). +Thus, expanding S, S†, S(1), S(2) [Eq. (B4)], one can +evaluate all the coefficients in Eq. (13). +Finally, one +can compute the matrix Γ according to Eq. (16). The +matrix Γ for this example, with parameters chosen as +g = 0.1, ω0 = 1, ωc = 10, β = 1, µ = −0.5, ∆ = 1 is given +by +Γ = ϵ2 +� +����������������������� +0 +0 +0 +0 +0 +0 +0 +0 0 0 0 0 0 0 0 +0 +0 +1.542 + 3.428i 0 +0 +0.014 + 0.047i +0 +0 0 0 0 0 0 0 0 +0 1.542 − 3.428i +0 +0 −0.18 − 0.007i +0 +0.18 + 0.007i 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 +−0.18 + 0.007i 0 +0 +0 +0 +0 0 0 0 0 0 0 0 +0 0.014 − 0.047i +0 +0 +0 +0 +0 +0 0 0 0 0 0 0 0 +0 +0 +0.18 − 0.007i +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 +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 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 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 0 0 0 0 0 +0 +0 +0 +0 +0 +0 +0 +0 0 0 0 0 0 0 0 +� +����������������������� +(B7) + +15 +We see that the above matrix has the expected struc- +ture of Eq. (17), +Γ = +� +Γα,˜α<4 +Γα<4,˜α≥4 +Γα≥4,˜α<4 +0 +� +(B8) +and crucially, Γα<4,˜α≥4 ̸= 0. Therefore, as per the GKSL +theorem, this RE will not preserve complete positivity. +This example was computed using QuTiP [68, 69]. +Appendix C: Effective Lindblad equation satisfying +local conservation laws +In this appendix, we will show that Eq. (29) implies +Eq. (31). The condition for a QME preserving complete +positivity and obeying local conservation laws is given by +Eq. (29), which we recall to be +− i[OM, H′] + +d2 +M−1 +� +αM,˜αM=1 +�ΛαM,˜αM +� +g† +αM OMg˜αM +− 1 +2OMg† +αM g˜αM − 1 +2g† +αM g˜αM OM +� += 0, +∀OM, +(C1) +where we write +H′ = +d2 +M +� +αM=1 +νd2 +L,αM gαM , +(C2) +for convenience. To move forward we will use the opera- +tor vector correspondence from Ref. 9, where the vector- +ized version of the operator X is given by vec(X), and +can be constructed using linearity and +vec(|i⟩ ⟨j|) = |i⟩ ⊗ |j⟩∗ , +(C3) +where |j⟩∗ denotes the complex conjugate of |j⟩. We will +apply Eq. (C3) to Eq. (C1), using the identity (Eq. 1.132 +of Ref. 9) +vec(A0BAT +1 ) = (A0 ⊗ A1)vec(B). +(C4) +Applying the vec operation on both sides of Eq. (C1), +we obtain +− i vec(IMOMH′) + i vec(H′OMIM) + +d2 +M−1 +� +αM,˜αM=1 +�ΛαM,˜αM +� +vec(g† +αM OMg˜αM ) − 1 +2vec(IMOMg† +αM g˜αM ) +− 1 +2vec(g† +αM g˜αM OMIM) +� += 0, +∀OM. +(C5) +Eq. (C5) can be simplified using Eq. (C4) to obtain, +� +− iIM ⊗ (H′)T + iH′ ⊗ IM +d2 +M−1 +� +αM,˜αM=1 +�ΛαM,˜αM +� +g† +αM ⊗ gT +˜αM − 1 +2IM ⊗ (g† +αM g˜αM )T +− 1 +2(g† +αM g˜αM ) ⊗ IM +�� +vec(OM) = 0, +∀OM. +(C6) +Eq. (C6) is of the form M vec(OM) = 0 for all hermi- +tian OM. Since hermitian matrices (such as OM) form +a basis for the entire space of operators, this implies +M vec(X) = 0 for all operators X. This is because one +can expand X as a linear combination of hermitian op- +erators (such as OM). Now, M vec(X) = 0 ∀ X implies +M = 0. Therefore, Eq. (C6) implies +M = − iIM ⊗ (H′)T + iH′ ⊗ IM + +d2 +M−1 +� +αM,˜αM=1 +�ΛαM,˜αM +� +g† +αM ⊗ gT +˜αM − 1 +2IM ⊗ (g† +αM g˜αM )T +− 1 +2(g† +αM g˜αM ) ⊗ IM +� += 0. +(C7) +If M = 0, then Tr(M) = 0. Taking the trace of Eq. (C7), +and using the orthonormality of {gi} along with the fact +that Tr(gi) = δi,d2 +M , we obtain +Tr(M) = − +d2 +M−1 +� +αM=1 +�ΛαM,αM dM = 0 +(C8) +which implies +d2 +M−1 +� +αM=1 +�ΛαM,αM = +d2 +M−1 +� +αM=1 +d2 +L +� +αL=1 +�Γ(αL,αM),(αL,αM) = 0. (C9) +which is Eq. (31) in the main text. +Appendix D: The condition for thermalization +From fundamental principles of quantum statistical +mechanics, we expect the system to thermalize when cou- +pled to baths at equal temperatures. The exact condition +that QME’s must obey to satisfy thermalization has been +derived in Ref. 23. For the sake of completeness, we recall +that discussion here. +Let the total system Hamiltonian be given by H = +HS + ϵHSB + HB, where ϵ is a dimensionless parame- +ter controlling the strength of the system bath coupling, +and HSB is the system bath coupling Hamiltonian. We +proceed by obtaining an order-by-order solution to the +steady state of our QME. Any QME describing our setup + +16 +can be expanded in the so-called time-convolution-less +form [6], +∂ρ(t) +∂t += +∞ +� +m=0 +ϵ2mL2m(t)[ρ(t)], +(D1) +where L could in general be time-dependent operators +and L0(t)[ρ(t)] = i[ρ(t), HS]. For quantum master equa- +tions written to second-order in system-bath coupling, +the above summation can be truncated at second order. +Denoting L2m ≡ limt→∞ L2m(t), the steady state ρSS +can be given by +ρSS = lim +t→∞ et(L0+ϵ2L2)ρ(0), +(D2) +which is assumed to be unique. The steady state satisfies +0 = +∞ +� +m=0 +ϵ2mL2m[ρSS]. +(D3) +We can then perform an expansion of ρSS in the even +powers of ϵ as +ρSS = +∞ +� +m=0 +ϵ2mρ(2m) +SS +(D4) +Using Eq. (D4) in Eq. (D3), we can obtain an order by +order solution of ρSS. At the zeroth order in ϵ, we obtain +[ρ(0) +SS, HS] = 0. +(D5) +Assuming that the Hamiltonian has no degeneracies, +Eq. (D5) implies that ρ(0) +SS is diagonal in the energy eigen- +basis, +ρ(0) +SS = +� +i +pi |Ei⟩ ⟨Ei| . +(D6) +where |Ei⟩ is an eigenstate of the system. At second order +in ϵ (m = 1), we obtain the following two equations, +⟨Ei| L2[ρ(0) +SS] |Ei⟩ = 0, +∀i +(D7) +i(Ei − Ej) ⟨Ei| ρ(2) +SS |Ej⟩ ++ϵ2 ⟨Ei| L2[ρ(0) +SS] |Ej⟩ = 0, +∀i ̸= j +(D8) +Since ρ(0) +SS is diagonal in the energy eigenbasis, Eq. (D7) +determines the diagonal elements of ρ(0) +SS. +Having ob- +tained ρ(0) +SS, Eq. (D8) then determines the off-diagonal +elements of ρ(2) +SS. +Note from above equations that the +leading order diagonal elements of ρSS are independent +of ϵ. +It can also be shown that the leading order off- +diagonal elements of ρSS in the energy eigenbasis of the +system scale as ϵ2. As discussed in the main text, the +QME thermalizes if +lim +ϵ→0 ρSS = ρth +(D9) +where ρth is the Gibbs state of the system given by +ρth = +e−βHS +Tr[e−βHS]. +(D10) +We then conclude that the thermalization in this sense +is a statement about leading order diagonal elements of +ρSS. Substituting Eq. (D9) in Eq. (D7), we obtain the +following condition on L2 for the system to thermalize, +⟨Ei| L2[ρth] |Ei⟩ = 0 +∀i. +(D11) +Appendix E: Semidefinite Programming (SDP) +1. +Basic Theory +In this section, we present the theoretical framework of +semidefinite programming (SDP). We follow the defini- +tion of SDPs given in page 57 of Ref. 9. In what follows, +we will use Φ and Ψ to denote hermitian preserving lin- +ear maps. We will also use Φ† to denote the “adjoint +map” [9], which is defined as the unique linear map that +satisfies +⟨A, Φ(B)⟩ = ⟨Φ†(A), B⟩ +(E1) +where +⟨A, B⟩ = Tr(A†B) +(E2) +denotes the Hilbert Schmidt inner product. +An SDP is defined by the tuple (Φ, Ψ, A, B, C), where +Φ, Ψ are hermitian-preserving linear maps, and A, B, C +are hermitian operators. The “primal” problem of the +SDP is given by +maximize : +⟨A, X⟩ +w.r.t. X +subject to : +Φ(X) = B, Ψ(X) ≤ C, X ≥ 0, +(E3) +where the inequalities represent matrix inequalities. I.e, +A ≥ B is equivalent to A − B ≥ 0 and implies A − B +is positive semidefinite. We will use the notation Xf to +denote any “feasible” value of X that satisfies the three +constraints in Eq. (E3), and P to denote the maximum +value of ⟨A, X⟩ attained in Eq. (E3) (assuming there is +atleast one X which satisfies constraints). +For every “primal” problem, there exists a “dual” +problem given by +minimize : +⟨B, Y ⟩ + ⟨C, Z⟩ +w.r.t. Y, Z +subject to : +Φ†(Y ) + Ψ†(Z) ≥ A, +Y is hermitian, Z ≥ 0. +(E4) +We will use the notation (Yf, Zf) to denote any “fea- +sible” value of Y, Z that satisfies the three constraints +in Eq. (E4), and D to denote the minimum value of +⟨B, Y ⟩ + ⟨C, Z⟩ attained in Eqs. (E4) (assuming atleast +some (Y, Z) satisfies constraints). + +17 +FIG. 7. +Schematic representing weak duality for the SDP +given in Eqs. (E3) and (E4), according to Eq. (E6). +X(j) +f +and (Y (j) +f +, Z(j) +f ) represents any feasible input to the primal +and dual problems respectively. Any such inputs yield upper +and lower bounds on the solutions of the primal and dual +problems. +Semidefinite programs have a notion of duality asso- +ciated with them, which relates properties of the primal +and the dual problems. In particular, it can be shown +that +P ≤ D +(E5) +a property known as “weak duality”. +In most situa- +tions, it can be shown that P = D, i.e, equality holds +in Eq. (E5). This condition is known as “strong dual- +ity”. +By weak duality and the definition of our primal and +dual problems, using Eq. (E3),(E4), and (E5), we obtain +⟨A, Xf⟩ ≤ P ≤ D ≤ ⟨B, Yf⟩ + ⟨C, Zf⟩ . +(E6) +From Eq. (E6), any feasible choice of inputs to the pri- +mal and dual problem (Xf, Yf, Zf) leads to lower and +upper bounds on the optimal values of the primal and +dual problems [see Fig. (7)]. In particular, if we obtain +⟨A, Xf⟩ = ⟨B, Yf⟩ + ⟨C, Zf⟩, equality holds throughout +in Eq. (E6). This property can therefore be exploited to +obtain exact solutions to the primal problem of an SDP. +We will show in Sec. E 2 that the thermal optimiza- +tion problem (TOP) in Eq. (40) can be reduced to the +standard form of SDP [Eq. (E3)]. +2. +Reducing the thermal optimization problem +(TOP) to standard form +Recall that the TOP was given by [Eq.(40)] +minimize : τ +subject to : H(L) +LS is hermitian, Tr(Γ(L)) = 1, Γ(L) ≥ 0. +(E7) +See Eq. (39) for definition of τ and Eq. (38) for defini- +tion of H(L) +LS , Γ(L). In this subsection, we will show how +the TOP from Eq. (E7) can be reduced to the standard +form of an SDP. We note that the standard form of SDP +in Eq. (E3) is not yet suitable for this purpose. There- +fore, we replace A → −A, C → −C, Ψ → −Ψ, Y → −Y, +in Eq. (E3) and Eq. (E4), leaving Φ, B, X and Z un- +changed. Since maximizing any function is the same as +minimizing its negative, we obtain the new “primal” form +as +minimize : +⟨A, X⟩ +subject to : +Φ(X) = B, Ψ(X) ≥ C, X ≥ 0, +(E8) +FIG. 8. +Schematic representing weak duality for the SDP +given in Eqs. (E8) and (E9), according to Eq. (E10). X(j) +f +and (Y (j) +f +, Z(j) +f ) represents any feasible input to the primal +and dual problems respectively. Any such inputs yield upper +and lower bounds on the solutions of the primal and dual +problems. +where we use �P to denote the minimum value of ⟨A, X⟩ +obtained in Eq. (E8). The new “dual” form is written as +maximize : +⟨B, Y ⟩ + ⟨C, Z⟩ +subject to : +Φ†(Y ) + Ψ†(Z) ≤ A, +Y is hermitian, Z ≥ 0, +(E9) +where we use �D to denote the minimum value of ⟨B, Y ⟩+ +⟨C, Z⟩ obtained in Eq. (E9). Eq. (E6) is then transformed +into [see Fig (8)], +⟨A, Xf⟩ ≥ �P ≥ �D ≥ ⟨B, Yf⟩ + ⟨C, Zf⟩ . +(E10) +We will now show how to reduce the TOP from +Eq. (E7) to Eq. (E8), via a series of changes to the opti- +mization problem in Eq. (E8). We do so in three steps. +Step 1: Our first step is to write down a primal op- +timization problem whose solution (minimum value at- +tained) is equal to +||K||1 = Tr( +√ +K†K). +(E11) +for any hermitian matrix K. Let Πp and Πn be projectors +onto the positive and negative eigenspaces of K. In this +case, +||K||1 = Tr(ΠpKΠp) − Tr(ΠnKΠn). +(E12) +Let us now consider the optimization problem given by, +minimize : +�� +I 0 +0 I +� +, +� +P +. +. +Q +�� +subject to : +Ψ1 +� +P +. +. +Q +� += +� +P +0 +0 Q +� +≥ +� +K +0 +0 +−K +� +, +� +P +. +. +Q +� +≥ 0, +(E13) +where we use dots to represent arbitrary blocks of the +matrices which can always be set to zero without af- +fecting the objective function or constraints. Note that +Ψ1 in Eq. (E13) is a map that replaces the off-diagonal +blocks with null matrices, leaving the diagonal blocks un- +changed. It is easy to see that Eq. (E13) is of the form +Eq. (E8) (with Φ and B omitted i.e., no equality con- +straint). Thus Eq. (E13) is an SDP. + +D +(Yf),zf1) +8 +P +α(f1),zf1) +D +8 +(y(2), z(2) ++8 +p18 +The dual problem to the primal problem in Eq. (E13) +is given by +maximize : +�� +K +0 +0 +−K +� +, +� ¯P +. +. +¯Q +�� +subject to : +Ψ† +1 +� ¯P +. +. +¯Q +� += +� ¯P +0 +0 +¯Q +� +≤ +� +I 0 +0 I +� +� ¯P +. +. +¯Q +� +≥ 0. +(E14) +where Ψ† +1 turns out to be the same map as Ψ1 using +Eq. (E1). It can be seen that Eq. (E14) is of the form +Eq. (E9) (again with Φ and B omitted). +We will now show that the optimal values attained in +both the primal and dual problems in Eqs. (E13) and +(E14) is equal to ||K||1. To show this, note that setting +Pf = ΠpKΠp, +Qf = −ΠnKΠn +(E15) +(where Pf and Qf denote ‘feasible’ choices of P and Q +respectively) in Eq. (E13) allows us to obtain ||K||1 in +the primal objective function. Furthermore, setting +¯Pf = Πp, +¯Qf = Πn +(E16) +in Eq. (E14) allows us to obtain ||K||1 in the dual objec- +tive function. Thus, we have explicitly constructed fea- +sible choices of inputs to the primal and dual problems +of Eqs. (E13) and (E14) respectively, that yield ||K||1 +in the primal and dual objective functions. Therefore, +according to Eq. (E10), the optimal values attained in +the primal and dual problems are both exactly equal to +||K||1. +Step 2: In the first step we constructed an SDP whose +solution is equal to ||K||1, for a fixed K. We will now +construct an SDP which computes the minimum value of +||K||1, subject to some constraints on K. We will first +recast the problem in Eq. (E13) as +minimize : +Tr(P) + Tr(Q) +subject to : +P ≥ K, Q ≥ −K, P, Q ≥ 0. +(E17) +Eq. (E13) computes +||K||1 = +� +i +|Kii|. +(E18) +when K is diagonal. Let G be a linear, hermitian preserv- +ing map that always outputs a diagonal matrix. Then, +the optimization problem given by +minimize : +Tr(P) + Tr(Q) +subject to : +P ≥ G(R), Q ≥ −G(R), +Φ(R) = B, +P, Q, R ≥ 0, +(E19) +computes minR≥0,Φ(R)=B +� +i |G(R)ii|. We will now begin +to connect the above formalism to TOP from Eq. (E7). +We will show how R can be chosen to reflect the opti- +mization over H(L) +LS and Γ(L), and Φ can be chosen to +reflect the trace constraint on Γ(L). Then, we will spec- +ify a map G that takes R as input (i.e, H(L) +LS and Γ(L) +LS ), +and outputs a diagonal matrix such that the objective +function computes +τ = +� +i +|⟨Ei| L2(ρth) |Ei⟩| . +(E20) +Step +3: +In +the +thermal +optimization +problem +[Eq. (E7)], we have an optimization over Γ(L) ≥ 0, and +hermitian H(L) +LS . We use the fact that any hermitian ma- +trix H(L) +LS can be written as a H(L) +LS += S − T, where +S, T ≥ 0. Moreover S − T for any S, T ≥ 0 is always +hermitian. We will now replace the hermitian H(L) +LS with +the difference of positive matrices S − T. This is needed, +since semidefinite programs can only handle optimization +over positive semidefinite variables. Furthermore, let us +identify Γ(L) with some matrix U. Now consider the map +G that acts as follows : +G +� +� +S +. +. +. T +. +. +. +U +� +� ≡ G(S, T, U) ≡ +� +� +� +� +� +⟨E1| L2(ρth) |E1⟩ +0 +. . . +0 +0 +⟨E2| L2(ρth) |E2⟩ . . . +0 +... +. . . +... +... +0 +. . . +0 +⟨Ed| L2(ρth) |Ed⟩ , +� +� +� +� +� +(E21) +where the map constructs L2 according to Eq. (38) after +setting H(L) +LS = S − T, and Γ(L) = U. Now, we consider +a specific case of the optimization problem in Eq. (E19), +for the choice of G in Eq. (E21). We obtain, +minimize : +Tr(P) + Tr(Q) +subject to : +P ≥ G +� +� +S +. +. +. T +. +. +. +U +� +� , Q ≥ −G +� +� +S +. +. +. T +. +. +. +U +� +� , +Φ1 +� +� +S +. +. +. T +. +. +. +U +� +� = Tr(U) = 1, P, Q, S, T, U ≥ 0. +(E22) + +19 +Since G always outputs a diagonal matrix [see Eq. (E21)], +Eq. (E22) computes +min +� +i +������ +G +� +� +S +T +U +� +� +ii +������ +subject to : +S, T, U ≥ 0, Tr(U) = 1 +(E23) +Since H(L) +LS can always be written as S−T, and Γ(L) as U, +Eq. (E23) [and therefore Eq. (E22) ] is identical to the +thermal optimization problem in Eq. (E7). +Therefore, +Eq. (E22) computes τopt. +All that remains is converting Eq. (E22) to the stan- +dard form Eq. (E8). +Eq. (E22) can be obtained from +Eq. (E22) after choosing +A = +� +� +� +� +� +I +I +0 +0 +0 +� +� +� +� +� , +B = 1, +X = +� +� +� +� +� +P +. +. +. +. +. +Q . +. +. +. +. +S +. +. +. +. +. T +. +. +. +. +. +U +� +� +� +� +� , +C = 0, +Ψ +� +� +� +� +� +P +. +. +. +. +. +Q . +. +. +. +. +S +. +. +. +. +. T +. +. +. +. +. +U +� +� +� +� +� = +� +P − G(S, T, U) +0 +0 +Q + G(S, T, U) +� +, +Φ +� +� +� +� +� +P +. +. +. +. +. +Q . +. +. +. +. +S +. +. +. +. +. T +. +. +. +. +. +U +� +� +� +� +� = Tr(U) +(E24) +Recall that it is helpful to think of P, Q as variables +needed to compute the objective function τ [Eq. (E7)], +S, T are variables that give rise to H(L) +LS = S − T, and U +is a variable that encodes Γ(L). We have therefore shown +that the TOP [Eq. (E7)] is an SDP. +It is to be noted that it is not necessary to reduce +the TOP to the standard form of an SDP in order to +use CVX [33]. Infact, the TOP from Eq. (E7) can be +directly implemented into CVX, which itself handles the +construction of the dual problem automatically in the +background. +[1] F. Binder, +L. A. Correa, +C. Gogolin, +J. Anders, +and G. Adesso, eds., Thermodynamics in the Quantum +Regime (Springer, Cham, 2018). +[2] K. L. Hur, L. Henriet, A. Petrescu, K. Plekhanov, +G. Roux, and M. Schir´o, Comptes Rendus Physique 17, +808 (2016). +[3] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. John- +son, M. Kieferov´a, I. D. Kivlichan, T. Menke, B. Per- +opadre, N. P. D. Sawaya, and et al., Chemical Reviews +119, 10856–10915 (2019). +[4] D. D. Awschalom, C. H. R. Du, R. He, J. Heremans, +A. Hoffmann, J. Hou, H. Kurebayashi, Y. Li, L. Liu, +V. Novosad, J. Sklenar, S. Sullivan, D. Sun, H. Tang, +V. Tyberkevych, C. Trevillian, A. W. Tsen, L. Weiss, +W. Zhang, X. Zhang, L. Zhao, +and C. W. Zollitsch, +IEEE Transactions on Quantum Engineering , 1 (2021). +[5] J. Cao, R. J. Cogdell, D. F. Coker, H.-G. Duan, J. Hauer, +U. Kleinekath¨ofer, T. L. C. Jansen, T. Manˇcal, R. J. D. +Miller, J. P. Ogilvie, V. I. Prokhorenko, T. Renger, H.-S. +Tan, R. Tempelaar, M. Thorwart, E. Thyrhaug, S. West- +enhoff, and D. Zigmantas, Science Advances 6 (2020). +[6] H.-P. Breuer and F. Petruccione, The Theory of Open +Quantum Systems (Oxford University Press, Oxford, +2006). +[7] ´A. Rivas and S. F. Huelga, SpringerBriefs in Physics +(2012). +[8] H. Carmichael, Statistical Methods in Quantum Op- +tics 1. Master Equations and Fokker–Planck Equations +(Springer-Verlag Berlin Heidelberg, 2002). +[9] J. Watrous, The Theory of Quantum Information (Cam- +bridge University Press, 2018). +[10] V. Siddhu and S. Tayur, “Five starter pieces: Quantum +information science via semidefinite programs,” in Tu- +torials in Operations Research: Emerging and Impactful +Topics in Operations (2022) Chap. 3, pp. 59–92. +[11] D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004). +[12] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011). +[13] M. M. Wolf, J. Eisert, T. S. Cubitt, +and J. I. Cirac, +Phys. Rev. Lett. 101, 150402 (2008). +[14] T. S. Cubitt, J. Eisert, and M. M. Wolf, Communications +in Mathematical Physics 310, 383 (2012). +[15] A. H. Kiilerich and K. Mølmer, Phys. Rev. A 97, 052113 +(2018). +[16] V. Gorini, A. Kossakowski, +and E. C. G. Sudarshan, +Journal of Mathematical Physics 17, 821 (1976). +[17] V. Gorini, A. Frigerio, M. Verri, A. Kossakowski, +and +E. Sudarshan, Reports on Mathematical Physics 13, 149 +(1978). +[18] G. Lindblad, Communications in Mathematical Physics +48, 119 (1976). + +20 +[19] H. Weimer, A. Kshetrimayum, and R. Or´us, Rev. Mod. +Phys. 93, 015008 (2021). +[20] M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 +(1998). +[21] J. Dalibard, Y. Castin, and K. Mølmer, Phys. Rev. Lett. +68, 580 (1992). +[22] K. Mølmer, Y. Castin, and J. Dalibard, Journal of the +Optical Society of America B 10, 524 (1993). +[23] D. Tupkary, A. Dhar, M. Kulkarni, and A. Purkayastha, +Phys. Rev. A 105, 032208 (2022). +[24] R. Hartmann and W. T. Strunz, Phys. Rev. A 101, +012103 (2020). +[25] P. R. Eastham, P. Kirton, H. M. Cammack, B. W. Lovett, +and J. Keeling, Phys. Rev. A 94, 012110 (2016). +[26] S. Anderloni, F. Benatti, +and R. Floreanini, Journal +of Physics A: Mathematical and Theoretical 40, 1625 +(2007). +[27] P. Gaspard and M. Nagaoka, The Journal of Chemical +Physics 111, 5668 (1999). +[28] D. Kohen, C. C. Marston, and D. J. Tannor, The Journal +of Chemical Physics 107, 5236 (1997). +[29] S. Gnutzmann and F. Haake, Zeitschrift f¨ur Physik B +Condensed Matter 101, 263 (1996). +[30] A. Su´arez, R. Silbey, and I. Oppenheim, The Journal of +Chemical Physics 97, 5101 (1992). +[31] F. Nathan and M. S. Rudner, Phys. Rev. B 102, 115109 +(2020). +[32] H. Spohn, Rev. Mod. Phys. 52, 569 (1980). +[33] M. Grant and S. Boyd, “CVX: Matlab software for disci- +plined convex programming, version 2.1,” http://cvxr. +com/cvx (2014). +[34] D. Walls and G. J. Milburn, Quantum optics (Springer- +Verlag Berlin Heidelberg, 2008). +[35] T. Becker, L.-N. Wu, and A. Eckardt, Phys. Rev. E 104, +014110 (2021). +[36] T. Prosen and M. ˇZnidariˇc, Journal of Statistical Me- +chanics: Theory and Experiment 2009, P02035 (2009). +[37] J. J. Mendoza-Arenas, S. R. Clark, and D. Jaksch, Phys. +Rev. E 91, 042129 (2015). +[38] M. ˇZnidariˇc, T. c. v. Prosen, G. Benenti, G. Casati, and +D. Rossini, Phys. Rev. E 81, 051135 (2010). +[39] J. T. Barreiro, M. M¨uller, P. Schindler, D. Nigg, T. Monz, +M. Chwalla, M. Hennrich, C. F. Roos, P. Zoller, +and +R. Blatt, Nature 470, 486 (2011). +[40] P. Schindler, M. M¨uller, D. Nigg, J. T. Barreiro, E. A. +Martinez, M. Hennrich, T. Monz, S. Diehl, P. Zoller, and +R. Blatt, Nature Physics 9, 361 (2013). +[41] H. Weimer, M. M¨uller, I. Lesanovsky, P. Zoller, and H. P. +B¨uchler, Nature Physics 6, 382 (2010). +[42] T. L. Nguyen, J. M. Raimond, C. Sayrin, R. Corti˜nas, +T. +Cantat-Moltrecht, +F. +Assemat, +I. +Dotsenko, +S. Gleyzes, S. Haroche, G. Roux, T. Jolicoeur, +and +M. Brune, Phys. Rev. X 8, 011032 (2018). +[43] G. Garc´ıa-P´erez, M. A. C. Rossi, and S. Maniscalco, npj +Quantum Information 6, 1 (2020). +[44] C. W. Kim, J. M. Nichol, A. N. Jordan, and I. Franco, +PRX Quantum 3, 040308 (2022). +[45] T. Becker, A. Schnell, and J. Thingna, Phys. Rev. Lett. +129, 200403 (2022). +[46] G. McCauley, B. Cruikshank, D. I. Bondar, and K. Ja- +cobs, npj Quantum Information 6 (2020). +[47] U. Harbola, M. Esposito, +and S. Mukamel, Phys. Rev. +B 74, 235309 (2006). +[48] F. +Nathan +and +M. +S. +Rudner, +(2022), +10.48550/ARXIV.2206.02917. +[49] A. +Trushechkin, +Physical +Review +A +103 +(2021), +10.1103/physreva.103.062226. +[50] A. S. Trushechkin and I. V. Volovich, EPL (Europhysics +Letters) 113, 30005 (2016). +[51] E. Kleinherbers, N. Szpak, J. K¨onig, and R. Sch¨utzhold, +Phys. Rev. B 101, 125131 (2020). +[52] D. Davidovi´c, Quantum 4, 326 (2020). +[53] E. Mozgunov and D. Lidar, Quantum 4, 227 (2020). +[54] G. Kirˇsanskas, M. Francki´e, and A. Wacker, Phys. Rev. +B 97, 035432 (2018). +[55] A. Purkayastha, G. Guarnieri, M. T. Mitchison, R. Filip, +and J. Goold, npj Quantum Information 6 (2020). +[56] G. E. Fux, D. Kilda, B. W. Lovett, +and J. Keeling, +(2022), 10.48550/ARXIV.2201.05529. +[57] A. Purkayastha, G. Guarnieri, S. Campbell, J. Prior, and +J. Goold, Phys. Rev. B 104, 045417 (2021). +[58] A. +M. +Lacerda, +A. +Purkayastha, +M. +Kewming, +G. +T. +Landi, +and +J. +Goold, +(2022), +10.48550/ARXIV.2206.01090. +[59] A. Majumdar, G. Hall, and A. A. Ahmadi, Annual Re- +view of Control, Robotics, and Autonomous Systems 3, +331 (2020). +[60] G. Guarnieri, M. Kol´aˇr, and R. Filip, Phys. Rev. Lett. +121, 070401 (2018). +[61] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. An- +ders, AVS Quantum Science 4, 012301 (2022). +[62] J. D. Cresser and J. Anders, Phys. Rev. Lett. 127, 250601 +(2021). +[63] A. Streltsov, G. Adesso, +and M. B. Plenio, Rev. Mod. +Phys. 89, 041003 (2017). +[64] G. Francica, J. Goold, and F. Plastina, Phys. Rev. E 99, +042105 (2019). +[65] J. P. Santos, L. C. C´eleri, G. T. Landi, +and M. Pater- +nostro, npj Quantum Information 5 (2019). +[66] V. Narasimhachar and G. Gour, Nature Communications +6 (2015). +[67] D. +Tupkary, +https://github.com/dtupkary/ +SearchingLindbladians (2022). +[68] J. Johansson, P. Nation, and F. Nori, Computer Physics +Communications 183, 1760 (2012). +[69] J. Johansson, P. Nation, and F. Nori, Computer Physics +Communications 184, 1234 (2013). + diff --git a/39A0T4oBgHgl3EQfNf8t/content/tmp_files/load_file.txt b/39A0T4oBgHgl3EQfNf8t/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f0d5423dd249bed4f554ce6c55afa155c943cdc4 --- /dev/null +++ b/39A0T4oBgHgl3EQfNf8t/content/tmp_files/load_file.txt @@ -0,0 +1,1536 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf,len=1535 +page_content='Searching for Lindbladians obeying local conservation laws and showing thermalization Devashish Tupkary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' ∗ Abhishek Dhar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' † Manas Kulkarni,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' ‡ and Archak Purkayastha3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' § 1Institute for Quantum Computing and Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' University of Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Ontario,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Canada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' N2L 3G1 2International Centre for Theoretical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Tata Institute of Fundamental Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Bangalore 560089,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' India 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Indian Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Hyderabad 502285,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' India 4Centre for complex quantum systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Aarhus University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Nordre Ringgade 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 8000 Aarhus C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Denmark 5School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Trinity College Dublin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' College Green,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Dublin 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Ireland We investigate the possibility of a Markovian quantum master equation (QME) that consistently describes a finite-dimensional system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' a part of which is weakly coupled to a thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In order to preserve complete positivity and trace, such a QME must be of Lindblad form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' For physical con- sistency, it should additionally preserve local conservation laws and be able to show thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' First, we show that the microscopically derived Redfield equation (RE) violates complete positivity unless in extremely special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We then prove that imposing complete positivity and demanding preservation of local conservation laws enforces the Lindblad operators and the lamb-shift Hamil- tonian to be ‘local’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='e, to be supported only on the part of the system directly coupled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We then cast the problem of finding ‘local’ Lindblad QME which can show thermalization into a semidefinite program (SDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We call this the thermalization optimization problem (TOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' For given system parameters and temperature, the solution of the TOP conclusively shows whether the desired type of QME is possible up to a given precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Whenever possible, it also outputs a form for such a QME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' For a XXZ chain of few qubits, fixing a reasonably high precision, we find that such a QME is impossible over a considerably wide parameter regime when only the first qubit is coupled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Remarkably, we find that when the first two qubits are attached to the bath, such a QME becomes possible over much of the same paramater regime, including a wide range of temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' INTRODUCTION A small finite-dimensional quantum system, a part of which is weakly coupled to a macroscopic thermal bath, is expected to thermalize to the temperature of the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Describing this dynamics is relevant across various fields in quantum science and technology, including quantum information and thermodynamics [1], quantum optics [2], quantum chemistry [3], engineering [4] and biology [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In absence of coupling to the macroscopic thermal bath, the dynamics of the density matrix of the system is gov- erned by the Heisenberg equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This unitary evolution is Markovian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' When coupled to the macro- scopic thermal bath, the dynamics becomes non-unitary, described by a quantum master equation (QME) [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We investigate whether it is possible to have a physically consistent Markovian QME describing such dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In order to do so, we are led to introduce the “thermaliza- tion optimization problem” (TOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This is a semidefinite program (SDP), the output of which conclusively shows whether, for given system parameters and temperature, such a QME is possible, up to given precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' When- ever possible, the output also yields one possible form for such a QME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Whenever impossible, it means that, for such parameters, the dynamics cannot be described ∗ djtupkary@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='ca † abhishek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='dhar@icts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='in ‡ manas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='kulkarni@icts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='in § archak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='p@phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='iith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='in FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Schematic of the setup we consider: an arbitrary fi- nite dimensional system described by Hamiltonian HS, a part of which is weakly coupled to a thermal bath at inverse tem- perature β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The Hilbert space of the system, HS is divided into a part HL which directly couples to the bath, and the remaining part HM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' by any Markovian QME even at weak system-bath cou- pling, and therefore must have some non-Markovian char- acter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The SDP can be solved using standard packages in high-level computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We note that, while SDP is widely used in many branches of quantum information and communication [9, 10], and also in quantum chem- istry [11, 12], it has been combined with open quantum system techniques in only few previous works [13–15], in very different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' It was shown by Gorini, Kossakowski, Sudarshan, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='02146v1 [quant-ph] 5 Jan 2023 Hilbert space Hl Hilbert space HM Hilbert space Hs = HL HM, Hamiltonian Hs2 Lindblad (GKSL) [16–18] that any QME that preserves complete positivity and trace of the density matrix, and describes Markovian dynamics has to be of the form ∂ρ ∂t = i[ρ, HS + HLS] + D(ρ), D(ρ) = d2−1 � λ=1 γλ � LλρL† λ − 1 2{L† λLλ, ρ} � , γλ ≥ 0, (1) which is commonly called the Lindblad equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (1), ρ is the density matrix of the system, d is the Hilbert space dimension of the system, HS is the sys- tem Hamiltonian, HLS is the Lamb shift Hamiltonian, Lλ are the Lindblad operators, γλ are the rates, and D(ρ) is called the “dissipator” term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The preservation of complete positivity condition is enforced by demand- ing γλ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Lindblad equations have been extensively used in studying both theoretical and experimental se- tups [6–8, 19–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Given this enormous scope of application, it is of paramount importance to assess the conditions under which such a Markovian description emerges from a more microscopic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The standard way to microscopi- cally obtain a Markovian QME is to consider the global Hamiltonian of the system weakly coupled to baths, and to trace out the baths perturbatively to the leading or- der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Starting with this microscopic viewpoint, it becomes clear that only having an equation in the Lindblad form is not sufficient, there are some additional fundamental requirements for physical consistency [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In particu- lar, one must preserve local conservation laws, and if the steady state is unique, the system is not driven and all baths have same temperature, the system must thermal- ize to the temperature of the baths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' It would be useful to have a QME which, by construction, is of Lindblad form and satisfy these additional requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In this paper, we systematically go about searching for such a QME for a setup where a part of the system is coupled to a single bath (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This is done in three steps, each step having important consequences: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The microscopically derived quantum master equa- tion to the leading order in system-bath coupling is the so-called Redfield equation (RE) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The RE has been shown to preserve local conservation laws and be able to show thermalization [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Here, we provide an explicit, model independent proof that the RE necessarily violates complete positiv- ity unless the Redfield dissipator happens to act “locally”, meaning it is identity on the part of the system that is not directly coupled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Al- though the violation of complete positivity by RE has been previously demonstrated in specific exam- ples [24–30], we are unaware of any previous work with such a model independent explicit proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We then prove that enforcing complete positivity condition γλ ≥ 0 and preservation of local conser- vation laws necessarily requires the Lindblad oper- ators and the Lamb shift Hamiltonian to be “lo- cal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' That is, they must act only on the part of the system coupled to the bath, and be identity on the part of the system that is not connected to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This directly rules out the possibility of any so-called ‘global’ Lindblad equation, such as the eigenbasis Lindblad equation [6, 8], the Uni- versal Lindblad Equation [31] to be consistent with local conservation laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Given the restriction of the dissipator and the Lamb shift Hamiltonian to be “local’, we devise a numerical technique using SDP to check conclu- sively in a case-by-case basis whether such a QME can show thermalization in a particular situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We call this the TOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We use this method to study the case of a XXZ chain of few qubits with a part of it coupled to a bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' If the bath is coupled only to the first qubit, our method conclusively shows that over a large regime of system parameters and temperature, no such QME exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' However, re- markably if the bath is coupled to two qubits of the chain, up to a chosen precision, our method shows that a Marovian QME respecting all condi- tions becomes possible over a considerable regime of parameters, including a wide range of tempera- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Note that the RE for the XXZ chain does show thermalization and preserve local conserva- tion laws [23], even when only one qubit is attached to a bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' But it is not completely positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' II we ex- plain the setup studied in this work, derive the Redfield equation for our setup, and show that it will necessarily violate complete positivity, unless the Redfield dissipa- tor happens to act “locally”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' III we consider quantum master equations preserving complete positiv- ity and obeying local conservation laws, and show that such equations must have a dissipator and Lamb shift operator that acts only on the part of the system cou- pled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' IV we discuss the possibility of QMEs respecting complete positivity, local conservation laws and being able to show thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' We intro- duce the TOP, and use it in the specific case of the few site XXZ chain with one or two sites attached to bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' V we summarize our results, and discuss future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Certain details are delegated to the appen- dices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' THE MODEL AND REDFIELD DESCRIPTION Our setup is described schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The full Hamiltonian be given by H = HS + ϵHSB + HB, (2) where HS and HB are the Hamiltonians of the system and bath respectively, ϵ ≪ 1 is a small dimensionless pa- 3 rameter controlling system-bath coupling strength, and HSB is the system-bath coupling Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' At initial time, the system is considered to be in an arbitrary ini- tial state ρ(0), while the bath is in a thermal state with inverse temperature β ρtot(0) = ρ(0) ⊗ ρB, ρB = e−βHB ZB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (3) Starting with this initial state, the whole set-up of the system and the bath is evolved with the full Hamiltonian H, and the bath degrees of freedom are traced out to obtain the state of the system, ρ(t) = TrB � e−iHtρtot(0)eiHt� , (4) where TrB(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=') denotes trace over bath degrees of free- dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (4), by construction, is a completely positive trace preserving (CPTP) map from ρ(0) to ρ(t) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Without any loss of generality, we assume TrB(HSBρB) = 0, where TrB(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=') denotes trace over bath degrees of freedom [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The effective equation of motion for the system density matrix written to the leading order in system-bath coupling strength ϵ is the RE, given by, [6], ∂ρ ∂t =i[ρ(t), HS] +ϵ2 � ∞ 0 dt′ TrB[HSB, [HSB(−t′), ρ(t) ⊗ ρB]], (5) where HSB(t) = ei(HS+HB)tHSBe−i(HS+HB)t (6) and ρB is the state of the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In complete generality, we can write the system-bath coupling Hamiltonian as HSB = � l (SlB† l + S† l Bl), (7) where Sl and Bl are operators on the system and bath respectively, and l can be summed over as many indices as required for HSB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (7) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (5) and simplifying, we have ∂ρ ∂t = i[ρ(t), HS] + ϵ2� � l � S† l , S(1) l ρ(t) � − � S† l , ρ(t)S(2) l � + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content='c � , (8) where S(1) l = � m � ∞ 0 dt′ Tr � BlB† m(−t′)ρB � Sm(−t′) + � m � ∞ 0 dt′ Tr � BlBm(−t′)ρB � S† m(−t′) S(2) l = � m � ∞ 0 dt′ Tr � B† m(−t′)BlρB � Sm(−t′) + � m � ∞ 0 dt′ Tr � Bm(−t′)BlρB � S† m(−t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (9) Since the actual microscopic evolution is given by a CPTP map [see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (3), (4)], one might naively expect that the evolution obtained from the microscopically de- rived RE respects complete positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' However, as we prove in the next subsection in generality, unless in ex- tremely special cases, the RE violates complete positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Violation of complete positivity in Redfield equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Choosing an operator Basis As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 1, we consider only a part of the sys- tem is coupled to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Let us denote HL as the Hilbert space of that part of the system that couples to the bath, and let HM be the Hilbert space of the re- maining part of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' In mathematical terms, this means that any operator OM in the Hilbert space HM commutes with the system-bath coupling Hamiltonian HSB, [OM, HSB] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (10) The system Hamiltonian can then be written as HS = HL + HLM + HM, (11) where the Hamiltonian HL is in Hilbert space HL, the Hamiltonian HM is in Hilbert space HM, and HLM gives the coupling between the two Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Note that we do not consider this coupling to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Let the dimension of HL and HM be dL and dM re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Then, one can pick an orthonormal basis of operators {fi} and {gj} on HL and HM respectively, where 1 ≤ i ≤ d2 L and 1 ≤ j ≤ d2 M, and where or- thonormality is defined according to the Hilbert Schmidt inner product given by ⟨A, B⟩ = Tr[A†B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' One can al- ways choose this basis such that fd2 L = IL/√dL and gd2 M = IM/√dM, where IM and IL are the identity op- erators on those spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Such a basis is required by the GKSL theorem [6, 7, 16, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Taking the tensor prod- uct of these two basis, one can obtain an orthonormal basis {Fk} = {fi} ⊗ {gj} for operators on HS, with Fd2 Ld2 M = IS/ √ d, where d = dLdM is the dimension of the system Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Without loss of generality, the Lindblad equation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (1)] written in this basis is given by ∂ρ ∂t = i[ρ, HS + HLS] + d2−1 � α,˜α=1 Γα˜α � F˜αρF † α − {F † αF˜α, ρ} 2 � , (12) where complete positivity of ρ is preserved iff Γ is positive semidefinite [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (12) can be turned into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (1) by diagonalizing the matrix Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The complete positivity of RE can be checked by tak- ing the RE to the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (12) and checking if the corresponding Γ is positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' To do so, 4 let us relabel the indices so that Fi = fi ⊗ IM/√dM for 1 ≤ i ≤ d2 L − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' This allows us to expand the system operators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (8) as, Sl = d2 � α=1 alαFα, S† l = d2 � α=1 a′ lαFα, S(1) l = d2 � α=1 blαFα, S(1)† l = d2 � α=1 b′ lαFα, S(2) l = d2 � α=1 clαFα, S(2)† l = d2 � α=1 c′ lαFα, (13) where alα = a′ lα = 0, ∀ d2 L ≤ α ≤ d2 −1 since Sl and S† l are identity on HM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (13) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (8), we obtain ∂ρ ∂t = i[ρ, HS] − ϵ2 � l d2 � α,˜α=1 � a∗ lαbl˜α[F † α, F˜αρ] + c′∗ lαa′ l˜α[ρF † α, F˜α] + b∗ lαal˜α[ρF † α, F˜α] + a′∗ lαc′ l˜α[F † α, F˜αρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (14) Using some straightforward algebra (see Appendix A), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (14) can be simplified to ∂ρ ∂t = i[ρ, HS + HLS] + d2−1 � α,˜α=1 Γα˜α � F˜αρF † α − {F † αF˜α, ρ} 2 � , (15) where Γα˜α is a (d2 −1)×(d2 −1) hermitian matrix given by Γα˜α = ϵ2 � l (a∗ lαbl˜α + c′∗ lαa′ l˜α + b∗ lαal˜α + a′∗ lαc′ l˜α), (16) where x∗ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (16) denotes the complex conjugate of x, and the expression of HLS that appears in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (15) can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (16) is of the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (12), the condition for preserving complete positivity of ρ is equivalent to Γ being positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' The dissipator Let us now look at Γα˜α from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Recall from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' II A 1, that if α ≥ d2 L, then alα = a′ lα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39A0T4oBgHgl3EQfNf8t/content/2301.02146v1.pdf'} +page_content=' Therefore, Γα˜α = 0 when α, ˜α ≥ d2 L, and has the following structure: Γ = � Γα,˜α